«> N y.'^ \ \ '\ ♦> ^^^^ aDUC^'4lf1<'Ji^^ Ue^*"' (F 1 NORMAL METHODS OF TEACHING OOHTAIiriNO i BBIEF STATEMENT OF THE PRINCIPLE3 AND METHODS OF THE SCIENCE AND ABT OF TEACHING, FOB THE USE OF NORMAL CLASSES AND PBIYATB STUDENTS PBEPABINO THEMSELVES FOB TEACHBBS. BY EDWARD BROOKS, A. M., Ph.D., rORSCER PRINCIPAL OP STATE NORMAL SCHOOL, PENRSTLVANIA, AND AUTHOR OT A NORMAL SERIES OF MATHEMATICS, MENTAL SCIENCE AND CULTURE, PHILOSOPHY OF ARITHMETIC, ETC. *'That divine and beautiful thing called Teaching.'* **Tlke object of aU education is to teach people to think for themulvet.** PHILADELPHIA : NORMAL, PUBLISHING COMPANY. 1889. Copyright, 1879, By EDWARD BROOKS, A. M. BY THE SAME AUTHOR. I. The Normal Series of Arithmetics. 1. Tlie Standard Series: a full course. Four books : New Primary 22; Elementary 45 ; New Mental a5: New Written 80. 2. Tlie Union Series; a sliorter course. Two books; Union Part I, 25; The Normal Union 90. II. Normal Elementary Algebra, $1.10 III. Normal Geometry and Trigonometry, - - 1.10 IV. Normal Higher Arithmetic, 1.25 V. The Philosophy of Arithmetic. .... 2.2.T VI. Keys containing Methods and Models. SOWER, POTTS & CO.. Publishers. 530 MAUIvET STREET, PHILADELPHIA. Copies mailed on receipt of prices annexed, and introduced into schools for one-third less. ie«)ucA"^o?vi nEP^i iNQriRKR V. * P. :o . 8TKRKOTYVKKS i FKIHTXU, LANCASTER, PA. PREFACE. 'T'EACHIXG is a Science and an Art. It embraces a system of truths that admit of scientific statement and may be woven together with the thread of philosophic principles. As such it may be taught like other sciences and arts, and may thus be presented in a book to be studied. The present volume is the result of an earnest attempt to give a scientific statement to the subject in a form that can be used in the training of teachers. Object. — The work is designed as a Text-Book on TencMng. It is a work to be studied and mastered by those who are preparing themselves to teach in our public shools. Several good books on the subject have been written from the standpoint of the profession for professional readers ; this work is written from the standpoint of the class-room, for those who are being trained for the profession. Its design is not so much to adorn the literature of the profession as to aid in building up the profession. The aim has been to prepare a work suitable for the use of the classes in our Normal Schools, — a work that can be studied and recited like a text-book on grammar or arithmetic. Origin. — The work grew up in the class-room. The matter was originally prepared for my own "teaching classes," and has been used by them for many years. Primarily, it was given orally to the classes, the pupils being required to write down only the leading defini- tions and principles and an outline of the discussions. Recently it became necessary to divide these classes and place them in charge of assistant teachers. For the use of these assistants the notes were expanded into a little treatise which the pupils were required to copy and recite. So inconvenient was this, and so much valuable time was lost, that I resolved to comply with the request of teachers and pupils and put the matter in a form for publication. Importance. — The need of such a text-book cannot be questioned. (iiij 54 1J43 IV PREFACE. The subject itself demands it. Teaching is a science and an art, and as such deserves to be stated in a scientific form. The teachers and students of Teaching demand it. The time has gone by when talks and lectures on the subject of Teaching will meet the wants of Normal instruction. The pupils need a book which they can study and recite; they want to see that they are making actual progress in Teaching as in other branches. In our Normal Schools the science of Teaching must be placed alongside of the other branches for exactness of state- ment, if it is to be respected and appreciated by students. Close study of mathematics and general talks on Teaching will never give pupils a very high appreciation of tiie Science of Teaching. We need such a work also for the teachers not attending our Normal Schools. County Superintendents tell young teachers to read and study works on Teaching ; and the question is, what shall they study? "How can we prepare ourselves in the science of Teaching?" is the question that comes up from every part of the State. There are several excel- lent works on the subject ; but they do not seem to be adapted to their wants. Some are too profound ; and others are deficient in systematic thought and statement. There seems to be no elementary text-book in which they can find that systematic and concise statement of definitions and principles and detailed description of practical methods of teaching the branches, which they need. It is the hope of the author that the present work may meet this want, and supply this demand. Nature and Contents. — A work on Teaching should be moulded by the wants of the student of Teaching. The student needs a systematic and comprehensive view of the entire subject. He needs carefully pre- pared definitions and statements of the different divisions and topics. He needs a clear and definite statement of principles which he may fix in his mind, and which become as germs to his thought and practice in teaching. He needs to be able to give clear and logical discussions of the principles, and to show their application to the methods adopted. He needs to be able to state the various methods of teaching, give illus trations of these methods and show their adaptation to the different subjects. He needs to be drilled in the literatxtre of teaching, to PREFACE. acquire a vocabulary of educational expressions that convey to his mind definite ideas like the terms and definitions of grammar and arith- metic. The student-teacher should study these until they become part of his educational vocabulary, as a student of law studies Blackstone until he becomes habituated to legal forms of thought and expression. The work aims to meet all these requirements. It presents, first, a scheme of a complete system of education, the principles on which it is based, and the nature and laws of its two principal divisions, Culture and Instruction. It presents, second, a detailed description of the methods of teaching the different Irranches of study. In discussing these Methods, it gives, first, a description of the general nature of each branch, and second, the methods of teaching it. Tlie former divis- ion embraces a statement of the philosophical character of the branch and its historical development's, both of which are of great value, since the nature of a branch determines the method of teaching it, and the historic order of its growth often indicates the order in which it should be developed in the pupil's mind. The methods of teaching each branch include, first, the principles, which should guide the teacher in his work ; second, the several methods that may be employed, indi- cating the correct method ; and third, descriptions and illustrations for actual model lessons in the branches. The Style.— ks, the work is designed for a text-book, special pains have been taken to employ a simple, clear, and concise style. All rhetorical ornament and diffuseness of description have been carefully avoided, and the attempt made to reduce the matter to a scientific form, and to state it in brief and simple sentences suitable for recitation. The author has endeavored to keep his classes of pupils before his mind, and aimed to adapt his statements to their comprehension and powers of express- ion. Nearly every paragraph has been written in view of the thought. How will this answer for pupils to study and recite? The object has been, not to talk about the subject, but to embody the subject in Ian guage, and thus make a text-book on Teaching. Correctness of Methods.— li is believed that the principles and meth- ods presented are not mere theories ; nearly all of them have been Vi PEEFACE. tested by actual experience in the class-room. There is scarcely a method suggested that I have not either tested myself, or had tested by my teachers in our Normal or Model School ; and hundreds of teachers have introduced them into the public schools of the State and proved their correctness by successful teaching. Many of the methods are used by the leading teachers of the country, and are generally accepted as correct. For any methods presented which may seem novel, and in advance of popular practice, I ask an impartial consideration, believ- ing that if tried, they will also prove to be worthy of acceptance. Origin of Matter. — In the preparation of the work, no attempt has been made to be merely original. The object has been to present the subject as it lies in my own mind and as it is thought it should be con ceived by young people preparing to teach. The subject has devel- oped itself in its present form through years of reading, reflection, and experience ; and it is impossible to separate, even if it were necessary, ■what has been acquired from that which is the product of the author's own thought. Whenever I am conscious of following anything peculiar to another author, an acknowledgment is made ; and it is possible that credit should have been given in some cases where it has been with- held. The historical facts have been taken from various sources, and, in some cases, the language has been partially followed. In closing this preface, I desire to express the hope that the book, though written specially for my own pupils, may be of value to many of the young teachers of our country ; and that it may aid in lifting up the practice of teaching to a higher plane, and afford means for that professional culture now so generally demanded. Having written the work in the interests of teachers, I shall find my highest reward in the knowledge that it has proved a benefit to teachers, and done something towards building up one of the best and noblest interests of society— the Profession of Teaching. EDWARD BROOKS. Normal School, MUlersmlle, Pa, i/a^ 10, 1879. TABLE OF CONTENTS. Pbefacb ...•••• PAKT I. GENERAL NATURE OF EDUCATION. CHAPTER I. The Nature of Educatiox ..... CHAPTER n. Gexeral, Pbinciples of Education CHAPTEIi III. The Science of TEAcaixo CHAPTER IV. The Natuke of the Mind . CHAPTER V. The Nature of Culture CHAPTER VI. Methods of Cultivating Each Faculty . CHAPTER VII. The Nature of Knowledge CHAPTER Vm. The Forms of Instruction . CHAPTER IX. The Okdeu of Ixsthuction CHAPTER X. The Printtpt-es of Txptrt'ction. I. Principles Derived from the Nature of the Mind . II. Principles Derived from the Nature of Knowledge III. PriiK-ipKs Derived fiom the Nature of Instruction (vii) PASK . iii 13 . 18 86 . 31 87 . 42 48 . 53 ST . «3 67 . 72 vai CONTENTS. PART II. TEACHING THE BRANCHES. I, OBJECT I.ESSONS. CHAPTER I. The Nature of Object Lessons. I. Value of Object Lessons II. Preparation for Object Lessons III. Method of Giving Object Lessons . IV. Errors to be Avoided in Object LessoiiS V. Course of Instruction in Object Lessons L Lessons on Form 2. Lessons on Color 3. Objects and their Parts . 4. Qualities of Objects . • 5. Elements of Botany . . • 1 k . 79 • • 82 . 83 • • 85 . 85 • • • a5 . 86 * • • 89 . 91 • 92 II. LANGUAGE. CHAPTER I. The Nature of Language . . • • I. Spoken Language . . • • n. Written Language . . • ■ HI. Course in Language . • • CHAPTER n. Teaching a Child to Read. I. Methods of Teaching a Child to Read . n. The True Method of Teaching a Child to Read CHAPTER m. Teaching the Alphabet. I. The Nature of the Alphabet . . . II. Methods of Teaching the Alphab^st i CHAPTER IV. Teaching Pronunciation. I. Nature and Importance of Pronunciation . n. Methods of Teaching Pronunciation . m. Teachine Correct Pronunciation 9» 94 95 105 107 110 118 123 127 129 135 CONTENTS. IX CHAPTER V. FeACHIXG OuTnOnKAPHT. PAGE. I. The Nature of Ortlios:raphy . . . . • 146 II. Metliods ol Teaching Orthography .... 152 III. The Written Method of Teaching Orthography . . .155 IV. The Oral Method of Teaching Orthography . . .158 V. General Suggestions in Teaching Orthography . . .163 CHAPTER VI. TEACHiifG Reading ....... 168 I. Tlie Mental Element in Reading ..... 172 II. The Vocal Element in Reading . . . .176 III. The Physical Element in Reading . . . . .200 CHAPTER Vn. Teaching Lexicoi-ogt ....... 21i CHAPTER Vni. Teaching English Grammar ...... 221 I. General Nature of the Subject ..... 222 II. Methods of Teaching Primary Grammar .... 239 III. Methods of Teaching Advanced Grammar . . . 266 CHAPTER IX, Teaching Composition ....... 286 I. Preparation for Composition Writing .... 288 n. Language Lessons ....... 296 III. The Writing of a Composition ..... 301 III. MATHEMATICS. CHAPTER I. The Natitre of Mathematics ...... 819 324 CHAPTER n. The Nature of Arithmetic ..... I. The General Nature of Arithmetic . . . . .325 II. The Language of Arithmetic ..... 329 III. The Reasoning of Arithmetic ..... 334 IV. The Treatment of Arithmetic . , . . .339 V. The Course in Arithmetic ...... 342 345 CHAPTER m. Teaching Phimakt Aiutiimetic ..... I. Teaching Arithmetical Language . . . ' • 34:7 II. Teachino; Addition and Sutttraction .... 353 X CONTENTS. FAOS. III. Teaching Multiplication and Division .... 359 IV. Teaefiing Common Fractions ..... 367 V. Teaching Denominate Numbers ..... 374 CHAPTER IV. Teachixg Men'tai, Arithmetic. I. Importance of Mental Arithmetic .... 378 II. The Nature of Mental Arithmetic ..... 382 III. Methods of Teaching Mental Arithmetic . . . 3S6 CHAPTER V. Teaching Written Arithmetic. I. The Nature of Written Arithmetic . . . ,393 II. Methods of Teaching Written Arithmetic . . .398 CHAPTER VI. Teaching Geometry. I. The Nature of Geometry ..... .404 U. Teaching the Elements of Geometiy .... 409 III. Teaching Geometry as a Science . . • • . 422 CHAPTER Vn. Teaching Algebra. I. The Nature of Algebra 432 II. Method of Teaching Algebra ..... 440 IV. PHYSICAIL SCIBNCC CHAPTER I. The Nature of Physical Scien«ce • . . . . 449 CHAPTER n. Teaching Geography. I. The Nature of Geography . ..... 460 II. Teaching Primary Geography ..... 466 III. Teaching Advanced Geography . . • . . 479 IV. Teaching Physical Geography ..... 483 V. HISTORY. CHAPTER I. Teaching History. I. The Nature of History and the Course .... 485 n. Teaching the Elements of History . . • . 490 in. Teaching Advanced History ..... 49.5 PART I. INTRODUCTION. NATURE OF EDUCATION. I. The Nature op Education. II. General Principles of Education. III. The Science of Teaching. IV. The Nature of the Mind. v. The Nature of Culture. VI. The Culture of Each Facultt. VII. The Nature op Knowledob. VIII. The Forms op Instruction. IX. The Order op Instruction. X. The Principles of Instruction, NORMAL METHODS OF TEACHING. CHAPTER I. THE NATURE OF EDUCATION. EDUCATION treats of the developing of the powers of man and the furnishing of his mind with knowledge. The term education is derived from educare, to teach, which is from educere, to lead out, which is from e, out, and duco, I lead. The primary idea of education, as shown by the origin of the term, seems to be the developing or drawing out of the powers of the mind ; and it has been supposed that this was its earliest use. It is said to be doubtful, however, whether the Romans ever used the word in this sense, though most modem writers have so understood it. The term has, at the present day, so broadened its meaning as to embrace both the development of man's powers and the furnishing of his min:' with knowledge. Proble^n of Education.— The problem of education em braces several distinct elements, as will appear from the following analysis. First, there must be a being to be edu- cated ; this being is Man. Second, there must be something with which to educate man, some material to be used in the educational process; this material, consisting of ideas, facts, truths, and sentiments, may be called the flatter of education. (13) 14 METHODS OF TEACHING. Third, there must be some wa}' in which these two elements are united in the educational process ; this wa}^ (methodos, a way) gives rise to the 3Iethods of Education. The problem of education is thus seen to embrace three elements — 3fan, Matter^ and Method. Man is the subjective element ; Matter is the objective element ; and Method is the process by which these two are linked together in the attain- ment of educational results. The old problem of common school education has been facetiously called the problem of "the three B^s — readin', 'ritin', and 'rithmetic ; " the real problem of education ma}^ be seriously called the problem of the three M^s — Man, Matter, Method. branches of Education. — This analysis of the problem of education enables us to determine the fundamental branches of the science of education. Considering 3Ian, the first ele- ment of the problem, we see that he has susceptibilities and powers which may be trained and developed. The process of bringing forth these powers in activity, strength, and har- mony, we call Culture. This culture is not a thing of chance; there is a proper way in which it is to be given. The consideration of the manner in which this culture is to be imparted, gives rise to the first branch of the science called Methods of Culture, Considering the Matter^ the second element of the problem, we perceive that knowledge, which is a product of the inind, may be used in giving culture to the mind. That which came forth from one mind may be developed in other minds, calling into activity the faculties by which it was originally produced. This process of developing knowledge in the mind is called Instruction. The consideration of the manner in which in- struction may be imparted gives rise to a second branch of the science called Methods of Instruction. At first thought, since culture and instruction are seen to embrace all possible educational processes, it would seem that these two branches constitute the entire science of education. THE NATURE OF EDUCATION. 15 A little further analysis, however, gives rise to another branch closelj'^ connected with these two primary branches, and possibly contained in them, but so important as to be regarded by some as coordinate with these two, and requiring a distinct treatment. Thus, since culture and instruction are to be given to a number of pupils together, called a school, and this school is to be organized, governed, etc., there ai'ise other subjects not immediately embraced in, or at least not conveniently treated under, the two primary branches of the science. On account of tlie intimate relation of these several subjects, educators have treated them under one head, and regarded it as a distinct branch of the science, which has been appropriately named by Dr. Wickorsham, School Economy. The science of education is thus seen to embrace three branches — Methods of Culture^ Methods of Instruction^ and School Economij. This three-fold division of the science is not new, although it is recent. It is not so much of a dis- covery as a growth. It seems to have been gradually de- veloping in the minds of educators for raau}'^ years, and is now largely accepted by the profession as a logical and com- plete classification. Culture and Instruction. — Culture is the developing of the powers of man. It is the art of drawing out the different powers and training them so that they may act with skill and vigor. Instruction is the furnishing of the mind with knowledge. The mind may be furnished with knowledge in two ways ; first by putting knowledge into the mind, and second by drawing know- ledge out of the mind. In the fact studies, as history and geog- laphy, knowledge must be put into the mind ; in the thought studies, as arithmetic and grammar, knowledge can be un- folded in the mind. Instruction is thus the art of putting knowledge into the mind and also of drawing knowledge out of the mind. In other words, instruction is the art of developing knowledge in the mind, or of building up knowledge in the mind. These two divisions, Culture and Instruction,are logically dis- ]6 METHODS OF TEACHING. tinguislied. The one seeks to draw out the powers of the mind; the other seeks to furnish the mind with knowledge. The for- mer is purely subjective, working from within outward ; the lat- ter is partly objective and partly subjective, as knowledge is both put into the mind and drawn out of it. Each, of course, implies the other. To give culture, we make use of knowledge ; in imparting instruction there must be some growth of the mental powers. The two processes, however, are not identical ; and the laws and methods of each are different. They are in fact the complements of each other ; the two hemispheres of the science, which, united, give it symmetry and completeness. Nature of Teaching The act of affording this culture and imparting this knowledge is called Teaching; and the person who does this work is called a Teacher. The term Teaching is also used as the name of the science and art of giving cul- ture and instruction. Thus we speak of the Science of Teach- ing and the Art of Teaching. The term Teaching, it is thus seen, is a little more compre- hensive than the word Instruction. An Instructor, strictly speaking, is one who furnishes the mind with knowledge ; a Teacher is one wlio furnishes the mind with knowledge and, at the same time, aims to give mental culture. Other Terms.— The term Educator is popularly defined as one who educates or gives instruction. It is more appropri- ately used, however, to denote one who is versed in, or wiio advocates and promotes, education. The term Educaiionid is also employed in this latter sense, and by many is pre- ferred to the term Educator. The term Pedagogics, or Pedagogy (jjais, paidos, a boy, and agogos, leading or guiding), is used by quite a large number of writers as the name of the science and art of in- struction. The term is popular in Germany, and efforts have been made to introduce it into this country and England; but so far with but little success. It is somewhat awkward and unmusical, besides which, the term pedagogue is, in both of these countries, used as a term of reproach. THE NATURE OF EDUCATION. 17 The term Didactics, from didan/co, I teach, is often used as the name of the science and art of teaching. The subject has been divided into two parts : General Didactics, which presents the principles of teaching ; and Special Didactics, or Methodics, which applies these principles to the several branches of instruction. The term is appropriate and may in time be adopted, but the term Teaching seems at present to be generall}' preferred. Kinds of Education — Education is generall}^ divided into Physical Education, Intellectual Education, and Moral and Religious Education. Phj'sical Education is that which per- tains to the body. Its object is to train every power of the body for the attainment of the ends of health, strength, skill, anil beauty. Intellectual Education is that which pertains to the intellect. Its object is to develop all the mental faculties into their highest activity, and to furnish the mind with valuable and interesting knowledge. Moral Education is that which per- tains to the moral nature of man. Its object is the develop- ment of conscience and the subordination of the will to the idea of duty. Religious Education has reference to the de- velopment of the higher spiritual instincts and sentiments forming the religious nature. It is especially distinguished from moral education in that the former finds its motive in human relations, and the latter in the existence of a Supreme Being. Besides these there are also several subordinate or collateral divisions; as Esthetic Education, which refers to the culture, of the imagination and taste; Domestic Education, which refers to the education of children in the household ; Common School Education, which refers to the education obtained in a common school ; Popular Education, which refers to the education of the people ; and National Education, which refers to a system of education provided by the state. CHAPTER II. GENERAL PRINCIPLES OF EDUCATION. EDUCATION is not a matter of chance or haphazard pro- cedure. All development must proceed in accordance with some regular plan or order. There can be no organic grow-th without the control of principles determining and shaping the development. The plant grows in obedience to the laws of vegetable life ; and the development of mind, which is the object of education, must be controlled by the laws of its own being. A system of education must therefore be based upon certain broad and fundamental principles which express the laws of human life and development. These principles are not only the foundation upon which the system rests, but thej^ give shape and character to the entire superstructure. All the great writers on education have conceived some lead ins: ideas and endeavored to unfold a scheme of instruction growing out of these fundamental conceptions. From a xerj careful survey of these different schemes and a thorough examination of the problem of education itself, the following principles have been reached which seem to contain a complete s^'Stem of education. These principles, it ^is thought, embrace all the fundamental- ideas of education from Aristotle to Pestalozzi and Froebel. The design is to enumerate only the general laws of education ; the particular laws of culture and instruction will be presented in another place. These principles are presented in ten propositions, which we may call our educational decalogue. 1. The primary object of education is the perfection of the individual. The educator should understand the object (18) GENERAL PRLN'CIPLES OF EDUCATION. 19 for which he labors; for the object to a large extent de- termines the means and methods emploj-ed in the work. A correct end in view will lead to correct methods; a false object will vitiate both the means and the methods of usins; them. In education, especially, the end aimed at croMTis the work with excellence. The true object of education has not been generall}' under- stood by educators and parents. The ancient Greeks made a fundamental mistake when they based their s^'stem upon the I^erfection of the state rather than the individual. Parents to-day send their children to school to fit them for business or a profession, to enable them to make a good living in the world, or to occupj- an honorable position in soeiet}-. Teachers often seem to think more about the amount of knowledge they are imparting to the child than of the training of its mind and the development of a manl}' and virtuous character. All of these objects fall below the high ideal we should set before us, and degrade and injure the work of education. We should, therefore, remember that the true object of edu- cation is the perfection of the individual. We should aim for the perfecting of man in his entire nature, — physically, men- tally, and morally. The teacher should never forget that the highest object of his work is the fullest and most complete development of the immortal beings committed to his care : and that his work is not onlj' for time but for eternitj-. In other words, it should be remembered that the highest object of education is human perfection. 2. The perfection of the individual is attaitied by a har monious development of all his powers. Man possesses a multiplicity of capacities and powers, all of which contribute to his well-being and his dignity. A perfectly' developed man- hood or womanhood implies the complete development of every capacity and gift. These powers are so related that they may be unfolded in very nearly' equal proportions, and harmoniously blend in the final result. For the attainment 20 METHODS OF TEACHING. of our ideal such a development is required. The educational work should reach ever}^ power, and aim at a full and har- monious development of them all. This principle is limited by the existence of special talents and the demand for special duties. While a general scheme of education should seek to give culture to all the powers, we should not be neglectful of special and unusual gifts. Genius should be recognized, and our general system be so far modi- fied as to give opportunity for its highest development and achievements. An unusual gift for poetry, music, painting, mechanics, mathematics, etc., should be recognized, and oppor- tunity offered for its fullest development. We must remember also that duties are diverse as well as talents, and that special training is needed for the preparation of mankind to discharge these special duties. There must be farmers, and artisans, and physicians, etc., and they need special preparation for their work ; and educational systems must recognize this fact and provide for it. The principle of harmonious development has reference to that general educational preparation which all persons need for their own personal excellence, and as a preparation for a special course of instruction to prepare them for specific duties and occupations. The general scheme of education should therefore aim at a full and harmonious development of all of man's powers. 3. These poivers develop naturally in a certain order, which should be followed in education. Intellectual life seems to begin in the senses ; the child awakens into knowledge through sensation and perception. Then follows the action of the memory as a retaining and a recalling power, accompanied by imagination as the power of representation. After this come judgment and reasoning and the power of abstraction, seneralization, and classification. Still later we become con- scions of the intuitive ideas and truths, and learn to work them up into new truths by the power of deductive thought. GENERAL PRINCIPLES OF EDUCATION. 21 Last of all, the mind awakens to the consciousness of man aa a moral and religious being, bearing relations to his fellow man and to God. Finding in man such a relation of faculties and powers, we should learn the order of their development and follow that order in our work. TVe should first afford food for the growth of the mind through the senses. We should call the memory into activity, and afford means for the culture of the imagina- tion. We should lead the mind gradually from things to thoughts, and give activit}' to judgment and reasoning, and also to the powers of abstraction and generalization. Desires should be awakened and directed, the affections unfolded, and the will be subordinated to the ideas of truth and duty Though these powers develop in a certain order, it is not to be thought that the activity of one waits upon the full development of another. To a certain extent they are all active at the same time ; but they are active in different de- grees. The order given represents the relative activity, and thus indicates the relative attention required to be given them in the work of education. Such a relation should be clearly understood by the educator, and should guide him in his work. 4. The basis of this development is the self-activity of the child. Education is a spiritual growth, and not an accre- tion. It is a development from within, and not an aggrega- tion from without. For this growth there must be forces working within the child. This force is the self-activity of the soul, going out towards an object as well as receiving impressions from it ; gaining power in the effort, and work- ing up into organic products the knowledge thus acquired. The object of education is to stimulate and direct this natural activity. The teacher, therefore, should never do for the child what it can do for itself. It is the child's own activity that will give strength to its powers and increase the capacity of the mind. The teacher must avoid telling too much, or aiding the child too frequently. A mere hint or 22 "METHODS OF TEACHING. suggestive question to lead the mind in the proper direction is worth much more than direct assistance, for it not only gives activity' and consequently mental development, but it cultivates the power of original investigation. We should aim to cultivate a taste and desire for knowledge on the part of the child, so that this activity may be natural and healthful. To force the mind to the reception of knowl- edge is not education, it is cramming ; and tlie object of educa- tion is not cram but culture. For the attainment of the high end of education, therefore, we must depend on the self- activity of the child ; and it is the teacher's office to excite and direct this activity. 5. This self-activity has two distinct phases; from without inward^ — receptive and acquisitive; and from within out- ward^ — productive and expressive. First, the mind is re- ceptive of knowledge. Objects of the material world make their impressions upon the senses, and ideas and thoughts spring up in the mind. Knowledge thus comes into the mind from without through the senses. The contents of books also flow into the mind through written language, and are treas- ured in the memory. In all this the mind is receptive, the process is from without inward, and the result is acquisition, learning. The mind is also active in creating as well as in receiving. It has the power to reproduce as well as to receive. In its self-activity it can take the material thus acquired, and work it up into new products. It can also send it forth on the stream of clear and definite expression in audible or visible speech. It thus works from within outward, creating, and evolving what it creates. The mind in its receptive phase is said to be intuitive; that is, the knowledge comes directlv into the mind. The mind in the second phase is called elaborative, because it works up the material into new products. This distinction has also an educational significance. GENERAL PRINCIPLES OF EDUCATION. 23 6. These two phases, the receptive and productive, should go hand in hand in the work of education. This is evident from their natural correlation. The activity of the mind in receiving naturally creates the correlative activitT^ of produc- ing. The knowledge coming into the mind through the re- ceptive capacity excites the mind to a productive activity. It acts like food in the stomach, which excites the powers of digestion and assimilation. Besides, the knowledge gained by the receptive powers becomes the material for the produc- tion of the creative powers. This material is operated upon and worked up into new products. These two operations are not to be separated in education. Each gives life and vigor to the other. The receptive powers are stimulated by the activity of the productive powers, and the productive powers are set into immediate activity by the presence of receptive knowledge. They thus play into each other's hands, act as a mutual stimulus to each other, and should go hand in hand in the work of education. 7. There must be objective realities to supply the condition for the self-activity of the mind. The mind cannot act upon itself alone ; there must be food for the mental appetite. There must be an external world of knowledge to meet the wants ol the internal knowing subject. Such an external world is supplied. There is a world of knowledge suited to and correlating with the wants of the soul. The objective world of nature is found to be an embodiment of thought, and this thought developed into science meets the wants of the active spirit. There is also the great world of space and number, with its ideas and truths ; and also the loftier abstractions of the True, the Beautiful, and the Good. This world of knowledge is adapted to every power and capacity of the mind. This is evident, since knowledge is the product of the mind operating upon external realities. Knowl- edge as the product of one mind must be suited to the differ- ent capacities of all other minds. It is thus seen that, there 24 METHODS OF TEACHINQ. is abundant provision for the activity and growth of all of the powers of the mind. 8. Education is not creative ; it only assists in developing existing possibilities into realities. The mind possesses in- nate powers. These may be awakened into a natural activity. The desisrn of education is to aid nature in unfolding the powers she has given. No new power can be created hy edu- cation; the object is to arouse those which exist to a health- ful activity, and to guide them in their unfolding. In other words, the object of education is to aid nature in unfolding the possibilities of the child into the highest possible realities. 9. Education should be modified by the different tastes and talents of the pupil. All minds possess the same general capacities or powers. These powers are, however, possessed in different degrees. An unusual gift of any one or more powers constitutes genius. Tastes or dispositions for par- ticular branches of science or art also differ. Such differences should not be overlooked in a scheme of education. While all should receive a course of general cul- ture, opportunity should be given for the development of special tastes and gifts. It is these which enrich science and art, and add to the sum of human knowledge ; and the pro- gress of science and art demands that genius shall have the most abundant opportunities for its fullest and highest devel- opment. 10. A scheme of education should aim to attain the triune results — development, learning, and efficiency. Development relates to the culture and growth of the powers of the child. This is the fundamental idea of education, and is of primary importance. Education has reference also to the acquisition of knowledge. It aims to enrich the mind with the truths of science, to make a man learned, to produce scholars. A third object is the acquisition of skill in the use of culture and knowledge. It is not enough that the mind has well-devel- oped powers and is richly furnished with knowledge. There GENERAL PRINCIPLES OF EDUCATION. 25 should be the power to make use of this culture and knowl- edge. The educated man should be able to do as well as to think and know. This third design of the educator, the attain- ment of skill, should not therefore be overlooked. The true aim of education is thus seen to be the attainment of the three ends — culture, knowledge, and efficiency. 8 CHAPTER III. THE SCIENCE OF TEACHING, TEACHING, as a science, treats of the Laws and Methods of human Culture and Instruction. The terra is derived from the Saxon word tcecan, which meant to show, to teach, and is allied to the Greek deiknunai^ to show, and the Latin docere, to teach. Primarily, the word appears to have meant very nearly the same as the word instruction ; though even in its primary sense of directing or showing, it is suggestive of the act of develop- ing the mind as well as instructing it. At the present daj''. Teaching embraces both Culture and Instruction, — the bring- ing out and training of the powers, as well as the furnishing or the mind with knowledge. A true teacher seeks to culti- vate the minds of his pupils as well as to instruct them. Laivs and Methods. — The definition of Teaching embraces four distinct and prominent ideas, — Laws, Methods, Culture, and Instruction. Bv Laws we mean the principles that guide us in an operation. Thus, in grammar, the principle that the verb agrees with its subject in number and person, will guide us in speaking and writing correctl}'. So the principles of numV)ers enable us to operate with them correctly in applying them to the business transactions of life. B}' Methods we mean the manner of performing an opera- tion. Thus, in arithmetic we have the methods of subtracting, of finding the greatest common divisor, etc. The rules of arithmetic are statements of methods of operation. So also in education, there are methods of doing things or of obtaining certain results. There are methods of giving culture to the dilferent faculties, and also of teaching the dillerent branches. (26; THE SCIENCE OF TEACHING. 27 The relation of Laws and Methods should be clearl}^ under- stood. Principles are self-existent, or belong to the very na- ture of the subjects ; Methods are derived from principles ; they are the outgrowth of laws or principles. Principles are of more value than methods ; if you know the principle, you can derive the method, though you may know the method without understanding the principle. One who is familiar with principles is thus much more independent than one who knows only methods. These relations of principles and methods may be illustrated in arithmetic and grammar, and in other school studies. Cultiir'e and Instruction. — Culture is the developing of the powers of man. The term is derived from colo, I culti- vate, and derives its educational meaning from the act of tilling and enriching the soil. It has reference to the deA'^elopment and improvement of any of man's faculties or powers. To awaken the mind into activity, to call out and mould its vari- ous faculties, to train the e3'e to see, the memory to retain and recall, the understanding to think and reason, etc,— this is to cultivate the mind. Instruction is the furnishing of the mind with knowledge. It is the process of developing knowledge in the mind of an- other. The term is derived from i?i, into, and atruo^ I build, meaning, I build into. To instruct the mind is thus to furnish it with knowledge, to build up knowledge in the mind. The instructor takes the knowledge that is in his own mind, and puts it into the minds of his pupils ; or he dcA'elops knowledge in the minds of his pupils, and builds it up there, as an archi- tect erects a temple, in symmetry and proportion. The relation of Culture and Instruction should be clearly understood. The object of Culture is to strengthen and de- velop the mind ; the object of Instruction is to furnish the mind with knowledge. Culture gives a person mental power ; Instruc- tion gives him information or learning. They are both impor- t:iut ; but Culture is more important than mere Instruction. 28 METHODS OF TEACHING. To be able to acquire knowledge is worth more than the knowl edge we have acquired. The ability to originate knowledge is even more important. A person should know more than he ever learned ; and this is possible when his powers have been cultivated. The object of the teacher, therefore, should be not merely to impart knowledge, but to cultivate mental power. Teachbiff a Scieiice. — Teaching is both a science and an art. That it is a science, which has l)«.'cn questioned, will appear from the following considerations: To constitute a sci- ence we must have three things: 1. Knowledge; 2. Knowl- edge sj'stematized ; 3. Principles showing the relations of this knowledge, and binding it together into an organic unity. There is a knowledge of teaching, as is attested by the many works and articles written upon the subject. This knowledge can be S3^stematized as logically as the knowledge of grammar or arithmetic. There are also fundamental principles of teach- ing, which express the laws of culture and instruction. Hence, from the delinition of a science, we can claim that there is a science of teaching. Branches of Teaching. — The Science of Teaching is divided into three branches ; llethods of Culture, Methods of Instruction, and School Economy. This three-fold division embraces everything that pertains to teaching, and is therefore regarded as exhaustive. Indeed, as previously stated, since man can only be cultured and instructed, it would seem that there could be only two distinct branches. Methods of Cul- ture and Methods of Instruction ; but since this work is to be done with the pupils organized into a school, and since such an organization gives rise to special regulations and provis- ions, there incidentally arises a third division, called School Economy. Methods of Culture treats of the nature of the powers of man, and how to develop them. It embraces three general divisions: 1. The Nature of Man; 2. The Nature of Culture; 3. The Methods of Cultivating each Faculty. In a ftdl treat- THK SCIENCE OF TEACHING. 29 ise upon this subject, each one of these topics should be dis- cussed in detail. Methods of Instruction treats of the different branches of knowledge and how to teach them. It embraces three general divisions: 1. The Nature of Knowledge; 2. The Nature of Instruction; 3. The Methods of Teaching each Branch. In a full treatise upon the subject, each one of these topics should he discussed in detail. The three divisions of these two branches of Teaching, are seen to correlate. Thus the Nature of Man in the first branch correlates with the Nature of Knowledge in the second ; the Nature of Culture in the former corresponds to the Nature of Instruction in the latter; and the Method of Cultivating each Power is correlative with the Method of Teaching each Branch. As the two branches stand in the relation of the subjective and the objective, so do the corresponding divi- sions of the two branches. School Economy treats of the methods of organizing and managing a school. It embraces five things: 1. School Pre- paration; 2. School Organization; 3. School Employments; 4. School Government ; 5. School Authorities. This classifi- cation is that presented by Dr. Wickersham, and is regarded as logical and complete. The relation of the several branches of the Science of Teach- ing, together with a few of their practical divisions, is expressed in the following outline : {Nature of Man. Nature of Culture. Method of Cultivating each Faculty. (Nature of Knowledge. Nature of Instruction. Method of Teaching each Branch. School Preparation. School Organization. School Employments. School Government. School Authorities. 12; S5 a: o o 3. School Economy. 30 METHODS OF TEACHING. In this work we design to speak mainly of Methods of Instruction, but so necessary is a knowledge of the mind and the methods of training it that we shall give a single chapter to each of the three divisions of Methods of Culture; namely, The Nature of Mind, the Nature of Culture, and the Methods of Cultivating each Faculty. We shall then speak of the Nature of Knowledge and the Nature of Instruction, embrac- ing under the latter head, Forms of Instruction, Order of Instruction, and Principles of Instruction. Having laid this foundation, we shall proceed to the consideration of the Methods of Teaching each Branch of Knowledge. CHAPTER IV. THE NATLTRE OF THE MIND. THE Mind is that which thinks, feels, and wills. It is that immaterial principle which we call the soul, the s^'irit, or the intelligence. Of its essence or substance, nothing is known ; we know it only b}' its activities and its operations. The ditferent forms of activity which it presents, indicate diJferent mental powers, which are called Faculties of the mind. A Mental Facixtt is a capacity for a distinct form of mental activity. It is ilie mind's power of doing something, of putting forth some energy, of manifesting itself in some particular manner. The mind possesses as many faculties as there are distinct forms of mental activity. In order, there- fore, to ascertain the different faculties of the mind, we must notice carefully the various ways in which the mind acts. General Classification. — The mind embraces three general classes of faculties ; the Intellect., the Sensibilities, and the Will. Every capacity or power which the mind possesses falls under one of these three heads. Every mental act is an act of the Intellect, the Sensibilities, or the Will. This three-fold division of the mind is the latest teaching of philosophy. These three classes of faculties are not to be consid- ered, however, as parts of a complex unit, but rather as forms of manifestation of the spiritual entity which we call The 3Iincl. The mind is thus a tri-unity, — one substance with a trinity of powers or capacities. The doctrine of the Trinity of the Mind is thus a fundamental fact of Psychology. The Intellect is the power by which we think and know. Its products are ideas and thoughts. An Idea is a mental pro- (31) 82 METHODS OF TEACHING. duct which may be expressed in one or more wf)rds, not forming a proj)Ositi<)n ; a^, a man, an aiiimu/, etc. A Thouirht is a mental product consisting of the combination of two or more ideas, which when expressed in words, gives us a proposition ; as, a man is an animal. The Sensibilities are the powers by which we feel. Their products are emotions^ affections^ and desires. An emotion is a simjde feeling, as the emotion of joy, sorrow, etc. An affec- tion is an emotion that goes out towards an object ; as love, hate, envy, etc. A denire is an emotion that goes out to an object with the wish of possession ; as the desii'e of wealth, fame, etc. The Will is the power by w^hich we resolve to do. It is the executive power of tlie mind, the power by which man becomes the conscious autlior of an intentional act. The products of the Will are volitiotis and voluntary actions. It is in the domain of the Will that man becomes a moral and responsible being. The relation of these three spheres of activity may be illus- trated in a variety of ways. I read of the destitution and suffering in a great city and understand the means taken for their relief; this is an act of the intellect. I feel a deep sym- pathy with this suffering ; my heart is touched with pity, and I experience a strong desire to aid in relieving their distress; this is an act of the sensibilities. I desire to express my feelings of pity and follow my sense of duty, and resolve to aid them by sending a contribution or going personally to their relief; this is an act of the will. The Intellect. — The Intellect embraces several distinct faculties; Perception, Memory, Imagination, Understanding, and Intuition, or the Reason. This classification of the Intel- lect is how almost universally accepted, though writers occa- sionally differ in the terms they use to name the different powers. Perception is the power by which we gain a knowledge of NATURE OF THE MESTD. 33 external objects through the senses. It is the faculty b}^ which we gain a knowledge of objects and their qualities. Its pro- ducts are ideas of external objects and of the qualities of ob- jects. The ideas which we possess of persons, places, things, etc., are mainly given b)^ perception. Memoky is the power by which we retain and recall knowl- edge. It enables us to hold fast to the knowledge we have acquired, and also to recall it when we wish to u?e it. These two offices of the Memory are distinguished as lietention and Recollection. By some writers these are regarded as separate faculties; and others again discard the element of retention. Besides these, the memory also gives us a representation of that which it recalls, and recognizes it as something of our past experience. Imagination is the power by which we form ideal concep- tions. It is the power of forming mental images by uniting different parts of objects given by perception, and also of creating ideals of objects different from an3'thing we have per- ceived. Thus, I can conceive oi a, flying horse by uniting my ideas of wings and a horse ; or I can imagine a landscape or a strain of music different from anything I have ever seen or heard. Imagination is thus the power of ideal creation. The Undebstandinq is the power by which we compare ob- jects of thought and derive abstract and general ideas and truths. It is the elaborative power of the mind; it takes the materials furnished by the other faculties and works them up into new products. Its products are abstract and general ideas, truths, laws, causes, etc. Intuition, or the Reason, is the power which gives us ideas and thoughts not furnished by the senses nor elaborated by the Understanding. Its products are called primary ideas and primary truths. The Primary Ideas are such as Space, Time, Cause, Identity, the True, the Beautiful, and the Good. The Primary Truths are all self-evident truths, as the axioms of mathematics and logic. 2* S4 METHODS OF TEACHING. The Understanding. — The Understanding embraces sev- eral distinct faculties or forms of mental activity. These are Abstraciicn, Conception, Judgment, and Reasoning. This division is now almost universally adopted, and the same terms are employed by nearly all modern -writers. Abstraction is the power of forming abstract ideas. It is the power by which the mind draws a quality away from its object, and makes of it a distinct object of thought. Its pro- ducts are abstract ideas, such as hardness, softness, color, etc. The naming of abstract ideas gives us abstract terms. The term Abstraction is derived from ab, from, and traho, I draw, and signifies a drawing from. Conception is the power of forming general ideas. By it we take several particular ideas, and unite their common prop- erties, and thus form a general idea which embraces them all. The products of Conception are general ideas, or ideas of classes; as horse, bird, man, etc. The naming of general ideas gives us common terms. This faculty is often called generalization ; but the term Conception is more appropriate, and is the one generallj- adopted by logicians. The term Conception is derived from con, together, and cajno, I take ; and signifies a taking together. Judgment is the power of perceiving the agreement or disa- greement of two objects of thought. It is the power of com- parison. It compares one object directly with another, and gives us a proposition. A proposition is a judgment ex- j tressed in words. Thus, a bird is an animal, is a judgment expressed. The term Judgment is applied to both the mental facult}^ and its product. Reasoning is the power of comparing two ideas through their relation to a third. It is a process of indirect or medi- ate comparison. It deals with three objects of thought and requires three propositions. Thus, suppose I wish to com- pare A and B, and perceiving no relation between them, see that A equals C, aad B equals C, and thus infer that A equals B ; such an inference is an act of reasoning. NATURE OF THE MIND. 35 The form in which reasoning is expressed is called a Syllih yism. A Sj'Ilogism consists of three propositions so related that one of them is an inference from the other two. Two of these propositions are called the i^remises and the third the conclusion. Thus, in the above example the two propositions, "A equals C" and "B equals C," are the premises; and "A equals B" is the conclusion. Reasoning is of two kinds; Inductive Seasoning and De- ductive Reasoning. Inductive Reasoning is the process of deriving a general truth from particular truths. Thus, if I find that heat expands several metals, as zinc, iron, copper, etc, I may infer that heat will expand all metals. Such an inference of a general truth from the particular facts is called Induction. Inductive Reasoning proceeds upon the principle that ichat is true of the many is true of the whole. Deductive Reasoning is the process of deriving a particu- lar truth from a general truth. Thus, from the general propo- sition that heat expands all metals, I may infer by Deduction that heat will expand any particular metal, as silver. Deduc- tion proceeds upon the principle that what is true of the whole is true of the parts. Other Forms of Mental Activity. — Besides the faculties now named, there are two other forms of mental activity, or mental states, called Consciousness and Attention. These are not regarded as specific faculties of the mind, but as condi- tions or accompaniments of these faculties. Cotisciousness. — Consciousness is the power or attribute of the mind by which it knows its own states or actions. The term is derived from con, with, and scio, I know, and means a knowing with the mental acts or states. It is regarded as an attribute of the mind, involved in the very idea of mind, and not as a mental faculty. Thus, to know is to know we know, to feel is to know we feel, to will is to know we will. The expressions, "I know that I know," "I know that I feel," etc., are equivalent to I am conscious that 1 know, I am con- S6 METHODS OF TEACHING. scious that I feel, etc. Coiisciousness is a kind of inner light by which one knows what is going on within his mind ; it is a revealer jf the internal phenomena of thought, feeling, and will. Atteiition. — Attention is the power of directing the mind voluntarily to any object of thought to the exclusion of others. It is the power of selecting one of several objects, and concentrating the mental energies upon it. The term is derived from ac^,to, and tendo^ I bend, which was probal ly suggested by the attitude of the body in listening attentively to a sound. Attention is not a distinct form of mental activity, but is involved in and underlies the activity of all the faculties. The voluntary operation of any of the mental powers, as Per- ception, Memorj^, etc., carries with it an act of attention. It is not the power of knowing, but of directing that which ma;' know. It has no distinct field or province of its own, yet, without it the faculties would be of little use to us. It works with them and through them, increasing their efficiency, and giving them a power they would not otherwise possess. Conception. — The term Conception is often used in a gen- eral and popular sense, meaning that power which the mind has of making anything a distinct object of thought. In this sense it is intimately related to all the mental faculties. Thus I can conceive of a tree or a horse which I have seen, a land- scape which I may not have seen, a proposition in geometry, a truth in natural philosophy, etc. Some writers have used the term in a more specific sense, as the power of forming an exact transcript of a past perception. In Logic the term is restricted to the power of forming general ideas, as we have previously defined it. CHAPTER V. THE NATURE OF CULTURE. CULTURE, as alread\^ defined, is the developing of the powers of man. It aims at the unfolding and growth of all the powers, and the training of them so as to attain their highest activit}'^ and fullest development. As in the culture of land we him to improve the soil, so in human cultui-e we aim to enrich the soil of the mind and cause it to bud and blossom and bring forth its richest harvests of thought and sentiment, of science, art, and character. Culture is usually divided into three distinct branches; Physical Culture, Intellectual Culture, and Moral and Relig- ious* Culture. Besides these there are also Social Culture, ^Esthetic Culture, Spiritual Culture, etc. These are, however, but varieties or special forms of those before mentioned, which are the ones generally embi-aced in a scheme of education. Physical Culture. — Physical Culture is that which relates to the cultivation and development of the physical powers. It embraces the culture and attainment of Health, Strength, Skill, and Beauty. A full discussion of the subject would include a consideration of the conditions, the laws, and the methods of securing each one of these objects. The first object of physical culture is Health. To a large extent man's health is in his own keeping. We can be sick or well as we choose. Sickness is the penalty of violating physical law. Death, except in old age, is a curse entailed upon man by his transgressions. A proper physical culture would banish disease and premature death from the land. It would increase the average term of life from thirty-three to, at least, threescore and ten j-ears. Physical culture seeks to (37) 38 METHODS OF TEACHING. ascertain the laws and methods by which these results are secured, and to present a sound body as a condition of a sound and vigorous intellect. The second object of physical culture is Slretiyth. By cul- ture a man can double or treble his natural strength ; and not transcend the limits of health. A proper physical culture would remove the bodily weakness which we find so prevalent in society. It would give muscular fibre and endurance where we now find flabbiness and debility. It would give physical power to our professional and business men, and enable them to endure much more fatigue and to accomplish nnuch more than they can at present. It would transform the delicate and frail-looking women of to-day, who cannot go upstairs without palpitation of the heart, or see a spider without faint- ing or shrieking, into women of muscular power and endur ance, such as were the women of Sparta, and as nature intended women to be. The third object of physical culture is Skill. This is also an object worthy of attainment. To use the muscles with dexterity, either for pleasure or business, is in itself laudable. There is a merit in being a good g3'mnast, or a good cricketer or base ball player. To walk far, run fast, jump a good dis- tance, etc., are not unworthy attainments. It is told to the credit of Washington that he could leap twentj'-four feet on a running jump. To possess manual skill and be able to use our hands for some useful purpose is especiallj^ desirable. A knowledge of a use of tools is of great value to every person. " Every man his own carpenter" is worth as much as " every man his own lawyer." Education should therefore aim to cultivate muscular skill and dexterity'. A fourth object of physical culture is the attainment of Beauty. Deformity, like siclaiess, is the result of vioUited phj'sical law. Had sin never entered Paradise, men woukl be as handsome as Adam, who was no doubt a model in phj-sical proportion ; and womer. would still be as lovely as NATURE OF CULTURE. 39 Eve, who was, it is believed, the perfection of womanly beauty. Let the race keep nature's laws, and we would return towartl the primitive beauty fashioned by the Divine hand. Art does much to restore what we have lost ; but culture is the best panacea for ugliness. The best coloring for the cheek is pure, rich blood; the best enamel for the neck and arms is the flush of health. Such beauty does not rub off nor come and go with the touch of art — " 'T is ingrain ; 't will endure Vvind and weather." Intellectual Culture Intellectual Culture is that which relates to the development and training of the intellect- ual powers. The object of intellectual culture is the normai growth and highest activity of all the intellectual faculties. A full consideration of the subject would present the laws and methods b}- which each susceptibility^ and power may be properl3' trained and developed. Only a few thoughts will be here presented. Intellectual Culture aims to cultivate the powers of Obser- vation. It enables man to see what is going on around him, and to acquire a knowledge of facts and phenomena. It makes him sharp-eyed and ready to drink in knowledge at every pore. It makes him an original observer of nature and society, obtaining his knowledge first hand, instead of de- pending on others for it. It thus gives him independence in his own ideas of things, and enables him to make contribu- tions to the sum of human knowledge. Intellectual Culture increases the power of the Memory It gives strength of retention and readiness of recollection It makes man a treasury- of knowledge, — a walking library of information. It aims to overcome the habit of allowing things to fade away from the memory, and trains the mind to hold what is worth knowing as a permanent possession. It aims to bring the memory up towards the old standard of power when men could repeat volumes of manuscript, or "call by name the twentv thousand citizens of Athens." 40 METHODS OF TEACHING. Intellectual Culture aims to give activity and direction to the power of Imagination. It leads it to delight in ideal creations, to enjoy the works of fiction, to wander with pleas- ure among the images of poetr}^, to linger delighted amid the romantic events of history, to awaken into activity in view- ing the varied beauties of earth, sea, and sky, and to revel among the works of art where the pencil of the painter or chisel of the sculptor has made a name immortal. It aims also to develop the creative power of artistic genius, and to stimulate those who have the gift divine to emulate the achieve- ments of the masters in poetry, fiction^and fine art. Intellectual Culture embraces the training of the power of Thought. It aims to make man a thinker, to enable him to draw true conclusions from the facts he observes, to exercise correct judgment in the affairs of life, to investigate and ascer- tain the laws of nature and society, to read the truths which God has written upon the pages of earth and sky, to build up the sciences and apply their principles to the advancement of truth and the improvement of the world. It aims to develop the power of thought by which man lifts himself into a higher civilization, makes the elements servants of his will to pro- mote his comfort and happiness, arms himself with the power to predict the events of the far off" future, and stands at the head of created beings, crowned with the triumphs of science and philosophy. JEsthetic Culture. — J^sthetic Culture embraces the culti- vation of the aesthetic nature. The aesthetic nature includes the activity of the Reason and the Sensibilities as pertaining to the beautiful. The Reason apprehends beauty ; the Sensibili- ties admire, appreciate, and enjoy it. Esthetic culture seeks to develop this nature to the fullest appreciation of the element of beauty as found in the works of nature and art, to lift the soul upward to the enjoyment of the refined and artistic, to refine and elevate the taste, and thus add to man's happiness and lend an influence for the growth of his spiritual nature. NATURE OF CULTURE. 41 Moral Culture. — Moral Culture embraces the training of the moral nature. The moral nature includes the activity of the entire spiritual being; it involves the activity of the Intellect, the S'^nsibilities and the Will. The Reason appre- hends the Right and the obligation to do the Right; the Sensi- bilities feel the obligation to act in accordance with an appre- hension of obligation ; and the Will puts forth the executive volition in obedience to the spiritual imperative. The M%- thetic nature consists of idea and feeling; the Moral nature consists of idea, feeling, and will. In mathematical phrase- ology, the ^Esthetic nature=the Reason, plus the Sensibilities ; the Ethical nature=the Reason, plus the Sensibilities, plus the Will. Moral Culture embraces the full and complete de- velopment of this nature. Jieligious Cultm^e. — Religious Culture embraces the train- mg and development of the religious nature. The religious nature is the highest form of the ethical ; it is the ethical acting in relation to the Supreme Being. It implies the con- secration of all our powers to God, and requires their fullest and highest activity. The highest operation of the Reason is Faith ; the highest operation of the Sensibilities is Love ; the highest operation of the Will is Obedience. The elements of religion, therefore, are Faith, Love, and Obedience ; Faith in God and salvation ; Love to God and man ; Obedience, the complete subordination of the human will to the Divine. Here w^e reach the crowning excellence of man's being, the keystone of the spiritual arch. Religious culture thus aims to cultivate faith in God, love to God and man, and complete obedience to the Divine will. CHAPTER YI. METHODS OF CULTIVATING EACH FACULTY. HAVING attained a knowledge of the nature of the mind and the general nature of culture, we are prepared to apply these to the training of each faculty' of the mind. Only a few brief suggestions can be made ; the subject would require a volume to treat it with any degree of completeness. Perception. — The Perceptive Powers should be cultivated in early childhood. This is indicated by Nature, who gives active senses to a little child. Teachers haA'e been entirely too neglectful of their duty in this respect. Children have not been trained to use their eyes and their other senses as they should have been. They have been taught to read the text-books of the school-room ; but, to a large extent, the "book of Nature" has been a sealed volume to them. The Perceptive Powers may be cultivated by training chil- dren to a habit of observation. The following suggestions will indicate to teachers the method of cultivating the per- ceptive faculty : 1. To cultivate the Perceptive Powers, require pupils to observe things for themselves. Bring objects into the school- room for them to see and examine. Send them out into the fields and woods to gather facts for themselves. Teach them to read the book of nature as well as the books of the school- room. 2. To cultivate the Perceptive Powers, require pupils to describe objects. In order to descrilje an object, it must be very closely observed. The attempt to describe will lead pupils to see the necessity of examining an olvject with atten- tion, and will give quickness and accuracy to the perceptive powers. ^ ^2 ^ CULTIVATING EACH FACULTY. 4b 3. To cultivate the Perceptive Powers, train pupils upon a well graded system of object-lessons. Give the pupils lessons on form, color, size, etc., and they will learn to notice these elements in the objects that they see. The sense of vision will thus become sharp, delicate, and accurate. 4. Require pupils to draw outlines or sketches of objects. In order to draw an outline of an object it is necessary to examine it very minutely. The practice of drawing Avill thus cultivate the habit of close and minute observation. Such exercises will train pupils to the habit of using their perceptive powers, and habit is nearly everything in educa- tion. Teachers should also impress upon the minds of their pupils the importance of using their eyes, and not going through the world blind to its most interesting facts. The 3Iemory. — The Memory should be carefully trained in youth, so that it may firmly hold the knowledge acquired and readily recall it. Minds differ in natural power of mem- ory, but much can be done to strengthen a weak or quicken a sluggish memory. A neglect of the proper use of this faculty leads to habits which weaken it, and make it slow to acquire and unreliable in recalling its knowledge. The followinsr suo;o;estions will indicate to the teacher how he may cultivate the memory of his pupils : 1. To cultivate the Memory , require pujnls to attend closely to xohatever subject they are considering. Attention is a necessary condition of remembering. A heedless mind soon forgets what it sees, hears, or reads. The mind must be con- centrated upon the object of thought that it maj- be indelibly impressed ui)on the memory. 2. To cultivate the Ifemory, lead jmpils to feel an interest in ivhat you wish them to remember.- An interested mind is open to receive the deepest impression. An incident which excites the mind is never forgotten. A pupil who takes delio;ht in what he is learning will have little difficultv in acquiring it, and will retain it permanently. 44 METHODS OF TEACHING. 3. To cultivate the Memory^ we should require a frequent review or repetition of that which the pupil has learned. Repetition seems to fix a subject more firmly in the memory. It acts like the die on a waxen tablet ; every repetition seems to make the impression more durable. The subject most fre- quently recited is the most readily recalled, and remains the longest in the memory. 4. To cultivate the Memory^ we should require pupils to commit many extracts of prose and poetry. This will fix words and forms of expression in the mind, and cultivate a memory for language. Practice of this kind will give great facility in committing, while a neglect of it will so enfeeble the memory that it will be almost impossible to commit any- thing. 5. -To cultivate the Memory, we should lead the pupils to connect their knowledge by the laws of association. This is the way in which the memory naturally acts, and in which it acts with the most readiness and accuracy. The pupil should associate similar facts in geography, events of the same date in history, or those related as cause and effect. Such a habit will give a strong and reliable memory. Tiie Imagination. — The Imagination of children should be carefully cultivated. This faculty is usually very active in childhood, and needs guiding and refining. When it is sluggish, it should be excited and aroused into activity; when it is too active, it should be restrained and directed. The judicious training of this faculty will be of great value to the pupil. It will be a source of pure and refined pleasure, and will exert an elevating influence on the character. 1. The Imagination may be cultivated by observing beauti- fd, grand, and picturesque scenery. The spreading land- scape, the flowing river, the wide extended ocean, the arching sky, out of whose deep blue the golden stars are shining, the moon in her beauty and the sun in his splendor — all these tend to give activity and culture to the imagination. CULTIVATING EACH FACULTY. 46 2. The Imagination may he cultivated by filling the memory ivith beautiful pictures of natural scenery. The beautifuJ objects we have seen should be brought before the mind as pictures upon which it delights to look. Each mind may thus be a gallery where pictures of beauty hang upon the walls of memory, exciting the imagination to activity and furnishing it with pure and lofty ideals. 3. The Imagination may be cultivated by reading poetry^ fiction^ and other imaginative compositions. Such produc- tions are the embodiments of the imaginings of others, and awaken our own imaginations into activity. The figures of the poet, the characters and incidents of fiction, linger in the memory and stimulate us to create for ourselves such images of beaut}'' and incidents of life. 4. The Imagination may be cultivated by hearing music, visiting galleries of painting, statuary, etc. Here we have the embodiment of imaginative beauty in color and form, which pleases and excites the fancy. That which was once in the imagination of the creator awakens a similar activit}' in the mind of the beholder. There is thus cultivated a pure and refined taste, and a natural and lively activit}^ of the Imagination. 5. The Imagination may be cultivated by creating imagin- ary scenes, incidents, etc. The creative power of the Imag- ination is its highest office, and such exercise gives it ihe highest culture. The pupils can be led to create and describe ideal landscapes or incidents of human action. They may be required to write and relate imaginary or fictitious events, as allegories, parables, novelettes, etc. Poetical composition and the creating of figures of rhetoric aflford valuable culture in this respect. The Understanding. — The Understanding of children should also be carefully trained. Pupils should be taught to think, as well as to see and remember. Care should be taken that the memor}" be not required to do that which the under- i6 METHODS OF TEACHING. standing of the child should perform. Perhaps the greatest mistake of school work is made jnst at this point. There ia often too much cramminor, and not enouah thinking in our schools. 1. The Understanding may he cultivated by the study of thought studies, as ]i[ental Arithmetic, Written Arithmetic, Grammar, Geometry^ etc. These studies require pupils to think, and pupils learn to think b}' thinking. Care should be taken that the pupils study with the understanding and not merely with the memory. 2. The Understanding may lie cultivated by working out original problems, parsing and analyzing sentences, etc. These exercises require the pupils to employ the power of original thought. They lead the mind to compare, and the process of comparison lies at the foundation of thinking. The judgment must be exercised to apply the principles and rules, and to see the relation of the conditions of the problem or the elements of the sentence. 3. The Understanding may be cultivated by writing compo- sitions and trying to think out and express something new. Such exercises bring into activity the inventiA-e powers of the mind. They require the pupil to elaborate his knowledge, to work it up into new forms, to think out something new for himself. Writing original compositions is thus a most excel- lent exercise for the cultivation of thought-power. 4. The Understanding may be cultivated by the study of the mathematical and physical sciences. The three best studies to develop the power of thought are Mental Arithmetic, for the young student ; Geometry, for students from fourteen to eighteen j-ears ; and Mental Philosophy and Logic, from eighteen years and upward. For inductive thought, the nat- ural sciences should be studied ; as Botany and Natural Philosoph3\ The former teaches pupils to generalize and clas- sify ; the latter to investigate the causes and laws of things. 5. The Understanding may he cultivated by reading the CULTIVATING EACH FACULTY. 47 icorks of the great thinkers. To follow thought, as ex})rcssed in language, will stimulate to thinking. By reading the works of Plato, Aristotle, Bacon, Hamilton, etc., the mind becomes familiar with great thoughts and is aroused to think for itself. 6. The Understanding mai/ be cultivated by thinking. We learn to think by thinking, thinking, thinking. Attention. — The power of Attention should be carefully trained in childhood. It is one of the most important of the mental powers, for upon its actiA'it}' depends the eflicieney of each one of the specific faculties. Mental power is, to a large extent, the power of attention ; and genius has been defined as " nothing but continued attention." The following suggestions will indicate to the teacher the methods by which the power of attention can be cultivated : 1. Have pu^Dils observe objects closely. 2. Require them always to study with close attention. 3. Read long sentences and have pupils write them. 4. Read quite long combinations in mental arithmetic, and have pupils repeat them. 5. Mathematical studies are especially valuable in cultivat- ing the power of attention. The following suggestions are made to aid a teacher in securing the attention of his pupils : 1. Manifest an interest in the subject you are teaching. 2. Be clear in your thought, and ready in your expression. 3. Speak in a natural tone, with variet}^ and flexibility of voice. 4. Let the position before the class be usually a standing one. 5. Teach without a book, as far as possible. (). Assign subjects promiscuously, when necessary. 7. Use the concrete method of instruction, when possible. 8. Yary your methods, as A-ariety is attractive to children, 9. Determine to secure the attention at all hazards. CHAPTER VII. THE NATURE OF KNOWLEDGE. IN order to give instruction skillfully, a teacher should have an idea of the general nature of the different branches of knowledge and their relations to one another. He should see clearly the elements of which the different branches are com- posed, the relation of these elements to the human mind, and the manner in which the sciences are developed. We shall, therefore, present a brief discussion of the Nature of Knowledije. Common and Scientific. — All knowledge may be embraced under two general divisions , Common Knowledge and Scien- tific Knowledge. Common Knowledge consists of unsystem- atized facts, ideas, and truths. It is a knowledge possessed by the common people, and is the basis of Scientific Knowledge. Scientific Knowledge consists of facts, ideas, and truths, sys- tematized and expressed in the form of laws and principles. It enables man to interpret the facts and phenomena of nature, to see the great laws by which the universe is governed, and to previse and predict the events of the future. General Division of Science. — Scientific knowledge has been divided into two general branches; the Empirical Sciences and the Rational Sciences. This classification is based upon the relation of their subject matter and methods of development to the human mind. The Empirical Sciences are those which are founded on the knowledge derived through the senses: they are developed oy Generalization, Classification, and Inductive Reasoning. Geography, Botany, and Natural Philosophy are examples of the empirical sciences. The facts of these sciences are givea (48) THE NATURE OF KNOWLEDGE. 49 by Perception: these facts are classified by Generalization, and their laws and causes are derived !);>• Induction. The Rational Sciences are those which are founded on tlie knowledge given by Intuition or the Reason: they are devel- oped b}' Deductive Reasoning. Arithmetic, Geometry, Logic, etc., are examples of the rational sciences. The fundamental ideas and axiomatic truths of these sciences are given by Intuition, and their dei'ived truths are obtained by Deduction. Schemes of Clas.sificafion. — There have been many at- tempts made to classify knowledge ; but no scheme of classi- fication has yet been presented which has been universally accepted. Comte, the celel)rated positive philosopher, classi- fies the sciences with respect to the matter of which they are composed. His classification i^ as follows : Mathematics, Astronomy, Physics, Chemistry, Physiolog3% and Social Phvsics. Dr. Hill classifies the branches according to the order of their development. His classification is Mathesis, Physics, History, Psychology, and Theology. Dr. Wicker- sham groups the sciences together under the following heads: The Elements of Knowledge, Language, The Formal Sciences, The Empirical Sciences, The Rational Sciences, The Histori- cal Sciences, and The Arts. Author's Classification Without discussing these sev era! schemes o.f classification, we sl?all present one which we think best suited to the training of young teachers. Knowl- edge may be classified into seven principal divisions: 1. Lan- guage, 2. Mathematics, 3. Physics, 4. History, 5. The Arts, 6. Psychology, T. Theology. This classification is simple, and has the advantage of employing the names of the branches as generally used in our schools. These general branches and their subdivisions are not always entirely distinct from one another. They often over- lap one another and intrnde upon one another's territory. It is impossible to draw a line, in every case, marking just where one branch ends and another begins. This ig tru« witli 50 METHODS OF TEACHINa. respect to every classification that has been attempted. The scheme here presented seems more satisfactory, for the pur- pose of teaching, than any that we have met. Laiiffuage. — Language is the instrument of thought and the medium of expression. The term is derived from lingua, the tongue. Primarily, Language is the means of communi- cating knowledge : it enables one mind to ti'ansfer its thought to another mind. It is also found that language is the means by which we think, as well as the medium by which we com- municate our thoughts. We cannot think to any great extent, if at all, without language; and the more perfect our language the more powerful mir thought — as in algebra, arith- metic, etc We therefore embrace these two uses of language in our definition, and define it to be the instrument of thought and the medium 0/ expression. JIatheniatics — Mathematics is the science of Quantity. The term is derived from mathematike, meaning science. It investigates the relations of quantity, and unfolds the trutlig and jjrinciples belonging 10 it. It is based on intuitive ideas and truths, and developed by deductive reasoning. The three principal branches are Arithmetic, Geometry, and Algebra. Arithmetic is the science of Number; Geometry is the science of Space ; Algebra is a general method of investigating all kinds of quantity by means of sj^mbols. J*/il/.sics. — Physics is the science of the material world. The term is derived from phusis, nature. It consists of facts and phenomena, and the laws and principles which control them. It begins with the observation of facts, compares and classifies them, and ascertains the causes which give rise to them and the laws which control them. The principal branches are Geography, Natural History, Natural Philoso- phy, Astronomy, Chemistry, Geology, etc. Geography treats of the facts relating to the surface of the earth, classifies them, and investigates their causes, and the laws which govern them. Natural history treats of the three THE NATURE OF KNOWLEDGE. 51 kingdoms of nature, — the mineral, the vegetable, and the ani- mal, — ascertaining the nature, structure, etc., of the indi- vidual objects, and classif}- ing them. It includes Mineralogy, Botany, and Zoology. Natural Philosophy treats of the facts and phenomena of nature, and ascertains their causes and the laws which govern them. It includes Mechanics, Optics, Acoustics, etc. Astronomy treats of the facts and truths relating to the heavenly bodies. Chemistry treats of the na- ture and properties of the elements of bodies. Geology treats of the origin, development, and structure of the earth. History. — History is a systematic description of the past acts and condition of mankind. It embraces the Facts of History and the Philosophy of History. The Facts of History embrace the events that have occurred in the life of individu- als and nations. The Philosophy of History endeavors to ascertain the causes which have contributed to produce the different changes in society and nations, and thus to predict the future condition of the face. In other words, it endeavors " to solve the problem of man's condition and destiny." Art. — Art is the application of knowledge or power to effect some desired object. It is the outgrowth of practice, and may be defined as practice guided by principle. The Arts are divided into two general classes ; the Fine Arts and the Use- ful Arts, The object of the Useful Arts is the attainment oi the end of utility; the object of the Fine Arts is the attain- ment of the end of beauty. These two, though primarily dis- tinguished, are often combined in the same production ; as in the manufacture of glass and pottery ware, in architecture, engraving, etc. Psychology. — PSYCHOLOGY is the science of the human mind. The term is derived from psyche^ the soul. It is sometimes divided into Empirical Psychology and Rational Psychology. Empirical Psychology treats of the nature of the mind as revealed in the experience of consciousness. Rational Psjx'hology treats of the nature of the mind as de- termined b}' the necessary principles given by the Reason. 52 METHODS OF TEACHING. TheolofTU — Theology is the science which treats of God. The term is from Theos, God, and logos, a discourse. It has been divided into Natural Theolog}^ and Revealed Theology. Natural Theology endeavors to ascertain the nature of God through his works, by the light of philosophy and reason. Revealed Theology seeks a knowledge of the Divine Being through his revealed word. Ofherr Distinctions. — It is often convenient to speak of the Inductive and the Deductive Sciences. The former in- clude all those branches of knowledge wliich begin in facts aiid are developed b}^ generalization and inductive reasoning ; as geography, botan^', natural philosoph}', etc. The latter include all those branches of knowledge which begin in ideas, and are developed by the process of deductive reasoning ; as arithmetic, geometry, etc. A division of the Rational Scien- ces is often made, called the Formal Sciences. The Formal Sciences may be defined to be those sciences which treat of the necessary forms in which truth presents itself. They include Mathematics and Logic ; Mathematics treating of the form in which quantity is presented, and Logic of the form in which thought presents itself. NoTK. — In respect to Geography, it should be stated that it is not a dis- tinct science, but a combination of several sciences. Thus in its higher departments, it embraces the elements of Astronomy, Natural Philosophy, and Geology; while that which relates to man and his works is historical in its nature. Some writers in view of the latter fact class it among the historical sciences. Political Geography seems to belong to Historj', and Physical Geography to the Physical Sciences ; and it maybe classed there- fore in either of these two divisions of knowledge. CHAPTER VIII. THE FORMS OF HSrSTRUCTION". INSTRUCTION, as previously defined, is the art of fur- nisliing the mind with knowledge. It is the art of develop- ing knowledge in the mind of another. By it we are enabled to build up in the mind of the learner a knowledge of the sciences, as an architect erects a building. Under the direc- tion of a teacher, a science is developed in the miad in sym- metry and beauty as a temple is erected under the guiding genius of a skillful architect. Knowledge may be developed in the mind in different ways ; these different ways we call Forms of Instruction. There is a certain order in which knowledge should be developed in the mind ; this order we call the Order of Instruction. There are certain laws which should guide a teacher in developing knowl- edge ; these laws we call the Principles of Instruction. In discussing the Nature of Instruction we shall speak of the Forms of Instruction, the Order of Instruction, and the Prui- ciples of Instruction. The Forms of Instruction are the various wa^'s in which we may develop knowledge. The principal Forms of Instruction are the Analytic and Synthetic, the Concrete and Abstract, the Inductive and Deductive, the Theoretical and Practical. We will define and illustrate each one of these forms. Analytic and Synthetic. — Analytic Instruction is that form of teaching which proceeds from wholes to parts. Thus, if I take a watch and separate it into its parts, and teach the name and office of each part as I take it to pieces, the process is anal3^tic. So in grammar, if I begin with the sentence and separate it into its parts, I am using the analytic process. If in geography we begin with the globe as a whole, and separate (53) 54 METHODS OF TEACHING. it into land and water, and come down from continents and oceans to the smaller divisions, the process is anal3^tic. Synthetic Instruction is that form of instruction which pro- oeeds from parts to wholes. Thus, if we take the parts of a watch as separated, and putting them together, teach the name and use of each part, we are teaching synthetically. If in grammar we begin with the words as parts of speech, and put them together to form sentences, we are teaching by the s^-n- thetic method. So if we begin with the geography of the school grounds, go out to that of the township, the county, and the state, and thus at last cover the entire surface of the earth, the method is sj'nthetic. Concrete and Abstract-. — Concrete Instruction is that form of teaching which employs objects and illustrations. Thus, object lessons, or the use of pictures and diagrams, are examples of concrete instruction. In Arithmetic, the teaching of the fundamental operations by means of the numeral frame, of fractions by means of illustrations, of denominate numbers by means of the actual measures, of banking by establishing a bank in the school, are examples of concrete instruction. Grammar taught from language, rather than from the rules of the text-book, is also concrete teaching. Abstract Instruction is that form of teaching which does not employ objects and illustrations. In Arithmetic, counting, addition, etc., taught without any objects or illustrations, denominate numbers by merely repeating the tables, per- centage by the definitions and rules without illustrating the actual business transactions, etc., are examples of abstract instruction. Grammar taught from the definitions of the text-books, instead of from language in which we find the principles embodied, is abstract instruction. Teaching Geog- raphy from the book, rather than from natural objects, is an example of abstract instruction. Inductive and Deductive. — Inductive Instruction is that form of teaching which proceeds from particulars to generals. THE FORMS OF IXSTRUCTION. 55 The leading oi" |nipils b}^ appropriate questions and examples to the apprehension of an idea or [irineiple before it is stated, is a process of inductive teaching. Thus, in Arithmetic, if by presenting particular examples we lead tlie pupil to see the principle or rule before stating it, wc teach inductively. If in Geometr}', In' approi)riate examj^les, we lead the pupil to a geometrical idea or principle, and then require him to express it, we are teaching inductively. In Grammar, teach- ing inductively, we would lead a pupil to the idea of a part of speech before we named and defined it; or lead him, as we often can, to the name of a part of si>eech, without his learning it from a book or the teacher. Deductive Instruction is that fomi of teaching which pro ceeds from generals to particulars. If we first state the gen- eral principle and then lead to the i^arti«ular applications of it, we are teaching deductively. Thus, in Arithmetic, we ma}' teach the pupil the principles of fractions, and then have him apply them; or in Grammar we may teach the words of a definition, and then illustrate its meaning: in botli cases we are teachins: deductively. Deriving ideas from definitions, methods from ])rinciples, particular methods from general laws, are all deductive methods of procedure. The Inductive and Deductive methods ma}' be distinguished even in stating definitions. Definitions ma}' be stat-ed either in an inductive or a deductive form. If we begin with the term to be defined and pass to its explanation, the form is deductive ; but if we beixin bv o:ivin2r the idea, and ^nd by naming the term, the form is inductive. Thus " Addition is the process of finding the sum of two or more numbers," is in the deductive form ; and " The process of finding the sura of two or more numbers is called Addition," is in the induc- tive form of stating a definition. Theoretical and Practical. — Theoretical Instruction is that form of teaching which deals principally with the laws and principles of a suiiject. Teaching the tlieory of arithmetic 66 METHODS OF TEACHING* without making an application of it to practical problems, is an example of theoretical teaching. The so-called practical problems of arithmetic, are sometimes purely theoretical, never occurring in actual life. Teaching the definitions and principles of grammar without apphing them — a fault not uncommon — is also an illustration of theoretical instruction. The teaching of geometry without any application of its prin- ciples to practical problems, a very common fault, is also an example of theoretical instruction. Pravticol Instruction is that form of teaching which deals jwincipally with the application of the laws and princii)les of a suV)jcct. When pupils are required to apply the principles of arithmetic to actual problems, and the students of grammar are taught to use the princii)les of language in their own speech and writing, we have an illustration of practical teach- ing. To open a counting-house in the school-room and show by actual transactions what the business problems of arith- metic mean, is practical instruction. The application of the principles of geometr}' to actual problems that may occur to a business man, and also to surveying and engineering, fur- nishes an example of practical instruction. Application. — Several of these forms may be used m teach- ing the same subject ; and sometimes one form is preferable and sometimes another. The concrete and inductive forms should be used with children ; the abstract and deductive forms are more suitable to older pupils. Analysis and syn- thesis are often emploj^ed in teaching the same subject ; though, as a rule, the analytic form should precede the sjm- thetic. All instruction should be practical, though at certain stages the abstract element may predominate. It is not our ])uii^ose to point out the use of these forms here, but merely to make the pupil familiar with the forms themselves, Their"^ use and special application will be indicated in the chapter on the Principles of Instruction, and in the methods of teaching the particular branches of study. CHAPTEE IX. THE ORDER OF INSTRUCTION. rEIE school-time of life has been divided into four periods ; Infancy, Childhood, Youth, and Manhood. Infancy em- braces the period from the birth of the child to the age of five years ; Childhood, the period from five to ten years ; Youth, the period from ten to sixteen years ; and Manhood, the period from sixteen to twenty-one years. These are not definitely fixed periods, as some persons mature very much earlier than others. Girls from twelve to sixteen years of age are usually much more mature than boys of the same age. The distinctions are suflSciently definite, however, for the purpose in view. The inquiry is, What is an appropriate course of study for each one of these periods ? How much of the several branches — Language, Mathematics, Physics, History, the Arts, etc., shall be taught in each one of these periods ? Several writers treat of this subject under the head of a Graded Course of Study, in which they attempt to fix the kind and amount of knowledge suitable for the various grades of a public school. Dr. Hill, who has a very complete discussion of the subject, divides the school time into five distinct grades; the first, or Sub-primary school, from five to eight ; the second or Primary school, from eight to eleven ; the third, or Grammar school, from eleven to fourteen; the fourth, or High school, from fourteen to seventeen ; and the fifth, or College period, from seventeen to twenty-one. This is practi- cal ; but as grades in different places var^', it has been thought best to discuss the subject in general under the four heads named, as is done b}- Dr. Wickersham in his Methods of In,' 3* (57) 58 METHODS OF TEACHING. strnction. Any teacher who understands the order presented will have no difficulty in arranging the studies of a graded school. Infancy- — During this period a child learns to talk. It may also learn a few written words, and the letters of the alphabet. In Mathematics, it may learn some of the figures of geometry', to count as far as twenty-five or fifty, and per- haps to add and subtract a few of the smaller numbers with objects. It will acquire a large number of facts in botany and zoolog}-, and also man}- of the elementary facts and phenomena of natural philosophy-. It may also become familiar with a few facts of history, learn to sing little songs, and to use a pencil and draw a little. This instruction should be given at home or in a kindergarten. Childhood. — During childhood, the child should learn to read, to spell, to pronounce correctly, and to express itself w^ith considerable correctness and facility, both in speaking and in writing. It should receive a systematic course of in- struction in Language Lessons, including orthography, the construction of sentences, the use of capitals, punctuation marks, etc. There should not, however, be any formal in- sti'uction in Grammar. If circumstances will permit, the child may learn to speak one or two modern languages, and even elementary instrivction in Latin could be given. Instruction in Arithmetic should embrace numeration and notation, — the n?ming, writing, and reading of numbers ; the fundamental operations of addition, subtraction, multipli- cation, and division ; the elements of common fractions, decimals, and denominate numbers. In Geometry, he should become familiar with all the ordinary figures, both plane and solid ; learn to construct and point out their different parts and elements ; and perhaps learn a few of the elementary truths of the science. In the facts of the Physical Sciences, his course should be quite extensive. He should become familiar with the leading THE ORDER OF INSTR'UCTION. 59 facts of descriptive geography, and be able to locate the prin- cipal countries, cities, rivers, mountains, etc., of the world. In Botan}', he should become familiar with the ordinary trees of his neighborhood, the principal flowers of the garden and meadows, be able to name many of the forms of leaves and corollas, etc. He should also learn the names of the principal animals, domestic and wild ; many of the ordinary insects, and some of the more common fishes. He should also learn the common minerals of the neighborhood ; as quartz, limestone, sandstone, granite, etc. Manj^ of the simple facts and phenomena of Natural Philosoph3', and the causes of the same, vnay also be learned, and some of the simpler experi- ments of the science may be presented to him. During this period a child can learn, by oral instruction, man}- of the leading facts of the History of the world, and of his own country. He is able also to read works on biography and histor}', if written in a simple and interesting st3'le, and should be encouraged to do so. In the Arts, he should be taught to write, to draw, and to sing ; and if he has any musical taste, ma}- ])egin to learn to play some instrument. Boys should learn the use of a knife and other tools, and girls the use of tlie needle, scissors, etc. Youth. — During the period of youth, the pupil should con- tinue the study of Language, increasing his vocabulary and acquiring skill in the use of his mother tongue. He should have a careful drill in orthography, pronunciation, and read- ing. He should also begin the study of grammar and the elements of rhetoric, learn to use the dictionarj-, and have constant exercises in composition writing. He should be required to read, commit, and recite choice extracts of prose and poetr}- for the cultivation of a literary- taste. He should also begin the stud}- of Latin and Greek, and perhaps one or two foreign languages. An extensive course of reading in poetry and prose would also be of advantage. In Mathematics, he should go through an ordinary- text- 60 MBH'HDBS &F TK ACHING. book on mental and written arithmetic ; begin and in many eases complete an elemcHtary work on algebra; and if he has had a good opportunity for mathematical study, should com- plete an elemeatary text-book on geometry. Of the Physi- cal Sciences, he should conitinue his course in descriptive geography, and also study physiology, botany, natural philoso- phy, astronom3',and physical geography. During this period he can complete the elements of these branches as they are presented in the ordinar}' elementary' text-books. Some of feiie elements of zoology and mineralogy should also be in- cluded in the course of studies arranged for pupils from te« to sixteen years of age. During this period he may complete the ordinary text-book on the history of his ovax country, and even a small text-book on general history. He should also read such works as the Rollo Books, Abbott's Histories, ajid other works of bitogra- raphy, travels, voyages, and explorations. The historical stories of Miss Yonge and Miss Strickland are especially recommended to pvi})ils of ten or twelve years of age. In respect to the Arts, the pupil should learn, during this period, to write a good hand, t© draw with considerable skill, to read musie by note, to sing, and if he has musical talent, to play one or two instruments. Girls should learn to sew, mend, darn, cut and fit garments, and receive instruction in housekeeping, cook- ing, etc. Boys skould become familiar with the use of tools, and acquire some of the elements of the mechanic arts. If there is special talent for the niechanic or fine arts, an opportunity should be aflTorded for additioaal oulture in these branches. Some of the elements of Mental and Moral Philosophy might be learned during this period, but it is thought that any formal study of these branches should be usually postponed until after the age of sixteen. Manhood. — During this period the Language studies of the previous period should be continued into their higher depart- ments; in addition to which there should be a thorough course in Rhetoric, Genesal Literature, Philology, etc. There should THE ORDER OF INSTRUCTION. 61 also be an extensive course of general reading of the poets and prose writers, and a close and careful study of some of them as models of style and expression. There should also be much practice in composition, and the pupil should become a good writer and speaker. The Mathematical studies of this period should include higher arithmetic, higher algebra, higher geometry, trigonom- etry and surveying, analytical geometry, difierential and integral calculus, and the philosophy of mathematics. If there is time, some of the recently developed branches of the science may also be studied ; and the pupil should be encour- aged to push his investigations beyond any of the ordinary text-books on the subject. The course in Natural Science should include a higher course in Natural Philosophy, embracing mechanics, optics, acoustics, etc. ; a course in theoretical and practical Astron- omy ; a full course in Chemistry, Anatomy, and Physiology; and, if possible, quite a thorough course in Natural History. The student should also begin the investigation of the facts and phenomena of the material world for himself. The course in History should include the reading and study of a complete history of one's own country, a complete course in general history, a careful reading of the detailed history of England, France, Gei-raany, Spain, etc., a study of the works on the philosophy of history, as Guizot, Buckle, Draper, etc. The effort should be to fix permanently in the mind all the great and leading events of history, and to learn the causes which have contributed to the rise and fall of empires and nations, and thus to learn the laws w^hich control the growth of civilization. The course in the Arts may be continued for a year or two, ac- cording to the taste and circumstances of the pupil. Girls may continue lessons in the household arts, and boys may acquire con- siderable skill in working in wood aud iron. When there is musical taste, it may include the culture of the voice, singing, instrumental music, thorough-bass, musical composition, etc. 62 METHODS OF TEACHING. When there is taste in drawing, it may include sketching from nature, perspective drawing, painting, etc., and the history and philosophy of art. Instruction in moulding figures out of clay or plaster will be valuable to both boys and girls. A course in Architecture and Landscape Gardening will also be of interest and value to the student if there are taste and time for it. The student is now prepared tor what are called the Meta- physical studies. During this period, he should take a course in Mental Philosophy, Moral Philosoph}^, Logic, Political Economy, .Esthetics, International Law, and the Evidences of Natural and Revealed Religion. The works of the great thinkers, Plato, Aristotle, Bacon, Locke, Kant, Hegel, Fichte, Hamilton, and the writers on the relation of modern science to philosephy and religion, ma\' be studied. Many of these studies, however, can be merely begun at the age of twenty- one, and should be continued through life. The course suggested will be found to be just a little in advance of the capacity of the average bo}^ and girl, as we find them in our families and schools ;_but if the pupil have a careful systematic training from the beginning, he will be prepared for the studies named. The object is to present an ideal of what the course should be, and of what we should aim to make it. CHAPTER X. THE PRINCIPLES OF INSTRUCTION. rpHE Principles of Instruction are the laws which guide JL the teacher in imparting instruction. These principles are derived from three distinct sources ; the Nature of the Mind, the Nature of Knowledge, and the Nature of In- struction. The principles derived from the nature of the mind have reference to the proper culture of the mental faculties ; those derived from the nature of knowledge have reference to the order in which knowledge shall be presented to the mind ; and those derived from tlie nature of instruction have reference to the manner in which knowledge shall be taught. We shall present ten principles of each class, which may be called the Teacher^s Decalogue. Principles Derived from the Nature of Mind. The following ten principles are derived from the nature ot the mind, and indicate the laws which should govern the teacher in imparting instruction so that the mind may be properly trained and developed : 1. The primary object of teaching is to afford culture. In educa- tion culture is more valuable than knowledge. Culture gives the power to acquire knowledge, and this is worth more to the pupil than the knowledge he has already acquired. Culture also gives one the power to originate knowledge, to invent new ideas and thoughts. Without culture the mind is a mere receptacle of ideas and tlioughts ; with it the mind is an active energy that can trans- form its knowledge into new products. Knowledge makes a learned man ; culture m^ikes a wise man ; and wisdom is better than learning. This primary object of teaching should never be '(63) 64 METHODS OF TEACHING, forgotten. The teacher should carry in his mind a clear conception of the faculties of -his pupils, and keep constantly before him the thought whether his work is adapted to the growth and culture of these faculties. He should know the relation of each branch of study to the minds of his pupils, see clearly what faculties are brought into activity by it, and be sure that his work is giving, not merel}' knowledge, but intellectual power. In other words, he should measure his work, not merely b}' the knowledge he is imparting, but by the mental power he is cul- tivating. The neglect of this duty has warped and stunted man^' a 3'onng mind. 2. Exercise is the great law of culture. This law is univer- sal, applying to l)oth mind and matter. A muscle grows strong by exercise. The arm of the blacksmith and the leg of the pedestrian acquire size and power by use. So ever}- faculty of the mind is developed b}' its proper use and exercise. The power of perception grows by perceiving, the power of mem- ory b}' remembering, the power of thought by thinking, etc. Hang the arm in a sling and the muscle becomes flabby and almost powerless ; let the mind remain inactive and it acquires a mental flabbiness that unfits it for any severe or prolonged activity. An idle mind loses its tone and strength, like an unused arm ; the mental powers go to rust through idleness and inaction. 3. The teacher ahould aim to give careful culture to the perceptive powers of the child. The perceptive powers are the most active in childhood. Mental activity begins in the senses. A little child almost lives in its eyes and ears and fingers; it delights to see and hear and feel. Its eyes are sharp, its ears are quick, and its fingers so busy as to be con- tinually in what people call "mischief." The teacher shouhl direct this activity, and give the child food for the senses. He should provide objects for its instruction, and give it facts to satisfy this craving mental appetite, rather than attempt to feed it upon abstract ideas and thoughts for which it has no taste or capacity. THE PRINCIPLES OF INSTRUCTION. 65 4. The teacher should aim to furnish the memory of the child with facts and ivords. Tlie memory of children is es- pecially strong for facts and words. Every object of nature comes through the senses with such a freshness to the mind that it stamps itself indelibly on the memory. Facts seem to stick as naturally to the young mind, as burrs to the dress. Its memory for words is no less remarkable than its memory of things. A new word, once heard, is usually a pernmnent pos- session. A child will learn to speak three or four languages in a year, if it has the opportunity of doing so. The teacher should remember these fticts, and conform his work to them. He should give the child an opportunity to furnish its mind with the facts of nature and science, and also to add to its stock of words and acquire a rich and copious vocabulary. 5. The memory should be trained to operate by the laws of association and suggestion. The mind in retaining and recall- ins: knowledae works in accordance with a certain law of mental operation. It ties its facts together by the thread of association, or arranges them in clusters like the grapes of a buncn. This tendency is called the Law of Association. The principal laws of association are the law of Similars, the law of Contrast, the law of Cause and Effect, and the law of Con- tiguity in Time and Place. The teacher should understand these laws and require the pupil to link his knowledge together b}^ means of them. In geography he should have pupils asso- ciate similar facts in respect to cities, states, etc.; in history he should require them to make use of the law of contiguit}'- in time and place, and lead them to associate events as related by cause and effect. A.11 the knowledge taught should be so systematized that it may be readily recalled by the law of logical or topical relations. 6. The power of forming ideal creations should be carefully cultivated. The faculty of ideal creation is the Imagination. This i)ower is awakened into action through the medium of perception. The facts of the senses touch the lancy, ami d6 METHODS OF TEACHING. arouse it into activity. Tlie forms and colors of nature, the arching skj^ and the spreading landscape, linger in the mem- ory as forms of beauty, and excite the imagination to modify aiid create such forms for itself. This tendency is sometimes so strong, that fact and fancy })ecome so interwoven in tlie mind of a child that it is ditlicnlt to discriminate betwoon them. The teacher should encourage the activity of this faculty, and ti'ain it to a healthy and normal development. 7. The mind should he gradually led from concreie to ab- stract ideas. The young mind begins with the concrete, with olvjects and their qualities. Its first ideas are perce2:)tions of objects, of things that it can see and hear and feel. Its ideas of quality are not abstracted from, but rather associated with, objects. These concrete qualities it begins to conceive inde- pendently of the objects in which they are found, and thus it gradually rises to abstract ideas. From hard objects it gets its ideas of hardness, from kind parents and friends it obtains its notion of kindness, etc. This natural tendenc}'^ should be noticed and aided, so far as possible, by the teacher. Espe- cially should he be careful not to lift the pui)il up into abstractions too soon. He should present concrete exa,mples of that which he is teaching, that the pupil may have a defi- nite idea of the subject to be presented before he attempts to consider it abstractly. He should aid the child to rise from things to thoughts. 8. A child should be gradualhj led from particular ideas to general ideas. The young mind begins with the particular. Its first idea is of particular objects, not of general notions, A man, to the 3^oung mind, is a particular person; a bird is a particular bird. Gradually it rises from the particular object to the general conception, from a percept to a concept. Tlie teacher should watch this natural tendcnc}' and aid it. The process should not be forced, it should not be attempted too earlj'; but when the pupil is ready, he can gradually be lifted lip from the concrete into tlie s])licre of abstract and general THE PRINCIPLES OF INSTRUCTION. 67 conceptions. It should be tlie«special aim of the teacher to aid the mind in rising from the particular to the general. 9. A child should be taught to reason first inductively and then deductively. The child's first thoughts are the facts of sense. From these particular facts it gradually rises to gen- eral truths. By and by, after the mind has attained to some general principles through Induction, it begins to reverse the process and infer particular truths from such general princi- ples. It also begins to apply the self-evident truths to reach- ing conclusions that grow out of them. This natural activity of the mind should be understood by the teacher, and the work of instruction be done accordingh'. Especial care should be taken not to require deductive thought too early. In all things the law of nature should be implicitly followed. 10. A child should be gradually led to attain clear concep- tions of the intuitive ideas and truths. Mental life begins in the senses ; the child's first ideas and truths are those which relate to the material world. But, by and by, intuition awak- ens into activity, and in it begin to dawH the ideas and truths of the Reason. The teacher should watch this natural activit}'^, and be governed by it. He may aid the child in developing the ideas of Space, Time, Cause, the True, the Beautiful, and the Good, by presenting suitable occasions. He may also aid the pupil in reaching the self-evident truths which spring out of these several ideas, by particular examples and suitable qiiestions. Some of the axioms of number and space are quite early awakened in the mind ; and the teacher can aid their development. Principles Derived from the Nature of Knowledge. The principles of the first class are drawn from a considera- tion of tlie nature of the mind. The principles of the second class are derived from the consideration of the nature of knowledge. The following ten principles are regarded as among the most important: 68 METHODS OF TEACHING. 1. The second object of teacJy,tig is to imparl knowledge, a person sliould not only know how to obtain knowledge, but he should possess knowledge. He should not only know how to use his memory in acquiring knowledge, but he should have it stored with interesting and useful facts. He should not only know how to think, but his mind should be filled with facts and truths both as the materials for and the results of thought. Though culture, which trains to the use of the faculties, may be better than learning, learning is very much better than ignorance. The teacher should therefore aim to fill the minds of his pupils with the tacts of history, geography, natui'al science, etc. He should hold up before them a high ideal of scholarship, and create in them an ambition for wide and extensive learning. 2. Things should he taught before ivords. This principle is in accordance with the natural development of knowledge. The object existed and was known before a name was given to it; the word was introduced to designate the object. This natural order in the genesis of knowledge should be folloAved in the imparting of knowledge. The principle is also in accord with the natural laws of mental development. This principle is very frequently disregarded by the teacher. It is violated by requiring pupils to commit words without definite ideas of their meaning, and to repeat definitions with- out understanding them. Such a course is most pernicious in its influence on the mind. It leads the pupil to acquire wrong habits of thought, to be satisfied with the expression without a knowledge of the idea or fact expressed ; and deludes him with the idea that words, the symbols, are the realities of knowledge. 3. Ideas should be taught before truths. This law is also iii accordance with the natural law of acquisition and mental development. The mind has ideas before it puts them to- gether in judgments or thoughts. Thus it has an idea of a chair and the ^oor before it thinks the chair is on the floor THE PRINCIPLES OF INSTRUCTION. 69 So iu science, as in aritlimetic and geometry, the ideas pre- sented in the definitions are learned before the truths which pertain to them. This principle is also manifest from the na- ture of the mind. Ideas are given b}' perception and concep- tion; thoughts are the result of judgment and reasoning; and the acts of perception and conception precede those of judg- ment and reasoning. This order should be followed in instruction. The effort of the teacher should be to fill the mind of the pupil with ideas, both concrete and abstract, and subsequently to teach the truths which belong to them. 4. Particular ideas s/wuld be taught before general ideas. This principle is in accordance with the genesis of knowledge and the natural activity' of the mind. Our first ideas are of particular objects, derived through the senses; following these come the abstract and general notions given b}^ the under- standing. • Thus a child has an idea of a particular bird before it can conceive of a bird in general, or of a class of birds ; and the same is true of other notions. This order, fre([uently violated in education, should be carefully followed. To depart from it is to invert the law of mental activity and injure the mind, as well as retard the acquisition of knowledge. The motto should be, — from the particular notion or idea to the general. 5. Facts^ or particular truths, should be taught before prin- ciples, or general truths. A fact is a truth in the domain ui sense; a principle is a truth in the domain of thought. The former is concrete; the latter is abstract; and the concrete should be taught before the abstract. The former results from an operation of perception and judgment; the latter from an act of reasoning; and an act of perception precedes an act of reasoning. Again, facts are particular truths; principles are ger.eral truths; and the particular should precede the general. The principles in natural science are a generalization from facts; and the mind must be familiar with the facts before it can generalize from them. It is thus clear that facts, or par- 70 METHODS OF TEACHINQ. ticuliir truths, should be taught before i3rinci2:)les, or general truths. 6. In the physical sciences causes should be taught before laws. In the«i)hysical sciences we proceed tVora facts and phenomena to the laws and causes relating to them. In pre- senting these, the law of mental growth indicates that we should teach the causes of things before presenting their laws. The idea of cause is very early awakened in the mind. One of the first questions of a little child is, " Mamma, what makes that?" The ascertaining of the laws wliich control facts and phenomena is a later consideration. The same conclusion appears from the genesis of knowledge. The causes of physi- cal phenomena were sought for long before an inquiry was made for their laws. The ancients early made inquiiies after the causes in natural philosophy and astronomy ; the attempt to ascertain the laws is of much more recent date. Besides, too, the law often flows from a correct idea of the cause, as in gravitation, optics, etc. It is thus clear that in teaching the ph^'sical sciences, the causes of facts should be considered before their laws. T. In the physical sciences, causes and laws should he taught before the scientific classificy the addition of white. The different varieties of Brown are Chocolate, Kusset, Snuff, Drab, and Tan. Craj/is composed of black and white. wit> a OBJECTS AND THEIR PARTS. 89 gliglit mixture of red, j-ellow, or black. The different varieties are, Slate, Pearl Gray, Steel Color, and French Gray. Tertiiiry colors are formed by mixing two secondary colors, or three primary colors in the pniportion of two parts of one and one part of each of the other two colors. The tertiary colors are Citrine, Olive, and Russet. There are several varieties of colors, indicated by the terms Shade, Tint, Hue, and Tinge. A Shade is formed by mixing black with any color, so as to make it darker than the oriffinal color. A Ti7U is formed by mixing white with any color, so as to render it lighter than the original color. A Hue is formed by combining two colors in unequal proportions ; as, a little yellow mixed with pure red gives scarlet, a h7ie of red. A Tirge is a slight coloring or tincture added to the principal color; thus, green, if it has a slight coloring of yellow, is said to have a titige of 3'ellow. Two colors which, when united, produce white light, are said to be Complementary. Thus, red and green, orange and blue, yellow and purple, are complementary colors. By the Harmony of Colors, we mean that relation of certain coiors, which gives special pleasure to the eye. The complementary colors are liarmonious. Since two colors are harmonious, which when mixed together produce white light, for harmony of color we must have one primary and one secondary color. The teacher may show the pupil that in the scale of prismatic colors, the harmonious colors stand to each other in the relation of fourths, like one of the richest chords in music. The teacher may show the application of the harmony of colors, bj asking questions about ladies' wearing apparel, furnishing a room, arranging a bouquet^ etc. III. Objects and Their Parts. Pupils should have lessons on Objects and their Parts They should be taught to know and name the parts of objects. For this purpose, teachers should have a suitable collection of objects in the school-room. The information concerning these objects, the teacher can obtain in various ways, as here- tofore explained. The following outlines, selected from Shel- don's Object Lessons, will suggest a course to the young teacher : 90 METHODS OF TEACHING. 1. Shaft. 2. Rins:. 3. Barrel. 4. Lip. 5. Wards. .6. Grooves. -• fl. Body "1 L 2. Spire '1. Mouth. 2. Lip. 3 Beak. ,4. Canal. ' 1 . Whorls. 2. Sutures. 3. Apex. a 1. Wood 2. Lead. 3. Head. Poiut. Number 6. Maker's Name, or Trade Mark. n. Blade. 2. Bows. 3. Limbs. 4. Rivets. 5. Edj^es. 6. Back. 7. Point. 8. Shaft. '1. Surface. 2. Faces. 3. Eds:es. 4. Milling:. .5. Impression. ft. Imaije. 7. Superscription. 8. Date. 1. Bowl. 2. Handle. 3. Upper Rim 4. Lower Rim. .5. Bottom. 6. Inside. 7. Outside. 8. Edges. a ( I. Head. S -I 2. Point. 3. Shaft. ( I -o t f 1- Front. 1. Posts ] 3 g^^^ (1. Front. 2 2. Rounds • (3. 1/; 3. Back. 4. Seat. 5. Pillars. 6. Spindles. 7. Slats. 8. Balls. 9. Beads. 10. Scallops. 11. Brace. 3. 4. 5. 6. 1. Upper. 2. Sole. Heel. Tip. Eyelets. Binding. 7. Seams. 8. Tongue. 9. Lining. 10. Insole. I 11. Counter. I 12. Shank. 1 13. Welt. 14. Strings. ' 1.5. Buttons. ^ 10. Vamps. Side. Back. Ph 1. Stem 2. Peel. 3. Pulp. 4. Juice. .5. Veins. 6. Dimples. 7. Eye. 8. Core. 9. Seeds. 10. Seed-case. 1. Bail. 2. Handle. 3. Ears. 4. Body. 5. Staves. 6. Hoops. 7. Bottom. 8. Rivets. 9. Chime. 10. Crole. a 1. Handle 2. Cup 3. Tctogue 1. Nut. 2. Catch. . 3. Shaft. 4. Ferule. 5. Number. 1. Border. 2. Rim. 3. Edge. 1. Loop. 2. Clapper. a o o < C4 r 1. Handle •a r I. Shell. I 2. Kernel. 1. Nut -I 3. Point. 4. Scar. [5. Membrane. (1. Scales. 2. Cup \ 2. Edo:es. I 3. Stem. 1. Rivets. 2. Frame. 3. Heel. 4. Sides. 5. Back. 6. Spring. 7. Grooves. 8. Plate. 2. Joint -I Pivot. 3. Blade -I fl. Edge. 2. Point. 3. Back. 4. Notch. .5. Sides. ,6. Maker's name. QUALITIES OF OBJECTS, 91 Every teacher of a primary or public school should go to work and collect facts concerning other objects, and prepare outlines for giving lessons upon them. No teacher should be without a copy of Sheldo7i^s Object Lessons, published by Scribner, Armstrong & Co. IV. Qualities of Objects. Pupils should be taught to distinguish and name the Quoli- ties of Objects. These qualities should be taught, not ab- stractly, but in connection with the objects in which they are found. The pupil should be led to perceive the quality in the object, and thus obtain a clear idea of it, and then its name may be presented and fixed in the memory. The following list of qualities will suggest to the teacher his work in this respect : Hard, Brittle, Round, Woven, Soft, Flexible, Square, Cellular, Rough, Pliable, Angular, Tubular, Smooth, Elastic, Triangular, Netted, Stiff, Ductile, Rectangular, Fibrous, Limber, Malleable, Cylindrical, Porous, Light, Buoyant, Spherical, Twisted, Heavy, Sonorous, Concave, Indented, Solid, Fusible, Convex, Crystallized, Liquid, Volatile, Spiral, Membranous, Transparent, Natural, Saline, Translucent, Artificial, Odorous, Opaque, Durable, Aromatic, Brilliant, Compressible, Edible, Adhesive, Pulverable, Tasteless, Tenacious, Soluble, Pungent, Amorphous, Insoluble, Emollient, Inflammable, Impervious, Sapid, Ck)mbustib] e. Serrated, Nutritious. V. Elements of Botany. Object lessons on the Elements of Botany may embrace the flower and its parts, the leaf and its parts, the names of leaves from their /or??is, the names of leaves from their ma,r- gins, the names of plants, trees, etc. 92 METHODS OF TEACHING. The following outline will course in botany: Calyx -[ Sepals. Corolla -{ Petals. f Filament. -! Anther. suggest Parts of Flower Stamens ( Pollen r Style. Pistils < Stigma. ( Ovary. Peduncle. Parts of Leaf Margins of Leaves Entire. Serrate. Dentate Crenate, Repand . Lobed. Bases of Leaves ' Cordate. Reniforra. Auriculate. Hastate. Sagittate. Shape of Leaves Oblique. Tapering. Clasping. Connate. Decurrent, Orbicular. Rotundate. Elliptical. Oblong. Linear. Acicular. Deltoid. Ovate. Lanceolate. Cordiform. Hastate. Sagittate. Peltate. Runcinate. Pedate. to the teacher a short r Blade. Midrib. Vein. Veinlets. Parenchyma. Margin. Apex. Base. Petiole. Stipule. ' Acute. Acuminate. Obtuse. Truncate. -j RetTise. Obcordate. Emarginate. Mucronate. Cuspidate. Apices of Leaves Petal f Limb. I Claw. Cruciferous. Rosaceous. Liliaceous. f Banner. Corolla -j Papilionaceous ■< Wings. I Keel. Rotate. Campanulate. Salver-form. Funnel-form. . Labiate. . Lyrate. In every public school there should be charts containing diagrams of all these forms, and the teacher should obtain specimens of them from nature. A work recommended for the teacher is Miss Youmans'si^irs/ Book in Botany, published by Appleton & Co. There should also be a course of instruction on insects, birds, and other animals, which may be given by colored engravings, specimens, etc. LANGUAGE. CHAPTER I. THE NATURE OF LANGUAGE. LANGUAGE is the instrument of thought and the medium of expression. The term is derived from lingua^ the tongue, and meant primarily that which came from or was moulded by the tongue. The primary idea of language is that it is the means of expressing our ideas and thoughts. It is the means b}' which we convey ideas and thoughts from one mind to another. It is seen, moreover, that language is necessary to thought ; that we think by means of language. Sir William Hamilton and other philosophers hold that there can be no thinking without thought sj-mbols ; that is, without words. If we add this further use of language to the primary idea of expressing thought, we have the definition given above. Language, as it now exists, means also the embodiment of thought in words. It is thought expressed, as well as the power of expressing thought ; it is thought made tangible to the senses of sight and hearing. Human language has been figuratively called the outward type or forvi which thoughts and the laws which regulate them, impress on the material of sound. Plato sa3"s, "reason and discourse are one," the former being the conA^ersation of the soul with herself, with- out the intervention of sound ; the latter being this conversa tion made audible by sound. Max Miiller says, — " Language and thought are inseparable. Words without thoughts are (93) 94 METHODS OF TEACHING dead sounds ; thoughts without words are nothing. To think is to speak low ; to speak is to think aloud. The word is the thought incarnate." Language is of two kinds ; Oral and Written. I. Spokex Language. Definition. — Oral Language consists of a combination of articulate sounds to express ideas. An articulate sound is literally a jointed sound, and is thus distinguished from a continuous sound, as a cry, etc. The sounds which are united in the formation of spoken words are called elementary sounds, and consist, in our language, of about forty. The ex- act number has not been definitely determined by orthoepists. Or if/in of Ltuiguage. — There are two general theories for the origin of spoken language, — the theory of Divine Origin^ and the theory of Human Origin. The theory of a divine origin assumes that God gave man a language when he created him, by which he could immediately communicate his ideas and thoughts. In favor of this theory, it is argued that God pronounced His work to be perfect, and tkat man would not have been perfect without the gift of a language. It is also claimed that man must have had a language, or he could not have conversed with God, as he is represented doing in the Garden of Eden. The theor}' of a human origin assumes that man had origin- all^' no language, but merely the power to form a language. He had the gift of speech as he had the gift of reason, and he formed his own language as he has formed the other arts and sciences. In favor of this theory, it is claimed that it is natural to suppose an analog}^ between the development of language and the development of the arts and sciences. Man was not created with a knowledge of the science of geometry, but with powers by which he could originate it. Language was an evolution from man's capabilities, the same as the sciences. NATURE OF LANGUAGE. 95 It is also claimed that the history of languages shows a growth and development from rude beginnings to a more fin- ished form. It is farther held that the Bible presents this view, for it says that the animals were brought before Adam to see what he would call them, " and whatever Adam called them, that was the name thereof." It is further held that so strong is this power of speech that children at the present time, if placed where they never heard a word spoken, would form a language of tlieir own. Instances are recorded in which the children in'a family have actually formed a language for themselves. It is now the general belief of writers upon the subject that language is of human, rather than of divine origin. Theories of Origin. — Assuming that language is of human origin, the question arises, — How or in what way was it formed ? Several theories have been offered as the answer to this question, which have been distinguished as the theories of Imitation, Interjections, and Verbal Roots. The first and second are also called the Mimetic and Exclamatory theories, and the last the theorj' of Phonetic Types. Theory of Imitation. — The theory of Imitation assumes that words originated in the imitation of the soiinds of nature Thus man heard a dog say bow-woiv, and he called it a how-woio. He heard a sheep say baa, and he called it a baa. He heard a bee buzz^ and he imitated the sound, and buzz became the name of a bee. It was supposed that by this principle of onomatopoeia originated many such words as crash, hiss, roar, crack, thunder, etc. This theory, which was once popular, is now generally dis carded by philologists. It is probable that very few words originated in this way. Many words which were supposed to have thus originated, have been traced to quite a diflerent origin. Thus, squirrel, which was supposed to be an imita- tion of the noise made by the animal, has been found to mean a "shade tail ;" cat, or the German hatze, which was supposed 96 METHODS OF TEACHING. to represent the noise made by the cat, comes from an expres- sion meaning "an animal that cleans herself;" thunder^ which was supposed to represent the rolling noise of the clap, comes from tan, signifying to stretch. A few words, as whijjjjoor- will, cuckoo, etc., had their origin in this way ; but such words are sterile, have no reproducing power, and thus are not con- sidered to be true words. Theory of Interjections. — The theory- of Interjections as- sumes that all words originated from primar^^ utterances of emotions. Thus all races emit certain similar ejaculations to express similar feelings of pain or jo3^ The cry, the groan, the laugh, etc., are common to all mankind. These natural utterances are supposed to have been the basis of language. There is no authority, however, for the theory, and it has now no supporters. Max Miiller calls this the Pooh-pooh theory. He also calls the theory of Imitation the Bow-wow theory. Theory of Verbal Roots. — The theory of Verbal Roots assumes that man was primarily endowed with a " linguistic instinct" by which he gave origin to verbal utterances. These primary utterances were very numerous ; many of them per- ished in the struggle for life, but those which remained became the parents of all the other words of the language. These expressions were verbal in their character, and hence are called the verbal roots of the language ; and the theory is known as the theory of Verbal Roots. In favor of this theory, it may be argued that a large part of the language can be traced back to verbs. If we open the dictionary for the etymology of a word, we usually find that it is derived from a verb. The preposition except was origi- nally the past participle of the verb to except, etc.; and even the conjunction if had its origin in a verb gif, to give or grant. The importance of the verb, which means the word (verbum), is also a consideration in favor of the theory. The Chinese call verbs living words, and all others, dead ivords. The theory, however, is not generally accepted. Whitney NATURE OF LANGUAGE. ^7 ridicules it,callino; it the Dirt g-dnng theory ,a-n epithet derived from MuUer's illustmlioii, tluit evervihiuii .struck riiiiis, and that the mind of the primitive man. «Ikmi stiiu-k liy the (diJL'cts of nature, rang out with a sound. The theory was tirst pro- posed b}- Heyse, and advocated by Miiller, who, however, now discards it. The True Theory. — The true theory for the origin of lan- guage is, that it is a natural outgrowth of man's mental and voeal powers. Man was gifted with the power of thought and feeling, and the faculty of exjjression. He was moved b}- his desires and impulses to embody- his thought in vocal utter- ances. These utterances were made partly bj' chance, aided perhaps by the imitation of the sounds of nature. They graduall}^ developed into more and more perfect forms, through the necessity and pleasure of communication, and the progress of the race in refinement and intellectual culture. In this evolution, thought and language, on account of their intimate relation, must have gone hand in hand. Which pre- ceded the other, has been a question among philosophers. G-eiger holds that man was guided in his utterances b}' that which he saw, and that the use of language, in a measure, pre- ceded and produced reasoning. Prof. Whitney and others maintain that thought is anterior to language, and independ- ent of it ; and that thought need not be internally or externally expressed to be thought. In fact, however, the two must have developed together, and langtiage not only expressed thought but aided it in its origin and growth. The Primitive- Language. — Which was the primitive lan- guage is not positively known; the question, however, is a very old one. Herodotus tells us that Psammitichus, King of Egypt, to ascertain the most ancient nation, gave two new- born children to a shepherd to be brought up so as never to hear any words spoken. When they were about two years old, they held out their hands for bread and cried " Becos," which they continued to use for the same purpose. This being 98 METHODS OF TEACHING. reported to the king, he inquired what people called bread " Beeos" ; and discovered that it was the Phrygians, and thus inferred tliat the Phrygian was the primitive language. James IV. of Scotland, in order to ascertain the primitive language, placed a deaf and dumb woman with two infant children, on the solitary island of Inchkeith, to see what language they would use when they came to the age of speech. A Scotch histo- rian, who gives the account, naively remarks, "Some say they spoke good Hebrew; for my part, I know not, but from report." Tlic JJii(//i.s/i Lfuif/inif/e. — The historical origin and devel- opmiMit of the P^nglisii language is well known. The island of Britain was originally settled by the Celts, a branch of the great Indo-European race, which had moved west, on the wave of emigration, from Central Asia. Remains of the same race are found all along the Atlantic coasts of Europe, though they were mainly congregated in Spain, Gaul, Britain, and the adjacent islands. In the 3^ear 55 B. C, the Romans under Julius Csesar, who liad previously conrpiered Gaul, passed over into Britain and s^ubdued and held possession of it for nearly five centu- ries. Very few Latin words, however, were introduced into the language of Britain during the Roman occupation, per- hai)s not more than a dozen. A few names of places derived from ca,s(ra, a, camp, remain; as Chester, Westchester, Chi- chester, Winchester, Lancaster, which indicate that the mili- tary camps of the Romans became centres of trade, and grew into towns. About the fifth centurj' the northern barbarians invaded Southern Europe, and threatened the overthrow of the corrupt and imbecile Roman provinces. Rome, to defend herself, was obliged to withdraw her forces, and leave Britain to contend with the tribes that surrounded her. In the j^ear 451 the Saxons, a Teutonic tribe from the southern shores of the Bal- tic, undtT the lend of the two brothers, Hengist and Horsa, cami' over and settled on the shores of tent. Swarms of the NATURE OF LANGUAGE. 99 sann! tribes followed from time to time, and drove the Celts into the mountains of Wales and Cornwall. In the year 827, about four centuries after the invasion, seven independent kingdoms had been established, known as the Saxon Hep- tarch3\ The most important of these Teutonic tribes were the Jutes, the Angles, and the Saxons. The Angles, who seem to have been distinguished for their energy and intelli- gence, though small in numbers, gave their name to the island, Eno-land beinfr a modification of Angle-land. The Saxon Ian- guage thus became the language of the island, a few Celtic names being mixed with it. During the ninth and tenth cen- turies, the Danes, a Scandinavian tribe, made incursions and conquests, and introduced a few words into the Saxon language. In 1066, "William, Duke of Normandy, invaded England, and, by the decisive battle of Hastings, established himself on the English throne. He divided the island among his fol- lowers, and determined to incorporate the Saxons with the Normans, and introduce the Norman language as the language of the island. To effect this, he ordered that the youth in the schools should be instructed in the Norman language, that the pupils of the grammar schools should translate Latin into French, and that all conversation in them should be carried on in one of these languages. Pleadings in the courts were to be in French, deeds were to be drawn in this language, no other tongue was used at court and in fashionable society. So great was this influence, that English nobles themselves affected to excel in the foreign dialect. The mass of the peo- ple, however, at first resisted this change ; but finally, as the two peoples intermingled, their languages intermingled also, and the English language is the result, — the basis of it being Saxon, and about one-third of it being from the Norman French. The Norman French was mainly a Latin tongue. The Nor- mans, or Northmen, were originally the inhabitants of ancien* 100 METHODS OF TEACHING. Scandinavia — Norwa}', Sweden, and Denmark. Fnder Rollo, about 912 A. D., they had conquered and settled in a province of France, where in time they adopted the religion and lan- guage of the French. The French was a corrupt form of Latin, formed by the mixture of the Latin introduced into Gaul by the Romans, and the hmguage of the Germanic tribes who afterwards conquered it. The Norman conquest thus introduced a large element of Latin into the ?]nglish lan- guage. A large number of Latin words were subsequently introduced b^' Latin scholars ; and in the same way the Greek element of our language originated. Words of other lan- guages have been introduced by business, commercial rela- tions, etc. The English language is thus a composite tongue, its basis being the Anglo-Saxon, with about one-third Latin, a sprinkling of Greek, and a few words from other tongues. Classification of Lnnfjnnge. — Attempts have been made to classify tha ditferent languages, but no scheme has been given which is universal!}- adopted. Max Miiller speaks of four distinct stages in the growth of language : the first being the epoch of roots; the second being the epoch of juxtaposi- tion and concentration, as in the Chinese ; the third being the agglutinative stage, represented by the Turanian tongues ; and the fourth being the inflexional stage, or stage of amal- gamation, represented by the Semitic and Ar^^an languages. An arrangement based on outward differences of form, that will give quite a clear idea of the subject, divides languages into three classes ; the Monosyllabic, the Agglutinated, and the Inflected. The Monos3'llabic class contains those lan- guages which consist only of separate unvaried monosyllables. The words do not naturallj- affiliate, and the scientific forms or principles of grammar are either wanting or ver}' imperfect. It includes the Chinese and Japanese languages and also the dialects of the Xorth American Indians. In the Agglutinated languages, the words combine only in a mechanical way ; they have no elective affinity and manifest no capabilities of a > ■> NATURE OF LANGUAGE. lOl living organism. Prepositions are joined to nouns and pro- nouns to verbs, but not so as to make a new form of the original word, as in the inflected tongues. This class is called Turanian, from Turau, a name of Central Asia. The principal varieties of this family are the Tartar, Finnish, Lappish, Hun- garian, and Caucasian. The Inflected languages haA^e a complete interior organiza- tion, with mutual relations and adaptations. They differ from the Monosyllabic as organic from inorganic forms ; and from the Agglutinated as vegetable growths from mineral accre- tions. This class includes the culture of the world, and in their history lies embosomed the history of civilization. To this class belong two great families, the Semitic and the Indo-European. The Semitic embraces the languages native to Southwestern Asia, supposed to have been spoken by the descendants of Shem. It includes the Hebrew, Aramiean, Arabic, the Ancient Egyptian or Coptic, the Chaldean, and Phoenician. The Semitic languages differ widely from the Indo-European in their grammar, vocabulary, and idioms. On account of the pictorial element in them, they may be called the metaphorical languages, while the Indo-European may be called the philosophical languages. The two principal languages of the Indo-European stock are the Aryan and the Graeco-Italic, or Pelasgic. The word Aryan (Sanskrit, Arya) signifies we.ll-horn, and was applied by the ancient Hindoos to themselves in contra-distinction from the rest of the world, whom they considered base-born and contemptible. The Pelasgic comprises the Greek family and its dialects and the Italic family-, the chief subdivisions of which are the Etruscan, the Latin, and the modern lan- guages derived from the Latin. The other Indo-European families are the Lettic, Slavic, Gothic, and Celtic, with their various subdivisions. The Indo-European languages are noted for their variety. ' flexibility, beauty, and strength. They are remarkable for 102 METHODS OF TEACHINQ. vitality, and possess the power of regenerating themselves and bringing forth new linguistic creations. They render most faithfully the various workings of the human mind — its wants, its aspirations, its passions, its imaginings — and em- body and express the highest products of its thought and philosophy. Through them, modern civilization, by a chain reaching through many thousand years, ascends to its primi- tive source. II. Written Language. Written Language is the art of expressing ideas and thoughts by means of visible sj-mbols. It is the embodiment of mental products in a form by which they may be transmitted to the mind through the eye, as spoken language communi- cates them through the ear. Written language may be either ideographic or phonetic. Ideographic writing may be either pictorial, representing objects by imitating their form, or symbolic, indicating their nature or proportions. Phonetic writing may be syllabic or alphabetic; in the former each character represents a syllable, in the latter an elementary sound. Origin of fFritten Language. — Of the origin of written language, but little is positively known. The Egyptians ascribed it to Thoth, the Greeks to Hermes or Cadmus, and the Scandinavians to Odin. The first step toward writing was probably the rude pictorial representation of objects, the next step was the application of a symbolic signification to some of these figures. Pictures, abbreviated for convenience and by constant use, gradually became conventional signs; and at last these characters became the sj^mbols of the sounds of spoken language. Systems of Written Language. — There have been four distinct systems of written language; the Ideographic, the Verbal, the Syllabic, and the Alphabetic. These bear certain historic relations to each other, the last being an outgrowth NATURE OF LANGUAGE. 103 of the first through the intermediate stages of the other two. A brief description will be given of each. 7'//^ fdeographic. — The Ideographic system (idea, idea, and grap/io, 1 write), represented things by pictures and symbols. Concrete objects were indicated by their pictures, and abstract ideas by their symbols, etc. Thus, the sxn was indicated by ft circle with a dot inside, 0, the moon, by a crescent, with a line inside,!^ , a mountain by three peaks, side b\- side, Wf\, rani by drops under an overarching line, /s^ , a child thus, ^ . These symbols could be combined to represent other obje;ts. Thus loater and eye combined represented tears; an ejr and a dixyr represented hearing ov understanding. Actions would be rL'i)resented b}- objects in the attitude of the act, Sisfiying by a picture of a flying bird, ascending by the picture of a person walking ap a hill, etc. Some charac- ters were used symbolically, as a /< a /k/ to indicate a workman, tivo values of a shell-fish to denote friends. Relations could also be represented, as above, bj' a dot over a horizontal line, below by a dot below a horizontal line, right by the symbol 1^ , and left by "| , etc. These illustrations are taken from the Chinese system of written laiii^iiage. The system of writing among the Egyptians was hiero- glyphic, and is consireciate what we read. Both of these are necessary conditions for correct and effective reading. The Vocal Element is that which pertains to the voice. It embraces Pronunciation and Modulation. Pronunciation is the art of giving correct utterance to individual words. It TEACHING READING OR ELOCUTION. 169 embraces Articulation and Accent, both of which have been discussed. Modulation has reference to the variations of the voice in reading and speaking. It embraces Quantity, Com- pass, Quality, and Time, each of which has its appropriate subdivisions. The Physical Element is that which pertains to the body and its members. It includes Breathing, Posture, Gesture, and Facial Expression. Frinciples of Teaching — There are several fundamental principles that will be of advantage to teachers of reading. The most important of these are Natural Expression, Imitation, Principles, and Correcting Errors. 1. Natural Expression. — The fundamental principle in teach- ing reading is that of natural expression. The constant effort of the teacher should be to have the pupils read naturally, or to read as they talk. The following ideas should be kept constantly before the pupils' minds. Talking is the natural expression of one^s own thoughts ; read- ing is the natural expression of written or printed thought. Written or printed thought should be expressed in the same way as one would express it if it were his own thought. Good conversation is thus the basis of good reading. Good reading is reading as one talks. To read well a person should express himself just as he does in natural conversation. If his conversational style is faulty, the first step is to correct and im- prove it. To read naturally, the pupil must make the thought of the author his own thought, and then express it just as he would if he had originated it. The reader must re-create the ideas of the author and stamp them with his own personality, and then ex- press them as if they were his ovm and not another's. 2. Imitation. — Reading is an art, and like other arts must be taught partly by imitation. We learn to talk by imitating our parents and other members of the household ; and we learn to write by imitating written or printed forms. So in order to 8 170 METHODS OF TEACHING. learn to read well, we must hear good reading. The teacher, therefore, should read for his pupils and have them imitate his reading, being careful to avoid all mannerisms that may vitiate their style or interfere with natural expression. The teacher should be a good reader, that he may present a correct model for his pupils. He must often lead them to cor- rect expression by having them imitate his own reading of a sentence or selection. This is the more necessary from the fact that there are many things in the reading book so different from the ordinarv topics of conversation that pupils need the model of the teacher's voice and manner to guide them. 3. Principles. — There should also be some general principles to guide a pupil in reading. By a principle of reading is meant some general law which can be readily applied to the particular forms of discourse we meet with in literature. Rules of reading have been criticised, and properly so, fur no one can learn to read correctly by rule. A principle, however, is more flexible than a rule, and will be fi)und of very great value, with the more advanced pupils, in learning to read. 4. Correcting Errors. — The teacher must also rely on the cor- rection of errors for instruction in the art of reading. He must notice carefully the errors of pupils, and correct them. He should not merely call attention to these mistakes, but should train the pupils in correcting: them until they have overcome the old habit and acquired the new one. It is sometimes well to imitate the mistake of the pupil ; his attention being thus called to it, he will usually correct it himself. Teaching Primary Reading. The course in Primary Reading includes such instruction in the art of reading as is required by the majority of the children in our public schools. Suggestions for this course will be pre- sented under the three general divisions named. 1. The Mental ElExMENT. — The Mental Element lies at the basis of good reading. The mind thinks the thought, and in TEACHING KEADINQ OR ELOCUTION". 171 correct reading the voice should express just what is in the mind. The pupil should, therefore, understand that good reading is merely having something in his mind and telling it. All the principles of reading have their origin in the mind, and are applied by it. The most important of these principles, which may be regarded as the conditions of good reading, are those of Comprehension, Appreciation, and Conception. 1. Comprehension. — The first law of good reading is that of comprehension. The pupil must be led to see that reading is not calling words in the hook, but merely telling what he thinks and Jeels. He must be taught to read from his thought and not from his book. In order to do this he must be trained to the habit of getting the thought of the selection he is reading. 1. See that the pupil understands the meaning of the words. Go over the sentences and paragraphs and call attention to such words as the pupil may not understand. Have the pupil use the words in sentences, to see that they are understood. 2. See also that the pupils understand the thought expressed in the sentences. Have them state the thought in their own words. Require them to look at a sentence and grasp it as a whole before attempting to give it expression. 3. Require pupils to analyze each sentence and paragraph, and point out the prominent ideas, so that they may know where to place the emphasis. When they do not see the prominent ideas call attention to these ideas by appropriate questions. 4. Require pupils to study their reading lessons. Examine them on the lesson to see that they understand it before permit- ting them to read. Explain such things as are not understood, especially figures of Rhetoric, such as similes, metaphors, person- ifications, historical and classical allusions, etc. 5. Do not go through the book too rapidly. In teaching read- ing it is a good maxim to "make haste slowly." Keep pupils at a selection until they are quite familiar with it. The better they know it the better they can read it. Let the first aim be to make the pupils thoroughly comjjrehetid what they are reading. 172 METHODS OF TEACHING. 2. Appreciation. — The second law of good reading is that of appreciation. Pupils should be led to appreciate the sentiment of what they read. The voice should manifest the feeling as well as the thought; the heart should speak in the voice as well as the head. Reading without feeling in it is a cold mechanical thing without beauty or power. 1. To awaken an appreciation, see that there is a full and complete comprehension of the subject read. What is not un- derstood cannot be very well appreciated ; a clear idea in the mind naturally awakens some corresponding feeling in the heart. 2. Try to make the appreciation so full as to result in a com- plete assimilation of the thought or sentiment. Lead the pupil to make the thought or sentiment /m own, as if it were the pro- duct of his own mind and heart ; and he will then read it as if he were telling something he had thought or felt. 3. To secure this condition of appreciation and assimilation usually requires careful culture. It is a matter of taste, and the culture of taste is often a slow process. Try to lead the pupil to see what is beautiful and admirable in thought and sen- timent ; to have his heart throb responsive to the beautiful image or touch of pathos expressed in the author's lines. 4. Do not allow pupils to read subjects that are not suited to their appreciation. Such sentiments as " Contentment," " Patri- otism," "Melancholy," "Aristocracy," etc., are foreign to the heart of a child, and such subjects should not be given him to read. He can appreciate " the pleasures of coasting," " sorrow at the loss of a pet bird," etc., and his voice will throb in unison with his beating heart as he reads of these things. 3. Conception. — Pupils when reading should form a clear and vivid conception of the subject. Young children describe what they have seen with graphic effect, because the picture of what they are describing stands before their mind as they are talking. Lead them to picture, in the same way, what they read, and they will also express it vividly and naturally. 1. Require pupils to form mental pictures of such things as can TEACHING READING OR ELOCUTION. 173 be represented by the Imagination. If they read, " I see a bird in a tree," they should form in the mind a picture of the tree and the bird in it. If they read of "a boy fishing," they should see the water, and the boy in the act of catching fish. If the lesson is about "a horse running away," require them to picture the horse running just as they would if they had seen it and were describing an actual run-a-way. 2. With the more advanced pupils, take such selections as "A Leap for Life," by Colton, or " The Day is Done," by Long- fellow, or " Abou Ben Adhem," by Leigh Hunt; and require the pupils to form pictures in the mind as they read or recite these selections. Test the power to picture by asking them to describe what is in the mind when they read. 3. Where a mental picture of the subject can not be formed, try to make the abstract conception as clear and real as possible. See that the thought or sentiment is distinctly conceived. When the conception is distinct and real, the heart will respond to the thought, and the voice will instinctively and truthfully portray the sentiment. 4. This exercise of vivid conception will be found of great value in teaching reading. It gives a reality to the subject in the pupils' minds which makes their reading not a mere calling of words, but a real relation of the thought or incident expressed by the author. It may be stated as a maxim that vividness of conception is a golden key to truthful and effective expression. II. The Vocal Element.— The next step is to attain a proper use of the voice in delivery. First, there should be exer- cises to train the voice in the correct utterance of sounds. Second, care should be taken that all the words be correctly pronounced. Third, the form of expression of words in sentences should be correct and pleasing. These three points will be considered under the heads of Exercises, Pronunciation, and Expression. 1. Exercises. — Pupils require some exercises to give flexibility and precision to the voice. These exercises train the ear to a delicacy of perception that will enable the pupil to correct his 174 METHODS OF TEACHING. errors and improve his utterance. They will also give such a control over the voice that it can be readily adapted to the differ- ent selections read. The fullowino: exercises are sussrested. 1. Train the voice in respect to force, pitch, and rate. Use the vowel sounds (vocals) a, a, a, a, e, c, etc., for this purpose. Unite these vocals with the consonant sounds (sub-vocals), as ha, hd, ha, etc. Drill also on special words ; as arm, gold, etc. 2. For exercises in Force, require the pupils to repeat the sounds with varying force, from soft to loud. Have similar exei- cises on words and on sentences appropriately selected. 3. For exercises in Pitch, have the pupils repeat the vocals on different degrees of the musical scale from low to high. Have them sing the musical scale, and use it in exercises on pitch. Drill on slides or inflections, both rising and falling. 4. For a drill in Time, use the vocals and words, repeating them with shorter and longer time. Have them also read sentences with different degrees of time. Djill also on pauses. 5. For a drill in Emphasis, use properly selected sentences containing emphatic words. Sentences containing contrasted emphasis will be of special use in this exercise. Lead them to see that the prominence of the idea determines the emphasis. 2. Pronunciation. — The pupil should be able to pronounce readily and correctly all the words in the reading lesson before he begins to read. Bad reading and bad habits in reading often result from the pupils stumbling over unfamiliar words. 1. See that pupils are able to pronounce words at sight. Re- quire them to know the words at a glance, so that they can speak them in reading without hesitation or stammering. 2. It is often well to go over the sentence or paragraph and have the pupils pronounce the words before they attempt to read it. They may sometimes begin at the latter part of the para- graph and " pronounce the words backward." 3. With the more advanced classes, before reading anew les- son, go over it and have the pupils pronounce the unfamiliar or difficult words. Some of these may be written on the blackboard to aid the pupil in remembering them. TEACHING READING OR ELOCUTION. 175 4. Careful attention should be given to articulation and accent Let the teacher be particular to secure clear and distinct enun- ciation. Do not permit a drawling tone in the utterance of words, nor a slovenly, careless or unrefined pronunciation, 3. Ex2)ression. — The proper use of the voice in vocal utterance is the final step in reading. Thi.s is a high accomplishment, and demands great care for its attainment. AVhat has been previ- ously explained is all ])reparatory to this final object, but a few special suggestions on expression will be of value to the teacher. 1. The fundamental principle of expression is that the v(nce exactly express the thought uhich is in the viind. To secure this, see that there is comprehension, appreciation and conception; and then that the force, pitch and rate are such as the sentiment requires. 2. See that the piipils read naturally, as they ivould talk, pro- vided they talk correctly. Let the natural expression of the pupil be the basis of his method of reading. If he does not read natur- ally, require hira to look off his book and tell you the subject. 3. Be careful to secure a proper variety in the tone of the voice, as in good natural conversation. Do not allow children to use the stilted and mechanical tone so common in our schools, nor the monotonous sing-song in which young persons often read. Dis- card by all means the well known "school-room tone." 4. See that the emphasis is properly placed, as misplaced em- phasis is one of the common faults of reading. Be sure that the pijpil fully understands the subject he is reading, and sees which are the important ideas. Lead the pupil to see what ideas are most important, and he will give correct emphasis. 5. Notice with care that the pauses be properly placed, and are of the proper length. Lead the pupils to see that it is the thought, and not the marks of punctuation, that determines the place and length of the pauses. Show them also the value of the pause after and before the emphatic word. 6. See also that the slides or inflections are properly used. Lead pupils to see that the sense will determine whether the slide 176 METHODS OF TEACHING. is downward or upward. Call attention, when they are in doubt about the slide, to the manner in which thev would naturally express themselves if they were telling the subject. Do not allow the use of the circumflex where it is not required by the sense. 7. See also that there is proper natural melody in the use of the voice. Be careful that there is no jerkiness or abruptness in the use of the voice ; but a natural melodious flow of tone that gives a sense of musical beauty to their expression. III. The Physical Element. — The Physical Element in reading is that which pertains to the body. It is of special value in recitation and oratory, but needs little attenti(»n in ordinary reading. Only a few suggestions will therefore be presented under this head. 1. Have pupils stand erect, with the book in the left hand, so that the right hand may be free to turn the leaf when needed. While reading the right hand should hang at the side. 2. See that the feet are in a natural, easy position, and that the body is erect with the shoulders thrown gently back to give free- dom to the organs of the chest. 3. Permit no lounging or leaning upon the desk or against the wall, or standing in any awkward or ungraceful attitude. Finally, teachers, if you see that your pupils stand in a proper attitude; that they comprehend, appreciate and conceive what they read ; that they read naturally, with correctness of force, rate, pitch, emphasis, slides, pauses, and melody; you will be a successful teacher of reading in our public schools. II. Teaching Advanced Heading. The course in Advanced Reading shows the principles upon which the art is based, and aims to inculcate the practice from the theory. It embraces a brief treatise upon Elocution, and prepares for recitation and declamation as well as reading. It is presented under the three elements already named; viz.: I. The Mental Element. II. The Vocal Element. III. The Physical Element. TEACHING KEADIXG OK ELOCUTION. 177 I. The Mental Element in Reading. The Mental Element in reading is that by which we under- stand and feel what we read. It includes the Intellectual and Emotional elements. The Intellectual Element is that by which we understand what we read; the Emotional Element is that by which we feel what we read. Both of these will be brietly considered. The Intellectual Element — A pnj)il should understand what he reads. No one can read correctly what he does not fully comprehend. He may pronounce the words correctly, but unless he comprehends the thought he is endeavoring to present, it will be merely " calling words," not reading. This condition of good reading is frequently neglected. Pupils are allowed to read without having any idea of the meaning of what they are reading. Pupils sometimes speak pieces with- out any clear conception of the ideas and sentiments ex- pressed. The artificial and unnatural st3de in Avhich young persons read is largel}^ due to the neglect of this principle. Most ridiculous mistakes are sometimes made b}' pupils in endeavoring to read that which they do not understand, or which they misunderstand. Pupils should be required to prepare their reading lessons as they do other lessons. Every pupil should study his read- ing lesson. He should see that he knows the meaning of the words, the idea intended to be expressed by the author, the general character of the sentiment, the meaning and force of the prominent allusions, rhetorical figures, etc. It will be well to go over the lesson and mark the emphasis, slides, varieties of voice, etc., appropriate to the different parts of the piece to be read. If a portion of it were entirely or partly committed to memory, it could be read much more readily and correctly. It is said that the great orators studied their addresses so carefully that they knew just what words they were to emphasize, where to make a gesture, etc. 8* 178 METHODS OF TEACUING. Teachers should examine their pupils to see that they under- stand the reading-lesson. They should ask them questions upon the meaning of words, upon the thought intended to be presented, upon the figures and allusions that may be used, upon the historical or biographical references, upon the gen- eral sentiment of the piece, and upon the style or character of the composition. Teachers who have not been accustomed to such an examination will be utterly surprised at the ignorance and thoughtlessness of pupils in this respect. Some very amusing and ridiculous mistakes could be given, illustrating the necessity of such questions. Pupils may often be required to give the sense of a passage or paragraph in their own lan- guage, to see if they understand it. Be especiall}' careful in their reading of poetry, that it is not a sing-song of words, without any true conception of the meaning. The teacher should explain what the pupil does not under- stand. He should explain the meaning of words, sentences, allusions, figures of rhetoric, etc., which the pupil has not understood. When the pupil meets such expressions as the " Archimedean lever," or the " Palladium of our liberties," as found in Washington's address, or the "Niobe of nations," as found in Childe Harold., etc., the teacher should explain the historical fact or mythological stor}^ from which they are derived, and show the force and beauty of the figure. So in reading poetr3'^ ; when he comes to such passages as " The darkness falls from the wings of night," or " Ai^d Wind, that grand old harper, smote his thunder harp of pines," or" The Morn in russet mantle clad, walks o'er the dew of yon high eastern hill," etc., let the teacher call the attention of the pupil to the beauty of the image and make his imagination picture it before the mind as it was seen by the poet who wrote it. The heart of the learner can in this way be thrillerl with the emotion of beauty, the imagination be trained, and the literary taste be cultivated The reading books should be adapted to the pupils. Foi TEACHIXG READING OK ELOCUTION. li'J young pupils, we need simple descriptions, lively narratives, and interesting conversations or dialogues ; for more ad- vanced pupils, essay's, reflections, discussions, orations, etc., are approi)riate. This principle is freciueutly disregarded. ]\Liny authors have completely' failed in the ada[)tation of the reading matter of their books to the capacity and taste of the puj)!!. Only a few seem to have accomplished the difficult task of entering into the sphere of child-life, and adapting their writings to children. Teachers must also be careful to grade the books properly for tile pupils. The general fault is that the books are too difficult for the classes using them. The pupil is often in the Fourth Reader when he should be in the Second or Third Reader. In such cases the pupil should be put in a lower book if possible. If this cannot be done, the easier pieces should be selected, and the pupil drilled on them until he is familiar with all their difficulties. The more familiar a pupil is with a piece the better he can read it. The reading teacher should be a good scholar. In no class does a teacher require so much general culture as in reading. He needs a knowledge of history, mythology, rhetoric, etc., in order to explain the references, allusions, rhetorical con- structions, etc., in the lesson. The reading class, properly taught, can be made the most interesting and profitable class in the school. More can be done for literary culture here than in any other study. Indeed, many a person has received his first impulse to literary culture in the reading class as taught by some earnest and enthusiastic lover of literature. The Emotional Element. — A pupil should not only un- derstand what he reads, but he should also feel and appre- ciate it. Literature appeals to the heart as well as to the head. The reader should be susceptible to all the various phases of sentiment, and feel them when he is reading so that he may make others feel them. If the subject is pathetic, his ueart should be touched with pity; if it is humorous, he should 180 METHODS OF TEACHING. appreciate the humor; if it is grand and sublime, be bbould feel the emotion of grandeur stirring in his soul. This point is of great importance in all the higher departments of read- ins, and demands the teacher's attention. Pupils do not usuall}' feel or appreciate what they read. They will read one st3'le of composition in just about the same tone and pitch as another, so that if you judged the composi- tion by the manner of reading, you could not tell whether the}' were reading a funeral sermon of Bossuet, or a humorous description bj' Mark Twain. There is no response to the touch of pathos or beaui 7, no heart-throb to the poet's line, or the orator's sentiment ; indeed there is often no more feel- ing than if a talking machine were repeating the words of the reading-book. The teacher should call the attention of the pupils to the sentiment, and endeavor to awaken an appreciation of it. By appropriate questions and explanations, he should endeavor to open the eyes of the pupil that he may see, and unseal his heart that he may feel, those touches of beauty and humor and pathos which throb in the poet's line, or live in tiie orator's phrase. He should give illustrations of the different kinds of sentiments, and show how the voice and mannei should be adapted to express them. In a word, be should train his pupils so that they may feel what they read, as well as understand it. Reading books should be adapted in sentiment to the age of the pupils. The grander sentiments of sublimity, patri- otism, etc., are not suitable to children. They cannot be expected to be much moved by a description of the " Sublimity of the Starry Universe," or the "Enjoyments of Content- ment," or the " Remorse for Neglected Opportunities." The pathetic and many forms of the humorous, however, will be readily appreciated. The narration of interesting events, of dangers in field or forest, of hairbreadth escapes, of the rob- bing of a bird's nest, of sorrow at the loss of a mother or TEACHING KKAlUNG OK ELOCUTION. 181 sister, etc., will awaken their little hearts to intense feeling. The compilers of text-books on reading should bear this in mind, and govern themselves in their work accordingly. The teacher should not only be a good literary sciiolar, but he should also possess a cultivated taste. Refinement of mind, a heart to feel and appreciate the beautiful and good, will enable a teacher of reading to touch the hearts of his pupils and cultivate in them a refinement of taste which will improve both their character and their reading. The teacher of reading should therefore take special pains, by the study of the fine arts and the cultivation of that which is beautiful and noble in human character, to acquire such refinement of taste and feeling as shall fit him for the highest attainments in his hisrh art. O' IT. The Vocal Element in Eeading. The Vocal Element in reading is that which pertains to the voice. It is the fundamental element of the art of readinir. The Mental Element is merely a condition for good reading, and the Physical Element an accompaniment of it ; but the Vocal Element is that which is immediately concerned in reading. It is the basis upon which the art is established. The importance of vocal culture in reading cannot be over- valued. The excellence of reading depends mainly upon the character of the voice. When the voice is harsh or hard and inflexible, it is impossible to read with artistic eflfect. A full, rich, musical voice will chain the attention of an audience, independently of the sentiment expressed; and when em- ployed in the expression of noble and soul-stirring sentiments, its influence is irresistible. Much of this excellence can be acquired by judicious cul- ture. Though some voices are by nature richer and more musical than others, yet careful training will remove many defects and impart flexibility and sweetness in a remarkable degree. Nearly every one is familiar with what culture and 182 METHODS OF TEACHING. training will do for a singer; and vocal culture is as necessarj' and useful to the reader as to the singer. The human voice, in the hands of a master, will attain to a wondrous strength and richness of tone. Practice also will give a person such a command over his voice and enable him to use it with such skill, that he can hold the attention of an audience by the music of his utterance, and thus deepen tlie impression of the sentiments he may express. The Vocal Element embraces four things ; Quantidj, Com- pass, Quality, and Time. These elements are usually included under the head of Modulation. Each of them will be consid- ered somewhat in detail. I. Quantity. — Quantity, as employed in reading, has refer- ence to the amount or volume of the voice. It is used by some elocutionists to mean the time occupied in pronouncing a word or syllable; but this is not the best or most accept- able use of the term. Quantity in reading is a general term including Force, Emphasis, Stress, and Slur. Quantity of voice is an important element of expression. Each sentiment has its appropriate quantity, and the quantity, if properly used, will indicate the sentiment. Thus, joy is expressed in a full tone, sorrow in a subdued tone ; modesty, humility, shame, doubt, mystery, etc., require soft and sub- dued tones. Anger declares itself in loud tones, confidence asserts itself with a full voice, secrecy softens the tone and speaks with muffled voice or whisi)ered accents. Force. Force is the quantity of voice used in reading or speaking. It is quantity as applied to vocal delivery. As used here it has reference to the standard force of the voice in reading or speaking. There are three degrees of Force; Soft, Moderate, and Loud. . Moderate Force is the ordinary force of the voice in reading and speaking. Soft Force is less force than the ordi- nary quantity ; and Loud Force is more force than the ordi- nary quantity. These are not fixed degrees of force, but TEACIIIXG READING UK ELOCUTION. 183 merely' relative distinctions. Let the pupil l)e careful not to confound loud and soft with high and luw, which are degrees of i)itch. A mistake of this kind often leads the reader, when he designs to increase his force, to raise his voice to a higher pitch, thus giving a higher instead of a louder sound. Hoio Tedch. — We should teach reading with respect to force In- Exercises, Imitation, and Correcting Errors. Exercises. — The Exercises recommended to cultivate force are as follows : 1. A frequent drill on the elementary sounds; 2. A drill on sentences selected for the purpose; 3. Physical exercises to develop the general health and strength. In the drill on the elementary' sounds, we should begin M'ith a moderate degree of force, and then increase the force gradu- ally to the limit of loudness, being careful not to strain or overtax the voice. Having reached the louder tones, pass gradually from these to the softer tones. After some prac- tice in this wa}', the pupil ma}' begin at the loud tones and pass to the softer ones ; or he may practice striking at once different degrees of force until he can give with ease and pre- cision any degree of force, from whispering to shouting. Similar practice with well-selected sentences is also valuable. Let the same sentence be given with varied degrees of power; and let sentences be selected requiring variety of force for their natural expression. Such exercises, continued for a few- months, will greatl}'- enlarge the quantitj- of the voice and giA'^e the reader a command over it by which he can readily adapt it to the requisites of reading or speaking. In case of weakness of voice, arising from ill health or lack of i)hysical strength, a course of gymnastics is recommended. The weak voices with which many clergA-men are troubled, coidd be cured on the base ball ground or in the gymnasium Theological students, or those preparing for public speaking, should take special pains to secure a vigorous constitution. Many a sermon could be rendered more eloquent and effective in this wu}', and many a case of bronchitis avoided. 18-i METHODS OF TEACHING. J'nndple. — Determine the standard force by the general spirit of the piece. If the general spirit is unemotional, the standard force is moderate ; if the general spirit is bold, noble, dignified, etc., the standard force is loud ; if the general spirit is grave, subdued, pathetic, etc., the standard force is soft. The pupil who grasps this principle and applies it intelligently, will find it of great value in reading. Correct Errors. — Some pupils read too softly or with too little force. This is often the case with young ladies. The admiration of the " low voice in woman " is carried to such an extent with many, that it is regarded as unladylike to read in public so as to be understood. It is an error, however, and one that should be corrected. To correct the error of reading too softlj'-, the teacher must notice its cause. Reading too softly is sometimes the result of a weak voice, sometimes of timidity, sometimes it is merely an affectation, and sometimes an unconscious habit. Correct the first by strengthening the voice, the second by aiding the pupil to acquire confidence, the third by a little judicious rid- icule, and the fourth by showing the pupil the defect and inducing him to overcome it. A pupil who reads too softly may be placed at a distance from the teacher in reading. Such pupils may read dialogues, standing on opposite sides of the school-room, or at some convenient distance from each other. Some pupils read too loud. Boys often make this mistake. Loud reading was formerly considered the best reading ; and boys would read almost as loud as they could shout. We can correct this error by showing them how unnatural and inap- propriate it is, and thus lead them to a natural method of expression. Most pupils do not adapt the force to the sentiment. This arises from the fact that they do not understand that to read anything is to express it naturall}'. This error needs the teacher's most careful attention. The pupil must be led to TEACHING UEADIXG OR ELOCUTION. 185 see that reading is natural oral expression, and that the force of the voice must be adapted to the sentiment expressed. Emphasis. — Emphasis is particular force applied to one or more words of a sentence. Its object is to give prominence and distinction to the important ideas. It brings out the meaning of an author, makes his thoughts and sentiments im- pressive, and gives beaut\- to expression as the play of light and shade does to a picture. A true emphasis keeps the attention of the listener in active s^-mpathj- with the thoughts of the speaker, gives full effect to all he utters, and makes a leep and lasting impression on the memorj'. There are two kinds of emphasis; Absolute and Antithetic. Absolute Emphasis is that which is applied to the prominent ideas of a sentence without an}- particular comparison with other ideas. Antithetic Emphasis is that which is used in contrasting ideas; as, "I said an elder soldier, not a better." Hoiv Teach. — We teach Emphasis by means of Exercises, Imitation, Principle, and Correcting Errors. Exercises, etc. — For Exercises, drill the pupils on well selected sentences containing emphatic words. Dialogues will be found most suitable for young pupils. Repeating the ele- mentary sounds, emphasizing at intervals, is a good drill exercise. The teacher should also read for the pupils, placing the emphasis correctly, and require the pupils to imitate him. Some of the chapters in the Bible, as that of the Prodigal Son, so often incorrecth' read, might be selected as an example. Principle — Xo specific rule can be given for emphasis ; it is a matter of judgment and taste. The principle of Prof. Bailey will be of great advantage in applying emphasis. This prin- ciple is closely related to that of Force, and, as he gives it, is a part of the former. It is as follows: Having determined the standard force for the unemphatic ideas, give more force to the emphatic ideas according to their relative importance. Correct Errors. — The teacher should constantly watch and correct the errors of pupils with respect to emphasis. The 18G METHODS OF TEACHING. most common errors are those of incorrect and random em ])hasis. Sometimes the emphasis is wrong because the pupil mistakes the sense ; this is corrected by calling attention to the important word. More freqnenti}' , the emphasis is api)lied at random, without any thought as to the prominent ideas. This error is often heard in the reading of the Bible and sacred hymns. It ma}- be corrected by calling attention to the proper use of emphasis, and the reader's disregard of it. A very common fault in emphasis is the use of the circum- flex upon the emphatic word instead of the slide, as will be subsequently explained. The faulty emphasis of the circum- flex can be removed b}- drill on appropriate examples, and by expedients adapted to individual cases. Another fault, often met with, is that of stiflf and excessive emphasis, which can be remoA'ed by practice, the study of good models, and the culture of taste. Stress. — Stress is force applied to particular parts of mono- syllabic words or s^dlables. It is an unequal distribution of force on a S3'llable, and gives variety in the expression of a single word, as emphasis does in the expression of a sentence. There are five kinds of stress; Radical, Vanishing, Median, Compound, and TJtorough. Radical Stress is force applied to the first part of a mono- syllabic word or of a syllable. It may be illustrated In- pro- nouncing the words eat, out, etc. It is used in expressing anger, command, positive assertion, and in energetic senti- ments of all kinds. By it animals are awed into submission, and audiences are often startled, thrilled, and swayed. Median Stress is force applied to the middle of the word or syllable; as ma^- be heard in pronouncing gold, fa?-, leap, etc. It is used in expressing dignity, grandeur, solemnity, supplication, plaintiveness, etc. Median Stress gives beauty and expression to delivery. It is the natural utterance of thoughtful sentiment, and the swell is more or less prolonged as the feeling is moderate, or deep and full, lofty and sublime TEACHIXG READING OR ELOCUTION. 187 It gives music to poetry, the spirit of devotion to sacred coin- position, and the touch of eloquence to orator3-. Vanishing Stress is force ap{)lied to the latter part of a word or syllable ; as may be illustrated in pronouncing hell^ low, ring, etc. It is the expression of intense feeling deferred ind accumulated upon the latter part of a word, as a child says, / iconH, I shatiH. This stress is used in expressing earnest purpose, determination, stern rebuke, contempt, aston- ishment, horror, etc. It is not so much an element of dignity as the median stress, yet it is an essential condition of hiy unremitting drill. 1'auses.— Pauses are cessations of the voice in reading and speaking. They are the intervals between the utterance of word-i, clauses, sentences, and jviragraphs, which correspond with and mark the divisions of meaning. Then' are two kinds of pauses ; the Grammatical and the IMietoricil. The (Jrammatical Pauses are those which indicate the logical or gramm:i,tical relation of the different parts of the discourse ; tlu'v are represented by the punctuation marks. Tlie Ulictorical Pauses are those which are recpiired to bring out the sense or exjjress the sentiment of a discourse; they 200 METHODS OF TEACHING. are not marked, but are determined by the sense of the piece and the judgment of the reader. Pauses are of great importance in reading and speaking They are required both for ease of utterance and for clear and emphatic expression. They are useful to both the reader and the listener. To the speaker the rhetorical pause is necessary for breathing after uttering a succession of sounds embracing at least one word which demands a great impulse of the organs, and which partiall}' exhausts the suppl}- of breath. The rhetorical pause is specially important to the listener. A proper pause at the end of a sentence, rests the mind of the hearer, and gives it time to dwell a moment upon the idea or sentiment presented. A pause after an emphatic word gives the mind an opportunity to linger on the idea and receive the full impression from it. As some one remarks, it gives time for the idea " to soak in" the mind of the hearer. It is thus true, in more senses than one, thai " a pause is more eloquent than words;" and that though "speech may be silvern, silence is golden." How Teach. — Pupils should be drilled in exercises to ac-' quire the right use of pauses. Show them the necessity and use of the rhetorical pause. Make them see that the length of the pause depends on the sense, and not on the punctua- tion. The old method of counting one at a comma, two at a semicolon, etc., is entirely objectionable. Show them also the use of the emphatic pause, and drill them in using it. Let the teacher give correct models, and correct all errors. Principle. — The length of the pause depends on the nature of the discourse spoken or read. In unemotional composition the pauses are moderate ; in energetic and impassioned utter- ance the pauses are long in order to give impressive emphasis; in strong and excited utterance they are often short and irreg- ular. Awe, solemnity, grandeur, etc., require long pauses, both at the end of sentences and for emphasis. Patfse in Poetry. — The measured character of verse requires TEACHING READING OR ELOCUTION. 201 certain pauses not used in prose. These are called the Poetical or Harmonic pauses. The Final pause is a short pause often used at the end of a line to mark its close. The Ctesural pause is that which is used to divide a line into equal oi- un- equal parts. The Demi-cassural pause is a short pause which is sometimes used to divide the parts of the line already' divided bj- the ctEsura. The rhetorical and ciBsural pauses usuall}' coincide. When no pause is required, either b}- the punctuation or the sentiment, the harmonic pau^e should not be observed. Reading Poetry. — The chief faults to be avoided in reading poetry are the following: 1. Too rapid utterance, b}- which the effect of the verse is lost to the ear ; 2. A jiiain and dr^'^ articu- lation, which, though it may bring out the meaning, does not indicate the beaut}^ of the sentiments and the rhythm ; 3. A mechanical observance of the harmonic pauses, without regard to the meaning; 4. A mouthing and chanting tone, pro- ducing the effect of bombast and mock solemnit}-, 5. A sing- song stjde, as frequently heard in the school-room. Poetry should be read a little more slowly than prose, with a moderate prolongation of vowel and liquid sounds, a slight degree of musical utterance, and with an exactness of time, as indicated by the nature of the verse and the emotion ex- pressed. The utterance should indicate the metre but should never render it prominent. IV. Quality. — Quality of tone has reference to the kind of voice used in reading and speaking. It is one of the most im- portant elements of vocal expression. The tone itself, inde- pendent of the words used, is expressive of thought and feel- ing. Tone is the language of the heart ; the soul can be thrilled by the utterance of melodious and varied sound. A rich, sweet voice will hold the attention of an audience, even when there is no especial interest in the thought expressed. A pleasing voice will cast a charm of feeling and intei'est around the dullest composition. 202 METHODS OF TEACHIXQ. All the varied tones which can be uttered b}' the human voice have been embraced under six classes; Pure, Orotund, Tremulous, Aspirated, Guttural, and Falsetto. These difter in different persons in accordance with the natural quality of the voice, j'et they represent distinct characteristics of the voice of each individual. Pure Tone is a pure, clear, round tone of voice. It is the ordinarj' tone of a good natural or well-trained voice. All the breath is vocalized, and the tone is produced by a very slight resonance in the head. It is appropriate to all kinds of dis- course not strongly emotional. Orotund is a full, deep, round, chest tone of voice. It is produced by a greater resonance in the head and chest, and requires a depression in the larj'nx, an opening of the throat, extension of the mouth, and expansion of the chest. It is appropriate to the expression of sentiments of dignity, grandeur, etc. It is emplo3'ed in reading epic and dramatic poetry, and is indispensable in oratorj'. Orotund qualit}' admits of three degrees ; Effusive, Expul- sive, and Explosive. Effusive Orotund is used in the utter- ance of sentiments of solemnity and pathos, when mingled with grandeur and sublimit3\ It is also the appropriate tone of reverence and adoration. Expulsive Orotund belongs to earnest and vehement decla- mation, to impassioned emotion, and to any sentiment uttered in the form of shouting. Explosive Orotund is the language of intense passion. It is heard when the violence of the emo- tion seems beyond the control of the will, as in a sudden ecstasy of terror or anger. Tremulous Tone is a vibratory tone of voice. It consists of a vibration of the pitch of the voice in the utterance of a word. It is used in expressing pathetic sentiments, in grief, pity, sympath}", tenderness, etc.; in suppressed excitement; in the trembling tones of old age, and occasionally in the exuberance of joy. A slight tremor often adds a charm to TEACHING READING OR ELOCUTION. 203 utterance, as the tremula in singing and violin plajang does to music. It siiould, however, be used with discretion, being careful that it is not overdone, when it savors of affectation. Dropping in now and then unexpectedly on expressive or tender words, it produces a very tine effect. Aspirated Tone is a whispered articulation, or a speaking by articulating the breath rather than the voice. It is used to give increased intensity to the utterance of the various emo- tions. It imparts an air of mystery to a subject, and is thus used in expressing wonder, fear, and in circumstances where the voice is awed into silence. It is sometimes used in givino- utterance to scorn, contempt, rage, etc., where the intensity of feeling seems to choke or destroy the power of vocal utter- ance. The Guttural is a deep throat tone of voice. It is a depth of utterance so low as to pass beyond the range of pure tone. It is used in expressing hatred, contempt, loathing, etc. The Falsetto is that peculiar tone heard in the higher degrees of pitch after the natural voice breaks or apparently transcends its range. It is used in the expression of extreme surprise, mockery, etc., and in the emphatic scream of terror, pain, etc. The most common use of it is with men in imitat- ing female voices. Hoiv Teach — Drill the pupils on exercises until they can readily give all the various kinds of tone. Lead them to see the adaptation of the tone to the sentiment. Correct all errors in respect to the quality of the voice. If there are any natural defects of quality, point out the errors and endeavor to have the pupils correct them. Have them imitate the teacher, and apply the principles. The princii)les are given in the statements of the natural relation of the quality to the sentiment to be expressed. 204 METHODS OF TEACHING. III. The Physical Element in Reading. The Physical Element is that which pertains to the body and its members. It is, as it were, the addition of visible lan- guage to oral expression, and is thus used to give emphasis and impressiveness to the spoken words. It includes Breath- ing, Posture, Gesture, and Facial Expression. I. Bredithing. — In order to read or speak well one m ist know how to breathe correctly. It is an element of gv^at importance, and one which has been greatly neglected. Many public speakers ruin their voices merely because they do not know how to breathe. Teachers' voices " give out" because thej^ make the muscles of the throat do the work of the sides and waist. Preachers are on the retired list with bronchitis who might have preached half a century, if they had known how to breathe properly. We present the following sugges- tions upon this subject: Hoiv to Jiveaihe. — Breathe deeply. Some people breathe merely with the upper part of the lungs. Let the entire lungs be brought into action. Breathe all the way down to the waist. Let the diaphragm be lowered, let the muscles of the back and the sides be brought into action, and let the waist be enlarged, even at the sacrifice of tight clothing and a false ideal of beaut3% Such an exercise will be of great value to weak lungs as well as to weak voices. Use no more breath in speaking than is needed. Very little breath is vocalized in speaking or reading, as may be seen by holding a piece of tissue paper hung by a silken thread, befoi-e the mouth, when speaking. The paper will scarcely move except in uttering the aspirates. Let the breath, therefore, be used with economy to insure ease and freedom of utterance There is no need of pupils getting out of breath in reading oi speaking, and the puffing and blowing of some speakers is not only unnecessary but ridiculous, reminding one of the spout- ing of a porpoise. TEACFIING READING OR ELOCUTION. 205 Be careful not to mix the breath with the voice. This is a fault occasional!}' met with among 3'oung pupils, and is a serious error in delivery. "Every tone," says Madam Seller, " requires for its greatest possible perfection, only a certain quantity of breath, which cannot be increased or diminished without injury to its strength in the one case, and its agreea- ble sound in the other." The use of too much breath mars the beauty of utterance and exhausts the reader. In breathing, the air should be inspired thi'ough the nose, and not through the mouth. A speaker who takes in air through his mouth will find his throat becoming dry by the evaporation of the mucus, or natural moisture with which nature lubricates the vocal organs. Besides, if there are any irritating particles in the air, they will produce an irritation and titillation in the throat. n. Posture. — Posture has reference to the position of the body and its members. The position of a person in reading or speaking is a matter that should not be overlooked. A person's appearance before an audience has much to do with the attention with which people listen to what he says. Anything awkward, clownish, or affected in the attitude, will naturally prejudice an audience against a reader or speaker. Elements of Posture. — Posture includes the position of the feet, the hands, the head, and the body. The Feet. — The feet should be placed at an angle with each other, the weight of the body resting on one foot instead of on both. The foot not sustaining the bod}' should be thrown slightly forward of the other, in such a position that if drawn towards the other, the heel of it would come to the hollow of the other. The foot wliich sustains the weight should be so placed that a perpendicular let fall from the pit of the neck would p.iss through its heel, the centre of gravity of the body being, fir the time, in that line. The sustaining foot is to be planted firmly, the leg braced but not contracted, the other foot an 1 limb being relaxed and resting for change. Thf 206 METHODS OF TEACHIXG*. weight should be occasionally changed from one foot to the other, care being taken that the transition be gently and easily made. The characteristics of a good attitude are thus firm- ness, freedom, simplicity, and grace. Another position of the feet is that in which the toes are on a line, the feet being slightly inclined to each other, the toes turning outward. There are persons with some peculiarity of the shai)e of the legs or feet, to whom this position is more suitable than the one previously described. This position is usuall}' more becoming to short than to tall persons, and is especially suitable to children. The errors of position are: continually changing the weight of the bod}' from one foot to the other ; swinging to and fro ; jerking the body forward at regular intervals, or after every emphatic word; crossing the feet or the legs; turn- ing in the toes ; standing with one foot on a stool or chair round, etc. An over-nicety in regard to position that attracts attention is also objectionable. Care should be taken to avoid all those errors. The posture should be equally removed from the awkwardness of the rustic and the affectation of the dancing-master. It should be natural, free from any bad liabits ; and will thus be both easy and graceful. Tlie Hands. — The hands should hang naturall}' and easil}- down at the side, except when they are being used for ges- tures. The fingers should be slightl}- bent and just touch each ("ther, and the thumb should be parallel to the fingers. Gentlemen sometimes place one hand at the waist, supported by the vest or buttoned coat, and ladies often read and recite with one or both hands at the front of the waist. These posi- tions are perhaps not ver}- objectionable, but are regarded as less elegant than when the hands are at the side. "When a book is used in reading, it should be generall}' held in the left hand so that the right is free to turn the leaf. The errors in the position of the hands are those of place and form. The errors of the first class are, — putting the TEACHING READING OR ELOCUTION. 207 hands in the pockets, placing them on the hips, playing with a button or the watch-guard, or Avith any portion of the dress, frequent changes as if tiie person did not know what to do with the hands, etc. The errors of the second class are, — spreading the fingers, closing the fingers too tightly, sticking out the thumb, straightening out the hand, closing the hand into a fist, etc. The teacher should carefullj^ note and correct all errors, and secure a natural, easy, and graceful position of the hands. Tlie Head. — The head gives the chief grace to the person, and is an important element in deliverj-. The position of the head should be erect and natural. It should not droop, which indicates humility or diffidence ; nor be thrown back, which indicates arrogance and pride ; nor be inclined to one side, which indicates languor, indifference, or clownishness ; nor be held too stiff, which indicates a lack of ease and self-pos- session. The Body. — The position of the body should be erect, easy, and natural, with the breast fully fronting the audience. The shoulders should be thrown gently back, so as to give the full- est freedom and capacity to the organs of the chest. The errors to be avoided are, leaning forward or backward, round- ing the shoulders, leaning to one side, and being too rigidly erect. How Teach. — In teaching posture, the teacher should him- self be able to present a model for imitation. He should be careful to correct all errors of feet, body, head, arms, hands, etc. He should also make his pupils familiar with the principles of posture. There is a natural language of posture, a language common to all times and races. For these principles, see works on Elocution. TTT Gesture. — By Gesture is meant the movement of the body and its members. It is a visible manifestation of thought and sentiment which accompanies its oral expression. Gesture is one of the most important concomitants of elocution. Some 208 METHODS OF TEACHING. writer remarks, "In the natural order of passionate expres- sion, looks are first, gestures second, and words last." De- mosthenes, when asked what are the requisites of an orator, replied, " Action, action, action." Gesture is the natural lano-uage of thouijht and sentiment. It is a universal language, understood by all people. No matter what their speech, all know the meaning of gestui'es, and can communicate with one another thereby. An entire play can be presented in pantomime so as to be fully understood. Ges- ture is visible language, apparent to the eye as the spoken sound is to the ear. When combined with speech, it is thus easy to see how it enforces the sentiment expressed. Indeed, it was a matter of dispute between Roscius and Cicero which could produce the greater effect, the former b3^ gesture, or the latter by spoken words. Gestures may be divided into three classes ; those of Loca- tion^ Illustration, and Emphasis. Gestures of Location are designed to indicate the position of the object or idea referred to. Gestures of Illustration are designed to show the way in which something appeared or was affected. Gestures of Emphasis are designed to give greater intensity to the mean- ing of words or sentences by physical movements. Eleuienfs of Gesture. — Gesture, in its fullest sense, in- cludes the Bow, and the position and movement of the Head, the E^-es, the Arms, the Hands, the Body, and the Legs and Feet. The Bow. — The Bow of a speaker should be graceful, easy, and dignified. It should be free from a careless, jerking abruptness, and from a formal, unnecessary flourish. It should not be too low, so as to seem overdone, nor too short, so as to seem trifling or disrespectful. It should not be a mere nod of the head, but the entire body should be slightly included in the movement. The bod}- should be bent directly forward, and not on one side. The foot ma}' be slightly' drawn back, or not, as is preferred. Some teachers prefer that there TEACHING READING OR ELOCUTION. 209 shall be a step backward subsequent to the bow ; but this is a matter of taste, and is not essential. The Hands. — The Hands are the most important members In gesture. As Quintilian remarks, these almost speak them- selves. " By them we ask, promise, call, dismiss, threaten, supplicate, detest, fear; display jo}*, sorrow, doubt, acknowl- edgment, penitence, manner, abundance, number, time." "So that amid the great diversity of language among all races and nations, this appears to me to be the common speech of all men." The Form of the hand in making gesture should be natural and unconstrained. The fino;ers should lie near one another, slightly curved, the thumb being pai'allel with the fingers. The gesture with the forefinger is sometimes appropriate, and is very expressive when the finger is long and slender. A gesture with the fist is very seldom allowable. The errors of gesture are, fingers straight and rigid, too much apart, too closely pressed together, thumb projected from the hand, etc. The Position of the hand in an ordinary gesture of empha- sis, should be a little above the waist, between the waist and shoulder. In referring to anything above one, or to grand and lofty sentiments, it should be elevated ; in referring to anything situated low, or to any low, debased sentiment, etc., it should be below the ordinary position. The Movements of the hand should be srraceful and in ffood taste. The hand should be raised in curved, and not in straight lines; and the movements should also be in gently curving lines. Gestures will thus embod3' the elements of grace and beautj*. Care should be taken that the movements and transitions be not abrupt or angular. After a gesture, the hand should fall gently and naturally to its place, and not go down with a jerk, or with an awkward restraint. The Arm. — The Arm, when not used in g-esture, should hanar naturally at the side. In gesture, the elbow should be slightly bent, except in the most emphatic gestures, when it may ofteii 210 METHODS OF TEACHING. be rigid and straight. Care should be taken not to exhibit an angle at the elbow. The Eyes. — The Eyes, -which are an important element of expression, should generally be directed as the gesture points, except when we wish to condemn, refuse, or require any object to be removed. The eye should rest upon the audience, not with a familiar stare, but with a kindly, modest, and dignified expression. To show a modest confidence in your audience goes very far to secure their confidence and sympathy. How Teach. — The teacher should be able to present a model in gesture worthy of the imitation of the pupil. He should also make the pupil familiar with the general principles which express the natural relation between the sentiment and the gesture. He should also be careful to correct all awkwardness of manner, in- appropriateness of movement, etc. Principle. — The first principle of gesture is that it should be natural and appropriate. The second principle is that it should be graceful, moving in fluent and connected lines, and not abrupt and desultory. A third principle is that strong, bold, determined, and abrupt expressions require straight lines; while all beautiful, grace- ful, grave, grand, and exultant sentiments require curved lines. We determine the force and extent of the gesture by the senti- ment expressed. If the sentiment is unemotional, as in ordinary conversation, the gestures are moderate, the movement being mainly from the elbow. If the sentiment is earnest, lofty, and sublime, as in oratory, the gesture is strong and wide, the arm moving mainly from the shoulder. If the sentiment is highly impassioned, as in dramatic composition, the gestures are still more vigorous and extended. Correct Errors. — Do not allow too many gestures. Excess of gesture is like redundancy of language, in bad taste and tiresome. Too few gestures are better than too many. Inex- pressive or meaningless gestures should be avoided. No ges- ture should be made without a reason for it. Some speakers accompany nearly every word with a bodily motion, which TEACHING REAInXG OR ELOCLTTIOX, 211 fatigues the eye and offends the taste. A gesture that illus- trates nothing is worse than useless ; it destro^-s the effect at which it aims. When a gesture has been assumed, there should be no change from it without a reason. Tne habit of allowing the hands to fall to the side immediately- after a gesture, produces an ungraceful and restless effect. IV. Facial Expression. — The face is the mirror of the mind. By nature it reflects promptly all changes of senti- ment and feeling. It is therefore one of the most important elements of expression. A voice may be artistic in its mod- ulations, it ma3- attune itself harmoniously to language, l)ut if the soul of the speaker docs not shine out from the coun- tenance, much of the power of expression is lost. All the great speakers and writers on oratory have under- stood the power of facial expression. Quintilian sa3"s, " The face is the dominant power of expression. With this we sup- plicate; with this we soothe; with this we mourn; with this we rejoice; with this we triumph; with this we make our sub- missions ; upon this the audience hang ; upon this the}' keep their e^'es fixed ; this the}' examine and study even before a word is spoken." Elements of Facial Expression. — The principal features in facial expression are the eyes and the mouth, though the brow and cheeks aid in exj)ression. The Eyes. — The eye is the window of the soul. Out of it the soul seems to shine, and the heart can be read by peeping in the eyes. " When there is love in the heart," says Beecher, " there are rainbows in the eyes." " The eye," says Tucker- man, " speaks with an eloquence and truthfulness surpass- ing speech. It is the window out of which the winged thoughts fl}' unwittingly. It is the tin}' magic mirror on whose crystal surface the moods of feeling fitfully play, like the simlight and shadows on a still stream." Many writers speak of "the mute eloquence of a look ;" and Byron sings of eyes which "looked love to eyes that spake again." 212 METHODS OF TEACHING. Tlie Mouth. — The mouth is even more expressive than the eyes. The peculiar character of the face is largely due to the size and shape of the mouth. A small mouth indicates secretiveness ; a large mouth, open-heartedness and good humor ; parted lips iudicate listlessness or stupidity ; com- pressed lips are a sign of firmness and decision of character ; etc. The expression of the mouth is due principally to the corners of the mouth. We draw up the corners of the mouth in laughing, and depress them in crying. "To be down in the mouth" is an expi-essive phrase for low spirits. In a picture the same face may be changed from laughter to weep- ing by merely making a change in the corners of the mouth. How Teach. — In facial expression nature must be our guide. The soul must feel the sentiment to be expressed, and the countenance must be the mirror of the soul. The play of features must respond to the sentiment stirring in the heart. The following propositions will indicate the general principles of facial expression. Unemotional sentiments require the countenance to be in repose. Sentiments of good humor, happiness, etc., require a pleasant and smiling countenance. Bold, grand, and noble sentiments require dignity and animation of countenance. Humorous sentiments require the play of humor in the face ; sad and pathetic sentiment should be accompanied with a dejected and softened expression ; shame requires the averted eyes and blush of guilt. Determination, anger, and a spirit of defiance are expressed by a contracted brow and com- pressed lips ; in scorn we elevate the upper lip and nose ; in fear, surprise, and secresy, the brow is raised, the eyes are opened, and the lips parted. General Suggestions. — We close this chapter on Reading with two or three philosophical and practical suggestions. Reading is an Art, and the basis of all Art is Nature. The object of culture in Elocution is therefore natural expression. It aims not to eliminate, but to train and improve the natural TEACHING READING OR ELOCUTION. 213 expression. Everything artificial in expression is regarded as inartistic and distasteful. The reader who "shows his elocution" in his reading, offends good taste, and shows his shallowness of mind and the imperfection of his art. In elocution especially, we should endeavor to attain that excellent standard of culture in which "the highest art conceals art." We are to look for natural expression in conversation. Con- versation is the simplest and most common form of human ex- pression. " It contains the germs of all speech and action," and thus constitutes the basis of all correct delivery. The importance of cultivating correct habits of voice and manner in conversation cannot be over-estimated. Conversation is a beautiful art, and deserves culture for its own sake, and also as a basis of elocu- tionary culture. The standard by which we judge of good reading is a cultivated taste. Man possesses an sesthetic nature, which when properly cultivated by the influence of natural expression in art, enables him to sit in criticism upon the productions of the artist. Where, through personal idiosyncrasies, tastes seem to differ, we are to be controlled in our decision by the opinions of the majority of cul- tivated persons. In conclusion, we urge teachers to remember that elocution is a beautiful art, and worthy of the highest culture. Voice and speech are divine gifts, and should be trained to their highest excellence. As Prof. Shoemaker so well remarks, " It is only the voice that has reached its best, and the eye that beams from the soul, and the hand of grace, and the attitude of manhood and womanhood, that can convey the immortality that has been breathed upon us." As God manifests His glorious attributes in the expression of Nature and the Bible, and above all in the Eternal Word, so may we show the image of divinity in our souls by a pure, natural, beautiful, and artistic expression. CHAPTER VII. TEACHING LEXICOLOGY. LEXICOLOGY treats of the meaning of words. The terra is derived from lexicon, a dictionary, and logos, a dis- course. It is usually emplo^'ed to embrace the origin and significance of words ; but it is here used as relating only to the meaning and proper use of words. The meaning of words is largely taught in all the branches of language. The subject, consequently, does not need a lengthy treatment b}' itself. No formal study of the subject is suggested for the ordinarj'^ common school; but much can be done in all the studies to lead pu2^ils to notice new words, learn their meaning, and fix them in their memory. In the higher classes, oral lessons might be given on the subject; and in advanced schools there should be a regular course of study to teach the meaning and use of words. A few suggestions will be made to guide the teacher in his work. B]f their Use — The meaning of words is taught by their use in conversation and speaking. The child first learns the meaning of words from the mother and father and other mem- bers of the household. The words he uses have never been explained to him; no definitions have been given him ; but he uses them correctly because he has heard them so used. Usage is his guide in using language. If he has been accustomed to hearing a correct and refined vocabulary, he will express himself with correctness and refinement. It is of inestimable advantage in linguistic culture to listen to the conversation ot intelligent and cultivated people. It is said that the Gracchi obtained the elegant use of language from their accomplished mother Cornelia ; and Aristotle imbibed from (214) TEACHING LEXICOLOGY. 215 his mother " that pure and sweet Atticism which ever^wliere pervades his writings," By lleuduxj. — The meaning of words is learned from read- ing. This is one of the most practical ways in which such a knoAvledge is acquired. In literature we see the correct use of the word, which we cannot alwa^'s tell from the definition. We also learn to appreciate those nice shades of meaning which cannot be stated in a definition. Pupils who read most have usually the largest vocabularj' and the best use of words. Young children will often be heard using the words in their conversation which they have met with in some book recently read; and, if properly taught, their compositions will show the same thing. Children sliould therefore be encouraged to read extensively and to read the best written Avorks. Teachers should call attention to the meaning and use of words in the reading lesson. Tiiey should require pupils to put the unusual and difficult words into sentences to see that they know how to use them. In tliis way the word is fixed in the memory-, and the child's vocabulary enlarged. The reading class presents one of the very best opportunities for teaching the meaning of words. By Illustrations. — With 3'oung pupils, the meaning of words may be taught hy means of objects or illustrations. Thus, the meaning of the word transparent may be illustrated with a piece of clear glass ; the meaning of the word translu- cent, by a piece of ground or painted glass ; the word opaque. by any object which does not permit the light to pass through it. The best way to teach the meaning of the word bone is to show the pupils a bone ; and the same may be said of calyx, corolla, stamen, pistil, etc. Most of the terms of the natural sciences may be taught in this way, and many of those in the abstract sciences, as the names of the figures in geometry. Object Lessons are especially vahiable in this respect. By Definitions. — The meaning of words may be taught l)v incans of popular definitions. The unknown word may be 216 METHODS OF TEACHIN^Q. made known hy comparing it with oa^ already understood, or b}' the use of several words which explain it. Care should be iaken, however, that the term used in the definition is simpler or better known than the word defined. This is not always the case with the definitions given in our text-books, especially tliose found in some of our school readers. To define shorten as abbreviate, or correct as rectify, or buying as purchasing, or belong as appertain, etc., gives the pupil another word for the same idea, but does not give him any new idea of the first word. The Dictionary. — The ineaning of words can be taught by a careful use of the dictionary. Pupils, as soon as they are old enough, should be required to make frequent use of the dictionary. This should become a habit with them. The great masters of language made the dictionary their constant compani(jn. Rufus Choate, so eminent for his scholarly use of the English language, was a constant and thorough student of the dictionary. In the stud}- of definitions, it should be remembered that we cannot always know how to use a word from its definition. Thus abandon means to forsake, to give up, etc.; but it would not be correct to say we "forsake a study" or even "abandon a bad habit," etc. Abbreviate means to shorten, but we would not appropriately speak of abbreviating a dress or a string or a stick of timber. We must notice the use of words in sen- tences in order to understand the nice distinctions between them; and definitions should alwa^'s be accompanied by sen- tences illustrating the proper use of the term defined. A pupil should acquire the habit of marking down every new word which he meets, or every word which he thinks is not a part of his practical vocabulary. He should keep a list of such words, frequently refer to them, and make use of them in speaking and writing. He will thus enlarge his stock of words, and learn to use them with readiness and precision of meaninof. TEACHING LEXICOLOGY. 217 From St/nonf/tna. — The meaning of words may be taught by the study of synonyms. By synonyms we mean words of the same general significance, 3-et with slight shades of ditter- ence in their meaning. They are words which, with great and essential resemblances of meaning, have, at the sauie time, small, subordinate, and partial ditferences. These ditferences may have originally inhered in them, or thev mav have ac- quirad them by general usage, or some earh' and latent mean- ing ma}' have been awakened by the special usage of some " wise and discreet master of the tongue." The English language is especiall}^ rich in sj'non3'ms. This arises from its being a composite language, words for the same thing being derived from ditlerent sources. Many of these in time became differentiated and now constitute our syno- nyms. Thus motherly and maternal, fatherly and paternal, happiness a,nd felicity, daily and diurnal, poicerful and poten- tial, etc., are pairs of words meaning very nearly the same, the first in each case coming from the Anglo-Saxon and the second from the Latin. The study of synonjms is especially' valuable in learning to use words correctly. It enables the pupil to see those nicer and more delicate shades of meaning by which words are dis- tinguished. It enables them to see in what cases words may be used interchangeably, and where they cannot be; thus we may say force of mind or strength of mind, but not strength of gravitation. It is onl}' by a careful comparison of words that a pupil can use such words as the following correctly: invent said discover ; only and alone ; enough and sufficient , avow, acknowledge, and confess; kill, murder, and assassinate Crabb's Dictionary of Synonyms is an excellent work for such a study, though the subject is quite full}' presented in Webster's and Worcester's large dictionaries. Loffical Definitions. — The meaning of words maybe taught by means of logical definitions. A logical definition is one which defines by means of the class and specific ditlerence, 10 218 METHODS OF TEACHING. called genus and differentia. Thus, a triangle is a polygon of three sides and three angles. Here polygon is the genus^ and Uiree-nidedness the sjtecijic difference. The practice of study- ing logical definitions tends to sharpen our conceptions of the distinction of words, and to cultivate the habit of careful discrimination in the use of language. Many terms will not admit of a logical definition. Such a definition is only possible when the genus and specific differ- once can both be stated. Terms expressing simple ideas can- not be logieall}' delined, because they cannot be i-esolved into their elements, and are thus without gen as and differentia. Thus, truth, space^ being, etc., will not admit of a logical defi- nition. Some terms, though belonging to a genus, cannot be defined on ace. unit of our being unable to state the differentia. Thus, in the statement red is a color, color is the genus, but who can give the differentia, the difference that separates red from the other colors? Latin and Greek. — The meaning of words may be learned by the Htudy of Latin and Greek. The practice of looking in the dictionary to find the English words which correspond to the words in other languages, makes the pupil familiar with the meanina: and use of the English words. The constant use and comi)arison of words, iiecessar3' in translation, give linguistic accuracy and a facility in their use. The process of translating cultivates that fine literary sense by which the delicate shades of meaning among words are perceived and appreciated. From Etijmoloffij — The meaning of words may be taught by the study of Etymology. A knowledge of the origin of a word sometimes aids us in understanding its meaning and use. Thus it adds to our idea of the word Education to know that it means to draw out, e and duco, and also subtraction, to know that it means to draw from under, sub and traho. The etymology often enriches and enlarges the meaning of a word, and puts an exjn-essiveuess in it by the image it brings before the mind as we use it. TEACHING LEXICOLOGY. 219 We cannot alwaj's use a word correctly, however, by know- ing its etymology. Indeed, the etymology of a word would usually lead us astray in its use. Thus the word subtraction, even, could not be used in its literal etymological sense of dr awing from under. The same may be said of right, wrong, conduct, normal, and a multiplicity of words which could be named. The principal use of et^^mology, aside from the inter- est and intrinsic value of the knowledge, is that it puts into the mind a concrete image which seems to add force oi em- phasis to the meaning of a term. There are two methods of teaching etymology; the Ana- lytic Method and the Synthetic Alethod. The Analytic Method begins with the word as a whole and separates it into its ety- mological parts, showing the meaning of the parts, and thus the meaning of their synthesis in the word. Thus, after the child is familiar with the word subtraction, it may be shown that it consists of the three jDarts, sub meaning under, tract from traho, I draw, and ion, the act of. A large number of words may be analyzed in this way as they occur, and a knowledge of the elements be reached through the words. The Synthetic Method begins by teaching a list of prefixes and suffixes and roots, and then unites them in forming words. Thus, after committing elements, the pupil may be shown that sub and tract, a modification of traho, and ion, give the word subtraction. In actual practice, there is a sort of analysis of each word into the elements which have been previously' learned ; but the spirit of the process is synthetic, since it passes from the elements to the word containing them. Of these two methods, the analytic is the better for begin- ners. It is the more interesting method ; the committing of a list of roots is rather dry work. It is also in accordance with the law of instruction, from the known to the unknoivn; while the synthetic method inverts this law. It also begins in the concrete, while the other is abstract. For advanced 220 METHODS OF TEACHIXQ. pupils the synthetic method may be preferred, as it is more formal and thorough in its procedure. Teachers should take pains to call the attention of pupils to the etj-mology of words. Even some incidental instruction of this kind will give the pupil a knowledge of the elements of a large number of words; and, what is better, cultivate a taste for et3'molog3-. They should not restrict their instruc- tion to Latin and Greek elements, but should call attention to the Saxon elements also. Such words as England^ '^'^f^-, husband, knave, heathen, etc., will be full of interest to chil- dren. Every teacher should have a coj)}' of "Trench on the Study of Words," and besides this it would be well for them to read Max Miiller, Whitney, Scheie de Vere, etc. CHAPTER VIII. TEACHING ENGLISH GRAMMAR. GRAMMAR is the science of sentences. English Grammar is the science of the English sentence. It treats of the relation and construction of words in sentences. In other words, grammar is the science of the sentential use of words. The term grammar seems to have been derived from gramma^ a letter, which came from grapho, I write. Grammar has sometimes been defined as " the science of language." This definition includes too much, for there are several other branches of lansruage coordinate with gram- mar, as Rhetoric, Etymology, Philology, etc. It is sometimes defined as "the science which teaches us to speak and write the English language correctly ;" but this also includes too much, as other branches aim at the same result. A sentence may be grammatically correct and still be incorrect in regard to other departments of language. Besides, it is not proper to define a science as " that which teaches" something. There is so close a relation between grammar and the two branches, Rhetoi-ic and Logic, that it is difficult to state clearly the distinction between them. Logic is the science of thought; but since this thought must be expressed, Logic deals also to some extent with the expression of thought. Rhetoric also treats of the manner in which thought and senti- ment are expressed. Popularly we may say, — Logic teaches clearness of expression ; Grammar, correctness of expression ; and Rhetoric, elfectiveness of expression. Fowler, in attempt ing to distinguish these three branches, says : " Logic deals with the meaning of language; Grammar with its construction ; and Rhetoric with its persuasiveness. Logic plans the tern- (221) 222 METHODS OF TEACHING. pie; Grammar builds it; Rhetoric adorns it." It is clear thai since thought determines expression, the science of logic is very intimately related to a full understanding of the subject of Grammar. The term Grammar was formerly used in a broader sense than at the present day. In its widest acceptation, and this was its primary use, it included all verbal expression of the products of the mind. Trench says, "Grammar is the logic of speech, even as Logic is the grammar of reason." It has also been used to signify a treatise on the elements or princi- ples of an}' science; as, a "grammar of geography," a "gram- mar of arithmetic." The terra has, however, become differ- entiated so as to be now restricted to the sentential use of words. I. General Nature of the Subject. I. Nature op Grammar. — To aid the student in understand- ing the methods o/ teaching grammar, we shall present a brief statement of the nature of the science. A conception of the subject of grammar may be presented in two ways ; first, b^'' considering the office of the individual words in a sentence ; and second, b}^ resolving the sentence into the thought elements which enter into its structure. The former is called the Etymological view of grammar ; the latter is called the Logical view of the subject. Etynioloffical Elements. — Language is made up of indi- vidual words. These words are all embraced under a few general classes, some eight or ten, called Parts of Speech. Each one of these parts of speech performs a certain office in a sentence, and some perform two or three offices. Parts of Speech. — The first and simplest class of words are those which are the names of objects, called Nouns. There are also words expressing some action or state of the objects named by these nouns, which are called Verbs. Then there is a cJass of words, usually expressing qualities, which are added TEACHING ENGLISH GRAMMAR 223 to the nouns to distinguish the objects referred to b^' the noun ; these are called Adjectives. Tlien we have a class of words used to distinguish the actions expressed by the verbs, called Adverbs. The words used to distinguish the qualities exin-essed by adjectives and by adverbs are also called adverbs. Then there is a class of words used for nouns, called Pro- nouns. There is also a class of words used to connect other words and show the relation between them, called Preposi- tions. We have also words which connect words and sen- tences without showing any relation between the words connected, which are called Conjunctions. There are words also which express feelings or emotions, which on account of their being thrown into the sentences formed by other words, are called Interjections. Properties of Parts cf Speech. — These parts of speech have certain relations to one another and to the things which they express, that give rise to certain changes in their form or meaning. These changes in form are called Inflections, from flecto, I bend, since the form of the word is changed, as in bending an object we change its form. Words which admit of such changes are said to be declinable^ from c?e, down, and clino, I lean or incline. In many cases in the English lan- guage there is no change of form to indicate the relation, though the relation really exists, and is thought if it cannot be seen. These are all embraced under the head of the Prop- erties of the parts of speech. The pi'operties of the Noun are Number, Person, Gender^ and Case. The properties of the Verb are Mode, Tense, and Voice, and also Person and Number derived from its subject. The change in the adjective and adverb is called Compar-ison. In some languages the adjective has the properties of number and case, which it seems to have derived from the noun. CVfls-ses of Parts of Speech. — These Parts of Sj^eech admit of various divisions into classes, which give us what are called the Classes of the Parts of Speech. Thus, Nouns are divided 224 METHODS OF TEACHING. into Proper and Common, etc.; Verbs into Regular and Irreg- ular^ Transitive and Intransitive, Qlc; Pronouns into Per- sonal, Relative, Interrogative, eta. \ Conjunctions into Coor- dinate and Subordinate ; etc. Rules of Construction. — From the consideration of tlie rela- tion of these words to one another, and a careful examination of the usage of cultivated men and women, we derive certain laws of construction, which constitute the Rules of Grammar. Some Offices. — Then we have certain offices ascribed to the words as limit, modify, govern, etc. One word is said to limit another when it limits its application to a part of the class of objects which it represents ; thus, in the expression blue birds, the word blue limits the word birds to only a part of the general class of birds. The term modify means very nearly the same as limit, one word modifying the application of another word ; as in red roses, the i-ed modifies the applica- tion of the word roses. By government in grammar is meant the power that one word is supposed to exercise over another word to cause it to assume some particular form or meaning. Logical Elements. — In this statement we have a brief out- line of the nature of grammar, derived from the consideration of the individual words in a sentence. There is another method of conceiving the subject, however, which consists in deter- mining the elements of language by regarding the sentence as a unit, and analyzing it into the necessary parts of which it is composed. We state briefly the results of such an analysis. Principal Elements. — A sentence is an assertion of some- thing about something. Every sentence thus contains two necessary elements ; that about which an assertion is made, and that which is asserted. These two elements are distin- guished as the Subject and the Predicate. The Subject may consist of a single word, or of a collection of Avords not form- ing a proposition, called a phrase, or of a collection of words containing a proposition, called a clause. Similarly, the Pred- icate may consist of a word, a, phrase, or a clause. TEACHING ENGLISH GRAMMAR. 225 Subordinate Elements. — Continuing the analysis, we find that some elements are used to limit, modify, or describe other elements, and these we call modifying or limiting ele- ments. When they limit the meaning or application of words used as the names of objects, they are called adjective ele- ments; when they limit the meaning or application of words used to express actions or qualities, they are called adverbial elements. These elements are often called adjuncts, because they are joined to the elements which they limit. To distin- guish them from the subject and predicate, they are called subordinate elements, the subject and predicate being called principal elements. These subordinate elements are, with respect to their form, of three classes ; words, phrases, and clauses; and with respect to their use thej^ are also of three classes; adjective, adverbial, and objective. Connective Elements. — In addition to the principal and sub- ordinate elements, there are also words used to connect the other elements, which are called connective elements. We also often find in language words that have no logical connec- tion with the other words; such words are called independent elements. This method of looking at a sentence and reaching; its elements may be called the logical method, in distinction from the other methcd. which niay be called the etymological method. II. Origin of Grammatical Elements. — Having pointed out the grammatical elements of language, the questions naturally arise, — How did these elements originate? Why have we just so many parts of speech ; and wh}' are they such as they are? We shall endeavor to give a brief reply to these questions. There are two theories upon this subject; one drawn from the consideration of the operation of the several faculties of the mind, and the other that presented by the writers on logic. These two views, for want of better names, we may distinguish as a New and the Old theory. A New Theory. — Language is the product of the human 10* 226 METHODS OF TEACHING. mind The thought went out into expression, and thus gave form to the language. In order, therefore, to understand the growth and nature of the grammar of the language, we must look at it through the laws of mental activity. Parts of Speech. — The faculty of the mind which first awakens into activity is Perception. Perception cognizes individual things, and forms particular ideas. These ideas we express in particular words; hence our first words are names which we call nouns, from nomen, a name. These names are of individuals and are thus proper nouns, or have the force of proper nouns. The mind also sees these objects act- ing or doing something, which it expresses in the form of action or doing xoords. These words are called verbs, from verbum, a word, because they are regarded as the most import- ant words in a sentence. These verbs, like the nouns, at first express particular actions. The mind, at first, cognizes objects as wholes, without dis- tinctly noticing their attributes ; but it soon begins to analyze them and to distinguish their qualities ; the naming of these qualities in their relation to the objects, gives us words to distinguish objects, which on account of their being added to nouns, we may call adnouns; or, since they are thrown to nouns, they have been called adjectives, from ad, to, a.ndjacio, I throw. The mind also compares actions and notices their differences ; the naming of these differences in relation to the action, gives us a class of words to distinguish actions; which, on account of their being added to verbs, are appropriately called adverbs. The mind in comparing objects, notices these similarities, and brings the similar objects together under a common name; it thus forms general ideas which give rise to general terms or common nouns. In a similar manner, the verb, which was at first the name of some particular action, becomes general in its application to a class of similar actions. The adjective and the adverb also become more general as our experience enlarges. TEACIIIXG ENGLISH GRAMMAR. 227 llavins: obtained general notions, the mind begins to com- l)are these general notions, and perceiving a relation between them, forms judgments, which when expressed, give ns the proposition. These judgments or propositions need a connect- ing or athrming word, which gives rise to the copula or neuter verb as " nr.an is an animal." The affirmation of an attribute of a general notion (regarding the intension of the concept) also requires the use of the copula, as " man is mortal." As our progress in thought and language continues, it is found convenient to avoid the too frequent repetition of nouns, which we do by the introduction of a class of words to be usedybr nouns, which we call for-nouns, or pronouns. If it were riecessar}' to have a class of words to avoid the too fre- quent repetition of the verb, we should have a class of for- verbs or pro-verbs also, which we seem to approximate in the peculiar use of the word do; as "John studies, and so do I." In order to unite and show the relation of some of the words we use in the construction of sentences, it was neces- sary to introduce words expressing relations, which we may call relation-words ; or, since the}'^ are placed before the word to be related to some other word, they are called prepositions, from prae, before, and pono, I place. Words used merely to conjoin words and sentences were also necessary, and were called conjoining words, or conjunctions. Words expressing emotions were also needed, and since these words had no relation to the rest of the sentence, but were thrown in abruptly between other words, they were called interjections, from inter, between, and Jacio, I throw. The Properties. — We may also account for the origin of the inflections or properties of the parts of speech in a similar manner. It was necessary to distinguish between the use of a noun as meaning one or more than one object, and this was con- veniently done by a change of termination in the nouns to in- dicate this meaning, which gave rise to the property of number For all practical purposes two forms were sufficient ; hence 228 METHODS OF TEACHING. we have only two numbers, singular and pIitraL Some nations, however, seemed to find it convenient to distinguish between one, two, and more than two things ; and thus arose a third form, called the dual number. This dual form is sup- posed to have been caused by the duality of the parts of the human body, as the eyes, the hands, etc. Since there were two objects of the same class of animals distinguished by sex, it is natural that words should be changed in their form to distinguish the sex of the object named ; and thus arose the property of gender. Since a noun could represent the three persons, the speaker, the person spoken to, and the person spoken of, there naturally arose a change of form in the noun to indicate the person ; which gave rise to the property, called the person of nouns. The dirt'er- ent relations that an olyect may sustain to an action or to another object, caused a change of termination to indicate the relation meant ; and this gave the property of case, six in Latin and eight in Sanskrit. In our language these relations- are principally exi)ressed by prepositions, leaving us only three cases for pronouns, and, some say, only two cases for nouns, the nominative and the possessive. All these proper- ties of nouns would, of course, belong to pronouns as their representatives. The properties of the verb originated in a similar manner The fact that a verb could be used in commanding, or inquir- ing, or simply declaring an action, gave rise to the property of the manner or mode of the verb. The idea of time and the fact that the action expressed by a verb could take place in ditferent times, gave rise to a change of the verb to indicate these times of an action, which produced the property of tense. It was natural for the form of the verb to vary as the nutibei and person of its subject varied and this gave rise to the number and person of the verb. The number and person of a verb are not intrinsic, but deriv- ative properties of the verb ; ^nd by some grammarians arc TEACUING ENGLISH GRAMMAR. 229 not regarded as properties at all. A certain writer saj's that to attribute person and number to a verb is "as anomalous as to assign gender and number to adjectives. Most languages fall into this error, which is, however, susceptible of a ver^' easy historical solution. It arose, doubtless, from the original custom of annexing the pronoun to the termination of the verb, and continuing the use of the inflection after its import had been forgotten, and when the pronoun had been formed into an independent part of speech." It seems to have been natural, primaril^^, to express the relation of words by an affix or prefix to the radical portion of the word ; these changes seem subsequently to have been replaced by particles. The earlier stage of a language is usu- ally richer in teriBinations ; which drop off as the faculty of abstraction becomes habitual. In a manner similar to that now explained, we can account for every grammatical distinc- tion by a development from the natural psychological opera- tions which give form to language. The different Classes of parts of speech arise from the different offices performed by words of the same general class, and the Rules of Construc- tion grew out of the laws impressed upon language by thought, modified by the circumstances of fashion, etc., which inti'o- duced changes into the language of the people. The Old Theory. — We have thus indicated what we con- ceive to be a correct idea of the development of grammatical elements from the natural operation of the human mind. There is, however, another view of the subject, drawn from logic rather than from psychology. We will briefly indicate this view. It is held that the first class of words are subslantives, so called because they are conceived as standing under (sub- stans) certain qualities. These qualities may also be consid- ered as substantives, as whiteness, greenness ; but when con- sidered in relation to the substances of which they are proper- ties, the}' constitute a second class of words, adjectives or 230 METHODS OF TEACHING. noun-adjectives. A. vonceptiou, or general notion, when formed, is capable of being resolved back into its constituent parts or qualities; and the attribution of a quality' to a sub- stance leads to a, judgment ; as "Snow is white," the sign of the attribution being called the copula. When the quality is combined with the copula, a third class of words is pro- duced, wiiich we call verbs. Thus, instead of saying, "the sun is bright," we may say " the sun shines." A verb is thus regarded as a compound part of speech, consisting of an adjective and a copula or affirmation. These three parts of speech — the substantive, the adjective, and the verb, are called the primary or essential parts of speech. The adverb, it is said, derives its existence from the difficulty of defining by one word the precise qualit}' of a particular object. Words are needed to indicate the degree of the quality expressed. The primarj- use of the adverb, it is thus seen, is to modif}'^ the quality or attribute expressed b}'^ the adjective and the verb. Prepositions are said to express relations be- tween substances, objective relations; while conjunctions may be regarded as expressing subjective relations, or those exist- ing between judgments, whether of mere succession, of infer- ence, or the lilce. The other grammatical elements would be derived ver^- nearly in the manner previousl3'^ explained, III. Origin of Grammar. — Grammar originated among the Greeks. It seems to have had its origin about the second century B. C, among the scholars of Alexandria. Many o^ them were engaged in preparing correct texts of the Greek classics, especially of Homer. The manuscripts differed, and the correct form was determined by a comparison with the language of Homer. They were thus forced to pa}'' attention io gi'ammatical structure, and to observe the laws of con- struction. The first real Greek grammar was that of Diony- sius Thrax, a pupil of Aristarchus. He went to Rome as a teacher about the time of Pompej', and wrote a practical grammar, it is supposed, for the use of his pupils. This work TEACHING EXGLISII GKAMMAR. 231' was the foundation of grammar. Later writers have improved and completed it, but have added nothing really new and original in principle. The earliest scientific investigations of language among the Greeks were not strictly grammatical, but discussed the rela- tion of thought to expression. The distinction of subject and predicate, and even the technical terms of case, number, and gender, were first used to express the nature of thought, and not the forms of language. The early Greeks had a very slight knowledge of grammar proper. Plato knew the iiou7i and verb, as two component parts of speech. Aristotle added con- junctions and articles, and observed the distinctions of num- ber and case. The word article with him, however, meant a socket in which the members of the sentence moved, and in- cluded many more words than at present. Before Zenodotus, 250 B. C, all pronouns were simply classed as sockets or arti- cles of speech. He was the first to introduce a distinction between personal pronouns and mere articles or articulations of speech. Aristotle had no technical terms, as singular or plural, and does not allude to the dual. Zenodotus seems to have been the first to observe the use of the dual in the Homeric poems, and changed many plurals into duals. The first attempt at an English grammar was FauVs Acci- dence, an English introduction to Lill3''s Latin grammar, written by Dr. John Colet in 1510. Lill^^'s grammar received the sanction of ro3'al authority and was the exclusive stand- ard in England for more than 300 years. The first book treating exclusively of English grammar was written by Wil- liam Bullokar in 1586. During the next century, several works on grammar were written, among which are mentioned one by Ben Jonson (1634), one b}^ Dr. John Wallis (1653) in Latin, and one by William Walker (1684), the preceptor of Sir Isaac; Newton, also in Latin. In 1158, Bishop Lowth pul)lislied his celebrated grammar, an excellent work from which Lindley Murray' drcM" most of his materials. Lindley Murray published 2o2 METHODS OF TEACHING. his first grammar in 1195, and his Abridgement in 1797, a work which has been extensively used in this countr}' and in England. The annual sale of the book in England has been estimated at 50,000 copies. This popular work was largely derived from Lowth and Priestley, and owed its popularity to its practical adaptation to the work of the school-room. The number of grammars published in this country is legion; the ablest and most celebrated is that of Goold Brown. The first to develop and give prominence to "grammatical analy- sis," was Prof. S. S. Greene, of Brown University. IV. The TEACHiNa of Grammar. — Having spoken of the nature of grammar, the origin of the grammatical elements, and the historical development of the subject, we shall now call attention to the manner in which it has been taught, and the different methods of teaching it. Gratnniar Poorly Taught. — Grammar has been more poorly taught than an}- other branch in the public schools. It has been made too abstract and theoretical. It has been taught as a matter of memory, and not of judgment and understanding. It has been a committing and repeating of definitions, and not a study of the relation of words in sen- tences. It has been a study of text-books on grammar instead of a study of the subject of grammar. It has been a memor- izing of abstract definitions and rules, instead of a practical application of them to the improvement of a pupil's language. It has been a worrj' and a waste of time and patience ; and a labor barren of adequate results. We believe we are correct in saying that more than three-fourths of the time spent in the stud}'^ of grammar in the public schools, has been worse than wasted. The result of such teaching is that the pupils of our com- mon schools go out with a much better knowledge of arithme- tic, geography, etc., than of grammar. Besides this, the methods of teaching have given pupils wrong ideas of the sub- ject and incorrect methods of studying it. Taught b}' reciuir- TEACHING ENGLISH GRAMMAR. 233 ing pupils to commit and recite definitions, tliey have come to look at the grammar of lan2:uao;e throiis-h the definitions rather than at the definitions through language. Pupils thus taught not ouly obtain confused notions of grammar, but often acquire a dislike and even a disgust for the subject. These errors in teaching grammar arise from two sources ; the defects of our text-books and the incompetency of teachers. The books have been defective on account of their beginning with definitions instead of exercises to lead to definitions. They have presented the matter too ab- stractly. They have not aimed to lead the pui)il to apply his knowledge of the subject. They have not been pro- perly graded ; and have introduced difficulties before the pupil was prepared for them. They have been constructed on the deductive method of teaching, instead of the inductive method, as all primar}' grammars should be. A change, however, is taking place in this respect; some of the more recent text-books on primary grammar being a great improve- ment on the old ones. The incompetency of teachers, stated as the second cause of this poor teaching, has been not so much in their imperfect knowledge of grammar as in their defective methods of teach- ing it. Teachers of the public schools usually know enough grammar for their work, but they do not know how to teach it. Having been incorrectly instructed themselves, and hav- ing received no instruction in the true method of teaching it, the^^ reproduce the same faulty methods in their own work, and thus the evil is perpetuated. The difficulty which pupils experience in learning grammar is entirely unnecessary. When properly taught, grammar is one of the easier studies of the common school course. In- trinsically, the elements of grammar are less difficult than the elements of arithmetic : a knowledge of grammar, such as is contained in an ordinary common school text-book, is much more readily acquired than the same amount of arithmetic 234: METHODS OF TEACHING. Grammar can also be made one of the most interesting studies of the public school, by teaching it according to a proper metliod. I have never seen children more interested in any classes than in the primary grammar class when correctly taught. Methods of Teachinfj. — There are two distinct methods of teaching grammar; the Synthetic or ^Etymological Method, and the Analytic or Logical Method. The Syutlietic or ¥A\- mological method begins with the word^^, regarded as hdiUoJ language, and proceeds to sentences. It regards the words as parts of speech, denoting objects, actions, qualifies, etc., and not as logical elements of thought. It is called Syutlietic be- cause it proceeds from words to their combination in sen- tences. It is called Etymological because it deals with the parts of speech as words. The Analytical or Logical method of teaching grammar begins with the sentence as i\xQunit of language, and analyzes it into its thought elements. It considers the sentence as con- sisting of two principal elements, the subject and predicate, passes from these to subordinate and connective elements, and at last reaches the words as parts of speech. It first regards words not as parts of speech, but as expressing the logical ele- ments of which a sentence is composed. It is called Analyt- ical, because it passes from the sentence as a whole to the parts composing it. It is called Logical, because it deals with the logical elements out of which sentences are composed. The ditierence between these two methods is radical and important. Thus, by the former raetliod, a yioun is taught as a, name; by the latter method it is regarded as expressing that of which something is said. By the former method a verb expresses an action or doing ; by the latter, it expresses ivhat is affirmed or asserted of the subject. An adjective, liy the former method, is the name of a quality of an object; by the latter method it is regarded as a word which limits the extent of a general conception or the application of a general TEACHING ENGLISH GRAMMAR. 235 term. Thus, by the logical method, good, in the expression^ good boys, is not regarded as expressing the quality or kind of boys, but as limiting the concept boys to a portion of its extent, or the term boys to part of the class. So the adverb limits the general action to some particular action : thus, in the sentence. The bird fies swiftly, the flying, which, without the adverb swiftly, would include all kinds of flying, is here limited to a particular kind of flying ; namely, swift flying. By the Etymological method, the preposition is taught as expressing the relation of objects ; by the Logical method it is taught as the connecting part of an adjunct or subordinate element. It should be observed that the two methods are not dis- tinguished merely by one beginning and the other not begin- ning with a sentence. We may begin with a sentence and teach by the etymological method, by regarding the words of the sentence as parts of speech. In teaching by the synthetic method we should use the sentence as well as by the analytic method. The essential diflerence is not in the use or non-use of the sentence, but in the manner of using it. In one case we begin with the words as parts of speech ; in the other case we bcin with the loijical elements of a sentence, and come down to the words as parts of speech through these logical elements. The Correct MetJiod. — In teaching grammar, neither one of these methods should be followed exclusively, but they should be judiciously combined. Both are needed to give a complete knowledge of grammar, and each will aid the other in giving clearer ideas of the subject than can be obtained by either one alone. From a generalization of the use of words as parts of speech the pupil is naturally and easily led to grammatical analysis ; and from some of the distinctions in grammatical analysis much clearer notions of the correct use and relation of words as parts of speech can be presented. Both methods are, therefore, essen- tial to a complete system of grammatical instruction, and they should go hand in hand in unfolding the subject in the mind of the learner. 236 METHODS OF TEACHING. The etymological method will serve as a valuable introduction to the logical method. The use of words as the elements of sen- tences will prepare for the use of collections of words as express- ing these elements. Thus from the use of single words as parts of speech, the pupil is easily led to see that phrases and clauses may perform these same offices. From a word used as a notui or an adjective or an adverb, etc., it is readily seen how a phrase or a clause may be used as a noun, adjective, or adverb, etc. The logical method will also be of great advantage in under- standing the use of words as parts of speech. Thus the concep- tion of a phrase as a modifier of a noun or a verb indicates the antecedent term of the relation of a preposition, which is not ahvays readily seen without this conception. Much clearer ideas of indirect objects, adverbial objects, relative pronouns, etc., will be obtained by the logical analysis of sentences. Indeed, analysis will aid in giving clearer ideas of nearly every part of the subject than can be obtained by the etymological method alone. The correct order of these two methods, it would seem, is to begin with the etymological and pass gradually to the logical method. Several reasons can be given for this order. First, the etymological method is simpler in thought than the logical jaethod, and is much more easily understood by young pupils. Second, it coincides with the natural method by which they learn language ; first words, and then sentences. A little child begins language with words as the names of objects rather than with sentences or propositions. Its adjectives are at first the names of qualities rather than limiting elements of general conceptions. This order is also to be preferred because it ibllows the law, from the particular to the general. "Grammatical Analysis" is, to a large extent, a generalization of the principles of etymological grammar. Thus, at first, we see that a single word is a part of speech, as an adjective; and later we learn that a phrase or a clause may be used as an adjective, and is thus an adjective element of a sentence. The same is true in respect to the noun, the adverb, etc. ; in each case there is a generalization Irom the use of words, TEACHING ENGLISH GRAMMAR. 237 to tte similar use of phrases and clauses. The order also cor- responds with the historical development of the subject, for "gram- matical analysis" is of comparatively recent origin and was a development of etymological grammar. The proper combination of these methods, it seems to me, is as follows : First, give the pupils an elementary knowledge of words as parts of speech. Second, give the pupils a general notion of the logical analysis of sentences. Third, then present a detailed treatment of the parts of speech including their classes and properties. In connection with this third division have constant exercises in parsing, analysis, and the correction of false syntax. Such a combination of the two methods will produce the happiest results in teaching English grammar. Principles of Teaching Grammar, — In teaching gram- mar by either of these two methods, the teacher should be guided in his work by the following principles of instruction: 1. Teach first by means of oral exercises. Do not l)egin bj' having pupils stud}' definitions from a text-book. No gram- mar-book is needed for several months, with a class l)Oginning grammar. In an ordinary common school, I should use no text-book on the subject for at least six months or a year. A school reader may be used for examples of parts of speech, for parsing, etc. A text-book in grammar is a positive dis- advantage to a beginner. It seems to stand as a partition wall between the pupil's mind and the subject. It causes him to " see through a glass" very darkly that which is simple and clear without the book. 2. Teach grammar from language and not from definitions. The old way was to begin with definitions ; the correct method is to begin with language. In this way the pupil will see and understand the grammatical use of words, while by the old method he recited their use without understanding it. By the former method, he depends on what the book says ; by the latter method, he learns to depend on his own judgment in determining the nature and relation of words. In one case, he looks at grammar through the definitions ; in the other case 238 METHODS OF TEACHING. he looks at grammar through the nature of the sulyect itself In the former case, grammar is too much a matter of memory ; in the latter case, it is a matter of the judgment and the un- derstanding. Let grammar, therefore, be taught from lan- guage, as it was originally developed by those who first inves- tigated it. An additional reason for teaching primary' grammar from language without a text-book, is that the proper study of the subject is especiall}' an act of the judgment. There are very few things to commit to memory in elementar}* grammar. There are a few technical terms which are readily remembered when the pupil has the ideas which they express. What we especially need is to examine language and notice the rela- tions of words ; and not to commit and recite definitions. There is no other study in the public school that so little needs a text-book as the first lessons in grammar ; and assuredly there is no study in which the text-book is so much of a hind- rance to the beofinner as this. Some of the most successful teachers of advanced classes in grammar use an edition of some of the favorite poems of our eminent authors as the text- book for the lesson, while the real text-book is used only as a work of reference. 3. Make the sentence the basis of grammatical instruction. Though we begin with words, we should pass as soon as pos- sible to sentences, and study the words with respect to their relations in sentences. Grammar treats of the sentential use of words, and it is only by viewing their relations in sentences that we can understand their grammatical meaning and use. In teaching grammar, therefore, the sentence is to be regarded as the unit of reference. But though we make use of the sen- tence in instruction, we are to consider, first, not the logical use of the words in it, but their etj'mological use. The words in the sentence are to be regarded as etymological elements expressing objects, actions, etc., and not the logical elements of wliich sentences are composed. TEACHING ENGLISH GRAMMAR. 239 4. Make the subject practical. We should require the pupils to use good grammar ; to apply what they learn in moulding and correcting their own speech. We should excite an interest among pupils in the use of correct lan- iTuaafe, and in correcting their mistakes. Have them bring in false sj-ntax heard in the school-room and on the play- ground. Let the teacher be careful to use correct language himself, as an example to his pupils. 5. The course in grammar should be preceded by a course of instruction in Language Lessons. The basis of instruction in grammar is language, and a pupil should have some lessons in language before he begins the subject of technical grammar. Such a course of lessons is indicated under the methods of teaching Composition. Time to Uegin. — The time to begin grammar depends upon the manner in which it is taught. If presented Induc- tivel}', with oral exercises, the pupil may begin the study at nine or ten years of age. The average age for the pupils of our common schools is probably about ten or twelve. If, however, grammar is taught by the old method from the text- book, it should not be commenced before the age of fifteen or sixteen. Division of Subject. — For the purpose of instruction, grammar may be divided into Primary Grammar and Ad- vanced Grammar. For Primary' Grammar, the sj'nthetic or etymological method of teaching is employed as the basis of instruction ; in Advanced Grammar the analytic or logical method should be made more prominent. We shall indicate a course of instruction in both. II. Method of Teaching Primary Grammar^ By Primary Grammar is meant such a course of instruction in grammar as shall present the fundamental facts and princi- ples of the science. It is designed to lay the foundation of grammatical knowledge, but does not extend to the higher 240 METHODS OF TEACHING. philosophical principles of the science, nor discuss the anoma- lies of construction, etc. Principles of Instruction — There are several principles of instruction that should be made especially prominent in a primary course in grammar. They are principles that have been previously announced ; but so important are they, and so often are they violated in grammatical instruction, that we repeat them here. 1. Teach first the idea and then the expression of it. This principle is of especial importance in teaching grammar. The old way was to teach the expression first, and often the pupil did not get the idea at all. Both teachers and pupils have used the expressions, "govern," "relates to," "qualifies," "modifies," etc., for years, without ever thinking what they meant. The majority of teachers of whom we have inquired, What do you mean by " prepositions govern the objective case ? " could give no intelligent explanation of its meaning. Do not, therefore, begin witli the definition as the statement of the idea, but present the idea first, and then lead the pupils to the expression of it. 2. Teach pupils to discover the idea you wish to express. The old way was to tell the pupil everything ; the better way is to allow him to discover all he can for himself. This is the inductive method of instruction, and grammar is one of the very best studies in which to apply the inductive method. It will make the pupil a thinker in grammar, independent of the teacher or text-book. 3. Let the primary aim be grammatical ideas rather than grammatical expressions. Care not so much for the defini- tions as for the idea to be defined. Do not require definitions until the idea is clearly developed in the mind of the learner, and let the definition flow from the natural expression of the idea. 4. Do not burden the memory with grammatical forms. A general fault in teaching grammar is that the subject is made TEACHING ENGLISH GRAMMAR. 241 too formal. Too much attention is paid to the manner of expressing grammatical ideas. In Primary Grammar, the forms of parsing and anal\'sis shoukl be very simple. We should depend more upon asking pupils questions in language than upon their giving any set forms of expression. The Order of Instruction in this subject, in accordance with these principles, is as follows : 1. The Idea ; 2. The Name ; 3. The Definition ; 4. Exercises. The Course of Instruction. — The Course of Instruction in Primary Grammar includes the following things : 1. The Parts of Speech; 2. The Properties and Inflections; 3. The Classes of the Parts of Speech ; 4. The Rules of Construction ; 5. The Elements of Parsing; 6. The Elements of Analysis ; 7. Correcting False Syntax. These are to be presented somewhat in the order named, but not entirely so. We should begin with the Parts of Speech, but the Classes and the Inflections may be taught somewhat together ; and the Elements of Parsing, Analysis, and the Correcting of False S^^ntax, should be introduced gradually as the pupils are prepared for them. Tlie above is a logical division, showing what is to be taught rather than the order in which the several things are to be presented. In presenting the subject, we shall first describe the method of teaching, and then follow the description with an inductive lesson, indicating how the pupil is led to the idea and its ex pression, by appropriate questions. I. Parts of Speech. — We begin the instruction in grammar, by teaching the Parts of Speech. In order to prepare for this, we should give to the pupil a clear idea of an object and a word. This can be done by showing them an object, asking its name, and calling attention to that which we hear spoken, or see written. The lesson suggested is as follows: Model Lesson.— TeacJier, holding up a book, a knife, etc., says, What is this? Pupil. K book. 7'. This? P. k knife. T. What do we call all these things we can see, touch, etc.? P. Objects. T. What do we call 11 242 METHODS OF TEACHING. those things we hear when we speak ? P. Words. T. Are there any words besides those we liear f P. Yes, words we can see. P. What shah we call the words we can see ? P. Seen icords. T. Since we write these words, what may we call them ? P. Written words. T. What may we call the words we hear? P. Heard words. T. Since we speak them, what kind of words may we call tliem? P. Spoken words. T. How many kinds of words then have we? P. Two kinds; spoken words and uTritten words. The Noun. — To teach a Noun, present several objects to the pupil, have him name them, write these names on the board, and lead him to call them object-words; then to define an object-word, and then give him the term not(7i as meaning the same as object-ivord, and have him define a noun. Then give exercises, requiring him to select nouns from a book, and to give examples of nouns. Then teach that the names of per- sons, places, etc., begin with a cajiital letter. Jfiidd Lesson. — Tearher. What is this I hold in my book. hand? Pupil. A book. T. What is this? P. A knife. ^"^f''- T. This? P. A pencil T. I will write these names on f„V'S\ n the boird; what are these in my hands? P. Objects. T. What are these on the l)()ard? /-'. Words. T. Wliat are these words the names of? P. The names of objects. T. Since they are the names of objects, what kind of words may we call them? P. Object- words. T. What then h im object- word? P. An object-word is tJie name of an object. Tliey have thus been led to the idea of an object-word, to name it themselves, and to make their own definition of it. The next step is to require them to name and write object- words, and to have them point out object-words in tlie reader. After they are familiar with object-words, then introduce the name noun. The exercise is as follows : Model Lesson. — Teacher. What do we call the names of objects? Pupil. Object-words. T. I will give you a shorter word that means the same as object-word; it is noun; what is it ? P. Noun. T. What then is a noun ? P. A noun is an nbjecf-irord. T. And what is an object-word ? P. An object- word is the name of an object. T. Wliat then is a noun? P. A aoun is the name of an object. Ttie Veih. — To teach the Verb., we call the pupil's attention runs. plays. sings, eats. drinks. strikes. TEACHING ENGLISH GRAMMAR, 243 to the actions of some object, write a list of words expressing action upon the board, lead the pupils to call them action- icords, and define an action-word^ and then, after they are familiar with the idea and name, introduce the term verb as moaning the same as action-word, and lead them to define it, and give them a drill on the verb and the noun. Mod-el Lesson. — Te/iof the book to the table now? P. On. above. T. What little word shows the relation of the b(X)k to the beside. table now? P. Under; etc. T. Here we have a list of words which do what? P. Show the relation of objects. T. What shall we call these words that show the relation of objects? P. Belation-words. T. What then is a relation- word? etc. T. The word preposition is used for relation-tpord ; what then is a preponition f etc. Ttie Interjection, — The Interjection should be taught as a feeling- or emotion-ivord. Ask what words we sometimes use ■when we feel very sad, or very glad, or wlien feeling surprised, etc., and get a list of words like oh, ah, aln.s^ hurrah, pi^haw, etc. Then, since these express feelings or emotions, they may be cvi\\Q(\ feelinrj- or emotion-jvords. Tlien lead to the defini- tion, etc. At last lead to the idea that they are interjected, or thrown in between other words, and may be called interjec- tions, and lead to the definition ; and then drill them on exer- cises on all the parts of speech, similar to the manner pre- viously suggested. II. Properties of Parts of Speech. — By the Properties of the parts of speech are meant those things which belong to or are peculiar to the different parts of speech. They include the Number, Person, Gender, and Case of nouns and pro- nouns ; the Number, Person, Mood, Tense and Voice of verbs ; and the C/)mparison of adjectives and adverbs. These proper- ties are also called Inflections, because there is a bending or change of the word from its original form or meaning. Idea of Property — The first thing to teach under Proper- ties, is to lead a pupil to a clear idea of what is meant by a property of a part of speech. Nine-tenths of the pupils in grammar who use the term property, never stop to think what it means. To present the idea of propertxj, take two objects, as a pencil and card., call attention to their qualities, lead to TEACHING ENGLISH GRAMMAR. 249 the idea that these qualities belong to the objects, lead them to tell you that that which belongs to any person is his property, and hence those things which belong to words and distinguish them are called properties of those words. Mo'lel Leason.— Teacher. What are these objects? Pupil. K pencil and card. T. Name some of their qualities. T. To which object does the quality icliite belong; to which object does the quality black belong? T. The things which belong to your father— his farm, horse, etc. — are called his what? P. His property. T. What then may we call tliose things that belong to objects? P. The properties of objects. T. What may we call the things which belong to words? P. The properties of words. T. What then is a property of a part of speech? etc. We can teach the meaning of an inflection by taking a word like abbot, which expresses a man, and show that it changes to abbess to express a woman ; that box changes to boxes to express the plural ; etc. Then lead them to see that such changes or bendings of words to express a change of thought, are called bendings or inflections. Let the student-teacher show the method by a lesson. Properties of Nouns and Pronouns. — We shall first con- sider the properties of Nouns and Pronouns, including Num- ber, Person, Gender, and Case. Number To teach Number, we lead the pupil to see that words have one form for one thing and another form for more than one thing ; and then since one, two, three, etc., are num- bers, this propert}'^ of nouns may appropriately be called the Number of nouns ; and the pupils may be led to define it. We then lead them to see that there are only two numbers, since there are onh' two forms, one form for one thing and another form for more than one thing. We may then lead them to call one single number and the other many number, from which we pass to singular and plural. We may then lead to the Rule for number, by leading them to see that we sometimes add an s to the singular and some- times es, and sometimes change the form of the word, in form- ing the plural. Model Lesson. — Teacher. What have I in my hand? Pw;n7. A book. T. How many books ? P. One book, T. How many have I now ? P. 11* 250 METHODS OF TEACHING. Two books. T. How many now ? P. Three books. The teacher will then write on the board, "I have one book," " I have two books," " I have three boolcs," etc. T. Which is the noun in these sentences? T. How many forms lias it ? P. Two forms, book and books. T. What is its form for one thing? T. What is its form for more than one thing? T. We have then discovered this property of a noun, — that it has one form for one thing and another form for more than one thing — let us see now what we shall call this property. T, What are one, two, three, etc., in arithmetic. P. Numbers. T. What might we call this property of a noun by which it has one form for one thing and another form for more than one thing? P. The?mm/>erof anoun. T. What then is the number of a noun ? P. Number is that property of a noun by which it has one form for one thing and another form for more than one thing, T. Let us now see how many numbers nouns have. T. How man}' forms has the noun book ? P. Tico forms. T. How manj' numbers then are there ? P. Two numbers. T. If there were three forms, one form for one thing, another foim for two things, and another form for more than two things, how many numbers would there be? P. Three numbers. T. Let us see now what we shall call these two numbers. T. When a horse is hitched up alone, what kind of a harness do you use? P. A single harness. T. When there is one thing alone, then what may you call it? P. A single thing. T. What may we then call this number of a word which represents a single thing? P. Single number. T Very well ; that is right ; now let us see what we shall call the other numbt-r. T. When a boy has a "whole lot" of marbles, he would say he had a great what marbles ? P. A great many marbles. T. More than one thing then may be called what ? P. Mmig tilings, T. This number, then, that means more than one, maj' be called what number? P. Many number. T. What are the two numbers then? P. Single number and many number. The pupils maj' then be led to define each, and sub- sequently the words singular and plural may be introduced. They may then be led to see that in words like hook we add s to form the plural, and state it as a rule. They may then be led to see that in other words, as in box, we add es to form the plural, and state it as a rule. Thej' may then be led to see that we sometimes change the form of the word to form the plural, as man, men, o.r, oxen., etc. The pupils should then be drilled on forming plurals; and also make a list of the per- sonal pronouns classed -with respect to number, as these will be needed in some of the exercises which follow. TEACHING ENGLISH GRAMMAR. 251 - Taught in this way, the pupils will see that though there are many numbers in arithmetic, there are only two numbers in grammar, since there are onl^' two forms of words to distin- guish the number of objects. The}' can be told that in some languages, as the Greek, there is one form for a single thing, another form for two things, and another form for more than two things, giving three numbers in grammar — the singular^ the dual, and the jjlural. „ I'erson. — To teach Person, we first lead the pupils to see that a noun may represent three distinct persons — the person speaking , the person spoken to, and the person spoken of. We tlien lead them to call this property of nouns person ; then lead them to a definition of the person as that property of a noun b}- which it represents the person speaking, the person spoken to, and the person spoken of. We then lead them to see that there are three grammatical persons, and that they are appro- priatelj- distinguished as first person, second person, and third person. To do this, we lead them to see that the first thing necessary for something to be said is a jierson speaking, the second condition is some one to speak to, and the third con- dition is some one or something to speak of; hence the name of the speaker may be called the first person, the name of the person spoken to, the second person, and the name of the per- son or thing spoken of, the third person. The pupils should then be required to point out the person of nouns and pronouns, use nouns and pronouns of a given person in constructing sentences, and be drilled on exercises similar to those already suggested. They should also be re- quired to make a list of pronouns arranged according to person. Let the student of this book be required to translate the above description into an inductive lesson, sucli as is given under the previous subjects. The following three sentences may be used in giving the lesson: "I, John, am Jiere;" "John, come here;" "John is here." 252 METHODS OF TEACHINQ. C(ise. — The subject of Case is regarded as very difficull for young pupils; but, if properly presented, it is quite readily understood. We should first teach the nominative and object- ive cases together, then the possessive case, and then the objective case after the preposition. We shall describe the lesson briefly. The teacher will write on the board, John strikes William, call attention to the action, ask who does the action, a\ ho receives the action, then ask if they both bear the same ri lo- tion to the action; what relation John bears to the action, having them say he is the doer of it; what relation William sustains to the action, leading them to say the object of tiie action; then tell them that this property of words sustaining ditferent relations to an action is called Case, and lead th«.m to define case ; then lead them to call John the doer case a.id William the object case, and define each ; after which he van introduce the terms nominative and objective. The jyossessive case can be easily taught by the relation of ownership or possession. The next step is to lead to the dif- ferent case forms of the personal pronouns. A list of tht'se should be made, classed according to case ; and the pupil be drilled on them until he knows them by sight, independently of their relation in the sentence. The next step is to teach the objective case after the prepo- sition. This needs especial notice, as it is not at all apparent to the learner that in the sentence, " He gave it to John," John is in the objective case. Indeed, if the teacher should take the two sentences, " John has the book," and " I gave the book to John," and ask what case is John in the first sen- tence, and then what case is John in the second sentence, the pupils would say nominative in both. This shows that it ia not evident to a beginner that prepositions require the object- ive case. We should, therefore, teach the objective case after a preposition by the use of the pronoun. Let the pupil see that in the sentence, " I gave the book to John," we cannot say TEACHING ENGLISH GRAMMAR. 253 "to /je," nor "to /*i.s," but Jire required to say, "to /im," which is the objective form; hence John, which is represented by him^ must be in the objective case. Let the pupil then l)e drilled on case by i)oiutiiig out the case of words in sentences, constructing sentences with given cases, etc., as before sug- gested. The student-teacher should be required to present this description in an inductive lesson, like those previously given. Gender. — Gender is easily taught. We first call attention to the dirterence of .sej? in animals, and the absence of sex in other objects. We then show that some words change their form to express males and females, which property is called the gender of nouns and i)ronouns. Then lead them to define gen- der, to see that there are two genders, since there are two sexes, and lead them to name and define each. Then lead them to see that the words which apply to objects that are neither male nor female, are said to be in the neuter gender ; and also that those words whicli are common to both males and females may be said to be in the common gender. The main point of difficulty is to distinguiiih sex, which is the attribute of objects, from gender, which is a property of words. Give abundant exercises as before suggested. The student-teacher may be required to give the lesson like the models presented. Properties of the Verb. — We shall now show how to teach the properties of the Verb to beginners in grammar. These properties are Number, Person, Mode, Tense, and Voice. The properties of Number and Person are derived properties, properties which the verb acquires from its sub- ject. The other properties are intrinsic, belonging to the verl) per se. Mode of Verbs. — To teach Mode^ write on the board, " John studies his lesson," " John, study 3'our lesson," and " John can study his lesson." Ask which sentence declares the action, which commands it, which expresses its possibility, then ask which part of speech expresses these three things. 254 METHODS OF TEACHING. Ask in what manner the verb expresses the act in the first sentence ; have them say it simply declares the act. Ask in what manner the verb expresses the act in the second sen- tence ; requiring them to say it commands the act ; etc. We thus discover the property that a verb may express an action in different manners ; then inquire what we may call this property of the verb, and have them call it the manner of the verb. What then is the manner of a verb ? If we use the word Mode, which means the same as manner, what shall we call this property' of the verb? Ans. The Mode of the verb. What then is the mode of a verb ? etc. The next point is to name the modes. In how many ways did we express the action? How many modes then are there? The first simpl}' declares or indicates the act, what mode then ma}' we call it ? Ans. The declaring or indicating mode. From this we lead to the declarative or indicative mode. The second commands the act ; lead pupils to call it the com- manding mode, and then give them the term imperative. The third expresses the possibility of the act ; lead them to call it the possible mode, and then give them the term potential^ as meaning the same thing. The subjunctive mode is so nearly obsolete that it need not be taught ; and the infinitive may be taught by its form; or, what is better, be called an infinitive, and not regarded as a mode of the verb. The participle may be taught in the same manner. The student-teacher should be required to present the method of teaching mode in an inductive lesson. Tense of Verbs. — To teach Tense we first call attention to the kinds of time — present, past, and future. We then write on the board — "John studies grammar," "John studied grammar," "John will study grammar;" and ask what time is expressed by each form of the verb, and thus discover that tlfe verb can express the act as present, past, or future. We then call attention to this property of a verb by which it ex- presses different kinds of time, and lead the pupils to call it TEACniNQ ENGLISH GRAMMAR, 255 the time of the verb. We then introduce the word iense^ mean- ing the same as time ; lead them to call the propert}^ the tense of the verb, and then lead them to define tense. We then lead them to call the first, present tense, the second, past tense, and the third, future tense, and require them to define each. The other tenses ma}' also be easil}^ tanght. Show them that have studied, since it denotes the act as completed or perfected, may be called the completed or perfect teiise ; and since it expresses an action having a relation to the present time, it may be called the present perfect tense. Also that had studied, since it denotes an act completed at some past time, may be called the past perfect tense. Also that shall or will haoe studied, since it denotes an act completed at some future time, may be called the future j)erfect tense. The tenses of the potential mode may be taught arbitrarily by their forms, since they do not express the distinctions of time as named. The student teacher will put the above in an in- ductive lesson, Nitmber of the Verb. — The Number of verbs should be taught with reference to the number of their subjects, as the verb of itself has no number. It is a property derived from its subject, and should so be presented to the learner. To teach the number of verbs, write a sentence on the board, as " He reads the Bible," and under it " They read the Bible," and ask what change there is in the verb, and the reason for this change. Let the pupils see that the change in the num- ber of the subject causes a change in the form of the verb. They thus discover a property, that the verb changes its form when the subject changes its number; and they may be led to call this property the number of the vei-b. Then lead them to define the number of a verb. Drill the class on the singular and plural forms ; have them point out the forms in sentences, construct sentences with given numbers, correct mistakes heard in conversation with respect to the number of the verb, etc. Require them also \.o 256 METHODS OF TEACHING. derive aud state the rule of the agreement of the verb with its subject in number. Person of Verbs. — The Person of the verb should be taught with reference to the person of its subject, as the verb in itself has no person, but derives it from its subject. To teach the person of verbs, write on the board, "He reads a book," and under it, " I read a book," and call attention to the change in the form of the verb. Then lead them to see that the subject has changed, not its number or gender, but its person ; and that we have thus discovered a property of a verb, that it changes its form as its subject changes its person; and that this property may appropriately be called the person of the verb. Then lead them to define the person of a verb as that property by which it changes its form as its subject changes its person. Then drill the pupils on person, as pre- viously suggested. Voice of Verbs. — The Property of Voice, if it be taught at all, may be presented as follows : Write on the board, " John strikes William," and " William is struck by John." Lead the pupils to see that in the first sentence the verb expresses the subject as acting^ and in the second it represents the sub- ject as receiving the act. We thus discover a property of a verb, that it may represent its subject as acting or being acted upon. This property needs a name; what shall we call it? Call their attention to the fact that we express things with the voice, and that since the voice is a way of expressing things, this property of verbs by which they express the act in different ways may be called voice. Then lead to the name of the two kinds of voice. Since the first expresses the subject as active, it may be called the active voice. Since the second expresses the subject as re- r-eiving the action, it may be called the receiving voice ; or, since the word passive means just the opposite of active, and the verb expresses its subject as not active, but passive, this second kind of voice ma}' be called the passive voice. . TEACHING ENGLISH GRAMMAR. 257 Comparison. — The Comparison of adjectives and adverbs is ver}' easily taught, and we will not take space to present the subject here. Any teacher who has become thoroughly im- bued with the spirit of the concrete and inductive form of instruction used in the previous exercises, will have no trouble in presenting the subject, if he understands it himself. III. Classes of Parts of Speech. — The Classes of the Parts of Speech should next be presented. It might be thought that these should have preceded the Properties, but in several cases we need a knowledge of the properties in order to make the distinction of classes. In actual instruc- tion, they should be, to a certain extent, combined, which is left to the judgment of the teacher. It is more convenient to consider them separately in this work. Under each head we will describe the method of instruction, but the student- teacher should be required to present it in the form of an in- ductive lesson. The author of this work does not consider his pupils as prepared to teach any part of grammar until the}' can present an inductive lesson, showing just how the^' would proceed in their instruction. Classes of Nov lis. — The teacher, b^' appropriate examples and questions, will lead the pupil to see that some nouns apply to particular persons and things; as, John, Mary, Bos- ton, Washington, etc. Each of these objects has its parti- cular or projier name ; and hence such nouns may be called proper yiouns. Lead the pupil to see also that many similar objects have a name in common ; that the term horse, for instance, does not distinguish any particular horse, but is a term common to all horses ; and that it may therefore be called a common noun. Lead them in the same wa}', when it is desirable, to the abstract and collective noun, and also to the classification in respect to form, — Simple, Derivative, and Compound. Classes of Verbs. — Verbs may be classified in two ways: I. With respect to their object, as Transitive and Intransitive; 258 METHODS OF TEACHING. 2. With respect to their ybrm, as Regular and Irregular. The old eUissification into active^ passive^ and neuter, is being dis- carded by modern grammarians. It might be well to retain the term neuter for the verb to be, and regard other verbs as active and passive, instead of distinguishing them ])y voice. The passive verb seems a little simpler to the learner than the 79a.s-sii;e uotce of the transitive verb. All active verbs do not express action, neither do verbs in the active voice. Transitive and Tntransitive. — To teach the distinction of transitive and intransitive, lead the pnpil to see, by exatn|»k's and questions, that sometimes the action of the verl) pnsse.-^ over to an object, and sometimes it does not ; and that there are thus tivo kinds of verbs. Next lead them to see that the verb in which the action passes over or makes a transition to the object may be called a transition or transitive vc-rl), and that the others may be called intransitive. Tlien drill them on transitive and intransitive verbs, as found in sentences, and also in constructing sentences. Pupils should also be led to see that this distinction of tran- sitive and intransitive is not an absolute one, but that many verbs are used in both ways. Indeed, there is hardly a tran- sitive verb in the language that may not be used intransitivel}-, Regular and Irregular. — To teach the distinction between regular and irregular verbs, lead the pupil to see that some verbs form the past tense b}- adding ed, and others have no regular way of forming it, and that those whicli form it regu- larly may be called regular verbs, and that those which form it irregularly may be called irregular verbs. Pupils should then be drilled on the regular and irregular verbs. A list of the irregular verbs should be presented and carefully studied until the pupil is familiar witli their proper forms. Sentences should be constructed requiring tlie use ot the verb; and sentences erroneous in this respect, corrected. Verbs in the use of which there are frequent errors, as lay, lie, sit, sat, prove, drink, etc., should be carefullj' considered- TEACHING ENGLISH GRAMMAR. 259 Infinitives There are two forms derived from the verb, usually called the infinitive mode and the participle, to which attention is briefly called. These may be taught by the form, arbitrarily giving them the names applied to them ; or they may be taught by their use and meaning. The pupil may be led to see that the participle participates in the nature of a verb and adjective, and is thus appropriately called a participle. It may also be shown that the infinitive, as to go, having n(j nominative, is unlimited by person and number, and is thus indefinite in this respect, and may consequently be called an infinitive, which means unlimited. It may also be shown that the participle is also unlimited in person and number, and is thus also an infinitive; and that consequently there are two infinitives, the verb infinitive and the participle infinitive. The pupil should also be led to see that there are two partici- ples, the present and the past or passive. Classes of Prououus Pronouns may be divided into five distinct classes; Personal, Relative, Interrogative, Respon- sive, and Adjective. Authors are not fully agreed in this matter, but the classification given is convenient and as cor- rect as any we have noticed. Personal Pronouns. — In teaching Personal Pronouns, the teacher will lead the pupil to see that each one of these indi- cates by its form whether it is first, second, or third person, and may for this reason be appropriately called personal pro- nouns. A list of these should then be given, and the pupil may be required to commit them to memory. The student- teacher may present an inductive lesson on the subject. Relative Pronouns. — A Relative Pronoun may be taught in two ways ; etymologically or logically. By the first method, we would show that it is a pronoun, because it stands for a noun ; and that it is a relative pronoun, because it refers hack or relates to some noun already named. The personal pronoun can be used independently of the noun; but the rela- tive pronoun is always used in relation to a given noun. 260 METHODS OF TEACHING. By the logical method we would teach that it is a pronoun as before ; and then lead tlie pupil to see that it is a relative pronoun, because it connects or relates the clause which it in- troduces to some previous word or clause. It will be well for the student-teacher to put both methods into an inductive lesson. The other classes of pronouns may also be easily taught in a similar manner. IV. Elements of Parsing. — The pupil should begin to parse as soon as he begins grammar. As he leanis each part of speech, he should be required to point it out in sentences. When he has learned some of the properties, he should also be required to give them in connection with the parts of speech. This is the kind of parsing that should be required in the Primary Course. It should be informal, and often con- sist merely of the answering of questions which the teacher maj' ask on the parts of speech and their properties. There should be no formal parsing, that is, no models should be fol- lowed which burden the memory with details. The main object should be to teach grammatical ideas and relations, and not grammatical forms of expression. To introduce these forms of parsing too early, is to burden the mind with forms, and thus prevent it from looking at the grammatical relations of words. Y. Elements of Analysis. — In this Primary Course, there should also be some instruction in the elements of grrammat- ical analysis. This instruction should be presented as a gen- eralization of the offices of the parts of speech. Pupils should first be led to understand the subject and predicate of the sentence. This may be done by showing that the verb, which primarily was regarded as expressing action, is used in ex- pressing an assertion^ that the word in the nominative case is the subject of this assertion, and may be called the subject of the sentence, and that what is asserted is the predicate. It may then be shown that a collection of words may be used as the subject, and a collection of words as the predicate TEACHING ENGLISH GRAMMAR. 261 We should pass next to the subordinate elements of the sentence. The pupil may be led to see that the adjectives which originally were regarded as expressing qualities, mark out or limit the meaning of nouns, and may be called limiting words. We should then pass from a single word as limiting a noun to see that a phra.se and a clause may be used in the same manner, and may then also be regarded as limiting elements. In the same way the pupil may be led to see that the phrase and clause may also perform the office of an adverb, etc. The aim of this instruction is to teach the ideas of anal3^sis, and lead the pupils to see and understand these logical rela- tions : but no formal anal3'sis should be required of them. They may be required to answer questions and point out ele- ments ; but they should not be required to commit and follow any set forms of statement, as is properly required of advanced pupils in grammar. VI. False Syntax. — Simple examples in False Syntax should be made use of from the beginning. Common errors in language should be presented, their faults pointed out and corrected. Mistakes heard on the pla^'ground should be brought in and corrected. Pupils should be encouraged to watch their own language and to endeavor to correct all their mistakes. No formal methods of correction should be re- quired, however, as would be appropriate for an advanced class. YII. The Logical Method. — After the pupil has attained a fair knowledge of the parts of speech, their properties, classi- fication, etc., with the elements of parsing and analysis, he is prepared to look at the subject of grammar from the stand- point of thought ; and we should then introduce the elements of analysis b}' what we have distinguished as the Logical Method of teaching grammar. In the Logical Method of teaching grammar, the sentence is made the basis of the instruction : the method beginning with the logical analysis of the sentence. This logical analysis. 262 METHODS OF TEACHING. instead of being built up b3' a generalization from the use of words, flows from the sentence as expressing a thought, and descends from the various elements as wholes to the parts of which they are composed. The pupil is taught to look at a sentence as a logical whole, and to study the logical elements of which it is made up. Language is regarded as the expres- sion of thought, and the structure of language is determined by the laws of thought. The principles of Logic are thus to be made use of in deter- mining the principles of language. Words are to be consid- ered not merely in their individual meaning, but as expressing, individually and collectively, the logical relations of the ele- ments of thought. The subject and predicate are regarded as expressing con- ceptions of the mind, the one being compared with the other, and the sentence expressing the relation betAveen them. In this the}' differ from nouns and verbs, which are usually re- garded not as expressing the mental product, but as the names of objects and actions. The subordinate elements are regarded as modifying elements, limiting the meaning or extent of the subject and predicate conceptions. In this they differ from the adjective and adverb etymologically considered, which express qualities of objects and actions. The connective ele- ments are those which unite the other elements into a unity of structure. Method of Teaching. — In teaching by the logical method, we should begin by giving pupils a clear notion of an idea and a thought^ and also of a sentence^ as expressing a thought. We should then lead them to see that some ideas are par- ticular and others are general, and that these general ideas embrace many individuals. We should then lead them to see how these general ideas are limited in their extent by other elements which, in comparison with the principal elements, va?Lj be called subordinate elements. We should then teacb them to see the different classes of subordinate elements, etc TEACHING ENGLISH GRAMMAR. 268 An Idea We ma}' lead pupils to a knowledge of an Idea by having them look at an object, then think of the object when not looking at it, noticing the product in the mind, and telling them that this mental product is called an idea. The exercise we suggest is as follows : Model Lesson.— Teacher. Look at this book. Can you think of this book when you do not see it ? Can you Imagine you see this book when your eyes are closed? Do you seem to have a picture of it in your mind? Such a mental picture is called an Idea. What then is the dif- ference between an object and an idea? Is the object in the mind? Is the idea in the mind? Where is the object? Pupil. Outside the mind. T. Where is the idea? P. Within the mind. Let there be a drill also to show that there are general ideas and terms, and to show the differ- ince between general and particular ideas and names. A Thought. — In order to teach a T/jo^/gf/!.^, have the pupils form two ideas, compare them, and think the relation between them. This mental product, in whicli one idea is affirmed of another, is called a thought. The lesson is somewhat as follows : }fodel Lesson.— Teaclier. Think of something, as a robin.; the mental product is what? Pupil. An idea. T. Think of something else, as a bird ; the mental product is what ? P. An idea. T. Can you think of any relation between these ideas? csvn you unite them in any way? P. Yes, sir, — a robin is a bird. T. This mental product is called a thought. A thought is the relation of two ideas in such a way that one is asserted of the other. T. Compare the two ideas, a horse, and an animal, and affirm the one of the other. P. A horse is an animal. T. This is also a thought. What is the difference between an idea and a thought? How many ideas are necessary to a thought ? The Sentence. — In teaching a sentence, we merely show that it is the expression of the thought, either in oral or written words. Take one idea or object of thought, and affirm some other idea or object of thought of the former; write the expression on the board ; this will be a sentence. Be careful that the pupil sees that such combinations as nweet apples, etc., are not sentences. Teach also the different kinds of sep^ tences. The student-teacher ma}- give the lesson. 264: METHODS OF TEACHING. Subject and Predicate. — To teach the Subject and Predicate, take a sentence, call attention to the two parts, showing that one is the name of that about which something is asserted, and the other is the name of that which is asserted ; lead them to call the first the subject, from the subject of a composition; and the latter predicate, because the teacher saj's that is its name. Model Lesson. — Teacher. In the sentence, Boys run, how maoy parts are there ? Which is the part about which something is said? Which is tlie part tliat tells what is said of boys ? Let us see what we shall call these parts. When you write a composition, what do you call that about whicli yoa write t Pupil. The Subject of the composition. T. Very well; what shall we call boys, about which something is said in the sen- tence, boys run? P. The subject of the sentence. T. What then is the subject of a sentence ? The word ra/^s does what? P. Tells or asserts something of boys. T. What may it be called ? P. The telling or as- serting word. T. Well, suppose predicate means the same as asserting word, what shall we call runs ? P. The predicate, etc. Subordinate Elements. — The Subordinate Elements may be taught somewhat as follows : Take a sentence like the follow- ing; "Man}' bright flowers fade quickly," and have them show what words can be omitted and still have a sentence; and lead them to call the necessary- words, ^^ou-ers and fade, being more important than the others, the principal elements. The words many, bright, and quickly, being less important, are subordinate in rank, and may be called subordinate elements. Let the student-teacher give this in a lesson. Limiting Elements. — The next step is to teach that a sub- ordinate element limits the meaning or extent of the principal elements. This is peculiar to the logical method of teaching grammar, for by the etymological method, the adjectives ex- press the quality of the objects, and the adverbs the quality of the actions. In order to develop the idea of limitation, use the subject first in its full meaning, then unite a word with it that restricts or limits it to a portion of its full meaning, and lead the pupil to see that the oflace of a subordinate element TEACHING ENGLISH GRAMMAR. 265 is to diraiuisli, or restrict, or limit the meaning of the general idea or term. Model fA's.son. — Teacher. When I s:iy, " Girls study," how many girls may I iiieaa? Pupil. All girls, or any number of girls. 7'. SuiJpose I say, "Good girls study," do I mean all girls? P. No, sir, only a part of girls. T. What word is it that restricts or liiyiits the meaning o( (jirls to only a part of girls? P. The word good. T. What kind of an ele- ment may I caWgood which limits the meaning of girls? P. A liiniting element. T. When I say, "Good girls study," do I mean any particular studying? P. No, sir. T. When I say, "Good girls study hard," do I mean any particular kind of studying ? P. Yes, sir, hard studying. T. What word limits the meaning of study to hard studying? P. The word hard. T. What kind of an element then is hard ? P. A limiting element. Kinds of Subordinate Elements. — The different kinds of subordinate elements are words, jarts of speech. Thus it may be seen that the noun, which was primarily a name, is often the subject of an assertion, and that the verb, which was primarily an action- word, is used to assert or predicate something of the subject It may then be shown that several words ma}' express the subject of the assertion, and also that several words may express the predication. Again, it may be seen that the adjective, which expresses primarih' a quality of an object, may be used to limit the meaning of the noun, and also that the adverb may limit the meaning of a verb ; and rising from this idea, we may see that a collection of words may perform the office of an adjective or an adverb, and thus become a limiting element. In this way the learner may reach a clear idea of the logical elements of sentences. It is recommended that the elements of analysis be pre- sented as early in the course as pupils are prepared to under- stand it. After the pupil is familiar with the parts of speech and their genei'al offices, he may be led to the idea of a subject and a predicate, and to see that collections of words perform the same office in the construction of sentences as individual words. He may thus be led gradually into the generalizations of grammatical analysis. At a certain stage of grammatical instruction it is recom- mended that a purely ''logical" method of treating the sen- tence be presented, the pupil being taught to look at the TEACHING ENGLISU GRAMMAR. 277 Structure of a sentence through the thought. This will give him additional power in the analysis of language, as it enables him to look at the grammatical constructions through the medium of thought, which gave it existence and moulded it into its present form. He may be led to a clear idea of ideas, of their comparison giving rise to judgments, which expressed, give the ■proposi- tion ; and then learn to distinguish the subject and predi- cate of the proposition. He may then pass to the idea of the limitation of the extent of a concept, and thus of a limiting element; and see that these may consist of words, phrases, and clauses. In this way he can reach the details of analysis, passing down until it meets its complement, parsing, in the etymological use of the individual words of which a sentence is composed. This logical analysis may be presented less subjectively by regarding the words as denoting objects, classes, etc., in- stead of ideas; and the subordinate elements as pointing out or distinguishing particular individuals or classes, instead of limiting the ideas; or as limiting the application of the term rather than limiting the idea or concept. This latter method is more objective than the former and probably a little sim- pler; but it does not seem so closely related to the laws of thought as the logical method previously described. It would no doubt be more acceptable to the "nominalist" than the former method. Methods of Analysis The logical analysis of a sentence may be presented in two distinct ways, which may be distin- guished as the analytic and synthetic forms. By the former method, we first name the sentence as a whole, then sei)arate it into its parts, naming the entire subject and the entire predicate, then pass from the entire subject to the simple or grammatical subject, and name its limitations, and proceed to an analysis of each of those elements; and then analyze the predicate in the same manner. 278 METHODS OF TEACHING. B3' Uie other method, we first name the sentence as a whole, then name the simple subject, then the subordinate elements which limit it, giving an analysis of these stibordi- nate elements, then put the simple subject and its subordinate elements together and name the com[)lete or logical subject ; and then proceed in the same manner with the predicate, passing from the simple to the entire or logical predicate. These two methods are very nearly opposite in form, though they do not differ in spirit. Some teacliers prefer one method and some the other, though it is difficult to tell which is preferable. The synthetic is probabl}' a little easier, as it gives a little more time to see what the full subject or predi- cate of the assertion is. It should be noticed that the syn- thetic forui of statement is just as much an exercise in logical analysis as the anal3'tic method ; the sjjirit is the same, the only difference is in the form or order of statement. JToruis of Analysis. — We shall now present some forms for oi'al and written analysis. The necessity of such forms is clear from what lias been said in regard to forms for oral and written parsing. The forms for analysis should possess the same attributes as those for parsing ; that is, they should be simple, clear, and logical. The forms presented are nearly the same as those used by Prof. E. O. L^-te in his Grammar and Composition. To illustrate, we take the sentence used to represent the forms of parsing, — " The man who came yester- daj', gave me a pair of beautiful sleeve-buttons." Ordl Analysis. — This is a complex declarative sentence. Man is tlie subject; it is limited by the, an article, and 'r/m cuine i/esterdai/, au ad- jective clause; irJio is tlic subjccl of the claiise ; it is used also as a subordinate connective ; nnne is the predicate ; it is limited by i/es- terdny, an adverb. Hare is the predicate of the sentence; it is limited by to me, an adverbial phrase ; to, understood, is a preposition, connect- ing gave and me ; me is the object of to; gave is also limited by pair, its object ; pair is limited by a, an article, and of sleere-huttons, an adject- ive phrase; <>/ connects pi (tV and sleem-battons ; sleeve-buttofis is limited hi' beautiful, an adjective. TEACHING ENGLISH GRAMMAE. 279 WRITTEN ANALYSIS, OR OCTIaLNE. man « The"-^ adj who » " cameP yesterday **• gave p (tojPme" adv pair" o(P sleeve-buttx)ns» "''■'■ beautiful"* Some teachers prefer a slightly different method of stating the analysis. Thus, instead of saying, "it is limited by who came yesterday, an adjective clause," they say, " it is limited by who came yesterday, a clause used as an adjective." Still another method is, " it is limited b}' the adjective chiuse who came yesterday,'''' which we like about as well as the model we have given. It is a question whether we should name the parts of speech in analysis; thus, whether we should say, "it is lim- ited b}' the, an article," or rather, "it is limited by the, an ad- jective element," or "by the adjective element the.'''' Logical analj'sis may be complete without mentioning or even Ivuowing the parts of speech ; though it is convenient to use the name of the part of speech when the element is a single word. Mixed Method. — There is also a method of disposing of sentences that combines analysis and an abridged method of parsing, which may be called a mixed method or grammatical de'icription. This is a valuable practical method, and is re- commended for the use of pupils who are familiar witli the el "iments of parsing and analysis. The method in:iy l)u illus- trate'd with the sentence, " The man whom I saw yesterday lives in Boston." TVe present it in two different foiin.s ; oiu' b^ing somewhat synthetic and the other somewhat asialytic. Firat Form. — The man trhom I sair i/extenJtii/ Uccs in Boxton. This is a ^•implex, declarative sentence. The is an article ; it is used to iiioditV •aaM. Man is a noun; it is used as the subject oolites. Wnun is a rcU- 280 METHODS OF TEACHING. tive pronoun, its antecedeut is man; it is used as tlie direct object of saiD ; it introduces the clause iclinyn I saw &n& joins it to man. Whom I saw is a clause used as an adjective; it modifies man. /is a pronoun; it is usod as tlie subject oi saw. Saw is a verb; its subject is /. Yesterday is an adverb; it is used to modify aaw. J is the subject of the clause, and saw wlujin is the entire predicate. Liccs is a verl); its subject is man. In is a preposition; it is used to introduce the phrase m i^o^^iort, and join it t« lices. In Boston is a phrase used as an adverb; it modifies lives. Boston is a noun; it is used as the object of in. The man whom I saw yesterday is the entire subject of the sentence, and lives in Boston is the entire predicate. Second Form. — The man whom I saw lives in Boston, is a complex sen- tence. The man lives in Boston is the principal clause, and whom I saw is the subordinate clause. The man whom I saw is the entire subject; and lices in Boston is the entire predicate. 3Ian is a noun used as subject of lices. The is an article, used to modify 7nan. Whom I saw is a clause, used as an adj ecti ve to modi fy imm. / is a pronoun, used as subj ect of saw. Saw is a predicate verb; its subject is 7. Whom is a relative pronoun; as a pronoun it is used as object of saw, as a relative or subordinate connect- ive it introduces the clause whom I saw, and joins it to man. Lices is a predicate verb; its subject is man. In Boston is a phrase used as an adverb to modify lices. In is a preposition used to show the relation of Boston to Vices. Boston is a noun used as the object of in. Errors in AnalysL^. — Errors in analysis consist of two classes: first, errors in stating the classification and elements of a sentence; and second, errors of expression. Errors of expression include the misuse of terms, such as clause for phrase, sentence for clause, or member; the needless repeti- tion of terms, such as "of which," the use of unnecessary terms, such as "elements of the second class," " elements of the third class," etc. A very common error in forms of analy- sis is the use of long and involved sentences in which tlie thought becomes obscured in the construction. The different points should be simply and directly stated. In written analysis, the commonest errors are, — errors of arrangement, and errors in writing the abbreviations. The teacher will be careful to guard against the following mis- takes : Drawing the lines too long, or in an oblique direction; TEACHING ENGLISH GRAMMAR. 281 failing to write the modifying words and the connectives in the proper places ; writing the predicate too far below the sub- ject; failing to write the proper abbreviations in the right place, and in a smaller hand than that used in writing the sentences. Diagrams for Analysis. — Several efforts have been made to devise some form of written analysis which will 'picture the grammatical relations to the eye. The most prominent of these methods is that of Prof. Clark, called the "diagrammat- ical method," given in Clark's grammar. A method of graphic analysis that looks well upon the board is that given in Reed and Kellogg's grammar. It is supposed that such a represent- ation aids the learner in grasping the grammatical relations of words, on the principle that the abstract idea ma}^ be seen through the pictured form. The objection to some of these methods is that it often requires more ingenuit}' to prepare the diagram than to understand the grammatical relations. If used at all, they should not be made prominent, or a pupil will become so dependent upon them that he will be unable to see the grammar of a sentence except through the medium of a diagram. Used occasionally for illustration, they may be of value to the student ; but when employed as a regular method of recitation, we believe them to be objectionable. IV. CoRKECTiNG False Syntax. — By False Syntax we mean constructions in language which violate the laws and usages of grammar. The principle upon which the correction of false syntax is based in teaching grammar, is that we learn the true by seeing the false ; as the Spartans taught their children temperance by showing them the silly actions of the Helots when intoxicated. The object of correcting false s}- ntax is twofold. First, it gives a clearer knowledge of the rules of syntax, and their application to language ; second, it impresses the correct form of the sentence, and leads us to avoid the errors with which we are thus made familiar. Its importance in a course of 282 METHODS OF TEACHING. grammatical instruclion is thus apparent : it aids tlie pupils in obtaining a more thorough knowledge of the rules of gram- mar, and trains them to acquire correct habits in the use of language. The exercises selected should in the main be such as actu- ally occur in conversation and writing, and not all sorts of impossible eruors. It is hardly worth while to manufactui'e errors such as may never be heard or seen in language, as enough actual mistakes ma}- be found to illustrate every rule. The errors of common conversation should be made especially prominent, as " Please let John and I go home," " Who did you see," " Who were 3'ou with," etc. We should also have examples of the mistakes involving the nicer distinctions of grammar, as the use of shall and ivill, the forms of the irreg- ular verbs, and the popular tendencies to depart from the strict rules of s^'ntax. The slips of eminent writers will be found useful to impress upon the minds of pupils the necessity of being careful in writing. The exercises in false sj-ntax should be used in connection with parsing and analysis. They may be given along with the etymological exercises of the book after the correct forms have been explained. Some authors, as Goold Brown, give a large collection of such exercises under the detailed discussion of the rules of syntax, which is a ver}' convenient method of con- sidering the subject. Some teachers recommend that the ex- ercises be graded, following the order of the sentence, proceed- ing from the simplest form of the sentence in the first step to the most complicated form in the last step ; but, though such a treatment would be logical, it is a question whether it would possess any practical value. Forms of Corrccfhu/. — With beginners, as already stated in the primary course, no special form is to be used in recita- tion, the object being to call attention to the error and correct the practice. With the advanced course, some definite fonn should be used by the pupils in recitation. This form, as in TEACHING ENGLISH GRAMMAR. 283 parsing and anal3'sis, should be as simple as is consistent with a clear and complete statement of the nature of the error and its correction. The teacher ma}' have his pupils use a full form until they are familiar with it, and then pass to an abbreviated form. Frequentlv in the recitation, all form should be dispensed with, the pupil mercl}' being required to state the error and the correction. We present several forms, as suggested by Prof. Byerly, and used by him in his classes. First Form. — The first method of correcting false syntax embraces five distinct things: 1. The pupil states that the sentence is incorrect ; 2. He shows wherein the rule is vio- lated ; 3. He quotes the rule violated as authority ; 4. He states what should be omitted, supplied, substituted, or changed ; 5. He gives the sentence in its correct form. To illustrate, take the sentence, " Who went ? Us girls." Illustration.— "Who went? Us gtrW This sentence is incorrect; because "us," a pronoun in t!ie objective case, is used as the subject of "went"; but according to Rule I., A noun or a pronoun used as the subject of a finite verb must be in the nominative case ; therefore, instead of "us," "we" should be used; and the sentence should be, "Who went? We girls." Second Form. — Another method is that which gives the result first and the reamn afterward. It differs from the first, as the last two steps of that are made the second and third in this. We illustrate with the same sentence. Illustration.— "WJio went f Us girls." This sentence is incorrect; instead of "us," " we" should be used ; and the sentence should be,— " Who went? We girls"; it is incorrect because "us." a pronoun used as the subject of " went" understood, is not in the nominative case ; bnt according to Rule I., etc. Other Forms. — Other methods, less formal than these, may also be used. Thus, we may navie the error, then the c<)?-rec. tion, and then give the reason for the correction by q noting the rule. Another form, which we should often use, is that 284 METHODS OF TEACHING. m which the pupil reads the sentence as given, and then simply reads it as corrected. Errors in Corrcctiug. — There are several erroneous ex- pressions to which pupils are liable in correcting false syntax, which should be avoided. First, the pupil should not be al- lowed to say " 'us' should be changed to 'we' ", as that cannot be done. Second, the pupil should not say " The sentence should read"; but rather "the sentence should be." Written Exercise. — There may also be a written exercise in correcting false syntax. The teacher may dictate the sen- tences and have them written on paper or on the blackboard, and then have them corrected by drawing a line under the incorrect word and writing the correct word below it. Or the teacher can write several sentences on the board, numbering- them in the order in which the^' are written, and require the pupils to write them correctly, indicating them by the proper numbers. "When written on paper, they may be read or handed to the teacher to look over out of class, and be re- turned at the next recitation. Some of the sentences assigned should be correct, some should contain an error to be corrected, and some should have introduced into them some error, easily detected, to hide, as it were, some other error not so easily observed. The teacher, of course, should inform the class that some of the sentences are correct, some contain one error, some two errors, etc. It will be well to introduce all kinds of linguistic errors in these exercises. Thus the sentences presented may contain errors in spelling, in the use of capitals, in punctuation, in the use of words, etc. Exercises in false syntax are usually found in the text-book, and may be studied by the pupil before coming to the recita- tion. The teacher may also prepare a list of such incorrect sentences as seem to him likely to be used, and also of such as he has met with in his reading or has heard in the vicinity of the school. He should encourage his pupils to prepare a TEACHING ENGLISH GRAMMAR. 285 list of incorrect sentences which they may hear used, and also to examine the books they are reading to see whether they can detect any errors in grammar. Such an exercise will make their grammatical sense very susceptible and accurate, and lead to great care in their own use of language. In conclusion, we remark that, with the more advanced classes in parsing and analysis, we should not restrict our- selves to the mere technicalities of grammar, but should extend the exercise so as to cover the whole subject of lan- guage. We ma}' call attention to the meaning of words, to the peculiarities of their use, to the etymology of prominent terms, to idiomatic constructions, to the allusions of history and mythology, to the use of capitals, punctuation marks, etc. We should combine the elements of rhetorical parsing with grammatical parsing, and so conduct the exercise as to give the pupil a knowledge of the correct use of language in its widest sense, and cultivate a critical and appreciative literary taste. In this way an exercise in parsing and analysis ma} be made one of the most interesting and valuable exercises in the entire course of study. CHAPTER IX. TEACHING COMPOSITION'. COMPOSITION is the art of expressing our ideas and tliouglats in words. It is the art of telling what we know, or of embodying our knowledge in language. This knowledare may consist of facts which we have observed, heard, or read; or of thoughts which we may have acquired by conversation and reading, or developed by thinking. Importance. — Composition is one of the most Important branches taught in our schools. It does more to prepare a pupil for success in many departments of life than almost anj^ other branch. It also attbrds valuable culture to the mind, for it reijuires closeness of observation, fullness and readiness of memor}', and the power of original thought and generaliza- tion. It is valuable for its own sake; the art of correct and elegant expression is an accomplishment to be highly prized. It also cultivates a literary taste that enables one to appreci- ate the works of literature, and thus becomes a source of the most refined and exquisite pleasure. • Composition is also, when properly taught, one of the most interesting and delightful of the common school branches. The popular dread of composition writing is due to the fact that it has been so poorly taught in our schools. There can be no intrinsic repulsiveness in writing compositions. Chil- dren love to talk, the}' delight in expressing their ideas and feelings; and if they are taught to understand that composi- tion is merely writing what they know and think, as they would talk it, pupils would take delight in writing composi- tions, and long for " composition day " more than they now dread it (286) TEACHING COMPOSITION. 287 Errors in Tettchiuff. — The errors in teaching composition are numerous. Our methods give i)upils a wrong idea of the nature of composition writing. Many pupils seem to have the idea tliat writing a composition is tr3'ingto express what they do not know, or the stringing of words together after some mechanical model, instead of merely writing simply and natu- rally what the}' know or think about something. Pupils have been required to write compositions without any instruction or })reparation for the exercise, and allowed to write blindly without any assistance. The subjects assigned are often un- puited to pupils, being too abstract and difficult. Teachers have made the subject too formal, and thus taken all the life, freshness, and zest out of it. Such teaching has given the pupils of our public schools a dread of composition-writing. They regard it as the " bug- bear " of the school-room ; and think of " composition day " wiLh a shudder. They perform the allotted task without any interest, merely because the}' are compelled to do so. They l)ut it oif to the last moment, and slip out of it whenever they can. They copy their compositions out of books, or get some older pupils to write for them. They acquire stilted and arti- ficial forms of expressing themselves, instead of writing in that natural and interesting stj'le in which the}' converse. There is great need of reform m this respect, and this need seems to be widely felt. It is an oft-repeated question. How shall we improve our methodsof teaching composition ? Our educational periodicals are crowded with criticisms of the old methods and suggestions for improvement. Authors are turning their attention to the subject, and text-books are multiplying upon it. Our grammars are growing more prac- tical, and text-books on Language Lessons, designed to teach expression, are becoming abundant. Division of the Subject. — In the discussion of the snbject, we shall speak first of the Preparation for Composition Writ- ing, and secondly, of the Methods of Teaching Composition '28S METHODS OF TEACHING. The Preparation for Composition will include a statement of those conditions and that culture which prepare a pupil for writing. Instruction in Composition will embrace first that primary instruction which is designed to prepare a young pupil to express himself in writing with correctness and free- dom. These exercises are now popularly known as Language Lessons. Under the second head we shall present some formal directions for Writing a Composition. I. Preparation for Composition Writing. Conditiojis. — The fundamental conditions of composition are, — first, something to say, and secondl}-, how to saj' it. In other words, composition-writing includes the matte?' and the expression. The matter consists, in a general waj', of ideas and thoughts. For the expression of these, we need a large and choice vocabulary of words^ and a finished and accurate style of expression. The first requirement in writing composition is, that there shall be something to say ; when there is nothing in the mind, nothing can come out of it. Here is the mistake of many teachers, who expect children to express ideas on a subject when they have no ideas to express. Ideas, thoughts, knowl- edge in the mind, it should be remembered, are the necessary antecedents to expression. In the second place, there must be something with which to express what we know. Our knowledge must flow out in the form of words ; and we must be familiar with individual words and know how to use them. The third condition is that we shall acquire a clear and cor- rect method of expressing our thoughts; and cultivate, so far as possible, those graces of style which give beauty and finish to expression. Let us inquire how each one of these condi- tions is to be attained. Sources of Material. — The materials of composition, as already stated, are ideas, facts, thoughts, sentiments, etc. There are several sources of these materials. The principal TEACHIXQ COMPOSITION. 289 sources of our ideas and thoughts ai'e Observation, Reading, Judgment, Imagination, and Reflection. Observation. — Man}' of our ideas come from the observa- tion of the ol)jects of the material world. The facts which we express are drawn largel}' from our experience of things and persons. Nearly all the great writers have been close ob- servei's of nature and human nature. Homer was in deep sympathy with the material world, and drew some of his finest figures from his observation. Shakespeare was a devoted lover of nature, and gives us hundreds of pictures like " The morn in russet mantle clad, walks o'er the dew of yon high eastern hill," showing how close and accurate was his obser- vation. Dickens drew manj' of his characters from actual persons whom he knew, and whose peculiarities he had care- fully studied. Pupils should, therefore, be taught to observe closely and accurately. Objects should be presented to them to examine and describe. They should be required not only to observe the principal features, but also to notice the minutise of things. Observation should be analytic, descending to the minor and less obtrusive parts of objects. Trained in this way, a pupil will acquire accurate ideas of things, and be able to point them out and to describe what he has seen with ease and accuracy. Reading. — We can also obtain ideas and thoughts b}'' Read- ing. In books we find facts, ideas, sentiments, opinions, figures of rhetoric, etc., which remain in our memor}^ and may be used i?\ their original form, or become types for creations of our own. In books are embalmed the choicest productions of the master minds ; and they enrich the mind of the reader, and give wisdom to his thought, and grace to his utterances. Young persons should cull in their reading the finest pas- sages, and write them down and commit them, The\' should also take note of the interesting and important facts in their bearing" on the subject, and fix them in the memory. An 13 290 METHODS OF TEACHING. effort should be made to become familiar w.th the opinions and noble sentiments of the great thinkers, for in this way thoucht will be enriched and expression beautified. Judgment. — Pupils should be taught to exercise the Judg- ment as well as the eyes and ears. They should be taught to compare things, to see their relations, and to draw inferences from them. They should be required not only to see, but to think about what they see ; and to form opinions concerning it. It is this observing with the jutlgment that makes the philosopher. By it Copernicus attained to the true idea of the i)lanetary system, and Newton reached the great law of universal gravitation. Imagination. — Pupils should be taught also to exercise the Imagination. Every form of nature not only embodies an idea, but may be perceived as the symbol of an idea. The things of the material world are typical of the things of the spiritual world ; they are often the symbols of ideas and sentiments and feeliniis. Here is the source of personifications, similes, met- ai)hors, etc. The flower looks up into our eyes, the streamlet bathes the brows of the drooping violets, the stars are the forget-me-nots of the angels, etc. It is the office of the Imag- ination to catch these analogies, to transmute the material thing into the immaterial thought, and "give to airy nothing a local habitation and a name." The Imagination may thus be taught to leap from the visi- ble form to the invisible image. Things may become the ladder by which it rises to the sphere of beautiful and poetic thoughts. Thus, Shakespeare gives us the figure " How sweet the moonlight sleeps upon this bank;" Alexander Smith says, " The princely morning walks o'er diamond dews ;" and Long- fellow gives us the picture of a " silver brook" which "bab- bling low amid the tangled woods, slips down through moss- gnnvu stones with endless laughter." The attention of the learner should be called to these and similar creations, and he should be encouraged to create images of his own. TEACHING COMPOSITION. 291 Reflection. — Much of the material of compositions comes from Thinking. We must therefore learn to think in order to learu to write. It is not enough to acquire the thoughts of others ; we must learn to evolve thoughts for ourselves. We must cultivate a reflective and creative cast of mind that seeks for the idea l^'ing back of the fact, that searches for the cause of the phenomena, and is ever inquiring what these facts prove, or what principle they illustrate or establish. We should en- deavor to originate new forms of expression, new figures of rhetoric, and to form ideas and opinions of our own on many subjects. Sources of Words. — The second condition of becoming a good writer is the acquisition of words. In order to write, we must not only have ideas and thouglits, but we must have lan- guage in which to express them. The thought is to be incar- nated in speech. Ideas and thoughts existing in the mind, intangible and invisible, are to be transmuted into audible or visible forms. Nature, as it were, goes into the mind through the senses, and reappears in the form of language. Form and color and tone in the natural world, give form and color and tone to expression. The freshness of spring, the brightness of summer, the rich tints of autumn, and the silver habit of winter, all give freshness and beauty and glory to the litera- ture and language of a people. These words may be acquired in several ways. Instinct. — Words are derived partly by an instinctive habit. We pick them up in conversation without any conscious etTort. A child will often be heard to use words which it but a short time before heard some one else make use of. Chil- dren seem to have an instinct for language, and new words cling to their memory like burrs to the garments. A child of four years of age ma}-- be able to speak three or four different languages if it has had an opportunity to hear them spoken. It is, therefore, of great advantage to a child to hear a large and expressive vocabularj' used in the household. 292 METHODS OF TEACHING. Conscious Effort. — "Words should also be consciously acquired. Tliere should be a special effort made to enrich the vocabulary. We should notice the words in our reading, and make a list of new words, or of those which we may think do Hot belong to our practical vocabulary. Such a list may often be reviewed until the mind becomes familiar with it. We should also make use of these words in our conversation and in writing. It is surprising how rapidl}' we would improve in expression by the adoption of this method. Our vocabulary, which is often small, smaller than we think, will l)ecome en- larged; and we will learn to speak and write with a copious, rich, and elegant expression. The Dictionary. — The pupil sliould form the habit of study- ingr the Dictionarv. The dictionary has sometimes been used as a text-book in schools, but this is not recommended; it should, however, be a student's constant companion. It should lie on every student's table, and be frequently consulted. This has been the habit of some of the most accomplished scholars and writers. Charles Sumner was a most assiduous student of the dictionary. He had several copies in his library in constant use, and usuallj^ carried a pocket edition with him; and they were found, after his death, to be the most thumbed of an}^ of his books. Lord Chatham went twice through the largest English dictionary, studying the meaning of each word and its various uses. General Reading. — An extensive course of general reading is valuable in acquiring a large and choice vocabulary of words. Such reading should be largely confined to our best authors, those who use words with correctness and artistic skill. The finished and thoughtful writer often puts a mean- ing in a word which we never noticed before, and thus stamps it upon our memory. It is only in this way that we can ac- quire that nice and delicate sense in the use of words which distinguishes the refined and scholarly writer. Ancient Languages. — The study of the ancient languages TEACHING COMPOSITION. 293 IS especially valuable in this respect. It was formerly thought that a knowledge of Latin and Greek was necessary in order to understand the English language ; but this claim is now seldom made. The great value of their study consists in the constant use of English words in the translations, and in the comparison and weighing of the sense of the various words given in the definitions to see which will express the meaning of the text the most accuratel3\ If the student should forget every word of Latin and Greek the year after he leaves college, the linguistic culture he has received is a permanent posses- sion, and will enrich his expression. Small Words. — In the choice of words, young pupils should be careful not to select merely the large words. The large words attract the attention and are the most liable to be re- membered. It is the little words, however, that are the most expressive, and are the most artistic in use. The good old Anglo-Saxon basis of our speech contains a richer and more expressive meaning than the larger Latin and Greek deriva tives. Our best writers delight in the skillful use of the small words ; and this is an especial characteristic of Shakespeare and our English Bible. This caution is the more necessary, as young persons have an idea that large words indicate learning and profund- ity of thought. Goethe refers to this when he makes Mephis- topheles say to Faust, " For that which will not go into the head, a pompous word will stand j'ou in its stead." This is quite a general opinion among the uncultured. The man who came to his minister, frightened at a strange appearance of the sun, was entirely satisfied when he was told that it was "only a phantasmagoria." Hazlitt, referring to the use of large words, says, "I hate anything that occupies more space than it is worth; I hate to see a load of empty bandboxes go down the street, and I hate to see a parcel of big words with- out anj'thing in them." Leigh Hunt gave a fitting reply to a lady who asked the question, "Will 3^011 venture on an 294 METHODS OF TEACHING. orange?" by his answer, "No, thank you, I fear I should fall off." Let the pupil, therefore, not select the large words, but learn to use the little words, the language of the heart and home, with skill and artistic effect. Style of Ex^ivession. — We not only need ideas and thoughts, and a rich vocabulary in which to express them, but we need also to know how to put these words together to produce the best results. We need to acquire a good st3le of expression. We need to acquire that ease and elegance of expression and that artistic skill in the use of language, which distinguishes the cultivated writer. In order to aid the pupil in this, several suggestions are made. Read Extensively. — First, we remark that pupils should read extensively. Reading not only gives words, but it gives facility in the use of words and the expression of ideas. Pupils who have read most are usually- the best writers. We often find in school those who are deficient in the more diffi- cult studies, yet who write excellent compositions ; and upon inquiiy, learn that they have read a great deal, perhaps merely novels. The best scholars in the school branches are often very poor writers, because they have done but little reading. By reading, we become familiar with the style of an author, and form a stjle of our own. Many distinguished men have formed their style b^^ reading a few books very thoroughly, Lincoln received his language culture ver}^ largely from read- ing the Pilgj'im^s Progress. Kossuth's masterly knowledge of English was acquired by the study of Shakespeare and the English Bible. The unique and expressive language of un- cultured men, derived almost entirely from reading the Bible, has often been a surpi-ise to us and demonstrated the utility of reading in acquiring a style of expression. Copy Productions. — Pupils should be required to coyy lite- rary productions. Copying an author will make a deeper impression than even a careful reading of one. Sight strikes deeper than sound ; to execute form stamps it upon the TEACHING COMPOSITION. 295 nemory like a die. To go over a production, word by word md sentence by sentence, writing it out, will impress the style of the author deeply upon the literary sense. I would therefore require pupils to "copy compositions." If a para- gra[)h could be written every day on the slate or on paper, it would greatly aid the literary growth of the pupil. Many eminent writers have practiced copying the productions of the masters of literature. Demosthenes copied the history of Thucydides eight times, in order to acquire his clear, concise, and elegant st3'le. Commit Extendvell|.—V\\^^\\^ should be required to commit extensively, both prose and poetry. Committing will make a deeper impression than either reading or copying. It will tend to fix the words and deepen the channels of thought and expression. It will, a'^ it were, give one literary moulds in which to run his own thoughts, or dig out literary channels in which our thoughts and sentiments may flow. This has been the practice of all who have obtained excellence in the use of language. Burke and Pitt cultivated their wonderful powers of oratory by committing the orations of Demosthenes. Fox committed the book of Job, and drew from it his grandeur and force of expression. Lord Chatham read and re-read the sermons of Dr. Barrow until he knew many of them by heart. Declamation.— l^ha old practice of " declaiming pieces" was of very great value to students in the culture of literary power. It gave them models of style and stimulated expres- sion. Indeed, it often did more to give a command of English than the whole college course. We have noticed the style of young men after their graduation at college, and could, in several instances, trace it back to the culture derived from their declamation pieces. All this preparation for writing requires time and patience. It cannot be acquired in a few months or a year, but is a matter of gradual development. Literary skill is the result of literary growth. A student can master a text-book in 296 METHODS OF TEACHING. geometiy or algebra in a few months ; but literary culture is the work of a life-time. It is an organic product, like the development of a tree. The exercises should be continued day by day, and the result will crown the work. We shall now proceed to the second division of the subject, — The Methods of Teaching Composition. We shall divide the sub- ject into two parts ; Language Lessons and Composition Writing. II. Language Lessons. The preparatory exercises required for young pupils in learning to understand and use the English language with skill, have, by common consent, received the name of Lan- guage Lessons. By Language Lessons we mean such element- ary training in the use of language as shall enable a pupil to understand and appreciate language, and to use it with cor- rectness, ease, and elegance. Niiture ami IiiiportttHce. — Of the importance of such les- sons there can be no doubt. The primary object of education in language is to learn to use language. In order to leara the correct use of language, we must notice and use language. The use of language is an art ; and we learn the art by imita- tion and practice. In order to learn to talk well, we must hear good talking and practice talking. In order to learn to write well, we must notice good composition and practice writing ourselves. A system of Language Lessons conforms to nature's method of teaching language. The little child, prattling in its mother's arms, is engaged in its first lessons in composi- tion. The simple name, the quality and action word, the short sentence, etc., all come in the natural growth of the power of expression. In teaching, we must observe nature's method and follow her golden rules. A correct system of language lessons is founded upon the way in which a little child naturally learns oral and written language. TEACHING COMPOSITION. 297 A system of language lessons will also teach a child to acquire and produce knowledge as well as to express it. It cultivates the habit of observation and comparison ; and thus leads a child to think as well as to express thought. Subjects should be assigned that require attentive examination, that call the judgment into activity, and that lead the pupil to investigate and discover facts, and thus gain knowledge for himself. The pupil will also be taught to classify the knowl- edge obtained from reading, to sift its true meaning, and to express in his own words the thoughts of the writer he has studied. The fundamental principle of these lessons is that pupils are to be taught the practical use of language by the use of language rather than by a study of the principles of lau- o-uaffe. There should be an imitation of models, and a free and spontaneous expression of ideas, without any thought of the grammatical rules or principles involved. For example, the pupil should express himself in sentences without any thought of the subject and predicate of a sentence, and use the ditferent parts of speech without any knowledge of them as parts of speech. He should use nouns and verbs without knowing that they are nouns and verbs; form plurals without any rules for numbers; use cases, modes, tenses, etc., without knowing that there are such things as cases, modes, tenses, etc. The system of language lessons aims to teach the use of language by imitation and practice rather than by the study of rules and definitions. The object is to give children a knowledge of the uses of words and the power to express their ideas, without clogging their memories with grammatical terms which are to them often only abstract sounds without any content of meaning. The pupils are brought into contact with living language, and not the dead dry skeleton of gram- matical definitions and rules ; and this living spirit becomes engrafted on their own language until it becomes a part of their nature. 13* 298 METHODS OF TEACHING. According to this principle, a knowledge of language should precede a knowledge of grammar. This is the historical order of development. The ancients knew language and could use it in literature, but they had very little knowledge of grammar. Homer sang in immortal verse, and probably hardl}^ knew a noun from a verb. The Iliad embodied the rules of grammar, without the author being conscious of them ; the rules of grammar were derived from the study of the Iliad. This is also the natural order, — practice precedes theory, the art comes before the science, — and should be fol- lowed in the early lessons on language. Another principle is that language lessons should lead to and be the basis of grammatical instruction. Most of our tert-books on language lessons invert this order by basing the lessons in language on grammar. This is a ver^- gi-eat mistake, and vitiates the whole course of instruction. The largua2:e lessons should prepare for and lead up to grammar. Grammar may then return the favor and aid in the correct use of language. Thus art gives birth to science, and science reciprocates the faror and gives perfection to art. The study of grammar, therefore, should not be begun until such a course in language lessons, as is suggested, has been completed. Such lessons should be begun as soon as the child can write. Before this it should be required to commit and recite little poems and pieces of prose. If it can hear good models of conversation, it will be of very great advantage in the culture of con-ect expression. Course of Lessons We shall now present an outline for a course of Language Lessons suitable for beginners. This is a mere outline and is to be filled out by the teacher in actual instruction. \. Require pupils to write the names of objects. Write the names of ten objects ; the names of objects in the school- room; objects in the house; objects they can see h\ looking out of the window; objects they saw in coming to school, etc. TEACHING COMPOSITION. 299 2. Require pupils to write the navies of actions. Write tiie actions of a child ; of a bird ; of a dog ; of a cat ; of a fish ; of a horse ; of a cow; of a cloud; of a river, etc. 3. Require pupils to write the names of objects with the names of actions, foi'miug a sentence. Give the name of the object, requiring them to give the name of the action ; also give them the name of the action, requiring them to give the name of the object. 4. Lead pupils to an idea of a sentence, as asserting some- thing of something. Lead them to see what is a telling or declarative sentence, an asking or interrogative sentence, and a commanding or imperative sentence. 5. Teach them that each sentence begins with a capital let- ter ; that a declarative or imperative sentence ends with a period, and an interrogative sentence with an interrogation point. Drill them in writing sentences and correcting sen- tences which violate these rules. G. Have them write sentences introducing adjectives^ ad- verbs, pronouns, inte7Jections, etc. The teacher will give the word, and have them form the sentences. Of course the pupils are not to know an^'thing about these words as parts of speech. 7. Show the difference between particular and common names, and teach the use of capitals for particular names. Teach also the use of capitals for I and 0. Have them write exercises involving these things, and correct sentences which violate their correct use. 8. Give two words, and have pupils write sentences contain- ing them both; give also three words to be put in a sentence, four words, etc. The pupils ma}' also be allowed to select the words which the}' are to unite in a sentence. 9. Give pupils sentences, with words omitted, and require them to insert the correct words. Such sentences can be dic- tated to them, the missing word being indicated by the word "blank." If they are written upon the board for them, the 300 METHODS OF TEACHING. missing words may be indicated by a dash; as, " I saw a building a in a tree." The teacher should select and prepare a large list of such sentences for the use of his pupils. 10. Have the pupils look at an object and describe it. Have them descinbe a school-mate, a horse, a cow, a cat, a pig, the school-house, a barn, a church, etc. A very interesting exer- cise can be had in describing one another, and other persons whom they know. 11. Have pupils look at a picture, and tell you all they see in it, and then write it out on their slates or on paper. Pic- tures can be found in the primary readers, or the teacher may bring a large picture to school for the pupils to look at, or pupils may bring some pictures from home. 12. Show them how to arrange lines of poetry, and that each line begins with a capital letter. Dictate poetry to them, and have them copy it, getting the lines and the capitals right. After pupils are familiar with correct formS; they may be allowed to criticise incorrect forms ; as Mary had A little Lamb. its Fleece was wiglit As snow I and Every Where that marry Went ? The Lamb ; was shure To go : 13. Have pupils talk about something, and then write down what they have said about it. Let thom learn to write their talk. Take such subjects as a knife, a chair, a boat, a pin, a needle, a cat, etc. Parts of the body, as the eyes, the nose, the mouth, the tongue, the hands, the feet, etc., are easy and interesting subjects for children to talk and write about. 14. Call out a child's knowledge of an object by asking questions about it, and then have him write down what has been said, in distinct sentences. Children often know more about an object than they can think of. Questions will also lead them to discover new things about the object that they had not noticed before, and teach them how to look at things and gain a knowledge of them. TEACHING COMPOSITION, 301 15. Talk to the children about something, have them repeat what you have said in their own words, and then write it out on their slates, or on paper. They will thus see that writing a composition is merely telling in writing what they know and can tell in talk. 16. Teach them the use of the hyphen, as connecting com- pound words; and also its use at the end of a line, in con- necting one syllable with the S3'llable beginning the next line, 17. Teach the use of the comma, as placed after the name addressed; as, "John, come here;" and also as connecting three words of a series; as, '' He saw a boy, a girl, and a dog." 18. Teach the use of the period after abbreviations; and make pupils familiar with the common abbreviations; as, Mr., Dr., Rev., Hon., Esq. Drill them on LL.D., so that they will not make the common mistake, " L. L. D." 19. Teach the use of quotation marks. Show that the in formal quotation is set otf by the comma; as, Mary said, "John, come here." Show also that a divided quotation has two commas; as, "To be good," says some one, "is to bo happy." 20. Teach also the use of the colon before a quotation In- troduced formally by such expressions as "the following,'' "as follows;" as, He spoke as follows: "Mr. President, .he gentleman is mistaken in his facts," etc. 21. Teach the use of the apostrophe in denoting possession; as, John's book. Also, its use in denoting omission of letttrs; as. Ne'er, 'T is, I 've, etc. 22. Teach the use of the exclamation point after interiec- tions; as, Oh! Alas I Pshaw! Hurrah! etc. 23. Let the teacher read a narrative and ask questions on it, and then have the pupils reproduce it orally and in writing. 24. Write sentences on the board, and have the pupils imi- fate them in other sentences. Write also faulty sentences for them to correct. Include errors upon all the things that hav been presented in these Language Lessons. 302 METHODS OF TEACHING. 25. Give related simple sentences, and require pupils to unite them into compound sentences. Tlius, "John stood up;" "John spoke to his father," changed into "John stood up and spoke to his father." Let them also decompose compound sentences into simple ones; as "John and Mary went home," changed into "John Avent home," and "Mary went home." 26. Give them some little proverb, and have them write out nn explanation of it; as, "Little children should be seen and not heard;" or, "Birds of a feather flock together;" or, "A rolling stone gathers no moss." 27 Require them to express sentences in different ways; as, " The flowers bloom ver3' sweetly in the spring of the year," changed to " In the spring of the j'ear, the flowers bloom verj' sweetly." 28. Require them to change poetry into prose. Write a stanza on the board, and have them express the same thing in prose; as, "The day is done, and the darkness Falls from the wings of Xight, As a feather is wafled downward From an eagle in his flight," Changed to "When the day is done, the darkness falls arouud us as gently as a feather which falls from the wing of an eagle flving above us." 29. Exercise them on misused icords and incorrect con structions; as, "I expect j'ou had a good time;" " Let Mar}' and I go out ;" etc. Make a full list of the incorrect expres- sions in common use, and drill the pupils in their correction. 30. Present the elements of Letter Writing. Teach the cor- rect form of the Date, Address, Introduction, Close, Super- scription, their punctuation, and the correct use of the capitals which occur in them. The teacher who does not understand the subject will find it explained in Westlake's How to Write Letters. 31. Require pupils to write letters of different kinds; as TEACHING COMPOSITION. 303 Business Letters, Notes of Invitation, Notes of Acceptance, Excuses for Absence from Sciiool, Receipts for Monej', Due Bills, Notes, etc. 32. Have them -n-rite a letter to a teacher, to a friend, to their father, to their mother, to a school-mate, etc. They will be interested in writing a letter to a dog, or a horse, or a bird, etc., imagining that the animals can understand thcni. Give them forms of letters as models for them to imitate. 33. Teach them a few of the simple figures of rhetoric^ as the Simile, the Metaphor, Personification, etc.; and require them to point them out in sentences and to form sentences containing such figures. Have them change metaphors into similes, and similes into metaphors, etc. 8-4. Have them write little newspai^er paragraphs, as an account of a fire, of a party, of a runawa}', of a railroad acci- dent, etc. Bring a newspaper into school and read such items of news as will interest them, and have them write little items in imitation of those in the paper. 35. During all this time, have them committing and reciting choice selections of prose and poetry. Do not allow them to repeat these mechanicallj' without understanding their mean- ing, but ask questions to lead to a clear idea of what is ex- pressed. This will cultivate a literary taste, which lies at the basis of all artistic excellence in the use of language. 36. Give them suitable subjects and require them to write little compositions. Let the subjects be simple, and of per- sonal interest to them. Indicate the method of treatment. Ask questions to lead them to what should be written. En- courage the timid and diffident. Suggest how to state facts, to sa^- bright little things, to express ideas and sentiments, etc. Lead them to write naturally, expressing what they think ami feel. Correct kindly and gently, and sti'ive to make them love to write compositions. The above presents a ver^' complete outline for instruction in Language Lessons. It is, howeA'er, merel}- an outline, and 304 METHODS OF TEACHING. needs to be filled out for actual use in the school-room. The teacher should take this outline and write out a list of exam- ples or exercises under each head, suitable for the use of liis pupils. No text-book in the hands of the pupils is needed for this work, if the teacher is properly qualified himself; but each teacher will find it of advantage to write out a little text-book for his own use in giving instruction in language lessons. To aid the teacher in preparing these lessons, we recommend the following works : Hadley's Lessons on Lan- guage, Llo3'd's Literature for Little Folks, Bigsby's Ele- ments of the English Language, and Swinton's Language Lessons. In following this outline, the teacher should make the exer- cises very full and complete. Do not be afraid of having too much under each head, for we are most liable to err by not giving practice enough. Let the motto be Make haste slowly. Give variet}' to the lessons, and pupils may be kept for a long time on each exercise suggested. Keep up a constant review by introducing parts of the previous exercises into each sub- sequent exercise. III. The Writing of a Composition. We shall now speak of teaching a pupil to write a composi- tion. The previous exercises have been designed for begin- ners, and are mainly imitative in their character; older pupils should depend more upon themselves, and be required to con- struct formal compositions. We shall speak of the subject under three heads: first, the Principles to guide a teacher in the instruction ; second, the Method of Writing a Composi- tion; and third, some General Suggestions on the subject. I. Principles of Composition Writing. — In teaching pupils to write a composition, the following principles should be l>orne prominentl}' in mind : I. Composition is to he regarded as the expression of what a child actually knows. The importance of this principle is TEACHING COMPOSITION. 305 enhanced by the fact that it has been very generally ignored by teachers. Many pupils go to work at their compositions as if they were expected to tell wliat they do not know. The exercise is not a spontaneous production of what they think, but a reaching out and striving after that which they have never thought. Tliis will account, to a large extent, for the general distaste for composition writing, and the frequent de- ception in respect to their authorship. Teachers, in assigning subjects, seem to have been oblivious of this principle, often giving subjects that are entirely beyond the reach of the pupil's experience and range of thought. 2. Pupils should begin with oral compositions. They should be required to talk about objects before writing about them. We should begin by having pupils talk compositions before they write compositions. Subjects can be assigned the same as for a written composition, time being given for preparation or not, as the teacher may prefer. Many of our eminent editors and literary men talk their literaxy productions, and have them copied by an amanuensis. 3. Pupils should be led to see that writing a composition is writing their talk. This is the key to composition writing with young pupils. This principle clearly understood, would be like a revelation to many a pupil; it would open up the way and remove the ditliculties that so often seem to rise up mountain high before them. Many persons who talk well seem to grow dumb when they take a pen in hand; what they need to learn is to write their talk. 4. Do not be too critical at first. Severe criticism tends to discourage the pupil, and create a distaste for the subject. There is no exercise in which criticism wounds so deeply or discourages so soon as that of composition writing. Pupils need encouragement as well as direction. We should com- mend that which is worthy of praise; and, in a kindly manner, point out the mistakes and suggest where improvements can be made. 306 METHODS OF TEACHING. 5, 3Iale the subject intereating. Cultivate a love for the expression of tlionght. Be an inspiration to pupils by writing for them and with them. Start a little newspaper in the school, and have them contribute to its columns. Make them feol that composition writing is a delightful task; the most delightful exercise in the school. They will thus long for "composition day," instead of regarding it with dread or in- ditference. Remembering these principles, the teacher's way in teaching composition will be much smoother than it has been, and the results will be much more satisfactory. Indeed, the teacher who catches the spirit of these principles, ami ap- plies them properly, can make the pathway all bright and fragrant with blossoms of interest, both for himself and for his pupils. Some of the author's pleasantest recollections of school life are associated with his classes in composition. II. WuiTiNG A Composition. — In the writing of a eamiiosi- tion, there are four things which call for special attention: 1. The Subject; 2. The Matter; 3. The Analysis; 4. The Amplijlcation. Each of these is modified by the kind of composition to be written. The principal kinds of composition are as follows: 1. Description; 2. Narratives; 3. Essays; 4. Discourses; 5. Fictions; 6. Poems. The first and second of these consist mainly of a description of facts. The Essay is a presentation of thought or opinion upon some subject: in a large sense it may include Editorials, Reviews, and Treatises. Discourses are productions designed to be read or delivered: they in- clude Lectures, Sermons, Addresses, and Orations. Dis- courses usually contain both thought and description. The Subject. — The Subject of a composition is one of the most important parts of the production. To select or invent a good subject often requires more thought and talent than to write the composition. The merit of a literary production often depends very largely on the selection of a happy and suggestive topic. TEACHING COMPOSITION". 307 It is usualh' best for the teacher to assign the subject to the pupil. He can better adapt it to the taste and capacity of the pupil than the pupil can himself. Besides, the pupil may not only select au inappropriate subject, but will often spend more time in making the selection than in writing upon it. It also secures more variety in subjects for the teacher to select them, and thus gives a wider culture in writing. It also removes, to a great extent, the temptation to plagiarize, as the pupils cannot so readily find access to an article on a given topic as when they choose the topic. At times, how- ever, pupils should be required to select and invent topics for themselves, as it is an excellent exercise for their ingenuity, and tends to cultivate independence and self-reliance of thought. Pupils who have always depended on the teacher for subjects, become very helpless when placed in circum- stances where the}' must make their own selection. In assigning the subject, the teacher should be careful to adapt it to the pupil. Do not give abstract or lofty subjects about which the pui)iU have no ideas or knowledge. What, for instance, does a little child know about Contentment, or Immortality, or Government, or The Sublimity of Thought, etc.? Let the subject be one that appeals to the pupil's experi- ence. For young pupils, subjects like going to school, swim- ming, fishing, skating, coasting, etc., would be appropriate; older pupils should write on subjects requiring more maturity of thought and experience. In all cases, let the subject be interesting to the writer, if possible, and one upon which he may express what he really knows. Subjects should be so varied as to give practice in various st3-les of composition Pupils should be required to write descriptions. of oljects. pLices, i)ersons, natural scenery-, etc.; the}' should be required to relate incidents of their observa- tion or experience; to write little fictions, allegories, orations, dialogues, etc. ; and, with many pupils, an exercise in writing poetr}' will also be of real value. 808 METHODS OF TEACHING. The subject must also be determined by the kind of compo- sition to be written. If the composition is designed for a public audience, it should be of popular interest and suited to the intelligence of the audience. The subject should possess unity, and be clear and fresli. The statement of it should be simple, not too figurative, but happy in expression, and, if possible, striking. The manner of stating a subject often gives popularity to a production. A book frequently owes a large share of its popularity to its title. The title, That Husband of 3Iine, sold many more coi)ies than the story itself merited, and became a model for the naming of a score of other works. The Material. — When the subject is selected, the first thing is to acquire the material for the production. There must be something to say before we attempt to say anything. We cannot draw water from a drj' well. This getting the material is called Invention; and it is the most ditticult part of the process of composing. It is not eas}' to sliow how it can be done. Some hold that it is not a thing to be taught, that "it is a part of one's native endowment," an original talent and not a power to be acquired. A few suggestions can be made, however, which are thought to be valuable. The material of a composition consists of facts and thoughts. Facts embrace such things as have been observed b}' the writer or by others. The thoughts embrace opinions, senti- ments, figures of rhetoric, etc. This material may be obtained from at least five different sources; Observation, Conversation, Reading, Imagination, and Reflection. These are treated quite fully under Prepatration for Composition Writing, and need not be discussed here. They are more or less prominent in supplying the material, according to the character of ohe subject upon which one is writing. Observation. — If the subject is descriptive or narrative, a writer should draw first from his own observation. That which is stamped with a writer's personality, is far more in- TEACHING COMPOSITION. 809 teresting than what he gives at second hand. Some one happily remarks, "Do not go to Homer for a sunrise when you can see one every morning." In the second place, the writer should draw from the experience of others, which may be done by conversation or by reading. Much can be picked up in conversation that will be fresh and interesting. In the use of books, select only those things that are most attractive, and endeavor to express these facts in your own language. When the material derived fi-ora these several sources is abundant, make use of that which seems to possess the most novelty. Imagination. — Try to throw the light of fancy around this material. The plain fact is not of so much interest as when it is made to glow with the touch of imagination. Let the fact awaken an image in the mind, if possible ; draw from it a simile or a metaphor ; endue it with the life of a personification, etc. Many writers, like Scott and Dickens, weave the most beautiful fancies into their statements of facts and cast a charm over the descriptions of the most ftimiliar objects. Reflection. — If the subject is reflective in its character, the ma- terial will consist principally of thoughts and opinions. These thoughts and opinions are attained by thinking, by reading, and by conversation. A writer should first try to think out all he can for himself. And here the question arises, how shall we evolve or create thoughts by thinking? No rule can be given, but a few sug- gestions will be ventured. First, we should put ourselves in a reflective mood ; we should fix the mind on the subject and think about it. Newton said he made his great discoveries by continually thinking about them. We should surround the subject of thought with questions. Asking questions is one of the doors to all great dis- coveries in science or inventions in art. We should try to answer our own questions. This will give activity to our thoughts, and afford us something to say on the subject. Thus, if the subject is, "The Stars," we may inquire, — What are stars? Whence do they come ? Why do they shine at night ? Why 310 METHODS OF TEACHING do they twinkle ? With what have they been compared ? etc The answering of these questions will give a large amount of material for a composition on "The Stars." Many subjects should be developed around some leading tiiought, and we should endeavor to find this leading thought, which gives unity to the treatment. The leading thought of a discourse is the germ from which it is developed. It is the living principle from which it grows; the parent idea which becomes the source of its life and growth, and without which the words will be but a dead letter. When the germ- thought appears in the mind, let the understanding brood over it, and it will develop into a living organism of thought and expression. This leading thought once in the mind, will give rise to many other thoughts connected with it, and which grow out of it as the branches shoot forth from the main stem of a tree. If this general conception does not occur at first, fix the mind on the ideas that do occur, compare them and see what principal thought they suggest or lead to, and thus reach the germinal princii)le of the composition, going from the parts to the whole. It is proper also to think out some figures of rhetoric, some com})arisons, similes, or metaphors to be used in the ami>lifi- cation of the material. Many such thoughts will occur to us in writing, and they are usually most appropriate when thus suirofested ; but some of our best writers mark down their happy thoughts to be worked into their productions as they are needed. Reading. — The writer may also read books written upon or touching upon the subject. Some of these ideas may be taken and used as presented, by giving credit to their author. Many of the thoughts can be worked up into new forms, so that they will be, in a certain sense, one's own property. Such an exer- cise will be of great value to a young writer, in teaching hirn how to think. In reading, however, one should digest and assimilate what he reads, so that it will appear with the stamp TEACHING COMPOSITION. 311 of his own mind upon it. It will then become his own prop erty and can be used at his will. Another suggestion in obtaining the material by reading, is to read authors who have written on the subject or a kindred one, and mark down the ideas which their thoughts suggest to the mind. Many authors are very suggestive of ideas. They seem to deal in seed-thoughts which fall into the mind and produce other thoughts in abundance. As we read, an idea seems to spring up in the mind by a sudden illumination, as the spark darts from the flint when struck b}"^ the steel. Thus Emerson and Carlyle can be most profitably read with a pencil in the hand, marking down the ideas which spring up in the mind as the eye passes over the printed page. The facts of biograph}-, history, etc., should be rallied around the leading ideas to support or prove the position taken. These facts may be culled out from the store-house of memory, or we may go to books and gather the material needed for illustration or proof. It is well for the student to liave a "commonplace book," and mark down such incidents and historic statements as he thinks may be of use to him in writing. Collect Material. — This material should be written down on paper, as it presents itself to the mind. It is well to have a blank book and jot down the thoughts as they may occur to us, without respect to vmy particular order. This can be done at odd times as the thoughts present themselves, so that when the time comes to write composition, there wiU be a fund of material to make use of. The Analtjsis. — The material having been acquired, the pupil should examine it, see what is most interesting or most pertinent to the subject, bring together those parts that are similar, and make a complete outline of the method and order of treatment. This is called forming the plan, or the Analy- sis: and is an important part of the composition. As a rule, it should never be omitted ; the pupil should always have 312 METHODS OF TEACHING. some general idea of the composition before he begins to write. In a kind of fancy writing, we may give free rein to thought and imagination, and allow them to play with the ideas that may chance to present themselves. The light and gossipy essays of Addison and Lamb could never have been written from an outline, though even in many of these there is a leading idea that gave shape to the production. It is an excellent exercise for the pupil to take ditferent subjects and merel}' prepare outlines of their treatment. In forming the analysis, the composer should have in his mind an idea of what he wants to present. If the object is description, he should see clearly the order in which the facts should be stated to secure the interest of the narrative. If the production is reflective, he should knov; what he desires to prove or to impress, and arrange the points in such a way as best to secure this object. Care should be taken that there be no abrupt breaks between the parts, but that one part flows naturally out of and into another. It will be well sometimes to tr\' diff"erent arrangements, and see which seems best. A writer will often change the whole plan of his essay while he is writing it out, as a general changes his plan of attack on the field of battle ; but this is alwaj's inconvenient and haz- ardous. A very great deal of good judgment may be shown in the analj'sis of subjects, and the success of a lecture or ad- dress is often largel}' due to the arrangement of its parts. The Anipliflcation. — Having formed the plan of the com- position, the next step is the Amplification. The facts and thoughts are to be presented in an orderlj' manner; care is to be taken that the sentences are clear and correct, that the matter is properly connected, that the style is suited to the ■subject, etc. Many new ideas and illustrations will present themselves in the course of the ami^lification; and when ap- propriate should be wrought into the composition. There are three parts of a literarj' production that require esj>ecial attention. These are the Introduction, the Body, and TEACHING COMPOSITION. 813 the Close. In an ordinarj'^ short school essay, these divisions are not so marked ; and 3'et the}' are not to be overlooked even there. Every production should have a fitting opening and closing thought; it should neither open nor close with an awkward abruptness. In lengthy essays, lectures, orations, etc., these divisions should be distinctly marked. The Introduction — The Introduction should be modest, appropriate, lively, and interesting. It should not promise too much, or the expectations it raises may be disappointed. It should grow naturally out of the subject, and be a natural in- troduction to what is to follow. It should be suited to attract r.otice, and to prepare the mind to listen with attention and an expectation of pleasure to what follows. An interesting incident, an apt illustration, a humorous remark, etc., are often used b^' good writers and speakers as an introduction. In an oration, the introduction should li:ive an air of candor and modest}- ; it should be calm and moderate, and not antici- pate the main points of the discussion. Cicero laid down the rule that the Introduction should be written last, though he did not always follow his own precept. He was accustomed to prepare introductions and lay them aside to be used when needed. On one occasion, he inadver- tently used the same introduction twice ; and upon being in- formed of it by Atticus, he confessed his error, and prepared a new one. The Body. — In the Body of the essay, the subject should be formally developed. The leading idea should be kept care- fully in mind, and the etfort be made to unfold it. There should be an organized growth of thought and expression in ■which all the subordinate ideas are gathered around the prin- cipal one. A thread of related thought should run unbroken through the entire exposition, binding all the parts together in symmetry and unity. In view of this, the ditferent points to be presented should be arranged in the best order. If there are objections to be 14 314 METHODS OF TEACHING. answered, it is usually best to attend to them first. Having cleared the way of these, the direct arguments may then be presented, throwing the less plausible ones in the middle, and thus giving the stronger ones first and last. The exhortation and appeal to the feelings come appropriately toward the close, but an incidental appeal may be made at different times as the occasion offers. If humorous passages occur in a spoken discourse, they should not come in too near the beginning, or they will unfit the minds of the hearers to listen to what follows. So also it is not well to touch the feelings with pathos near the early part of the discourse, for it will be difficult to hold the inter- est after the reaction of feeling takes place. It is well also that the production should increase in majesty and grandeur of exi)ressi()n towards its close. Care should be taken that the thoughts be expressed in an attractive and pleasing form. The language should be simple, clear, and impressive. When suitable, it may be adorned with fio;ures of rhetoric and pictures of the imagination. Nothino; should be introduced, however, for mere ornament, and that does not contribute to the main purpose of the essay. Much self-denial is often retjuired to avoid putting words or figures into a production when their only claim is their beauty. It is in this that the difference between a culti- vated and an uncultivated writer is readily noticed. The Conclusion. — The Conclusion, or Peroration of a dis- course, like the Introduction, requires especial care. The object is to leave as deep an impression on the mind of the reader or listener as possible. This is sometimes done by re- serving the strongest or most impressive head of the discourse for the last, and ending with it. Sometimes the writer or speaker gives a brief and striking summary of the whole dis- course, bringing it all. in rapid succession, again before the mind. In this way the conclusion becomes a kind of burning- glass which gathers into a focal point all the separate rays ol the production. TEACHING COMPOSITION. 315 The conclusion ma}' often consist of an exhortation or appeal to the feelings, in view of what had been stated. Ac- cepting the views of the writer or speaker, the reader or lis- tener is prepared to sympathize with his feelings and to share in his emotions. In every case, where there is a formal con- clusion, it should seem to flow naturally out of the discussion and be appropriate to it and the subject. Dr. Hart says, " The main thing to be observed is to hit upon the precise time for bringing the discourse to a point. If this is done too abruptly, it leaves the hearers expectant and dissatisfied. If, when the discourse seems ended, and the hearers are looking for the close, the speaker continues turn- ing round and round the point, without coming to a pause, the audience becomes restless and tired. There are, indeed, very few speakei'S that know how or when to stop." In this discussion of composition writing, we have lifted the subject up into the plane of preparing a lecture or an oration; but it will be seen that the suggestions given nearly all apply to the writing of an ordinary school composition. A compo- sition, if thoughtful, is a little lecture or a little oration, and is designed to prepare for these larger productions ; and the same methods and principles that apply to one apply also to the other, the difference being one of degree only. We close the subject with a few general suggestions. III. General Suggestions. — There should be frequent ex- ercises in writing composition. In many schools pupils are required to write once in two weeks. It would be better, however, to have them write every week; and still better to have the exercise more frequently. Paper, Writing, etc. — The pupils should be required to write on paper of a uniform size. The large-sized letter- paper, known as " Bath post," is perhaps the most convenient. The subject should be written at the top of the page on the middle of the first line ; and a blank line left between the heading and the composition. There should be a margin of 816 METHODS OF TEACHING. about one inch on the left-hand side of each page, to allow room for corrections. The first line of each paragraph should be indented about one inch. The writing should be neat and legible, with no flourishing or fancy writing ; and care should be taken with respect to the paragraphs, etc. The signature should be written on the next line below the close of the com- position, near the right-hand edge; and tiie name of the place and date on the next line below the signature, near the left- hand edge. The compositions should all be folded alike, in three divisions; and the name of the writer, the subject, and the date, be written on the back. If an outline is required, it may be written either at the beginning or close of the composition. Corrections. — The compositions should be handed in promptl^"^ at the time appointed, for correction. The correc- tions, as a rule, should be made b}' the teacher; though at times the essiA^s may be distributed among the members of the class for correction, under the general supervision, how- ever, of the teacher. The corrections may include errors in orthography, punctuation, use of capitals, hyphens and apos- trophes, construction of sentences, figures of rhetoric, style of expression, general development of the subject, etc. The closeness of the correction should be adapted to the age and ability of the pupils. Severe criticism will tend to discourage young pupils, who are especially sensitive in respect to their own compositions. It will be best for the teacher, as a rule, to indicate the errors, rather than to correct them, requiring the pupils to make the correction. This will make a deeper impression upon the mind than when the teacher makes the corrections for them, and will lead them to be more careful not to repeat the mistake. In order to indicate the errors, some system of notation should be used. A line may be drawn under each error, and the sj-mbol indicating the nature of it be written in the margin. TL.e notation used in our own school, is as fol- TEACHING COMPOSITION. 317 lows: ^ for analysis; 0, orthography ; (?, grammar; TT, wrong word ; S, sentence ; P, punctuation ; etc. For a fuller statement of the system, see Westlake's Three Thousand Practice Words. The teacher may sometimes take the compositions into the class and call attention to the errors, withholding the name of the writer, if he chooses, and invite corrections and sug- gestions. The pupils may also be required to read the errors marked, and correct them orallj^, or write them upon the board with their correction. Some teachers require i)upils to copy the composition in a book provided for the purpose, with the mistakes all corrected. Reading Compositions — There should be a time set apart for the reading of compositions. This is a very useful exer- cise, and may be made the means of a great deal of literary culture. In this exercise, each pupil may read his own pro- duction, or one may read for another, as the pupils or teacher may pi'efer. After the reading of a composition, remarks and criticisms may be made, first by the pupil and then by the teacher. Care should be taken that the attitude, expression, etc., of the reader be free from error. Pupils may often be required to commit their essays and recite or declaim them, those designed for declamation being written in the style of an oration. The exercises of "composition day" may be made very in- teresting and instructive by varied literary exercises. A paper, with an appropriate name, to which the pupils contrib ute, will give variety to their productions and be of great interest to the pupils. It may contain short essays, editorials, items of news, amusing incidents, wit, humor, poetry, adver- tisements, etc. Some orations, recitations, dialogues, and debates will also give additional interest to the occasion. The class may occasionally be resolved into a literary society with regular officers and a programme of exercises, consisting of an inaugural address, orations, recitations, essays, answera to referred questions, a paper, etc. 318 METHODS OF TEACHING. In convjlusion, we remark that the teacher should spare nc pains to create an interest in literary culture. No greater intellectual benefit can be conferred upon a pupil than to cul- tivate in him a literary taste and train him to an appreciation of literary productions. That teacher achieves a great success and accomplishes a valuable work, who makes composition writing a pleasing task and composition day to be regarded with interest and delight. MATHEMATICS. CHAPTER I. THE NATURE OF MATHEMATICS. MATHEMATICS is the science of Quantit3\ It seeks to ascertain the relations and truths of quantity, and to de- rive unknown quantities from other quantities that are known. This definition indicates the general nature of the subject, tliough it is not entirely free from objections. Many attempts have been made to frame a philosophical definition of mathematics, but none has yet been presented which is gen- erally acceptable. The term Mathematics is derived from the Latin mathemat- ical or the Greek mathematike, which comes from mathesis, learning. The use of the word in the plural form indicates that this department of knowledge was formerly considered not as a single branch, but as a group of several branches, similar to our use of the phrase, the mathematical sciences. Previous to the present centuiy, nouns ending in ics, as optics, mechanics, etc., were construed with a verb in the plural ; but they are now generally regarded as singular. The fundamental branches of mathematics are Arithmetic and Geometry. This classification arises from the nature of the two kinds of quantity considered. The two general divisions of quantity are Number and Extension : the science of number is Arithmetic; the science of extension is Geometry. These two branches have also been distinguished with respect to their relation to Time and Space. Extension has its origin in Space, and number in Succession, which is only possible in Time. Hence, Geometry has been called the science of Space, (319) 320 METHODS Oy TEACHING. and Arithmetic the science of Time. Geometry has also been called the science of Form, since it treats of the possible forms of space. If we inti'odnce general symbols for numbers, and develop a science with them, we iiave another branch of mathematics, called Algebra. If we use these general symbols in investi- gating geometrical magnitudes, we obtain another branch called Analytical Geometry. If we investigate quantity by considering the infinitesimal elements of which it is com- posed, we obtain a branch called Differential and Integral Calculus. There is another method of conceiving the subject of quan- tity and reaching a division of the science. Quantity is of two kinds; discrete and continuous. Discrete quantity is that which exists in separate parts, forming quantity of multitude or number; as a number of men, trees, etc. Con- tinuous quantit}' is that which daes not exist in separate parts, or is that in which the parts are connected together in one whole, as length, time, etc. Discrete quantity is immedi- ately expressed in numbers, and gives rise to the science of arithmetic. Continuous quantity cannot be immediately ex- pressed in numbers; a part of the quantity must be taken as a unit of measure in order to express it numerically. One form of continuous quantity, that which belongs to space, gives rise to the science of geometry. There is no general agreement among writers in respect to the philosophical division of the science of mathematics. Comte, the most celebrated writer on the philosophy of mathematics, divides the science into two parts; Concrete Mathematics and Abstract Mathematics. Under Concrete Mathematics he includes Geometry and Rational Mechanics ; under Abstract Mathematics he includes the Calculus, which embraces Arithmetic, or the Calculus of Values, and Algebra, or the Calculus of Functions. The latter, called also Analy- sis, embraces ordinary algebra, in which the equations arc THE NATURE OF MATHEMATICS. 321 directly established between the magnitiidos under considera- tion, and the Transcendental Analysis, in wliich the desired equations are derived by invariable analytic methods from equations between quantities indirectly connected with those of the ppoblem. These are distinguished as the Calculus of Direct Functions and the Calculus of Indirect Functions. 3Iaterials. — A knowledge of mathematics consists of Ideas and Truths. The Ideas of mathematics represent the different forms of quantity which present themselves for con- sideration. The Truths of mathematics are the relations that exist between the quantities. When we conceive and examine the different forms of quantity, we perceive some truths that ai'e self-evident; such truths are called axioms. By means of these axioms we compare the different quantities, and attain to other truths; these truths are said to be derived by reasoning. The ideas and axioms are thus the basis upon which, b}' the process of reasoning, we build up the science of mathematics. The Ideas. — The Ideas of mathematics are not merel}^ ideas or products of the mind. The}^ represent realities, things which have an objective existence. The}' are not ideas of material things, there is no tangible reality corresponding to them, but they are real forms of space and number. The forms of geometry are pure forms, forms not filled with con- tent; and the numbers of arithmetic are pure numbers, inde- pendent of any association with material things. But in both cases the quantities are realities which admit of application to the objects of the material world. Definitions. — The description of the ideas of mathematics in clear and exact language gives rise to the Definitions of the science. The Definitions of mathematics may thus be re- garded as the precise description of its ideas. The ideas are antecedent to the definitions, and are the basis of them. In teaching the science, definitions are employed to lead the mind of the learner to clear conceptions of the ideas. On account of the intimate relation be' ween the idea and the definition, 14* 322 METHODS OF TEACHING. some writers state that the foundation of mathematics is definitions and axioms, rather than ideas and axioms. Axioms. — The Axioms of mathematics are the self-evident truths of the science. They are intuitive truths which arise in the mind immediately on the contemplation of the various forms of quantity, without any process of reasoning. They express a self-evident and necessary relation between quanti- ties, and thus involve a comparison or a judgment. Given the several conceptions, and the truth is immediately per- ceived by a direct comparison, without any intervening pro- cess of thought. Axioms are the basis of mathematical reasoning. Without some self-evident truths as a starting point, no process in thought is possible. By some they are regarded as general truths which contain the particular truths of the science, and from which the particular truths are evolved b^' reasoning. It seems more correct, however, to regard them as laws which direct or govern the comparisons of the reasoning process. Thus the truth that "things that are equal to the same thing are equal to one another," is a law to guide us in comparing quantities, rather than a general truth that contains the other truths of mathematics. Reaaoning. — The Reasoning of mathematics is deductive. It deals with necessary truth, and derives the relations by universal and necessary' laws of inference. The basis of the reasoning is the definitions and axioms, or in other words, the conceptions and the self-evident truths arising out of these conceptions. Thus having a conception of a triangle and a right angle, we may b}' comparison, in accordance with the laws of inference, derive the truth that "the sum of the angles of a triangle equals two right angles." So in arith- metic, having a conception of some subject, as the greatest common divisor, we can derive a method of obtaining the greatest common divisor of two numbers, gi iding the inA'csti- gation by the self-evident truths that pertain to the subject THE NATURE OF MATHEMATICS. 323 Value of Matltematlcs. — Mathematical studies have iu all ages been valued for the mental discipline they atford. There is probably no single study pursued in the schools which develops the mind in so man}' wa^'s, and is so well adapted to every stage of mental growth as mathematics. Mathematical studies give some culture to perception and memor}', faculties which it has been thought the}- almost entirely' neglect. They require the most complete mental concentration, and thus aftbrd the highest culture to atten- tion. Dealing with the relations of quantity, they give con- stant exercise to the judgment, and train it to the closest discrimination of similarity and ditferences. Every derived truth is a logical deduction from premises, and is reached by the continued operation of the power of reasoning. The first truths are axiomatic, and are comprehended only in an act of intuition, which gives exercise to the Reason. The Imagination is also active in geometry in picturing the parts of the figures upon which we reason, and in creating diagrams to discover new relations. All the definitions are "logical definitions," and as such train to the nicest percep- tion of the relation between ideas and their expression. In fact there is no one science that brings so large a number of the faculties of the mind into so constant and forcible an activity, and especiallj^ those faculties which give strength and dignit}' to the intellect, and glory to scientific achieve- ment. That it does not train to habits o^ probable reasoning, and does not give facts for induction and for oi)inions on social and political questions, is admitted; but that it does more than any other school study to give mental power and logical habits of thought, must also be admitted. Eleinentary Urauches. — The three elementary branches of mathematics are Arithmetic, Algebra, and Geometry. These are taught in all our graded and high schools, and should, in their elements at least, be taught in the ordinary common schools. We shall in this work discuss the methods of teach- inji these three branches. CHAPTER II. THE NATURE OF ARITHMETIC. THE science of arithmetic is one of the purest products of human thought. It was aided in its growth by the rarest minds of antiquity', and enriched b\' tlie thought of the pro- foundest thinkers. Over it P^'thagoras mused with the deepest enthusiasm; to it Plato gave the aid of his refined speculations ; and in unfolding some of its truths Aristotle emplo3'ed his peerless genius. In its processes and principles shines the thought of the ancient and modem world ; the subtle mind of the Hindoo, the classic culture of the Greek, and the practical spirit of the Italian and Englishman. It comes to us adorned with the offerings of a thousand intel- lects, and sparkling with gems of thought received from the great minds of nearly every age. Like all science, which is an organic unity of truths and principles, the science of arithmetic has its fundamental ideas, out of which arise subordinate ones, which themselves give rise to others contained in them, and all so related as to give sj-mmetry and proportion to the whole. These funda- mental and derivative ideas, the law of their evolution, and the philosophical thread that runs through them and binds the parts together into an organic unity, should be understood by the teacher. To aid the teacher in acquiring a more philosophic concep- tion of arithmetic than he obtains from the text-book, we shall speak of the General Nature of the science, its Language, Reasoning, and Methods of Treatment. We shall then pro- ceed to consider the Methods of Teaching the subject. (324)' ■ NATURE OF ARITHMETIC. 325 I. The General Nature of Arithmetic. Definition. — Arithmetic is the science of numbers and the art of computing with them. The term is derived from arithmetike, which is from arithmos^ meaning number. This is the definition usually given, and is sufficiently correct for all practical purposes. There are some writers, however, who hold that Arithmetic is only one of the sciences of numbers, Algebra and Calculus being also regarded as sciences of num- ber Some French writers call the general science of numbers Numerique ; and divide it into Arithmetic, the science of special numbers, and Algebra, the science of number in gen- eral. Sir Isaac Newton called algebra Universal Arithmetic. The Nature of Nuuiher. — The basis of arithmetic is Number. A number is usually defined as a unit or a collec- tion of units, a definition derived from Euclid. This definition is liable to the objection that a number and a collection are not quite identical in meaning. Many definitions of a Number have been attempted, but none has yet been given which is entirely satisfactory to mathematicians. The simple idea of a number is that it is the how-many of a collection of objects, and it might be so defined. The definition first given is, however, the one generally preferred by writers on arithmetic. There are three fundamental classes of numbers ; Integers, Fractions, and Denominate Numbers. These three classes arc practically and philosophically distinguished, and constitute the basis of a threefold division of the science. Logically the distinction is not entirely without exception, since a fraction may be denominate, and a denominate number may be inte- gral; but the division is regarded as philosophical, since these three classes of numbers not only differ in character, but re- quire distinct methods of treatment, and give rise to distinct rules and processes. Integers. — An Integer is a number of integi'al units. These 326 METHODS OF TEACHING. units are regarded as iudividual or whole, and hence an Inte- ger is called a whole number. There is no relation of the things numbered to an^- other thing regarded as a unit; but simply the relation of the collection to a single thing of the collection. An integer is thus a pure product of synthesis. Fractions. — The Unit, as the basis of arithmetic, may be multiplied or divided. A synthesis of units, as we have seen, gives rise to Integers; a division of the Unit gives rise to Fractions, Dividing the unit into a number of equal parts, we see that these parts bear a definite relation to the integral unit, and name them from this relation. These parts mHy be regarded as individual things, and constitute a particular class of units called fractional units. The collection and numbering of these fractional units give rise to a particular class of numbers called Fractions. Denominate Numbers. — Quantity is of two kinds — (juantity oi multitude and quantity oi magnitude — called also discrete and continuous quantity. Discrete quantity exists in indi- vidual units and is immediately estimated as how many; continuous quantity exists in the mass, and is primarily esti- mated as how much. Thus we say how many apples, how viany trees, etc., while we say how much money, how much land, etc. Quantity of magnitude does not primarily admit of numerical expression ; to thus express it, we fix ui)on some definite part of the quantity as a unit of measure, and express the quantity by the number of times it contains the unit of measure. Continuous quantity thus becomes expressed as discrete quantity-; the how much is reduced to the how many ; and a new class of numbers arises, called Denominate Num- bers. These units being of difii'erent sizes and bearing ditier- ent relations to one another, require a special method of treatment which gives rise to a distinct department of arith- metic. With the adoption of the metric S3-stem, this part of arithmetic will lose the distinctive character of its operations. Logical Outline of A I'ilhmctic. — The science of Arithme- THE NATURE OF ARITHMETIC. 327 tic is based upon and is developed from these three classes of numbers. Its several parts are evolved from the possil)le operations upon these numbers. A consideration of tliese possible operations will give us a Logical Outline of Arith- metic. All numerical ideas begin at the Unit. The Unit is the origin, the basis of arithmetic. The Unit can be multiplied and divided; hence arise Integers and Fractions. Each Integer is a synthetic product derived from a combination of units ; each Fraction is an analytic product derived from the division of the unit. Having obtained numbers by a synthesis of units, we may unite two or more numbers, and thus obtain a larger number by means of synthesis; or we may reverse the operation and descend to a smaller number b}^ means of analy- sis. Hence the two fundamental operations of arithmetic are S^aithesis and Analysis. To determine when and how to unite and separate numbers, we employ a process of reasoning called comparison. This process compares numbers and de- termines their relations ; it is the thought process of arithme- tic, as analysis and synthesis are the mechanical processes. Comparison directs the original processes of arithmetic, mod- ifies them so as to produce from them new ones, and also itself gives rise to other processes not contained in or growing out of the original ones. Comparison is thus the process by which the science is constructed ; it is the key with which the learner unlocks its rich store-house of interest and beaut^'. Syyithesis. — A general synthesis is called Addition. A special case of the sj'uthetic process of addition, in which the numbers added are all equal, is called Multiplication. The forming of Composite Numbers hy a synthesis of factors, which we call Composition, Multiples formed by a synthesis of particular factors, and Involution, a synthesis of equal factors, are all included under Multiplication. Hence the process of Addition includes all the synthetic processes to which numbers can be subjected. 328 METHODS OF TEACHING. Analysis. — A general anal3'sis, the reverse of Addition, is called Subtraction. A special case of Subtraction, in which the same number is successively subtracted with the object of ascertaining how many times it is contained in another, is called Division. A special case of Division, in which many or all of the makers or factors of a number are required, is called Factoring ; a special case of Factoring, in which one of the several equal factors is required, is called Evolution; and a case in which some common factor is required, is called Com- mon Divisor. Hence the process of Subtraction includes all the analytic processes to which numbers can be subjected. Comparison. — By comparison the general notion of relation is attained, out of which arise several distinct arithmetical processes. By comparing numbers we obtain the idea of Ratio ^ arithmetical and geometrical. A comparison of equal ratios gives us Proportion. A comparison of numbers differ- ing by a common ratio gives us Arithmetical Progression and Geometrical Progression. In comparing numbers of difierent units, we observe we may pass from one to another of differ- ent species under the same genus, and thus have the process of Reduction. In comparing numbers, we may assume some number as a basis of reference, and develop their relations in regard to this basis ; when this basis is a hundred, we have the process of Percentage. In comparing numbers, we dis- cover certain relations and peculiarities which give rise to the Properties or Principles of numbers. Remarks. — We thus derive a complete outline of the science of numbers. Arithmetic is seen to consist fundamentally of three things ; Synthesis, Analysis, and Comparison. Synthe- sis and Analysis are fundamental mechanical operations; Comparison is the fundamental thought-process which con- trols these operations, brings out their potential ideas, and also gives rise to other divisions of the science growing out of itself. The whole science of pure arithmetic is the out- growth of this triune basis, — Synthesis, Analysis, and Com- THE NATURE OF ARITHMETIC. 329 parison. The rest of arithmetic consists of the solution of problems, either real or theoretical, and may be included under the head of Applications of Arithmetic. This outline of the science grows out of the idea of pure number, independent of the language of arithmetic. These fundamental processes are modified by the method of nota- tion adopted to express numbers. With the Roman or Greek methods of notation, the methods of operation would not be the same as with the Arabic sj-stem. The method of adding by " carrying one for every ten," of subtraction by "borrowing," a portion of the treatment of common and deci- mal fractions, the methods of extracting roots, etc., are all largely due to the system of notation adopted, and many of them have their origin in the Arabic S3Stem. It may be remarked, also, that the power of arithmetic as a calculus depends upon the beautiful and ingenious system of notation adopted to express numbers. II. The Language of Arithmetic. The expression of the fundamental ideas of arithmetic givetj rise to Arithmetical Language. This language is both oral and writt-en. The oral language is called Numeration ; the written language is called Notation. Numeration is the method of naming numbers and of reading; them when ex- pressed in written characters. Notation is the method of expressing numbers in written characters. Numeration. — In naming numbers we do not give each number an independent name, but proceed upon the principle of naming a few numbers and then forming groups or collec- tions, naming the groups, and using the first names to num- ber the groups. This ingenious, though simple and natural, method of naming numbers by forming groups or classes, seems to have been adopted by all nations. It has the advan- tage of emplo3'ing but a few names to express even very large numbers, and of enabling the mind, by the principle zf 330 METHODS OF TEACHING. classification, to conceive quite readily of a numlter otherwise entirely bej'ond its powers of conception. Thus, after naming the numbers as far as ten, we regard the collection ten as a single thing, and count oheration with the Arabic system is recommended. II. Secondary Operations. — The Secondary Operations are Composition and Factoring, Common Divisor nnd Common Multiple, Involution and Evolution. Tlie new division, called Composition, seems necessary for scientific completeness, that each analytical operation may have its corresponding syn- thetical operation. It is also convenient in naming certain operations for which we formerly had no appropriate term. Definitions. — Composition is the process of forming com- posite numbers out of the factors. Factoring is the process of finding the factors of composite numbers. A common divisor of two or more numbers is a number that will exactly divide each of them. A common multiple of two or more numbers is a number which is one or more times each of those numbers. It is usually defined as a number which contains these numbers, but this does not include the idea of multiple. Treatment. — In treating these subjects, we first establish some general principles, and then derive the methods of op- eration from these principles. It will be well to require the student-teacher to show the method of development of each division, and to point out the philosophy of the method of treatment. III. Common Fractions. — Integers originate in a synthesis of units, fractions in a division of the unit. A fraction in- volves three things: first, a division of the unit; second, a comparison of the part with the unit; third, a collection and numbering of the parts, A Fraction is thus a triune product — a- result of analysis, comparison, and synthesis. A fraction may also arise from the comparison of numbers. Definition. — A Fraction is a number of equal parts of a unit. This seems to be an improvement on "one or more TREATMENT OF ARITHMETIC. 341 equal parts of a unit." Since the parts of a unit are num- bered, these ma}' be called fractional imits, and a fraction may be defined as a number of fractional units. Among many of the incorrect definitions, we mention, — "A fraction is a part of a unit ;" " A fraction is an expression for one or more of the equal parts of a unit;" " A fraction is nothing more nor less than an unexecuted division." Cases. — The cases of fx-actions are all included under ^\n- thesis, Analj'sis, and Comparison. To perform the synthetic and analytic processes, we need to change fractions from one form to another; hence Reduction enters largely into the treatment of fractions. The comparison of fractions gives us several cases called Relation of Fractions. The student- teacher may state the cases. Treatment. — There are two methods of developing com- mon fractions, known as the Inductive and Deductive Meth- ods. By the Inductive Method, we solve each case by analy- sis, and derive the rules by inference or induction. By the Deductive Method we first establish a few general principles, and then derive rules of operation from these principles. The Inductive Method is simpler for 3'oung pupils; the Deductive is more satisfactory for older pupils. The student-teacher may illustrate both methods. Principles. — The deductive method is based on certain principles which express the law of multiplying or dividing the terms of a fraction. These principles can be demonstrated either by the principles of division or independently as frac- tions. The latter method is by far the better. There should be a real demonstration, and not some loose statement such as we often find in arithmetics. lY. Comparison. — We have not room to indicate the treat- ment of comparison, but refer the student to the author's Philosophy of Arithmetic. A review of the subjects of arith- metic, pointing out the philosophy of its methods of treat- ment, would be of advantage to the student-teacher. 342 METHODS OF TEACHING. y. The Course in Arithmetic. Arithmetic, for the purpose of instruction, may be divided into two parts; Mental Arithmetic and Written Arithmetic. In Mental Arithmetic the problems are solved without the aid of written characters. In Written Arithmetic the opera- tions are performed with the aid of written characters. Oral Arithmetic. — Many educators divide the course into Oral and Written Arithmetic; and at first thought such a division seems plausible and natural. Language is of two kinds, oral and written ; when we solve problems without written characters it is naturally called oral arithmetic; when the operations are performed with written characters it is naturally called written arithmetic. Such a division is, how- ever, a mistake, and results from a superficial view of the subject. Written Arithmetic is just as oral when recited as Mental Arithmetic ; and Mental Arithmetic is no more oral when not recited than Written Arithmetic. Both are oral when recited ; neither is oral when not recited. Intellectual Arithmetic Nearly all authors of Mental Arithmetic call their works Intellectual Arithmetic. The term Intellectual is, however, objectionable, as it does not accord with popular usage. No one thinks of calling a " men- tal solution" an " intellectual solution ;" or would say, " he solved it intellectually," but rather, " he solved it mentally." Practical Arithmetic. — Many authors call their works on written arithmetic, Practical Arithmetic. This, however, is a misnomer; all arithmetic should be practical, mental arithme- tic as well as written arithmetic. The proper term is ivritten, to indicate that we employ written characters. The term " Practical" may do very well as a " trade mark " but it should not pretend to any scientific accuracy. It was sug- gested by Orontius Fineus, in 1535, in a work entitled Arith- metica Practica ; and first used by Joseph Chapman in 1732 in a work entitled " Practical Arithmetic Compleat." TREATilEXT OF ARITHMETIC. 343 True Division The natural division of the subject is, therefore, into Mental Arithmetic and Written Arithmetic. There are several considerations in favor of the term Mental. First, it is in accordance with the popular usage, for all per- sons would say of a solution without the aid of written char- acters, " he solved it mentally," and not " orally" or " intellec- tually." Second, the distinction is philosophical. Both meth- ods of solution employ the mind, and one employs the written characters also, and it is appropriate to distinguish the two methods by this distinguishing characteristic, calling that which employs written characters Written Arithmetic, and the other, which is purely mental. Mental Arithmetic. One is purely mental and the other mental and wj-itten, a.nd it is nat- ural and convenient to distinguish them b}' names which indi- cate these distiuguishinir characieristics. School Course. — The common school course of arithmetic may be divided into two parts ; Primary Arithmetic and Ad- vanced Arithmetic. The Primary Arithmetic is designed to teach a child the elementar}^ ideas and processes of arithmetic; the Advanced Arithmetic is designed to present as full a knowledge of the science as should be taught in our public schools. In some schools a course in Higher Arithmetic may also be required, including the more abstruse principles of the science and a more extended application of them. 2f umber of Books. — If mental and written exercises are combined after the Primar}- Arithmetic, the entire course may be embraced in two books, which ma}- be called the Primary Arithmetic and the Union Arithmetic. If it is thought best to separate the mental and written exercises after the first book, we shall have three books in the course, which may be distinguished as the Primary Arithmetic, Mental Arithmetic, and Written Arithmetic. In schools of a certain grade there has been a demand for a book between Primary and Advanced Arithmetic, svhich has been met by an Elementary Arithmetic. Union Arithmetic. — Mental and written exercises should 344 METHODS OF TEACHING. be combined in the Primary Arithmetic, and many teachers advocate this union throiijjliout the entire course. The reasons are: 1. Economy of time; 2. Econom}' in the purchase of books; 3. One aids in learning the other ; 4. The sole object of Mental is to aid in the study of Written. The objections are ; 1. Their object is different ; the object of Mental is discipline in analysis, the object of Written is skill in calculation ; 2. Their spirit is diverse ; one being analytic and the other more synthetic ; 3. They cannot be properly coordinated ; 4. Hence to combine is to neglect Mental. The present desire for combination is an example of history repeating itself, as may be seen in the works of Smith, Emerson, etc. Whether this demand will be permanent, or, like a new fashion, change again in a few years, time will decide. Ejctent of Course. — Many teachers think the present com- mon school course in arithmetic too extensive, but we doubt it. The child needs a thorough drill in arithmetic for the thought- power it imparts; and for the practical value of arithmetical knowledore. Every child coming out of our public schools should have a good practical knowledge of nearly every sub- ject treated in the ordinary common school arithmetic to prepare him for the practical duties of the business world. Course in Arithmetic. — We shall discuss the subject under three heads; Primary Arithmetic, Mental Arithmetic, and Written Arithmetic. These three parts are not entirely distinct ; to some extent they run into and overlap one an- other; but a clearer idea of the principles and methods of instruction can be given by such a division. The several sul> jects embraced in the course are as follows : — 1. Ideas of Numbers; 2. Arithmetical Language ; 3. Operations of Arith- metic ; 4. Reasoning of Arithmetic ; 5. Definitions of Arith- metic ; 6. Rules of Arithmetic ; 7. Principles of Arithmetic ; 8. Applications of Arithmetic. In the next three chapters we shall consider the methods of teaching Arithmetic. CHAPTER III. TEACHING PRIMARY ARITHMETIC. niHE course in Primaiy Arithmetic should embrace the X Ideas of Numbers, Arithmetical Language, the Funda- mental Operations of Synthesis and Anah'sis, the Elements of Fractions, and the Elements of Denominate Numbers. In other words, the course should embrace the elements of Numeration and Notation, the elements of Addition, Subtrac- traction, Multiplication, and Division, the elements of Frac- tions, and the elements of Denominate Numbers. Principles of Teachivff. — In presenting these subjects to the learner, it is believed that the course of instruction should be based upon the following principles : 1. The first lessons in arithmetic should be given by means of oral exercises. Such instruction is needed for several reasons: First, pupils can learn arithmetic before they can read; and hence, of course, before they can use a book. Second, even with pupils who can read, such exercises are a very valuable preparation to the study of the subject in the text-book. Third, more thought can be developed, more in- terest awakened, and much more i-apid and thorough progress can be made with such exercises. These exercises should be continued thi-oughout the entire course in arithmetic. Every subject, even in the more advanced parts of written arith- metic, should be introduced b}^ such exercises. 2. The first lessons in Primary Arithmetic should be given by means of sensible objects. Such exercises will give distinct ideas of arithmetical quantities. Children's numerical ideas are often vague and indefinite. The names of numbers are often merely abstract terms to them. The denominations ounces, gills, pints, cords, etc., are often mere words without 15* (545) 346 METHODS OF TEACHING. any concrete meaning to them. The ideas and processes of fractions cannot be clearl}' understood by children without such illustrations. The objects used ma}' be marbles, grains of com, beans, peas, little blocks, etc. Dr. Hill says the whole science of arithmetic may be taught with a pint of beans. The most convenient object for man}' of the processes is the Numeral Frame, or Abacus. This should be in every public school, and should be in constant use. Large ones, three or four feet square, are used in the primary schools of Sweden, Germany, etc. Mauy authors give pictures of objects, marks, stars, etc., in their books; but these do not seem to be necessary, as the objects themselves are better than the pictures of objects. Besides, no pupil should begin to study arithmetic in a text -book, who needs pictures to aid in the primary operations. 3. In Primary Arithmetic^ the order of instruction is, — first the method, then the reason for it; first the mechanical part, then the rational. This is the natural order of develop- ment with children; fin^t the hoio and then the why. Methods of doing should be taught before the reasons for doing; ideas should be taught before the expression of them; operations before rules describing operations. Principles should follow problems; not precede them. Much precious time has been wasted in primary instruction hy the violation of this prin- ciple; and minds have been dwarfed by being led to incorrect habits of study and thought. 4. In Primary Arithmetic, the method of teaching should be inductive. The pupil should be led to each new idea and process by appropriate questions and illustrations. The defi- nition should be drawn from the ideas, rather than the ideas from the definition. The pupil should be led to see the prin- ciple clearly before he is required to state .it ; and rules or methods should be derived b}' inductive inferences from analytic solutions. The child should be led to see the propri- TEACHING PRIMARY ARITHMETIC. 347 ety of a new term for the expression of a new idea, when pos- sible, for he will then see the meaning of it. 5. Mental and Written Arithmetic should be united in Pri- mary Arithmetic. This is indicated by the logical relation of the subjects. As soon as the pupil can express arithmetical ideas oralh", he is ready to learn to express them in written lan2:ua6 , METHODS OF TEACHING. Such exercises should be continued day after day, in connection witb the lessons which precede and follow this lesson, until great facility ia acquired in the operations. Fourth E.cercise. — Give a pupil a number and require him to separate it into two parts; then inio three parts, etc. Thus 6=34-3 ; 6^4-)-2; 6=5+1. Also 6=2+2-h3; 6=4+1 + 1; 6=3+1+2 ; etc. Fifth Exercise. — Let the teacher write two columns of figures on the board, as indicated in the margin. Call the firet column additive, and the second subtractive. The teacher then with the + — pointer will indicate the number, the operation being indi- cated by the column. When he points to a figure in the first column, the number which it indicates will be added, but when he points to a figure in the second, the number indicated will be subtracted from the result which the pupils have pre- viously obtained. If the Arabic characters have not been given, numerical words may be written in columns, instead of the figures. These exercises may be conducted sometimes in concert and sometimes singly. While one is adding alone, let the others keep careful watch for errors ; a good degree of interest may thus be created, each pupil trying to obtain the largest sum before making a mistake. Written Addition. — While the pupils are learning to add and subtract mentally, they should also perform work on the slate and blackboard with written characters. These exer- cises should first extend as far as the elementary sums, that is, to about " 12 and 12 are 24." So far written addition and subtraction should go together with the mental exercises ; but subsequently it is more convenient to separate them, teaching first addition and then subtraction. Cases. — Written Addition should be presented in two cases ; first, where there is nothing " to carry ;" and second, where there is something to carry. In both cases, we should first require the pupils to learn to perform the operation without giving an}- reason for it. The primaiy object is to make them familiar with the mechanical operations. Subsequently they may learn to explaiu ihe woik Course of Liessons, — The course ol lesaonb iii W niiti. Addition is as follows ; TEACHING PRIMARY ARITHMETIC. 357 1. To write the elementary sums of the addition table. 2. To add single columns of numbers, in which the sum exceeds nine. 3. To add numbers of two, three, four, etc., terms, in which there la nothing to carry. 4. To add two columns in which there is something to carry, then three columns, etc. Explanation. — The explanation may be given in what is called the Simple Foi"m or in what is called the Full Form. B}' the Simple Form, the pupil will merely state the method without giving the reason for it. By the Full or Complete Form, the puj^il gives the logical solution, which states each step in the process and the reason thereof. Young pupils should use the simple form, as they are not prepared to understand and state the reasons for the various operations. Much time has been wasted, and man}- j-outhful minds injured, by the attempt to have children give logical forms of explana- tion before they were prepared for them. With more ad- vanced pupils, we should require full explanations, in concise, simple, and logical language, showing the reason for every step of the process. For the two forms, see the author's Primary and Elementary Arithmetics. The student-teacher will illustrate the subject in model lessons. Written Subtraction. — After the pupil is somewhat familiar with Written Addition, he should begin Written Subtraction. The subject should be presented under two cases: 1st, To subtract without "borrowing;" 2d, To sub- ti'act by "borrowing." The first of these cases should be taught before the pupils take the second case of addition ; the second should follow the second case of addition. The pupil should be first taught the mechanical method of doing the work, without being shown the reason for it or being re- quired to give any explanation of the process. A general idea of "borrowing" 10 from the next term of the minuend, and "carrying" 1 to the next term of the subtrahend, may be given to pupils who seem prepared to understand it ; but QO logical solution should be required at first. METHODS OF TEACHING. Course of Lessons. — The course of lessons in Written Sub- traction is as follows: 1. To write the elementary differences of tlie subtraction table. 3. To subtract numbers of two, three, etc. terms, when there ia lotliing "to borrow." 3. To subtract numbers of two terms, then of three terms, etc., when there is something "to borrow." 4. To subtract when there are two or more ciphers in the minuend. Illustration. — The method of subtraction may be illus- trated by bunches of little sticks, representing tena, hundreds, etc., and showing how one bunch of a higher denomination represents ten of a lower, and how we can " borrow" one of the higher and unite it with the lower denomination. Little heaps of pebbles, or of beans, or of grains of corn, and boxes, real or imaginary, may also be used, or pictures of them on the board representing the same. These may be of some aid to the beginner; but if the notation has been thoroughly tauirht, there will be very little need of concrete illustration. Let the student-teacher give a lesson on the subject. Explanation. — Young pupils should first be taught to do the work, and afterwards be required to explain it. The ex- planation should at first be in a simple form, merely indicating the steps of the mechanical process. Older pupils should give a full logical explanation. There are two methods of explaining the second case, known as the "method of borrow- ing" and the " method of adding ten," either of which may be employed according to the preference of the teacher. It is difficult to decide which is the simpler, though teachers generally prefer the method of "borrowing;" and when there are no ciphers in the minuend, it is probably the simpler. When there are several ciphers in the minuend, the method of "adding ten" is simpler. The latter method is much pre- ferred in practice. We should teach a pupil to subtract, in practice, by adding ten to the upper term, and adding one to the next higher term of the subtrahend, though many pupils arc practising the opposite method at the present day. TEACHING PRIMARY ARITHMETIC. 859 III. Teaching Multiplication and Division. As soon as the pupils can add and subtract a few of the smaller numbers, they should begin multiplication and division. The old way of teaching multiplication was to begin by put- ting a " table-book " in the hands of the pupils, and requiring them to commit the multiplication table. What the table meant, where it came from, what it was for, or where it was going to or going to lead them, they knew no more than they did of the origin or nature of a sunbeam. Principles of Teaching. — There is a better, easier, and more natural way; and this way we will suggest in the follow- ing general principles: 1. Maltiplication should be taught as concise addition. Thus the pupil should be taught that two 2's are 4, because 2 + 2=4; or that two 3's are 6, because 3-|-3=6, etc. Instead of the pupil's entering upon multiplication as a new and in- dependent process, he will thus see the nature of the subject and its logical evolution from a general synthesis. He will see the origin and meaning of the multiplication table ; and not regard it as a mere collection of abstract names and num- bers to be committed to memory. He w'ill be able to make the multiplication table for himself, and will see the reason for committing it to memory, that he may not have to derive the products every time he wishes to use them. 2. Division should be taught as reverse multiplication. Division can be taught in two ways ; as concise subtraction, or as reverse multiplication. That is, the elementary quotients may be obtained by a process of subtraction, or by reversing an elementary product. Thus, if we wish to show a pupil how many times 4 is contained in 12, we can subtract 4 suc- cessively from 12 three times to exhaust 12; and thus infer that 4 is contained in 12 three times. We can also derive the quotient from the consideration that, since three 4's are 12, 12 contains three 4's, or 12 contains 4 three times. 300 METHODS OF TEACHING, Both of these methods are legitimate, but the method of roverse multii)lication is preferred for two reasons. First, it is the more convenient in practice, since the elementary '/iiofients can be immediately derived from the elementary /irodi/cts. By the method of concise subtraction, we should liave to derive each elementary quotient by performing several subtractions, which would often be very tedious. Second, it avoids the necessity of committing a division table. If we derive the elementary quotients by subtraction, it would be necessary to arrange them in a table and commit them, as we do the elementarj^ products ; but if we obtain the quotients by reverse multiplication, we can derive them from the mul- ti[)lication table, and will not need any table of division. 3. Multiplication and Division should be taught sim,ul- taneously. This is suggested by the logical relation of the two subjects. The two ideas are so intimately related that one grows directly out of the other. Every s^-nthesis sug- gests naturally its opposite, an analysis. A multiplicative synthesis can hardly be made without the intimation of its opposite, a divisionative analysis. Thus, as soon as the pupil learns that /bur times five are twenty, he is prepared to see that twenty contains five, four times. The method suggested is thus founded upon and indicated by the laws of thought. It is also much more convenient to present the subject in this manner. The same fact, a product, answers a double purpose ; the additive process which determines a product, gives also the materials for a quotient. In practice we should have the pupil commit the whole of " one column" of multi- plication before we have them derive the quotients. This principle applies to mental multiplication and division, and to the simple written exercises which represent these operations. In written multiplication and division proper, it is more con- venient to teacli the processes separately. Multiplication should be taught first, and after the pupils are quite familiar with the process, they should pass to division TEACHING PRIMARY ARITHMETIC. - 8ol 3IufHplirafion Tab^e. — The Multiplication Table is a table of the elementary products. These [)roducts have to be com- mitted to memory. This is not an easy task; indeed, it is one of the most ditticult tasks with which the ^'oung i)U()il meets. It requires months and sometimes years for the child to become thoroughly familiar with it. In the early history' of arithmetic in Europe, operations were often pei'formed in such a way as to require only a portion of the multiplication table, on account of the extreme ditlicult}' of committing the products to memor}'. How shall we teach a child the table? The method of for- mer times is not easily forgotten. The book was put into the child's hand, and sometimes the birch upon his back, that the products might be put into his head. Who does not remember the toil and the trouble ; how we dreaded the result of a treacherous memory ; how we rejoiced in " five times " and •' ten times ;" how we " stuck" on ** 9 times 7," and " 7 times 8," and confounded " 1 1 times 1 1,'* and *' 11 times 12;" and how, at last, through great tribulation, we scaled the mount and stood victor of a hard-fought battle at the top? Is there a better wa^- than the old way? The Method. — First, pupils should make the multiplication table for themselves. They will then see the nature and use of such a table, and will study it with more interest and com- mit it with greater ease. Second, when thus formed, study, recitation, frequent repetition, are necessary to fix it in the memory. The pupil must repeat it over and over, and be drilled upon it until he knows it. Third, writing it frequently on the slate or blackboard, will assist the pupil in committing it to memor}-. The seeing of it will tend to fix it upon the visual memory, which is often better than attempting to fix it in the oral memory. The eye will aid the ear in making the acquisition. Fourth, reciting the table in concert will also aid in learning it. It gives animation and zest to the recitation, and deepens the impression through the increase of interest. 16 362 METHODS OF TEACHING. The duller pupils will thus learn also from the brighter pupils The frequent hearing of the names associated together will at last make a permanent connection between the factors and the [)roducts, so that as soon as we think of one, the other will occur to us. Fifth, the singing of the table to a little tune is also recommended. This has been practiced by many teachers, and with good results. It is a pleasant exercise, and the pupil is learning a lesson while amusing himself with a song. Tliere are several little tunes to which the words may be fitted ; among the best is one known as "Sparkling Water," or " Old Dan Tucker," a coarse name for a beautiful melody. Division Table Should a division table be committed to memory' ? It has been the custom of many teachers to have their pupils study and commit a table of quotients after they have committed a multiplication table. This, however, is not necessary, if division is taught as reverse multij^lication. The multiplication table gives also the quotients, the product being regarded as the dividend, and the two tactors as divisor and quotient. If, however, division be taught as concise sub- traction, it will be necessary for a i)upil to commit a division table, as he has no method of determining a quotient but bj' subtracting, or remembering it from a table. Course of Lessons. — The course of lessons in Primary Multiplication and Division is as follows: 1. Lead the pupil to a clear idea of "times," and then of a numbei taken several times. 2. Lead the piii)il to make the table of "two times," and have him coniinit it . '6. Apply the table of "two times," in solving little problems like "If one orange costs 3 cents, what will 2 oranges cost?" 4. Lead the pupil to derive quotients from "two times," and to apply tliese quotients to solving little problems. 5. Proceed in the Siime way with "three times," " four times," etc., 'ip to " twelve times," deriving the quotients from each multiplication column. 6. Drill the pupils on M'riting and reciting the multiplication table until they have committed it. TEACHING PRIMARY ARITHMETIC. 363 Model Lesston. — How many times do you recite in a day? How many times does the clock stril umber 5. — The exercises on Five are as follows : 5=l+l+etc.; 5=4-hl ; 5=3+2; 5=2+3; 5=1+4; etc. 5-1=4 ; 5—2=3 ; 5 -3=2; 5-4=1 : 5—5=0. 5=,5X1 ; 5=1X5; 5^1=5 ; 5^5=1. The Number 6. — Tlie exercises on Six are as follows : 6=l+l+etc.; 6=5+1 ; 6=1+2; 6=3+3 ; 6=2+4 ; 6=1+5; etc. 6—1=5; 6-2=4; 6—3=3; 6—4=2; 6—5=1 ; 6—6=0. 6=6X1 ; 6=1X6; 6=3X2; 6=2X3. 6h-1=6; 6h-6=1; 6h-2=3 ; 6^3=2. The Number 7.— The exercises on Seven are as follows : 7=l+l-retc.: 7=:=6+l; 7=.5+2 ; 7=4+3 ; 7=3+4; 7=2+5; etc. 7—1=6 ; 7-2=5; 7—3=4; 7—4=3 ; 7—5=2 ; 7— «=1 ; 7—7=0. 7=7X1 ; 7=1X7 ; 7-1=7 ; 7^7=1. TEACHIXG PRntARY ARITHMETIC. SGiT Similar exercises are presented on tlie numbers in regular order as far as 100 or 144. comprising the entire multiplication table with the corresponding quotients. Experienced teachere of this method suggest that beginners should complete the exercises as far as 10 the first year; as far as 20, the second year; as far as 100 the third year, etc. Some teachers combine the fractional parts of numbers with the above exercises. The nature of the Grube Method may be more clearly seen by comparing it with the Xormal Method previously given. 1. The Gi-ube Method makes number the basis of arithmetical in- struction : the Xormal Method makes ajjerations on numbers the basis. The leading object of the former is to comprehend numbers ; tlie object of the latter is to use or operate with numbere. •2. The Grube Method proceeds by analysis from a number to its parts; the Normal Method proceeds by synthesis from the ijarts of the number to the number. 3. The Grube Method requires llie learner to comprehend and operate with the four fundamental rules from the beginning; the 2s ormal Metliod teaches the two correlative processes of addition and subtraction together; and subsequently the two correlative processes of multiplication and divi.sion together. 4. The exercises of the Grube ^lethod u^ed in teaching addition, subtraction, etc.. are used in the Xormal Method after the pupil learns to add. subtract, multiply and divide. Note. — Space does not prrrnit a discussion of the relative merits of the two methods. An enthusiastic teacher who has faith in his method, will succeed in teaching the elenient^iry sums, ditfereuces, products, and quo- tient? with either. By the Xormal Method these may be taught in a little more than a year ; teachers of the other method usually devote three years to the subject. lY. Teaching Common Fractions. After the pupils are somewhat familiar with the funda- mental operations, they are ready to begin the subject of Com- mon Fractions. In teaching Fractions, the teacher should be guided by the following general principles : Principles. — 1. The first lessons in fractions should be given orally. So text-book is needed in teaching the pri- mary ideas of the subject. The teacher should drill the pupils for several daj-s before taking up the subject in the i)ook. C)(j3 METHODS OF TEACHING. 2. Mental and written e.rercines nhonld be combined in the first let^sons. The order is, first the idea, then the oral ex- pression of it, and then the written ejrjiression of it. As soon as the pupil has an idea of an operation, he should be taught to express it in written characters. The seeing of the opera- tion will help to make it clear to the understanding and fix it in the memory. 3. The elements of fractions should be taught by means of risible objects. The pupil should he led to see the fractional idea and relation in the concrete, Ijefore he is required to con- ceive it abstractly. The olvjects to be employed are apples, lines, or circles on the blackboard, etc. An arithmetical frame, with long rods cut in sections, is used in the schools of Sweden, Prussia, etc. 4. The operations in written fractions should be taught to young pupils mechanically. They sliould be drilled upon the operations until they are thoroughly familiar with them, even before they understand fully the reason for such operations. This is in accordance with the principle that, with young pu- pils, practice should precede theory. 5. The Methods or Rules in fractions should be derived by analysis and induction. Special problems should be given for solution, and the rules or methods of operation be inferred from the analysis of these problems. The principles of frac- tions should be first illustrated rather than demonstrated. These principles should be committed, and the pupil should learn to apph' them readily. Tfiinfjs to be Taufiht The several things to be taught in fractions are as follows : 1. Tl)e idea of caA fraction. 2 The fractional parts of numbers. 3. Solution of problems requiring the fractional parts of numbers. 4. The notation of fractions. 5. Analysis of concrete problems. 6. The cases of reduction of fractions, and their analysis. 7. Addition, subtraction, multiplication, and division of fractions. S. Th^ rules and the principles. TEACHING PRIMARY ARITHMETIC. o(J'J 1. Idea of a Fraction. — First, give a lesson on one-half ; then on one-third ; then on one-fourth^ etc., as tar as one-tenth. The method is shown in the following model lesson : ' Model r,f'ssi}n. — If I divide an apple into two equal parts, what is one part called ? What are two parts called? How many halves in a whole apple? What Is one-half of anything? Ans. One-half of ajiylhing is one of the two equal parts into which it may be divided. 2. Applij to Numbers. — The next step is to apply the fractional idea to finding the parts of numbers. This is a logical step in adv%ance: the first step was to get a part of a unit ; now we pass to finding parts of collections of units. To illustrate, supi)ose we wish to obtain the one-half of 6. Show the pupil that one of the two equal parts of 6 is 3, hence 3 is one half of 6. We then proceed to obtain one-half of other numbers, from 2 to 24; then get one-third and two- thirds of numbers, also one-fourth.^ two-fourths^ etc., of num- bers, etc. The next step is to find the fractional parts of numbers which do not give exact parts ; as ^ of 7, ^ of 11, etc. After the pupil has the idea, he should be required to give a simple solution of the process. Two forms of solution are suggested, one resting on multiplication, the other on division. The latter will be more convenient in practice, as in finding one-half, one-third, etc., of large numbers, we must divide them by two, three, etc. The thought is, that to find one- half of a, number, we divide the number into two equal parts. Model Lesson. — Problem. What is one-half of 6? Solution. One- half of 6 is 3, because 2 times three are 6. Sol. 2d. One-half of 6 is 3, because 6 divided by 3 is 3. Illustration. 6=34-3; hence 3 is one of th^ two equal parts of 6, and 3 is therefore one-half of 6. We should also get two, three, etc., fractional parts of numbers. Prob. What are two- thirds of 6 ? Sol. One-third of 6 is 2, and two-thirds of 6 are two times 2, which are 4 ; therefore, two-thirds of 6 are 4. Prob. What is i of 7 ? Sol. 7 equals 6+1 ; J of 6 is 3, ^ of 1 is ^, etc. 3. Concrete Problems. — The next step is to appl}^ these fractions to concrete problems. Thus, " If A has 6 apples and B has one-half as many, how many apples has B?" The pupil should be required to give a clear and simple solution. 16* 370 METHODS OF TEACHING. lllustratLon. — Prob. If A has 6 apples and B lias one-half as many, how many apples has B ? Solution. If A has 6 apples and B has one- half as many, B has one-half of 6 apples, or 3 apples. Prob. If I have 9 marbles and give 2 thirds of them away, how many will I give away? Solution. If I have 9 marbles and give 2 thirds of Iheni away, I give away 2 thirds of 9 marbles; one third of 9 marbles is 3 marbles, and 3 thirds are 2 times 3 marbles, or 6 marbles. 4. The Xotation. — The next step is to present the notation of fractious. This, in practice, may be done in connection with some of the previous exercises. The notation may be presented in two ways. The teacher may simply state the method of writing the numerator and denominator, and drill the pupils in writing until they can read and write fractions readily. By this method pupils will see no reason for the method, and, indeed, will not think to inquire after any reason. It will be purely arbitrary and conventional to them. Another method is to lead the pupil gradually to the nota- tion, somewhat as we may supix)se it was reached historically. Thus, we ma}'' write some fraction, as 3-/ourths, then abbre- viate it to 3-4ths, then still further abbreviate by omitting the ths, giving 3-4 ; then represent it by separating the 3 and 4 by an oblique line, as V4 ; ^"cl then let this line crowd the 4 down under the 3 and leave |. Or, we might have the name written under the numerator, as, __? — ; then abbreviate ' ' fourths ' it into -?_, and then as:ain into f. Let the teacher illustrate. 5. Atmlysis The next step is to apply the fractions in the analysis of two or three classes of concrete problems, as follows: 1. "If 2 apples cost 4 cents, what cost 3 apples?" 2. " What will 3 yards of ribbon cost, if | of a yard cost 8 cents?" 3. "Six is | of what number?" etc. lllustratian. — 1. Prob. If 2 apples cost 4 cents, what cost 3 apples? Sol. If 2 apples cost 4 cents, 1 apple costs i of 4 cents, which is 2 cents, and 3 apples cost 3 times 2 cents, which are 6 cents. 2. Prob. What will 3 yards of ribbon co.st, if | of a yard cost 8 cents? Sol. If f of a yard cost 8 cents, | of a yard costs i of 8 cents, or 4 cents, and f, or 1 yard, costs 3 times 4 cents, or 12 cents; and if 1 yard costs 12 cents, 3 TEACHING PRIMARY ARITHMETIC. 371 yards cost 3 times 12 cents, or 36 cents. 3. Prob. Six is f of what num ber? Sol. If 6 is | of some number, 1 of the number is ^ of 6, or 3, and |, or the number, is 3 times 3, or 9. 6. liediution of' Fractionts. — Tho next step is to present some of the simpler cuses iu the reduction of fractions. The several cases of R-eduction are: 1. A number to a fraction; 2. A fraction to a number; 3. To higher terms; 4. To lower terms; 5. Compound fractions to simple; 6. To common denominator. We should present these casies first concretely by illustration, and then require the pupils to give a simple solution of the problems. We shall illustrate with a few of the simpler cases. Illustration. — Take the problem, "How many sixths inf ?" We may illustrate this bj circles or lin-es on the board. In the first circle, we have three equal parts, and two q:' them are two-tldrds. Dividing these thirds into two equal parts, we see that we have six equal parts, and each part is one- sixth ; and we see that the two-thirds contain four sixths; hence, we see that | equal 4. The same thing is shown with the two lines, in which the distance be- tween the two parallel horizontal lines iudiciites the unit. By reversing this, we may illustrate how to reduce //•<>/« higher to lower terms. By a similar illustration we Gin show how to reduce a compound frac tion to a simple one. To illustrate, take the problem, " What is i of ^? " Divide the circle into four equal parts; each part is one-fourth. To ob- tiiin k of one-fourth, we must divide each fourth into two equal parts. Doing this, we find we have eight parts in the circle; hence each part is ()/>.c-fi^7A^/i; hence one-Imlf oi one- fourth is one-eighth. The line may also be used; the line in the margin illustmtes finding one-half oi one-third. The teacher should make constant use of these illustrations, and require the pupils also to illustrate the problems. 7. Analysis in Fractions. — The pupils should learn the anali/ses of these cases of fractions. These analyses should be simple and concise. They are designed to state the steps of the judgment in obtaining the results required. In these an- 372 METHODS OF TEACHING. alyses, the unit is made the basis of reasoning; it is a centre around which the reasoning revolves. We present the analy- sis of a few of the eases. Case I. is to reduce a number to a fraction. Prob. How many fourths in 2|? Sol. In one there are 4 fourths, and in 2 there are 2 times | or J ; and I plus f are J^. Case II. Is the reverse of this. Pkob. In V how many oneal Sol. In one there are |, hence in ^- there are as many ones as 4 is contained times in 11, which are 2|. Another solution of this second case is as follows : In one there are four fourths; hence \ of (he number of /owr^/io equals the number of onfs; } of 11 equals 2|. '1 lie first of these is much more easily understood by cliildren. Case III. is to reduce to hUjher terms. Pkub. In | how many sixths? Sol. In one there are % and in i there are \ of °, or I, and in | there are 2 times f , or I. Case IV. is the rever.se of this, to reduce to lower terms. Prob. In % how many thirds? Sol. In one there are f, in one-third there are \ of%, or two-sixths, hence in | there are as many thirds as | are contained times in ^, or |. This case also leads to the reducing to a common denominator, in which the analysis is likethatjust given. Case V. is the reducing a compound fraction to a simple one. Puob. What is I of I ? Sol; One-third is one of the three equal parts into which a unit is divided; if each third is divided into two equal parts, three thirds, or the unit, will he divide 1 into three times two, or six equal parts, and each part will be one-sixth of a unit; hence J of | is ^. Another Solution. One third equals |, and \ of | is i; hence \ of \ is ^. The first solution is a little diflScult for a beginner; but it involves precisely the mental process of obtaining i of }, and is the one which should be used when a child is ready for an analysis. The second solu- tion does not show why 4 of | is \, though it obtains the result. It may be used with the beginner, hut it should be afterward followed by the other solution. A celebrated author of Mental Arithmetic gave the fol- lowing solution : "One-half of one is \, and if i of one is i, i of | is ^ of \, which is ^." The error in this logic is that to explain what is "i of |," the author assumes he knows what "^ of ^" is, the more difficult thing of the two. 8. The Other Cases. — All the other eases of Fractions, — Addition, Subtraction, Multiplication, Division, and Relation of Fractions, — should be solved by analysis, as the pupils become able to understand them. The student-teacher should be required to show how to give the instruction. Illustrate both the inductive and deductive methods. TEACHtXG PRIMARY ARITHMETIC. 373 9. The Rules. — The pupil needs to be able to derive results without going through tlie anal3'Sis each time, and for this purpose Rules should be drawn from these analyses. These may be derived by inference or induction. Thus, in reduc- ing 2| to fourths, since in the analysis we take the product of 4 and 2 and add the 3, for the number of fourths, we may infer the rule, "To reduce a mixed number to a fraction, we multiply' the integer by the denominator of the fraction, and add the numerator to the result," etc. The other rules of fractions may be derived in the same way. Another method is to derive them from the principles of fractions. The in- ductive method is easier for learners, and is prefei-red in primary arithmetic. With beginners, however, it may be well to teach the method of doing the work, without giving any reason for it; and subsequently, when they are familiar with the rules, the}' may learn to derive them. Let the student- teacher give examples of each case of fractions, and show how to analyze the problems and derive the rules. 10. The Principles. — After pupils are somewhat familiar with these fundamental ideas and processes, they should be taught the pi'inciples of fractions. These principles may be illustrated so that the pupils may have a general idea of the manner in which they are derived, or pupils may be required to commit and apply them without any idea of how the}' are derived. A simple solution like the following may be given: " Multiply the denominator of | by 2." Sol. — If we multiply the denominator of | by 2, we have 3 eighths, which is one- half as much as 3 fourths, since eighths are only half as great as fourths; from which we infer that multiplying the denom- inator by 2 divides the fraction by 2. When the pupil is ready to understand the demonstration of the principles, a real demonstration should be given, and not some loose, in- definite statement, such as we find usually presented. For the more general demonstration, see the Treatment of Fractions in arithmetics. 374 METHODS OF TEACHING. The Student-teacher should now be required to outlice the course of instruction in Fractions in primary arithmetic, and show by model lessons how he would teach them. V. Teaching Denominate Numbers. Pi'iuciples of Insfrnction. — In giving instruction in De- nominate Numbers, teachers should be governed b}- the fol- lowing principles: 1. Denominate Ninnhers should be taught concretely. The teacher should have the actual measures to illustrate the sub- ject. If they are not in the school-room, the teacher can pro- cure them at a trifling expense. In some text-books on pri- mary arithmetic, we find pictures of the measures; but the measures Ihemselces are worth much more than the pictures of them. In fact the picturr^s give an inadequate idea of the measures, and often an incorrect one. The neglect of this principle is very common. Most teachers have the pupils re- peat the tables without any illustration of their meaning. The result is, that these "weights and measures" are to manj- pupils merely so many words without any corresponding defiuite ideas. 2. The teacher should require the pupils to make a practi- cal application of these measures. He should drill them on measuring and judging of the length of rooms, the height of ceilings, the area of surfaces, the volumes of solids or vessels, the amount of land in fields, the amount of plastering in a room, the amount of carpet required to cover a floor, etc., etc. These measures will thus become actual and practical realities, and not mereh* a lot of names to be committed to memory. Measures of 3[oney. — In teaching the measures of Money, the teacher should show the pupils the cent, dime, dollar, eagle, etc. Every school should be provided with a collection of coins to illustrate the subject. There should be specimens also of the English penny, half-penny,t,he shilling, sixpoice, florin, etc. In teaching French money, there should be TEACHING PRIMARY ARITHMETIC. 375 specimens of the franc, half-franc^ five centimes or sou. ten centimes or two sous. In teaching German money we shoultl present tathe pupil the mark, the thaler, the groschen, etc. The tables are also to be committed, written on the board, and repeated. Jleasures of Weight. — In teaching the table of Weights, the pupils should be shown the ditferent weights, — the ounct, the pound, etc., — ■ and be required to examine and handle them until they are entirely familiar with them. They should see and handle the pennyweight, the ounce, the pound, Troy; also the scruple, dram, ounce, and pound, Apothecaries. There should be a pair of scales in the school-room to weigh objects. Pupils should also be required to " heft" diflerent objects, as a book, a chair, etc., to learn to judge of the weight of objects. The tables of weight should be studied and committed. We should also have specimens of the gram, decagram, and kilogram. Measures of Length. — The teacher should give the pupil definite ideas of all the measures of length. There should be the foot and yard rules, divided into inches, half-inches, etc.. Have the length of a rod marked on the wall or floor, show the pupils the distance of a mile, a half-mile, etc. Have a meter properly divided, show its relation to the 3'ard, and give definite ideas of the decimeter, centimeter, etc. Pupils should also be drilled in estimating the length of objects, dis- tances, heights of ceilings, of trees, etc. MeasHves of Surface. — The teacher should mark on the board a square inch, square foot, and square yard, to show what is meant by these surfaces, and also to give definite ideas of them. Show also the reason why 9 sq. ft. make one square yard, and 144 sq. in. equal a square foot. Measure o f a square rod out in the field, and also an acre, and have pupils judge of the number of acres in a field. Have pupils remem- ber that a square about 209 feet, or 70 paces on a side, is an acre. Teach the surfaces in tlie metric system in the same 376 METHODS OF TEACHING, way. Have the pupils study and recite the table of square measure. Measures of Volume The teacher should show the pupils a cubic inch and a cubic foot. ^ He should draw them and also the cubic yard upon the blackboard. He should also, as clearl}- as possible, show the relation of them, — that is, that 27 cu. ft. equal a cubic yard^ and 1728 cu. in. make a cubic foot — by a figure on the board, or by blocks prepared for the purpose. Give them an idea of a cord by taking a lot of little sticks 4 inches long, and making a pile 8 inches long and 4 inches high, and show them that a cord contains 128 cu. ft. To give them an idea of a cord foot^ measure off 1 inch of the little cord, and run a thin stick or a piece of wire down, cutting off a part of the pile 1 inch long, which will represent a cord foot; they will thus see that 8 cord feet make a cord. Liquid Measure. — In teaching Liquid Measure, have the measures in the school-room, — the gill^ the pitit, the quart^ and the gallon. Show them by actual trial that 4 gills will fill a pint, 2 pints a quart, etc. Barrels and hogsheads can be seen at a store. We should also have samples of the Apothe- caries' liquid measures in the school, — the miiiim^fluidrachms, fluidounces, etc, Dry Measure. — In teaching Dry Measure, the pint and quart at least should be in the school-room. Have the pupils call at the grocer's to see the peck and bushel, or examine these measures at home, if their parents have them. They should also be led to compare the liquid quart and dry quart, etc. The metric system of measures should also be in the public school, and the pupils be drilled on them. Measures of Time. — Time will be quite easily taught, aa il:e measures are in such constant use. We should begin with the day as the most natural unit, and pass to the other 11 easures. We should explain how nature fixes the day, and n onth, and year, and give the meaning of these terms. By TEACHING PRIMARY ARITHMETIC. 377 means of a clock, we can teach the number of hours in a day, the number of minutes in an hour, and of seconds in a minute. The number of days in each month is best taught by the stanza, " Thirty days hath September," etc. This can also be remembered by the hand, the fingers representing January-, March, May, etc., and the spaces between them representing Februar}^ April, June, etc. We should also show them that tiie calendar begins one day later each year, and two days later after a leap year, and explain the reason for it. When they are prepared to understand it, we can explain the reason for leap year, etc. Circular Measure. — In teaching Circular Measure, draw a circle on the board, and teach the different parts — circum- ference, semi-circumference, quadrant, arc, etc. Then explain the division into 360 equal parts, each called a degree; that the semi-circumference contains 180°, and the quadrant 90°. Then show that the degrees are divided into 60 equal parts called minutes, and the minutes into 60 equal parts called seconds. Show that all these are parts of the circumference, that they are not of a fixed length, but differ in size with different circles. Call attention also to the difference between minutes and seconds of circular measure and of time measure. A drill like this in Denominate Numbers will give the pupils definite ideas of what they are committing, and will make these tables a reality to them, and not a mere collection of abstract names. It will make them interesting to pupils and much more easily' remembered than when taught in the usual abstract method of our schools. When the classes are more advanced, the many interesting facts concerning the tables — the origin of their names, of their units, etc. — may be presented. CHAPTER IV. TEACHING MENTAL ARITHMETIC. AFTER completing the course in Primiirv Arithmetic, tin pupil may take a complete course of Mental Aritiinieti'- in one book, and a complete course of Written Arithmetic in another book; or these two courses ma}' be combined in one book, as the teacher prefers. In this chapter we shall speak of the Importance of Mental Arithmetic, its Nature, and the Methods of Teaching it. I. Importance of Mental Arithmetic. — Mental Arithmetic has become one of the most popular studies of tlie public school; in many places it has been the idol of the school- room around which have centered the affections of teachers, pupils, and parents. This preference is not a mere whim, but is founded on the intiinsic value of the subject, wliich we shall briefly consider. The value of Mental Arithmetic is two-fold; first, as a mental discipline, and second, as a means of cultivating arithmetical power. Mental Discipline. — The science of numbers before the introduction of Mental Arithmetic, was far less useful as an educational agenc}' than it should have l)een. Consisting mainly of rules and methods of operations, without leading the pupil to see the reasons for these operations, it failed to give that high degree of mental discipline which, when prop- erly taught, it is so well adapted to atford. By the introduc- tion of Mental Arithmetic a great change has been wrought in this respect; the spirit of analysis has entered into tlu; science; and now the science of numbers presents one of 1 he- best, if not the very best, means of discipline in the curricu- lum of the common school. . 1. Mental Arithmetic given culture to the reasuiiing facul- (318) TEACHIXG MENTAL ARITHMETIC. 879 ties. No study in the scliool equals, surelj- none surpasses, Mental Arithmetic in giving exercise and development to the power of reasoning. It is a S3'stem of practical logic ; all its processes are in accordance with the laws of thought ; every step is a judgment direct or indirect; and the entire subject is permeated with the principles of logic. Its processes are analytic, and it thus trains the mind to the most rigid and severe analysis. Every truth is bound to every otlier truth b}^ the thread of related thought; and the mind of the pupil becomes habituated to following a chain of logically connected judgments, until it reaches a desired conclusion. It is thus clear that Mental Arithmetic must be very valuable in sivincr culture to the power of thought. 2. Mental AiHthmetic cultivates the power of attention. When properly taught, no study compares with Mental Arithmetic in this respect. The problem, as read \>y the teacher, must be repeated by the pupil, each number is to be remembered in its proper place, and each condition properly related; and this can be done only by the most careful atten- tion. Pupils trained in this way acquire the abilit}' to repeat long and complicated problems with ease and accuracy. Such discipline enables them to fix their minds upon a discourse and reproduce much of what they hear. 3. Mental A rithmetic gives culture to the memory. Memory dejJeiKls upon the power of attention: we remember that which we fix in the mind by close attention ; we forget that to which we are inattentive. Few persons, after hearing a sermon or discourse, can tell 3'ou anything definite concerning it, because they are careless and inattentive listeners. Any- thing that trains the mind to habits of close attention tends to give strength and reliability to the memory. Mental Arithmetic, therefore, in its discipline of the attention, is an important means of training the memory to habits of readi- ness and accuracy. 4. Mental Arithmetic cultivates exactness of language. It 380 METHODS OF TEACHING. is SO rigidly exact in its processes of thought that it requires corresponding exactness in its language. The right word must be used in the right place, or the reasoning will be at fault. The language of Mental Arithmetic is simple, clear, and precise; and the mind, becoming habituated to such forms of expression, will naturally incline to use them in the con- sideration of subjects not mathematical. 5. Mental Arithmetic sharpens and strengthens the mind in general. The s^-stcm of rigid analysis gives point and penetrating power to the mind, and enables a person to pierce a subject to its core and discern its elements. In this respect, Mental Arithmetic is a sort of mental whetstone which gives edo-e and keenness to the mind. Old Robert Recorde called his work on arithmetic the "Whetstone of Witte;" had he lived until the era of Mental Arithmetic, he would have seen the full meaning of his words, for mental arithmetic is indeed a whetstone of wit, a sharpener of the mental faculties. It also strengthens the mind as well as sharpens it. The mind, like a muscle, grows tough by hard work; we toil foi strength in study, as we do upon the playground or in the o-ymnasium. Mental Arithmetic is a mental gymnastics ; through it the mind grows strong and tough, taking hold of difficulties with a will, laughing at obstacles, and rejoicing in the investigation of the intricate and profound. 6. Mental Arithmetic prepares a pupil for extemporaneous speaking. In solving a problem the pupil must stand up be- fore his class, hold the conditions of his problem clearly in his mind, and proceed to develop the matter under consideration in logical forms of thought and expression. This is precisely the discipline needed to make a good extempore speaker. It also tends to correct the habit which many speakers have of talking without saying much. The good speaker is one who utters thought, and not words merely ; and the study of men- tal arithmetic tends to cultivate speakers who think and uttex thought. TEACHING MENTAL ARITHMETIC. 381 Arithmetical Power. — The influence of Mental Arithmetic has been no less marked upon the science of arithmetic itself. Consisting heretofore of mechanical methods for finding re- sults, it was drj^, uninteresting, and difficult. Few pupils attained any excellence in it ; and many acquired a positive distaste for the subject. But these things have passed away; a new era has dawned upon the science of numbers ; a " royal road" to arithmetic has been found ; and it has been so graded and strewn with tlie flowers of reason and philosophj' that it is now full of interest and pleasure to the youthful learner. The agent that has produced this change is the method of analysis which we know as Mental Arithmetic. 1. The study of Mental Arithmetic gives the pupil the power of independent thought in arithmetic. The spirit of mental arithmetic is analysis. It is not merel}^ oral arithmetic ; it is analj'tical arithmetic ; and in this consists its power. Bj^ it pupils become able to investigate for themselves, and are no longer bound down to the dictation of rules. " The rule says so," is no longer the touchstone of the science or the key to the result ; but a careful comparison of the conditions of the problem will enable the pupil to make his own method and derive his own rule. By it he becomes, not a mere arithmet- ical machine, h\it an original thinker, understanding what he does, and prepared to make new investigations and new dis- coveries in the science. If we were obliged to choose be- tween a course in mental and one in written arithmetic, we should take a complete course in mental in connection with the fundamental rules of written arithmetic ; and we would turn out better-trained thinkers in arithmetic than if we had drilled them in the usual course of written aritlimetic. 2. The study of Mental Arithmetic is an excellent prepara- tion for Algebra. Arithmetic and Algebra are intimately related, algebra being a kind of general or s^-mbolic arithme- tic. The analysis of mental arithmetic is especially similar to the elementary reasoning of algebra, the main ditference be- 382 METHODS OF TEACHIN-Q. ing that the latter employs s^'mbols which render it more concise and general. The one insensibly glides into the other by the snbstitution of a symbol for a word ; and it is thus evident that the study of mental arithmetic is a most valuable preparation for the study of algebra. Its Great Value No words can convey a full appreciation of the importance of mental arithmetic. Onl}' those who experienced the transition from the old methods to the new, can full}' realize the supreme value of the study. Indeed, we believe that the method of mental arithmetic is the great- est improvement in modern education ; and the world owes a debt of gratitude to Warren Col burn, its author, which it can never pay. Though there has been a recent reaction in public sentiment against the subject, we believe that it is merely a wave of opinion and cannot be permanent. Mental ai-ithmetic is the great source of discipline to the power of thought in our public schools. When properly taught, it gives quickness of perception, keenness of insight, toughness of mental fibre, and an intellectual power and grasp that can be acquired by no other primary study. To omit, there- fore, a thorough course in mental arithmetic in the com- mon schools, is to deprive the pupils of one of the principal sources of thought power. II. Nature of Mental Arithmetic. — In order to teach Mental Arithmetic properly, or to appreciate its value as an educational agency, its nature should be clearly understood. It is a popular view that mental arithmetic is merely the working of problems in the mind, and this is the opinion of many who oppose it as a distinct study; but this is a mistake, and one that should be corrected. The genius of mental arithmetic is not merel}'^ the " working of problems in the head," but the analytic and inductive treatment of the science of numbers. We shall attempt in a few words to explain its nature. General Nature. — A system of Mental Arithmetic is de- • TEACHIXG MENTAL ARITHMETIC. 383 veloped upon the principles of Analysis and Induction. The reasoning j)rocesses are purely analytical, not demonstrative ; and the methods of operation should be derived from these analyses by inference or induction. Each problem is resolved into its simple elements, and the relation of the elements, lead- ing to the desired result, determined by comparison. When we wish to derive rules to apply to other problems of the same class, we notice the process generated by the analj'sis, and generalize this process into a rule. This brief statement shows the philosophy upon which a system of mental arithmetic is founded. It is purel}' analytic and inductive ; and not synthetic and deductive, like written arithmetic. Anal^^sis determines the process in fxny particu- lar case, and Induction derives the method that applies to all problems of the same class. Analysis and Induction are the golden ke^'s which unlock the various complex com- binations of numl)ers ; they are the magic wands Avhose touch unfolds tiie m3-sterious and beautiful combinations of numbers. Analysis. — Arithmetical analysis assumes the Unit to bo the fundamental idea of arithmetic, and comprehends all uum bers and their relations through their relation to the unit. It compares numbers and the effects produced b}'^ a number of equal causes through their relation to the unit or the effect of a single cause. It comprehends a fraction by a clear apprehension of the relation of the fractional unit to the inte- gral unit; and thus develops the principles and methods of fractions. In this manner the whole science is evolved, pre- senting one of the most beautiful examples of pure logic that can be found in any science. The simplicity and beauty of this process is seen in the fact that the unit is the fundamental idea of arithmetic. Arith- metic begins with the unit ; all numbers arise from a repeti- tion of the unit ; fractions have their origin in the division of tlie unit. Hence, in the comparison of numbers the unit nat- 884 METHODS OF TEACHING. urally becomes the basis of the reasoning process. We reason to the unit, from the unit, and through the unit. The unit is the foundation upon which we build ; it is the stepping-stone in the transition of thought ; it is the centre around which the process of reasoning revolves. In it we have an illustration of the general principle that the One lies at the basis of all tilings. All science is a striving after the One which con- tains the All ; the Cause which contains the phenomena, the Law which contains the facts, the one principle that binds all variety into unit}'. Comparing Integers. — In applying this analysis to num- bers, we have three cases: First, where we pass from the unit to a number; second, where we pass from a number to the unit; and third, where we pass from a number to a number. In the first and second cases, the transition is immediately made, since the relation is immediately apprehended, being given in the genesis of numbers. In the other case, the com- parison is not immediately seen ; it must therefore be made by the intermediate comparison of each to the unit. That is, in passing from a collection to a collection, or from one num- ber to another, we first pass to the unit and then yrom the unit. Thus, take the problem, " If 4 apples cost 12 cents, what will 5 apples cost?" Here the cost of 4 apples is the known quantity, the cost of 5 apples is the unknown quantity ; the object is to determine the unknown by comparing it with the known. This compai-ison cannot be made immediately, since the mind does not readily perceive the relation between fve and four; we therefore pass from /our to one, and then from one to five. Thus the analysis is : "If /our apples cost 12 cents, one apple costs \ of 12 cents, or 3 cents; and if one apple costs 3 cents, /ue apples cost 5 times 3 cents, or 15 cents." This pi-oblem also illustrates the first and second cases of comparison. Comparing Fractions. — With Fractions the same law TEACHING MENTAL ARITHMETIC. 386 holds as with Integers, though the existence of fxoo units, the integral unit and the fractional unit, somewhat compli- cates the process. There are three distinct cases as in integers: (1) yia.-ismgfvoma.n integer to a fraction; (2) passing from a. fraction to an integer; (3) from a fraction to a frac- tion. In the first case, we pass to the unit, then to the fractional unit, and then to the collection of fractional units. In the second case we pass to the fractional unit, then to the integral unit, and then to the collection of integral units. In the third case we pass to the fractional unit, then to the integral unit, then to the other fractional unit, then to the c:)llection of fractional units. We give a problem of the third class, which includes also what in both of the others differs from the case of integfers. Take the problem, "If f of a j'ard of cloth cost 8 cents, what will I of a j-ard cost?" The solution is as follows: "If 2 tliirds of a yard cost 8 cents, one-third of a yard costs ^ of 8 cents or 4 cents, and three-thirds, or one yard, cost 3 times 4 cents, or 12 cents; if one yard cost 12 cents, one-fourth of a yard cost | of 12 cents, or 3 cents, and three-fourths of a 3'ard cost 3 times 3 cents, or 9 cents." Here the ol)iect is to compare | with §, which we do by the intermediate relations of the units. It is as if one stood at A and wished to . A pass to E. The mind cannot step di- zFL j-T L. t-Ha/' rectlv over from A to E, so it first .s ]-J- — i steps two steps down to B, then three steps up to C, then /our steps down to D, then three steps up to E. Application of Analysis. — These analyses represent the spirit of Mental Arithmetic. Such processes of reasoning run through the entire science. The subject of Fractions, present- ing many interesting cases, is beautifully unfolded by it. It can also be api^lied to prolilems in Simple and Compoimd Proportion Partitive Proportion, Medial Proportion, etc., 17 386 METHODS OF TEACHING. giving simple and elegant solutions. The subject of Percent- age and Interest is also developed by analysis with great simplicity and elegance. Induction — The office of Induction in Mental Arithmetic is to derive methods of operations or rules from the analyses. The object of these methods is to enable us to reach the result directly by a mech8.nical operation, instead of going through the process of analysis ever}' time we need a result. Thus, suppose we wish to find a method of reducing fractions ^o lower terms; by analysis we reduce some fraction to lower terms, as yV equals §; and then, by examining the process or b}' comparing the two fractions, we can derive the rule for reducing a fraction to lower terms. The same thing can be done for all the many cases which arise in fractions. Such inferences are necessary in Mental Arithmetic if we would attain any methods of operation, indei)endent of the analyses. In Written Arithmetic these rules ma}' be derived by demonstration ; but no demonstration is appropriate to the spirit of Mental Arithmetic. To introduce demonstration in Mt'utal Arithmetic would destro}' or mar its anal3'tic spirit, which is the distinctive characteristic of the branch. B}' the use of induction, the analytical spirit of the science is preserved, while it becomes practical in its methods and con- cise in its operations. III. Methods of Teaching Mental Arithmetic. — The course in Mental Arithmetic is so definitely laid down in our text-books, and the methods of instruction so clearly indicated, that but little need be said with respect to methods of teach ing the subject. Only a few suggestions will be presented. Pupils' Preparation. — Pupils in preparing their lessons should be careful to go through the form of analysis, making the clear expression of the reasoning the test of their knowl- edge of the lesson. To perform the mechanical operations necessary to attain the results is not sufficient. They may aid themselves, however, with pencil by writing out the solu- TEACHING MENTAL ARITHMETIC. 387 tion, where it is long and complicated. The reducing of the solution to writing requires exactness of thought, and tlie seeing of the analysis will aid in fixing it in the under- standing. Pupils should be especially careful to depend upon them- selves in solving the problems. The habit of a few pupils in the class working out the more difficult problems for the others, deprives the pupils assisted of the principal benefit of the study. A pupil should never be allowed to take the solu- tion of another pupil or of the teacher and commit it to memory. It is better not to know how to solve a problem than to solve it with the memor}'. The Recitation. — At the recitation, the teacher should read the problem and require the pupil to arise, repeat it, and give the solution. The pupil should not be allowed to use the book during recitation. The practice of some teachers of allowing the pupils to read the problems and solve them from the book is a needless and a pernicious one. The book is not needed in recitation by the pupils ; a very little practice will enable them to reproduce long problems and hold the condi- tions in the mind with entire ease. More than half the benefit of the study is lost when the pupils solve with the book in their hands. Pupils may be required to wi'ite out their solutions on paper or on the blackboard. This is especially convenient when the class is large, some being busy writing out the solutions, while others are reciting orally. The solutions as written should be not merely the operations, as in written arithmetic, but a complete anal^'sis of the problem. Where the solution makes equational thought prominent, the form of writing may approximate that used in algebra. Great care should be taken that the language of the solu- tion be concise and accurate. The pupil should be required to say just what he means. The teacher should not accept his " 0, that's what I meant," when he said something quite 388 METHODS OF TEACHING. different. The singular and plural should be used as accu- rately as they can be in the language of arithmetic. We should insist also upon a uniformity of tenses in a solution, for pupils incline to get their tenses very much mixed in their forms of statement. Methods of Hecitdtion. — There are several different meth- ods of recitation in mental arithmetic, which wc shall name and describe. Some of these are preferred to others, but all may be used occasionally with advantage. Common Method. — By tliis method the problems are assigned promiscuousl}-, the pupils not being permitted to use the book during recitation, nor retain the conditions of the problems by means of pencil and paper, as is sometimes done. The pupil selected by the teacher arises, repeats the problem, and gives the solution, at the close of which the mis- takes that may have been made should be corrected by the class and the teacher. Silent Mel/iod. — By this method the teacher reads a prob- lem to the class, and then the pupils silently solve it, indicating the completion of the solution by the upraised hand. After the whole class, or nearly the whole class, have finished the solution, the teacher calls upon some member, who arises, re- peats the problem, and gives the solution, as in the former method. In this method the whole class solves every problem, thus securing more discipline than b}^ the preceding method. It, however, requires more time than the former method ; hence, not so many problems can be solved at a recitation. We prefer the first method for advanced pupils, and the second, at least a portion of the time, with younger pupils. It may also be used now and then for variety. Chance Assignment. — This method differs from the first only in the assignment of the problems. The teacher marks the number of the lesson and the number of the problem upon small pieces of paper, which the pupils take out of a box passed TEACHING MENTAL ARITHMETIC. 389 around by the teacher or some member of the class. The teacher, then, after reading a problem, instead of calling upon a pupil, merely gives the number of the problem, the person having the number, arising, repeating, and solving it. By this method the teacher is relieved of all responsibility with reference to the hard and easy problems ; and it is also be- lieved that better attention is secured with it. It is particu- larly adapted to reviews and public examinations. Double Assignment. — By this method the pupil who receives the problem from the teacher arises, repeats it, and then as- signs it to some other pupil to solve. It may be combined with either the first or second methods. The objects of this method are variety and interest. Method by Parts. — By this method, different parts of the same problem are solved by different pupils. The teacher reads the problem and assigns it to a pupil ; and after he has given a portion of the solution, another is called upon, who takes up the solution at the point where the first stops ; the second is succeeded in like manner by a third ; and so on until the solution is completed. The object of this method is to secure the attentiou of the whole class, which it does very effectually. It is particularly suited to a large class consist- ing of young pupils. Unnamed Method. — By this method the teacher reads and assigns several problems to different members of the class be- fore requiring any solutions, after which those who have re- ceived problems are called upon in the order of assignment for their solutions. There are several advantages of this method. First, the pupil having some time to think of the problem, is enabled to give the solution with more promptness and accuracy ; and, second, the necessity of retaining the numbers and their relations in the mind for several minutes affords a good discipline to the memory. In regard to these methods, the first, second, and third are probably the best for the usual recitations ; but the other 390 METHODS OF TEACHING. methods can be emploj-ed very profitably with younger classes, or, in fact, with any class, to relieve monotony and awaken interest. With advanced pupils we prefer the first method, or the first combined with the third. Errors to be Avoided. — There is a large number of errors to which pupils in every section of the country are liable, a few of which we shall mention. There are many words which pupils in their haste mispronounce, and also many com- binations, which by a careless enunciation make ridiculous sense, or nonsense. We call the attention to a few of them, suggesting to the teacher to correct these and others he may notice. ''And'' is often called " an;'' ''for" is called "fur;" "of" is pronounced as if the o was omitted ; words commencing with wh, as when, which, where, etc., are pronounced as if spelled "wen," "wich," "were," etc. "Gave him" is called "gavim;" "did he" is called "diddy;" "had he" is called "haddy;" "give him" is called "givim;" "give her" is called "giver ;" "ivhich is" is often changed into " witches ;" and "how many" is fre- quently transformed into " hominy." " How many did each earn" is often rendered " hominy did e churn." A very common error, and one exceedingly ditficult to cor- rect, is the improper use of the and are ; as in the following solution : " If 2 apples cost 6 cents, one apple will cost the i of 6 cents, which are 3 cents." Here "the" is superfluous, and " are" is ungrammatical. Pupils are so determined upon tlie use of "the" that we suggest the placing of a " big the" upon the board, and allowing the class to point to it every time the mistake occurs. The following is a frequent error: "If one apple cost 3 cents, for 12 cents you can buy as many apples as 3 is contained in 12, which are 4 times." The objections are, first, 3 is not con- tained any apples in 12 ; secondly, the result obtained is times, when it should be apples, or a number which applies to both times and apples. The solution should be, " You can buy as i TEACHING MENTAL ARITHMETIC. 391 many apples for 12 cents as 3 is contained times in 12, which are 4," With regard to is and are, it is not easy to determine which should be used in some cases in Arithmetic. It may be that it would be better to use the singular form always, whether the subject is an abstract or a concrete number ; thus, 8 is 2 times 4, and 8 apples is 2 times 4 apples. But since custom sanctions the use of" are" with a concrete number as a sub- ject, it is necessary to adhere to that form. There is some authority for using "is" in the " Multiplication Table," and it would be at least convenient if the singular form were uni- versally adopted. Pupils have some difficulty in knowing how to read such expressions as $|. They object to saying " | dollars,'''' since there are not enouter we shall speak of the Nature of the Course, and the Methods of Teaching the subject. I. Nature of Written Arithmetic. — Written Arithmetic differs from Mental Arithmetic in several respects. The ob- ject of Mental Arithmetic is the analysis of numbers; the object of Written Arithmetic is tho attainment of skill in cal- culation. Written Arithmetic is a oalculus, and the primary object is to learn to work with the Arabic system. The second object is the attainment of practical methods of operation, and the acquisition of readiness and accuracy in the use of these methods. Method of Trenttnent. — The method of treatment in Writ- ten Arithmetic should be more deductive than that of Mental Arithmetic. The definitions which in Primary Arithmetic and Mental Arithmetic are given in the inductive form, should here be presented deductively. In the previous course the rules should be derived by induction from the analyses ; but in Written Arithmetic the deductiA'e method must also be emplo^'ed. Here many things are to be demonstrated, and demonstration is a deductive form of reasoning. While analysis and induction are often used, yet the spirit of the science is deductive and demonstrative rather than anal3'tic and inductive. Arrange tnent. — The arrangement of the subjects in the (392) TEACHING WRITTEN ARITHMETIC. 3f 3 text-book used should be both scientific and practical. By a scientific arrangement is meant such an order as the logical de- velopment of the subject suggests. By a practical arrange- ment is meant such an order as is best adapted to the wants of pupils in pursuing the study. A merely scientific arrange- ment, however satisfactory to the accomplished arithmetician, would not be sufficiently progressive to meet tho purpose of instruction. A merely practical adaptation of the easy and difficult parts to suit the young learner, mighl completely Ignore the logical relations of the science, and thus fail to give that mental discipline which the logical evolution of truth imparts. These two methods should run together; the work should be practically adapted to instruction, and at the same time the philosophical spirit of the science should be preserved. The Gradation, — The course in Written Arithmetic should be carefully adapted to the ditferent classes of pupils who use it. It should be simple enough for 30ung pupils, and yet sufficiently advanced for those of more mature minds. This adaptation may be accomplished in two or three different ways. The first part of the work should be very simple, the difficulties gradually increasing as the pupil acquires strength and culture. The teacher ma}' omit certain subjects with elementary classes until review, or until the pupil is pre- pared for them. Thus the more difficult matter will be left for the pupil until he will have become somewhat familiar with the easier principles and rules, and will have gained mental strength to cope with the greater difficulties. Another object gained by this plan, is the interest that new matter gives to a review. The Reasoning. — In Written Arithmetic, as previously stated, the methods of reasoning are more S3'nthetic and de- monstrative than in Mental Arithmetic. Thus, many subjects which in Mental Arithmetic we treat analj'tically, in Writ- ten Arithmetic we should treat by demonstration; as may be Been in Fractions, Percentage, etc. Besides this, there are 17- 391 METHODS OF TEACHING. many subjects in Written Arithmetic which are purely deduc- tive and demonstrative in their nature; as Proportion, Pro- gressions, Evolution, etc. Hence, the pupil will be required to learn demonstrative reasoning as well as arithmetical analysis. The Princijiles. — Arithmetic as a science involves, and as an art is based upon, certain principles; and the most import- ant of these should be distinctly stated and clearly demon- strated. The form of statement should be deductive; nnd, when not too difficult, the method of demonstration should be deductive also. In other cases the truth may be shown inductively, suggesting to tlie pupil, however, that it is sus- ceptible of rigid deductive demonstration. Where the principles are essential to the development of a subject, the}' should be given at the beginning of the treat- ment of it ; in other cases, they may be stated at the close of the subject. Thus, in Least Common Multiple, Greatest Com- mon Divisor, Common Fractions, Proportion, etc., the princi- ples are given first, and the development based upon them; in the Fundamental Rules, etc., a knowledge of some of the prin- ciples not being essential to the development of the subjects themselves, may be given after them. The importance of principles in written arithmetic should not be overlooked. Until within a few years, American text- books and American instruction almost completely ignored the principles of the science, making arithmetic to consist entirely in the solution of problems. This is a great error, and one most pernicious in mental discipline. Especially is attention to principles important in Normal instruction, where the pupil expects to teach others. No matter how hard a problem he can soh e, if he cannot give neat and clear expla- nations, he is unfit to be an instructor of others. It should be rememben^d also that a clear knowledge of the principle makes a problem, otherwise difficult, comparativeh' eas}'. fhe Problems. — Problems are of two kinds, abstract and TEACHING WRITTEN ARITHMETIC. 395 concrete. Abstract problems are designed to illustrate the principle, or fix the rule in the mind. They serve to make pupils read}' and accurate in the mechanical operations. Such problems should be suited to the rule the}' illustrate and the capacit}' of the pupil, being simple at first, and graduall}' in- creasing in difficult}'. Concrete problems are the api)lication of the abstract principles to something that either does or may exist in actual life. These problems should also be adapted to the subject and the capacity of the learner. Simple at first, they should be gradually complicated until the pupil needs to think closely to unravel the complication and attain the result. Number of Problems. — There should be a large number of problems in the course in Written Arithmetic. Principles and methods are fixed in the mind by their application, and prob- lems are intended for such application. In this respect there is a great difference between the French and English works. The French have many i)rinciples and few problems; the P]ng- lish fewer principles and more problems. The true method is principles and problems, enough of the former, the more the better of the latter. Especially should there be a large col- lection of problems under the fundamental rules, as the first object in the study of arithmetic is to acquire skill in the mechanical processes of adding, multiplying, etc. Variety of Problems. — Problems should be so varied that the solution of one cannot be directly and mechanically applied to all the others of the same class. This is an impor- tant i)oint. Many teachers who condemn the faults of the old schoolmasters in working everything by rule, fall into a simi- lar error by requiring pupils to solve everything by " model solufioyis.^" To give a pupil a solution of one of a class of problems, and then have him apply it to a dozen others of the same class, without any A'ariation or new complication of the conditions, so as to require original thought on the part of the pupil, is not much better than to solve by the old 396 METHODS OF TEACHING. method of " /Ae rule says so.^^ Problems should, therefore, be varied so as to give the pupil opportunity for original thought and investigation, that he may become an independ- ent reasoner and not a mental parrot. Practical Character. — The practical character of the prob- lems should be a prominent feature of them. They should represent the actual business of the day, and not the scholar's idea of what business might be. The problems and proces^'^es should be derived froii. actual business transactions, and the teacher should endeavor to make this one of the leading char- acteristics of his instruction. Solutions and Demonstrafions. — The solutions and demon- strations should be simple and clear, that they may be readily understood, but at the same time concise and logically accu- rate. A solution may be too concise to be readily inider- stood ; and it may also be too prolix, the idea being smoth- ered or concealed in a multiplicity of words. Both of these errors should be avoided. There is a language of arithmetical science, simple, clear, and concise, as appropriate to the sci- ence of numl)ers as the language of geometry is to the science of form. This language is the natural ex])ression of the logi- cal evolution of the subject, and should be employed even in the most elementary ])rocesses of arithmetic. The teacher should always remember that the highest science is the greatest simjilicity. The Rules. — The rules of arithmetic are statements of the methods of operation. These rules should be expressed in brief and simple language, and in a form easily understood by the learner. The statement should not be too general in its terms, but should indicate each step in its natural order. In most cases the rule should be derived from the solution of a problem, that the pupil may see the reason for it, and be able to dei'ive it himself, as an inference from the solution. In some cases it is more convenient to state the rule first and then de- monstrate it; and this should be done wherever it is seen to TEACinXG WRITTEN ARITHMETIC. 397 be preferable. Youii^' pupils should not be required to commit the rule to memory, ])ut they should be thoroughly drilled upon the methods of openition. Older pupils should be required to describe the methods of operation, and the study of the rules will aid them in doing this. Definitions. — The definitions should be clear, concise, and accurate. There are two methods of givinsr definitions, which are distinguished as the Inductive and Deductive methods. By the Inductive method we pass from the idea to the word ; by the Deductive method we pass from the word to the idea. Thus, by the Inductive method we would say, "The process of finding the sum of two or more numbers is Addition;" by the Deductive method we would say, " Addition is the process of finding the sum of two or more numbers." In the course in Priraaiy and Mental Arithmetic, the Inductive method is preferred; in the Written Arithmetic, the Deductive method should be used. Answers. — The question is often raised whether a text- book on Written Arithmetic should contain the answers to the problems. We believe that most of the problems should have no answers given in the text-book. In case of any pecu- liarity in a problem by which pupils would be liable to obtain an incorrect result, the correct answer should be given ; in other cases it would l>e better to omit them. In practical life, our problems are without answers ; we must determine the correct results for ourselves. Education should be disciplin- ary for life, hence the pupil should learn to rely upon himself in studying his text-book. We have no answers in Mental Arithmetic, and get along well without them; could we not do as well without them in Written Arithmetic? These views, however, confiict with the popular view and |)ractice. Nearly all teaeluM's prefer having answers to the problems in the text-book ; and with elementary classes they may be of some practical advantage, to both pupil and teacher. Tliere are some teachers, however, who will not use an arith- 398 METHODS OF TEACHING. meticwitli answers; and several authors publish two editions of their works, one with and the other without answers, so as to meet the wants of all in this respect, II. Methods of Teaching Written Arithmetic. As con- ditions for thorough instruction in Written Arithmetic, each pupil should be provided with an arithmetic, slate, and pencil. In latter times book-slates and scribbling paper have in many places superseded slates, and are in some respects preferal)le to the old-fashioned slate. The school-room should also l;e furnished with a blackboard of suitable size and quality. Tlie necessity of a blackboard in the school-room is imperative. No good teaching can be done without it, especially in mathematics. Assif/iiiiient of the Lesson. — The lesson should be assigned at the close of each recitation, that the pupil may have time to prepare it for the next recitation. In assigning the lesson the teacher should be definite as to place and extent, stating just where a lesson begins and where it ends, so that there can be no doubt about it by the pupil. The extent of the les- son should be adapted to the ability of the class, care being taken that neither too much nor too little be assigned. Tiiis point is important, for if too little be given, the pupils become lazy; if too much, they will become discouraged and disgusted with the study. Attention should also be called, to' prominent points or unusual difficulties, that they may receive special attention in the preparation of the lesson. Preparation of the Lesson. — In the preparation of the lesson the pupil should be thrown, as far as possible, upon his own resources. The teacher should give him no assistance, or, at least, very little ; and he should prevent, as far as possi- sible, his obtaining any from other memljers of the class or the more advanced pupils. The habit of running to the teacher with every little difficulty is a most pernicious one, and de- structive of invigorating mental discipline. Independence of thought and bold self-reliance are indispensable traits of man TEACHING WRITTEN ARITHMETIC. 899 hood, and should be cultivated in the studies of youth, and especially in the study of mathematics, which is particular!} adapted to give such training. This point cannot be too stronglj' urged ; its neglect has been productive of much mischief. We have known pupils who, for a whole session, scarcel}' ever solved a problem for themselves, but prepared their lessons with the aid of otliei pupils. At other times they have obtained notes from those who had previously' passed over the same subject, and have used these notes to the utter neglect of self-thought. It is needless to sa}' that much time was thrown awa}^ and that such stud}' is worse than useless. Let the pnpil, in the pre- paration of his lesson, depend mainly upon himself; and what he fails to get out in this way, the teacher can explain to him at the recitation. The Recitation. — The Recitation is the great instrument of instruction. In it the teacher comes in contact with the mind of the pupil, calls out its energies and moulds it to his will. Mind meets mind here — the pupil's mind and the teacher's mind — thought is evolved, and mental activity stimulated. It is here that the teacher shows his power as a teacher, rousing up dormant faculties, directing mental activ- ity, and creating interest and enthusiasm in that which was before dry and repulsive. The method of recitation must be determined by the several objects to be attained. These objects, briefly stated, aj-e: 1. To find out what the pupil knows of the subject ; 2. To fix the subject clearly in the mind ; 3. To cultivate the power of accu- rate expression ; 4. To impart instruction. These objects should be kept clearly before the mind of the teacher to direct and inspire his work. Method of Recitation. — The lesson being prepared and the hour of recitation having arrived, the class take their seats in the recitation-room for the purpose of reciting. The teacher calls the roll to see if all are present, and then proceeds with 400 METHODS OF TEACHING. the recitation, the most important points of which will be briefly specified. Preparation of the Blackboard. — The first step is the prepa- ration of the blackboard. The teacher saj-s, " Prepare the board," and the pupils arise and pass orderly to the board, erase what work there may be on it, and then divide it into equal spaces hy vertical lines, each pupil drawing a line to his right, and writing his name at the upper part of the space. This done, as quietly as possible, the pupils turn and stand with their backs to the board, and face towards the teaclier. As.ngn7nent of Problems. — The next step is the assignment of problems. With young classes the same problem should be assigned to all the pupils, or at least one problem to four or five pupils ; with advanced classes each pupil should receive a different problem. If the class is not too large, the problem should be read by the teacher, and the pupil be required to copy the conditions as he reads. When the class is large, the pupils, or at least a part of them, may be permitted to copy the problem assigned from the book. They should be required, however, to close their books as soon as the conditions are written. Writiug the Problem. — The next step, or one co-ordinate with the above, is the copying of the problem upon the board. Wlien the problem is abstract, merely requiring an operation upon abstract numbers, the pupil will copy such numbers as the teacher reads. If the problem is concrete, involving sev- eral cbnditions, the pupil should be required to mark the conditions by a sort of short-hand or abbreviated process which he can write rapidly, and which will be readily under- stood. He should also write the page and number of the problem at the upper part of his space, so that the teacher may readily refer to the problem in the text-book. Working the Problem. — The next step is the working of the problem upon the board. In this the pupil should practice neatness and exactness. The figures should be plainly and TEACHING WRITTEN ARITHMETIC. 401 neatly made. The lines drawn beneath any part of the work should be straight and horizontal. The work should srener- alh' be written in the form which a person would employ in actual calculation. It may sometimes be written in an ana- lytic form, the operations and general form of solution being indicated by the form of writing. The pupil should be exact in such expressions, and not write one thing and mean another. Ever}' point, symbol, etc., should be written in its proper l)lace, the teacher not being satisfied by the pupil's saying he meant so and so, when he had written something else, or neglected writing some essential part. Having completed the solution of the problem, the pupil should take his seat and retain it until called upon by the teacher for his explanation. roitition at the Board. — In reciting, the pupil should stand in an erect and easy attitude, with the pointer in one hand and the other hanging down by the side. His side, and not his face, should be turned towards the board, so that he can see both the solution and the teacher. The teacher should be particular upon these points, allowing no awkwardness or clownishness of attitude, but endeavoring to cultivate an easy and graceful carriage. At West Point, one of the best mathematical schools in the country, they are ver}' particular upon such points as these, and the effect of it is seen in the progress and attainments of the pupils. Explanation. — The explanation of the problem should be given in a full and natural tone of voice, with great care in re- spect to clearness of thought, accuracy of expression, and dis- tinctness of enunciation. Those who speak too low should be encouraged to speak louder; and those who speak in a loud and declamatory- style should be taught to speak in a lower and more natural tone. If the form of solution is analytical, each point should be clearlj^ stated as it follows a preceding one, care being taken that the whole chain of anahsis be kept complete. If the solut^ion is deductive, the different steps being based upon principles previousl}'^ explained, these 402 METHODS OF TEACHIXQ. principles should be referred to in their proi)er order and eon nection. The exphination should be clear and full in all its parts, and complete as a logical whole. Crificifyyns. — The pupils who are not engaged at the boaiil should be rei[uired to observe closeh' each explanation, notii- ing carefully all mistakes in solution, expression, etc. At the close of the solution, the class should be called upon fni correction of erroi*s, suggestion of improvements, etc. The character of the solution, whether incorrect, too long, or no' surticiently clear, tlie form of statement upon the board, po-^i tion at the board, st^le of expression, etc., are all legitimal*' subjects of criticism. After the pujjils have given their criti- cisms, the teacher should present any other suggestions or corrections which may be required. At the close of such criticisms, the pupil \vho explained will erase his work upiui the board, and receive another problem, or take his seat, ami the next pupil jiroceed to explain. Teacher's Exftlaiuttiou. — It is often necessary for the teacher to cx|)lain some principle or problem to the class ; the proper time of doing this will be suggested by circumstances. A principle that several meraViers of the class do not uneople ignorant of the simplest principles of mechanics. Such expressions heard among mechanics as a " long square," a " slanting square," a ''square triangle," a "long circle," etc., show the defects of our common schools in respect to this branch. The common schools are fitting persons for every avocation ; and they should give pupils at least the fundamental principles that enter into so many of the practical atfairs of life. Third, inMrucfion in the elements of geometry lies at the basis of drawing. The simplest figures of the drawing lesson are the geometrical figures. Drawing should, therefore, begin 18 o 410 METHODS OF TEACHING. in geometry; and the elements of geometry may be made a stepping-stone to the introduction of drawing into the public schools. Fourth, lessons in geometry will be of value in school disci- pline. Pupils should be required to draw figures on their slates, and this will give employment to both minds and fin- gers, and keep them out of the mischief that comes from idle- ness. In this manner the teacher can reduce mischief into geometry, and thus interest and instruct little minds, and keep pupils obedient and quiet, because they are busy and happy. Principles of Teaching There are several principles that determine the order and methods of teaching the elements of geometry, which we state briefl}'^ : 1. The elements of geometry should precede the elements of arithmetic. It has been customary to defer geometry until the pupil is quite familiar with the elements of arithmetic, but this is a great error in education. The elements of geometry are much easier than the elements of arithmetic. The ideas of number are much more abstract than the ideas of form. The child of four years of age can acquire but a very small knowledge of arithmetic, while it may learn to distinguish and name nearly all the ordinary geometrical forms. 2. The reasoning of geometry should follow the reasoning of arithmetic. Though the ideas of Geometry are simpler than the ideas of arithmetic, the reasoning of arithmetic is much simpler than the reasoning of geometry. The former is often a mere succession of intuitive judgments, each com- parison bearing its evidence in itself; while the reasoning of geometry is syllogistic, depending on a principle of inference. For this reason the reasoning of geometry should not be in- troduced until the pupil has made considerable progress in arithmetic. 3. The method of teaching the elements of geometry should TEACHING GEOMETRY. 411 be concrete. The pupil should see the forms, rather than learn to define them. Figures cut from pasteboard, models made out of wood, diagrams on the board, etc., should be exten- sively used in these instructions. Even the truths should be illustrated or presented in the concrete, rather than by ab- stract demonstration. 4. The method of teaching should be inductive. The pupil should be led to the idea of the different figures and to the dif- ferent truths. He should be led to see the distinguishing characteristics of figures, the reason why they are named as the}'' are ; and in many cases he can be led to apply the appro- priate term himself by appropriate questions. I. The Geometrical Ideas. — The fundamental Ideas of ge- ometry are those of the Line, the Surface, and the Volume. These elements may be reached in two ways, analytically or synthetically. We may begin with the idea of a volume, and pass from it to the surface and line as elements of it; or we may begin with a point, pass to the idea of a line, from the line to a surface, and from the surface to a volume. The former method is analytic ; the latter is synthetic. Analytic Method, — We ma}'^ present the elements of geo- metrical quantity analytically as follows: The teacher may take some regular form, as a box, and call attention to it. He then takes a rule, and leads the pupils to see that it can be measured in three directions ; in length, breadth, and thick- ness. He then tells them that these measurements are called the dimensions of the box, and leads them to see that it has three dimensions, length, breadth, and thickness. The next step is to lead them to call it a solid. He leads them to call water, because it flows, afltiid; and because the hand will not move through the box, as through the water, we call the box a solid. He then leads them to conceive of the form of the box in space, and shows that the hand can move through this, therefore, this form is not a solid ; from which they may see that the better term is volume. They 412 METHODS OP TEACHING. may thus be led to conceive of form in pure space ; which is the ffeometrical volume. The next step is to teach the idea of a surface. The ceacher leads them to call a side of a box the surface^ and then measuring it, shows that a surface has length and breadth. He then asks how far they can see into the surface, and thus leads them to the idea that it has no thickness, but merely the tico dimensions, length and breadth. He then leads them to see that where two surfaces meet, since neither has anj' thickness, the edge will have no l)readth nor thickness, but merely length ; and that this is a line. In a similar manner, he may show that the end of a line has no length, breadth, or thickness, and is called a point. The stu- dent-teacher may be required to put this description into an inductive lesson. Synthetic Method. — By the Synthetic Method, we should have a pupil conceive a point in space ; then cause this point to move, and its imaginary pathway would be a line; then conceive this line to move in the direction opposite its lent^th, and it will form a surface ; then conceive this surface to move in a certain waj', and its motion will form a volume. This method is a legitimate one ; the principle of it is employed in geometry in the case of the cylinder, cone, and sphere. The analytic method is preferred, however, for sev- eral reasons. It is more concrete than the synthetic method, as it begins with that which can be seen, and not merely con- ceived. The synthetic method begins with the most difficult geometrical conception, a point, which has no dimensions, but position only. Lines. — The pupil has now the general idea of a line; the next step is to teach the three kinds of lines, the straight, the curved, and the broken line. To do this, take a small twig to represent a line ; put it into different forms, leading them to name the forms, and then drawing lines to represent these forms, have them appl}^ the names to the lines. TEACHING GEOMETRY. 413 Modd Lesson. — Teacher. "When I pull this stick out sti-aigliU-whafkmd of a stick is it? Ptqyil. A straight stick. T. If I draw a line like this on the board, what kind of a line is it? P. A straight line. T, bend- ing the stick, says, What am I doing to the stick? P. Bending it. T. When I have bent it, what kind of a stick is it? P. A bent stick. T. I will place this against the board and draw a line of the same shape; what kind of a line is it? P. A bent line. T. Very well; another name for this line is curved line T., breaking the stick, says, What am I doing with the stick? P. Breaking it. T. When I have broken it, what kind of a stick is it? P. A. broken stick. T. I will place it against the board, and draw a line like it on the board; what kind of a line shall we call it ? P. A broken line. The Angle. — The next step is to give the pupils an idea of an angle., and of the several kinds of angles. This maj' be done by taking some object, as a knife, opening it, then plac- ing two straight sticks side by side, and making an opening like an angle, leading the pupils to call it an opening ; and then giving the correct name, have the pupils define an angle. Lines on the blackboard may also be used. Modd Lesson. — Teacher, taking a knife and opening it, asks. What am I doing ? Pupils. Opening your knife. T. The space between the blade and the handle may be called what? P. The opening. T. I will lay two sticks, the one on the other, and open them; what is the space be- tween them cjilled? P. An opening. T. Yes, that is right, but there is another name for it; this opening is called an angle. T. What then is an angle? P. An angle is the opening beticeen two lines. The teacher will then make angles and require the pupils to make angles on the board. Kinds of Angles. — The teacher will then lead the pupils to notice the difierence between angles, to see that some are sharp and others blunt; and that these may be called acute and ob- tuse. Then lead them to see that there is one neither sharp nor blunt, and which, like a boy who is neither too sharp nor too blunt, is just right, and may therefore be called a right angle. The student-teacher will put this into a model lesson. Parallels, etc. — We next teach j^orallel lines, oblique lines, converging lines, diverging lines, perpendicular lines, and ho7-izontal lines. The method is simple ; the student-teacher iLay describe it and give a model lesson. 414 METHODS OF TEACHING. The Triangle. — To teach the Triangle^ give the children some little sticks, and have them make "little pens" with them. Tell them to make a pen with Jive sticks, then with four, then with three, then with two; and thus lead them to see that three lines is the least number that will enclose a sur- face. Then call attention to a fiarure made with three lines : ask how man}^ angles it has; lead them to call the lines sides; then lead them to call it a '■'■ three-side,'''' and then a ^Hhree- angle,'''' and then introduce tri, and lead to the name tri- angle. Kinds of Triangles. — Then lead them to see that triangles differ, and that the different kinds can be named from their angles and their sides. Then lead them to name the right-an- gled triangle, the obtuse-angled triangle, and the acute-angled triangle. Lead also to the different kinds of triangles with respect to their sides, and give them the names equilateral, isosceles, and scalene. Have them draw them on the board, and drill them until the}' are entirely familiar with them. Teach also the base and altitude of the triangle. The student- teacher will give an inductive lesson on the triangle. The Quadrilateral Have pupils make a four-sided figure, lead them to name it from its angles a. four-angle, give the word quadra, lead to quadrangle, its proper name. Then lead them to name it from its sides a four-side; introduce lateral for side, and quadra for four, and lead to quadri- lateral. Then lead them to discover the three classes of quadrilaterals; and give the names parallelogram, trapezoid, and trapezium. Then lead them to discover the several kinds of parallelograms ; the rectangle, square, rhombus, and rhom- boid. The subject will admit of a beautiful inductive devel- opment, which the student-teacher will give. J*olygons.^We should then give a general lesson on Fol;/- gons, including the pentagon, hexagon, heptagon, etc. We should teach the meaning of perimeter, area, regular and irregular polygons, their division into triangles, etc. TEACHING GEOMETRY. 415 The Circle. — We should next teach the Circle^ including the circumference^ semi-circumference, quadrant^ arc, diam- eter, radius, chord, sector, segment, tangent, etc. We should show pupils how to construct the circle, and require them to draw and name the different parts. Attention may be called to the difference between the circle and the circumference, which are often confounded. The use of the circumferent-L- in measuring angles may also be explained, and the division of the circumference into degrees, minutes, and seconds. Pupils may also be taught to inscribe squares in circles, and circles in squares, etc. The}' ma^- also be shown how to in- scribe a regular hexagon by taking the radius as a side ; and also how to form an inscribed triangle from the inscribed hexagon. The student-teacher will give a model lesson ou the circle. Volumes. — Among the Volumes we should first teach the cube, the pyramid, the cylinder, the cone, and the sphere^ We next teach the prism, and the different kinds of prisms, named from the form of the bases. We should next teach the oblique and right pj'isms, the parallelopipedons, rectangular parallelopipedons, the frustum of a pyramid, frustum of a cone, etc. We should have models of these different volumes, and also draw them and show pupils how to draw them on the board. Round Bodies. — We ma}' then give a more detailed lesson on the three round bodies, the Cylinder, the Co7ie, and the Sphere. We may show that the cylinder can be generated by the revolution of a rectangle ; explain which is the base, the altitude, and the convex surface. We may show how a cone can be generated b}- the revolution of a right-angled triangle about one of its sides, and explain the 6a.se, altitude, slant height, and convex surface. We ma}' show how a sphere can be generated by the revolution of a semicircle around the diameter, and explain the diameter, radius, convex surface, small circles of the sphere, great circles, spherical triangles, spherical polygons, the lane, etc. 416 METHODS OF TEACHING. We mention in detail the things to be taught, so that 3 oung teachers may have a clear conception of the course sugg isted. They should be prepared on the subject themselves, and then know how to present it in an interesting manner. Let the student teacher be required to present a model lesson on each one of the figures. II. The Geometrical Truths, — Children maj' also learn many of the truths of geometry as well as the ideas. The truths of ireometrv are of two kinds; those that are self-evi- dent, called axioms, and those that are derived by demonstra- tions, called theorems. Many of the self-evident truths of geometry are readily understood by young pupils. Many of the theorems may be illustrated or presented by what might be called a concrete demonstration. An abstract or logical demonstration of them would be too difficult for children, and nothing of the kind should be attempted. Some of the other truths which cannot be illustrated may be taken on faith ; the pupils accepting them, not because they can see a reason for them, but because the teacher tells them they are true. Self-evidetit Truths. — Little children may readily be led to see that "A straight line is the shortest distance from one point to another." Unite two points with a straight line, a curved line, and a broken line, and they will see by intuition that the straight line is the shortest route. To make it inter- esting, have them suppose three little boys start from the same point to travel on three lines, and they will readily see which has the shortest road to travel. The ancients used to say that a donkey knew that one side of a triangle was shorter than the sum of the other two sides, for he would go straight across from one comer of a field to the other, rather than follow the two sides of the field. They may also be taught to see that " two right angles are equal to one another." Care should be taken that they see that the size of the angle depends on the extent of the open- TEACHING GEOMETKY. 417 ing, and not on the length of the sides. They may also read- ily see that "the diameters of the same circle are all equal;" that " the radii are all equal ;" that "the radius is half the diameter," etc. In fact, they may be taught nearly all the geometrical axioms. The student-teacher will present the lesson. Truths by Concrete Demonstration. — Many of the truths of geometry can be taught by concrete deniondrilion. That is, they may be illustrated in such a way that pupils can be assured of their truthfulness without depending upon the statement of the book or the teacher. We will give a list ol such theorems, and suggestions for their illustration. 1. If one straight line meet another straight line^ the sum of the two adjacent angles equals two right angles. Take two straight sticks, A and B ; place the end of A near the middle of B, perpendicular to it ; then will be formed two right an- gles. Then incline the stick A, and the pupil can see that one angle loses what the other gains, and that they both just fill up the space of two right angles, and hence are always equal to two riarht angles. Illustrate the same also on t-he board. 2. All the angles formed on one side of a straight line by drawing lines from the same point, are equal to two right angles. This can be shown as in the previous theorem, and the pupil may illustrate it on the board. 3. The sum of the three angles of a plane triangle is equal to two right angles. To illustrate this, cut out a triangle from stiff paper of any form;- then cut off two of the angles, and place one on each side of the third angle, and it will be found that they Just fill up the angular space of two right angles. 4. If two triangles have two sides and the included angle of one respectively equal to two sides and the included angle of the other, the two triangles are equal. To show this, cut out of paper a triangle of any shape ; then mark out on another piece of paper two sides and an included angle equal to those of the given triangle, then draw a straight line uniting the 18* 418 METHODS OF TEACHING. extremities of the sides, cut out the triangle, and compare them b}' placing one on the other, and it will be found that they exactl}' coincide. 5. The area of a r'ectangle equals the number of units in the base multiplied by the number of units in the altitude. Take any number of square blocks, as foe, and pile them up in three rows of fve each, forming a rectangle. The whole surface of the rectangle is formed by the one side of the square blocks, and since there are 5 in a row, and 3 rows, there are 3 times 5, or 15, in all; hence the product of the number of units in the base multiplied by the number of units in the height, will give the whole number of square units in the sur- face. Illustrate it also on the blackboard. 6. The area of a parallelogram is equal to the product of the base and altitude. Cut out a paper parallelogram, cut off one corner vertically across ; put this triangle on the other end of the parallelogram, and it will become a rectangle. Now the surface of this rectangle is precisely the same as the sur- face of the parallelogram, and its base and altitude are the same. But the area of this rectangle is equal to the product of the base and altitude ; hence the area of the parallelogram is equal to the product of the base and altitude. 7. The area of a triangle equals the product of the base by half the altitude. Cut out a parallelogram; then divide it into two triangles, cutting across from one corner to the other. These two triangles are equal, and hence equal to one-half of the parallelogram, and hence to one-half of the product of the base multiplied by the altitude. 8. The area of a trap)ezoid is equal to the sum, of the two parallel sides multiplied by half the altitude. This can be shown by cutting out a trapezoid, dividing it into two tri- angles, showing that the area of each equals its base into one- half of its altitude, and that their sum will be the sum of the two bases into one-half of the altitude. 9. The square on the hypothenuse of a right-angled tri- TEACUING GEOMETRY. 419 antfiti is equal to the sum of the squares on the other two siaes Make a right-angled triangle on tlie board, one side 3 and the other side 4, the liypotheuuse will be 5; construct squares on each, and divide them into small squares; the square on one side will co; tain 9, that on the other 16, and that on the hj'pothenusi' 25; and 25 we see is the sum of 9 and 16. Here we see that '.he squai-e on the hypothenuse is equal to the sum of the squa es on the other two sides. Many otLer truths can be taught in this way ; and such a concrete consideration of the subject will be a valuable prep- aration for the stud^' of the subject abstractl3'. Let the student-teacher give a lesson on each one of these, using paper and the blackboard. III. TinJTHs TO BE Taken on Faith. — We should teach the pupils of the common school some truths that cannot be illus- trated to them. Such truths they may take on faith ; pupils believing them as they do the facts of geography and histoiy, because the teacher states them as true. This instruction may extend to curves not treated of in ordinary geom- etry, including the Parabola, the Ellii)se, the Hyperbola, the Cycloid, the Catenaiy, etc. J/t Ordinary Geometry It will be well to teach the more advanced pupils how to find the circumference of the circle by multipUjing the diameter by 3.1416, to find the area by mitlti- plyiruj the circumference by half [he radius; that an angle at the centre is measured by the arc included between its sides: that an angle at the circumference is measured by one-half the arc included between its sides; how to find the volume of a prism, the convex surface and volume of a cylinder, the volume and convex surface of a pyramid and a cone, the sur- face and volume of a sphere. These should be introduced as they are prepared for them, the pupils being drilled on their application, but no attempt being made to explain the reason for them. Tlie Parabola. — If a cone be cut by a plane parallel to its 420 METUODS OF TEACHING, slanting sides, the section formed is a beautiful curve called a Parabola. This is a very interesting curve. Every stone that a little bo3' throws at an object forms a parabola in its flight. In a snow-balling match, all the balls form parabolic curves ; and in a battle, shot and shell go hummingr and screechin2 through the air in parabolic arcs. It may be well to show that the area of a section of this curve is two-thirds of the bane muHiplied by the altitude. This area may be compared with the area of a rectangle and triangle of the same base and alti- tude. The method of constructing a parabola should also be given. The Ellipse. — If a cone be cut by a plane making an angle with the base l.ess than tliat made by the side of the cone, the result will be a closed curve that looks like a circle drawn out. Such a curve is called an Ellipae. This curve can be made by driving two pins in a board, and tying a string at each end to one of these pins, and then putting a pencil point inside the string, stretching it out and moving it round. A doubled string passed around the jjins is still better. The two points at the pins are called the foci of the ellipse. The point be- tween these is the centre^ a line through tlie foci is called the vxajor axis ; and a line perpendicular to this throug'i the cen- tre is the minor axis. The ellipse is also an interesting curve. The earth in its march around the sun follows a pathway of the form of an ellipse, the sun being in one of the foci of the elliptical orbit. The moon moves around the earth in an ellipse, and all the planets and satellites move in the same curve. To find the area of an ellipse, we multiply the half of the two axes toijcther, and that product by 3.1416. Another interesting fact is that if we had a mirror in the form of an ellipse, a light placed at one focus would have its rays all reflected in the other focus; and if we had a whispering gallery in this form, a whisper at one focus would be distinctly heard at the other focus, T/ie Hyperbola. — If a cone be cut by a plane making a TEACHING GEOMETRY. 42] larger angle with the base than the slanting side makes with it, the curve formed is an Hyperbola. If we tie strings at dif- ferent points of a horizontal v.ire, and draw them all through a point below the wire and cut them off at the point, when they hang down straight their ends will form an hyperbola. If we tie threads to each link of a hanging chain, and cut olf their ends in a level line, and then draw the chain out hori- zontal, the lower ends of the threads will form an hyper- bola. There are many interesting truths concerning the hyperbola. The Cycloid. — If a wagon wheel roll on a level floor, a nail in the tire or rim will make a series of curves, each called a Cycloid. A boy can make a cycloid by fastening a pencil to a spool and rolling the spool slowly against the inside of the frame of his slate. There are several interesting properties of the cycloid. 1. The height of the cycloid at the middle is equal to the diameter of the wheel or circle which formed it. 2. The length of the straight line joining the two ends of the curve, called the bane, is equal to the circumference of the generat- ino- circle. 3. The leng^th of the curve is four times the diam- eter of the generating circle. 4. The area of the curve is equal to three times the area of the generating circle. When the circle is at the middle of the cycloid, the curious looking three-cornered figures on each side of the circle are each exactly as large as the circle itself. 5. If a cycloid is turned upside down, a ball will roll down it quicker than on axvy other curve : for this reason the cycloid is called the curve of swiftest descent. If a hill were hollowed out in the form of a cycloid, a sled would run down it faster than if it were of any other shape. 6. Another curious property is that if several balls start at different points on the curve at the same moment, they will all reach the bottom at the same time ; so that it is also the curve of equal descent. 422 METHODS OF TEACHING. The Catenari/. — When a chain hangs from two posts, it makes an interesting curve, called a Catenary. A jumping rope, a clothes-line, and a gate chain, all hang in the form of a catenary-. The curve was first noticed by Galileo, who thought it was the same as the parabola. Its true nature was first demonstrated b}' James Bernoulli. This curve has also several curious properties. 1. If the chain were made of a great man}' short pieces of wood or metal hinged together b}' rivets, like the inside chain of a watch, and we could turn it up in the same form it has when it hangs, it would stand up without ftiUing in, and be a catenary upside down. This is the only curve that possesses this property. 2. If we wish to make the strongest possible arch for a Drid^c, we should make it in the form of a cate- nary. For other facts in the elements of geometry, see First Les- sons in Geometry, by Dr. Thomas Hill, a valuable little book. II. Geometry as a Science. The previous course in the elements of Geometry is de- signed as an introduction to the stud}' of the subject as a science. By means of it, pupils will become familiar with the leading ideas and truths of geometry, and thus be prepared for a more intelligent study of the science when of a suitable age. Pui'ils may begin the stud}' of geometry' as a science s-hen abo. * thirteen or fourteen 3'ears of age. I. The Nature of the Study. — The study of Geometry as a science includes Definitions^ Axioms^ Fodulates, Theorems, Demonstrations, Problems, Solutions, and Apjdications. We shall speak of the nature and methods of teaching each of these, and also of the method of hearing a recitation in geometry. Definitions. — The Definitions of geometry are statements of the ideas of the science, or a description of the quantities upon which we reason. They are examples of what are known TEACUING GEOMETRY. 423 as logical definitions ; tliat is, they define by genua and di(fer- en^m, or specific difference. Tliiis, in the definition, "A tri- angle is a polygon of three sides," polygon is the generic term, and three sides is the differentia or specific difference. No science presents so many fine examples of logical defini- tions as geometry. These definitions shonld be expressed in the deductive form; that is, we shonld begin with the term to be defined, and pass to genus and differentia. The definitions should be stated positively, not negatively, telling what a thing is, and not what it is not. Thus, the old definition, "A straight line is one that does not change its direction," etc., is not so satis- factory as the positive one, " A straight line is one that lies in the same direction," etc. How Teach. — In teaching the definitions, the first requisite is that the pupils have a clear notion of the tiling defined. They should be required to give an illustration of each defini- tion in which there may be the least difficulty. This point is important, as pupils are often found trying to reason from a definition when they have no clear idea of the quantity de- fined. It is especially necessary with pupils of good memory, who are apt to rest satisfied with a form of words without taking the trouble to see clearly what is meant by them. Care should be taken to see that the definitions, as given by the pupil, are strictly accurate. The language of the author should be insisted upon, unless the teacher or pupil can im- prove the definition, which, in such a science as geometry, will be seldom possible. Most of the definitions are classic with culture and age, and have become fixed in form, and will not admit of improvement. It is an excellent exercise to show, by question and illustration, the importance of the prominent points of a definition, and how any departure from the statement will vitiate the correctness of the definition. A proper study of the definitions of geometry may be made a source of excellent mental discipline. 424 METHODS OF TEACHING. Axioms. — The Axioms of geometry are the self-evident truths which pertain to the subject. These truths lie at the basis of the science; they are the foundation upon which all the other truths rest. They express the fundamental and necessary relations of quantity, and depend for their existence on no truths which lie behind them. These truths are intu- itive; they are not the result of reasoning. The mind is so constituted that it knows them to be true upon the mere an- nouncement or contemplation of them, and neither asks nor needs an}' proof of them. Two Kinds. — There are two kinds of axioms in geometry; those which pertain to quantity in general, and those which grow out of the particular quantity considered. Examples of the former class are, " Things that are equal to the same thing are equal to one another;" "If equal quantities be equall}' increased or diminished, the results will be equal." These apply to arithmetic and algebra, as well as to geometry. Examples of the second class are, "All right angles are equal to one another;" " The radii of a circle are all equal." These arise out of the particular kind of quantity considered, and apply only to geometry. Their Use. — The use of axioms in reasoning, as usually stated, is that they are general truths which contain all the particular truths of the science. According to this view, the geometer needs onl}' to anah'ze the axioms, and he will find in them all the truths of the science. In reasoning, he onl}- un- folds these general truths and evolves the special truths which he finds contained in them. This view of the subject admits of question. It maj- be pleasant for one to suppose that when he knows the axioms of a science, he has in his mind, potentially if not actually, the entire science; but it does not seem to express a scientific truth. A general formula ma}' truly be said to contain many special truths which may be derived from it; but no axiom in this sense can be named that contains the other truths of geometry. TEACHING GEOMETRY. 425 Another view is that axioms are the laws which guide us in reasoning: they are the laws of comparison or inference. Thus, if we wish to compare A and B, seeing no relation directly between them, we may compare each to C ; and prov- ing that they are both equal to C, we infer that they are equal to each oiher. The law that governs this comparison, and enables us to make the inference, is the axiom. Things that are equal to the same thing are equal to one another. So in comparing parts of the circle, we must always bear in mind the truth that the radius is half of the diameter ; but it can- not be truly said that this axiom contains other truths. It is also true that an axiom may be one of the premises of a syllogism from which a conclusion is drawn. Thus in a dem- onstration we may see a line A equal to a radius B of a circle, but radius B is equal to radius C of the same circle ; there- fore, this line A is equal to radius C. In this case the axiom of equal radii is neither a general truth containing other truths nor a law of reasoning. Axioms may thus perform several offices in a demonstration ; but they are always first truths, beyond which we cannot go in thought. How Teach. — In teaching the axioms, the pupil should be required to give an exact statement in the language of the book, unless it can be improved. No awkward or half-way statement should be accepted as satisftictory. He should also be required to illustrate the axiom, that the teacher may be sure he has a clear conception of the truth he is stating. Postulates. — An axiom may be defined as a self-eviden' theorem. A self-evident problem is called a Postulate. Tha« it will be granted that "a straight line may be drawn froir one point to another," or that " two lines may be constructed equal to each other." The postulates bear the same relation to problems that axioms do to theorems. The same remarks will apply to the teaching of them that we have already made with respect to teaching axioms. Reasoning. — All reasoning is the comparison of two ideas 4.26 METHODS OF TEACHING. through their relation to a third. Thus, suppose I see no re- lation between A and B,but upon looking at atiiird quantity, C, I perceive tliat A equals C, and also that B equals C ; and I can then infer that A equals B, I thus compare A and B through their common relation to the tliinl quantity, C ; C thus stands intermediate between A and B, and the process is called a process of mediate or indirect comparison. This is the general nature of the reasoning of geometi;. In its application to geometry reasoning assumes two diriii- ent forms, wluch may be distinguished as the analytic and synthetic methods. The analytic method is adapted to the discovery of truth; the synthetic method is used in proving a truth when it has already' been discovered. Synthetic Method— The Synthetic Method of proving a truth already known is called demonstration. Demonstration begins with self-evident truths or truths already proved ; and passes, step by step, to the truth to be proved. There are two distinct methods of demonstration. The simplest form Is that in which figures are directly compared by applying one to another. This is called the method by superposition. It is used in proving the equality of polygons and also of some of the volumes. The more general form of demonstration is that in which truths are proved by a reference to the defini- tions and axioms, or to some principle previously proved. Analytic Method.— The Analytic Method begins with the thing required, and traces the relation between the various elements, till we arrive at some known truth. It is a kind of going back from the result sought, by a chain of relations, to what has been previously established. In the synthetic method, we pass through every step, from the simplest self- evident truth to the highest truth of the science. In the process of analysis, we pass over the same path, descending from the higher truths to the simpler and fundamental truths. Analysis is the method of discovery; synthesis is the method of presentation. The one has for its object to find TEACHING GEOMETRY. 427 unkuown truths; the other to prove known truths. Fre- quently both methods are employed simultaneously, when the object is to discover new theorems, or to find the solution of new problems; but when we wish to prove to others the truths already- known, the synthetical method is usually preferred. Reductio ad Abffui'duvi. — There is a form of reasoning which is analytic in its character, known as the reductio ad absurdum. It consists in supposing that the proposition to be proved is not true, and then showing that such a hypothi?- sis leads to a contradiction of some known truth. This proves a theorem to be true by simplv showing that it cannot be false. The method is frequently used to prove the converse of a proposition, when there is no good direct method ; it is also used in treating incommensurable quantities. This method of reasoning is also called a demonstration , and is called the Indirect Method^ to distinguish it from the other, which is called the Direct Method. The indirect method is not considered as satisfactory^ as the direct method, and should never be used except when no good direct method can be found. Errors in Reasoning. — There are two errors 4n reasoning into which 3'oung geometricians are liable to fall. The first is called Reasoning in a Circle; the second is called Begging the Question. We reason in a circle, when, in demonstrating a truth, we employ a second truth which cannot be proved with- out the aid of the truth we are tr^aug to demonstrate. We are said to beg the question, when, in order to establish a proposi- tion, we emplo}' the proposition itself. Practical Problems. — A radical defect of most of our text-books on geometr}- is that the}- present the subject so abstractly that when the pupil has completed his course, he is often unable to make any practical application of what he has learned. This defect can be supplied by requiring the i)upils to apply the principles of the science to practical examples. Such applications will show them the use of the principles. ^.2^ METHODS OF TEACHING. and they will thus understand it better and remember u longer. They will also place a higher value on the science on account of their being able to apply their knowledge to some practical purpose. These applications will also add an inter- est to the study that it cannot possess by the purely abstract method. Every text-book in geometry should be supplied with a large collection o^ practical examples. Undemonstrated Theorems. — Another defect in the teach- ing of geometry has been the lack of matter for original thought. The study as usually pursued does not give train- ing to the inventive powers of the student. He is required to learn the demonstrations of the text-book, but he has no undemonstrated theorems to test his own geometrical powers and to train him to reason independently of the text-book. To remedy this defect, he should be given a collection of theorems for original thought, and be required to try his powers of reasoning in finding out the demonstration for himself. These theorems should be easy at first, and gradually in- crease in difficulty as the pupil gains strength for the work. They may lie mingled with the propositions of each book (geometry is usually divided into a number of books), or they may be placed at the close of each book. The latter method is preferred with most pupils, as they should be quite familiar with the propositions of any given book before they are pre- pared to apply these principles to the investigation of other truths. One original theorem each day to apply the prin- ciples gone over, in connection with two or three theorems of the following book, will make a very interesting exercise. At the close of the text-book, there should be a large «umber of miscellaneous theorems for original thought. This is the method used in arithmetic and algebra, and it seems surprising that it has not been more generally em- ployed in geometry. Several authors seem recently to have realized the importance of such exercises, and have occasion TEACHING GEOMETRY. 429 ally given some practical problems, and, in one or two in- stances, a collection of undemonstrated theorems. In the author's work on geometr3^, such problems and theorems are a prominent and essential part of the plan. II. The Recitation. — The several things to consider under the recitation in geometry are: 1. The assignment of the theorems ; 2. The construction of the diagrams ; 3. The dem- onstration ; 4. The criticism; 5. Xew matter. Assignment. — The theorems may be assigned to the pupils in various waj's. They ma}- be given out at random, without any reference to the ability of the class; or, if there are some in the class who are not very strong in the branch, the easier propositions may be given to them. The best way probably' is to assign by chance, which may be done by writing the numbers of the propositions on small pieces of paper, and re- quiring the pupils to draw these papers. It is suggested that at least one da3''s review lesson should be included in each recitation, the class taking three or four propositions in ad- vance, and the same number in review. Construction, — The pupil haxing received a theorem, should be required to go to the board and construct the dia- gram without any reference to the book. The lines should be drawn by free hand, and not with the aid of a ruler. The letters of the diagram should be placed at random, and dif- ferent from the order in the book, in order to preA'ent a recita- tion from memory. Figures in place of letters may often be used in marking the diagrams. It will add interest also for one pupil to construct the diagram for another pupil, each thus constructing the figures of one proposition, and demon- strating another. Demonstration. — In demonstrating the theorems, the pupil should stand at the board in an erect and easy attitude, his face turned partly toward the class, and the pointer being in the hand next to the board. The theorem should first be stated clearlj' and precisely, and in the language of the book, 430 METHODS OF TEACHING. unless it can be eriualed or improved. The deraonstration should be clearl}' and logically presented, the definitions and axioms referred to by number or, with beuinners, by repeti- tion, and previous theorems referred to by number of book and theorem. When the demonstration involves several pro- [jortions, these may be written out on the board and be pointed at in the demonstration. It will' be well also for the pupil to write out an analysis of the course of reasoning involved in a demonstration. Some- times an analysis merely of the references or dependent truths may be written. Sometimes the pupil may be required to write an analysis of all the principles involved in the demon- stration, tracing each truth all the way back to the definitions and axioms. Such an exercise will be found most valuable in giving pupils a thorough knowledge of the subject. Criticism. — At the close of tlie recitation of any pupil, the members of the class who have observed any errors may be called upon to point them out. These may consist of the omission of necessary links in the chain of reasoning, the omission or misquoting of references, etc., etc. Pupils who have a shorter or better, or even a different method, may be called upon to give it. Errors unnoticed by the pupils, may then be pointed out b}' the teacher. Quesfioniuf/. — The teacher should quiz the pupil on his demonstration. He should ask questions like the following: What kind of demonstration is it? Why do you begin as you do? Why do you prove such a thing equal to such a thing? What relation does this proposition bear to the preceding proposition ? What application can you make of this truth ? Show its application, etc. Outlines. — At the close of a book, the pupil should be re- quired to give an outline of the book; show the design of it; show what propositions reach final truths, and what prop- ositions were merely auxiliary: show the relation of each proposition to the chain of logic, and how the chain would TEACHING GEOMETRY. -iSl be broken by the omission of any proposition ; etc. By fol- lowing these suggestions, the teacher will make geometry a delightful study to his pupils, and a most valuable means of mental culture. New Matter. — If the teacher has any new matter, it may he presented at this time. He may give a discussion of the general nature of the lesson, show the excellence or defect of the method of development made use of, and make a compar- ison between the method of treatment used In* the author and that of other authors. He should then assign the nest les- son, and present any suggestions concerning it that may seem advisable. Conclusion. — In conclusion, we would urge teachers to introduce the elements of geometry into our public schools. A little less arithmetic, if need be, in order to present some geometry, would be an advantage. We trust that teachers may realize the importance of the subject, and endeavor to awaken a deeper interest in the beautiful science of form — a science over which the ancient sages mused with such deep enthu- siasm, and to which the achievements of modem art and invention are so largely indebted. CHAPTER VII. TEACHING ALGEBRA. ALGEBRA is that branch of mathematics which inves- tigates quantity by means of general characters called symbols. The term originated with the Arabs, and comes from al-gabr,a reduction of parts to a whole. The definition given states the general character of the subject, though it is diflicult to give a definition that fixes precisely its province and object. Relation to Arithmetic. — Algebra in its elements is closely related to arithmetic. It had its origin in arithmetic, and its fundamental ideas and operations are arithmetical. Its sym- bols of quantity were at first merely general symbols of num- bers, and its fundamental operations of addition, subtraction, etc.. were entirely similar to those of arithmetic. On account of this relation, algebra has been called a kind of general arith- metic. Newton called it Universal Arithmetic. D'Alembert regards it as a special branch of tl^e general science of num- bers ; and divides arithmetic into Numerique, special arith- metic, and Algebre, general arithmetic. Wider View. — This view of the nature of algebra is now too narrow. Algebra has ti-anscended the bounds of its ori- gin. It reaches from arithmetic over into geometry, including continuous as well as discrete quantity. From the generality of its symbols, also, man}^ ideas and processes arise which have no meaning or use in arithmetic ; as negative and imaginary quantities, the solution of higher equations, etc. Another important difference is, that iu arithmetic the com putations being made as they arise, all traces of the interme- diate steps are lost, and the result is applicable to a single case only ; wliereas in algebra the result is general, and con- (482 > TEACHING ALGEBRA. 433 tains implicitly the answer to all problems of the same general class. The combination of alo-ebraic symbols leads to expres- sions called formidas, in whicli the operations are indicated rather than performed, and which admit of interpretation. These formulas often express a general truth corresponding to a theorem, which arithmetic can verify in particular cases ; as (a-{-b)(a — b) = a'^ — 6^, and x= — p±'^q-\-p^. Conife's View, — Comte divides mathematics into geomefri/ and analysis or calculus. Calculus embraces algebraic calcu- lus, or algebra, and arithmetical calculus, or arithmetic. Al- gebra is defined " as having for its object the resolution of equations,''^ which signifies " the transformation of implicit functions into equivalent explicit ones." Arithmetic is defined as the science which " ascertains the values of functions.''^ "Algebra is the calculus of functions ;'''' and "Arithmetic is the calculus of values." Sir William Rowan Hamilton, the author of Quaternions, defines algebra as the science of time, which De Morgan changes to the calculus of succession. Symbols. — The symbols of algebra are of three general classes ; Symbols of Quantity, Symbols of Relation, and Sym- bols of Operation. The Symbols of Quantity are of two kinds; symbols of knoivn quantities and symbols of unknown quantities. They include also the two limits of quantity, zero, 0, and infinity, oo. The Symbols? of Operation include the signs of all the o[)erati()ns to which quantity can be sub- jected. The Symbols of Relation include the s^^mbols which arise in comparing quantity; as, =,:,::,>■•<, etc. The symbols of quantity apply to continuous as well as dis- crete quantity. Thus a and b may represent two lines as well as two numbers. If these lines have a common unit, then a and b may be regarded as representing the lines numerically ; but when the lines have no common unit, a and 6 denote them as continuous, and not as discrete quantity. Generalization. — The spirit of generalization in algebra is the source of many of its ideas and processes. From this we 19 434 METHODS OF TEACHINa. have the negative quantity, the fractional and negative expo- nent, the imaginary quantity, etc., each of which admits of explanation and leads to new conceptions in the science Thus, the sign of subtraction is primarily used to denote that a Quantity is to be subtracted ; but if we subtract a from the quantity a — b, we have a remainder of — b, the interpretation of Avhich gives us the idea of a Negative Quantity. The Fractional Exponent originates in the same way. Hav- ing agreed to indicate a power b}' an exponent, b}' generaliza- tion we have a"; and since n can represent anv quantit}-, it may represent a fraction, as |, and we have a*. This expres- sion being interpreted, we find means the third power of the fourth root of a. Or, having the rule that the root of a quan- tity may be obtained by dividing its exponent, in extracting 3 the 4th root of a^ we reach the same result, a^. The Negative Exponent has a similar origin. Since the general exponent may represent any quantity, it may repre- sent a negative quantity, and we may thus have a"** ; a new idea which needs interpretation. Or, if we divide a" by a-** according to the general rules of division, we also reach the expression a-^ ; and this we find denotes the reciprocal of a" , 1 or that a—^= — a« The Imaginary Quantity arises b}' a similar process of gen- eralization. In the general expression v^a, n ma}' be even and a may he negative, which gives us such exjjressions as v^-4,v -8, v^_l6, etc. Or, given general methods of solving quadratic equations, imaginary expressions ma}- arise from the solution of such equations, as x'^=: — 4, or x"^ — 2j:= — 5. This expression must also be interpreted. 'In the same way, other ideas arise in algebra from the generality of the notation and of the methods used I>ivisioH of Subject. — The science of algebra admits of the same fundamental divisions as arithmetic. These pro- cesses are all included under the three heads ; Synthesis, Analy- TEACHING ALGEBRA. 435 sis, and Comparison. The fnndamenta! operations are Addi- tion, Subtraction, Multiplication, and Division. The deriva- tive or secondary processes are Composition, Factoring, Common Multiple, Common Divisor, Involution, and Evo- lution. Comparison gives rise to the Equation, Ratio, Proportion, the Progressions, etc. Each of these processes, on account of the generality of the symbols and operations, gives rise to processes and expres- sions not found in arithmetic. In respect to the new process called Composition, we remark that its scientific necessity is seen from the fact that each analytic process has its corre- sponding synthetic process. Thus addition is synthetic, subtraction is analytic, multiplication is synthetic, division is analytic, etc.; it follows, therefore, that there should be a syn- thetic process corresponding to the analytic process of Fac- toring. This process we have called Composition; and its value is especially apparent in algebra, on account of the seveval interesting and practical cases which it embraces. The Equation. — The fundamental process of comparison in algebra is that of the Equation. The equation makes its api>earance in arithmetic, but is not of sufficient distinctive impoi'tance to be regarded as a distinct part of the science. In algebra, however, it is of fundamental importance ; and gives the science its principal value. Indeed, so largely does it enter into the subject, that it would not be very far from the truth to say that algebra is the science of the equation. The principal use of the equation is to compare unknown quantities, variously involved, with known quantities, the object being to find the value of these unknown quantities. In the effort to disengage the unknown quantity from the known and find an expression for its value in known terms, we discover methods of procedure called the solution or reso- lution of the equation. The solution of the equation gives rise to several processes, among which are Transposition, Substitution, Completing the Square, etc. 4,Z^ METHODS OF TEACHINQ. The solution of the general equation has never been deter- mined, and is no doubt impossible. The solution of the cubic and bi-quadratic is attended with difficulties that render the present methods not entirely satisfactor3'^ ; and the solution of the general equation beyond the fourth degree has never been accomplished and is believed to be impossible. But though no solution of the general equation has been found, many pro- perties have l>een discovered that enable us to know much about their roots. These properties embrace some of tlie most beautiful things in the science of mathematics, such as Descartes' Rule, Sturm's Theorem, etc., and confer immor- tality upon their discoverers. Besides these, we have in Horner's Method a general method of solving all numerical equations that have real roots. Reasoning. — The reasoning of algebra is essentially de- ductive. The comparison of quantities is usually that of equals, the relation being expressed by the equation. This equation is operated upon in various ways, all the operations being controlled by the axioms of the science. All the ope- rations of addition, subtraction, transposition, substitution, etc., are governed by axiomatic principles, and this makes the reasoning deductive. . Induction. — Though algebra is a deductive science, it is possible to derive some of its truths by induction. Indeed, man}' of the first generalizations of its symbols are inductive in their character. Several of its leading truths were discov- ered by an inference from particular cases, and were after- ward demonstrated. Newton's Binomial Theorem was derived in this way ; and it is presented in this manner to the students of elementary algebra. The divisibility of a" — 6" by a—b ma}- be inferred from the truth of the several cases a^ — b"^, a^—b^, a* — &*, etc. Mathematical Induction. — There is a method of reason- ing in algebra called mathematical induction., which differs from pure induction. Mathematical induction derives a gen- TEACHING ALGEBRA. 437 eral truth by showing that what is true in n cases is true in At+l cases ; while pure induction proceeds upon the principle that what is true in many cases is true in all. The principle of mathematical induction is used by many writers in proving that o" —b^ is divisible by a— 6, and also in giving a general demonstration to the Binomial Theorem. History of Algebra. — The first known treatise on algebra is found in the Arithmetic of Diophantus, written in the fourth century. Though not presenting a complete treatise on algebra, it lays an excellent foundation for the science. It contains the first enunciation of the rule that " minus multi- plied by minus produces plus ;" solves such problems as " Find two numbers such that the sum or difference of their squares are squares;" and then proceeds to the solution of a peculiar class of problems which belong to what is now called indeterminate analysis. It is supposed that some of the principles were known be- fore the time of Diophantus ; but he greatly enriched the science with new applications. He shows great skill in the subject, presenting some elegant solutions, and is regarded as the author of Diophantine Analysis. The celebrated Hypatia composed a commentar}'^ on Diophantus, which is now lost. The work of Diophantus was discovered at Rome, in the Vatican library, about the middle of the sixteenth century, having probal)ly been brought there from Greece when the Turks captured Constantinople. Algebra was introduced into Europe by the Arabs, who had carefully collected the writings of the Eastern mathemati- cians and written commentaries upon them, A copy of an Arabic original is preserved in the Bodleian Library at Oxford, bearing a date of transcript corresponding to the 3'ear 1342. This work is supposed to have been derived from the Hin- doos. Very few additions to the science seem to have been made by the Arabs, though they cultivated it with great enthusiasm. The science of algebra was introduced into Italy 438 METHODS OF TEACHING. by Leonardo, a merchant of Pisa, who had travelled exten- sivel}' in the East, in a work composed two centuries before the invention of printing. He could solve equations oi" the first and second degrees, and was particularh' skillful in tlie diophantine analj^sis. Like the Arabian writers, his reason- ing was expressed in words at length, the use of symbols being a much later invention. The earliest printed book on algebra was corajjosed by Lucas di Borgo, a Minorite friar. It was called Summa de Arithmetical Geometria, Proportioni, et Proportionalita^ and was published in 1494 and again in 1523. It followed Leon- ardo very closel}' ; but the mode of expression was very im- perfect, the symbols employed being a few abbreviations of the words or names which occurred in the process of calcula- tion, — a kind of short-hand arithmetic. The application was also limited, being confined to the solution of certain problems about numbers. It included the solution of equations of the first and second degrees, the latter being divided into cases, each of which was solved by its own particular rule, many of which were derived from geometrical constructions, and expressed in Latin verses to be committed to memory. Up to the fifteenth century, the science was limited to the solution of equations of the first and second degrees. In 1505 Scipio Ferreus, a professor of mathematics in Bononia, dis- covered the solution of a particular case of an equation of the third degree. Ferreus communicated his discovery to a favorite scholar, Florido, who challenged Tartaglia, a noted mathematician, to a trial of skill in solving questions. Tar- taglia had, however, discovered the solution of four cases of cubics, and came off victorious. Cardan, Professor of Mathe- matics at Milan, made great efforts to obtain the rules of Tartaglia, who finallj' consented to show his method, which Cardan, in violation of an oath of secrecy' exacted by Tartag lia, published with some improvements, in a work he was then preparing. Lewis Ferrari, a pupil of Cardan, soon after- TEACHING ALGEBRA. 439 wards discovered the solution of an equation of the fourth degree. In 1572, Bombelli, an Italian mathematician, pub- lished a work in which he explained the natnre of the irre- ducible case of cubic equations, which had perplexed Cardan. In 1540 Recorde published his famous Whetstone of WUte, in which the sign of equality first appeared. Vieta (1540- 1G03) was the first to employ general characters to represent known quantities, which was a great step in advance. He also improved the theory of equations and gave the first method of solving them by approximation. Albert Girard (1629), a Flemish mathematician, was the first to speak of Imaginary Quantities; and inferred also by induction that any equation has as many roots as there are units in the num- ber of its degree. Thomas Harriot made the important dis- covery that every equation may be regarded as formed by the product of as many simple equations as there are units in the number expressing its order. He also made several changes in the notation, and added several signs, so that as it came from his hands it differed very little from its form at the pres- ent time. Descartes (1637) made one of the greatest improve- ments by the application of algebra to curved lines, which resulted in a new branch. Analytical Geometry. The science was subsequently enriched by Newton, who dis- covered the binomial theorem, and by Euler, who made exten- sive applications of it. Lagrange was the first to prove that every numerical equation has a root, which had previously been only assumed. Gauss, 1801, developed the subject of binomial equations; W. G. Horner, in 1819, published his celebrated method of solving numerical equations ; and in 1829 Sturm made known his beautiful theorem for assigning the position of the real roots of an equation. The latest improvement is the development of the subject of Determinants. The germ of this theory is found in the writings of Leibnitz. It was revived more than fifty years afterwards by Cramer, and was extended by Gauss and others. 440 METUODS OF TEACHING. It has received its latest and fullest development at the hands of two great English mathematicians, Cayley and Sylvester. Method of Teaching Algebra. We shall now give a brief discussion of the method of teach- ing algebra. We shall present several principles to guide the teacher in the instruction, then show how to teach some of the elementary portions of the subject, and then close the article with a few general suggestions to the teacher. Principles of Instruction. — There are several general prin- ciples which should guide the teacher in presenting the sub- ject of algebra to the beginner. 1. We should lead the jitipil to make the transition from arithmetic to algebra; algebra grew out of arithmetic. This is in accordance with the genesis of the science. It is also indicated by the law of thought from the particular to the general, algebra being a kind of general arithmetic. We should introduce algebraic methods while teaching arith- metic. Mental arithmetic, especially, may be made to flow naturally into mental algebra. Algebraic methods may also be introduced into written arithmetic, as in percentage, /)= br ; also in interest, as i=ptr ; and also in the progressions, etc. In advanced arithmetic, many of the subjects should be generalized and presented in algebraic notation. 2. We should begin algebra with concrete problems, and not with the abstract operations of the science. This is also in accordance with the laws of thought. It is also the historic order; algebra was an outgrowth of the attempt to solve con- crete problems. It makes the subject much easier for pupils, as they catch the spirit of the algebraic method, and are thus better prejjared to understand the abstract operations of the science. The more recent writers on elementary algebra make a great mistake in omitting such exercises as an intro- duction to the subject. 3. The pupil should have a thorough drill in the practice r TEACHINQ ALGEBRA. 44] of algebra. Algebra is a calculus, and the pupil needs to be- come skillful in algebraic manipulations. It la discouraging to nave pupils in analytical geometr}- and calculus, who are constantly making mistakes in the algebraic operations. There should be a large collection of examples in the funda- mental rules, fractions, equations, radicals, etc., to attbrd the means of acquiring this skill. The teacher of elementary algebra should select and prepare two or three times the num- ber of examples found in any ordinary text-book on algebra, and drill his pupils on them. Course of Instruction. — The course of instruction in ele- mentary algebra should include the following things: 1. An Introduction, including the solution of concrete problems and the introduction of the algebraic symbols ; 2. Algebraic No- tation ; 3. Explanation of the Negative Quantity ; 4. Funda- mental Operations; 5. Secondary Operations; 6. Fractions; 7. Simple Equations ; 8. Solution of Problems, etc. 1. Introduction. — To introduce the subject of algebra, take a simple problem in mental arithmetic, and write out the analysis upon the board, and then transform this analy- sis into the abbreviated method of algebra. To illustrate, take the problem, " William has 3 times as many apples as Henry, and both have 24; how many has each?" Illustration.— By arithmetic we solve the problem as follows : Henry's number, plus three times Henry's number, equals 24; Hence 4 times Henry's number equals 24 ; And once Henry's number equals \ of 24, or 6, etc. Now, if we represent the expression, " Henry's number," by some character, as the letter x, the solution will be made shorter, as seen in the margin. If f P'^^ ^ times z equals 24, „ ^ i.,,.,.- >> J hence 4 times X equals 24; we now use 3x to represent 3 tunes x, and ^^^ ^^^^ ^ ^^^^{^ g^ 4x to represent "4 times .r, " the symbol = for the word "equals," and the symbol -\- x+3x=24; for the word "plus," the solution will be ^^^ still shorter, as seen in the margin. This solution is purely algebraic, and is a type of the entire method of algebraic reasoning. 19* 442 METHODS OF TEACUING. The pupil will see that the last solution is the same as the first, except that we use characters instead of words. These characters are called symbols. The pupil may then be shown that 2x, Sx, etc., means " 2 times x," " 3 times a;," etc.; that " one-half of a:," " 2 thirds of .r," are expressed thus : ^x, fx, X 2x or—, — , etc. He may also be told that an expression like 2 3 x+3j:=24, is called an equation. The pupil should then bre drilled on the solution of concrete problems until he is familiar with the algebraic idea, and the fundamental principles of notation. Problems may be selected in which all the simple elements of notation may be gradually introduced. Symbols for known quantities may also be used. For classes of prob- lems, see author's Elementary Algebra. 2. Algebrnic Xotatioii. — The pupil is now ready for a for- mal explanation of algebraic notation. The various symbols should be presented, and the pupil quite thoroughly drilled in reading and writing algebraic expressions. It will be well also to drill the pupil \i\ finding the numerical value of alge- braic expressions b}' substituting numbers for letters. 3. Negative Quantity. — The next step is to explain the meaning and use of the negative quantity^ as this will be needed in understanding the fundamental operations. We first show that a positive quantity means an additive quantity, and denotes that something is to be increased by it ; and that a negative quantity is a subtractive quantity, and denotes that something is to be diminished by it. We next lead the pupil to see that, since positive and negative are opposite in meaning^ the}' may be used to represent quantity considered in opjjosite directions or senses. Thus, if we use -f- to represent a per- son's gains in business, we may use — to represent his losses : north latitude va.ay be denoted b}' -f- and south latitude by — ; future time by + and past time by — , etc. It will thus be seen that the symbols -|- and — may indicate the nature of quantity, as well as tiie operations to be performed on it. 1 TEACHIXQ ALGEBRA. 443 Principles. — We next establish some pi'inciples pertaining to the negative quantity. Thus, since $8 united with $5 gain and $5 loss leaves $8, we infer that uniting -{-5 and — 5 makes nothing, or that uniting a poi,etc.; hence,-6, taken any +"^ ~^* ~^^ ~+a? number of times, as a times, is — ab. Third, b multiplied by — a, means that b is to be taken subtractively a times; b taken a times is nb, and taken subtractively is —ab. Fourth, —b multiplied by —a, means that —b is to be taken subtract ively a times ; —b taken a times is —ab, and used subtractively is — ( — ab), which, by the principles of subtraction, is -\-ab. Hence, we infer that t?ie product of quantities having like signs is PLUS, and having unlike sigreg is minus. 1. Division. — Division is conveniently treated under two cases: 1. To divide by a monomial; 2. To divide by a poly- nomial. There are four things to be considered, as in multi- l>lication: 1. The Co-efficients; 2. The Literal Part; 3. The Exponents; 4. The Signs. The explanation of the division of the co-efficients and letters depends on the principle, Taking a factor out of a quantity divides the quantity by that factor. The explanation of the exponents depends on tlu" 446 METHODS OF TEACHINQ. principle, The e.^ portent of a term in the quotient equals ith *',xponent in the dividend minus its exponent in the divisor. Tlie law of the fi^ns is derived from the law of the sioriis in multiplication. 8. Coinposition and Factorituj — For the treatment of Composition and Factoring, see the author's Elementary Algebra. Attention is called especiall}- to the demonstration of the theorem concerning the divisibility of a" — 6" by a — 6, and theorems siaiilar to it. The usual method is that of mnlhc- viatical induction; the method we have given is much simpler. It is suggested chat the student be thoroughly drilled in Fac- toring, as it lies at the basis of algebraic analysis. The student-teacher may be required to give an outline of the several cases, and show how to teach them. 9. Fractiona — Fractions, in algebra, are to be regarded as the expression ol one quantity' divided by another. The principles are established by demonstration, and then are to be applied in deriving the rules of operation. Let the student- teacher give an outline of the cases, and explain the method of their treatment. Show also what difficulties pupils usuallj- meet with, and how to explain them. 10. Equations. — Tlie elements of equations are simple and readilv taught. Some teachers illustrate transposition by a pair of scales or balances, showing that if anything is put into or taken from one scale an equal quantity must be i)ut into or taken from the other scale. Such an illustration is not needed, however, as a pupil readily gpsps the axiom that if equal !i be added to or taken from equals, the results will be equal. The pupils should be thoroughly drilled on the solu- tion of equations until they are familiar with the general methods and all the special artifices that appl^- to particulai cases. 11. Solution of Problems. — Pupils should have an extensive drill on the solution of concrete problems. The solution of such problems consists of two parts; the forming of the TEACHING ALGEBRA. 447 equation, and the solution of the equation. The first is called the concrete part, the latter the abstract part of the solutioi;. The pupil should have wide and extensive experience in hotl> of these, for it is only in this way that he can become a skill- ful algebraist. He maj'- also be encouraged to make new problems for himself and schoolmates to solve. General Suggestions. — We close the subject with soii;c general su22:estions to the teacher. Literal Notation The teacher should be careful to see- that the pupils have a clear idea of the literal notation. First, they should see clearl}' that a letter represents a general num- ber, and that this number may be integral or fractional. Second, that as involved in an expression, each letter is a factor. There should be a drill with figures, as 3x4x5, and then changing to a x6xc, and then to a6c, until this idea is clearly developed. Positive and Xi^yative. — The pupil should be led to a clear idea of the positive and ?ie9a43 UNIVERSITY OF CALIFORNIA LIBRARY