REESE LIBRARY OF T^ UNIVERSITY OF CALIFORNIA. Accessions No.u w^. Cl,us No -vl NOTES SCIENCE AND ART EDUCATION. WILLIAM NOETLING, PROFESSOR OF PEDAGOGY, STATE NORMAL SCHOOL, BLOOMSBURG, PA. NEW YORK AND CHICAGO: E. L. KELLOGG &-CO. Copyright, 1^95, by E. L. KELLOGG & CO., NEW YORK. PREFACE. I HAVE for some years dictated my instructions on the science and art of teaching, in the form of notes, sugges- tions, and hints, to the junior class of this school ; but a desire has been expressed by students and others to have the work in a more convenient shape for preservation and use, and to enable the many teachers who do not have the advantages of normal instruction to avail themselves of its helps. In compliance with this desire I have prepared the notes for publication: , They do not, however, constitute a methodical or a complete treatise upon pedagogics, but only thoughts and suggestions for prospective teachers and for beginners in school-room work. Every subject has been treated with as much fullness, as well as brevity, as my experience has shown necessary. To subjects in which beginners need most help I have given more space than to others ; this accounts for the dis- proportion in the number of pages devoted to different subjects. The matter will be found in harmony with the best in education and teaching, and presented, it is hoped, with a sufficiency of explanation to make it intelligible to be- ginners. Repetitions occur here and there, wherever it is believed they will be serviceable to those for whom the book is in- tended. WM. NOETLING. STATE NORMAL SCHOOL, BLOOMSBURG, PA., Dec. 19, 1894. 3 VALUABLE BOOKS ON METHODS OF TEACHING. PARKER'S TALKS ON PEDAGOGICS PARKER'S TALKS ON TEACHING CALKINS' How TO TEACH PHONICS . SINCLAIR'S FIRST YEAR AT SCHOOL SEELEY'S GRUBE METHOD OF NUMBERS PAYNE'S LECTURES ON TEACHING . HUGHES' MISTAKES IN TEACHING DEWEY'S How TO TEACH MANNERS ' JOHNSON'S EDUCATION BY DOING ETC., ETC., ETC. $1.50 1.25 0.50 0.75 1. 00 1. 00 0.50 0.50 0.50 *** Catalogue describing all of our publications, over 400 in number, sent free. E. L KELLOGG & CO., NEW YORK. CONTENTS. PACK AUTHOR'S PREFACE 3 INTRODUCTORY CONSIDERATIONS 7 PART I. THE CARE OF THE BODY '. 8 PART II. THE MIND n Chapter I. The Intellect n Chapter II. The Feelings 33 Chapterlll. The Will . 38 PART III. IMPORTANT OBSERVATIONS AND INFERENCES 42 PART IV. OBJECT LESSONS 45 (a) Their Design (b) The Plan or Method of a Lesson PART V. PENMANSHIP 46 PART VI. PRIMARY READING 48 5 6 Contents. PART VII. PAGE ADVANCED READING 57 PART VIII. NOTES AND SUGGESTIONS ON TEACHING THE ENGLISH LANGUAGE. Chapter I. General Considerations 64 Chapter II. I. Oral Language 68 2. Written Language 71 PART IX. SUGGESTIONS FOR TEACHING NUMBERS 87 PART X. GEOGRAPHY. 173 PART XI. HISTORY 185 PART XII. THE HUMAN BODY 189 PART XIII. CIVIL GOVERNMENT 193 PART XIV. DRAWING 194 NOTES ON THE SCIENCE AND ART OF EDUCATION. INTRODUCTORY CONSIDERATIONS. 1. The science of education embraces the principles, laws, or knowledge in accordance with which the education of a child must be carried on ; and the art, the carrying on of the process. In other words, the science is the knowing, and the art the doing. 2. To conduct the education of a child intelligently and successfully, the teacher must possess a thorough knowl- edge of its constitution, physical and mental. 3. Understanding the physical constitution implies a knowledge of its mechanism, the function each organ per- forms, and the relation the various organs bear to one another. 4. Enjoyment is one of the most important elements in our existence upon earth ; and this depends upon health the health of the body and of the mind. Preserving health, or securing it if not possessed, is therefore one of the first things, if not altogether the first, in the education of a child. 7 THE CARE OF THE BODY. 5. To preserve the body in a healthy state, it must have a proper supply, as fast as needed, of the elements that enter into its composition ; it must be kept clean, properly clothed, must have rest and exercise-; and be surrounded by suitable light and proper conditions of atmosphere. 6. The body is composed of a number of kinds of mate- rial, and the quantity of each necessary at any time to pre- serve the health and strength of every organ and power depends upon the kind of activity in which the person is engaged, or, in other words, upon the amount of activity of each organ and power. For example, if the muscles chiefly are exercised, the material of which they are formed must be supplied in sufficient quantity to make up for their waste the amount destroyed by use ; if the bones, the material of which they are made must be supplied in suffi- cient quantity to make up for their loss by use ; and if the brain and nerves, the kind of material that builds them up must predominate in the nourishment taken. 7. The proper nourishment of the body does not, how- ever, alone depend upon the quantity of any kind of food taken, but more, perhaps, upon the quality and upon the preparation it has received. 8. Another important condition of food is variety. An unvaried diet destroys both appetite and health. Walker, in his Physiology, says : " The system craves a varied diet, and living for a length of time on even an abundance of food, if it be unvaried from day to day, will generally result The Care of the Body. in loss of appetite and in disease. * * * When a variety of articles cannot be obtained, varied methods of preparing and cooking the limited supply should be resorted to. ' Good cookery means economy; bad cookery means waste/ " On the other hand, however, there may be such a thing as too great a variety, and this also will destroy the appe- tite." " It is a dictum of mental as well as physical hygiene that it is far better to stint one's self along any other line rather than deprive ourselves of food of needed quality and quantity. I say stint, for it is not- economy. Poor food means poor blood and not enough of it, and this in turn means a brain starving for oxygen. Such a brain is always a weary brain, slow to re- spond and erratic in its activities; and this fatigued, poisoned brain can pever sustain mental processes of high character or strict integrity. Therefore, I say, that in treating of the re- ciprocal influence that obtains between body and mind, the question of diet is one of special importance to those having the care of children, and should be discussed at great length by educators in order that it may receive in every quarter the attention it so richly deserves." Krohn, Practical Lessons in Psychology. 9. Exercise, too, at proper times, and suitable in kind and amount, is a necessity to health and comfort. But exercise of any part of the body tears it down in other words, gradually wears it out, consumes its tissues ; and if this process is continued too long or faster than new ones are formed to take the places of those that have been de- stroyed, pain, in form of weariness or fatigue nature's warning ensues, and is a sign that the safety-point has been passed and that rebuilding is necessary. 10. Rebuilding of brain requires more time than that of muscle or of bone ; brain-workers, therefore, demand more sleep than those who chiefly exercise their muscles. But sleep should be sound, unbroken. Students, when per- mitted to do so, frequently study at hours of the night when they should sleep, and afterwards try to sleep when, from exhaustion or worry, they are unable to do so. But wake- (UNIVERSITY) V 0,,,' ul. Science and Art of Education. fulness, when the physical system needs repairs through rest, implies some kind of functional disorder, and un- doubtedly is a premonition of brain-exhaustion. Dr. J. L. Corning, an authority on cerebral diseases, says : " Derange- ment in the function of sleep is an infallible sign that the proper relation between waste and repair of brain-tissue no longer exists; and that, unless the undue expenditure of brain-force be made to cease, cerebral bankruptcy is im- pending. * * * The injury produced upon the thought and emotional centres by a high degree of worry, conjoined with undue intellection, it is almost impossible to over- estimate; indeed, a very large percentage of cases of brain- exhaustion is directly traceable to this baneful combination of causes." 11. When sleeplessness, headache, loss of appetite, languor on rising in the morning, and general debility manifest them- selves, and drugging begins, the danger-signals are clearly in view, and, if not heeded, a wreck is sure to follow. (For a fuller treatment on securing and preserving health, see Walker's Physiology and Martin's The Human Body.) THE MIND. CHAPTER I. THE INTELLECT. 12. Method of Studying It. What mind is we do not know, in fact cannot know ; for it is not a material thing, a thing that can be cognized by the senses. Like electricity and magnetism, it can be studied only through its manifesta- tions or effects. This method of learning is that of in- ference, inferring causes from their effects learning the promptings of the mind by the appearances and muscular activities that follow them, and that seem to be their effects or that we infer to be their effects. Mental states and activities must therefore be studied inductively. Generaliza- tions must be cautiously made, and no conclusion hastily drawn. Facts, sufficient in number, must be observed, noted, and classified, before a valid conclusion can safely be reached. Nor will one person's facts serve any purpose for those of another ; each must make his own observations, classifications, and draw his own conclusions. This is what each must learn to do in order to become a student of mental phenomena. Neither the teacher nor any one else can do it for him ; all the help that any one can give him is to show him how to study himself and others, and how to test his conclusion. 13. The Body the Servant of the Mind. The mind manifests itself through the body, performs all its observable Science and Art of Education. or visible work through it ; and one of the objects of educa- tion is the training of the body to become the obedient and skilful servant of the mind. REMARK. Under no circumstances must the mind become the mere servant of the body, or the body be allowed complete control of the mind. 14. Forms of the Material World and How Cognized. The material world presents itself to us in six different forms: i. Colors and figures ; 2. Sounds ; 3. Heat and cold ; 4. Hardness and softness, smoothness and roughness; 5. Tastes; 6. Smells. And the organs through which these forms are cognized are the senses, the feelers seeing, hearing, touch- ing, tasting, and smelling. It is, however, the mind that sees, hears, touches, tastes, and smells, and not the organs which it uses to do so. 15. When an object affects a sense-organ, the nerves con- nected with the organ convey the impression to the mind ; in other words, make the mind conscious of it. If the object is of sufficient interest or importance at the time, the mind observes it and takes, so to speak, an impression or image of it. This impression is termed a percept. REMARK. A percept remains before the mind, or in con- sciousness, only as long as the object which gives rise to it affects the sense-organ. 16. Attention. In order that the mind may be in the most favorable state for the reception of impressions, certain conditions are necessary. One of the most important of these is attention, or the concentration of the mind upon the object examined or subject studied. 17. For educational purposes, attention may be divided into two kinds, apparent and real ; the former being only the appearance of attention, the latter the reality. 18. Real attention may be either attracted or directed, and divided or undivided. The Intellect. 13 REMARK. Attracted attention is also called non-voluntary (without an effort of the will), and directed, voluntary (by an effort of the will). 19. Apparent and divided attention, having no value in education, need no further consideration here. 20. Attracted attention is that which is given from interest or novelty in the subject examined or studied, and depends upon the teacher's ability to make the subject of instruction interesting or attractive. REMARK. Attracted attention is the only kind that can be expected from children. 21. Directed, or voluntary, attention is that which comes from an effort of the person giving it. This kind can be expected from persons of sufficient age and judgment to appreciate the value of subjects of study, but not from children. Voluntary attention must, however, ultimately change to non-voluntary. If it fails to do this, the teaching is defective a failure. 22. Attention cannot be given equally well under all cir- cumstances. Its most successful efforts depend upon certain conditions ; the most important of which are : (i) good health, (2) good light, (3) pure air, (4) proper temper- ature, (5) comfortable seats, (6) absence of distracting objects, (7) proper position, and (8) close classification. 23. Conditions of attention, however important, do not imply attention ; they imply simply that everything neces- sary, so far as the pupils and their surroundings are con- cerned, has been supplied. The next thing to do is to secure attention, and this not all teachers can do with equal success. As a requisite, to begin with, the teacher must have the confidence of his pupils. He should be : (i) master of the subjects he teaches; (2) unhampered by the use of a text-book during recitation; (3) cheerful; (4) interested; (5) in earnest. He should begin where the children's knowl- edge ends ; and should excite curiosity. 1 4 Science and Art of Education. REMARK. The teacher should not abruptly pass from one subject to another, but should by gradual steps lead his pupils to it. Inattention is frequently caused by abrupt transitions. 24. Attention must not only be secured, but must be continued or held. The following suggestions, if carried out in the proper spirit, not merely in a mechanical manner will aid the teacher in keeping the attention of his pupils : i. Do not attempt to teach anything that is beyond the ability of the pupils ; 2. Keep curiosity aroused ; 3. Let the age of the pupils determine the length of the lesson ; 4. Let the advancement determine the subject and the lesson ; 5. In teaching children, sense-wholes should be taught be- fore their parts ; 6. Go from the known to its related un- known ; 7. As a general rule, tell nothing to pupils which they can find out themselves or be led to find out ; 8. Vary your mode of instruction, your mode of presenting subjects ; 9. As far as possible, give every pupil a share in the reci- tation ; 10. Use illustrations and apparatus; n. Do not teach longer than you have the attention of your pupils ; 12. Utilize the instincts activity, curiosity, imitation, etc. REMARK ON CONDITION 10 OF THE FOREGOING. If you have apparatus, teach with it. Requiring pupils to prepare lessons from imperfect descriptions or poorly-made illustrations or pictures, when a school has the apparatus with which the subject may be studied, is inexcusable. REMARK ON SECURING AND HOLDING ATTENTION. To secure and to hold attention, something new must be presented, or the mode of presentation must be new. No one can give continued attention to that which yields nothing. 25. Like the other powers of the mind, attention may be cultivated. The following are some of the means that may be used for this purpose : i. Read or relate something of interest and value to the pupils, to be reproduced by them ; 2. Require them to reproduce from memory a con- versation, lecture, address ; 3. Require the reproduction of a problem read to them or by them ; 4. Require the repro- duction of a paragraph, article, or selection read by them. The Intellect. 15 REMARK. Special periods for the purpose of cultivating the power of attention are not to be recommended ; for the same ends may generally be attained in the regular classes with the daily work. 26. Attention Depending upon Pupils* State of Mind. Besides the foregoing conditions of attention, there are others which, on account of their relation to successful school-work, are of sufficient importance to merit separate consideration, i. The pupils may be tired ; 2. Their minds may be occupied with something to them of more interest than that which the teacher is endeavoring to introduce ; 3. They may be in a so-called neutral or indifferent state ; 4. They may be in a state of expectancy waiting, with inter- est, for the lesson to be commenced. In the first case, they should either be excused from fur- ther work, or given some such exercises as the making of forms of different kinds of material ; work in which they can be interested and which will not tire them. The second state may be one of antagonism or opposi- tion, one in which they refuse to take part or interest in the subject which the teacher desires to introduce. In this case the only remedy is to prepare the way, step by step, by the introduction of something which shall cause them to forget their former thoughts and thus prepare them for the introduction of the lesson for the period. The third condition needs the same kind of preparatory treatment as the second, the only difference being that, as a general thing, it requires less effort on the part of the teacher. In the fourth condition the pupils are eagerly waiting for the lesson to begin, and consequently need no preparation for it. 27. Perception Observation Reflection. As already stated, the impression which an object makes upon the mind through the senses is called a percept ; in other words, 1 6 Science and Art of Education. a percept is a mental construction of an external object a construction in the mind by the mind itself. A percept is a product, and the operation or act that gives rise to it is perception. Percepts are of two kinds, original and acquired. REMARK i. Some psychologists apply the term perception to a complete mental picture of an external object, and percept to a single element of it obtained through one of the senses. REMARK 2. Perceptive knowledge, or that obtained from the direct or immediate apprehension of an object, is also called presentative knowledge, and its revival or representation from memory, representative knowledge. 28. Our knowledge begins with experience ; it depends, consequently, upon the completeness and clearness of our percepts ; and the quality of our percepts depends upon observation upon attention to all the particulars or char- acteristics of an object. Careful observation therefore lies at the foundation of education. 29. Not only can the mind observe the outer, the world of matter, but it can also turn itself in upon itself, so to speak, and scan its own states and activities. This process observing what the mind itself is doing is called intro- spection, reflection ; sometimes, internal perception. 30. Memory. Memory (figuratively speaking) is the mind's treasure-house, the power of preserving and recalling the facts of consciousness. When attention is withdrawn from an object, its percept passes from consciousness to the memory. 31. For the convenience of study, memory may be con- sidered under two distinct heads, retention and recollection. Retention is the mind's power of holding or preserving the facts of consciousness, and recollection that of bringing them forth when they are wanted. REMARK. James Mark Baldwin, in his Elements of Psychol- ogy, says : " In considering the entire mental function which we call The Intellect. 17 memory, we find that it involves several factors or stages, which are sometimes treated as distinct operations, but may properly be considered, as we find them, together. Together they constitute a chain of events whereby the mental life of the past is retained and utilized in the present. First, there is the permanent possibility of the revival of a past experience when its first circumstances are repeated; this is called Retention. Next, there is the actual return of the image to consciousness : Reproduction. Third, this image is known as having already been presented in our past experience : Recognition. And finally there is, in most cases, an immediate reference to the exact past time of its first experience : Localization in time. These, taken together, constitute a finished act of memory. Accordingly, memory may be defined as a mental revival of a conscious experience" 32. As before remarked, when an object of perception no longer affects or stimulates a sense-organ, no percept of it is in consciousness ; recollection therefore does not bring percepts before the mind, but transcripts, mental represen- tations, images, ideas, or concepts of them. REMARK i. The term concept is preferable to that of idea for recalled mental pictures, because idea is used in so many senses as to lead to confusion in the minds of students of men- tal phenomena. Trench says : " The word idea is, perhaps, the worst case in the English language. One person, for example, has an idea that the train has started, another had no idea that the dinner was so bad." REMARK 2. Conception is a constructing process, and its products are concepts. REMARK 3. All forms in which past experiences may be re- called and brought before the mind will in these pages be con- sidered as representations or mental pictures, and all mental pictures, except percepts, as concepts. 33. The mind's power to retain facts depends greatly upon the state or condition in which it is when it receives them. Among the most important conditions of retention are the following : i. Healthy and fresh state of body and mind ; 2. Undivided attention ; 3. Thorough, clear, and distinct comprehension ; 4. Lively and sincere interest ; 5. Determination, or force of will ; 6. Repetition ; 7. Suffi- cient time for the impression to be made. 1 8 Science and Art of Education. 34 An examination of the foregoing conditions of reten- tion reveals the fact that retention is not an active power of the mind, and that it is cultivated only through the ac- tivity of the other powers ; its highest degree of success depends, therefore, upon the perfection of the activity of the other powers. 35. Mental images play an important part in remember- ing. Kay, in his book on the memory (page 208, etc.), says : " The subject of mental images is one that has hitherto re- ceived but little attention, and yet it is one of the deepest in- terest, and calculated to throw light upon many obscure mental phenomena. Whenever a sensation or an idea is presented to the mind, a mental image or conception of it must be formed in order to its being perceived or understood. In proportion to the clearness and distinctness of the image will be the under- standing of it by the mind, and the hold taken of it by the memory. " As there are different kinds of sensations and different classes of ideas, there exists a like variety among mental images ; and some minds excel in some, others in other. Thus, some may excel in the formation of visual images, others of auditory ones. The former will remember best those things that are presented to the eye, and of which they can form visual images ; the latter, such as are addressed to the ear, and form auditory ones. The former will take in and remember what they read, the latter what they hear ; the one will learn a language most easily by the eye, from books ; the other by the ear, from conversation. Some, in listening to a discourse, image every word they hear as it appears to the eye ; while others, with the auditory faculty largely developed, will image what they read as if it were addressed to the ear. Others, again, in reading or in listening to a discourse, will attend only to the sense or meaning, and form sense-images. These can give the substance of what they have read or heard with great accuracy, though they may not perhaps be able to recall any of the words. In each case it is of importance to ascertain in what direction the image-forming power of the mind chiefly lies. " Further, not only are there images of the eye and ear, and of the other senses, but there are also images of muscular movements, as of the tongue and hand. Some may not re- member much of what they see or hear, but remember readily what they say or do. Hence some children learn best by re- peating aloud, others by writing down what they wish to remem- The Intellect. 19 her. Most persons have probably observed, in writing a word in regard to the spelling of which they are sometimes in doubt, that if they write it at once, without thinking about it, they usually spell it correctly; but if they doubt and hesitate and think, they become uncertain, and most probably spell It wrong. The reason is that the mental image which directs the hand is, in this instance, a surer guide than that furnished by the intel- lect. In such cases the more the mind is engaged in thought the less able is it to listen to those inner promptings of our na- ture the muscular images of past movements, on which so much that is finest and most delicate in our action depends. " But not only are there in the mind mental images of sensi- ble objects, and of muscular movements, of what we feel and what we do, but every thought, however abstract or apparently disconnected from sensible objects, has its image in the mind. We can only conceive an abstraction by having an image of it. The abstract idea of a triangle, which is not any particular tri- angle, but represents the properties common to all triangles, has as much its image in the mind as any individual triangle that may have been before it. Further, we mast regard each abstract idea as having a physical state corresponding to it ; and hence we can localize abstract ideas and recall the occa- sions when they were present." 36. Not only must facts be retained in the mind for future use, but they must be recalled when wanted. Minds differ much in their recalling power ; some readily recall one kind of facts, others another kind ; but, in either case, the ease with which they can be brought forth depends upon the manner in which they were stored away in the memory. If they are associated, or presented to the mind, in a systematic, related manner, they will be returned in the same manner, every link in the chain of ideas suggest- ing its neighbor. 37. Concepts or ideas may for educational purposes be associated in four ways : i. By contiguity; 2. By similarity; 3. By contrast; 4. By cause and effect. Contiguity means adjoining, contact ; similarity, likeness, resemblance ; con- trast, dissimilarity ; cause, that which produces a change, an effect ; and effect, that which has been brought about or produced by a cause, Science and Art of Education. REMARK. Other modes of association besides the foregoing" are sometimes given by writers on psychology, but as they are of no service to teachers, they are here omitted. 38. The underlying principle of contiguous association is that concepts or experiences which occur together or in immediate succession afterwards tend to revive one another Observation, too, seems to teach that the mind integrates or " completes any process upon which it enters, if it has performed the same process before." 39. Contiguity may be in time or in place. One thing suggests another that appeared before the mind with it or immediately before or after it. Things that were before the mind either at the same time or in successive time, or that were before it together in space (place), are revived together or have a tendency to reappear together. REMARK. A number of dissimilar objects may be placed in a fixed order and remembered or recalled in that order, by as- sociating them in the mind invariably in the same order. 40. An object or concept will bring before the mind others that bear a resemblance or an analogy to it. When a relation of some kind can be discovered among objects or concepts, and this relation is made the basis of a sys- tematic arrangement of the objects or concepts, so that any one in the series, in consequence of the relation, will sug- gest the next, the recalling process is much easier than when the association is arbitrary, mechanical, without re- lation. 41. By the association of similars we are enabled to form classes of objects or concepts, on account of their resem- blance in form, material, or quality, and to recall them. By their resemblances, also, we trace the relations of words and thus discover their meanings. The study of languages is greatly shortened by the association of similars. 42. For the purpose of firmly fixing anything in the mind and readily recalling it when it is wanted, association by The Intellect. contrast is more serviceable than association by resemblance or analogy. Tate, in his Philosophy of Education, says : " Associations of resemblance are rarely so vivid as those of contrast ; and hence it follows that scenes or events which are in contrast with each other are more likely to* be remembered than those which have a resemblance. Con- trast, like light and shadow, makes the objects more promi- nent." REMARK. Only such things as have some points of similarity can be contrasted. Those which cannot be compared cannot properly be contrasted. 43. Landon, in his School Management, says : " The natural relationship which links ideas by means of cause and effect renders them eminently suggestive of one another ; and where the connection exists, to fix it clearly in the mind is one of the most powerful means of associa- tion at our disposal. It is especially valuable in lessons from the physical or natural sciences ; and should be much more generally employed in the teaching of such subjects as history than it appears to be." Tate, the author above quoted, says : " The minds of children are so constituted that they most readily remember effects in connection with their causes ; for example, they readily associate the light of day with the presence of the sun ; storms with winds and clouds ; the heat of summer with the long days of sunshine ; the improvement of the mind with application to study ; misery with crime, and happiness with virtue ; and so on. Associations of this kind are most interesting and instructive ; one idea be- comes the nucleus of a whole series, and idea becomes so linked with idea that we are enabled to form a con- tinuous chain of them. Thus we readily remember the following chain of associations : Rain falls from the clouds ; the clouds are chiefly formed by winds and mountains ; the cold on the tops of the mountains condenses the moisture 52 Science and Art of Education. in the air, and thus clouds are formed ; the cold on the tops of the mountains is caused by the thinness of the air, etc. ; thin air is colder than dense air, because it has greater capacity for heat, and so on." 44. From the foregoing statements concerning the link- ing of concepts it will readily be observed that there are really but two forms, or modes, of association, arbitrary or mechanical, and philosophical or logical. Association by contiguity is arbitrary or mechanical, and the association of concepts by such a relationship that one calls into con- sciousness another bearing a resemblance of some kind to it, or belonging to the same logical connection, is properly termed philosophical or logical. 45. Association by contiguity, of necessity, has its place in the acquisition of knowledge; but philosophical or logical relationships are more valuable in the development of the mind, because they call into activity the higher mental powers. No philosophical or logical bond can be dis- covered without thought. REMARK. Children and the illiterate depend chiefly upon mechanical associations ; older people, and especially the edu- cated, use philosophical or logical relationships. 46. There is only one mode of acquiring facility in the suggesting, or recalling, process, and that is intelligent, per- sistent practice. 47. The following are some of the subjects that may be taught by resemblance and contrast. A. RESEMBLANCE. a. Geography. North and South America ; South America and Africa ; California and Spain or France ; Italy and India ; Australia and Cuba ; Jorth America and Europe ; Pennsylvania and Missouri or West Virginia ; New York and Boston or Chicago ; Baltimore and Philadelphia ; Richmond and Albany ; Atlantic Ocean and Pacific Ocean ; products of East Indies and West The Intellect. 23 Indies ; climate and products of southern U. S. and those of India ; etc. b. History. Settlement of Massachusetts and of Virginia, of New York and of Massachusetts, of Pennsylvania and of Maryland, of Ohio and of Connecticut, of California and of Kansas, of New Jersey and of Georgia ; Washington's and Jefferson's administrations, Washington's and Lincoln's ; Columbus and Captain Cook, Lincoln and Garfield, Grant and Napoleon, Bacon and Newton, Pestalozzi and Horace Mann, Bryant and Longfellow, Daniel Webster and Jrhn C. Calhoun, Horace Greeley and John Bright ; etc. B. CONTRAST. a. Geography. The Old World and the New ; the two hemispheres (eastern and western); London and New York ; Spain and Italy ; coast of Europe and of North America ; eastern coast of North America and west- ern coast of same ; valley of the Mississippi and that of the St. Lawrence ; North and South America ; Africa and South America ; Philadelphia and Chicago ; etc. b. History. Settlement of Jamestown and of Massachu- setts Bay Colony ; colonists of Virginia and of Massachu- setts ; Dutch of New York and Quakers of New Jersey and Pennsylvania ; colonists of Maryland and of Virginia ; Washington and Jefferson ; Clay and Webster ; etc. c. Physical Geography. Surface ot Pennsylvania and of New York ; climate of Pennsylvania and of New York or Massachusetts ; products of Pennsylvania and of Massa- chusetts, Maine, or Virginia ; climate and productions of North America and of South America ; valley of the Danube and of the Mississippi ; climate and productions of Russia and of France, of Ireland and of Australia, of Arabia and of India, of Siberia and of British America, of Southern Africa and of the southern part of South America, of North Temperate and South Temperate Zone ; etc. C. MISCELLANEOUS. Octagon and circle ; degrees of hardness and softness ; resemblances of color ; light and Science and Art of Education. heat ; cylinder, cone, and sphere ; forms of letters of the alphabet ; spelling of words of similar forms and sounds; etc. 48. The learning of words depends upon association. The child associates the name of a word with the form of the word, and either one of them suggests the other. So also in acquiring knowledge are the names of things asso- ciated with the things themselves, and thus remembered and recalled. REMARK. The work of the primary teacher consists largely in helping the children to form permanent associations. 49. Repetition and reviews constitute the main reliance for forming lasting associations ; but the repetitions and reviews must not be allowed to become monotonous. Pleas- urable emotions, or interest, can be excited and kept up in no other way than by variety and novelty. 50. If memory is found weak in any particular direction, in recalling names, dates, etc., for example, and any one's duties demand much work of this kind, the only remedy is daily exercises of the kind required. It is wonderful to what an extent the memory can be trained in any special direction by persistent efforts. 51. Sully says : " Committing anything to memory is a severe demand on the brain-energies, and should, so far as possible, be relegated to the hours of greatest vigor and freshness. The morning is the right time for learning. In addition to selecting the best time, every resource should be used to make the subject as interesting as possible." REMARK. A ready memory is undoubtedly an invaluable possession, yet it needs to be carefully kept within its legitimate bounds ; it should not be permitted to supplant any of the other powers perception, imagination, judgment, etc. 52. Imagination. Imagination is the power which forms or constructs mental images. It is divided into constructive, reconstructive, and productive invention. The Intellect. 2$ 53. Construction may take place from observation or from description from what we ourselves observe or from what we learn from others. Perceiving is a constructing process ; it is forming images in the mind ; and though not generally called imagination, it deserves as much that name as any other picturing process performed by the same power. 54. In the study of all branches of knowledge in which, in the absence of the objects or phenomena, mental pictures are required, the imagination constructs and paints them. In the study of geography and history the use of the con- structive imagination is indispensable. 55. Reconstruction takes place in recollection or from memory ; it is rebuilding that which at some previous time had been built or formed ; it is constructing according to a former pattern or model. 56. Reconstruction is employed in all recitation. The pupil reconstructs his former constructions, and the teacher and the class observe whether they are correct or reasonable. REMARK. The criticisms made by the teacher and the pupils should enable the one who recites to correct and complete his constructions. Simply pointing out errors, without indicating how they may be corrected, has some value ; but pointing them out in such a way as shall enable the student to see them and correct them is the proper way to make the corrections. Criti- isms should always be helps, not hindrances. 57. Production, or invention, has reference to original combinations or constructions. The ability to make these cannot be directly taught ; it can only be encouraged, not learned ; it can only be acquired by those who have the talents and the patience by continued study and practice. One who succeeds in the highest forms of original construc- tion is said to be a genius. 58. The teacher should not only allow his pupils the exercise of originality, but should encourage it whenever it is practicable to do so. Throwing them as much as possible 2 6 Science and Art of Education. upon their own resources is one of the best ways of doing this. 59. Currie, in his Principles and Practice of Common- School Education, says : " Observation is limited by very narrow boundaries of time and space ; to whatever extent we pass these, it must be on the wings of imagination. Accordingly, descriptions of natural scenery and scenes from life, real or ideal, are the field in which this mode of intelligence must be exercised ; and both are very rich in materials. " When the pupil has observed the elements of the landscape at home, he is required to carry these abroad and, by modifica- tion, interchange, and amplification, to construct another land- scape there; the hill, rivulet, meadow, and wood of his own native district become the snow-clad peak or volcano, the mighty river, the far-spreading desert or cultivated plain, the trackless forest, of other lands ; from the summer's heat and winter's ice, whose effects he observes at home, he passes to the heat of tropical, and the cold of arctic, regions, with the lux- uriant vegetation of the one and the stunted growth of the other; the plants and animals of his own country, with their interesting habits, serve as a standard by which he may estimate their representatives in other countries; and the notions of the adaptations of labor, and the modes of life, which he forms from what he sees around him, are drawn upon to construct pictures of industry and the habits of other races of his fellow- men. Then it is largely by the imagination that a knowledge of life is gained, whether of individuals or of communities. The life of home or school, and the life of society so far as the child sees it, are limited in their incidents ; yet the teacher hesitates not to tell or read the story of human life, in its spheres and with its diversified enjoyments, in the belief that the pupil's experience, narrow as it may be, will enable him to realize the emotions portrayed, and gather up the lessons suggested. Biography and history are the natural sources of supply for materials of this sort; narratives of adventure by sea or land ; descriptions of manners and customs ; incidents in the life of men or societies which embody the virtuous emotions of our nature. Ideal life may come in to increase the store of materials ; it is equally rich in instruction with the life recorded in history itself." 60. The happiness of childhood depends almost wholly upon the use of the imagination. The plays of children are nearly all imitations of the real work of older people, and The Intellect. 27 serve as a preparation for the later actual duties of life. Thus they make dolls, dress them, feed them, put them to bed and rock them to sleep ; they cook, bake, set tables, wash dishes, clean house, make parties, receive callers, build dams, mills, houses, railroads, forts ; ride horses, keep store ; act soldiers, doctors, preachers, teachers, etc. REMARK. The statement is sometimes made that the imagi- nation is more active in childhood than it is at any later period of life, but such an opinion rests upon superficial observation or investigation. The imagination is not stronger, or more active, in early life than it is at a later period ; on the contrary, it is weaker, but its activities are more noticeable, because nearly all of them are, of necessity, imitations of the actual work of grown people. 61. The use of the imagination is required in every calling in life. Nothing can be done intelligently and skilfully without a mental picture as a pattern. The mechanic, the dressmaker, the baker, the cook, the farmer, the doctor, the lawyer, the orator, the preacher, all must use it to meet with success in their vocations. 62. To feel with a person and for a person, it is necessary to imagine ourselves in his place. The chief reason why some persons seem to be unsympathetic is that their imaginative powers are sluggish or dull; they cannot place themselves in the position of those who are in sorrow or distress. 63. The imagination is the faculty which constructs forms of beauty ; it is therefore one of the powers that give culture to taste and refinement. 64. James Freeman Clarke says : " No man can be wholly unhappy who is accustomed to look for beauty in nature and in human life. His is a joy which never wearies. " All mere drudgery tends to stupefy the imagination ; and all work is drudgery which is done mechanically, with the hand and not with the mind; when we are not trying to do our work as well as possible, but only as well as necessary. Science and Art of Education. Such work stupefies the ideal faculty, quenches the sense of beauty. ' No matter how lowly the labor may be, if a man performs it as well as he can, he is an artist.' But when a man tries to shirk his work, when he does it in a slovenly manner or way, not as well as he might, then he becomes a drudge, even though his work be that of a poet or a sculp- tor. He ceases to exercise his ideal faculty, and stupefies it. Then the sense of beauty dies out of his mind. " If men are taught to look for beauty in all that they see, to embody it in all that they do, the imagination will be both active and healthy. Life will then be neither a drudge nor a dream, but will become full of God's life and love, and we will be brought into the love of that divine beauty which is above all, through all, and in us all." 65. Explanation of Terms i. Analysis consists in separating a complex whole, whether material or mental, into its elements the parts of which it is composed. 2. Synthesis consists in forming a whole of its parts. 3. Ab- straction is mentally withdrawing certain characteristics or qualities from objects. It may also be regarded as the withdrawing of the mind from all the other properties of objects except those under special consideration. 4. Gen- eralizing is finding the general or common characteristics in a number of objects or concepts. 5. Classification is grouping similars into classes, and embraces generalization. 6. Comparison is simultaneously giving attention to several objects to discover their agreements or differences. 7. General concept is the term applied to a class, or to a group of all the common properties of the objects that compose the class. A general concept, therefore, cannot be imag- ined, it can only be thought. 8. To apprehend is to seize, to take possession of ; to comprehend is to understand. 9. A thing is known when it is assigned to its proper class. 10. Apperception is the appropriation of the new by the The Intellect. 29 old ; it is an organizing process, n. Elaboration is an- other name for thinking. 66. Judging. When the mind compares two objects or concepts directly with each other to determine their rela- tion, the process is termed judging, and the result is a judgment. The objects compared may both be physical (material) or mental, or one may be physical and the other mental ; that is, we may compare one object (physical or mental) or act with another, or we may compare it with our concept of it or of what it should be. The concept or image is then taken as the standard, or measure, of the comparison. Determining whether a thing is sour, sweet, or bitter ; hard or soft ; cold or warm ; black or white ; weak or strong ; coarse or fine ; suitable or unsuitable ; good or bad ; right or wrong, are all acts of judgment. " We judge whenever we affirm or deny one thing of an- other. Everything we know, or think we know, involves an element of judgment, and, when it becomes distinct knowledge, can be explicitly set forth in a proposition. "An expressed judgment is a proposition." Sully. 67. Without the use of judgment, no advance step can be taken in the acquisition of knowledge. Judgment is among the earliest powers exercised by children ; every act of discrimination requires it is an act of judging. Ideation employs it in completing its images. REMARK i. Thinking is a general term applied to discover- ing relations ; and its three successive stages are conception, judging, and reasoning. REMARK 2. The several intellectual processes through which the material of knowledge passes from the concrete to the com- plete general concept are also usually given as three : i. Com- parison ; 2. Abstraction ; 3. Generalization. REMARK 3. The last step in the conceiving process is that of denomination, giving a name to the concept. 68. Judgments may be explicit or implicit ; that is, we may judge consciously or unconsciously. 30 ' Science and Art of Education. 69. Reasoning. Not all objects or concepts can be di- rectly compared with each other. Whenever direct com- parison is impossible, a third object or concept must be found with which each of the others can be compared and their relation determined. REMARK. Frequently more than one intermediate term of comparison is necessary to determine the relation of the two objects or concepts under consideration. 70. When the relation of two objects or concepts is de- termined through on or more intermediate ones, the pro- cess is called reasoning. Examples of reasoning : i. Given the cost of 4 Ibs. of butter, to find the cost of 3 Ibs. Here the intermediate term, or number, is i Ib. 2. To find the number of men that can build 80 rods of wall in 16 days, if 12 men can build 50 rods in 15 days. Here there are two intermediate terms, rods and days ; and the terms of comparison are one rod and one day. 3. A house and lot costing $6750 were sold at a gain of 12^ per cent. ; how much was received for them ? Here the term of compari- son is i per cent. 4. On butter sold at 40 c. the gain was 25 per cent. ; what was the cost ? Here the term of com- parison is i per cent. 5. Stealing is a violation of law ; a violation of law is a crime ; therefore stealing is a crime. Here the intermediate term is a violation of law. 6. Per- sons who tell falsehoods are not believed when they tell the truth ; Alias tells falsehoods ; therefore he is not be- lieved when he tells the truth. Here the inteumediate term is falsehoods. 71. Kinds of Reasoning There are two kinds or modes of reasoning, the inductive and the deductive. When a general truth is inferred or discovered from the examination and comparison of a number of individual facts the process is called induction ; and when a particular truth is found through a general truth it is called deduction. The appli- cation of general principles to cases that come under them The Intellect. 3 1 or are included in them is also deduction. When the child finds that a hot stove, a gas-flame, a hot coal, etc., burn its fingers, and that fire is the general cause of the heat, it comes to the conclusion that fire burns ; and it reaches this conclusion inductively. Afterwards when the child keeps its fingers from hot objects, it does so because it rea- sons deductively that all hot objects burn. 72. All the known laws that govern the material universe have been learned by induction. The naturalist cannot make nature's laws ; he can only discover them. 73. Leading pupils to discover their own rules, principles, and definitions, from the examination of a number of spe- cial cases, is inductive teaching. The deductive method is the reverse of the inductive ; by this method pupils are taught to apply rules, principles, and definitions. REMARK. The inductive is the method of acquiring knowl- edge ; the deductive, of using it. 74. The Quantity of General Concepts Concepts, be- ing aggregates, or syntheses, of attributes or properties, are said to have quantity. By the intensive quantity of a con- cept is meant the sum of the qualities which constitute it and distinguish it from other classes of concepts ; and by the extensive quantity is meant the sum of the concepts that may be classed under it. Thus, a quadruped is an animal having four feet ; four feet constitute the intensive quantity ; and the number of animals having four feet, the extensive quantity of the term quadruped. 75. Formal Reasoning. Every logical form of argument (syllogism) or reasoning consists of three propositions, judgments, or statements, called, respectively, the major premise, the minor premise, and the conclusion. The sub- ject and the predicate of a proposition are called its terms ; the major term being that of widest scope, or extent, in the major premise ; the minor, that of least extent in the minor premise ; and the middle, that found in both the others, 3 2 Science 'and Art of Education. and through which they are compared. The middle term includes the minor and is itself included in the major. Ex- ample : A is B (is included in B). C is A (is included in A) ; therefore C is B (is included in B). Or, taking a concrete example : Man is mortal ; Smith is a man ; there- fore Smith is mortal. 76. Systematization. Arranging concepts or classes of objects according to some order or plan is called systema- tizing. Arranging the classes or the grades of a school, the goods in a store, our daily work, etc., are examples of sys- tematizing. 77. Intuition. Intuition means immediate beholding. We can immediately behold facts of matter and of mind, and we can immediately behold, or cognize, necessary and universal principles. The senses behold facts of matter and of mind ; and therefore give us empirical intuitions, or percepts ; the Reason beholds necessary and universal principles, and gives us rational intuitions. 78. Of the correctness of no other knowledge can we have less doubt than of our rational intuitions. For exam- ple, we can for a certainty know that things which are equal to the same thing are equal to each other ; that the whole of anything is greater than any one of its parts ; that every effect must have a cause ; that no object can exist without time or space. REMARK. A principle or truth is necessary when we cannot think its contrary; it is universal when it is believed and true throughout the universe when it has no exceptions. 79. Sources of Knowledge As has already been shown, the mind has three sources of knowledge, perception (em- pirical intuition), thought (logical conclusions), and the Reason (rational intuitions) REMARK. Perception, memory, imagination, comparison, abstraction, generalization, conception, judging, reasoning, sys- tematizing and rational intuition, belong to the intellect, the knowing power, CHAPTER II. THE FEELINGS. 80. Numerous divisions of the feelings have been made by psychologists ; the following, from Dr. A. Schuyler's Empirical and Rational Psychology, are among the simplest, i. Physical: sensations, instincts, and appetites. 2. Vital: feelings of rest, as fatigue, of vigor or languor, and of health or sickness. 3. Psychical : emotions, affections, and desires. REMARK. Sensations are feelings. A sense is an organ through which we feel ; hence the senses are feelers. 81. Our feelings may also broadly be divided into pleasures and pains, or into physical feelings and mental feelings. Physical feelings come from affections o( the physical or- ganism, and mental feelings from cognitions and thoughts. Mental feelings, in general, without regard to the usual subdivisions, will alone be considered here. 82. Mental feelings are both causes and effects ; as causes they move the mind to action, and as effects (as be- fore stated) they result from cognitions and thoughts. 83. Our pleasures and our pains, our joys and our sor- rows, depend upon our feelings ; or, perhaps better, are themselves states of feeling. Whatever produces pleasant or agreeable feelings is done with comparative ease ; and whatever causes painful or unpleasant feelings is wearisome and exhaustive. Whatever excites the feelings pleasurably is said to be interesting ; and whatever is interesting strengthens us and hence aids us in the performance of duty. Discouragements, worry, or whatever else depresses the feelings, reduces strength and is a hindrance to the performance of work, Anticipation or hope of success 33 34 Science- and Art of Education. excites pleasurable feelings and consequently lightens labor. Expectant enjoyment as the fruit of success strengthens us and enables us to work with ease. 84. With children, the time that intervenes between the performance of duty and its resultant enjoyment must be short ; their experiences are too limited to expect much enjoyment from that which is far off and, to them, of doubtful realization. 85. That which affords immediate enjoyment is its own stimulus to interest and exertion ; but that which gives only anticipated, distant realization of success and enjoy- ment needs to be invested with novelty and interest to ex- cite curiosity a desire to know as a condition to lively attention and exertion. 86. The motives that produce pleasurable emotions in some natures do not do so in others. The love of praise urges some to action ; hope of success, others. A student will study hard, day in and day out, early and late, to at- tain the end he has in view prosperity or eminence. A man will perform disagreeable as well as severe physical labor if it will minister to his ultimate comfort or enjoy- ment. It is evident, therefore, that our feelings play an important part in the performance of all kinds of work, especially in that of education and that the motives which excite them and urge the student to action should be care- fully studied by every one who desires to become an intel- ligent and a successful teacher. Garvey, in his Human Culture, says : " To interest pupils in their studies is the great secret of success in teaching ; but the interest of the pupils is best awakened by exciting their curiosity, by hold- ing out to them, in a pleasant manner, whatever of strange, new, and wonderful the proposed study contains to reward their perseverance and attention. * * * The self-sus- taining power of pleasurable feeling ought to be a grand lever in the hancls of the educator. If we once make the The Feelings. 35 subject of study a source of pleasure, we may safely leave the mastery of it to the spontaneous energy of the pupil's mind. * * * By exciting the emotions we excite the inner forces of the mind, which cause it to expand and unfold its faculties jijst as the influence of the seasons excites the dormant forces of the plant." REMARK. Excitement, as here used, does not mean passion, but such interest or pleasurable feeling as shall carry the stu- dent through his work with comparative ease, such as shall arm him with strength and urge him onward to the reward of his labor. 87. Character. What a man really is constitutes his character ; what others think of him, his reputation. Character depends largely upon the feelings ; and since character is the man makes him either worthy or unworthy of the confidence and respect of his fellow-men the moral feelings cannot be overlooked in a course of instruction. 88. The feelings cannot be educated ; they are not think- ing powers ; they can only be trained, and they must be so trained that right conduct, right and worthy motives, shall afford pleasure ; and wrong and unworthy motives and con- duct, displeasure and pain. 89. The character of the teacher has much to do with the moulding of that of his pupils. He must be consistent ; he must himself be and do what he would have his pupils be and do. He must not only hold up for their admiration that which is noble and good, but must exemplify it in his own daily life. He must show not only displeasure, but positive abhorrence, of that which is ignoble, wrong, mean, or debasing. Young people, especially children, are imita- tive beings, and both consciously and unconsciously acquire the habits of those with whom they are frequently associ- ated. They are in the presence of the teacher five to six hours a day during not fewer than six months of the year, and, too, during the most pliable period of their lives ; that his influence must be the main factor in shaping their 36 Science and Art of Education. thoughts and habits must therefore be evident to all who properly consider the matter. Hence his character should constitute one of the principal elements of his qualification. 90. Good conduct comes from good thoughts, and good thoughts come from associations and surroundings that in- spire them or prompt them. The mind is ever active ; it must have something to act upon. Those who have the charge of children must provide this material. If they fail to do so, the children will make their own selections ; and, owing to their inexperience their inability to judge cor- rectly as to what is best for them and the natural tenden- cies to prefer present and temporary enjoyment, to future permanent good, they will, with the rarest exceptions, choose that which will in the end prove harmful destructive of success and happiness. 91. Beautiful surroundings, such as well-arranged and well-kept school-grounds, neat and clean school-rooms, tastefully decorated walls, are important factors in the cul- tivation of good taste and right feelings. Nor must the teacher's own appearance be left out of the consideration. As has already been stated, but will bear repetition, chil- dren are imitative creatures, and both consciously and un- consciously adopt many of the habits and traits of those who are their daily associates. The teacher's influence may therefore justly be considered even greater than that of the parents ; for, besides being the instructor of the children, during their school days he is continually in their presence, and this causes them to be impressed more or less firmly not only with his appearance, but even with his modes of speech and action. 92. Children should be trained to conscientiousness ; they should be led to take pleasure in that which is right and good, and to hate that which is wrong and bad. Lan- don, in his School Management, says : "Although the germ of the conscience is born within us, it depends almost en- The Feelings. 37 tirely upon the kind and extent of the moral, religious, and intellectual training we undergo as to what strength and correctness of action it shall attain. We should therefore lose no opportunity of associating emotions of pleasure and satisfaction with right doing, of enlightening the children as much as possible as to the nature of various actions, of strengthening the judgment by suitable exercises, and of ennobling the sense of duty. The cultivation of the con- science so that it may be a sure guide to the children, and that they may readily obey its dictates, should engage the teacher's attention throughout the period of school life. This training is for the most part indirect, but it is none the less important and should be none the less sure." 93. The feelings cannot decide as to what is right or what is wrong ; decisions belong to the knowing powers. The feelings can only prompt the will to choose and to act in accordance with the decisions of the intellect. The feel- ings are moved by the intellect, and can be changed alone by it. Good thoughts produce good feelings ; bad thoughts, bad feelings ; sad thoughts, sad feelings ; pleasant thoughts, pleasant feelings ; angry thoughts, angry feelings. Angry feelings can be changed to good or pleasant feelings by changing the thoughts that give rise to them to others of a good or pleasing nature. 94. We are responsible for our thoughts and feelings. The fact that we can change them proves it. 95. Strong feeling, excitement, or passion prevents sound, sober thought. It is impossible to reason with an angry person. If we desire to reason with him, we must tell him an amusing anecdote or something else that will change the subject of his thoughts. The same (as before stated) applies to pupils in a school. If any of them are angry or sulky, they cannot study. The remedy is a change of thought, by means of an anecdote, an interesting talk or conversation, an illustration, or an experiment. REMARK. Conscience includes knowing and feeling. CHAPTER III THE WILL. 96. The only power of the mind whose operations remain to be considered is the will. This is the power that directs, chooses, and acts; it is the executive power of the mind. It directs the intellect gives it its work, sets it in motion, and waits for its conclusions. If the thing considered by the intellect is capable of ministering to our gratification or well-being, a desire or longing for it follows. It remains, then, for the will to be governed by this desire, or to refuse to do so, in view of other considerations. REMARK. One of the distinguishing characteristics of will is a conscious effort to attain a known end. 97. The manifestations of will may be classed under two heads, untrained and trained; in other words, brute and rational. An untrained will is one that is governed by im- pulse or passion. A person under the influence of such a will disregards the light of judgment and reason, and is therefore said to act contrary to reason, to be unreasonable, stubborn, obstinate, headstrong, ungovernable. This state of will is that of children and of the uncultured an evi- dence of an untrained mind. A rational will is one that is not hasty, waits for the decisions of judgment and reason, and is governed by the highest good of the individual. Such a will is said to be strong when it inflexibly ministers to the best interests of the person, as dictated by the de- cisions of the intellect. It is weak when it permits itself to be controlled by the appetites or by any unworthy motives. The glutton, the tobacco smoker and chewer, the opium-eater, the inebriate, etc., have weak rational wills. Like children, they are under the domination of their ap- 38 The Will. 39 petites, and the longer they permit themselves to be thus ruled the weaker the will becomes. 98. When our appetites and our passions gain the mastery over us, they become tyrants, dethrone the will, and reduce us to a state of imbecility pitiable objects ! No one need, however, be under the power of his appetite or controlled by any debasing motives. The will, like other powers, can be trained; but the training must begin early and continue as long as may be necessary for the individual to acquire self-control. 99. One of the first things a child should be taught is un- questionable obedience to those in authority over it. Besides obedience, neatness and cleanliness of person, truthful- ness, self-respect, respect for the rights of others, politeness, and kindness should early be impressed upon its mind. Permitting a child to have its own way is no kindness to it. On the contrary, it is doing it a lasting harm. The will is trained by rigidly requiring the right to be done and the wrong to be shunned. In this way right-doing becomes a habit, and this habit is a right will. 100. Since the feelings urge the will, the importance of the formation of right feelings or desires becomes appar- ent; for if our desires are in harmony with our best judg- ment, and their force is of sufficient strength to move the will in the same direction, we have our powers under perfect discipline or control. 101. The will is simply the person willing or exercising the power of willing. He is at liberty to select and to do that which will minister to his well-being, either physical or spiritual, or that which will prove destructive of it. The will, therefore, is an important factor in the formation of character. 102. The power to will, to select one of several duties, acts, or things, implies the ability or freedom to reject any or all of them, and consequently makes us responsible for -^t^f^ (UNIVERSITY Science and Art of Education. our acts. Responsibility does not, however, apply only to overt acts, but even to thoughts. This is evident from the fact that the will can change the subject of thought, or can direct what it shall be. 103. Since good conduct comes from good thoughts and bad conduct from bad thoughts, the importance of having the mind occupied with wholesome thoughts is too plain to require further proof. REMARK. Here, again, the importance of beautiful surround- ings, suitable associates, and wholesome literature or reading- matter becomes evident. 104. Bad thoughts can be expelled from the mind only by the introduction of good ones the good will drive out the bad, and similarly the bad will drive out the good. Here, again, we see that we are responsible for our thoughts. 105. As before stated, the will and the feelings depend upon the intellect. The feelings are excited by the thoughts, and the will is moved by the feelings. The feelings of all pers.ons are not, however, excited with equal ease. Some are moved very quickly, others very slowly, and between these extremes many grades are found. Those who are easily or quickly excited are said to be passionate, impul- sive; they act without proper consideration, without waiting for a complete report or decision from the intellect. Rash or impulsive persons are neither safe counsellors nor safe guides; for they frequently do things of which they after- wards are ashamed, and for which they are sorry. 106. In addition to the foregoing statements concerning the importance of training the will to depend upon the in- tellect for its course of action, it should be emphasized that this dependence must be formed into a habit. 107. HABIT. Habits control nearly everything we do, and it is only after we can do a thing from habit that we can do it with ease and pleasure. The use of good lan- guage, all mechanical executions, thinking good thoughts, The Will. . 41 controlling our feelings, and acting from pure motives must largely become matters of habit. Teachers must invariably insist upon such conduct from their pupils as is reasonable and right ; and this course must be continued until right- doing has grown into a habit until the pupils can be left to govern themselves. 108. However necessary outward means of government may be during the early periods of life, the fact must not be overlooked that they effect no radical change in the dis- position : they serve simply as a restraint. A change of disposition must come from within, and must give stability to habits of doing right. Hence the further pupils advance in intellectual and moral culture in their ability to take care of themselves the less care need be exercised over them by parents and teachers. 109. It cannot be too deeply impressed upon the minds of teachers that the best book upon psychology is the liv- ing, acting child. Child study, or experimental psychology, is bearing fruit for the teacher's guidance that speculative psychology never dreamed of. 110. Teachers who desire to read more to aid them in their studies than is contained in the foregoing notes would do well to consult Dr. W. O. Krohn's " Practical Lessons in Psychology " and Prof. William James' " Briefer Course in Psychology." Those who wish to acquaint them- selves with the Herbartian psychology and pedagogy should read Lange's " Apperception " (edited by Dr. Charles De Garmo), Dr. Charles A. McMurry's " General Method," or Rein's "Outlines of Pedagogics." IMPORTANT OBSERVATIONS AND INFERENCES. 1. Education begins and ends with life. 2. The object of scholastic education is to teach the pupil how to learn. Its methods should therefore be such as will enable him to carry on his own education pleas- antly and successfully. 3. Education cannot create powers or faculties : it can only develop those that exist. 4. The mind can develop what is in itself only by its own activity ; self-activity (exercise), therefore, educates. 5. The powers of the child can be exercised by no one but the child itself ; consequently, not what the teacher does for the child, but what it does itself, educates it. 6. The inner promptings of the child make themselves known by outer manifestations. 7. The child indicates its own mode of receiving instruc- tion: hence the teacher learns from the child how to teach it. 8. It is natural for a child to seek knowledge. 9. Gratifying a child's desire for knowledge stimulates that desire. 10. Knowledge begins with experience ; the concrete should therefore precede the abstract things should pre- cede their signs or names. 11. Facts and phenomena should come before laws and principles. 12. Thought should come before expression. 13. Each mind has its own rate of growth and develop- ment. 42 Important Observations and Inferences. 43 14. Substantial learning or attainments cannot proceed faster than the mind's rate of growth and development. 15. Attempting to go too fast, or going too slowly, weakens the mind. 16. There is an intimate relation between the body and the mind ; work of either reduces the power of the other. REMARK. Bain, in " Mind and Body," says : " The fact is now generally admitted that thought exhausts the nervous substance as surely as walking exhausts the muscles." 17. Rapidly growing children tire easily ; their rapid growth reduces their power of endurance. 18. "The time to acquire skill in the use of any power, mental or physical, is when the power is growing." 19. " The time to teach a thing is indicated by the arrival of the child's interest in it. Children's interests change with age. What interests them at an early period does not do so at a later." 20. Attention, memory, and imagination are not so many separate entities, but rather necessary accompaniments or aids to the other powers. We may have as many memories and imaginations as there are classes of things to remember and to image, each requiring its own peculiar training. 21. Before a teacher charges his pupils with inattention, he should ascertain the predominating kind of mental images they form, whether visual or auditory. 22. Poor remembering means in most instances poor images and understanding, for which teachers should largely hold themselves responsible. 23. Before the teacher begins to instruct, he should as- certain the contents of his pupils' minds, or he may build without a foundation. 24. The present must grow out of and upon the past. What the child has not experienced it cannot image or comprehend. 25. Every percept is made of present and past experi- 44 Science and Art of Education. ences. All our mental activities are performed with accu- mulated capital ; that is, every thought we think receives the benefit of all our past thinking. That is apperception. 26. Instruction aims at power and skill ; education at character. 27. Adaptation of the subjects of instruction to the pupils' growing needs is the key to success in teaching. 28. How a subject is taught is more important than what is taught. 29. The teacher's life should be an example for his pupils, and his success should be measured by his moral influence. 30. The end of school government should be self-control and character. Herbartianism. The five methodical steps which, ac- cording to Herbart and his disciples, must be taken in teach- ing a lesson : i. " The preparation [analysis] ; 2. The pres- entation [synthesis;] 3. The combination [association] ; 4. The recapitulation [system]; 5. The application." Lange's Apperception. OBJECT-LESSONS. A. THEIR DESIGN. a. The training of the senses observation. b. The gaining of knowledge. c. The development of the power of thought. d. The cultivation of expression. B. THE PLAN OR METHOD OF A LESSON. a. Have a well-defined end in view. b. Select a suitable object to be used. c. Determine the method or order to be pursued. d. Lead the pupils to make their own discoveries. REMARK. There is no better way of enlarging the children's vocabulary and of increasing their stock of knowledge than by means of lessons on objects (things). With such lessons they can be taught the names, properties, relations, and parts of ob- jects. By " object-lessons," however, is meant lessons with ob- jects, not simply about them. The children themselves must examine the objects. Knowledge acquired in this way is more certain and permanent than that which is obtained at second- hand from books or from teachers. Children could profitably spend months, upon lessons of this kind before receiving instruction of any other character. 45 PENMANSHIP. REMARK. The doing of anything is best learned by intelli- gently directed practice, and this applies to nothing more forcibly than to penmanship. Every one, unless paralyzed or deformed, can learn to write a neat hand. The wretched writ- ing found in most schools and among many persons otherwise claiming to be educated is an unmistakable sign of defective teaching. 1. Suggestions. Proper position of the body, correct penholding, and the best work the pupils are capable of doing, must be insisted upon ; and all their writing, until it is as nearly perfect as they can make it, should be considered practice in penmanship. 2. Every letter should invariably have the same form, that of the so-called standard letters. 3. Penholding. The penholder should be placed be- tween the thumb and the first two fingers, so that it may rest before the third joint of the forefinger. REMARK. The penholders and pencils for children should, if possible, be no more than an eighth of an inch in thickness. 4. The thumb and the fingers should be bent outward so as to bring the end of the thumb opposite the first joint of the middle finger. 5. To keep the pen at the proper angle, the thumb should be pressed a little under the holder. 6. The left side of the middle finger should support the holder just above the pen, and the forefinger close over the holder, 46 Penmanship. 47 7. The pen should be held as lightly as possible. REMARK. The suggestions on penholding apply equally to the holding of pencils. 8. Position of the Body. The body should be erect- not resting on the arm that carries the pen the head slightly inclined forward to see the writing. 9. The forearm should rest on the muscle in front of the elbow and the hand on the nails of the third and fourth fingers. The wrist should not rest on the desk or the paper. 10. If the hand is so held that the wrist is horizontal, both sides at equal distances from the desk, the penholder will point to the right shoulder and give the right slant to the writing. 11. To enable the children, in their early efforts, to write in straight lines, their tablets should be ruled with base or guide lines. 12. Penmanship should be commenced with copying the first lessons in reading, and that a good beginning may be made, the teacher should show the children how to hold the pencils (or the crayon, if they write upon the blackboard), and, when necessary, how to form the letters, sometimes guiding their hands. 13. Height of Letters and Spacing The propor- tionate height of the letters, and proper spacing of letters and words, should not be overlooked. 14. Flourishing. Until the pupils can write a neat, plain hand, no efforts at flourishing should be tolerated. 15. Charts. Penmanship charts should be placed upon the wall above the teacher's blackboard, where the pupils can at all times see the correct forms of the letters. REMARK. If the foregoing suggestions on penmanship be strictly carried out, neither copy-book nor classes in writing wiil be necessary, and better penmanship, in less than half the present usual time, will be the result, PRIMARY READING. 1. INTRODUCTORY REMARKS. Before an effort is made to give formal instruction to children, either in reading or any branch of knowledge, their confidence and good wilj must be gained. This may be done by engaging them in conversation upon something that is familiar to them and in which they take an interest. These conversations should be continued until the children's timidity has been overcome and they freely converse with the teacher. During these familiar talks as much knowledge should be drawn from them as possible. In this way, too, the teacher can learn the extent of their stock of knowledge, and this informa- tion will serve him as a basis for the superstructure of knowl- edge which, under his guidance and superintendence, they are to rear. 2. Reading is thinking not mere word-calling ; con- sequently the more thoughtful and intelligent the pupils are made by such preparatory lessons as those on objects, the better they will be prepared for reading. 3. Reading may properly be considered under two heads, impression and expression, or silent reading and audible reading. Silent reading has for its object the getting of thought, and audible reading that of conveying it to others. The getting of thought is the first and chief thing to be aimed at. When this end has been attained, the thought found, conveying it to others by means of audible reading becomes comparatively easy; for emphasis, pause, and in- flection take care of themselves the thought controls them. 48 Primary Reading. 49 4. Various methods of teaching beginners to read have from time to time been advocated and practised. Among them may be named the alphabetic, word, phonic, and sen- tence. But as no one alone of these methods is free from objections, it is found best to form a method that combines what is best in all of them. 5. The word method seems to be the simplest, and there- fore the best to begin with. This method follows that of nature, presenting wholes before parts. It is also philosoph- ical, for, in order to read, to get the thought, words must be recognized as wholes, each as a single picture. If either the names or the sounds of the letters, instead of the whole words, attract the attention of the pupils, they lose the thought, and merely pronounce the words. REMARK. Since, in order to read, children must recognize words, it is not only reasonable, but a saving of time, to begin with words. 6. Words with which the children are familiar should at first be taught. The words which they recognize through the ear they must now learn to recognize through the eye. REMARK. The length of time required to impress a word pamanently upon the mind and to make its name of ready recollection depends upon the interest with which the teacher presents it. To make this work a success lie must have a num- ber of devices at ready command. Suggestions for Teaching Primary Reading. i. Select a word that is the name of an object in which the children take an interest. Engage them in a conversation about the object, to interest them, to gain their attention. 2. Write the word upon the blackboard. Tell them what you have written (its name) and urge them to observe it carefully, so as to be able to recognize it when they see it again. 3. Erase it and write it among others vpon the black- board. Tell them to find it. 50 Science and Art of Education. 4. By means of a variety of exercises of this kind and of others, impress the word upon their minds, always associat- ing the form (written word) with its name. 5. Test the impression with word-cards, charts, and any other means which you may have at your command or which you may be able to devise. REMARK. Writing the word among others upon the black- board affords a good test of the impression. 6. To make the impression permanent, the children should be required or urged to copy their lessons -both upon the blackboard and into their tablets. The tablets used for this purpose should be ruled, and the lines should be not less than three eighths of an inch apart. As before stated, the pencils should be very thin one eighth of an inch in thickness to enable the children to hold them properly. 7. Every lesson should begin with a review of the pre- ceding lesson or lessons. Every word taught should be reviewed from day to day until it is permanently impressed upon the mind and can instantly be recalled. The recog- nition of words should be made automatic. 8. After the first word has been taught, others should be taught to combine with it, or should directly be combined with it, to form a phrase or sentence. 9. As many new words should be taught at each lesson as the pupils can well learn. 10. The teacher should keep a memorandum of every- thing given to the class, to be used as material for reviews. He should also prepare as many slips of paper with the words which the children have learned written upon them, as he has pupils. These slips may be kept in pasteboard boxes, and whenever the children are at leisure when they have performed their other tasks, or their assigned work they should take out the slips and see how many of the words they remember. 11. The sentences written upon the blackboard as fead- Primary Reading. ing exercises may also be copied upon slips of paper and kept in boxes for the children to read after their other work has been performed. 12. Lists of the words taught may be written upon the blackboard, where the children can, from their seats, see them. Of these they should be encouraged to make as many sentences as they can. The sentences should be brought to the class and kindly criticised errors pointed out and improvements suggested. These exercises train children to the correct use of language. 13. As before stated, the words of the reading-lessons should at first all be taken from the children's vocabulary, or stock of words in use. 14. The statements, or sentences, should as far as pos- sible be about something that interests the children. State- ments about themselves frequently prove interesting to them. 15. The teacher should have a good supply of toy ob- jects. At the recitation every member of the class should be given one, and the children encouraged or requested to talk about them. Their talks, or sentences, may be written upon the blackboard and read, or only the most suitable ones may be written upon the blackboard and read. 16. Number-lessons may also be made reading-lessons. The statements of facts which the children make may be written upon the blackboard and read, or they may write them, at their seats, into their tablets, bring them to class, and read them. 17. Lessons on size, form, weight, position, color, quality, etc., may be used in the same manner and for the same purpose as those on number. 18. The reading exercises should be varied from day to day ; that is, of the same words as many different sentences should be made as possible. Varying the exercises not only lends interest to the work, but helps to impress the words upon the children's minds, 5 2 Science -and Art of Education. 19. The sentences should at first all be short. Long sentences are too difficult for beginners ; their length pre- vents the children from readily grasping the contained thought. 20. The reading exercises, the words, phrases, and some- times sentences, may be taken from a suitable book, a Primer or First Reader. 21. The pupils should at first read only from the black- board ; afterwards also their own written work, from their tablets, papers, or slates. 22. After the pupils have learned from one hundred to one hundred and fifty words by sight, by the word method, they should be taught the analysis of words into their sounds. They should be shown that the names of all words are made of combinations of sounds. This may be done by taking suitable words and pronouncing them slower and slower each succeeding time until the separate sounds are heard. Words in which all the sounds may be pro- longed should at first be used. REMARK. Some teachers introduce the sounds of the let- ters and their signs almost from the first lesson in reading, and do it successfully. 23. The analysis of words into their sounds should be followed by the synthesis of the sounds. The children should be made acquainted with the letters that stand for the sounds, and should have practice in determining the pronunciation of words. NOTE. Teachers who desire a carefully worked-out system of primary reading, in general accord with the foregoing, will find " The Rational Method," published by Silver, Burdett & Co., New York, the most recent and the best. It is a thought- method, and the shortest to the mastery of words and intelli- gent reading. 24. All words that cannot well be taught by sounds should be taught by sight, or as wholes. 25. The English language contains about forty-three Primary Reading. 53 different sounds, and has only twenty-six letters by which to represent them ; some letters, therefore, stand for more than one sound. The particular sound for which a letter stands is indicated by a sign placed upon the letter, above it or below it. These signs are called diacritical marks. 26. Whenever a letter is taught as the representative of a sound, its mark should be taught in connection with it. REMARK. The diacritical marks enable the children to dis- cover the pronunciation of words. 27. Whenever a new word is introduced, its sounds should be indicated by their proper marks. In the dic- tionaries the pronunciation of words is indicated by dia- critical marks. In some of the first reading-books diacriti- cal marks are also used. Sounds of the vowels and consonants as given by Webster : VOWELS. i as in ape s as in n6te 1 " " at 6 44 " n6t *S * 4 " arm 3 44 44 ^r a 44 ' 4 ask 6 44 4 ' love 1 " " c3re P. 44 " mflve 9 " 44 ftll 9 44 " WQlf a " " what o " 4 ' obey a 44 44 senate 03 44 44 m6^r * 44 " me s in January. It happened on the tenth of the month. c. Wodin was the chief god of the Old English people. Wodin was by the Danes called Odin. He was the chief giver of valor. He was the chief giver of victory. d. John Wycliffe did a great work. This great work was a translation of the Bible. The translation was made by himself. He was assisted by several friends and followers. e. The English fearlessly boarded the ships. The ships were those of the enemy. They cut the rigging. They gained the victory. The victory was easily won. /. The next morning the battle began in terrible earnest. The next morning was the 24th of June. The battle began at break of day. g. Columbus returned to Spain in 1493. He had spent some 82 Science and Art of Education. months in exploring the delightful regions. These regions had long been dreamed of by many. These regions were now first thrown open to European eyes. Columbus had been absent seven months and eleven days. 29. Combine each of the following sets of simple sen- tences into a complex sentence. a. Tin is a metal. Ancient Britain was most famous for tin. The Phoenicians were first induced to visit Britain for tin. b. He spoke to the king like a rough man. I think this my- self. He was a rough angry man. He did nothing more. c. The ingenuity of man has made a lever of the mind. This lever spares him an immensity of toil. This lever is applied to machinery. d. The ships of Sesostris, the Egyptian conqueror, were formed of cedar. One of these ships was 280 cubits long. Ancient writers notice this. The gigantic statue of Diana in the temple of Ephesus was also formed of this timber. e. Andrew Douglas was willing to share the danger and the honor. Andrew Douglas was master of the Phcenix. He had on board a great quantity of meal from Scotland. /. Three or four bounds of the horse carried us out of reach of the enemy. The enemy quickly resumed his flight. The enemy had merely turned in desperate self-defence. g. God in his good has covered the earth with herbs and trees. We inhabit the earth. These herbs and trees furnish us with food, clothing, and other articles. These articles con- tribute to our comfort and luxury. h. At length the mystery of the ocean was revealed. The theory of the great navigator was triumphantly established. The theory had been the scoff of sages. He had secured to himself glory, This glory must be durable. The world itself is durable. 30. Change the following simple sentences to compound sentences : a. The Rhone, flowing into the Lake of Geneva, emerged from it at the town of the same name. b. These events, trifling doubtless in the estimation of the reader, were affecting to me in the highest degree. c. Snatching the handkerchief, he quickly wrapped it around the wounded part. d. The trees met overhead, forming an archway. e. On further consideration I have decided to remain. /. After a moment's reflection he proceeded to pass sentence. g. The king, a man of rare vigor, would allow no foreign prince to encroach on his rights. General Considerations. 83 //. In forwarding your own interests, do not impede those of others. /". The coral insect, barely possessing life, is hourly creating habitations for men. 31. Combine each set of the following simple sentences into a compound sentence : a. They next erected a crucifix. They prostrated themselves before it. They returned thanks to God. God had conducted their voyage to such a happy issue. b. He possessed quick perceptions. He observed accurate- ly. He was able to place his hand on the right animals. He did so without hesitation. c. The island at first seemed uninhabited. The natives grad- ually assembled in groups upon the shore. The natives grad- ually overcame their natural shyness. The natives received us very hospitably. They brought down for our use the various products of their island. d. The struggle was now at an end. The inhabitants were terror-stricken. They burst through the lines. They fled in every direction. e. They saw their leader fall. They thought him slain. They at once gave up the contest. This was in accordance with the practice of their ancestors. /. Steam has increased indefinitely the mass of human com- forts. Steam has increased indefinitely the mass of human en- joyments. Steam has rendered cheap the materials of wealth and prosperity. Steam has rendered accessible the materials of wealth and prosperity. It has done so all over the world. g. The sun then broke out. The sun dispersed the vapor and the cold with his welcome beams. The traveller felt the general warmth. The sun shone brighter and brighter. The traveller sat down. The traveller was overpowered by the heat. The traveller cast his cloak upon the ground. 32. Change the following compound sentences to com- plex sentences: a. You have asked me a question and I have answered it. b. The statement is false and he knows it. c. They did not know their lesson and so he kept them in. d. Finish this and then I will let you go. e. He was very ill, but still he tried to finish it. /. A boy had seen it fall and had picked it up. g. He tried several keys, but none of them would fit it. For a greater variety of exercises in the various kinds of sentences, see First Steps and Second Steps in English 84 Science and Art of Education. Composition, published by W. Stewart & Co., London, England ; Practical Exercises in Composition, and Exer- cises in English, by H. I. Strang, the former published by The Educational Publishing Co., Boston, the latter by D. C. Heath & Co., Boston. Kerl's Composition and Rhet- oric is also a good book for sentence and composition work ; published by American Book Co., New York. 33. Exercises in the discrimination of words should form part of the sentence work. Write sentences in which the following words shall be cor- rectly used : Raise, rise ; sit, set ; bring, fetch ; fly, flee ; flow, flew, fled ; shut, close ; board, feed ; hung, hanged ; lay, lie ; leave, let ; lend, borrow; lose, loose ; teach, learn ;. wring, ring; begin, commence; forsake, desert; load, burden; empty, va- cant ;' sleigh, slay ; expect, suspect, suppose ; believe, calculate ; may, can ; fix, repair, mend ; think, guess ; enjoy, possess ; be- tween, among; invent, discover ; handsful, handfuls; preacher, minister, pastor, clergyman ; sow, sew ; luck, success ; station- ary, stationery ; desert, dessert ; continual, continued ; compo- sition, essay ; there, their; hard, difficult; balance, remainder; advice, advrse ; all, awl ; aloud, allowed ; altar, alter ; ant, aunt ; ascent, assent; assistance, assistants; bail, bale; bait, bate; bald, bawld ; bear, bare ; base, bass ; beat, beet ; blew, blue ; bawl, ball ; burrow, borough ; bough, bow ; cannon, canon ; capital, capitol ; ceiling, sealing; cell, sell; cent, sent, scent; chord, cord ; site, cite, sight ; climb, clime ; council, counsel ; fare, fair; gait, gate; great, grate; holy, wholly ; gage, gauge; hail, hale; pale, pail; pane, pain; plane, plain; pray, prey; rain, rein, reign ; sale, sail ; strait, straight ; vane, vain, vein ; pare, pear, pair ; canvass, canvas ; tacks, tax ; claws, clause ; nought, naught ; feet, feat, leave, lieve ; meet, mete, meat ; peel, peal; pleas, please; seed, cede; seas, sees, seize; heard, herd; lesson, lessen ; miner, minor; petition, partition ; of, off. 34. The properties that characterize well-written sen- tences, paragraphs, and essays, under the usual titles of purity, propriety, precision, and clearness, strength, unity, and harmony, should not, as is usual, be postponed until a late period of a pupil's school life, but should as early as is possible be introduced by directing his attention to merito- rious as well as to faulty constructions, and demanding cor- [UNIVERSITY General Considera rections or improvements so far as he is capable of seeing their force and making them. 35. The only sure way of training pupils to the careful use of English is to begin as early as their years permit, and to demand that every exercise of theirs shall be as nearly perfect as they can make it ; and this course must be con- tinued to the end of their school days. 36. Essays. When pupils have acquired sufficient power of thought and skill in expression to write essays, subjects for the purpose may be assigned them. Care must, however, be taken that no abstract subjects, no subjects beyond their comprehension, nor anything in which they cannot be inter- ested, be given. 37. After a general subject has been selected or assigned, it should be considered in all its bearings, and some special line of thought or view of it decided upon. REMARK. Writing upon general subjects cannot end in any- thing definite. 38. Having determined upon the line of discussion, the next thing is to keep it before the mind until it has been thoroughly thought over and everything found that has a direct bearing upon it. This done, a careful, logical out- line of the main and sub-topics should be made, containing nothing not strictly in accord with the determined line of thought. REMARK. Finding the matter for the essay and making the outline constitute the most important and difficult part of the work. 39. Next comes the writing, the composition. A good plan to pursue is, to write a little essay, as it were, upon each main topic, combine them into one, look it ovec to correct errors, then lay it away for a few days or a week before re-examination and rewriting. An essay should be several times carefully examined and rewritten before it is handed to the teacher for inspection and suggestions. 86 Science and Art of Education. 40. Unless intended for a public audience, no essay handed to the teacher should be corrected by him ; only the place where an error exists should be indicated by some general mark or sign, and the pupil, at least at first, left to discover the fault himself. It is only by practice in discov- ering that discovering is learned. 41. An essay should be handed two, three, or more times to the teacher for inspection and criticism, and as many times rewritten. In short, it should be criticised and re- written until it is free from errors. In few things, if in any, do we find more failures in teaching than in composition, and the best book on essay writing is a competent teacher. The following books on composition, in addition to those already named, may be used with advantage by teachers : Longmans' School Composition, published by Longmans, Green & Co., New York ; The Foundations of Rhetoric, published by Harper & Brothers, New York ; Composition and Practical English, by William Williams, published by D. C. Heath & Co., Boston. For lower grades of schools, the following will prove serviceable : Stories for Composi- tion, published by Educational Publishing Co., Boston ; How to Write a Composition, published by Dick & Fitzger- ald, New York ; Primary Reproduction Stories, and Hall's Composition Outlines, published by A. Flanagan & Co., Chicago ; The Writing of Compositions, published by E. L. Kellogg & Co., New York ; and W. B. Powell's whole series of language books, published by E. H. Butler & Co., Philadelphia. Nearly all the foregoing books can be had of E. L. Kel- logg & Co., 6 1 East Ninth Street, New York. SUGGESTIONS FOR TEACHING NUMBERS. TEACH the concept (idea) concretely, with pebbles, beans, grains of corn, shoe-pegs (colored or plain), spools, squares, cubes, balls (spheres), cylinders, triangles ; in short, with any suitable objects that may be had. A variety of objects should be kept on hand for this purpose. Figures should not be introduced until the children can work well with objects, pictures, etc., and until they will not confound figures with numbers. Color and form should be taught in connection with numbers. LESSON ON Two. a. What the Pupils must Discover. 1. i -|- i = 5. 2 2 = 9. f of i = 2. I X 2 = 6. 2 -I- I = 10. \ Of 2 = 3. 2 X I = 7- 2 -r- 2 = II. f Of 2 = 4. 2 I = 8. i Of I = 12. f = b. For seat work trje following notation may be used with children that are not far enough advanced to write words or to use figures : I. i + i = 11 5- ii ii = o 9- f(i ) =1 = 2. 1X1: t = n 6. 1 1 -^i = II 10. i() =1 + 3- 11X1 = 11 7- ii -T-II = I ii. f (n) =f + 4- ii i = i 8. i (i) = i 12. f =i NOTE. i. The teacher should substitute some form of the concrete for the fractions in the seat-work, until the children can use figures. A short vertical line divided into two equal 87 Science and Art of Education. parts, with a nought covering the upper part, might represent \; both parts uncovered, f ; the line divided into three equal parts, with the upper part covered, % ; with the two upper parts covered, , etc. 2. Horizontal lines, squares, triangles, circles, and pictures of suitable objects may be used by the children in performing the fractional work ; the whole object or picture representing or being the unit, and the parts the fractions. 3. In solving problems in which a fractional part is required of a number consisting of several ones, or units, the children should first be taught to find the sum of the fractional parts of the separate units; later, after they can readily do this, they should be taught the usual way of obtaining the result. Problem 10 of the foregoing seat-work may be solved with two horizontal lines, one of the halves of each being covered with a nought or some other device to indicate that it is not to be counted. It may also be solved with squares, as follows : n+mmn. c. Suggestive Questions and Problems. 1. Give me a cube. Give me another. How many have I now ? (One and one are called two; or, simply, are two.) 2. Show me two fingers, two hands, two boys, a two of red cubes, a two of pencils, a two of anything else. REMARK. A two of anything means two things taken to- gether and considered as a unit. 3. One bird and one bird are how many birds ? 4. Show me a two. Show me a one. How many ones are in a two ? 5. If two birds are on a tree and one of them flies away, how many remain ? 6. What number is one more than one ? 7. What number is one less than two ? REMARK. Every operation should be proved or illustrated with objects or drawings of them. 8. How many twos are in two ones ? How many ones does it take to make a two ? 9. From a two take away two ones, and what have you left.? Prove it with yellow cubes. Suggestions for Teaching Numbers. 10. From two ones take away a two, and what have you left ? Prove it with blue cylinders. d. Suggestive Dialogue to Teach the Half. Teacher. If you wanted to give me half of an apple, how would you cut (or divide) the apple ? Pupil, I would cut it through the middle. T. Which one of us would get the larger piece ? P. Neither; one piece would be as large as the other. T. Make a picture-apple upon the blackboard, and draw a line through it where you would cut it. T. Here are two pieces of cardboard; which of them is the longer ? P. One is as long as the other. T. How much longer are the two pieces together than this piece ? P. They are just as long. T. If I should cut the long piece across the middle, which of the two parts would be the longer ? P. Neither piece would be longer than the other. T. What part of the whole piece would each of the parts be ? P. One half. T. What did I do ? P. You cut the long piece through the middle. T. How can you tell that I cut it through the middle ? P. By trying whether one piece is as long as the other. T. You may try it. P. One piece is just as long as the other. T. What part of the whole piece do I hold in my hand ? P. One half. T. One half of what ? P. Of the whole piece. T. Lay one piece at the end of the other, and see what the two halves make. Science and Art of Education, P. They make the whole piece. T. Since these two pieces make the whole piece, one vi them is what part of two ? P. One half. T. How do you know that each piece is one half of the two pieces ? P. Because it is one half of the whole piece, and the whole is made of the two pieces. T. Which of these two yellow cubes is the larger ? P. One is as large as the other. T. How do you know ? P. I have tried them. T. Now since one of the two pieces of cardboard is one half of the two pieces, what part do you think one cube is of the two cubes ? P. One half. T. One apple is what part of two apples ? P. If they are of the same size, it is one half of them. T. How many halves are in the whole of anything ? P. Two. T. How can I get one half of anything of an orange, for example ? P. By cutting it through the middle. T. Why .cut it through the middle ? P. To make one piece as large as the other. T. Why must one piece be as large as the other r P. If they were not of the same size they would not be halves. T. How can I get one half of two candies ? P. By breaking each one into two equal pieces, and tak- ing a piece of each of them. T. Cculd you get a half of the two in any other way ? P. If one of the candies is as large as the other, one of them would be a half of the two. T. Which one of them ? Suggestions for Teaching Numbers. 91 P. Either one. T. How can you tell ? P. Because one is as large as the other. T. Are all halves of the same size ? P. Yes, they are. T. You may draw two picture-apples upon the black- board, making one larger than the other. T. Draw a line through the middle of each, and then tell me whether the halves of the small one are as large as those of the large one. P. No; they are not. T. Do you still think that all halves are of the same size ? P. No; I do not. T. What halves, then, are of the same size ? P. Those of the same thing, or of things of the same size. T. In how many ways could you cut this card (parallel- ogram, i in. by 2 in.) into halves ? T. I will give you a card and you may draw a pencil- mark across where you would cut it, and then tell me in how many ways you could cut it ? P. I could cut it in two ways, lengthwise and crosswise. T. I will give you another card, and you may cut one of them lengthwise and the other crosswise, and then tell me whether all the halves are of the same size ? P. No, they are not ; the pieces of the card cut length- wise are longer than those of the one cut crosswise. T. Do you notice any other difference ? P. Yes, the short pieces are broader than the others T. Find out how much broader they are ? P. They are twice as broad. T. Now cut one of the short pieces into two equal pieces and see whether the two parts laid together make a piece as long as one of the long pieces. P. Yes, they do. T. What can you again say of the halves of anything ? 92 Science and Art of Education. P. That one is as large as the other. T. In how many ways did you cut your card into halves ? P. In two ways. T. What are they ? P. Lengthwise and crosswise. T. Can you cut them in any other way so that the two parts will be of the same size ? P. I think I can, but I am not sure of it. T. You may try it. P. Yes, I can cut them in another way I can cut them from one corner across the middle to the opposite corner. T. How do you know that the pieces are of the same size ? P. I laid one upon the other and it covered it exactly. T. Show me half of your cubes. T. Of how many do you show me a half ? P. Of two. T. How many halves does it take to make the whole of anything ? P. Two. T. Two what ? P. Two halves. T. To make what ? P. To make the whole of anything. e. Suggestive Dialogue to Teach the Applications in Two. T, What do I hold in my hand ? P. A two-cent piece. T. How many cents would you give me for it ? P. Two. T. How many cent-candies could you get for one cent ? P. One. T. Illustrate (show) it with pictures upon the black- ooard ; also with toy-money and crayons. Suggestions for Teaching Numbers. 93 T. How many cent-apples could you get for two cents ? P. Two. T. Prove (show, illustrate) it with picture-apples and picture-cents. T. If I should send you to the store to buy two slate- pencils that cost a cent each, how much money would you pay for them ? P. Two cents. T. Your sister sends you to the store with a two-cent piece to buy an orange that costs a cent; how much money will you bring back ? P. One cent. T. If you should go to the post-office with two cents to buy cent-stamps, how many would you get ? P. Two. T. Henry has two rabbits, and this is one more than Sarah has ; How many has Sarah ? P. One. T. John has half as many roses as his sister ; if he has one, how many has she ? P. Two. T. I know a number whose double is two, what is it ? P. One. T. What number is that whose half is one ? P. Two. LESSON ON THREE. a. What the Pupils must Discover. 1. 2 -f i = 8. 3-*- 3 = 15. f of 2 = 2. I + 2 = 9- 3 -f- I = l6. f Of 2 = 3. i X 3 = 10. 3 -s- 2 = 17. \ of 3 = 4. 3 X i = ii. i of i 18. f of 3 - 5. 3 i = 12. f of i = 19. of 3 6. 3 - 2 = 13. | of i = 20. f of 3 = 7. 3 3 - 14. \ of 2 = 21. | of 3 = 94 Science and Art of Education. REMARK. i. In 3 -s- 2, the question is, how many twos are in three ; and the answer is, one two and one one, and may be written thus, i (i), the parenthetic part being the remainder. 2. The yard with its parts in feet should be taught in connec- tion with the foregoing. b. For seat-work the following will serve as example : 1. 11 + 1 =IH 4. in xi = 11 1 7. m iii=o 2. I -f II=III 5. Ill 1= II 8. III-i~III = I 3. i xn 1 = 1 1 1 6. in 1 1 = 1 9. iii-7-i = in 10.111 -5- ii = i(i) 16. i(n) =1+1 = 11 11. i (i) =i 17- 12. |(i) =f 18. 13- 1(0 =l=i 19- i("i)=i+t+i=t= 14. i (")=*+*=! 20. f (in)= + |+|=ii 15- I (")=f +1=1* 21. | (Ill)=f + f + f = HI NOTE. The method of solution is indicated in all the exer- cises that follow the tenth. REMARK. i. As before remarked, instead of the foregoing notation, horizontal lines, squares, triangles, circles, and pict- ures may be used for performing all the fractional work. For examples, the 2oth of the foregoing may be solved thus, with squares : nitiMMD-iiD n a REMARK. 2. The pupils should be required to make stories of their exercises. Of No. i of the foregoing the following may be made : I had two cents and my sister gave me another; then I had three. Or, Sarah had two roses and her mother gave her another ; how many had she then ? c. Suggestive Questions and Problems. 1. Give me two red spheres (balls). Give me another. How many have you given me altogether? (Two and one are three). 2. Show me three fingers, three cylinders, three girls, three boys. 3. What number is one more than two ? 4. Three i$ one more than what number ? Suggestions for Teaching Numbers. 95 5. One is two less than what number ? 6. What number is less than three ? 7. What number is two more than one ? 8. To what number must I put (add) one to make three ? 9. One and one and one are how many ones ? 10. Show me a two, also two ones ; which is the larger? 1 1. How many ones are in a two ? In a three ? 12. How many twos are in a three, or in three ones? 13. Make three in all the ways you can. 14. Make three picture-boys on your tablet, three pict- ure-girls, three picture-horses, three picture-wagons, three picture-cats. 15. Under each of three trees John found an apple; how many apples did he find ? 1 6. Three mice were in a box and Charles killed all but two ; how many did he kill ? 17. Henry had three roses and gave all but one to Alice; how many did Alice receive ? 1 8. Frank has three horses and one saddle, how many more horses has he than saddles ? 19. Ella had three playmates and gave to each a pear; how many pears did she give away ? 20. Jacob caught three rabbits in three traps ; how many did he catch in each ? 21. If you should lay three grains of corn upon the floor and a mouse should come and carry one of them away at a time, how many trips would it have to make to carry all of them away ? 22. Place three grains of corn upon the floor ; now, if a mouse could carry away only one grain at a time, how many mice would be required to carry all of them away at once ? Make picture-mice and prove it. 23. Place three buttons upon the table ; take them away two at a time ; how many times did you take two away , ? 9 6 Science and Art of Education. 24. How many twos in three ? How many ones ? How many threes ? 25. Make two in all the ways you can. 26. Make ihree of ones ; how many does it take ? 27. How many ones in two? In three? How many twos in two ? 28. How many ones in one half of two ? In three halves of two ? 29. How many twos in two ones ? In three ones ? 30. Take three buttons and lay each one by itself ; how many separate buttons have you laid ? Count them. Each one of the three is what part of all of them ? (It is one of the three equal parts, or one third.) 31. Show me one third of three yellow squares ; two thirds of three buttons ; three thirds of three picture- apples. 32. How can you get one third of anything ? Can you get one third of one ? How ? Do it. 33. How would you get one half of anything ? 34. Which is the larger (or greater), one half or one third ? Prove it. 35. How many halves in one and one half? 36. If I cut an apple into two equal pieces, what is one of the pieces called ? What is each of the pieces called ? 37. Show me how you would get two thirds of anything. Two is what part of three ? One is what part of two ? 38. This stick (3 inches) is how many times as long as that (i inch) ? How can you find out ? Do it. 39. This stick (i inch) is what part of that (3 inches). How can you tell ? 40. This stick (i inch) is what part of that (2 inches.) ? d. Suggestive Practical Business Problems. We will now play store. John may be the merchant and the others his customers or buyers, Suggestions for Teaching Numbers. 97 1. Sarah may buy two buttons, at a cent apiece, and give him a three-cent piece for them. How much change will he give her ? REMARK. The more these transactions are made like real business the more interest the children will take in them. 2. Alice may buy a one-cent candy and a two-cent candy How many cents must she give for them ? 3. Henry buys three pencils at a cent each ; how much money will pay for them ? 4. Fannie wants three cents' worth of eggs, and the eggs cost a cent each ; how many will she get ? How many would she get for two cents ? Illustrate. ^ 5. How many two-cent rings can I get for three cents ? 6. How many nuts at a half-cent each could you get for one cent ? How many for a cent and a half? 7. Fill this pint measure with sand and pour it into that quart measure. Did it fill it ? Fill it again and pour it in. Is it now full ? What did I call this measure ? What that ? How many times this did it take to fill that ? How many pints in a quart ? In a half-quart ? 8. One pint is what part of a quart ? Two pints are what part of a quart ? We will call this water milk, and Annie may sell it at a cent a pint. 9. Alice wants a quart and gives Annie a three-cent piece ; how much change will she receive ? 10. Sarah wants two pints ; how much will they cost ? 11. Jacob wants a half-quart and has a two-cent piece with which to pay for it ; give him the change. 12. John may take this foot-measure and see how many such could be made of that yard-measure. 13. Henry, you may tell us how many foot-measures it takes to make a yard-measure, or a yard. Prove it. 14. How many feet make a yard, or are in a yard ? How many in one third of a yard ? In two thirds ? In one half ? Sarah may now sell tape to the other members of the class. Science 'and Art of Education. REMARKS. Let the children make tape of newspapers, by cutting them into narrow strips and sewing the ends together. 15. Alice buys a yard that costs two cents and a yard that costs one cent ; how much will it all cost ? 1 6. Anna wants a third of a yard of that which costs three cents a yard ; how many cents must she give for it ? Prove it ? 17. Fannie wants a cent's worth of that which costs three cents a yard ? How much will she get ? 1 8. Peter wants a yard and a half of that which sells at two cents a yard ; how much money will pay for it ? 19. Ella had three yards and sold one third of them ; how many had she left ? If it sold at three cents a yard, what did she get for what she sold ? 20. When a quart of- milk costs two cents, what does a pint cost ? 21. How many pints in a quart ? How many quarts in three pints ? 22. How many feet in a yard ? In a half -yard ? In a third of a yard ? 23. One foot is what part of a yard ? 24. If a yard of tape costs three cents, what does one foot of it cost ? What two thirds of a yard ? LESSON ON FOUR. a. What the Puils should Discover. i. 3 + 1 = 9-4-4= 17. |of4= 25. i qt. 2. 1+3= 10. 4 -f- 2 = 18. iof i = 26. 1 gal. 3. 2 + 2 = II. 4 -4- 1 = 19. f of i = 27. 4 gills 4. 2X2 = 12. 4-3= 20. f Of I = 28. i pt. 5. 1X4= 13, 4 -s-"4= 21. | Of 1 = 29. 4 pk. 6. 41 = 14. iof4= 22. i Of i = 30. i bu. 7- 4-2 = 15. |of4= 23. 4qts.= 31. i pk. 8. 4-3= j6. I of 4= 24. 2 qts. = 33, 4 weeks Suggestions for Teaching Numbers. 99 b. The following will serve as examples of what may be given as seat-work : 1. III + i i 10. iiii-*-ii = 18. i (i ) = 2. I + ni = II. ii u-i-i = 19. 1 (i ) = 3- II + n = 12. IIII-5-III = 20, | (i ) = 4- II X II = 13- 1111-4-1111 = 21. | (i ) = 5- I X I I I I = 14. (mi) = 22. ^ (in i)= 6, I I I I I = 15. \ (mi) = 23- I (nn)= 7- mi II = 1 6. I (mi) = 24. 1 (nii)= 8, 1 1 1 1 III = 17- 1 (mi) = 25- Hi ) = 9- i in IIII rs NOTE. In No. 25, the question is, What part of one is one half of one half of one ? and by dividing each of the two halves of a given line or square into two equal parts, thus, ^ ( ^ , t I ,the answer is found to be one fourth. REMARK. The pupils should also be required to write the foregoing exercises in words, either in the question or in the statement form or in both. This work will give them practice in penmanship, spelling, and language, in addition to that of numbers. Nos. i, 10, 14, 15, and 24, for example, maybe written as follows: (i.) Three and one are how many? or three and one are four. (10.) How many twos are in four? (14.) One half of four ones (or of four) is two ones (or two). (15.) What is one third of four ones? (24.) What are three fourths of four ones, or of four (ones being understood) ? c. Suggestive Questions and Problems. 1. Take a one and a two. How many ones have you taken ? How many threes ? 2. Take a three and a one. How many ones did you take ? (Three and one are four.) 3. Give me a two. Give me another. How many ones did you give me? How many ones are in four? How many twos ? How many threes ? How many fours ? 4. Show me one four, one three, one two. 5. How many twos in three ? Science and Art of Education. 6. One half is what part of one and one-half ? Of two ones? 7. How many ones in three halves ? In four halves ? 8. How many feet in a yard ? In a half-yard ? In two thirds of a yard ? 9. How many pints in a quart ? In a half-quart ? 10. How many quarts will fill this gallon measure ? How can you find out ? Do it. We will call this water milk, and sell it at four cents a gallon. 11. Sarah wants a quart and gives a two-cent piece to pay for it ? how much change shall she receive ? 12. Alice wants a half -gallon ; how many cents will pay for it ? 13. Jacob buys two pints ; what will they cost ? 14. Anna wants two half-gallons ; how much money will she need to pay for them ? How many two-cent pieces will pay for them ? 15. Here is a measure we have not yet used; it is called a peck, and is used for measuring apples, peaches, nuts, corn, wheat, etc., things not liquid and not sold by the pint or quart. Henry may fill it with saw-dust and empty it into that half-bushel. Does it fill it ? Peter may fill it and also empty it into the half-bushel. Is the half-bushel now full ? How many pecks make (fill) a half-bushel ? How many a bushel (whole bushel) ? We will now sell peaches, at a dollar a bushel. You may get the money (toy-money). John may sell them. He has no peaches,- but he may make believe (pretend) that he is selling and you may make believe that you are buying. 1 6. Jacob wants two pecks. He may show us the money he will give for them. How many quarter-dollars has he ? Has he enough money ? Prove it. How many half-dollars would pay for them ? Prove it. 17. Peter sells pears, at two dollars a bushel, and John. Suggestions for Teaching Numbers. buys a peck. He may get the money to pay for them. Did he get enough ? 1 8. Jacob wants a bushel and a peck of pears. He may get the money to pay for them. Has he enough money ? How can you tell ? 19. Three quarters of a bushel make how many half- bushels ? How many pecks ? 20. How many half-dollars will pay for three quarters of a bushel of pears ? What part of a dollar would pay for them ? 21. Alice wants a bushel and a half ; how much money will pay for them ? Prove that your answer is right. 22. Sarah sells Henry two and a half yards of cloth, at one dollar a yard ; how many dollars will pay for them ? How many dollars will three half -yards cost ? how many three quarters of a yard ? 23. What number diminished by two leaves nothing ? 24. From what number can I take one-half and have one and one fourth left ? 25. From what number can I take two and one half and have one and one half left ? 26. What number added to one makes four? 27. What number added to one and one half makes three ? 28. I think of a number whose half is two ; what is it ? 29. What number taken four times makes four ? 30. I think of a number whose fourth part is one ; what is it ? 31. What two numbers make four? What three num- bers? 32. What number taken three times, and ^one added, makes four ? 33. What number doubled makes four? What three ? 34. What number doubled, and one added, makes three ? 35. Make four in all the ways you can, mentally. 102 Science and Art of Education* 36. Make two in all the ways you can ; also three. 37. What is one half of one ? of three ? 38. What is one third of one ? of two ? of four ? 39. What are two thirds of one ? of two ? of four ? of three? 40. What are two fourths of one ? Three fourths of one ? Three fourths of two ? of three ? of four ? REMARK. Whenever it is possible stories should be made of abstract problems. Stories give reality and interest to the work. LESSON ON FIVE. a. i. 2. 3- 4- 4+1 = i + 4 = 3 + 2 = 2 + 3 = 1 8*. .|. .|. .,. .|. IO IT) XT) *O 2 3 4 5 = 29. 30- 31- 3 2 - *of iof fof *of 2 = 3 = 3 = 3 5- 2 + 2 + i = 19. tot 5 = 33- iof 4 = 6. 2 X 2 + 1 = 20. iof 5 = 34- fof 4 = 7- I X 5 s 21. fof 5 = 35- |of 4 = 8. 5 X i = 22. iof 5 = 36. *of 4 = 9- 5 i 23. Iof 5 := 37- iof 5 = 10. 5 2 = 24. iof i = 38- fof 5 = ii. 5 3 = 25- fof i = 39- I of 5 = 12. 5 4 = 26. fof i = 40. *of 13- 5 5 = 27. iof 2 = 41. iof 5 = 14. 5 -r i = 28. fof 2 = b. Examples of Seat-work. REMARK. For the seat-work of five and the following num- bers, until the children can use figures, the notation of the numbers preceding five, or the following, may be used. 1. four + one = 12. five four = 2. one + four = 13. five five = 3. three + two = 14. five -*- one = 4. two -f three = 15. five -f- two = 5. two + two -f one = 16. five *- three = 6. two x two 4- one = 17. five -*- four = 7. one + two x two = 18. five -*- five = 8. one x five = 19. one half of five = 9. five one = 20. one third of five = 10. five two = 21. two thirds of five = n. five three = 22. one fourth of five = Suggestions for Teaching Numbers. 103 23. three fourths of five = 33. one fifth of four = 24. one fifth of one = 34. two fifths of four = 25. three fifths of one = 35. three fifths of four = 26. five fifths of one = 36. four fifths of four = 27. one fifth of two = 37. one fifth of five = 28. three fifths of two = 38. two fifths of five = 29. four fifths of two = 39. three fifths of five = 30. one fitfh of three = 40. four fifths of five = 31. two fifths of three = 41. five fifths of five = 32. four fifths of three = c. Suggestive Questions and Problems. 1. Place four blocks upon the table. Place another on. How many are on now ? (You have placed five on.) How many did you place on first ? How many afterwards ? How many altogether? Then four and one are how many ? 2. Can you make five of ones? Try it. How many does it take ? 3. Can you make five of twos ? How many does it take ? Then two twos and one are how many ? How many ones ? 4. Can you make five of threes? How make does it take ? One three and one two are how many ? One three and two ones are how many ? 5. Can you make five of fours ? How many does it take? 6. Make five in all the ways you can (in your mind) without objects, and in every case tell the result. 7. How many threes are in four? How many twos? How many ones ? 8. What is one half of four? of two? of one? One third of three ? of four ? of one ? 9. What is one fourth of four ? of three ? of two ? of one ? One third of two? 10. Divide five pebbles into five equal parts. How many have you in each ? One is what part of five ? How many such parts are in five? How many fifths of five are in five ? How many fifths of one are in one ? (UNIVERSITY OF '' v IO4 Science and Art of Education. ii. How many times can you find two fifths of one in one? REMARK. Problems of this kind should at this stage of the pupil's progress be solved with diagrams. Solution of prob- lem u : The answer is 12. How often can you find three fifths of one in one? .1% Solution: i % JJJ * or rrrrrv m. 13. How often can you find four fifths of one in one ? Prove it. 14. How often can you find two thirds of one in one ? i H Prove it. Solution : 15. What is one half of five ? One third of five ? One fourth of five ? One fifth of five ? Solution of first : i + 4 + * + 1 +4 = 2*. 1 6. What are two thirds of five? Three fourths of five? Two fifths of five ? Solution of first : 4- -I = 1-k* I o Oo* 17. I know a number to which if I add three it will make five ; what is it ? 1 8. I know a number that contains two twos and one one ; what is it ? 19. I think of a number to which if I add four it will make five ; what is it ? 20. I know a number whose half is two ; what is it ? Suggestions for Teaching Numbers. 105 21. I know a number whose half added to it makes three; what is it ? 22. From what number can I take its third and have two left? 23. If to a certain number I add its two thirds, the sum will be five ; what is the number ? 24. What number added to one and one half makes four ? 25. I think of a number that is three less than five ; what is it? 26. What number is two less than three ? 27. If from a certain number I take its fourth, three will be left ; what is it ? 28. What number doubled makes five ? 29. Three times a number and twice the number make five ; what is the number ? 30. What number doubled and its half added makes five ? 31. What number increased by its fourth equals five ? 32. What number lacks two of being five ? 33. What two unlike numbers make five ? What unlike numbers make four ? What like numbers ? 34. I buy apples at two cents each and they cost me four cents ; how many do I buy ? 35. I bought candies at one cent each, and out of three cents received one cent change ; how many did I buy? 36. I paid four cents for two pencils ; how much did they cost me apiece ? 37. I bought two-cent oranges, and out of five cents re- ceived one cent change ; how many did I buy ? 38. Sarah had five two-cent pieces and gave one to each of three boys ; how many had she left ? How many cents could she get for them ? 39. A bird laid five eggs and hatched them all but four ; how many did it hatch ? 40. Two boys and three girls went coasting ; the number of boys is what part of that of the girls ? By what part of 106 . Science and Art of Education. their number should that of the boys have been increased to have made it equal to that of the girls ? 41. Jane has four ducks and three pigeons ; how many times as many ducks has she as pigeons ? The number of pigeons is what part of that of the ducks ? 42. Henry's book cost him three cents, and his pencil one third as much ; how much did both cost ? The differ- ence in cost of the two is what part of the cost of the book ? The cost of the pencil is what part of the difference ? 43. One is what part of two ? of three ? of five ? 44. Two is what part of three ? of four ? of five ? 45. Three is how many times two ? Three is what part of three ? of four ? of five ? 46. Four little birds were in a nest and one half of them flew away ; how many remained ? 47. Three boys are playing ball ; what part of their num- ber must be added to them to make it five ? 48. How many two-cent postage-stamps can you get for five cents ? 49. How many yards of tape, at three cents a yard, can you get for five cents ? What part of a yard for two cents ? 50. If your mother should send you to the store with five cents, to buy thimbles that cost four cents each, how many could you get ? How much change would you get ? 51. With five cents buy all the two-cent lemons you can? How many can you get ? Prove it. 52. A bushel of apples costs three quarters of a dollar ; how many bushels can you get for two dollars ? for three dollars ? For two and one half dollars ? 53. A bushel of pears costs two dollars ; how many bushels could you get for five dollars? for two and a half dollars ? What part of a bushel could you get for one and a half dollars? for three fourths of a dollar? for a half dollar ? How many pecks could you get for one and one fourth dollars ? Suggestions for Teaching Numbers. 107 54. A peck of plums costs a half dollar ; how many half bushels could you get for a dollar and a half. How many bushels for three dollars ? 55. When a half bushel of quinces costs one and one half dollars, how much does a peck cost ? how much a bushel ? What part of a bushel could you get for a half dollar ? for two dollars ? 56. When a gallon of honey costs two dollars, what part of a gallon can be had for a half dollar ? for a dollar ? How many quarts for a dollar and a half ? What part of a dollar would a pint cost ? 57. At five dollars a yard, what part of a yard of cloth can be bought for three dollars ? what for two and a half dollars ? for one dollar ? 58. A quince and a peach together cost three cents ; how much did each cost, if the quince cost twice as much as the peach ? 59. Two oranges cost five cents ; how much did each cost, if one cost a cent more than the other ? 60. Three times a number less twice the number equals one ; what is the number ? 6 1. Once a number and half the number equal three ; what is the number ? 62. There are three times as many geese in a pond as ducks; if there are three geese, how many are there alto- gether ? 63. Sarah has three canary-birds more than Anna ; how many have both, if Anna has one ? 64. In three halves, how many ones ? Three halves equal what part of two ? 65. In five thirds how many ones ? How many , halves ? REMARK. The second part of question No. 65 may be solved with the following diagrams : io8 Science and Art of Education. 66. In five fourths how many ones ? how many halves ? Solution of second by diagrams : 67. One and one half is what part of two ? Solution : mn *-*> 68. One and one fourth is what part of five ? Solution : 69. How many thirds in one half ? Solution : 70. One third is what part of one half ? Solution: One third contains two of the three equal parts of one half, and is therefore two thirds of it. 71. How many fourths in one half? 72. One fourth is what part of one half? 73. How many times is one half in two thirds? Solution: Examining the diagram, we find that one half of it contains three blocks, and two thirds four; the problem, therefore, is reduced to finding the number of times three blocks are found in four blocks. Ans. i. 74. One third is what part of two thirds ? 75. Two fourths are what part of three fourths ? 76. Three fifths are what part of four fifths ? 77. One half is what part of three fourths? Suggestions for Teaching Numbers. 109 LESSON ON Six. a. What the Pupils Must Discover 2. 3- 4- 5- 6. 7- 8. 9- i+5= 2+4= 3 + 3= 3x2= 2x3 = 1x6= 6-1 = n. 12. 13- 14. 15- 1 6. 17. 18. 19. 6-2 = 6-5= 6-6= 6-*-i = 6-5-2 = 6-5-3= 6-5-4= 6-5-5 = 6-5-6= 21. 22. 23- 24. 25. 26. 27. 28. 29. f of 6= iof 6= f of 6= iof 6= f of 6= fof 6= fof 6= jfof 6= iof 6= 31- 32- 33- 34- 35- 36. 37- 38. 39- iof 6= fof 6= * of 6= f ofi = Iof i = fof i = iof 2 = iof 5 = 41. 42. 43. 44. 45- 46. 47- fof 2: fof 4= |of 4 : I , f 5: fof 5 fof 6: fof 5= 10. 63= 20. iof6= 30. f of 6= 40. | of 2= b. Examples of Seat-work. 1. five + one = 8. one x six 2. one + five = 9. six one 3. four + two = 10. six three 4. two + four = ii. six two 5. three -f three = 12. six five 6. three x two = 13. six six 7. two x three = 14. six-*-one = 21. two halves of six = 32. 22. one third of six = 33. 23. two thirds of six = 34. 24. one sixth of six = 35. 25. two thirds of six = 36. 26. three thirds of six = 37. 27. four sixths of six = 38. 28. five sixths of six = 39. 29. one fourth of six = 40. 30. three fourths of six = 41. 31. one fifth of six = 42. = 15. six-*-two = = 1 6. six -s- three = = 17. six-^four = 18. six-5-five = 20. one half of six = three fifths of six = four fifths of six = one sixth of one = three sixths of one = four sixths of one = one sixth of two = one sixth of three = one sixth of four = one sixth of five = three sixths of five = five sixths of five = etc., etc. r.. Suggestive Questions and Problems. i. Take five cubes. Take another. How many alto- Science and Art of Education. gether have you taken ? (You have taken six.) Five and one are how many ? 2. Four and two are how many ? Three and two ? 3. Three and three are how many ones? how many twos ? 4. Six ones are how many twos ? how many threes ? 5. How many fours in six ? 6. Make six of ones ? how many does it take ? 7. Make six of twos ? how many does it take ? 8. Make six of threes ? how many does it take ? 9. Make six in all the ways you can mentally, and in every case tell the result. 10. Three twos are how many ones ? 11. Two threes are how many ones ? 12. Six ones are how many twos? how many threes? 13. I know a number whose third is one; what is it ? 14. I think of a number which doubled makes six; what is it? 15. I think of a number whose half added to it makes six; what is it ? 1 6. If to twice a number I add two, it makes six ; what is the number ? 17. From what number can I take its half and have one left? 1 8. What number diminished by its third leaves four ? 19. If to a certain number I add its fifth, it makes six; what is the number ? 20. From what number can I take half of four and have four left ? 21. If to a certain number I add a third of three, it makes two; what is the number ? 22. Three times a number less once the number is four ; what is the number ? 23. What equal numbers make six ? What unequal num- bers? Suggestions for Teaching Numbers. 24. How many fourths in a half ? in a fourth and a half? 25. How many sixths in one? in a half? in a third ? 26. A half and a third equal how many sixths ? 27. A half, a third, and a sixth are how many ones ? 28. Two thirds are how many sixths ? Three equal how many thirds ? how many halves ? 29. Four sixths equal how many thirds ? how[ many halves ? 30. Five sixths equal how many thirds ? how many halves ? 31. At two cents a yard, how many yards of tape could you buy for six cents ? 32. How much money will pay for four apples at a half- cent each ? at a fourth of a cent each ? 33. If plums sell at* a fourth of a cent each, how much would six cost ? 34. Sarah can walk two miles an hour and Helen three ; !.o\v long would it take each to walk to a town six miles distant ? If they should start together, how far apart would they be at the end of two hours ? If they should start to- gether but go in opposite directions, how far apart would they be in one hour ? 35. Alice went to the grocery with six cents ; she bought pears with one third of them, cherries with one half of them, and chestnuts with the remainder ; how much money did she give for each ? 36. If fish-hooks cost a half-cent each, how many can Charles buy for three cents ? 37. Henry bought rabbits at a half-dollar each ; Kow many did he get for two dollars ? 38. When hickory-nuts sell at four cents a quart, how Science and Art of Education. many pints can John get for six cents ? How many for two cents ? 39. Jennie and Cora went to the cellar for apples ; how many did each get, if Cora had twice as many as Jennie, and both had six ? 40. If a yard of muslin costs six cents, what does a foot of it cost ? 41. One sixth is what part of three sixths ? of two sixths ? of five sixths ? 42. One sixth is what part of one f half? Solution by diagram : 43. One sixth is what part of one ^ fourth ? Solution by diagram : (Tfj | ] i 1=% 44. What part of one is one half of one ^ third ? Solution : I I I I one of the blocks, being a half of of a third, is a |^l I ' sixth of one, or of the whole. 45. One half of one half is what part of one ? H\ ^ H. -t I or > 46. One third of one half is what part of one ? Solution : | \ \ \ \ \ |, or %. 47. What is one half of two thirds ? of three thirds ? REMARK i. It is not deemed necessary to indicate what the pupils should discover, nor to give examples of seat-work, further than the number six. 2. From here on figures may gradually be introduced. Suggestive Questions and Problems on Seven. 1. Put four cubes upon the table ; put three more on. How many ones have you put on ? (Four ones and three ones are seven ones.) 2. Three twos and one are how many ? 3. Five and two are how many ? 4. Two threes and one a.re how many ? Suggestions for Teaching Numbers. 1 1 3 Seven ones are how many twos ? how many threes ? 6. What number is three less than seven ? 7. What number added to five makes seven? 8. I think of a number whose double and one added make seven ; what is it ? 9. What number is that whose half added to it makes six ? 10. What is one seventh of seven ? What are three sevenths of seven ? five sevenths ? seven sevenths ? 11. Two thirds of three apples and three fourths of four apples are how many apples ? 12. Jane found one half of four chestnuts and Alvira two. thirds of six ; how many did both find ? 13. Of tape costing a half-cent a yard, Hannah bought one third of three yards, Eva three fourths of four yards, and Emily five sixths of six yards ; how much did they all pay? 14. O n e half of four and two thirds of six are how many ones ? how many twos ? how many threes ? 15. Four sevenths of seven and three fifths of five are how many ones ? how many twos ? how many threes ? 16. Make seven of twos ; how many does it take ? 17. Make seven of threes ; how many does it take ? 18. Make seven in all the ways you can mentally, and in each case state the result. 19. How many two-cent postage-stamps could you buy with seven cents ? 20. An orange costs three cents and a lemon two thirds as much ; how much do both cost ? 21. How many three-cent and four-cent pies could you buy with seven cents ? 22. How many pints in three quarts ? How many gills in a pint and a half ? 23. How many feet in two yards ? in two-thirds of a yard? 1 1 4 Science and Art of Education. 24. If two cakes cost six cents, what is the cost of each ? 25. John has six pears which he wishes to share with his two companions ; how many will he give to each ? What part of them will he keep ? REMARK. A square has four equal sides and four right angles. When the sides are one inch, it is a square inch ; when one foot, a square foot ; when a yard, a square yard, etc. 26. How many squares, an inch on each side, could you cut from a piece of paper two inches long and one inch wide? 27. How many squares, two inches on each side, could you make from a piece of paper four inches long and one inch wide ? 28. How many squares, a yard on each side, could you cut from a piece of muslin three yards long and two yards wide ? 29. Henry borrowed a book from the town library, at a half-cent a week, and kept it until the hire of it amounted to three and a half cents ; how long did he keep it ? 30. John has two three-cent pieces; how many quarts of milk, at two cents each, can he get for them ? 31. Mrs. Smith borrowed Mrs. Brown's sewing-machine, at a quarter of a dollar a week, and kept it a month ; how much did she owe Mrs. Brown for the use of it ? 32.' Mr. Stone hires out his horse at the rate of three dollars for two days ; how much would he charge for the use of it one day ? How much for three days ? 33. If two quarts of milk cost four cents, what will three quarts cost ? Suggestive Questions and Problems on Eight. i. How many ones in seven ? How many twos ? How many threes ? How many fours ? How many fives J How many sixes ? How many sevens ? Suggestions for Teaching Numbers. 1 1 5 2. Put one to seven and how many have you ? (Seven and one are eight.) 3. Place eight cubes upon the table and divide (separate) them into two equal groups. How many have you in each group ? What part of the whole ? 4. Divide eight cubes into four equal groups and tell me how many you have in each group, also what part of the whole. 5. Divide eight cubes into eight equal groups. How many have you in each ? what part of the whole ? 6. See how many groups of three you can find in eight ; how many of five ; of sixe ; of seven. 7. Five balls and three balls are how many balls ? 8. Eight less two are how many ? 9. Two and six are how many ? How many more than four ? How many more than seven ? 10. Two fours are how many more than three twos ? 11. One is how much less than one half of two ? 12. What is the difference between three and two thirds of six ? 13. See how many quarts of sand will fill this peck measure. 14. What is the larger measure, a peck or a gallon ? what is the difference ? 15. You may now find how many pecks of sand (or saw- dust) will fill this half-bushel. How many would fill a bushel ? How many half-bushels in a bushel ? 16. How many pints of chestnuts, at four cents a quart, can you buy for six cents ? How many for two cents ? 17. When milk sells at eight cents a gallon, how much would two quarts cost ? How much a pint ? 1 8. Helen bought a yard and a third of six-cent muslin ; how much did she pay for it ? 19. Margaret bought plums at four cents a quart, ho\v 1 1 6 Science and Art of Education. much did she pay for one fourth of a peck? for one eighth of a peck ? for a pint ? 20. How many eighths in one half? in one fourth ? in two halves ? in three fourths ? 21. How many more eighths in two fourths than in one half? 22. How many eighths in one half, one fourth, eighth ? 23. Which is the larger, one half or one third? What is the difference ? 24. Nora bought two thirds of a yard of lace at one store and one half of a yard at another ; how many yards altogether did she buy ? 25. How many months in eight weeks ? 26. Henry was hired to Mr. Carpenter at four dollars a month ; how much did he receive for three weeks ? for five weeks ? for three days ? for a week and a half ? 27. Mr. Clay hired Mr. Thome's horse at the rate of eight dollars for six days ; he used the horse four days. How much did he owe Mr. Thorne ? 28. Joseph borrowed five dollars of Philip, agreeing to pay a half-cent a month for every dollar borrowed ; if he kept the money two months, how much did he owe for the use of it ? 29. Alfred rented two books from the library; the first was worth one dollar, the second two dollars. If he paid a cent a week for the first and two cents for the second, how much did he owe at the end of two and a half weeks ? 30. One is what part of five ? of six ? of seven ? of eight ? 31. Two is what part of six ? of seven ? of eight ? 32. Three is what part of six ? of seven ? of eight ? 33. Four is what part of five ? of six ? of seven ? 34. Five is what part of six ? of seven ? of eight ? Suggestions for Teaching Numbers. 117 ' 35. Six is what part of seven ? of eight ? 36. Seven is what part of eight ? Seven is how many halves of eight ? 37. If you had one third of two yards of ribbon, what part of a yard could you make of them ? 38. Three fourths of two quarts of peanuts would make how many whole quarts ? how many pints ? 39. The surface of anything a piece of paper or board, for example is the outside of it. A square surface an inch on each side and all the angles of which are right angles, is I I || called a square inch ; one foot on each side, a square Mil foot, etc. A board three feet long and two feet wide contains how many square feet ? 40. How many square inches could you cut from a piece of paper three inches long and three inches wide ? How many from one four inches long and two inches wide ? 41. How many pieces of paper two inches long and one inch wide could be made from a piece three inches long and two inches wide ? 42. A piece of sheeting one yard long and two thirds of a yard wide contains how many square feet ? what part of a yard ? 43. How many square feet in a piece of cardboard a half-yard square ? REMARK. A block all of whose surfaces are equal squares is a cube. If the sides of the squares are one inch the block is a cubic inch ; if they are a foot it is a cubic foot, etc. 44. I have a cubical block of wood each of whose edges is two inches in length ; how many cubical blocks whose edges are one inch could be made of it ? How many such could be made of a block two inches long and wide and one inch thick? How many of one six inches long, one inch wide, and half an inch thick ? Scienct arid Art of Education. 45. A bushel of cloverseed costs four, dollars ; what is the cost of a peck ? of a quart ? 46. One fourth is what part of three fourths ? 47. Two fifths are what part of three fifths? of four fifths ? 48. One sixth is what part of five sixths ? of two sixths ? 49. Three sixths are what part of four sixths? of six sixths ? 50. Four sevenths are what part of six sevenths? One half is what part of four sevenths ? 51. One half is what part of five eighths ? Three eighths equal what part of one half ? 52. One fourth is what part of six eighths? of two thirds ? 53. One sixth is what part of one fourth ? Solution by diagram : 54. One seventh is what part of one sixth ? Solution by diagram : 55. One eighth is what part of one fifth ? Solution by diagram : -K. Suggestions /or Teaching Number $* tig Suggestive Questions and Problems on Nine* 1. Two threes and one two are how many? how many ones? 2. Three twos less two threes are how many ? 3. Two fours less two thirds of six are how many ? how many twos ? 4. Which is the larger, four twos or two fours ? 5. Four twos are equal to how many ones ? 6. If to eight ones you add one one, how many ones have you ? (Eight and one are nine.) 7. How many 'threes can you find in nine ones, or nine? 8. Nine contains how many twos ? fours ? fives ? sixes ? sevens ? eights ? 9. How can you get one third of nine ? What are two thirds of nine ? 10. Name the sums of the following numbers as quickly as you can : Three and four ; two and six ; five and three ; two and seven ; four and four ; one and four ; six and three; etc. 11. Name the differences between the following num- bers : Three and five ; two and six; one and four ; two and seven ; five and eight ; four and nine ; one and eight ; seven and nine ; etc. 12. From eight take two threes; from nine three take twos ? 13. I know a number which doubled and one added makes this nine ; what is it ? 14. Add one third of nine and three fourths of four. 15. Into what like numbers can you divide eight ? nine ? six ? 1 6. Three fourths of eight and one third of nine equal how many threes ? 17. Make nine in all the ways you can mentally, and in each case tell the result. Science and Art of Education. 1 8. In three yards how many feet? 19. How many yards in five feet ? in eight feet ? 20. What number is that whose third is three ? 21. If to one third of a number I add two, the sum will be three ; what is the number? 22. Three miles added to three fourths of the distance Samuel rode on his bicycle make nine miles ; how far did he ride ? 23. If from four times the number of quarts of chest- nuts Henry gathered three be substracted, five will be left ; how many did he gather ? 24. Five times the money Sarah received for making a dress added to two dollars equal seven dollars ; how much did she receive for her work ? 25. Three fourths of eight cents is two thirds of the price of a gallon of milk ; what is the price ? 26. It requires two thirds of nine yards of goods to make Ella a dress, and this is three fourths of the number re- quired to make one for Stella ; how many yards does Stella require ? 27. Elmer is twice as old as Carrie, and the sum of their ages is nine years ; required the age of each. 28. From four twos take two thirds of nine, and what is the result ? 29. How many square yards of carpet would be required to cover a floor nine feet long and six feet wide ? What would be the cost of the carpet at a dollar and a half a yard ? REMARK. The pupils should by this time have learned that the contents of surfaces are obtained by multiplying the length by the breadth, and that the quotient of the contents divided by one of the factors gives the other. 30. I want a piece of paper to cover the bottom of a box that is two inches wide and four and one half inches long ; how many square inches must it contain ? 31. The leaves of my book contain eight square inches ; how long are they if they are two inches wide ? Suggestions for Teaching Numbers. 1 2 1 32. I have a little cubical box whose inside is one inch in length, width, and depth ; how many half-inch cubes would fill it ? How many inch cubes ? Diagram. 33. How many inch cubes would cover the bottom of a box that is three inches long and two inches wide ? How many inch squares would cover it ? How can you ascertain without trying it ? 34. How could you find the number of square inches on the cover of my book ? You may do it. 35. Mr. Fox hired a carriage from the livery stable at the rate of three dollars for six days and used it four days; how much did he pay for the use of it ? 36. Dora is working for Mrs. Wilson for a dollar and a half a week (of seven days); how much will she receive for four weeks and three and a half days ? 37. How many two-cent pieces would you give me for eight cents ? for seven cents ? six cents ? five cents ? 38. As quickly as you can, give me the results of the following : Five less three ; eight less five ; seven less four ; eight less two ; six less three ; nine less six ; nine less four; seven less two ; three times two ; two times four ; three times three ; four times two ; eight less three ; five and four ; two and six ; twos in four ; threes in six ; fours in eight ; twos in three ; threes in four ; threes in nine ; etc. 39. What part of one is one third of one third. Solu- tion : 40. What part of one is one half of one fourth : Solu- tion : 41. What part of one is one fourth of one half? Solve by diagram. Science find Art of Education. 42. One eighth is what part of one fourth ? Solve by diagram. 43. Two eighths equal what part of one half. Solve by diagram. 44. Three eighths equal what part of five eighths ? Solve by diagram. 45. How many eighths in three fourths ? Solve by diagram. 46. How many eighths in two thirds ? Solution by dia- gram. 47. How many thirds in five eighths ? 48. How many fifths in two thirds. Solution : 49. How many fourths in three fifths ? Solution : 50. How many thirds in four ninths ? 51. How many halves in two thirds ? Solution : ^*h 52. One third is what part of one half? 53. How many eighths in one fifth ? Solution : H J-K, Suggestions for Teaching Numbers. 54. How many sevenths in one sixth ? Solution: 55. One eighth is what part of one third ? Solution : K. 56. One seventh is what part of one half? Solution: Suggestive Questions and Problems on Ten. 1. Three threes are how many ones? Add one to them and how many have you ? (Nine and one are ten.) 2. Four twos and two ones are how many ones ? how many twos ? 3. Make ten of fives ; how many does it take ? 4. Two fours are how many less than ten ? how many more than six ? 5. Six and how many make ten ? Four and how many make ten ? 6. How many threes in ten ? How many fours ? How many sixes ? 7. Seven and two are how many ? How many more than five ? How many less than ten ? How many more than eight ? 8. How many must be added to two to make ten ? 9. Make ten in all the ways you can mentally, and in each case state the result. 124 Science and Art of Education. 10. What is one half of ten ? One fifth of ten ? Three fifths of ten ? 11. To one third of nine add three fourths of eight, and what is the sum ? 12. What is the sum of three fourths of eight and two thirds of six ? 13. What is the sum of two thirds of three and one half of nine s ? 14. Ten less seven are how many ? 15. How many fifths of five must be added to five sixths of six to make, nine ? 1 6. I know a number which doubled makes ten ; what is it? 17. I think of a number which taken three times and two added makes eight ; what is it ? 1 8. What number diminished by its third leaves six ? 19. What number increased by its half equals nine ? 20. Six times a number diminished by five equals one; what is it ? 21. The sum of two numbers is seven and one of them is four, what is the other ? 22. What number added to one third of ten makes four ? 23. If you double a number and take four from it you have six ; what is the number? 24. Ten thirds equal how many ones ? 25. How many ones in ten fourths ? in ten fifths? 26. How many tenths in one fifth ? in one half? 27. Two fifths and one half equal how many tenths?. 28. Can you malce tenths of fourths ? Why not ? 29. What can you make of halves and fourths ? Why ? o.f halves and thirds ? Why ? of halves and fifths ? 30. What can you make of halves, thirds, and sixes? Solution by diagram : X Suggestions for Teaching Numbers. 125 31. One half and tsvo thirds equal how many ones ? 32. Can you add halves, fourths, and eighths ? How ? 33. Can . you add halves and fifths ? How ? Show by diagram. 34. In a piece of goods ten yards long and one yard wide how many square yards ? Prove it. 35. How many yards long is a ten-foot pole ? 36. Helen's apron is two feet in length and a half-yard in width ; what part of a square yard of goods does it con- tain ? 37. A board eight feet long contains four square feet; how wide is it ? What would its width be if it contained ten square feet ? 38. My table is a yard in length and five sixths of a yard in width ; how much will it cost to cover it with oilcloth at six tenths of a dollar a square yard ? 39. How would you find the number of square feet of paper required to cover the lower sash of that window ? Do it. 40. Here is a chest that is four feet in length, two feet in width, and one foot in depth ; how many blocks, each containing a cubic foot, would exactly fill it ? How can you tell ? Show by a diagram. 41. Henry bought a half-gallon of cherries at five cents a quart and paid for them with five-cent pieces ; how many did it take ? How many two-cent pieces would pay for them ? How many ten-cent pieces ? How many dimes ? 42. John sold nine pigeons at the rate of a half-dollar a pair ; how much did he receive for them ? How many quarter-dollars ? 43. When sugar is five cents a pound, what will one and cost two thirds pounds ? 44. Mr. Down loans money at a third of a cent on a dol- 1 26 Science 'and Art of Education. lar a month ; how much interest does he receive for the use of five dollars for three and a half months ? for six months ? 45. At a half-cent a month on a dollar, in what time would six dollars give nine cents interest ? 46. At a fourth of a cent a month, how many dollars (how much principal) would in four months give eight and three fourths cents interest ? 47. At what rate (how many cents on the dollar) will four dollars, in three months, give six cents interest ? 48. If Henry adds the interest of nine dollars, at a third of a cent a month, for two and a half months, to the princi- pal, what will the amount (sum) be ? 49. If five oranges cost nine cents, what is the cost of three of them ? 50. When two fifths of a yard of tape cost four cents, what does a half-yard cost ? 51. Three yards of lining cost nine cents ; at the same price, what would two and two thirds yards cost ? 52. At two thirds of a cent each, how many oranges can be bought with two cents ? Solution: i. If they had cost one third of a cent each, for two cents six could have been bought ; but since they cost two thirds of a cent each, only half as many can be bought as if they cost one third of a cent each, or three. 2. Since two thirds of a cent pay for one orange, one third of a cent would pay for one half of it, three thirds for three halves, and two cents for two times three halves, which are six halves, or three. 53. At three fourths of a cent each, how many yards of cord could you buy for two and a half cents ? 54. Mary bought six dollars' worth of cloth at two and a half dollars a yard ; how many yards did she buy ? 55. In ten pecks how many bushels ? In nine pecks how many? 56. In a half-bushel of cherries, how many gallons ? Suggestions for Teaching Numbers. 127 57. One is what part of ten ? of nine ? of seven ? of two and a half ? 58. Five is what part of ten ? of nine ? of six ? 59. What part of one is one third of one third ? NT1-.H. 60. What part of one is one fourth of one half? 6 1. One half of one fifth is what part of one ? Solve by diagram. 62. How many fourths in two thirds ? H( 63. How many ninths in one half ? -2%. 64. How many eighths in one third ? 65. One fourth is what part of one third ? 66. One fifth is what part of one half ? of one third ? Solve by diagram. 67. How many fifths in one third ? Solve by diagram. 68. One ninth is what part of pne third? 128 Science and Art of Education. 69. One eighth is what part of one third ? 70. One fourth is what part of two sevenths ? 71. One half is what part of three fifths ? 72. How many times can you find three fourths in six ? 73. How many times can you find two thirds in five ? 74. How many times can you find One half in three fourths ? 75. Two thirds are what part of four ? 76. If a tailor can make a pair of pantaloons in a day how long would it take two tailors to make them ? 77. Two boys can pick two quarts of berries in an hour , how long would it take one of them to pick them ? How long would it take three boys to do it ? 78. Four boys gathered two bushels of chestnuts in two hours ; how many boys could, in the same time, have gathered three bushels ? 79. If five cows can eat an acre of grass in one week, how long would it take one cow to do it ? How long would it take one cow to eat two acres ? 80. Emma can make a dress in two days and Jennie in three ; what part of it can each make in a day ? What part of it could the two together make in a day ? How long would it take them to make it working together ? Suggestions for Teaching Numbers. 129 81. A man and a boy together do a piece of work in two days ; what part of it does each do, if the man does twice as much as the boy ? How long would it take each alone to do it ? 82. Bessie and Elsie received nine cents for picking strawberries ; how much did each receive, if Elsie received half as much as Bessie ? 83. Henry can do three times as much work in a day as Samuel ; how long would it take Henry to do what Samuel can do in two days ? How long would it take Samuel to do what Henry can do in two days ? 84. Find the length and width of the smallest board that you could exactly cover either with two-inch or three-inch squares. 85. What is the smallest bag that you could make that could be exactly filled with a pint-measure or a quart- measure of chestnuts ? 86. What is the length of the shortest pole that could be exactly measured either with a two-foot measure or with a yard-stick ? 87. What is the smallest number of apples that you could all sell either by twos or threes ? 88. What is the largest measure with which I could empty each of two boxes, one containing a quart of berries, the other a gallon ? 89. Mr. Miller has two baskets of cherries, one contain- ing two quarts, the other three ; what is the largest cup with which he can exactly measure the contents of each basket ? 90. What is the largest measure that is contained in three feet, six feet, and nine feet ? in four feet and eight feet ? in four feet, six feet, eight feet, and ten feet ? I. Writing Numbers aboiw Nine. REMARK. The pupils are supposed to have learned to write and use numbers up to ten. 130 Science and Art of Education. 1. They should now be told that nine is the highest number that we can write with one figure, and that above nine the numbers are written as tens and ones. 2. Before the children are taught to write ten, they should have practice in finding the number of tens in a number of objects ; tying toothpicks, or other suitable objects, into bundles of tens, affords perhaps the best prac- tice in counting by tens. One bundle should be called one ten, two bundles two tens, three bundles three tens, etc. 3. If a pupil has more toothpicks given him than make an exact number of tens, the remainder should be con- sidered as so many ones (or units). For example, if he should have sixteen given him, after having made all the possible bundles, he would have six toothpicks, or ones, left. 4. To make the bundles, the children should sit or stand around a table, each having a small handful of splints or toothpicks before it, and rubber bands or threads to tie those of a bundle together. 5. After the bundles have been made the children should count both them and the single things or ones, and write the results in columns prepared for the purpose. After each child has written its results or sums in the proper columns, the columns should be added, also the bundles and single things, and the results compared. Example i : Suppose there are four children and each has made its bundles ; the first having 2 bundles and 4 single things ; the second, 4 bundles and 3 single things ; the third, i bundle and 7 single things ; and the fourth, i bundle and 2 single things. They now write Tens o/i 6 their results in columns, as here indicated, and add them. The first, or ones' column, gives i bundle and 6 single things. Writing the 6 under the ones' column and adding the bundle to the tens, gives as the result of both, 9 bundles and 6 single things, Suggestions for Teaching Numbers. 131 After the sums of the columns have been ascertained, the children should give their bundles and remaining tooth- picks to one of their number, who, after having made as many bundles as possible of the remaining single tooth- picks, should compare her bundles and remaining toothpicks with the sums of the columns. Example 2 : REMARK. This example, besides carrying the work another step forward, also introduces the nought. Let us suppose that there are six children in the class, each having made its bundles ; the first having 3 bundles and 5 toothpicks ; the second, 4 bundles ; the third, 2 bundles and 7 toothpicks ; the fourth, 3 bundles ; the fifth, 2 bundles and 8 tooth- picks, and the sixth, 2 bundles. Adding the columns and also the toothpicks, it is found that there are ten bundles and eight bundles; but since there can be no more than ten of a kind, the ten-bundles must be tied together, making a ten-ten bundle, and its number written at the foot of a line next to the left of the tens. 6. The children should work with toothpicks in connec- tion with figures until they can write and read numbers to one hundred at least. They should also be led to see that all numbers above ten are composed of tens or tens and ones; and instead of continuing to use the names tens and ones, the usual names should gradually be introduced. Thus, for ex- ample, instead of saying one ten and six, two tens and four, etc., they should learn the names sixteen, twenty-four, etc. REMARK. The nought, having no numerical value, is used to give the significant figures their proper places. 7. The pupils should be led to see that the value of a figure depends upon the place it occupies in a number ; that, in general, every ten of one place makes one of the next to the left, or higher, and also the reverse, namely, that overy place to the right is one tenth of the next to the left. 1 3 2 Science and Art of Education. 2. Suggestions on Teaching Numbers up to Twenty. The numbers up to twenty at least should be so thorough- ly taught that the pupils can instantly give the sum or pro- duct (less than twenty) of any two of them ; also the reverse ; and if a sum or product is given and one of the two numbers or factors that compose it, the other should instantly be upon the pupils' lips. 3 A Device for Oral Addition. Two numbers below ten, whose sum exceeds ten, may readily be added by taking the difference between the larger and ten from the smaller, adding it to the larger, and add- ing the remainder of the smaller to ten, the sum. Example : To find the sum of 6 and 8. The difference between 8 and 10 taken from 6, and added to 8, makes 10; and 4, the re- mainder of the smaller, added to 10, makes 14. REMARK. The teacher should lead the children to discover every device that will enable them to overcome their early difficulties and lighten their labor. 4. The Four Fundamental Processes Carried on Together. It should be borne in mind that addition, subtraction, multiplication, and division are to be carried on together. 5. Pupils May Construct the Tables. To familiarize themselves with the relations of number, the children may construct the tables, but in giving the results in class they must not think of the position of the numbers in the table, but must give them instantly as far as reasonable, automatically. 6. Suggestive Exercises for Seat Work. REMARK. o = what ? I. 2. 3. 4. 1. 1001 = 11 I. + 2=12 I. 1+0=13 1.10 + 4=0 2. 9 + 0=11 2.120=8 2.1301 = 12 2.1301 = 14 3. 01 = 10 3. + 3=12 3. 10 + 3=0 3. 140=9 4. 8 + 3=0 4. 1202=10 4. 130=9 4. 04-2=7 5.11-0=9 5. 0x3=12 5. 0+7=13 5.11+0=14 Suggestions for Teaching Numbers. '33 6. 0-:-I = II 7. 114=0 8. + 5 = 11 9. 7+0=11 10. + 0=11 n. 00=6 etc. 5. 6. 127=0 7. 705 = 12 8. 0x0=12 9. 0-0=7 10. O + O=I2 II. O-r-O = 6 etc. 6. 6. 13-5=0 7. 0+0=13 8. 00=3 9. 138=0 10. 1102=13 n. 130=6 etc. 7. 6. 7x2=0 7. 608 = 14 8. 0x0=14 9. 0+0=14 10. 00=5 II. 0-r-0=2 etc. 8. i. 0+14=15 i. 0+10=16 i. 10+7 =o I. 0-1 = 17 2. 150 =10 2. 16 o =9 2. 06 =11 2. I+0=l8 3. 906 =15 3. + 3 =16 3. 1708 =9 3. 1803 = 15 4. 5x0=15 4. 16 10=0 4. 15+0 =17 4. 15 + 3=0 5. 1510=0 5. 0+11 = 16 5. 17-3 =o 5. 0-5 = 13 6. 1503 =5 6. 16013=3 6. 0-14=3 6. 6+0=18 7. o-i =14 7. o+o =16 7. 17012=5 7. 18-7=0 8. oxo =15 8. oxo =16 8. 7+0 =17 8. 0+0=18 9. o+o =15 9. o-o =5 9. + 9 =17 9. 0x0=18 10. O-HO =3 10. o-*-o =4 10. o+o =17 10. 00=7 ii. o o =4 ii. 8 + 8 =o ii. oo =6 II. 0-5-0=2 etc. etc. etc. etc. 9. 10. n. 12. i. 0+3 =19 i. 19+0 =20 i. of 4=0 I. i Of =f 2. I9 O =15 2. 20 3 = 2. $ Of 8=0 2. 2 X O =1 3. 1 1 +8 =o 3. + 14=20 3. i of 6=0 3. o of 6 =4 4. 1906 =13 4. 20 o 11=9 4. of 9=0 4. X f =2 5. 0+7 =19 5. 3+0 =20 5. i of 5=0 5. i of o =| 6. 190 =6 6. 4+16=0 6. i of 4=0 6. f of 2 =o 7. 3 + 16=0 7. 208 =o 7. i of 8=0 7. i of 3 =0 8. 1908 =11 8. 0+6 =20 8. f of 9=0 8. | of 4 =0 9. 0-14=5 9. + =20 9. f of 7=0 9. of 10=0 10. o+o =19 10. 0X0 =2O 10. $ of 4= 10. f Of 2 =O ii. o o =8 ii. O-HO =5 ii. i of 7=0 ii. of 6 =0 etc. etc. 13. 14. 15. 16. i. 4 = | of o i. 4 x i=o i. 9= f of o I. | Of 12=0 2. 5 = 4 of o 2. o x 4=5 2. 8= $ofo 2. i of 3 =0 3. 6 = \ of o 3. i of 0=3 3. 3= 1 of o 3. \ of i=o 4- ox | = f 4. i of o=i 4. 2= 5 x o 4. of o =i 1 34 Science and Art of Education. 5. f of o = 3 5. O X = 2 5. 12= f Of 5. X $ =| 6. 8 = f of o 6. ^ x o=* 6. |= | of o 6. | of o =7 7. 9 = 1 of o 7. f ofo=f 7. 5= 3 x o 7- i of o =3* 8. 2 = f of o 8. o x 2=3 8. i= i x o 8. i of 2 =6 9. i = 3 x o 9. 2 X O=5 9. I O = i 9. 2 x o =i 10. 6 = f of o 10. | of 5=0 10. o + 3f = 5 10. o x 1^=4 II. f i Of II. o x 1=1 ii. fos = 2 ii. } of o = 17. 18. 19. i. Jofo = i i. f ofi =0 i. 5 = f of o 2. If X = 4 2. fof f=0 2. f Of 2 =4X0 3. X 2 = If 3. i of 1=0 3. f of 3 =3x0 4. f of 0=14 4. | of 4=1 of o 4. 18 = i x o 5. 0-1-4 = 1 5. fof 5=2 x o 5. 2 x o = of 3 6. if x o = 5 6. i x 2i=of o 6. $off = iofo 7. f x = 3 7. i= fofo 7. IO = 2 X ofo 8. \\ x o= i 8. i of 1=0 8. f =f of o 9. 2* x o = \ 9. i= fofo 9. fof = fofo 10. 3i = i f 10. = | Of 10. i Of $ = T x Of II. 2f X = | ii. f = i of o ii. 18 =41 x o 20. 21. 22. i. 9 = 9xf of o I. 12 -r- =9 i. o -4- 4f =si 2. $ Of 2=f Of 2. 9 = 12 X 2. 5 = | ofo 3. 2f x 0=18 3. 10 = f of o 3. f of o =9 4. o x 4=15 4. 15 = 16 x o 4. o x 18 =17 5. f Of 2= X 5. i of 5 =10x0 5. 14 = 15 x o 6. f of 2=f of o 6. 0x9 =6 6. 15=1 of o 7. I of f=fofo 7. 5 -T- o = 20 7. 13 = f of o 8. f of =f of o 8. 8-4-0 = if 8. 9x1^=0x2 9. f of 4=16 xo 9. f Of =2 9- 3f -*- 1 =o 10. 17 = 5f xo 10. 6xo = i| 10. 4| -5- o =3 II. ^ 2 = lf 11 | X O = ii. f -*- o = i 23. 24. 25. I. O-r-4 = lf I. 0x18=19 I. 2f -4-0=f 2. 3-0 =f 2. 4-5- 0= 8 2. f -4-0= If 3. 6 of =4 3- 5-*- 4= 3- o -f =f 4. I 2 =f 4. 0-5- $= 6 4. if +o=3f 5. f 02 = lf 5- f + o= 1 5. 2f Xf=0 6, 0-5-f =1 6. 13 x 0=17 6. 1^x0=17 Suggestions for Teaching Numbers. 135 7. f-J-o =3 ; 8. O-f =2 9. ^xi =o c 10. 2i = l K ii. o-*-4i=4i ii 26. I. o lf=2| Lem 2. 12-5-0 =3| 3. Of Ii=2 4. 3 -5- o =8 5. 4-5-0 =3 6. 9 x o =19 r. 12-1- O= 9 ). 9= OX 12 :>. 15-5- 0=20 . o-f- |=I5 27. ons. cts. Lemons. 2 3 3 2 I 4 4 3 2 4 7- f oi= 3 8. f =5x0 9. 2| X2=f Of ii. ii +o=3i 28. Oranges cts. Oranges. 4 5 2 7 3 5 6 3 7 2 7- x 17 = 13 5 3 3 6 2 8 7 -h- = 12 2 6 4 8 3 9- 1 1 X = 16 6 2 5 15 2 10 12 =8* 4 4 2 4 3 u. fof* =0 5 2 3 9 4 2 6 3 29. 30. 31. Apples. cts. Apples. Yds. cts. Yds. Yds. cts. Yds. 3 12 4 4 12 6 I 15 ^ 4 16 3 6 12 9 f 3 2i 2 IO 4 8 16 i I| 2 3 5 15 4 5 15 i 2i 4 1 2 4 8 2 16 i f ii f 7 7 5 i 2 3 f ii 2i 4 20 3 i 2 i 1 J T 4 6 12 7 i 2* 4 2 2i f 7 H 9 * 8 | | 2 t 9 18 8 f 9 I | f 2i 8 16 7 2 16 1 Ii 2 2* REMARK. To indicate how the exercises under 27-31 should be read, the first under 29 may be taken as an example. If 3 apples cost 12 cents, what cost 4 apples ? 7. Suggestive Problems for Oral and Seat Work. 1. If* 4 quarts of milk cost 8 cents, what will 8 quarts cost? 2. If 2^ yards of tape cost 15 cents, what will if yards DIVERSITY) v o,n ,<**. y 136 Science and Art of Education. 3. My table is 3^ feet in length and two feet in width ; how many square feet of oilcloth will cover it ? 4. A piece of cloth is one yard in length and the sajme in width ; how many square feet of paper would cover it ? How many square feet does it contain ? 5. How many square feet in a surface a foot square ? in one a yard square ? in one 3 feet square ? in one 4 feet square ? 6. How many square yards in a surface 6 feet long and 3 feet wide ? 7. I have a box that is 2 feet in length, i foot in width, and i-| feet in height ; how many square yards of paper would cover it ? 8. Show me an inch on this foot-measure. How many inches in length is the whole measure ? How many inches are in a foot ? in a \ foot ? in f of a foot ? in f of a foot ? in | of a foot ? in i feet ? in i feet ? 9. How many feet in 9 inches ? in 8 inches ? in 14 inches ? in 18 inches? in 3 inches? 10. How many quarts in 2 pecks ? in a half-bushel ? 11. How many pecks in 12 quarts? in 18 quarts? in 6 quarts ? 12. At 8 cents a gallon what would 12 quarts of milk cost ? What would 18 quarts cost? 13. Take the weights and find out how many ounces make a pound (avoirdupois). How many ounce weights are as heavy as f of a pound ? What part of a pound weighs as much as 12 ounces ? 14. How could you add into one sum 2^ feet and 4 inches? How 5 feet, f of a foot, and 8 inches ? How 3 yards, 2 feet and 18 inches ? 15. Take the yard-stick and measure 5-} yards from the platform through the middle of this aisle. How many feet did you measure ? 5^ yards, or i6j feet, are called a rod. 16. How many rods in the length of this room ? How Suggestions for Teaching Numbers. 137 many in the width ? How many feet in a | rod ? in \ of a rod ? in f of a rod ? REMARK. The pupils should make inch, foot, yard, -and rod measures and use them in measuring lengths and distances. The shorter measures may be made of wood, the longer of twine or cord. 17. Add into one sum i rod, i yards, i foot, and 8 inches ? 1 8. Find the sum of 3 gallons, 2 quarts, and i pint. 19. Find the sum of J of a pound and 7! ounces. 20. How many pints in f of a peck, 3^ quarts, and f of a pint ? 21. How many feet in of a rod, 2^ yards, 2 feet, and 9 inches ? 22. How many ounces in f of a pound and 2\ ounces ? 23. In 15 pints how many gallons ? One pint is what part of a gallon ? of 3 quarts ? Three quarts are what part of three gallons ? 24. In 18 inches how many feet? What part of a yard ? 25. Nineteen ounces equal how many pounds ? 26. Twelve ounces equal what part of a pound ? 27. In 17 feet how many yards ? how many rods ? 28. How many feet in 16 inches ? in 10 inches ? 29. In 15 weeks how many months ? in 17 weeks ? 30. How many weeks in 2^ months ? in of a month ? 31. How many days in 2^ weeks ? in f of a week ? 32. In 14 quarts how many pecks ? what part of a bushel ? 33. In 20 pints how many pecks ? what part of a bushel ? how many half-bushels ? 34. How many months in \\ years ? in if years ? in f of a year ? 35. How many years in 15 months ? in 8 months ? in 9 months ? 36. Find the interest of $2- at 6 per cent (6c. on a dollar Science and Art of Education. for a month) for f of a year? for 16 months? for 3 months ? 37. What is the interest of $4! at 3 per cent for if years ? for 4 months ? 38. The interest of $5 for 2 years is 2oc. ; what is it of $i for i year? 39. What principal, at 2C. on a dollar a year, will in 16 months give i5c. interest ? 40. In what time (how many years) will $6, at 2^ per cent, give i8c. interest ? 41. If by selling my knife for i6c. I gain of the cost, what was the cost ? 42. Sarah lost \ of its cost by selling her bird for i2c.; how much had she paid for it ? 43. By selling a book for ice. I lost J of its cost ; what should I have sold it for to have gained \ of its cost ? 44. One of two boys can do a piece of work in 3 days, the other in 4 days ; what part of it can each do in a day ? What part can both together do in a day ? How long would it take both together to do the whole of it ? 45. If it takes A 2 days to do a piece of work and B 5 days, how many days would it take them together to do it ? 46. Four men can do a piece of work in 2 days ; how long would it take one of them alone to do it ? Solution : oo + oo + oo + oo = 8. Let every o represent a day's work of "each man ; then all of them will represent 8 days' work of a man. 47. If 5 girls can make a certain number of dresses in 4 days, in how many days could two of them do the same work ? Illustrate by diagram or other form. 48. If 5 boys can pick 30 bushels of apples in a day, how long would it take i boy to do it ? How long would it require 3 boys to do it ? 49. If 10 is f of a number, what are of it ? Suggestions for Teaching Numbers. 139 50. If f of Henry's ducks equal -J of his turkeys, and he has 24 ducks, what is the number of his turkeys ? 51. A watch and chain cost $15 ; what was the cost of each, provided the chain cost f as much as the watch ? 52. If Jennie adds 8 years to of her age, the sum will be her age ; how old is she ? 53. One half, , and i of a certain number added to 8 make 21 ; what is the number ? 54. At f of a cent each, how many lemons can be bought for 4 cents ? 55. Alberta bought 4 oranges for 3 cents ; what was the cost of each ? How many did she get for i cent ? 56. At $f a yard, how many yards of cloth can be bought for $6 ? 57. A boy gave $3 to a number of beggars, giving to each $f ; how many beggars were there ? 58. If to Alfred's money you add 4 times his money, he will have 25c.; how much money has he ? 59. Find the difference between a square inch and an inch square ; between 2 square inches and 2 inches square. 60. What is the difference between a foot square and a half-foot square ? Illustrate with diagram. 61. How many 6-inch cubes could you make of a cubic foot ? how many 4-inch cubes ? How many 2-inch cubes could you make of a 6-inch cube ? Diagram. 62. How many cubic feet in a box that is 3 feet long, 2 feet wide, and i foot deep ? 63. Could you find how many square yards of carpet would cover this floor ? How would you do it ? If the carpet were f of a yard in width, could you find how long a piece it would require to coyer the floor ? Illustrate with diagram. 64. How could you find the number of square feet of paper required to cover the walls and ceiling of this room ? Could you find the number of square yards ? How ? Could 140 Science and Art of Education. you find how long a roll it would require if the paper were only a half-yard in width ? 65. How many foot cubes would cover the floor of this room ? how many would fill the room ? How many cubic feet in this room ? how many cubic yards ? 8. Diagrams and Figures. 1. The teacher should be careful that pupils do not mis- take fractional expressions for fractions ; fractions are parts of things or wholes, and fractional expressions, as , , etc., are the signs or language by means of which they are repre- sented. 2. Fractions may be introduced and to some extent taught by means of folding paper 01 by various forms of diagrams. The following illustrations, in addition to those already given, may prove helpful to teachers, and may be preferred by some. X X XX g X 3. After pupils have learned to work with real fractions (parts of objects) they should be taught the signs by which they are represented. 9. Adding, Subtracting, Multiplying, and Dividing by Diagrams. i. How many fourths in -J ? in and \ ? What H part of one is | of 4? i + i=? t+t=? i + i =? i+i + =? i-i =?*-*-*=? etc. 2. How many sixths in one ? in ? in \ ? What part of Suggestions for Teaching Numbers. 141 i is i of ^ ? of J ? k is what part of -J ? is what part of ? i is what part of ? How many sixths in ? How many | in f ? How many }(> -i=?i+i-H=?i + f=? . Si ? 5 1 >4 JL > _L 1 "> JL 1 - ~~ 6 1> 3 '3 2 * ~ "6 2 ~ "5 ' 3. How many -J in i ? in i ? in - ? ^ is what i of | ? -J of J ? How many i in f ? How many -J in ? What part of i is J of J ? of i ? How many i in $ ? How many times are f contained in |? --{-=? i + i=? f + i =? i + + i=.> i + |=? i-i=? i-t = ? i ^. | =? i -f- f =? i -f- i =? etc. 4- i of i is what part of i ? How many is what part of ? how many ^ in i ? i con- tains how many i ? how many i in f ? J -f =? t + i =? i-i =?f-i =?* + *=? i -i-t = ? etc. 5. i of i is what part of i ? How many \ in i ? How many ^ in -J ? how many ^ ^ ? is what part of ? ^V is what part of ? How many ^ in J ? How many in J ? How many times is contained In -^ ? How many times are ^ contained in 4- ? -ft- are what part ofm-ff + iV=?i-H=?i-|=?A- in in ? how many U H 7. What is i of } ? J of ^ ? J of J ? How many , 1 .> in I 4 2 Science and Art of Education. \ ? in \ ? in J ? in J ? How many \ in ? in i ? How many J in - ? J is what part / of 4 ? -J is what part of \ ? is what part ^( of f ? How many^ in f ? i is what part \ of \ ? How many J in f ? How many ^ in \ ? f equal what part of jj ? -J- -j- J -j-^ M % % & fcfc K 2 K 2 K 2 ^ 2 K, t - A =? t x i =? i x i =? * x t =? f-x-t =?? f 4-.i 8. How many ^ in ^ ? How many )& * i in J ? ^ is what part of \ ? -J is what , . I^nT7T77h7T\ r 1 -i 1 il i li 1 > K i M 716 TIC 716 716 \ part of ^ ? -|- -J- ^ + ^ -f" 'iV A ~r : ^ ) -A=? lT ^i= f? -5-f=? t^A=?i M H6_K6_K6_^ X i=? ^Xi=?iXi=?fX2=:? MJ^JM^J^ |Xi=? i X T V =? etc. 9. What is i of I ? i of i ? i of i ? I of i ? i + I + i =? Ms Ms Ms Ms Ms Ms Ms Ms Ms Ms Ms Ms Ms Ms Ms Ms Ms Ms 10. One fourth is what part of i ? - wna t part of is what part of J ? of is what part of^? of^ =?ix J=?t xi=? 4x^=? jxf =?A =? Ms Ms ' ;,. Ms Ms Ms Ms Ms Ms Ms to Ms Ms Ms to Ms Ms Ms Ms Ms Ms to to Ms Ms Ms to to to Ms Ms to to to to Ms Ms Ms to to to to Ms Ms to to Ks to Suggestions for Teaching Numbers. ii. How many .01 in i ? How many .1 in ^? in many .01 in i ? in i ? How many .1 in .50 ? i + .1 + .01 ? How .1 .1 .1 .1 .1 .1 .1 .1 .1 .1 Ho Mo Ka K Hi Ko Ki Ko Ko Ko .01 .01 (11 01 .01 01 .01 01 .01 01 .01 .01 01 01 .01 01 .01 01 .01 .01 .01 .01 .01 01 .01 .01 .01 01 .01 .01 .01 .01 .01 01 .01 .01 .01 .01 .01 .01 .01.01 .01 .01 .01 .01 .01 .01 .01 .01 .01.01 .01 .01 .01 .01 .01 01 .01 .01 .01.01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 .01 -f .05 =? .56 .4 =? X .10 = ?T*TF + -3 =?.i+.i = ?.!-. 01=?.! + ^=? .1 -T-.OI =?t -5--^=? .01 + .4 = ?. 4 + .0 7 =?.I2 + T*ir=? .7 + .6 -? .18 + .24 =? .06 -4- .02 =? .50 -f- .05 =? .75 -i- .3 =? .09 .5 =? .12^ -|- .14 =? etc. REMARK. If the foregoing work has been well done, the pupils will, to such an extent, have learned to make their own discoveries to help themselves that they will experience comparatively little difficulty with the higher numbers and their combinations. 10. Me, tal Addition of Two-figured Numbers. 1. Two numbers, each composed of tens and units, may readily be added mentally by first finding the sum of the tens and then adding to it that of the units. Example : Find the sum of 28 and 39. Two tens and three tens are five tons, or fifty, and 8 units and 9 units, or 17 units, (added) equal sixty-seven. 2. As soon as the pupils are far enough advanced to do so, instead of saying two tens and three tens are five tens, they should say twenty and thirty are fifty, etc. They should also be led to see that the same combination of units will invariably give the same units' figure. For example, 6 and 7, 16 and 17, 26 and 37, and so on, will all have 3 for the units' figure. 1 1 Subtracting a Greater from a Less. i. Subtracting one number from another when some of the figures of the minuend are smaller than those of the T44 Science and Art of Education. subtrahend to be taken from them, may be developed by 453 means of tooth-picks. Example : . Since 4 cannot 219 be taken from 3, one of the ten-bundles must be opened, its contents put to the 3 ones and the 4 taken from the sum. 2. The same thing may also be developed with dollars, dimes, and cents of toy-money. Taking the foregoing example, since 4 cents cannot be taken from 3 cents, one of the ten-cent pieces of the minuend. must be exchanged for cents and the latter added to the three to make the subtraction possible. One ten having thus been taken from the 5 of the minuend, 4 remain, from which the 3 of the subtrahend must be taken. 3. The teacher should lead his pupils to see that finding the difference between two numbers is the same as finding what must be added to the smaller to make the larger, and that the difference may therefore be found either by addi- tion or subtraction. 4. Methods of proof should also be developed, so that the pupils may have the means of testing the correctness of their work. REMARK. i. As far as possible and as long as necessary the solution of every problem should be illustrated either with objects, diagrams, or drawings. The drawings, at first crude, will with the teacher's assistance gradually improve and will create an interest in that kind of work. REMARK. 2. It cannot be too strongly impressed upon the mind of the teacher that a part of every lesson should be a review of some of the previously prepared lessons or subjects passed over. Daily reviews give not only clearness to concepts, but impress them firmly upon the mind. 12. Mental Multiplication. i. Mentally finding the product of any two numbers up to 100, besides affording a good exercise for the memory, enables the pupils to perform many arithmetical operations Suggestions for Teaching Numbers. 145 without resorting to pencil and paper. To illustrate the method of doing this when both factors are less than 20, let it be required to obtain the product of 14 by 14. Fourteen is the sum of 10 and 4. First, multiply 14, the multipli- cand, by the 10 of the multiplier (or simply annex a nought); next the 10 of the multiplicand by the 4 of the mul- tiplier ; add the two products ; finally, multiply the 4 of the multiplicand by the 4 of the multiplier and add the prod- uct to the sum of the previous products. Operation : 14 X 10 = 140 ; 10 X 4 = 40 ; 140 + 40 = 180 54X4 = 16 ; 180 -f- 16 = 196. 2. The foregoing method applies as well to unequal as to equal factors. For example, let it be required to find the product of 19 by 15. Operation : 19 X 10 = 190 : 10 X 5 = 50 ; 190 + 50 = 240 ; 9 X 5 = 45 ; 240 -f- 45 = 285. 3. To find the product of any two factors that are greater than 20 and less 100, multiply the multiplicand by 10, the product by the number of tens in the tens of the multiplier, the tens of the multiplicand by the units of the multiplier; add product to preceding produt; multiply the units of the multiplicand by the units of the multiplier, and add product to preceding sum of products. For example, multiply 48 by 36. Operation : 48 X 10 = 480 ; 480 X 3 = 400 X 3 -f- 80 X 3 = 1440 ; 40 X 6 = 240 ; 1440 -f- 200 + 40 = 1680 ; 8 X 6 = 48 ; 1680 -f 40 + 8 = 1728. REMARK.- As an aid to memory, whenever it is possible, round numbers (as in the foregoing operation) should be mul- tiplied and added. 13- Properties of Nine and Their Application. 1. If a number is divisible by 9 the sum of its figures is divisible by nine. Illustrative examples : 18, 27, 63, 81, 4896, etc. 2. The remainder, or excess, of the division of a number by 9 is the same as that of the sum of its figures divided by 9. Illustrative example : 5843 divided by 9 leaves 2 as 146 Science and Art of Education. the excess, and 20, the sum of its figures (5 + 8 + 4 + 3), divided by 9, leaves the same. 3. The excess of the division of the sum of two or more numbers by 9 is the same as that of the excess of the sum of their excesses. From this fact is derived one of the simplest and quick- est methods of proving addition and subtraction. 4. Application to Addition. RULE : Find the excess of each of the numbers added, add these excesses, find the ex- cess of their sum, and if the latter is the same as the excess of the sum of the numbers added, the work may be con- sidered correct. Illustrative example : 5873 5, excess of division by 9. 4652...... 8, 8763 6, 579 3^ " " " 19867... 4. 22.... 4. The sum (22) of the excesses, divided by 9, leaves 4 as the excess ; and the sum (19867) of the numbers divided by 9 leaves 4, the same excess. 5. Application to Subtraction. RULE : Find the excess of the minuend, subtrahend, and remainder or difference, add those of the remainder and of the subtrahend, and if their sum, or the excess of their sum, is the same as that of the minuend, the work may be considered correct. Illus- trative example : 8975 2, excess. 7487....^ " 1488 3, " Adding (3) the excess of the remainder and (8) that of the subtrahend and dividing the sum by nine, the excess found is 2, the same as that of the minuend. NOTE. The excess of the division of the product of two or more factors by 9 is the same as the excess of the product of the excesses of the factors. Suggestions for Teaching Numbers. \ 4 7 From this fact is derived one of the simplest and most expeditious modes of proving multiplication and division. 6. Application to Multiplication. RULE : Find the excess of the product and of each of the factors, multiply that of the multiplicand (or the reverse) by that of the multiplier, find the excess of their product, and if it is the same as that of the product of the factors, the work may be re- garded as correct. Illustrative example : 687. . 4IIH !$ 6 46984 35238 4Q3475 1 6 The excess of the product (4034751) is 6, that of the multiplicand (5873) 5, and that of the multiplier 3 ; and the excess of the product (15) of the latter two is 6, the same as that of the product of the factors. 7. Application to Division. RULE: Find the product of the excess of the divisor and of that of the quotient ; to it add the remainder, or its excess (if it has one) ; and if the excess of the sum is the same as that of the dividend, the quotient may be considered correct. Illustrative example : 26) 6547 (251 52 130 47 26 The excess of 9*5 of the divisor (26) is 8, that of the 148 Science and Art of Education. quotient (251), 8, their product is 64 ; to this add (3) the ex- cess of the remainder (21), and the excess of the sum (67) is 4, the same as that of the dividend. NOTE. The only case in which the foregoing proofs could fail would be when one error would balance another when, for instance, 56 would be written 65, or o written instead of 9; but since such errors are of the rarest occurrence, the proof by the rejection of the 9's may be considered of equal validity with any others. 14* Composition of Numbers. 1. The pupils should be led to see how numbers are com- posed. The following may serve to show how it may be done : 78964 = 70000 -f- 8000 -j- 900 -f- 60 -f~ 4. 9000 -J- 700 + 80 + 3 = 9783. 2. The numbers may also be written in vertical columns : 78964 9000 700 70000 80 8000 3 900 60 9783 4 15. Some Points in Multiplication. 1. When the multiplicand coiioists of several figures and the multiplier of but one, the pupils are usually instructed to begin at the units' place of the multiplicand and to multiply each of its figures in succession by the multiplier ; but this procedure is looked upon by the pupils as arbi- trary. If, however, taking the following example, 456 X 8, the- multiplicand be separated into 400, 50, and 6, and the pupils told to find the sum of the products of these numbers each multiplied by 8, and then be led to see that the usual method is an abbreviation of this, all thought of arbitrariness will disappear. 2. A development of multiplication like the following throws light upon points frequently not well understood : Suggestions for Teaching Numbers. 149 456 X 234 400 X 200 -|- 400 X 30 + 400 X 4 + 5 X 200 -f 50 X 30 + 50 X 4 -f 6 X 200 + 6 X 30 + 6 X 4 = 400 X 200 = 80OOO 400 X 30 = 12000 400 X 4 = 1600 50 X 200 = 10000 50 X 30 = 1500 50 X 4 = 200 6 X 200 = 1 200 6 X 30 = 1 80 6 X 4 = 24 456 X 234 = 106704 3. The following will show why the first figure Oi the product must be written under its multiplier: ( 456 X 4 = 1824 456 X 234 = j 456 X 30 = 1368/0 ( 466 X 200 = 912/00 456 X 234 =106704 4, Various ways in which the partial products may be written : (0 456 (2) 456 (3) 456 (4) 456 2 34 234 234 234 1824 912 912 24 i3 68 1824 1368 18 912 1368 1824 12 20 106704 106704 105404 15 IO 16 12 8 106704 Science and Art of Ed 5. Multiplication is a short method of finding the sum of a number of repetitions of the same number. ADDITION. MULTIPLICATION. 45 6 7 4567 4567 4 4567 4567 18268 18268 1 6. Some Points in Division. 1. In multiplication, the number to be repeated and that denoting the number of repetitions are given to find the sum ; in division, the sum and the repeated number are given to find the number of repetitions ; or the sum and the number denoting the repetitions may be given to find the repeated number. 2. Division may be regarded as a short method of per- forming a number of subtractions with the same subtra- hend. SUBTRACTION. DIVISION. 2193 243) 2I 93 (9 243 (l) 2187 195 2 43 ( 2 ) 1707 243 (3) 1464 243 (4) 1221 243 (5) 97 8 Suggestions for Teaching Numbers. 978 243 (6) 735 243 (?) 492 243 (8) 249 243 (9) 3. In multiplication two factors are given to find their product ; in division the product and one of the factors, to find the other. 17. Important Divisibilities of Numbers. 1. A divisor of several numbers is a divisor of their sum and difference. 2. A divisor of the sum of two numbers and one of the numbers divides the other. 3. A divisor of a number is a divisor of any multiple of it. 4. A number is not divisible by any number but its factors. 5. Dividing one of the factors of a number divides the number. 6. Multiplying the divisor divides the quotient. 7. Dividing the divisor multiplies the quotient. 18. Meanings of Division. Of the following indicated division, 8-5-4, three different cases may be assumed : i. How many 4*5 are in 8 ? 2. Four is what part of 8 ? 3. What is i of 8 ? Although the three answers are expressed by the figure 2, no two of them represent the same thing. 1 52 Science and Art of Education. 19* Long Division. 1. There is no real difference between short division and so-called long division; both aim at the same end and attain it by the same method. In short division most of the work is done mentally, but in long division the larger divisor makes it necessary to write it. 2. A few short-division problems solved by the long- division process forms one of the best introductions to long division. 20. Greatest Common Measures. 1. The G. C. M. of several numbers is the largest factor common to all of them. It may consist of a prime number or of the product of several prime numbers. 2. When the numbers are small, either of the following methods may be employed to find the G. C. M : 32 V 45 8, ^3X3X3 3 3 9 i5 2 7 45 = 3 X 3 X 5 5 9 81 =3X3X3X3 3 X 3 = 9, G. C. M. 3 X 3 = 9, G. C. M. 3. When the numbers are large the method by division must be employed. Example : 1679)7981(4 6716 1265) 1679 (i 1265 414)1265(3 1242 ""23)414(18 i 184 184 By the following reasoning the method by division may be proved. In the example given, since the smaller of the Suggestions for Teaching Numbers. 153 two numbers is not an exact divisor of the larger, it is not their G. C. M.; but since (17, i) the number sought cannot be greater 'than (1256) the difference of the two numbers, it may be this difference ; a trial, however, shows that it is not, that it is not an exact divisor of (1679) tne smaller of the two numbers. Continuing the same reason- ing, we find that the G. C. M. cannot be greater than (414) the difference between 1265 and 1679, and as a trial shows that it is not this difference, it maybe the difference between a multiple of this difference and 1265, and this is found to be correct. 21. The Least Common Multiple. , 1. The L. C. M. of several numbers is the smallest num- ber that contains each of them as a factor. 2. The following are the two methods of finding the L. C. M.; that by factoring being the more easily explained. 10= 2 X 5 18 = 2 X 3 X 3 56 = 2X2X2X7 75 = 3X5X5 2X2X2X3X3X5X5X7 = 12600. 18 56 75 28 75 i 9 28 15 28 2X5X3X3X28X5 = 12600. 22. Fractions. Though fractions have received considerable attention in the preceding pages, a few more thoughts concerning them remain to be given. i. A so-called compound fraction is an indicated multL plication of fractions and not a fraction. ie|4 Science and -A rt of Education. 2. What is called a complex fraction is usually an indi- cated division of a fraction by a fraction. 3. Multiplying a fraction by a fraction is taking such a part of the multiplicand as the divisor is of the unit. 4. Dividing by a fraction is taking the dividend as many times as the divisor is contained in the unit. REMARK. From 3 and 4 of the foregoing, we note that, generally speaking, multiplying by a fraction divides, and dividing multiplies. 5. A fraction may be reduced to its lowest terms by fac- toring both of its terms and cancelling the common factors ; or, when both terms are large, by dividing them by their G. C. D. 6. The following exercises may be used to show that the value of a fraction is not changed by multiplying or divid- ing both of its terms by the same number. * = * = *=*=*, etc.; J = f = A = A = A, etc.; .; J = f = A = A = A, .; t = A = A=A = A, REMARK. Exercises like the foregoing may also be used to show that fractions of unlike denominators may be reduced to the same denominator and added or subtracted. 7. Reducing several fractions of different denominators to the same denominator, by multiplying each numerator by all the denominators except its own, and all the denomina- tors together for a new denominator, multiplies both terms of each fraction by the same number. Illustrative example : 9 . 4. 6^2X5X7^4X3X7^6X3X5^70 1 8 4 - 90 357 3X5X7 5X3X7 7X3X5 105 105 105* Instead of finding the numerators by the preceding method, they may be found by dividing the common Suggestions for Teaching Numbers. 155 denominator by the denominator of each fraction and mul- tiplying the quotient by the numerator. 8. The numerator of a fraction may be divided by an in- teger (whole number) by first multiplying both terms of the fraction by the integer. Multiplying the denominator of a fraction by an integer divides the fraction by the integer, because it increases the ratio of the numerator to the denominator by the multiplier. 9. No reason can be assigned for the inversion of the divisor in the division of a fraction by a fraction, but its correctness can be shown by means of an analytic explana- tion or demonstration. For example, let it be required to divide \ by |. Explanation : One third is contained in i three times, and f (being twice as large) one half as often ; that is, the number of times J are contained in i ; but in \ they are contained one eighth as often, and in \ seven times as often as in \. Statement of analysis: fXiXjX^= f X . In this statement we see that the divisor has been inverted ; and since the same will invariably be the case, we must infer that inverting the divisor and then multiplying give the correct result. 10. The correctness of the foregoing may also be shown by means of a diagram. 23. Decimals. 1. Decimals are parts of things, or units, in which the division is made according to the scale of tenths. Decimal expressions, though not decimals, will here for convenience be treated as decimals. 2. Decimals, being derived from integers, should be de- veloped from them. This may be done by continuing the division as far below the decimal point as may be thought necessary. The following example will indicate some of the steps that may be taken to introduce the subject : 1 5 6 Science and Art of Education. 1000, 100, 10, I, , , . IOOO+ 100+ 10+ I H 1 1 . 10, 100 1000 ' 10 ' loo ' 1000 III I IOO I IO =.i; = .oi; = .ooi. = ; = . 10 IOO 1000 10 1000 IOO 1000 IOO IO , I III . I + .01 + .001 =.III. = . 1000 IOO ' 1000 1000 I.I. I IOO ,IO, I III H = = = . i + .01. + .ooi=. in. 10 ' IOO 1000 1000 IOOO ' 1000 IOOO 3. The pupils should be led to see or to discover (i) that the point is the distinguishing mark of decimals, and that it is placed before them to separate them and to distinguish them from integers ; (2) that as the value of integers de- creases from left to right and increases from right to left, so also does that of decimals ; (3) that the value of a decimal depends upon its distance from the point ; and (4) that for every place a decimal is moved to the left it is multiplied by 10, and for every place it is moved to the right is is divided by 10. 4. Practice should be given in writing and reading deci- mals until the pupils can readily do either. Decimals should be read as if they were common fractions, the name of the last place to the right given as the denominator. Ex- ample : In 0.00456, the last place to the right is that of hundred-thousandths, the fraction is therefore 456 hundred- thousandths. REMARK. Decimals may be written in four different ways : i. In words (three hundredths) ; 2. In the common fractional form ( T s) ; 3. With the per- cent sign (3^) ; 4. In the usual form (-03). 5. Place of Point in Multiplication. The following will show how the rule for the location of the point may be de- rived. Taking .5 X .4 as an example, if we discard the points and multiply 5 by 4, we obtain 20 as the product ; but the multiplicand is not 5, but ^ of it, hence the prod- uct is -jV of 20, or 2.0 ; and for a similar reason, since the Suggestions for Teaching Numbers. 157 divisor is not 4, but fa of it, the last found product is not 2.0, but -fa of it, or .20. An examination of the number of places in the product shows that it is equal to the sum of those in the factors. In the same manner may the rule be derived when the factors contain two or more decimals. 6. Place of Point in Division. If in .45 -f- .5 = .9, we dis- card the points and perform the division, we obtain 9 as the quotient ; but since the dividend is not 45, but T J-^- of it, the quotient must be T ^ of 9, or .09. This is the quotient obtained by dividing by 5, ten times the divisor, and is therefore -fa of the correct quotient, or .9. Here we observe that the number of decimal places in the quotient is that by which those in the dividend exceed those in the divisor. The same kind of reasoning will discover the rule when both terms contain several decimals. 24. Analysis. I. PROPORTION. The problems usually found under the head of proportion in books on arithmetic can, with more benefit to the pupils, be solved by analysis and cancellation. Simple problems, such as are found on page 128, should at first be given, and their length and difficulty increased as the pupils show themselves able to master them. Problem i. If in 9 days, of 8 hours each, 20 men can build a wall 40 feet long, 2 feet thick, and 6 feet high, how many men would be required to build a similar wall 60 feet long, 3 feet thick, and 5 feet high, in 15 days, of 12 hours each? REMARK. Before the analysis of a problem is commenced, a statement of the conditions of the problem should be made. It should also be observed that the first term of the analysis should be of the same kind as the one required. Statement of Conditions : M. d. hrs. ft. I. ft. th. ft. h. 20 9 8 40 2 6 ? 15 I? 6p 3 5 158 Science and Art of Education. Statement of Analysis : Analysis, or Explanation. Since the work can be done in 9 days by 20 men, it would require 9 times as many men to it in i day, and 3^ as many in 15 days as in i day. That is, if they work 8 hours a day ; but i hour a day would require 8 times as many men as 8 hours, and 12 hours a day -fx as many as i hour. That is, if they make it 40 feet long; but i foot long would require -fa as many men as 40 feet, and 60 feet 60 times as many as i foot. That is, provided they make it 2 feet thick ; but i foot thick would require one half as many as 2 feet, and 3 feet three times as many as i foot. That is, if they make it 6 feet high ; but i foot high would require one sixth as many men as 6 feet* and 5 feet five times as many as i foot, t Of every problem twice as many problems can be made as it has terms. Of the following eleven, made from the foregoing, only the statements of the conditions and of the analyses will be given. The problems can be read from the statements. Problem 2. Problem 3. ? x ! x _'_ I I 15 Problem 4. M. d. hrs. ft. 1. ft. th. ft. h. 15 15 12 60 3 5 ? 9 8 40 2 6 7 : < I X 1 >: *i: *~3 X I 5' < - = M. d. hrs. ft. 1. ft. th. ft. h. 20 9 8 40 2 6 15 ? 12 60 3 5 I 12 40 ' *$?: 2 I X 6 ) i M. d. hrs. ft. 1. ft. th. ft. h. 15 15 12 60 3 5 20 ? 8 40 2 6 12, I I 40. I 2 I N < 6 = F' 8 60 I 3 I 5" I Suggestions for Teaching Numbers. 159 M. d. hrs. ft. 1. ft. th. ft. h. Problem 5. 20 9 8 40 2 6 15 15 \ 60 3 5 -x-x- ) < 9 X x -X- x x -x -x,- xf = 12. I i 15 ' 1 15 40 I 2 i 6 M. d. hrs. ft. 1. ft. th. ft. h. Problem 6. 15 15 12 60 3 5 2O 9 ? 40 2 6 L 2 x l x - : < i_5 ) <-X -X^- x- z x 2 I X- = 8. i i 20' i 9 OO I 3 I 5 i M. d. hrs. ft. 1. ft. th. ft. h. Problem 7. 20 9 8 40 2 6 15 15 12 > 3 5 4 - X - X - 5 x-> <^> <5X'- 2 X-X i 6 x-- 60. I 20 I 9 8 i I 3 I 5 M. d. hrs. ft. 1, ft. th . ft. h. Problem 8. 15 15 12 60 3 5 20 9 8 ? 2 6 Q x - X - I X r 5 I : ?2 x r x^x i*f *-;= 40. M. 4 hrs. ft. 1. ft. th . ft. h. Problem 9. 20 9 8 40 2 6 15 15 12 60 ? 5 ? X -L X !1 X -X- _5_X- 12 4 x _ I 6 X- = 3, I 20 I 9 1 i 6 K> I 5 M. d hrs. ft. 1 ft. th, , ft. h. Problem 10. 15 15 12 60 3 5 20 9 8 40 ? 6 3X X -X x ? x _ ix?x 5?x- i 5 I __ 2. i 15 I 15 I i 2 I I A ^0 I 6 M. ' d. hrs. ft. 1. ft. th . ft. h Problem II. 20 9 8 40 2 6 15 15 12 60 3 ? 160 Science and Art of Education. M. d. hrs. ft. l. ft. th. ft. h. Problem 12. 15 15 12 6c 3 5 2O 9 8 40 2 p I " 15 ' * I " 15 I ' " 12 ' " I " I " 40 " I " 2 REMARK. Until the pupils can themselves see the reason for doing so, they may be told that the reasoning must begin and the units be taken in the horizontal line of the statement of the conditions in which all the numbers are given. As will be seen, the reasoning in all the foregoing begins in the upper line. The following problems can all be solved by the foregoing method, and in most cases with much less work than the usual method requires : i. If it takes 13,500 bricks, 8 inches long, 4 inches wide, and 2 inches thick, to build a wall 200 feet long, 20 feet high, and 16 inches (i feet) thick, how many bricks, 10 inches long, 5 inches wide, and 2^ inches thick, would be required to build a wall 600 feet long, 24 feet high, and 20 feet thick ? Brick. Wall. Bricks. in. 1. in. w. in. th. ft. 1. ft. h. in. th. 13500 842 200 20 16 = f ft. ? 10 5 2^ 600 24 20 ft. 373,248. 2. If 6 men in 4 months, working 26 days for a month and 12 hours a day, can set the type for 24 books of 300 pages each, 60 lines to the page, 12 words to the line, and an average of 6 letters to the word, in how many months of 24 days each, and 10 hours a day, can 8 men and 4 boys set the type for 10 books of 240 pages each, 52 lines to the page, 1 6 words to the line, and 8 letters to the word, 2 boys doing as much as a man ? M. mo. d. hrs. books, pages, lines, words, letters. (8 men -{-4 boys 6 4 26 12 24 300 60 12 6 = 10 men.) 10 ? 24 10 10 240 52 16 8 Suggestions for Teaching Numbers. 1 6 1 3. How many cords of wood in a pile 80 feet long, 12 feet high, and 4 feet thick ? Cords. ft. 1. ft. h. ft. th. i 8 4 4 ? 80 12 4 J x f x Y x* x V x * x f = gj 4. How many cubic yards of earth is taken from a ditch 120 feet long, 4 feet wide, and 9 feet deep? Cu. yd. ft. 1. ft. w. ft. d. 1333 120 4 9 |XiXH A X J -f i X4X^XiXf= 160. 5. How many perches of stone in a wall 36 feet long, 2$ feet thick, and 5 feet high ? Perches. ft. I. ft. th. ft. h. i i6J ij i ? 36" 2* 5 tX^X}X\-XiXfXiXfXf= 18.18 -f . 6. What is the number of bushels of wheat a bin 9 feet long, 5^ feet wide, and if feet deep. Bu. ft. 1. ft. w. ft. d. (i cu. ft. = i i ij i i bu., nearly.) ? 9 5^3$ |X-iXtXfXjXYXiXY = 144- 7. Required the number of gallons of water in a tank 12 feet long, 3^ feet wide, and if feet deep. Gal. ft. 1. ft. w. ft. d. (i cu. foot = i\ gal- 7^ i i i Ions, nearly.) ? 12 3^ if -xYxi>* = iW= A ; s = -db- = A ; i* = TO ; 9$ = A ; M = i; 7<>* = A; 6<# = f ; 4 o# = i; 33i# = -i; i6f* = i; 12^ = |; and (510 = A Many oral problems should be given to familiarize the pupils with the new terms and with the applications of per- centage. The analytic method of solution applies as well to per- centage as to proportion and other similar subjects. NOTE. Every number is 100 per cent (or the whole) of itself. Problem i : 4 is what % of 5 ? No. % (No. = number.) 5 100 ^ X -J- X f - 80. 4 ? Explanation. Since 5 is 100$, i is \ as many$, and 4, 4 times as many as i. Problem 2 : 4 is 80$ of what number ? No. * ? ioo \ X *V X H- = 5- 4 80 Explanation. Since 4 is 80$ of the number, i$ of it is -fa of 4, and 100$, or the whole of it, ioo times as many. Problem 3 : What is 80$ of 5 ? No. % 5 ioo \ X T h- X - 8 T - = 4- ? 80 Problem 4 : If 4 is 80$ of some number, what$ of it is 5 ? Suggestions for '1'eachtng Numbers. 165 No. % 5 ? V X i X f = ioo. 4 80 Problem 5 : By selling my cow for $40 I gained 25$ on its cost ; what did it cost me ? REMARK. Adding 25^ (the gain) to ioo# (the cost) and we have 125 % of the cost. No. % 40 125 Y-X ilr X^ = 32. ? IOO Problem 6 : On counting my money I found that $24 was 2o# less than I had when I left home ; how much had I spent ? REMARK. Subtracting 20% from ioo#, and we have 8o# ; what was left. No. % 24 80 VxA-xv=-6. REMARK. The pupils should, as soon as possible, be led to solve percentage problems by the fractional method. IV. TRUE DISCOUNT. In true discount the present worth corresponds to the principal in simple interest, the dis- count to the interest, and the debt, or sum discounted, to the amount. When the sum to be discounted, the rate, and the time are given, to find the present worth, the problem is the same as having given the amount, rate, and time, to find the principal. Problem : I owe $540, due in 4 years without interest, money being worth $5$ ; what sum would discharge the debt to-day ? Explanation. As $540 is the amount of the present worth of the debt due in 4 years at 5$, we must find the amount of $i for 4 years at 5 #. The interest of $i at 5$ for 4 years is 20 cents, and this added to the dollar makes '^Mfi^V NIVERSI1 \w CU; jFoaN'> 1 66 Science and Art of Education. $1.20, the amount. Now, considering it as a case in simple interest, and we have given two amounts and the principal of one, to find that of the other. As will be observed, therefore, two operations are required ; the first, to find the amount of $i ; the second, to find the present worth. The following are the statements for the second part of the so- lution : P. amt. (P. = present $1.00 $1.20 worth.) ? 540 ^ X Tib X -^f 2 -* = 45 required present worth. V. BANK DISCOUNT. Bank discount differs from true discount in two respects : i. In being the interest on the face of the note ; 2. In adding 3 or 4 days of grace to the required time. REMARK. The face less the discount equals the proceeds. Explanation. Since the proceeds are found by subtract- ing the interest (discount) from the face of the note, hence, when the proceeds are given and the face value is required, the proceeds of $i must be found; then the proceeds of the dollar bear the same relation to the dollar as the given pro- ceeds do to the required face of the note. Here, also, it will be observed, two operations are required : the first, to find the proceeds of one dollar ; the second, to find the re- quired face of the note. P. int. time. i. $1.00 6c. 36od. f X-j-5irX- 9 T 3 - .0155, discount on $i. $1.00 ? 93d. i.oo .0155 .9845, proceeds of $i. Proceeds. Face. $0.9845 $' $600 ? (0 t X TgW X ^ 609,446 + required face. (2) . i X ^ X UULJAO. = 609,446 + " REMARK. Number (2) is, perhaps, the simpler operation. Suggestions for Teaching Numbers. 167 VI. TIME PROBLEMS. Time relations are best repre- sented by the divisions of horizontal lines. Problem i : What time of day is it when the time to mid^ night is twice the time past noon ? Solution. N. p. t .M f IM. The time past noon and the time to midnight equal 3 times the time past noon ; hence, the time from noon to midnight, or 12 hours, is 3 times the time past noon, and once the time past noon, or the required time, is 4 o'clock p. M. Problem 2 : What time of day is it when the time past noon is of the time to midnight ? Solution. ty-^ F-^ i K^- N - The time to midnight and half the time to midnight, or f the time to midnight, equal the time from noon to midnight, or 12 hours ; hence, % the time to midnight equals $ of 12 hours, or 4 hours, and f of the time to midnight equal 8 hours, and the time is 4 o'clock P.M., the same as that of the previous problem. Problem 3 : Required the time of day when the time past noon equals J of the time past midnight. Solution. M -M'M . M . K^-K P- Since the time past noon is J of the time past midnight, the time before noon, or from midnight to noon, must be f of the time past mid- night, and this is 12 hours. If of the time past midnight is 12 hours, \ of it, the time past noon, is J of 12 hours, or 4 hours, and the required time is 4 o'clock P.M. Problem 4 : What is the hour of day when the time to noon is ^ of the time to 2 o'clock P.M ? Solution. 5-HN-H , K i ** i K2 p -^ If the time to noon is of the time to 2 o'clock P.M., then the time past noon, or 2 hours, must be f of the time ; and if of the time equal two hours, of it, or the time to noon, is i of 2 hours, or an hour, and the time required is n o'clock A.M. 1 68 Science and Art of Education. VII. AGE PROBLEMS. Age relations are best repre- sented by the divisions of vertical lines. REMARK. The difference in the ages of two persons re- mains the same as long as both of them live. This fact serves as key to the solution of age problems. Problem i : John is now 4 years of age and Frank is 10 ; in how many years will John be f as old as Frank ? Solution. The difference of their ages is 6 years, ^Frank and this, as the lines show, will be of Frank's age at the required time. As John's age will then ~^ be f of Frank's, it will be twice 6 years, or 12 years, and since he is now 4, the required time is the difference between 12 and 4, or 8 years. Problem 2 : Sarah is 1 1 years of age and Alice is 20 ; how long since Sarah was i as old as Alice ? ^. A lice Solution. The difference oftheir ages, 9 years, was \ of the age of Alice at the required time ; hence Alice was 18 years of age when Sarah was \ as old ; and this was two years ago. Problem 3 : Twelve years ago I was \ as old as Mr. Jones, now I am f as old ; what is my present age ? Solution. Looking at the divisions of the Jones lines, and we find that 12 years ago the dif- T 12 yea s ^ ference of their ages was -J of Jones' age, ^ " M i and now is $ of it ; but as the difference - - % - 56 H does not change, evidently \ of Jones' age % j ^ 12 years ago was ^ of what it now is, and f _L of it then f of what it now is. From this we see that 12 years ago Jones was f as old as he now is, and since he now is f, 12 years must be the i which he has in these years added. If 12 years equal of Jones' present age, he is now 36, and I am f x>f 36, or 24. Suggestions for Teaching Numbers. 169 Problem 4 : Sarah is now 3 times as old as Martha, but in 9 years will be twice as old ; how old is each now ? Solution. From the divisions of the lines Sarah ..- , . T 9 years -r we see that the difference of their ages is now twice Martha's age and in 9 years will be once her age ; but as the differences are the same, twice Martha's age now is once what it will be in 9 years, and once her age now is of what it will then be. Since Martha's age will then be f of what it now is, she must in 9 years add the other | ; and if 9 years equal -j- of her age then, it must be her present age, and Sarah's is 3 times 9, or 27 years. REMARK. If in the foregoing problem the question were, how old will each be then, the following would be the solution, differing only in one point from the preceding : Solution. From the divisions of the lines we observe that the difference of their ages is now twice Martha's age, and in 9 years will be once her age ; but as the differences are the same, 2 times Martha's age now is once what it will be in 9 years, and once her age now is of what it will then be. Since Martha's age will then be of what it now is, she must in 9 years add the other ; and if 9 years equal of her age then, f, or the whole of it, will be 18 years, and Sarah's will be 2 times 18 years, or 36 years. VIII. WATCH AND CHAIN PROBLEMS. Problem i : A man had 2 watches and only i chain. If the chain be put upon the first watch it wfll make its value twice that of the second ; and if it be put upon the second watch it will make its value 3 times that of the first ; if the value of the first watch is $30, what is that of the second, and of the chain ? REMARK. The equations are designed to indicate the rela- tions of the parts of the problem and thus to aid the solution. Science and Art of Education. Solution. As the first watch f. w. & ch. = 2 s. w.; 3 s. w. = whole. and chain are together worth ; s - w - = ^ of w |> ole ' & s. w. & ch. = 3 f. w.; 4 f. w. = whole. twice as much as the second, .-. f. w . = y of whole. if to this we add the second M+U = *;H- A = A- we have 3 times the second for the whole, and consequently the second is J of the whole. In the second condition we have the second watch and chain worth 3 times as much as the first ; if to this we add the first, we have 4 times the first for the whole, and the first is J of the whole. Since the first watch is worth $30, the value of the whole is $120, that of the second $40, and -fy, that of the chain, $50. IX. FISH PROBLEMS. Problem : The head of a fish weighs 10 Ibs., the tail weighs as much as the head and -J the body, and the body weighs as much as the head and the tail ; what is the weight of the fish ? Solution. Make a sketch of the fish, divide it into head, body, and tail, and upon each part place the number given to it in the problem. The sketch of the body of the fish shows that 20 Ibs. and i the body is the whole body, but whatever added to a makes the whole must be the other half ; therefore 20 Ibs. must be the other -J-. If \ the body weighs 20 Ibs., the \whole of it weighs 40 Ibs., and the fish head 10, body 40, tail 30 = 80 Ibs. .X. HOUND-HARE PROBLEMS. Problem i : A hare is 20 leaps before a hound and takes 4 leaps to the hound's 3, but 3 of the hound's leaps are equal to 6 of the hare's ; how many leaps must each take until the hare is caught? Solution : First make a graphic representation or illustra- tion of the conditions of the problem. Suggestions for Teaching Numbers. 1 7 1 An examination of the accompanying repre- hound, hare, sentation of the conditions of the problem 3 4 shows that while the hound takes 6 of the s = hare's leaps the hare takes but 4, and thus loses 2. If the hare loses 2 in running 4, to lose i it must run \ of 4, or 2, and to lose 20 if must run 20 times 2, or 40. When the hound runs 3 it gains the 2 which the hare loses; hence, to gain i it must run of 3, or ij, and to gain 20 it must run 20 times i, or 30. Problem 2 : A rabbit is 60 leaps before a hound and takes 9 leaps to the hound's 3, but 2 of the hound's equal 7 of the rabbit's ; how many leaps will each take until the rabbit is caught ? Solution. Make a graphic representation of hound, rabbit, the conditions, and then, as the numbers of the hound's leaps are unlike, multiply each of ~ the conditions by such a number as shall e = 21 make them alike, or the same. The remainder of the solu- tion is the same as in the previous problem. XI. ALLIGATION. Problem : How shall I combine sugars that cost me 6c, 7C, I3C, and i4c a Ib. so that I may be able to sell the mixture at gc a Ib ? NOTE. The only conditions to be observed in the solution of problems of this kind are (i) that the sum of the gains shall equal that of the losses, and (2) that every ingredient upon which there is a gain shall be combined with one upon which there is a loss. REMARK. The arrangement of the solution of the problem here given is the usual one, except the horizontal line that separates the losses and gains. Solution i. On a pound of 6c p fcet Diffi sugar sold for gc there there will be a gain of 3C, on a pound of 7c sugar Average sold for gc there will be a gain of 2c, 9 on a pound of I3C sugar sold for gc there will be a loss of 4C, and on a 172 Science and Art of Education. pound of 140 sugar sold for 90 there will be a loss of 50. Now, since the gains and losses must equal each other, if we take 5 pounds of that on which there is a gain of 3c and 3 pounds of that on which there is a loss of 5c, the two will balance each other;' and if we take 2 pounds of that on which there is a gain of 2C and i pound of that on which there is a loss of 4C, they will balance each other. Solution 2. If we take 4 pounds of that on which there is a gain of 3C and 3 pounds of that on which there is a loss of 4c, they will balance each other ; and if we take 5 pounds of that on which there is a gain of 2c and 2 pounds of that on which there is a loss of 5c, they will bal- ance each other. GEOGRAPHY. Introductory Considerations. Geography is generally classed among the dry subjects, but the dryness is not so much in the subject as it is in the teachers and the teach- ing. A real teacher can invest any subject with interest, but a lesson-hearer kills even the interest that naturally in- heres in a subject. Much of the matter that has in the past been taught as regularly belonging to geography, and not a little of that still taught to children, is not of the most inviting nature. No greater mistake can be made in teaching beginners than requiring them to shut their eyes to the world in which they live and to look into a book where all is strange and meaningless ; yet this is the method generally pursued. The children's experiences, which should be used as start- ing-points, are not only ignored, but regarded as of no value. Definitions, of whose meaning the children can have no idea, are frequently the food which their mental stomachs are given to digest. After they are supposed to have learned these, then, with their undeveloped imagination, they are expected, from the study of a globe, to form a conception of the earth as a whole, and afterwards to di- vide it into its so-called members, and these again into sub- members, and so on, until the smallest division has been reached ; this method of procedure being followed, upon the ground that in everything studied or conceived, that is composed of parts, one must begin with analysis with the 173 174 Science and Art of Education. whole and go to its parts ; but such a strain at universality, the law " from the whole to its parts," will not bear ; and a little reflection upon the manner in which the mind builds its spacial forms will show the erroneousness of the method. The suggestions which follow are a departure both in matter and method from what usually passes for geography, but are in harmony with the best in that line of instruc- tion. The study of geography should be commenced in the primary school, with what the children can observe, and should be largely conversational. Its object should be to awaken an interest on the part of the children in the boundless forms of nature that meet them on every hand. The work of the primary and others of the lower grades of schools should embrace the following topics : 1. Land. i. Its forms in meadows, uplands, plains, hills, and mountains. 2. The material of which land is composed, namely, soils and rocks. 3. The materials of which the different kinds of soils are composed, namely, loam, sand, gravel, and clay. REMARK. The children should examine the different kinds of soil. 4. The part each of the two soils (top and sub) performs in the growth of vegetation. 5. The kinds of soil certain crops demand, the prepara- tion and fertilization of the ground for the reception of the seeds, the time and mode of planting, and care of the plants. 6. The changes which land-forms are undergoing, and their causes. 7. The influence of land-forms upon climate. NOTE. The children should make models in sand of the forms of tend which they have observed ; this will enable them Suggestions for Teaching Numbers. 175 later on in their study, when observing a model, to look with their imagination beyond it to that which it represents. 2. Water. i. The forms of water ice, liquid, and vapor. 2. How ice and vapor are formed and how returned to the liquid state. NOTE. Whenever necessary and practicable, processes should be shown by means of experiments. 3. Uses of ice and vapor in nature and to man. 4. The formation of clouds, and their kinds. 5. How rain is produced ; its uses or benefits, especially in the growth of plants. 6. How springs, rivulets, creeks, rivers, ponds, and lakes are formed. REMARK. If no pictures are at hand that illustrate it, draw- ings can be made upon the blackboard to answer the purpose. 7. The uses of water for drinking, washing, cooking, highways, as a force, and as a cooler and moistener of the air. 3. Air. i. The properties of the air. 2. Its use in supporting animal and vegetable life. 3. How winds are caused, and their kinds. 4. The uses of winds in carrying vapor, purifying and cooling the air, propelling vessels, and turning machinery. 5. Changes of temperature, how caused, and how meas- ured. 6. Effects of changes of temperature upon animal and vegetable life. 7. Protection of animal and vegetable life against great changes summer's heat and winter's cold. 4- Heat. i. Its source or modes of production, 2. Its effect upon different substances, 3. Its uses. 176 Science and Art of Education. 5 Plants. i. Common varieties. 2. The roots, and the part they perform in the growth of plants. 3* The kinds of roots, also duration. 4. Roots used for food. 5. The kinds and use of stem. 6. Leaves, their forms and use. 7. Flowers, their forms, use, and beauty. REMARK. The children should be trained to draw and paint plants, including flowers. 8. The kinds of fruit or seeds, when they ripen, and their uses. 9. Buds, what they contain, when they begin to swell, and why they do so. 10. When the blossoms appear. 11. When the fruit ripens. 12. When the leaves begin to fall and what causes them to fall. 13. How roots are protected from the cold in the winter. 14. Food of plants, also cultivation. 15. Medicinal and food plants, also plants that are poi- sonous. 6. Domestic Animals. i. Varieties, also form or struc- ture. 2. Adaptation of structure to mode of life and subsist- ence. 3. Grass, grain, and flesh eating animals, and habits and use of each. 4. How fed, sheltered, and cared for. 7 Wild Animals. i. Varieties of form or structure. 2. Adaptation of structure to mode of subsistence and defence or protection. 3. Their habitat, or homes. 4. Grass, grain, nut, and flesh eating quadrupeds. Suggestions for Teaching Numbers. 177 (a) Those that are useful for food, skins, and furs, (b) How captured, also how tamed. 5. Flesh and grain eating birds, and varieties of each. (a) How they fly, and how they hold themselves to limbs of trees and other objects. (b) How and when they secure their food. (c) How they are captured ; also how tamed. (a) Which used as food, and why. 6. Fish, their varieties, also adaptation of structure to medium in which they live. (a) How they move themselves, and adaptation of form to mode of movement. (b) Their food, and how they secure it. (c) Which are used for food. (d) Methods of catching, preserving, and preparing for the table. Among animals should also be included insects and rep- tiles, the latter embracing lizards, turtles, tortoises, frogs, toads, and serpents. An examination and study of the nature and habits of these will create an interest in them in the children and will lead them to see that none of God's creatures are useless, but that all of them when prop- erly understood have their purpose, and instead of being our enemies are our friends. REMARK. Microscopes are necessary for some of the work referred to. 8. Man. i. His superiority to other orders of creation. 2. House and home life in comparison with those of other living beings. 3. Adaptation to a variety of occupations or employ- ments. 4. His genius in using the forces of nature in the per- formance of labor and in surmounting obstacles. 5. Modes of communication and travel. 178 Science and Art of Education. 6. Possibility and means of improvement. 7. Means or modes of enjoyment. 8. Pictures of the various races of the human family should be shown, and the race characteristics described. 9. The effect of climate upon character and disposition should be explained. REMARK. As before stated, the foregoing work should, as far as possible, be informal and conversational. 9- Study from Maps and Models. i. After the locality or community in which the children have their homes has been explored and studied as carefully and thoroughly as their age will permit, and a sufficient amount of experiences or apperceiving concepts stored away as constructive ma- terial, the children are prepared to extend, their vision, by means of the imagination, to the unseen. This they must learn to do by the use of pictures, maps, and models, and the best and most natural way to learn to understand a map or a model is to help to make one. The school-room is the most convenient and suitable thing to begin with in map- ping. 2. To enable the children to judge of distances and areas they should have practice in making measurements. A yard, rod, or mile, either linear or square, should convey something definite to them. 3. The cardinal points east, west, north, and south should be determined and marked or fixed upon the floor or elsewhere by means of lines connecting the opposite ones. 4. A map of the school-room may be made upon the floor, but better upon a piece of heavy paper, say a yard square, or larger, if necessary, painted with a mixture of shell-lac (dissolved in alcohol) and lampblack. 5. To make the map, the paper should be laid at some convenient place upon the floor. If the map is to be pro- Suggestions for Teaching Numbers. 179 portioned to the size of the room about an inch to a foot the pupils should make the measurements, determine the proportions, and otherwise assist the teacher in the work. Frequent questions should be asked of the pupils with regard to the direction of lines, the location of seats, teacher's desk, and other objects. 6. After the map has been made and the pupils can read it name any object upon it pointed to by the teacher or one of their own number, also its direction from some point named it may be hung upon the wall and again read. The latter reading will prevent the erroneous notion some- times formed by pupils that north is in a vertical line above south, or at the zenith. 7. After the children have become familiar with the map of the school-room, the surroundings of the school, and, if in a town or city, some of the principal streets and build- ings may be added. Next, if it contains enough objects of importance, the county may be drawn, then the state, and other states separately and in sections or groups, until the whole country has been built or mapped, and studied, and the children can in imagination see it or any part of it. 8. As will be noticed, the method here indicated is pro- gressive ; and to show more definitely how it may be carried out, the following suggestions are added : (a) If the school is in Pennsylvania, begin with that State ; and if the pupils have had little or no practice in drawing from memory, the teacher should make the first drawing, explaining his work as he proceeds. (b) For the first lesson, the pupils should prepare to draw the outline (of the state) and the rivers, and at the recita> tion make a sketch of their lesson upon the blackboard, and describe it, the teacher questioning them upon it. (c) For the second lesson, they should add to the first the mountains, principal towns, cities, and other objects of importance. After they have completed their sketches or 1 80 Science and Art of Education. maps they should describe them, and the teacher should ask questions about the proportions, the relative position of objects represented upon them, etc. (d) For the third lesson, the second should be repro- duced as a review, and the outline and the rivers of New Jersey added. REMARK i. While the pupils are learning the maps, the teacher should, with all the helps at his command, such as models, pictures, and descriptions, enable them, in imagination, to see the states or countries of which the maps are represen- tations. 2. Descriptions, questions, and reviews of as many previous lessons as may be necessary should constitute a part of every recitation. (e) The fourth lesson should review the third and add to it the mountains, cities, etc., of New Jersey. (/) The fifth lesson should include the fourth and add to it Delaware, with all in it of importance. (g) For the sixth lesson, add to the flfth the outlines of Maryland. (ti) For the seventh lesson, add to the sixth whatever may be considered of importance in Maryland. REMARK. The daily reviews should be spirited ; slow, sleepy work should not be permitted. The pupils should be trained to accurate rapid sketching or mapping. (/) If by this time the pupils have Pennsylvania well pictured in their minds, so that they can make a rapid and sufficiently accurate sketch of it and describe it, they may drop that state for a while and add to the others already drawn Virginia, then West Virginia, and so on, always, in the sketching, dropping those first made as soon as they are well fixed in, the mind. (J) A daily review, either oral or by sketching, is a neces- sity, in order to connect the work from the beginning into a mental picture of the whole and to impress it firmly upon the mind. Frequently all the states that have been studied Suggestions for Teaching Numbers. 181 should be sketched as a whole. It is of far more impor- tance that what has been learned should find a permanent lodgment in the mind, than that more should be added to what is already fading. (k) After New Jersey has been added to Pennsylvania, instead of taking Delaware next, New York may be taken, then Connecticut, Rhode Island, Massachusetts, Vermont, New Hampshire, and Maine. REMARK. If the pupils can do so, they may group two, three, or more states together as a lesson. (/) The teacher should begin with the state in which his school is located and build from that out. He may, of course, begin with some other state, but it is more natural to begin at home. (///) Only things that are important should be found in the sketches. (//) Maps in which no exactness is required should be proportioned by the eye, without construction-lines. How- ever, if a pupil finds it difficult to give the desired shape to his map while preparing his lesson for a blackboard sketch, dividing his paper into rectangles, proportioned as nearly as possible to those made by the parallels and meridians of the maps in his book, will give him all the points he needs for the required form. (o) While the pupils are doing the work here suggested they should also read the descriptive matter found in their books pertaining to the states which they are sketching, and should be questioned upon it by the teacher. What- ever of this matter may be considered of sufficient impor- tance to constitute a permanent possession of the mind should also enter into the daily reviews. (p) Climate and productions are not limited by political divisions, but belong to physical sections or regions, and, unless peculiar to a state, should be taught with the regions to which they belong. 182 Science and Art of Education. (q\ Definitions, instead of being the first thing presented to a learner, should generally be the last, and instead of being memorized from a book should as far as possible be drawn from examples or given from a knowledge of the subject. (r) To complete the map of North America, add to the United States the British possessions, Alaska, and Green- land, on the north, and Mexico, Central America, and the West Indies, on the south, grouping all into one picture. REMARK. Countries of which our knowledge is limited and, at best, inaccurate, should have only the outline, principal di- visions, rivers, mountains, and cities represented on the map. (s) South America may follow North America, then Africa, Europe, Asia, with its surrounding islands, Australia, and the more important islands of the Pacific Ocean. 9. After a whole country has been mapped and impressed upon the minds of [the pupils, its prominent or controlling physical features those upon which the climate, produc- tions, etc., depend should be studied on a relief map or on a model in sand. 10. The study of the influence of the physical features of a country should be followed by the position of the country upon the globe, also its position with reference to other countries; and its intellectual, moral, and commercial posi- tion or standing among the countries of the earth. 11. Countries should also be compared with each other, their resemblances and contrasts noted, as indicated in Part II, under Association. 12. When each country has its place assigned upon the globe, and a picture of the whole has been formed in the minds of the pupils, then they are prepared to study it in- telligently as a whole, with its lands, waters, motions, forces operative upon it, diversities, and possibilities of life, etc. This is a study of cause and effect, and suitable only for Suggestions for Teaching Numbers. 1 83 pupils of sufficient age and mental development to make broad generalizations; 10. The Sand-box. i. Elevations are best represented by means of models of sand, clay, putty, or paper pulp. A box of inch pine-boards, 3 ft. X 4 ft. X 3 in., painted on the inside with two or three thick coats of lead paint, placed upon trestles 2 ft. in height and containing about a peck or ten quarts of moulders' sand obtained at a foundry, is not expensive, and should be found in every school in which geography is taught. 2. That the sand may at all times be ready for use, it should, when not needed, be kept in a rounded heap, pounded together to hold the moisture, and once a day, or oftener, sprinkled with as much water with a sprinkling- can as will soak in without running off. 11. Relief -maps. i. Instead of purchasing expensive relief-maps, every teacher can, with the assistance of his pupils, make his own maps of paper pulp. The pulp may be made in the following manner : Take white waste paper, (other paper may be used) and tear it into pieces about an inch square, until enough has been prepared to fill a com- mon-sized wooden water-pail or bucket. Pour enough hot water upon the paper to cover it two or three inches deep, and let it remain on it six or eight hours, or overnight. When ready to make the pulp, pour nearly all the water off, leav- ing only as much as may be necessary for the proper moist- ure of the pulp. Pour a quart or more (if the pail is nearly full of paper) of flour starch upon the paper, and work or mix it well in with the hands. Now, after pouring half the mass into another vessel, the reduction being more easily and quickly made by taking a half-pailful at a time, let each of three boys take in each hand a stick of hard wood about three feet long, three-fourths of an inch thick, and pointed at one end, and, sitting around the pail, drive them down through the mass as rapidly as possible (frequently Science and Art of Education. stirring the paper up from the bottom), until the desired condition of the pulp has been attained. Treat the other half in the same manner, and when the pulp has all been made, put it into a stone jar or crock, and cover it well, in order that it may hold the moisture until it is wanted to make the maps. 2. To make the maps, take a piece of cardboard about an eighth of an inch in thickness, proportion the map to the one in the book used as a copy, making it two, two and a half, three, or four times the size of the copy. Cut the card at least two inches larger each way than the proposed map, to allow for an inch border all around. Next, draw the map upon the card, and when done, with the -fingers put the pulp on, not thicker than an eighth of an inch, pressing it down well to make it adhere to the card. It is best to begin putting on the pulp around the outline, and after- wards to cover the remainder. No pulp should be put upon places intended to represent lakes, nor should rivers be covered with it. Simply press the pulp down on both sides against the river line, but not together, and when it dries it will sepatate along the line and represent the river. 3. The maps can be made harder and more durable if, when dry, they are given a coat of gum-arabic. If desired, when the gum-arabic is dry the maps may be painted. 4. To make the pulp adhere more firmly to the card- board, the latter should also be given a coat of mucilage inside of the outline. If this is done the mucilage should be allowed to dry before the pulp is put on. HISTORY. 1. History is a record of human deeds, either of individ- uals or communities ; and since deeds, measured by moral standards, may be good or bad, the sfudy of history is the study of conduct, the study of morals. Viewed in this light it has an important bearing upon the formation or growth of character. 2. The greater the age a country has attained the larger its accumulation of historical matter at the command of the teacher. In wealth of material, it is true, the lands beyond the ocean surpass us ; but for the lessons that young Americans need to learn, our stores are not only ample, but superior to all others. Our soil has been consecrated to liberty. The heritage which our forefathers have left us is priceless. No other country upon the globe offers such opportunities for persons of ambition and worth to rise from the humblest walks in life to the highest. No law says, " thus far shalt thou go but no further." 3. But those who are soon to become actors in the drama of life must not only be taught to appreciate the value of the freedom handed down to them, but, above all, how to maintain it ; they must be taught that " righteousness " in the individuals "exalts a nation," and that sin is a re- proach to any people. That character or worth makes the man must therefore be impressed upon the minds of the children so as to become the ruling principle of their conduct and lives ; and, to accomplish this end, no other branch of study furnishes material equal to that of history. 185 1 86 Science dnd Art of Education. 4. Erroneous notions concerning the influence of certain kinds of matter prevail among teachers. For example, some labor under the delusion that the study of battles and bloodshed cultivates a spirit of patriotism and bravery. They do not distinguish between bloodthirstiness, and patriotism and bravery. Pupils nourished with the former diet soon eagerly devoured by boys frequently find the ordinary quiet walks of life too uneventful for " the fires that in them burn," and start on a career of lawlessness. The matter for class use for children should therefore be selected with care. 5. Children should be taught, and early, too, that might does not make right in the eyes of the civilized world ; and that, consequently, decisions rendered by the sword a relic of barbarous ages are not to be relied upon as founded upon justice, nor as compatible with Christian civilization. 6. We look to the past also for guidance in the future. For this purpose, however, much of the past is not only un- necessary, but useless ; hence teachers of history should exercise more judgment than is usually done in selecting that which has a direct bearing upon the points they are endeavoring to have brought out or discussed. When the pupils have reached a sufficient age and stage of mental development to do so, the selection of the matter should, as far as possible, be left to the exercise of their own judg- ment, the teacher simply stating the question to be dis- cussed. REMARK. A teacher of history should be free from all taint of political bias. A politician or partisan would on all occa- sions that offered themselves try to influence his pupils to become of his own sort, and thus defeat the object of the study. 7. When not confined to the dry boxes of the subject, such as uneventful administrations and the like, or to the packing of the memory with dates and less important facts, Suggestions for Teaching Numbers. 187 and when taught intelligently, history is a study of infer- ence, induction, one of the best thought, or logical, subjects, and should not fail to be intensely interesting. 8. A teacher of history should be a good story-teller. He should be able to put dry facts into the form of stories and invest them with interest. Children are fond of stories, hence this is the best way of presenting the subject to them. 9. If the formation of Christian characters is uppermost in the teacher's mind and his pupils are primarians, he can do nothing better than begin with Bible stories, those of the Old Testament, of Abraham, Isaac, Jacob, Joseph, Moses, Joshua, Samuel, Solomon, David, Elisha, Job, Isaiah, Jeremiah, Daniel, and, in the New Testament, Christ, whose life and teachings furnish endless material for building character. 10. Following Bible stories may come the history of our own country, beginning with that of the community town- ship or county in which the school is located, bringing in the Indians, then the discovery and settlement of the country by Europeans, the encroachment of the white man upon the hunting-grounds of the Indians, the resulting animosities and strifes, etc. Special stress should be laid upon the events and influences that have controlled our growth and strength as a nation. 11. With whatever matter the teacher may begin, whether Bible stories or the community, it should be given in the form of anecdotes and stories. No books should be used ; for there is no surer way to destroy interest in the subject and create a dislike for it than requiring the contents of books to be recited. However, when pupils are old enough to read understandingly they should be encouraged to con- sult books, and if the instructions they have received from the lips of the teacher have created the proper interest, they will gladly avail themselves of every opportunity to add to their stock of knowledge. 1 88 Science and Art of Education. 12. The matter for advanced pupils should frequently, if not generally, be given in the form of problems, or subjects for discussion, to give them practice in solving the perplex- ing problems that will meet them later in life and which confront the citizens of a country like ours. 13. Teachers of history should bear in mind that a good knowledge of their subject of instruction does not neces- sarily imply a large accumulation of facts, but the ability to use those at command to the best advantage. 14. A teacher of history should have all the appliances in the way of maps, charts, pictures, etc., that are neces- sary to give his pupils accurate and clear mental pictures of the events that present themselves in their lessons. It is more frequently on account of failures in imagining than of treachery of memory that lessons are unsatisfactorily prepared. The more real the teacher succeeds in making his instructions the better they will be comprehended and remembered. 15. In history fully as much as in anything else, if not more so, daily reviews constitute a necessity. This is the only way to connect the work and to give it a permanent place in the mind. part ?3I3J. THE HUMAN BODY. 1. Of all the studies pursued in the schools, that relating to the knowledge and care of our bodies is least under- stood. It is perfectly safe to say that we do not under- stand how to live well, and the little we pretend to know we disregard. 2. Our food is selected in almost total indifference of what the system craves in kind, proportion, and quantity. Some kinds come too frequently, others the reverse. Again, some that the system does not need, cannot use, and that therefore do harm, are provided, while others that are needed are not supplied. Neither the work performed nor the season is consulted. The same kind is frequently provided for all, whether suitable or unsuitable, whether their stom- achs can bear it or not. All are supposed to be alike. 3. Cooking is done by persons who are ignorant of the laws of health ; frequently they do not know that there are such laws ; and, in the preparation, either destroy the most nutritious part of the food or permit it to evaporate. 4. The amount of exercise needed is as little understood as the food required. Either too much is taken or not enough. It is scarcely known, and still less believed, that the human machine, like others not human, may be run too fast or too slow ; neither of which can long be done with impunity. The avenger invariably comes to call a halt. 5. Rest, in the form of sleep, seems to be, too much re- 189 1 90 Science and Art of Education. garded as a thing that can be put off from time to time until there is nothing else to do ; but rest is fully as impor- tant as food, and while some may take so much as to be- come tired of it, others, and many of them, do not take enough, especially those who sleep by the clock and not by what they need. Fortunately, no one needs a clock to tell him when he has had enough ; he has his guage within him. Those who claim, as some do, that time does not permit them to sleep until they feel that they have had sufficient, earlier or later pay the penalty for their indiscretion or imprudence. 6. Cleanliness of person is also a thing that needs more attention than it usually receives. Bathing, some think, should be done now and then, say once a week or month, just as there may be time to spare from other duties. But daily bathing, and during warm weather more frequent, is a necessity to comfort and health. The pores of the skin need to be kept open and active. 7. Ventilation is seldom overdone, but in ninety-nine cases in a hundred the reverse. Pure air, if a little cool, is considered dangerous, while the rankest poison thrown off from the lungs and skin is, with the utmost composure and ignorance, time and again returned to the lungs, until almost complete stupor ensues. 8. What are our schools doing to remedy'these defects ? What are they doing to teach right living well living ? Will memorizing the names and number of the bones, teeth, muscles, nerves, etc., usher in the wished-for millennium ? If it is supposed to do so, it has hitherto proved a signal failure ; and as the past has bee.n so will the future be, unless a wiser course be pursued. 9. It is not unfair to ask, how many persons who under- take to give instruction in this all-important subject are competent to do so? Not a few of them, judging from their frequent' ailments, dp not know their own bodies we!! Suggestions for Teaching Numbers. 191 enough to take care of them, yet do not hesitate to instruct others ho\v to take care of theirs. 10. The structure of the body is well enough understood to be taught with success. All that teachers need is to make themselves thoroughly acquainted with it. 1 1. With regard to hygiene the case is different. All that can be learned from books and teachers in the present state of knowledge of the subject is some very general facts, nothing more. Until we shall have something much more definite than we now possess, and perhaps ever after, every one must study himself to learn what his well-being de- mands. This study must be inductive, and must deter- mine what, under varying conditions, is healthful and what injurious. 12. Primary pupils should, as far as possible, be taught objectively, without the use of books. One of the best teachers' helps for giving such instruction is " Practical Work in the School-room," Part I The Human Body, published by A. Lovell & Co., New York. Lessons on The Boy, in " Systematic Science Teaching," a recent work, published by D. Appleton & Co., New York, will be found exceedingly helpful in showing how such and all other science instruction should be given. 13. For work above the primary classes the following or- der of presenting the subject may be followed : i. The framework or bones ; 2. The muscles ; 3. The skin ; 4. Di- gestion ; 5. Food ; 6. Circulation ; 7. Respiration ; 8. The nervous system ; 9. The senses. REMARK. In connection with each part or organ should also be taught, as far as possible or practicable, its care. 14. All unimportant and unnecessary details and scientific terms a burden to the memory should be omitted. It is more important that pupils should become interested in the study of their own bodies than that they should know all 192 Science and Art of Education. about it. The orcler of presentation should generally be, first the thing, then its name. 15. To teach the human body successfully, either a skel- eton and models of the various organs are required or a manikin ; without either of these it is impossible to give the pupils an accurate conception of the forms and position of the internal organs. CIVIL GOVERNMENT. 1. There is no reason why the children in the lower grades of schools should not be made acquainted with the elements of civil government. There is nothing difficult to understand about the subject, and if presented in an intelli- gent manner it is an interesting one. 2. The first lessons should be about their own community, whether township or borough, and should include the fol- lowing: i. What public duties arc, and why public rather than private ; 2. By what officers the duties are performed; 3. Whether officers are elected or appointed, and by whom, how, when, and for what length of term ; 4. How officers are installed, and when ; 5. How and by whom paid ; 6. Duties each is required to perform ; 7. Rights and duties of citizens ; 8. How rights are secured and wrongs re- dressed ; 9. What laws are, by whom made, for whose ben- efit, and why needed ; 10. What the laws forbid. Next should come the government of the county, and then as much of that of the State as the pupils are old enough to understand and to be interested in. 3. No subject should be presented to pupils before they have reached a period of life at which they can be inter- ested in it ; hence the study of the general government should be left for the high-school. 193 DRAWING. Drawing is a mode of expression that manifests itself early in children. Give one a pencil or a piece of crayon and he will try to express something, however crude or un- intelligible the performance may be. Since the desire or instinct exists, why not foster it ? What the children need at this period is encouragement, help. Hence, whenever anything presents itself in their lessons that admits of rep- resentation by lines or colors, let them try to draw it or paint it ; and, when necessary, show them how to improve their work. If this course be pursued, drawing and paint- ing will be taught in the most natural way, and without a special class or period for it '95 NEW BOOKS FOR TEACHERS. ILES' A CLASS IN GEOMETRY. By GEORGE ILES. "It cannot fail to give to the teachei of this science new enthusiasm and new ideas, and to all teachers the pleasure arising from following our ideal method." Limp cloth. Price 300. post-paid. KELLOGG'S ELEMENTARY PSYCHOLOGY. By AMOS M. KELLOGG, Editor of the School journal. A concise outline for Normal students and the home study of pedagogy. It will aid those who have found other works obscure. Limp cloth. Price 2jc. post- paid. ROOFER'S APPERCEPTION, or, "A Pot of Green Feathers," is a very simple book on psychology, strange as the title may seem. It dis- cusses perception and shows how it becomes percep- tion. Limp cloth. Price 2$c. post-paid. ROOPER'S OBJECT TEACHING makes plain this much-talked-of but little-understood subject both in its philosophical basis and its practice. Limp cloth. Price 2<$c. post-paid. HALL'S CONTENTS OF CHILDREN'S MINDS on Entering School, by G. STANLEY HALL, President of Clark University, gives the results of careful investiga- tions made by the writer and others to determine the amount and kind of knowledge possessed by the average child on entering school. Limp cloth. Price 250. post-paid. # % Large descriptive catalogue of Jive hundred books and aids fot teachers in all branches of school work free. E. L. KELLOQQ & CO., New York and Chicago. The Best Educational Periodicals. THE SCHOOL JOURNAL is published weekly at $2.50 a year and is in its 23rd jear. It is the oldest, best known and widest circulated educational weekly in the U. S. THE JOURNAL is filled with ideas that will surely advance the teachers' conception of education. The best brain work on the work of professional teaching is found in U not theoretical essays, nor pieces scissored out of othe* journals THE SCHOOL JOURNAL has its own special writers, the ablest in the world. THE PRIMARY SCHOOL is published monthly from September to June at $1.00 a yean It is the ideal paper for primary teachers, being devoted almost exclusively to original primary methods and devices. Several entirely new features this year of great value. THE TEACHERS' INSTITUTE is published monthly, at $1.00 a year. It is edited in the same spirit and from the same standpoint as THE JOURNAL, and has ever since it was started in 1878 been the most popular educa- tional monthly published \ circulating in every state. Every lino is to the point It is finely printed and crowded with illustra- tions made specially for it. Every study taught by the teacher is covered in each issue. EDUCATIONAL FOUNDATIONS. This is not a paper, but a series of small monthly volumes that bear on Professional Teaching. It is useful for those who want to study the foundations of education ; for Normal Schools, Training Classes, Teachers' Institutes and individual teachers. If you desire to teach professionally you will want it. Hand* some paper covers, 64pp. each month. The History, Science f Methods, and Civics of education are discussed each month, and it also contains ail of the N. Y. State Examination Ques- tions and Answers. OUR TIMES gives a resume of the important news of the monthnot the murders, the scandals, etc., but the news that bears upon the progress of the world and specially written for the school- room, It is the brightest and best edited paper of current events pub- lished, and so cheap that it can be afforded by every pupil. - Club rates, 25 cents. %* Select the paper suited to your needs and send for a free sample. Samples of all the papers 25 cents. * B. L. KELLOGG & CO. , New York and Chicago. BEST BOOKS FOR TEACHERS, Classified List under Subjects. To aid teachers to procure the books best suited to their purpose, we i ve below a list of our publications classified under subjects. The division \s sometimes a difficult one to make, so that we have in many cases placed the same book under several titles; for instance, Currie's Early Education appears under PRINCIPLES AND PRACTICE OF EDUCATION, and also PRIMARY EDUCATION. Kecent books are starred, thus * HISTOEY OF EDUCATION, GREAT EDU- CATORS, ETC. Allen's Historic Outlines ot Education, - - Autobiography of Froebel, browning's Aspects ot Education Bwt edition. " Educational Theories, Best edition. * EDUCATIONAL FOUNDATIONS, bound vol. '91-'92, " " '" Kellogg's Life of Pestalozzi, - Lang's Comenius, ------ Basedow, ------- Rousseau and his "Emile" - - - * ** Hoi-ace Mann, ------ * " G reat Teachers of Four Centuries, - * Herbart and His Outlines of the Science of Education. - Phelps' Life of David P. Page, - Quick's Educational Reformers, Best edition. ~ *Reinhart's History ot Education, - Retail. paper cl. cloth cl. paper cl. paper paper paper paper pacer Cl. .25 PRINCIPLES OF EDUCATION. pd. Our By Price to Mail Teachers Extra .15 50 .40 .25 .20 ,50 .40 .60 l.OO .15 .15 .15 .05 pd. pd. pd. pd. pd. .15 pd. .15 pd. .25 .20 .03 .20 .03 paper .15 pd. cl. J.OO .80 .08 cl. .25 .20 .03 Carter's Artificial Stupidity in School, - - paper .15 pd. *EUUCATIONAL FODN CATIONS, bound vol. '91-'92, paper .60 pd. * " " " '92-'93, cl. l.OO pd. Fitch's Improvement in Teaching, - paper *Hall (G. S.) Contents of Children's Minds, - cl. .25 .15 pd. .03 Huntington's Unconscious Tuition, - paper Payne's Lectures on Science and Art of Education, cl. Reinhart's Principles ot Education, - cl. 1.00 .25 .15 .80 .20 pd. .08 .03 *>pencer's Education. Best, edition. - - - cl. 1.00 .80 .10 Perez's First Three Years of Childhood, - - cl. 1.50 1.2O .10 *llein*s Outlines of Pedagogics, - cl. .75 .60 .08 Tau's Philosophy of Education. Best edition. - cl. 1.50 1.2O .10 *Teachers' Manual Senes. 24 nos. ready, each, paper .15 pd. PSYCHOLOGY AND EDUCATION. Allen's Mind Studies for Young Teachers, - cl. .50 .40 .05 A lien's Temperament in Education, - - - cl. .50 .40 .05 *Kelloj.'g's Outlines of Psychology, ... paper Perez's" Kirst Three Years oi Childhood. Best edition, cl. .25 1.50 .20 1.2O 03. .10 Hooper's Apperception, Best edition. - - cl. Welch's Teachers' Psychology, - el. Tniks on Psychology, - cl. .25 1.25 .MJ .20 l.OO .40 .03 .10 .U6 MANUAL TRAINING. Butler's Argument for Manual Training, - - paper .15 pd. "Larsaon's Text-Book of Sloyd, - - - - cl. 1.50 1.2O \15 Love's Industrial Education, - c\ 1.50 1.2O J2 nipham's Fifty Lessons in Woodworking, - cL .50 .40 .05 QUESTION BOOKS FOB TEACHERS. Analytical Question Series. Geography, - - cl. .50 .40 .05 8 " " P. 8. History, - cl. *.50 .40 .05 * * * 4 Grammar, - - cl. .50 .40 .05 * EDUCATIONAL FOUNDATIONS, bound vol. *91- 1 93, paper .60 pd. * " 92.'93. cl. l.oo pd. N. Y. State Examination Quest ons, - CL 1.00 .80 .08 *Shaw 's National Question Book Newly revised. 1.75 pd. Soutbwick's Handy Helps, ----- cl. 1.00 .80 .08 Southwick's Quiz Manual of Teaching. Best edttion. cl. .75 .60 .05 PHYSICAL EDUCATION and SCHOOL HYGIENE. Groff 's School Hygiene, - paper .15 pd. MISCELLANEOUS. Blaikie On Self Culture, - - - - - Fitch's Improvement in Education, - Gardner's Town and Country School Buildings, Lubbock's Best 100 Books, - - - - - cL pa cT paper cl. oL paper .25 2.50 .30 .50 .80 .20 .15 2.OO .20 .24 5.OO .40 .24 l.oo .03 pd. ^ pd. .03 pd. .05 .03 pd. Walsh's Great Rulers of the World, - - - Bas-Relief s of 12 Authors, each. - - SINGING AND DIALOGUE BOOKS. *Arbor Day, How to Celebrate It, paper .25 pd. Reception Day Series, 6 Nos. (Set $1.40 postpaid.) Bach. .30 .24 .03 Song Treasures. ------- paper .15 pd. *Rest Primary Songs, new ------- .15 pd. *Washington's Birthday, How to Celebrate It, - paper .25 pd, SCHOOL AFFAEATUS. Smith's Rapid Practice Arithmetic Cards, (32 sets), Each, .50 pd. " Standard 1 " Manikin. (Sold by subscription.) Price on application. "Man Wonderful" Manikin, - - - - 4.OO pd. Standard Blackboard Stencils, 500 different nos., from 5 to 50 cents each. Send for special catalogue. " Unique " Pencil Sharpener, - 1.50 .10 * Russell's Solar Lantern, ----- 25 OO pd, Standard Physician's Manikin. (Sold by subscription.) E^" 100 page classified, illustrated, descriptive Catalogue of the above and many other Method Books, Teachers' Helps, sent free. 100 page Cat- logue'of books tor teachers, of all|publishers, light school apparatus, etc., sent free. Each of these contain our special teachers' prices. E. L. KELLOQQ & CO., New York & Chicago. GENLKAL METHODS AND SCHOOL MANAGEMENT. J..UU 1ft .uo * Art of Securing Attention - paper .*15 pd. ** Lectures on Teaching, - cl. 1.25 1.00 .15 pd! Hughes' Mistakes in Teaching. Best edition. - cl. .50 .40 J05 * Securing and Retaining Attention, Best ed. cl. .50 .40 .05 " How to Keep Order. - paper .15 pd. Keliogg's School Mimagement. - cl. .75 McMurry's How to Conduct the Recitation, - paper * Parker's Talks on Pedagogics. cl. 1.50 .60 .15 1.20 .05 pd. .12 Talks on Teaching, - cl. 1.25 l.OO 1.2O .09 14 *Page's Theory and Practice of Teaching, - cl. .80 Patridge's Qumcy Methods, illustrated, - - cl. 1.75 Quick's How to Train the Memory, - paper '.64 1.4O .15 .60 '.08 .13 pd. 08 "iteinhart's Principles of Education, - cl, ^25 .20 ios * * Civics in Education, - cl. .25 .20 .03 *Rooper's Object Teaching, - cl. .25 .20 .03 Sidgwic-k's Stimulus in School, - paper Shaw and Donneli's School Devices, - cl. 1.25 Southwick's Quiz Manual of Teaching, - cL .75 .15 1.00 .60 pd. UO .05 Yonge's Practical Work in School, - paper .15 pd. METHODS IN SPECIAL SUBJECTS. Augsburg's Easy Drawings for Geog. Class, - paper .50 44 Easy Things to Draw, - - - paper .30 .40 .24 .05 .03 *Burnz Step by Step Primer, - Calkins' How to Teach Phonics, - - - cl. .50 .25 .40 Pdg Dewey's How to Teach Manners, - cl. .50 Gladstone's Object Teaching, - paper .40 .15 .15 '.05 pd. pd. *Iles' A Class in Geometry ----- .80 Johnson's Education by Doing, - cl. .50 .24 .40 .03 .05 *Kellojrg's How to Write Compositions - - paper 15 pd. Keilogg's Geography by Map Drawing - cl. .50 *Picture Language Cards, 2 sets, each, Seeley's Grube Method of Teaching Arithmetic, cl. 1.00 Grube Idea in Teaching Arithmetic - cl. .30 .40 .30 .80 .24 .05 pd. !03 Smith's Rapid Practice Cards, - - 32 sets, each .50 Woodhull's Easy Experiments in Science, - cl. .50 .40 '.05 PRIMARY AND KINDERGARTEN Calkins' How to Teach Phonics, - cl. .50 .40 .05 Currie's Early Education, ----- cl. 1.25 1.00 .08 Gladstone's Object Teaching, - paper .15 pd. A utobiography of Froebel, - - cl. .50 .40 .05 Floffman 's Kindergarten Gifts, - paper .15 pd. Johnson's Education by Doing, - cl. .50 .40 .05 *Kilburn's Manual of Elementary Teaching - 1.50 1.2O .10 Parker's Talks on Teaching, - cl. 1.25 l.OO .09 Patridge's Quincy Methods, - cl. 1.75 1.4O .13 Uooper's Object Teaching, ----- cl. .25 Seeley's Grube Method of Teaching Arithmetic, cl. 1.00 .20 .80 .03 J07 Grube Idea in Primary Arithmetic, - cl. .30 .24 .03 'Sinclair's First Years at School, : cL .75 .60 .06 ' gg^'B ALL ofebgfeS *6 fc B. L. KELLOGG & CO., NEW YORK & CHICAGO. Aliens Mind Studies for Young Teach- ERS. By JEROME ALLEN, Ph.D.. Associate Editor of the SCHOOL JOURNAL, Prof, of Pedagogy, Univ. of City of N. Y. 16mo, large, clear type, 128 pp. Cloth, 50 cents ; to teachers, 40 cents ; by mailed cents extra. There are many teachers who Know little about psychology, and who desire to be better in- formed concerning its princi- ples, especially its relation to the work of teaching. For the aid of such, this book has been pre- pared. But it is not a psj^chol- ogy only an introduction to it, aiming to give some funda- mental principles, together with something concerning the phi- losophy of education. Its meth- od is subjective rather than ob- jective, leading the student to watch mental processes, and draw his own conclusions. It is written in language easy to be comprehended, and has many VEROME ALLEN, Ph.D., Associate Editor Practical illustrations. It will of the Journal and Institute. aid the teacher in his daily work in dealing with mental facts and states. To most teachers psychology seems to be dry. This book shows how it may become the most interesting of all studies. It also shows how to begin the knowledge of self. " We cannot know in others what we do not first know in ourselves." This is the key-note of this book. Students of elementary psychology will appreciate this feature of " Mind Studies." ITS CONTENTS. CHAP. I. How to Study Mind. II. Some Facts in Mind Growth. III. Development. IV. Mind Incentives. V. A few Fundamental Principles Settled. VI. Temperaments. VII. Training of the Senses. VIII. Attention. IX. Perception. X. Abstraction. XI. Faculties used in Abstract Thinking. CHAP. | XII. From the Subjective to the, Conceptive. XIII. The Will. XIV. Diseases of the Will. XV. Kinds of Memory. XVI. The Sensibilities. XVII. Relation of the Sensibilities to the Will. XVIII. Training of the Sensibilities. XIX. Relation of the Sensibilities to Morality. XX. The Imagination. J XXT Imagination in its Maturity. XXII. Education of the Moral Seuae feklvb ALL ORDERS tfo ^. L. KELLOGG & CO., NEW YORK & CHICAGO, 3 Browning's Educational Theories. By OSCAR BROWNING, M.A., of King's College, Cambridge, Eng. No. 8 of Reading Circle Library Series. Cloth, 16mo, 237 pp. Price, 50 cents; to leacJiers, 40 cents; by mail, 5 cents extra. This work has been before the public some time, and for a general sketch of the History of Education it has no superior. Our edition contains several new features, making it specially valuable as a text-book for Normal Schools, Teachers' Classes, Reading Circles, Teachers' Institutes, etc. , as well as the student of education. These new features are: (1) Side-heads giving the subject of each paragraph; (2) each chapter is followed by an analysis; (3) a very full new index; (4) also an appendix on "Froebel," and the " American Common School." OUTLINE OF CONTENTS. I. Education among the Greeks Music and Gymnastic Theo- ries of Plato and Aristotle; II. Roman Education Oratory; III. Humanistic Education; IV. The Realists Ratich and Comenius; V. The Naturalists Rabelais and Montaigne; VI. English Humorists and Realists Roger Ascham and John Milton; VII. Locke; VIII. Jesuits and Jansenists; IX. Rousseau; X. Pes- talozzi; XI. Kant, Fichte, and Herbart; XII. The English Pub- lic School ; XIII. Froebel ; XIV. The American Common School. PRESS NOTICES. Ed. Courant. " This edition surpasses others in its adaptability to gen- eral use." Col. School Journal." Can be used as a text-book in the History of Education." Pa. Ed. News." A volume that can be used as a text-book on the His- tory of Education." School Education, Minn. " Beginning with the Greeks, the author pre- sents a brief hut clear outline of the leading educational theories down to the present time." Ed. Review. Can. "A book like this, introducing the teacher to the great minds that have worked in the same field, cannot but be a powerful stimulus to him in his work." BttNB ALL OKDMtS TO E. L. KELLOGG & CO., NEW YORK & CHICAGO, tl Curries Early Education. " The Principles and Practice of Early and Infant School Education." By JAMES CURRIE, A. M., Prin. Church of Scotland Training College, Edinburgh. Author of " Common School Education," etc. With an introduction by Clarence E. Meleney, A. M., Supt. Schools, Paterson, N. J. Bound in blue cloth, gold, 16mo, 290 pp. Price, $1.25 ; to teachers, $1.00 ; by mail, 8 cents extra. WHY THIS BOOK IS VALUABLE. 1. Pestalozzi gave New England its educational supremacy. The Pestalozzian wave struck this country more than forty vears ago, and produced a mighty shock. It set New Eng- land to thinking. Horace Mann became eloquent to help on the change, and went up and down Massachusetts, urging in earnest tones the change proposed by the Swiss educator. What gave New England its educational supremacy was its reception of Pestalozzi's doctrines. Page, Philbrick, Barnard were all his disciples. 2. It is the work of one of the best expounders of Pes- talozzi. Forty years ago there was an upheaval in education. Pes- talozzi's words were acting like yeast upon educators ; thou- sands had been to visit his schools at Yverdun, and on their return to their own lands had reported the wonderful scenes they had witnessed. Rev. James Currie comprehended the movement, and sought to introduce it. Grasping the ideas of this great teacher, he spread them in Scotland ; but that country was not elastic and receptive. Still, Mr. Currie's presentation of them wrought a great change, and he is to be reckoned as the most powerful exponent of the new ideas in Scotland. Hence this book, which contains them, must be considered as a treasure by the educator. 3. This volume is really a Manual of Principles of Teaching. It exhibits enough of the principles to make the teacher intelligent in her practice. Most manuals give details, but no foundation principles. The first part lays a psychological basis-Miie only one there is for the teacher ; and this is done in a simple and concise way. He declares emphatically that teaching cannot be learned empirically. That is, that one can- not watch a teacher and see how he does it, and then, imitat* ing, claim to be a teacher. The principles must be learned. 4. It is a Manual of Practice in Teaching. nALL ORDERS TO 0., NEW YORK & CHICAGO. 53 Standard Ulack ^Board Stencils. AIDS TO ILLUSTRATION. The need of illustration in the work of the school-room is felt by every teacher; but lack of skill in drawing is a great obstacle. To overcome this we are manufacturing an entirely new line of blackboard stencils, by which hundreds of ob- jects may be put on the blackboard quickly and handsomely by any teacher however inex- perienced in drawing. Indeed it can be done by almost any pupil. Our blackboard stencils beautify the school-room and make it attrac- tive. They give good models for drawing and writing lessons. They assist the teacher in illustrating Geography, Language, Botany, and History. No class-room is complete with- out these available aids. Our standard blackboard stencils are made of tough manilla paper of grait strength, made specially for us, on which the design is traced. These stencils will enable the teacher to put a handsome illustration on the blackboard in Language Lessons, Geography, Physiology, History, Botany, etc., etc., and thus attract and hold the attention of the class. These stencils can be used any number of times. Five to ten minutes will give a perfect map, or a drawing of an elephant, children playing, etc. A large and perfect map of Europe, 24x30 inches, showing all the prominent rivers, Lakes, mountains and large cities can be made in eight minutes. Each stencil can be used an indefinite number of times, and only requires a little pulverized chalk for im- mediate use. WHY THE BEST. 1. All our designs are new and of a high grade of artistic merit. 2. The nnimals. plants, children, birds, portraits, etc., etc., are put on paper 17x2-2 inches in size. The maps are usually 24x36 inches in size. No other stencils on the market compete with them in size. 8. The maps are from the recent surveys and are absolutely correct in outline. 4. Each figure and map is plainly numbered and named to correspond with the catalogue. 6. Many of these stencils are arranged in groups. Each group contains fire (5) Stencils, packed in a strong envelope. This envelope gives us a secure way of sending the stencils by mail, and the buyer a neat receptacle to pack each away when through using. SOLD IN SINGLE: NUMBERS as well as in groups. TWO SAMPLES FOR TRIAL. A simple map of South America and a design suitable for a language or drawing lesson will be mailed post paid for 10 cents. We will also send a complete catalogue. SEND ALL ORDERS TO 54 E. L. KELLOGG & CO., NEW YORK & CHICAGO. MAPS. These maps are made on special ma- nilla paper, size 24x36 inches. Price, 10 cts. each. Please order by number. 501 Eastern Hemisphere. 502 Western Hemisphere. 503 Mercator's Eastern Hemisphere. 504 Mercator's Western Hemisphere. 505 North America. 506 South America. 507 Europe. 508 Asia. 509 Africa. 510 Australia. 511 British Isles. 512 Mexico. 513 Canada. 514 West Indies. SEPARATE STATES AND TERRITO- RIES. 48 maps, 24x36 inches. Price, 10 cents each, as follows : Please order by num- ber. 524 Alaska. 548 Missouri. 525 Alabama. 549 Minnesota. 526 Arizona. 550 Montana. 527 Arkansas. 551 N. Hamp. 528 California. 552 N. Jersey. 529 Colorado. 553 N. Mexico. 530 Conn. 554 New York. 531 Dakota. 555 Nebraska. 532 Delaware. 556 Nevada. 533 Florida. 557 N. Carolina. 534 Georgia. S58 Ohio. 535 Idaho. 559 Oregon. 536 Illinois. 560 Penn. 537 Indiana. 561 R. Island. 538 Ind. Ter. 562 S. Carolina. 539 Iowa. 5fi3 Tenn. 540 Kansas. 564 Texas. 541 Kentucky. 565 Utah. 542 Louisiana. 566 Vermont. 543 Maine. 567 Virginia. 544 Maryland. 568 Wash. Ter. 545 Mass. 569 West Virginia. 546 Michigan. 570 Wisconsin. 547 Mississippi. 571 Wyoming. GROUPS OF STATES. Size 24x36 inches. Please order by number. Price, 10 cents each. 515 NEW ENGLAND, comprising Me., N. H., Vt., Mass., R. I., Ct. 516 MIDDLE ATLANTIC. N. Y., N. J., Pa., Del., Md., Va., and W. Va. 517 SOUTHERN STATES (three groups). No. I.-N. C., S. C., Ga., Fla., Ala., Miss., La., and Tex. 518 No. II.-W. Va., Va., N. C., S. C., Ga., Fla., Ala., and Miss. 519 No. III. Ark., La., Tex., and In- dian Ter. 520 CENTRAL STATES (two groups). No. I. Minn., Wis., Mich., la., 111., Ind., Ohio, Mo., and Ky. 521 No. II. Dak. Ter., Minn., Wis., Mich., Neb., la., 111., Ind.. Ohio, Kan., Mo., and Ky. 522 WESTERN STATES (two groups). No. L-Wash. Ter., Idaho, Mon. Ter., Dak. Ter., Oregon, Wyoming Ter., Neb., Cal., Nev., Utah, Col., Kan., Arizona Ter., N. Mex., Ind. Ter., and Tex. 523 No. Il.-Wash. Ter., Idaho Ter., Mon. Ter., Oregon, Wyoming Ter., Cal., Nev., Utah Ter., Col., Arizona Ter., New Mex. LARGE MAPS. These stencils make maps as large as the largest wall maps. 572 United States, 34x56 inches. Price, oO cents. 573 Mercator's Eastern and Western Hemisphere with Western Hemisphere repeated , 34x56. Price, 50 ce n ts. HISTORICAL MAPS. Please order by number. 600 Mercator's Eastern and Western Hemispheres with the Western Hemis- phere repeated, showing all the routes of the early voyagers to America ami around the world. Price, 50 cents. 601 Large map of the U. S. showing territorial growth. Price, 25 cents. FRENCH AND INDIAN WAR. Five maps, each 24x36 in. Price, 10 cents each. Set, 50 cents. 602 Map of Va. and Pa., showing Wash- ington's home, route taken in his jour- ney to St. Pierre, Ft. Duquesne. 603 Map of N. Y., showing all forts on the great lakes and L;ike Champlain. 604 Canada, showing all the principal places and Nova Scotia. 605 Map showing British possessions before the War. 606 Map showing British possessions after the War. WAR OF THE REVOLUTfON. ' Five maps, each 24x36 in. Price, 50 cents each. 50 cents a set. 607 Boston and vicinity. N. Y. and vicinity. 60S Phila., Trenton, Valley Forge, Monmouth. 609 Burgoyne's Invasion. 610 Yorktown and Southern Battle Fields. 611 Map showing Territory of U. S. at close of the War. WAR OF 1812. Three maps, size 24x36 in. each. Price, 10 cents each. 612 Great Lakes and vicinity, showing battle fields. 613 Washington and vicinity. 614 New Orleans. CIVIL WAR. Size, 24x36 in. Price, 10 cents each $1.00 a set. 615 U. S., showing territory seceded. 616 Washington and vicinity. 617 Richmond and vicinity. 618 Charleston Harbor. 619 Miss. River, New Orleans, etc 620 Gettysburg Campaign. SEND ALL ORDERS TO E. L. KELLOGG & CO.. NEW YORK & CHICAGO. 55 621 Sherman's March. 622 Battle Fields of Ky. and Tenn. Group Fourteen ANIMALS. 66 Wolf. 69 Kangaroo. 623 Battle Field of Va. 67 Fox. 70 Donkey. 624 Petersburg and Appoto . ax. C3 Hyena. MISCELLANEOUS. Group Fifteen FLOWERS. Size, 17x:& inches Price, singly, 5 1 Wild Rose. 74 Laurel Spray. 2 Calla Lily. 75 Pear Blossom. design, 25 cents. 3 Solomon's Seal. Group One CHILDREN . In a Swing. 4 Kite Flying. 2 Jumping Rope. 5 Skating. Group Sixteen FLOWERS. -6 Wood Violet, 79 Morning Glories. 7 Pond Lilies. 80 Fuchsias. 3 Leap Frog. 8 Roses. Group Two CHILDREN. 6 Feeding Doves. 9 On a Toboggan. 7 RollingtheHoop.10 Where am I ? Group Seventeen BIRDS. 81 Quails. 84 Stork. S2 Woodcocks. 85 Swan. 8 Blowing Soap Bubbles. 83 Eagle Flying. Group Three CHILDREN. 11 Two Lillfes. 14 Fast Friends. Group Eighteen OLD AND YOUNG. 86 Hen and Chick- 88 Dut-k and Duck- 12 Training Pussy. 15 Dance, Little ens, lings. 13 What Do I Care. Baby. Group Four-CHILDREN. 16 Oh, How High ! 18 " My Pony Loves 17 Naughty Tab Sugar." and Dash. 19 Can I Get Them? 87 Goose and Gos- 89 Owl and Owlets, lings. 90 Bird and Young. Group Nineteen BUILDINGS. 91 Lighthouse. 94 Bird House. 93 Castle. 95 Fort. :.'> Mud Pies. !>3 Wind Mill. Group Five-CHILDREN. 21 Saved From 2S Learning to Drowning. Read. 22 St. Bernard Dog 24 Who Broke the and Boy Window ? Group Twenty-PATRIOTIC LIST. '.Hi The American 99 The American 97 Llh't'-rty IM1. 100 Goddess of Lib- 25 The Milkmaid. IB U. s. Coat of erty. Group SIX-CHILDREN. Arms. 26 Wide Awake. 29 The Pet Squirrel. BORDERS. 27 Fast Asleep. 30 Learning to 101 Spiral Curves. 28 Have You Been Walk. 102 Greek Fret. Bathing ? 103 Triangular Combinations. Group Seveii-ON THE SEA-SHORE. 104 Greek Fret. 81 Star Fish. 84 Jelly Fish. 82 Hermit Crab. 35 Red Coral. 83 Lobster. 105 Greek Pattern Anthlmion. 106 Egyptian Lotos. 107 Ivy Leaf. Group Eight-PRESIDENTS. 86 Washington. 89 Lincoln. 108 Dog Wood. 109 Holly Leaf and Berries. 87 Jefferson. 40 Grant. 110 Holly Leaf and Berries. 88 Jackson. Group Nine POETS. 41 Whittier. 44 Bryant. 42 Longfellow. 45 Tennyson. ROLLS OF HONOR. Ill Script Letters, plain. 112 Script Letters, fancy. 113 Old English Letters. ^^roupTen-DOMESTIC ANIMALS. 46 Cow and Calf. 49 Camel. 47 Horse and Colt. 50 Reindeer. 48 Elephant and Baby 114 German Text. 115 American Eagle on Shield. 116 Excelsior. WRITING CHARTS. 117 Capitals and Small Letters. Group Eleven- DOMESTIC ANIMALS. 51 Dog. 54 Pig. 52 Cat. 55 Goat. The letters are nearly 6 in. high. Size of Stencils 9x36 in. The set contains 11 charts. Price, 50 cents a set. PHYSIOLOGY CHARTS. Group 'Twelve SMALL ANIMALS. Six charts, size 24x36 in. each. Price, 56 Rabbit. 59 Mouse. 10 cents each. Set 50 cents. 57 Bat. 60 Lynx. 118 Bones. 121 Lungs. 58 Rat. 119 Skull. 123 Liver. Group Thirteen LARGE WILD ANI 120 Heart. 124 Intestines. MALS. 61 Polar Bear. 64 Rhinoceros. 62 Lion. 65 Hippopotamus. 63 Lioness. NATURAL HISTORY CHARTS. Price each, 10 cents, except No. 12u Size 2-1x36 inches. 8 nos. And many others. Full catalogue on application. SEND ALL ORDERS TO 20 E. L. KELLOGG & CO., NEW YORK & CHICAGO. Hughes Securing and Retaining Atten- TTON. By JAMES L. HUGHES, Inspector Schools, Toronto, Canada, author of "Mistakes in Teaching." Cloth, 116 pp. Price, 50 cents; to teachers, 40 cents; by mail, 5 cents extra. This valuable little book has already become widely known to American teachers. Our new edition has been almost entirety re-written, and several new important chapters added. It is the only AUTHORIZED COPYRIGHT EDITION. Caution. Buy no other, WHAT IT CONTAINS. I. General Principles; II. Kinds of Attention; III. Characteristics of Good Attention; IV. Conditions of Attention; V. Essential Characteristics of the Teacher in Securing and Retaining Attention; VI. How to Control a Class; VII. Methods of Stimulating and Controlling a Desire for Knowledge; VIII. How to Gratify and Develop the Desire for Mental Activity; IX. Distracting Attention; X. Training the Power of Attention; XT. General Suggestions regarding Attention. TESTIMONIALS. S. P. Bobbins, Pres. McGill Normal School. Montreal, Can., writes to Mr. Hughes:" It is quite superfluous for me to say that your little books are admirable. I was yesterday authorized to put the * Attention ' on the list of books to be used in the Normal School next year. Crisp and attractive in style, and mighty by reason of its good, sound common-sense, it is a book that every teacher should know." Popular Educator (Boston):" Mr. Hughes has embodied the best think- ing of his life in these pages." Central School Journal (la.)." Though published four or five years since, this book has steadily advanced in popularity." Educational Courant (Ky.). "It is intensely practical. There isn't a mystical, muddy expression in the book." Educational Times (England)." On an important subject, and admir* ably executed." School Guardian (England)." We unhesitatingly recommend it." New England Journal of Education." The book is a guide and a manual of special value." New York School Journal," Every teacher would derive benefit from reading this volume." Chicago Educational Weekly. " The teacher who aims at best sue- c_ss should study it." Phii. Teacher." Many who have spent months in the school-room would be benefited by it." Maryland School Journal." Always clear, never tedious;" Va. Ed. Journal. " Excellent hints as to securing attention." Ohio Educational Monthly." We advise readers to send for a copy." ""icific Home and School Journal." An excellent little manual." YB 04999 /.fr A/7- WHVERSnY OF CALIFORNIA LIBRARY