•iu IMAGE EVALUATION TEST TARGET (MT-3) 1.0 I.I 'fl '" 1 2.2 110 III 2.0 1.8 1.25 1.4 1.6 .4 6" ► v2 signifie "A SUIVRE", le symbole V signifie "FIN". Les cartes, planches, tableaux, etc., peuvent dtre filmds d des taux de reduction diffdrents. Lorsque le document est trop grand pour dtre reproduit en un seul clichd, il est filmd d partir de Tangle supdrieur gauche, de gauche d droite, et de haut en bas, en prenant le nombre d'images ndcessaire. Les diagrammes suivants illustrent la mdthode. 1 2 3 32X 1 2 3 4 5 6 1 HINTS ON TEACHING ARITHMETIC BY H. s. Maclean, Assistant Principal Manitoba Normal School ; Author of The High School Book-keeping. TORONTO : THE COPP, CLARK COMPANY, LIMITED. M3 Entered according; to Act of the Parliament of Canada, in the year one thousand eight hundred and ninety-flve, by Thb Copp, Clark Ooupant, Limitkd, Toronto, Ontario, in the Offloe of the Minister of Aj^riculture. ~%oi^\ PREFACE, As indicated by the title of this maniml, no attempt lias been made to enter into an exhaustive discussion of tlie psychological principles underlying the teaching of Arithmetic. The aim has been, rather, to present in brief form, and in as practical a manner as possible, what liave appeared to the author to be the most important features of the subject, viewed from the teacher's stand- point. The early pages are devoted chiefly to the considera- tion of such fundamental questions as seem to have the most direct bearing on practical school work. An eflbrt is made, whether successful or not, to outline what may be done in the school-room during consecutive periods. I^astly, suggestions are offered on points where mistakes are likely to occur in teaching. This book, containing the substance of talks given at institutes, and in the Manitoba Normal School, is humbly submitted for the consideration of teachers generally, with the hope that it may prove to be of some little value to, at least, the younger members of the profession. CONTENTS. 1. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. PAOK. Importance as a Subject of Study 1 Relation to Other Subjects 2 Number a Purely Mental Conception 'S Development of the Idea of Number 4 What the Idea of Number Includes 5 Units of Mkasurement 6 Numerical Relations 9 Numbering by Ones 15 Numbering by Groups 17 The Use of Objects 18 The Fundamental Rules 21 Arithmetical Processes 23 Problems 26 Educational Value of Probiems 34 Expression , 38 Course of Study 41 Outline of Work for First Period 41 Outline of Work for Sec(md Period 40 Outline of Work for Third Period 56 Plan of Teaching Nos. 1-20 62 Seat Work for Junior Classes 69 [V] ^* CONTENTS. 18. T*RACTICAL SU(;OKSTI()NS ^^^J'. Reading Numbers q. Writing Numbers g2 Addition „„ Subtraction or. Multiplication o . Division o^ oo Fractions oq Decimals q- Percentage ^^^^ Bank Discount w.q Compound Interest jq^ ' . S<|uare Root ^^- 19. Review Exercises jqh PAOK. ... 81 . .. 81 ... 82 . .. 82 . .. 83 . .. 84 . .. 86 . . . 89 . .. 97 . .. 100 . .. 102 . .. 103 . .. 105 ... 107 ■J •. HINTS ON TEACHING ARITHMETIC. 1. Importance as a Subject of Study. Arithmetic is one of the few suV)jects whoso rifjht to a place on the Pu})lic School programme of studies has never l)eeii seriously questioned. Its adaptability to the different stages of mental development, it'* suitability as a means of training in concentration, accuracy, rapidity and logical ariangement of thought, its tendency to secure clearness, exactness and conciseness of expression, and its usefulness in the every day business of life are all so evident as to give it an undisputed right to a high position among school subjects. The material world presents objects of endless variety which invite the attention of man. The human mind reacting on these ascribes to them qualities of form, size, etc., and classifies them according to resemblances and differences of such quali- ties. But these objects manifest diversity of form and size, and they are acted upon by forces. Hence the necessity for accurate quantitative measurement arises. A standard of measurement is determined and the relation between it and the whole quantity measured is established. This quantity relation, which is the product of the action of the mind, is termed number. The study of number is, therefore, necessary in order that man may be brought into harmony with his environment. Besides this, a knowledge of number is essential in all the details of actual life, for there could be no progress in science, manufactures, or commerce, without exact quantita tive measurement. [1] I I HINTS ON TKArillNC} AinTMMKTK!. T]ui conditions of (;xiston(!o sinco tliti vory oarliosl, .'ulvancos wovo, )n;ui(i in civilization liavo ron(l(M'<'(l it iiniK'mtivc that inaten tlu^ teacher who advocates the con-elation of studies and the teacher who insists on teaching one thing at a time. J5oth are riglit ; they are simply viewing the matter from dillerent standpoints. 3. Number a Purely Mental Conception. Num))er is not an inh(M'ent (juality of the material ol)jects or groui)S of obj(>cts, forc(!s, etc., that an; mc'asured numeri cally. but it is the product of the mind's action in detei'niining their e.xact limitations as to quantity. This becomes obvious when we consider that the measure of any definite (piantity will depend altogether on the unit of measurement which tlu; mind chooses. Objects and groups of objects are perceived by the senses, but not quantity relations. The presence of o))jects is a necessary condition — but only a condition — ^for occasioning the mental activity that results in establishing a quantity relation between a unit of measurement and a whole or aggre- gate. The sense concepts corresponding to objects presented, furnish the material upon \vhich the mind operates in deter- mining numerical relations. Number is always abstract. There can be no such thing as concrete number. The unit may Im; particular or limited in its application, as in 10 pears, 10 yards, etc., :.)r it may be general t HINTS ON TEACHING ARITHMETIC. I : i or unlimited, pinent of utlined as rom other bhe atten- stics. this that age such any, few, compari- into like make up separat- t'formed. as first, complete •, as one, / If we accept the foregoing as correct, we are in a position to mark out the lines along which progress in number nmst be made. Beginning with an undetermined quantity which is seen to be capable of being measured, the child separates it — roughly at first but with sufficient accuracy to satisfy him at this early stage — into parts, and by counting these he defines the whole. To his original percept of the whole as a quantity extended in space he has added the new element of time. The undetermined whole hns become to him a numbered quantity. But as he gains strength intellectually greater exactness is demanded. The indefinite units by which he first measured the whole must become more and more definite, otherwise numbering can no longer afford a mental stimulus for him. Well defined, measured units now become a necessity, for without them the higher thought processes of number would be impossible. It is thus seen that the direction of progress in number is determined by the conditions of mental growth. There are many other intermediate steps which are more or less difficult to determine. According to Perez,* M. Houzeau says : " Children at first can only distinguish l)etween a single object and plurality. At eigliteen months they can tell the difference between one, two, and several. At about three, or a little sooner, they have learnt to understand one, two, and four — two times two. It is scarcely ever till ca later age that they can count the regular series, one, two, three, four; and here they stop for a long time." 4. What the Idea of Number Includes. In every complete idea of number there are three elements. (i) A unit of measurement, (ii) An aggregate or whole quantity measured, (iii) A quantity relation (number). Any two of these determine the other. If either the unit 'Perez, The Fimt Three Yearn of Childhood— yt. 186. ;i':: i !l:^: 6 HINTS ON TKACHINO ARITHMETIC. or tho quantity relation is unknown or not fully comprehended, the aggre<(ate cannot be definitely fixed. For example, 20 square feet will not convey its full meaning to a pupil who has not a clear conception of the unit square foot, even if he knows the measure (20) of the aggregate 20 square feet. The same may be said in the case where the unit is well understood, but the measure of the aggregate (number) is not. From these considerations we can easily see that in teaching arithmetic, units themselves, as well as the numbering and relating of units, demand the careful attention of the teacher. I'' II 5. Units of Measurement. By the term unit, in numbering, we mean one of the like things of a unity on which the attention is fixed as the basis of measurement. For example, when we speak of 12 pears the unit which we have in mind is one pear ; in 12 square feet it is one square foot, and in $^f it is one twenty-fifth of one dollar. Any aggregation of units considered as one whole is termed a unity. It is evident that if any unity be regarded as a standard of measurement for like things, it then })ecomes the unit. Three feet may be taken as the unit of 12 feet, in N/hich case the measure is 4 instead of 12. Again, if we take | of a foot as the unit, the measure of 12 feet becomes 48. The measure, therefore, of any quantity depends altogether 'on the unit which the mind fixes upon as the basis of measure- ment. Before an aggr-egate can be measured the unit nmst be clearly seen. Much of the work in arithmetic consists in transforming quantities so that they can be measured by par- ticidar units and rice versa. Sometimes it is convenient to take one unit and sometimes another. This will depend on the whole quantity measured or on the purpose for which the measurement is made. For UNITS OP MEASUREMENT. example, 1,000 is more easily thouglit of as 10 liundreds tlian as 1,000 ones. We can count 120 people in a room more readily by threes than by ones, and we can measure off G60 feet much more quickly by using a 16| foot pole, or G6 foot chain as the unit than we can by means of a foot rule. Again, it may be necessary to consider a part of a whole thing which we have in mind as the unit. We may wish to pur- chase I of an acre of ground. Here the unit is | of an acre. It is thus seen that the unit may be increased or diminished to any extent, at will, in order to meet the requirements of any particular case. The pupil should be given practice in scilecting different units by which aggregates may be measured. He should also be led to see the necessity for changing from one unit to another. Units may be divided into two kinds, concrete and abstract. A concrete unit is one which may be used to measure only a particular kind of quantity, as in 12 horses, 25 yards, etc. An abstract unit is one that is thought of as applicable to the measurement of any quantity whatever, as in 12. Concrete units may be undefined as to quantity, as in C books, 16 houses, etc., or they may be well defined standards of measurement, as in 12 yards, 100 ohms, etc. A fractional unit is one which is dependent for its value on its relation to some other unit. When we speak of | of one yard we have in mind the unit ^ of one yard, which is clearly dependent for its meaning on the unit one yard, to which it is related. In fact the fractional number 1 does notliing more or less than express the ratio between the fractional unit and the prime unit from which it is derived. What is the purpose of considering units as a factor in determining a course in number ] To discover their degree of detiniteness in themselves in 8 HINTS ON TEACHING ARITHMETIC. ii :' order that we may proceed from the vague to the definite in accordance with the natural progress of thought. The order then is, (a) Concrete units of single objects which do not repre- sent any fixed quantity, as block, pin, etc. An aggregate determined by such units gives rise to counting, and nothing more definite. (6) Units composed of numbered groups of such objects. Here the unit, though still somewhat indefinite, is limited to a degree as it is a numbered unity, (c) Fixed standards of measurement. These are well defined, and their use in measuring aggregates leads to the highest conception of measurement, viz., ratio, by directing the mind not only to the numerical value of the aggregate, but also to its quantita- tive value, as compared with that of the fixed unit, (d) Frac- tional units. Such units are completely defined in so far as their relation to the unit from which they are derived is con- cerned. The absolute definiteness of a fractional unit will depend entirely on the degree of definiteness of the related unit, as ^ of a line, ^ of 6 feet, etc. As the simplest fraction expresses a relation based on a comparison of quantities, it is clear that fractions should not be introduced until the pupil has made considerable progress in relating quantities. The fact that many good teachers advocate the introduction of fractions at the very commence- ment of a course in number is manifestly the result of a difference of opinion as to the meaning of the term fraction. In teaching a pupil to number, none but familiar objects, unattractive in themselves, should be used. To present objects as toys, flowers, etc., which possess striking characteristics may prove interesting to the pupil but the interest will be centered on the qualities of the objects, and consequently it will be foreign to the purpose of a lesson on number. As has already been shown, the pupil must lose sight of the distinctive qualities of the objects making up a group before he definite in t. Tlie order do not repre- A.n aggregate ;, and nothing ed groups of bat indefinite, y. (c) Fixed led, and their conception of 1 not only to y its quantita- lit. (d) Frac- 1 in so far as lerived is con- nal unit will )f the related n based on a is should not rable progress ood teachers ry commence- rei^ult of a m fraction. liliar objects, resent objects laracteristics erest will be tisequently it number. As sight of the group before ■■ KUMEKICAL RELATIONS. 9 they can be regarded as units by which the group may be numbered. The pupil should be conscious of space limitation sufficient to distinguish one object from another ami nothing more, in order that the best condition for numlxM-ing may be fulfilled. The less definite the unit is, so long as it is seen to be a unit, the more will the attention be fixed on numbering as a means of defining the aggregate. Tliat is the reason why it is easier for the pupil to count ol)jects by ones than by twos, threes, etc. As soon as the unit becomes definitely measured the difficulty of numbering is increased, because greater exactness of thought is demanded. This being the case, it is easily seen that units which themselves represent fixed measurements should not be presented until tlie pupil has gained some power in numbering. After he is once able to deal with such units, the measurement of aggregates by them will greatly increase his power, and his knowledge of number. In introducing such measured units as inch, foot, yard, pint, quart, gallon, ounce, pound, etc., the experience of the pupil is an important factor in determining the order in which these should be taught. By a judicious selection of such units, much may be done towards connecting the school life of the pupil with his home life. Facility of concrete presentation and simplicity of relation to other units will also be considered in this connection. 6. Numerical Eelations. Number is the product of the action of the mind in defining a quantitative whole. The first step is one of analysis, wheieby the whole is thought of as being made up of like parts. This gives rise to the idea of unit. It is a, process of simplification on which numbering is conditioned. If the mind could go no farther than this there could be no numbering or numei'ical to HINTS ON TEACHING ARITHMETIO. liiiii! i I! Ill 1 1 II i! I measurement. The like pjirts must be related to one another and to th(^ whole, in some degree at l(?ast, befoi'e any progress can he made in determining the whole quantity. The first efforts may, indeed, produce rather indefinite results, but they are the beginnings of greater things. The mind proceeds from the vague to the definite. In numbering, this has two appli- cations, first, as to the units themselves, second, as to the manner in which the units are related. It has already lieen hinted that there is a progressive order, from the vague to the definite, in attitude of the mind with reference to units as advancement is made in number. The units first employed by the child are regarded as alike, but in themselves they are quantitatively undefined by him. The relating of such units cannot rise higher than mere numbering. An aggregate composed of such can be defined in so far as the nnmbe^' of the units is concerned, but it cannot be defined as a whole quantity. For example, when we speak of 10 apples we think only of the number, the how many, we know nothing definite about the how much. If we wish to de- termine the whole quantity we must do so in terms not of one apple, but of some fixed standard of measurement, as one pound; We cannot say 3 apples! 15 apples; ; 1 ; 5, but we can say 3 lbs. of apples ; 1 5 lbs. of apples ; ; 1 ; 5. If we do make the former statement we have first to regard the units as quantitatively equal to one another, otherwise the state- ment is not necessarily true. Undefined units may be regarded as taken together and separated ; they may also be enumerated, but a direct comparison of quantities based on such is impossible. Even in the case of measured standard units it is not until the exactness of a mathematical concep- tion of them is gained, that it is possible for the mind to relate in the highest degree. This indicates not only the onler in which units should be presented, but also the order in which they should be related. i NUMERICAL RELATIONS. 11 Fioin these considoniti()iis wo see tliat tliere are two dis- tinct stages in relating — (i) Defining aggregates by numbering. — Number. (ii) Defining aggregates ])y comparison. — Ratio. As the evolution of number is nothing more or less than tlio evolution of thought in a certain direction, numerical i('latil)j«'c'i, Lessons leading up to nKjasurenuuit and comparison will bt- given from the beginning. They are necessary, for the reason that the child has already begun the work of measuring and comparing. But he has done it in a very indcliniti! mann(!r. The vague notii . underst li ilie prime factors of the numbers, he dis- -; . .^ *aie H. C. F. The whole process is clearly y tie pupil from beginning to end, for he has five no end 1(1 the very ise the pro :ht simply means he 50 that he )n and sub- erefore go le teaching from one hich a cer- nstrument ion of the )il to have 'ill enable lall num- Into their he dis- Is clearly he has ARITHMETICAL PROCESSES. 25 worked it out himself. He is then asked to find the H. C F. of 1908 and 2736. He proceeds thus: 1908 = 2x2x3x3x53. 2736 = 2x2x2x2x3x3x19. 2x2x3x3 is found to be the H. C. F. of the given numbers. The pupil is now required to scrutinize carefully and to state clearly what he has done in finding the H. C. F. required. He finds that he has retained all the common factors of the num- bers, and eliminated all the factors which are not common. The teacher then shows him a process — places an instrument into his hands — by means of which the common factors may be retained, and the factors which are not common may be got rid of more expeditiously than by the method formerly em- ployed thus : ■ 2)1908 1056 3) 252 216 36 2736( 1 1908 828(3 756 9/2 72( 72 The pupil understands fully the use of this new process — the work the instrument is capable of doing. He can, therefore, employ it intelligently. But he may not be able to compre- hend the fundamental principles underlying the process, much less explain them. This form may be given, but it affords no explanation, it 26 HINTS 01^ TEACHING ARITHttBTia i; merely shows clearly how the formal process of finding the H. C. F. serves the purpose for which it is intended. 2)2x2x3x3x53 2x2x3x3x46 3)2x2x3x3x7 2x2x2x3x6 2x2x3x3x1 2x2x3x3x76(1 2x2x3x3x53 2x2x3x3x23(3 2x2x3x3x21 2x2x3x3x2 (2 2x2x3x3x2 The points to be noted in teaching arithmetical processes are: (i) The pupil must feel the need of the process before it is taught. This means that he has done all he can do conveni- ently without it, and that he is fully cognizant of its purpose. Give the pupil plenty of opportunity to reason from first principles. (ii) The process is presented by the teacher and the pupil is immediately required to apply it. (iii) The process is examined when the pupil is able to deal with it, so that he may discover the principles on which it is based. To teach process, and reason for process together, is, generally speaking, a pedagogical error, because two difficulties have to be overcome at once, and further, because it is only after a pupil has become familiar with a process that he can be in the best position to investigate it. 12. Problems. During every stage of progress the pupil should be given ample opportunity of applying his knowledge practicaUy. Problems should therefore have a prominent place throughout the whole course in number. The greatest care should be exercised by the teacher ill selecting or making problems so that they may be well PROBLEMS. 27 adapted to the requirements of the pupil. To give prob- lems which present no stimulus is to waste time. This surely finds its application in the thousand and one problems which are so often given on the number 5. On the other hand, to ask a pupil to solve a problem requiring for its solution a pro- cess of thought higher than that to which he has attained, is to attempt the impossible. There is but one error greater than this one, and that is for the teacher to give an explana- tion of the problem and then try to " sense " the pupil into understanding it. Unfortunately, too many pupils have been subjected to such unprofessional treatment, and have had to suffer its evil consequences. Viewed from the standpoint of discipline, a problem is worth but little to a pupil unless he solves it himself. As to whether the solution is the neatest or best matters but little at first, provided that it is original. After the pupil lias discovered a path for himself, however, his attention may be directed to it in order that he may find out whether or not he might have chosen a better one. Short cuts taken at the proper time form a valuable means of discipline, as they necessitate vigorous thought. For example, the solution of the problem, / sold a Jiorse for $75 gaining 2n %, wimt % should I have gained if I had sold him for $80 ? may take several forms as, (i) |§§of cost = J •'• Thrs of co8t = Y^iy of $75. .-. i§§of co8t=}ogof $75. Therefore gain in second case is $80 -$60 =$20. The gain on $60 =$20. .-. The gain on $1 =^ of $20. .-. The gain on $100= VjP of $20. = $33j. The gain % in second case is 33^. t. 28 Hlir 3 ON TEACHING ARITHHETia m (ii) $75=125% of cost. ,\^1 =7?5of 125%ofco8t. .-. $80=^gof 126 %ofcost. = 133 J % of cost. The gain % is therefore in second case 33^, (iii) $75 = 125% of cost. .-. $80=f§of 125 %of cost. = 133 J % of cost. The gain % therefore in second case is 33^. (iv) $75=1 of cost. .-. $80=f§of f of cost. =f of cost. " The gain in second case is J of cost =33^ %. v Suppose that the first form (»f solution is the one which is obtained. It is roundabout, and therefore indicates a degree of weakness on the part of the pupil, but if it is his own, it has developed power. The pupil may be asked to attempt a shorter form of solution. A question to put him on the crack may be given, but nothing more. If he gets the second form, he shows a gain in relating, although there is still an indication of weakness in that he has not made a direct comparison between $75 and $80. The third and fourth forms here given are preferable to the others, in so far as they indicate a greater degree of ability in establishing a ratio between quantities. No form should be placed before the pupil in any case if the purpose is to secure discipline, and not merely to show how to solve the problem. To give a form of solution before the pupil has done the thinking correspond- ing to it is a delusion. To relate all quantities to unity when more difficult ratios can be established, is another source of weakness. PROBLEMS. 29 What arc the characteristics of an arithmetical problem ? (i) Certain conditions are given. (ii) A (Quantity is to be determined. (iii) Definite relations are expiv^ssed connecting the undetermined quantity with the given con- ditions. The difficulties which the pupil will meet in problem work are these : (i) Comprehending the data. (ii) Analyzing the given conditions and determining the relations they bear to the unknown quan- tity. (iii) Deciding upon the numerical operations that will correspond to and carry into effect the thought ' process of solving the problem. (iv) Performing the numerical operations necessary to evaluate the required result. These it may be well to consider somewhat in detaiL (i) The conditions given may not be fully comprehended. If the data are not understood, the pupil cannot even begin to do what is required of him. Indeed, the difficulty may not be an arithmetical one at all. Compare, for example, the problems. Find the cost of 10 desks at $Jf.87 each, and Find the cost of a hill of exchange for £10 at J^.87. The former of these is suit- able for almost a beginner, the latter might put some teachers to the test, and yet the arithmetical operations required for their solution are identical. The duty of the teacher is to lead the pupil by means of questions or illustrations to grasp the con- 30 HINTS ON TEACHING AltlTHMETIC. ditions given. If they arc l)eyond his reach, the problem is unsuitable, and should be withdrawn. The second of the foregoing problems indicates clearly that quantity relation is not the only thing to be dealt with in what is usually termed arithmetic. Often the greatest ditticulty which the pupil has to contend with arises from the qualitative elements of the data. He lacks the experience necessary to enable him to interpret the given conditions and to represent them to himself in such a manner as to be able to apply either his mental power or his knowledge of number in relating them. This shows the disadvantage of attempting to teach interest, discount, exchange stocks, etc., to pupils, before they have a proper conception of the business transactions ou which such divisions of the subject are based. (ii) The pupil may not be able to analyze the problem sufficiently to locate the difficulty definitely. In that case he does not know the exact point at which to concentrate his energies. For example, a pupil fails to solve, Fiml the num- ber of cords of wood in a rectangular jnle which is 7 feet hiyh, and which covers ^ of an acre. He knows what ^ of an acre means, and he can calculate the number of cords when the length, breadth and height are given. If asked why he can- not get the answer, he will probably say that the length and breadth are not given, but that is not the real difficulty at all. It is this, he does not see the connection between the linear measurements of length and breadth in one case, and the sur- face measurement in the other. The duty of the teacher is to question the pupil, so as to lead him up to the exact point of difficulty, and then leave him to grapple with it as best he can. To solve the problem and then try to get the pupil to see through it would be a serious pedagogical error. It is not to be inferred from what has been said that no yalue whatever attaches to following step by step a neat, logi- PROBLEMS. ai problem is I early that th in what > dirticulty qualitative (cesHary to ) represent iply either II relating ^ to teach ;fore they 8 ou which 5 problem it case he ntrate his ! the num- feet hiyh, •f an acre when the y lie can- igth and ty at all. he linear the sur- ler is to point of he can. il to see that no lat, logi- cally arranged solution worked out by another ; for, under certain circumstances such an exercise may prove a very effi- cient means of stimulating independent effort. This is a very different thing from gi^'ing assistance whenever a difficulty is met. We must not forget that self-reliance, determination, and strength are developed by overcoming difficulties and sur- mounting obstacles, and in no other way. The instructor who keeps this clearly in view will do much more for the pupil than teach him how to solve arithmetical problems. Again the difficulty may be that of determining the connection between what is given and what ia to be /ound. We may take, by way of illustration, the following problem : The amount of $100, bearing compound interest /or two years, payable annually, is $1^1. Find the rate per cent. The pupil can find the com- pound interest on a sum of money for a given time, but he fails to solve the foregoing. What is to be done ? Surely not to give him a form of solution and ask him to think it out, or to lead him by a series of questions* — perhaps logically arranged, but arranged by the teacher, not by the pupil — to arrive at the answer. Where is the real defect? It lies in the fact that the pupil did not secure a proper grasp of compound interest. He looked at the matter from one point of view only. What should be done is this : He should be asked to perform the operation of finding the compound interest over again, and to retrace the steps taken in finding the answer. If he is able to deduce from his work the relation A=:P (1 -»- r)^, he is in a position to solve the problem ; if he has not power to do this, it is too difficult for him.f (iii) It is unnecessary to illustrate the case where the pupil has difficulty in determining the operations to be performed. We frequently, hear such questions as this asked by beginners, * Proper questioning; at the right time ia a good thing, but the kind of questioning so often practised, whioh is merely an interrogative form of telling, is pernioioui. tSee page 104. 1 32 HINTS ON TEACHING ARITHMETIC. whether shall I multiply or divide ? Such is the result of defective teaching. The important fact that number is the product of mental action has been lost sight of. The pupil has been shown how to perform operations without learning how to make use of them. The remedy consists in getting him to apply whatever knowledge he may have, and to illustrate his work objectively when necessary. (iv) Lack of ability to calculate accurately and readily is not by any means uncommon. The weak points should be noted by the teacher, and suitable, well arranged exercises on the simple rules should be given. This matter is referred to elsewhere. One other difficulty may be worthy of consideration. There may be known facts bearing on the solution of the problem which the pupil cannot summon to his aid. Sometimes a mere suggestion may serve to lead him back to a related known fact or principle. For example, a pupil is asked to find the area of the right-angled triangle whose sides are 6 feet, 8 feet and 10 feet. He draws a diagram of it, and he attempts to find the required result, but he cannot succeed. He then informs the teacher that he does not know how to find the area of the triangle. The teacher asks him to say what he knows about finding areas. Probably this may be quite suffi- cient to enable the pupil to relate the area of the right-angled triangle to that of the corresponding rectangle. The recitation may take various forms, according to the object in view : 1. Stating in simple language the conditions of the ]}roblem. This is often a very necessary preliminary exercise. It awakens interest and it assures the pupil that he is in posses- sion of the facta on which the solution is to be based. tji PROBLEMS. 33 Confidence will thus be gained, and the pupil will engage in the work with a determination to succeed. 2. Analyzing the conditions carefnUy and relating them, to the required answer. This is what may be termed solving the problem. In the case of advanced classes it is generally better to make no use of figures at all, except in so far as they may be required to express relations. The object is to deal with the maiu thought process of the problem. 3. Stating the operations to be 'performed in arriving at the answer. This will be found to be an excellent exercise after the pupil has had sufficient time to think over the problem. It directs attention to the uses of the fundamental rules as instru- ments for serving the purposes of thought. In addition to this, it is a time-saving exercise. Probably ten problems can be dealt with in such a manner in the time taken by two or three worked out from beginning to end. At the close of the recitation, or some time afterwards, the pupils should be required to perform the operations so as to find the required result, as a quantity in its simplest form. The latter should often be an oral exercise. 4. Thinking out the solution^ finding the answer, and stating the ivhole process in written form. This is an excellent exer- cise, especially when large numbers are involved. It gives the pupil an opportunity of doing the whole work and of expressing it clearly and neatly. Tlie time given must be limited, however, so that the pupil will not fall into sluggish habits. This exercise, though valuable, should not be allowed to usurp the place properly belonging to other forms. 5. Thinking out the solution, finding the answer and stating it orally. This invaluable exercise should have a prominent place in every schoolroom in which number is taught. Mental arithmetic is highly advantageous for several reasons : (i) It J-' I 34 HINTS ON TEACHING ARITHMETIC. requires the pupil to hold all the conditions and relations of the problem in mind at one time, thus tending to give him a strong mental grasp, (ii) It leads to accurate and rapid calculation, (iii) It is the most practical in everyday life, (iv) It stimulates the pupil to put forth his best effort — an education in itself. There is far too much use made of the pencil in our schools at the present time. Twenty minutes devoted to vigorous mental work is worth more than an hour spent in leisurely performing written exercises. New principles should always be developed by means of oral exercises, using the very simplest numbers. In fact, the greater part of the whole time devoted to arithmetic should be taken up with mental work. Improvement in this direc- tion is greatly needed. 13. Educational Value of Problems. The value of problems as a means of intellectual training is so generally recognized that it is unnecessary to discuss it here at any great length. But the consideration of two or three points which are likely to escape notice may not be out of place. Number, in common with other school subjects, has its mechanical side. Fundamental facts must be remembered, processes must be learned, or there cannot be progress. In this there is always a danger that the pupil may become so confirmed in a habit of thought belonging to a lower stage of mental activity, as to prevent him from passing easily and naturally to a higher. Herein consist chiefly the evil effects of memoriz- ing facts before they are comprehended, of learning tables which are not understood, hi dealing with fractions which are not fractions at all (to the pupil), of becoming familiar with forms of expression which do not correspond to thought EDUCATIONAL VALUE OP PROBLEMS. d^ ns of the a strong culation. imulates in itself. p schools vigorous leisurely leans of fact, the ) should is direc- l-inmg IS it here •r three out of las its ibered, |In this ^firmed lental rurally kmoriz- I tables 3h are with lought activity, etc. The solving of problems is one of the best safe- guards against this, as the pupil is constantly required to relate new ideas to those already in his possession. By this means he is enabled to rise by easy gradations from one stage of advancement to the next, always making use of what he already knows. Two conditions are here assumed : (i) That the problems are suitably arranged in a progressive series. t (ii) That in solving them the pupil does the whole work for himself. But if these conditions be not fulfilled, problems may be a source of injury rather than benefit. By way of illustration take the problem, If 5 lbs. of sugar cost 30c., what is the cost of 10 lbs.? Supposing the pupil not to have grasped the idea of ratio, the problem is clearly beyond him. Although it appears simple, the knowledge which he has cannot be applied in solving it. But he may be shown how to get the answer thus : The cost of 5 lbs. is 30c. 1 lb. " } of 30c. 10 lbs. " 10 times ^ of 30c. = 60c. lems as : (i) A farmer tests his seed wheat and finds that 85 out of every 100 grains will grow. How much seed must he provide for 100 acres, allowing 1| bushels of good wheat per acre 1 (ii) A team travels at the rate of 1| mile per hour in ploughing a field of 26 acres. How many days of 10 hours each will be required for the work, supposing the width of the furrow to be 15 inches? (iii) Find the cost of 20 boards 16 w i' ; 38 HINTS ON TEACHING ARITHMETIC. f<;et long, ,14 inches wide and 1 inch thick at $2*2.50 per thousand, (iv) A macliine saws 40 cords of wood per day. How many days' work does a rectangular pile 80 feet long, 20 feet wide and 6^ feet high represent? Problems on marketing grain, purchasing goods, laying out and fencing fields and gardens, building barns, houses, etc., will be found particularly suitable for pupils of rural schools. Such problems as the foregoing say be easily defended on the ground of their practical utility. But this is not the only, or the strongest argument in their favour. The data of these problems, being closely associated with the everyday life of the boy, direct his attention to his surrorouljngs, and lead him to take a deeper interest in them, lie) i-''u^ '-linp a truer concep- tion of the world he lives in, and coi -eqiuiDtlj, of his duty in relation to it. • Much of the material employed in problems ciiOi ' ■ Ix, drawn from the other subjects of study, as geography, botany, physics, chemistry, etc. By this means the pupil's knowledge will gradually tend to become unified into a complete whole. Problems should be thoroughly practical. Power to think out relations is only one of the ends to be kept in view. Another important end is to train the pupil to do something use- ful. It must not be forgotten that the value of power depends altogether on how it is applied. Hundreds fail in life from a lack of training in doing for every one who fails merely for want of mental power. The aim should be to secure discipline as the result of doing what is in itself useful. 14. Expression. Language is the expression of thought. Number is the product of mental action. Numbering requires, in com- mon with all other forms of thought, language which appro- :|| EXPRESSION. dd priately expresses it. In order to adapt the moans of expres- sion to the thought to be expressed and to its purposes, tlie introduction of numerical symbols became necessary. This gave rise to the use of characters — letters and figures. Several systems of notation have been devised but only two of them are now generally known, viz., the Koman and Arabic sys- tems. In the expression of numbers, four things have to be taught : (i) Spoken words, (ii) Figures — Common notation, (iii) Letters — Roman notation, (iv) Symbols of operation. Owing to the interdependence and interaction of thought and language, it is imperative that the pupil, in the lowest as well as in the highest stage of progress, should be in a position to express his thought clearly and fully. Applying this general truth to the case of number, it is seen that the pupil must be able, from the beginning, to use either words, figures or letters in order to make advancement in the subject. Now the practical questions to be considered are, at what time and in what order should these different forms of expression be taught 1 Spoken words should be used from the beginning of the course. As tlie pupil becomes able to distinguish groups of two, three, four, etc., objects from one another he should be given the symbols one, two, three, four, etc., as spoken sounds. These will serve for him every purpose of thought and ex- pression for some time. Indeed, it is quite possible to make considerable progress in numbering without a knowledge of any other symbols. ' Figures may be taught as soon as the pupil has reached a true conception of number, that is, when he can think of a 40 HINTS ON TEACHING ARITHMETIC. I -;i whole quantity in relation to a unit. It must be kept in mind, however, that many vague determinations of quantities will necessarily precede the condition of thought here in- dicated. When a new symbol is introduced the pupil should be required to use it frequently until he can associate it readily with the idea for which it stands. In such exercises as these the teacher will place the figure carefully each time on the blackboard, using it instead of the spoken word. Take up 6 blocks. Bring me 6, the teacher pointing to the objects. How many ones in 6 ? How many more than 5 is 6 ? How many threes in 6 *? How many twos in 6 ? etc. It is highly important that the symbol be of as accurate form as possible. The primary teacher, above all others, should write accurately and neatly on the blackboard. The signs of operation should be presented one at a time as they are required for written exercises. As in the case of figures, they should take the place of words which are well understood by the pupil. Roman characters may be introduced gradually after figures and symbols of operation are well known. The decimal system of notation will present no difficulty until the number 10 is reached. It is best to show the pupil how to express 10, 11, 12, 13, 14, without any reference to the notation for these reasons : (i) No difficulty is experienced in doing so. The pupil is perfectly satisfied to know the how, the why does not concern him. (ii) By the time that, say, 14 is reached the pupil has gained some experience in relating these numbers to 1 0. He has a few facts at his command, and he is, therefore, in a position to make comparisons, 11=10+1, 12=10+2, 13=10+3, etc., I I. COURSE OF STUDY. il from tlieso he iniiiiediatoly gets the idea of 1 ton -{-one, 1 ten-f- 2 ones, 1 ten-|-3 ones, etc. The main thincr is, of course, to make sure that the pupil has the thought giving rise to thes<^ forms of expression. Plenty of practice should be given in both reading numbers and writing them after the system of notation is understood. The pupil should become thoroughly familiar with it as a pre- paration for performing numerical operations by means of figures. Much attention should be given to expression in arithmetic. In fact the sul)ject cannot be well taiight without this. It demands logical arrangement and continuity of thought, exactness in the use of terms and clearness of statement. This is too often lost sight of in actual practice. 15. Course of Study. Outline of Work for First Period. Numbering. A — Counting — from 1 to 20. B — Grouping — (i) Objects — from 1 to 10. (ii) Squares of equal size systematically arranged — leading up to comparison, as : DO D > DD DDDDDDD DDD DD DD DD , DD ODD DD- C — Separating and combining — from 1 to 20.* (i) E(jual groups of objects, (ii) Unequal groups of objects. m 5 See outline of plan of teochini;, Nos. 1 to 20, p. 62. 42 HINTS ON TEACHING ARITHMETIC. \k: ■« ' ii D — Addition. (i) Numbers from 1 to 20 — Addition Table, (ii) Numbers from 20 to 100 — Types of Exercises. («) 6 7 7 7 6 6 6 7 7 6 6 6 6 7 ~> ~j ~") ~> -, etc. ('>) 10 20 30 30 40 10 20 30 20 30 10 20 30 10 20 ~~") > 'j -, — , etc. (^•) 20 30 60 90 8 8 8 8 > > > — , etc. (d) 10 20 40 70 ' / 18 18 18 18 > > J — , etc. (e) 25 35 55 75 8 8 8 8 J > > — , etc. ;/) 25 35 55 75 s*^ / 18 18 18 18 ) ) > — , etc. (y) 10 20 40 60 15 15 15 15 . 18 18 18 18 ) J J — , etc. (h) 10 15 9 6 12 9 8 7 13 17 7 2 25 16 6 9 19 6 5 8 7 13 -, — , etc. COURSE OF STUDY. 48 While facility in addiug may be regarded as a highly important end in teaching addition, it is not the only thing to be aimed at. The pupil must be trained to apply his knowledge, he must comprehend, and be- come familiar with, the coniinou system of notation as a means of thought expression, and he must gain such mental jjower as will enable him to pass on to the next higher phase of relating (quantities. The question to be considered is, How are these results to be attained ? Although this question may be viewed from several stand- points, we are here concerned with only one aspect of it, viz., the character of the exercises which the pupil is required to perform. They may be indicated as follows : (i) Exercises requiring the pupil to use what he knows in discovering new facts. In («), as given above, the pupil is required to relate the known, C + 6, and C + 6 + 6, to 6 + 6 + 7, 6 + 7 + 7, and 7 + 7 + 7. After the pupil has performed the work, the teacher will question him in such a manner as to make him fully conscious of what he has done. (ii) Exercises specially intended to familiarize the pupil with the ten- unit. Every primary teacher is aware of the rapid progress that a pupil is capable of making after he is once able to relate by tens. Why? Certainly not because the ten-unit possesses any magic power in itself. The reason is simply this : the decimal system of notation is an arbitrary one, which cannot, therefore, be employed by the pupil as a natural means of expression until he has become familiar with the unit which forms its basis. But just so soon as the pupil can think in tens, the resulting unification of thought and expression manifests itself, not only in the increased power which it gives, but also in the pleasure derived from the consciousness of such power. Much practice should therefore be given in finding the sum of such addends as will direct attention to the decimal system of notation. See exercises given above. (iii) Exercises which pave the way for multiplication. Finding the sum of equal addends forms the most direct preparation for this. (iv) Exercises which are designed to emphasize particular combina- tions. This is referred to elsewhere. (v) General review exercises. Nothing should be said about "carrying" until the pupil has actually practiced it. If the exercises are properly graded and related to one another, the pupil will almost unconsciously do the work for himself. •'>' 44 HINTS ON TPJACIIINO ARITHMETIC. i if! At tli« propnr tiiiio, liowever, lio Hliould bo made fully aware of how iliu opcrutioiiH lit! has already purforinud are related to the notatioD. Before the formal process is taught the pupil should have a good knowledge of notation. E — Subtraction. (i) Numbers from 1 to 20. (ii) Numbers from 20 to 100. . — Multiplication. (i) Numbers from 1 to 20. (ii) Numbers from 20 to 100 — Types of Exercises. (a) The multiplicand a multiple of 10, as 10 x 2, 20 X 3, 20 X 4, 30 x 2, 30 x 3; etc. (6) The multiplier a multiple of 10, as 2 x 10, 2 X 20, 3 X 10, 3 X 20, 3 X 30, 4 X 10, 4 X 20. etc. (c) 12 X 2, 22x3, 22x4, etc. (d) 15x2, 15x3, 25x4, etc. (e) 2x 15, 3x 15, 4x 25, etc. (/) 24 X 2, 24 X 4, etc. The exercises should be related in such a way that the pupil will dis- cover the law of commutation. Objects should be used for this purpose ; a blocks X 5 = 5 blocks x 3. G — Division. (i) Numbers from 1 to 20. (ii) Numbers from 20 ^.o 100. Multiplication and division should be taught as related to each other. For example, 10 blocks x 3 = 30 blocks may be read as 10 blocks taken Pi bel 8h^ .; :ii J COUHSK OF STIDY. 40 .? times = 30 bloekH, nml it may he further interpretod a8 implying that .'{0 blocks contaiiiH 10 bloc-kH '.\ tiinoH, ^Multiplication an^l (livision aro complementary to each other, and they should from tlie beu:innin^ bo taught together. Kven in giving practice in the more advanceil work of a later stage it is well not to lose sight of this completely. . When a pupil has found that 1728 -f 12 - 144, he will understand it all the better for proving his answer. H— Multiplication Table— from 1 to 100. The pupil I make his own table as ho proceeds. Each number should be traced back to addition, at least for a time ; thus, 6 + 6 + ti + 6 + 6 = 30 = 5 sixes. Mkasurino and Comparino. A — Volume. (i) Comparative size of objoct.s — Estimating; and testing by means of displacement of water when possible.* (ii) Compai'ative size of rectanf^ular piles of cubical blocks, leading up to definite measurement. The blocks should be of uniform^ size. (ill) Relation of size to weight- Estimating ,nd B— Weight. testing. (i) Comparative weight of objects — Estimating and testing by means of balance when possible. (ii) Relation of weight to size — Estimating and testing. C — Surface. (i) Comparative area of plane surfaces of objects, as tables, desks, cubes, sheets of paper, pieces of cardboard, etc — Estimating and testing by superposition when possible. •I- 'I ji ""We are not certain as to the practicalDe&i of this. It is merely suggested. m i • lU If I* V i ■ if 46 BINTS ON TEACHING ARITHMETIC. (ii) Comparative area of coloured circles, squares drawn on V)lackboard. (The whole figure must be chalked over, not merely the out- line. Nothing should be said about linear dimensions here.) Estimating and testing by superimposing pieces of cardboard. D — Length. (i) Comparative length, breadth, height, etc., of objects — Estimating and testing. (ii) Comparison of length with breadth, etc. — Estimating and testing. (iii) Distance between objects — Estimating and test- ing by pacing. E— Value. (i) Measurement of sums of money as lOc, 15c., 20c., etc., using one-cent coins (afterwards five-cent coins, ten-cent coins, etc.). (ii) Comparative value of small coins. (iii) "Making change," etc. (iv) Easy problems. F— Time. (i) Comparative length of intervkls between suc- cessive acts, as ringing the bell, raising the hand, etc. (ii) Comparative length of time taken by different objects in falling, as hail and snowflakes, pebbles aiid feathers, etc. The main purposes of these lessons are (a) to give the pupil a clear conception of what is meant by voluit^e, surface, etc., so that he may be COURSE OP STUDY. 47 prepared to measure them ; (6) to direct attention to the idea of quanti- tative measurement ; (c) to train the pupil to observe closely, to measure with the eye, etc. Expression. A— Oral. (i) Names of numbers from 1 to 1 00. (ii) Language of number used in explaining or illus- trating operations performed. B.— Written. (i) Figures. (ii) Notation and numeration — Numbers from 1 to 100. (iii) Symbols of operation. (iv) Roman notation from 1 to 20. (v) Exercises. Problems. A — Addition. * (i) One-step — John has 4 marbles and James has 5 ; how many have both together. (ii) One-step — John has 4 marbles and James has 5 more than John ; how many has James ? (iii) Two-step — John has 4 marbles and James has 5 more than John; how many have both together? B — Subtraction. . , (i) One-step — John had 6 marbles and he lost 2; how many had he left ? hi ' 49 HINTS ON TEACHING ARITHMETIC. l' >r I ft J, ' i (ii) One-step — John has 8 marbles and James has 15; how many more has James than John ? (iii) Two-step — John won 12 marbles and lost 6. He then had 18 ; how manv had he at first. The problems given at this stage should be chiefly based on addition and subtraction. Until the pupil has grasped the idea of ratio he has but little power in the appUcation of the processes of multiplication and division to the solution of problems. Drill Exercises — Generally without objects. A — Adding to or subtracting from a given initial number by twos, threes, fours, etc. This forms an excellent exercise on the different combinations. B — Adding rapidly such as J ^ ^^ ^J ^J C — Adding columns. The columns should be arranged, generally speaking, so that difficult combinations would be repeated frequently. T> — Finding the value of a series of numbers connected by the signs of operation, as, 64 - 32 + 1 7 - 8 + 3 - 1 2 = 1 E — Adding columns purposely design- ed to give practice in all possible combinations within certain limits. For example, this will furnish 16 different exercises in addition, pre- senting nearly a.U the combina- tions of the first 6 numbers. F — Performing operations indicated as follows : 6 6 4 2 6 2 3 6 3 4 2 3 4 3 6 6 (i) 3 5 8 -f-14 (ii) X S i ■28 24J 20^ 12 (iii) 16 !> -=- 4 (iv) 24 = ni-r- 6x 48 -i- 3x 40^ 8x (V) 4 = J 16-^ 4x 28-=- 12 X 32 -f- 2x ll2- imes has m John ? it 6. He rst. d addition itio he has tiplication number excellent rranged, IS would Bcted by 2= ? i I I 1 2 6 3 5 /'24-f- 48-r 40-=- 28-=- 32 -f^ 12-^ COURSE OF STUDY. 4U iSuch exercises as the foregoing may be varied to any extent to suit the requirements of the pupil. They shouhl l»e performed accurately, and as rapidly as possiltle. In so far as the pupil is ahle, he should be required to relate results. Why is 24 e((ual to four tiuies 6, but cit/ht times 3? Why are 24 divided by six and 4S divided by twelve equal to each other ? It is an excellent plan for the teacher to prepare a set of number charts for the purpose of drill. Exercises carefully thought out are likely to bo much more effective than those hurriedly placed on the blackboard. The charts may be made of ordinary manilla paper, and if properly mounted they will last for a long time. Outline of Work for Second Period. Numbering — I to 1000. A — Addition, Subtraction, Multiplication and Division. Plenty of practice should ])e given in addition and subtraction ; but multiplication and division should become more and more prominent as the pujjil advances. B— Multiplication Table— From 1 to 144. Measuring and Comparing. A — Volume. (i) Measurement of quantities of sand, water, etc., using pint, quart, and gallon measures — one measure at a time. Comparison of whole quantity with the unit. Comparisons of quan- tities measured. Problems. (ii) Measurement of one standard by means of anotlier, as finding the number of pints in a gallon, quarts in a gallon, etc. Compai'isons based on such as 2 gallons = ? quarts, '2 gallons = ? pints. llelation of pint and quart without actual measurement, etc. Other standards. Problems. ^. (iii) Measurement of rectangular solids. ^ '<-^'^^ 50 HINTS ON TEACHING ARITHMETIC. ** li, ' ^li; ■ ■♦*■■■ Kectangular blocks of uniform size should be first used. These can be arranged together so as to represent rectangular solids. The aggre- gates may be measured, taking one block, two blocks, etc., as the unit, as the pupil may choose. * Comparisons will then be made. The great difficulty which is usually experienced with cubic measure arises from the fact that standard units of solidity are derived from correspond- ing units of length. For that reason nothing should be said about linear measurement, at least in this connection, until the pupil has grasped the idea of solidity, and until he is able to select units which are applicable to the measurement of solids. B— Weight. (i) Weighing quantities, using ounce and' pound weights. Comparison. Problems. (ii) Comparison of ounce, pound, hundred weight. Problems. (iii) Relation of weight to volume. A pint measure of water weiglis so much, what will a gallon weigh] etc. Problems. C — Surface. (i) Measurement of rectangular plane surfaces, using one square foot as the unit. Compari- son. Problems. j (ii) Measurement of a square yard of surface by means of the unit one square foot. Also the measurement of a square foot by the units, 1 square inch, 2 square inches, 36 square inches, 72 square inches, etc. Comparison. Problems. (iii) Relation of different standards to one another. Problems. A piece of cardboard exactly 1 foot square may be cut out neatly to re- present the unit. By this means the attention of the pupil will be fixed on the idea of surface measurement. The main thing is to lead the pupil to see that a surface can be measured only by some unit of surface. The *The Uaoher will observe ihat th^ operation Atai performed is one of aiialysis. Why? COURSE OF STUDY 01 measure form of the unit should be changed ; for instance, the square may be cut into two e(jual parts, and the parts placed so as to form an oblong 4 times as long as wide, etc. If one unit is thoroughly taught, and if the pupil once sees that it is applicable to the measurement of surface, the main difficulty of square measure is overcome. What has been said regarding cubic measuro applies equally to square measure. D — Length. (i) Measurement of linear dimensions of objects, as table, desk, floor, wall, blackboard, etc. Com- parison. Problems. (ii) Measurement of distances between objects, as two pickets placed in the ground 30 feet apart. Comparison. Problems. (iii) Relations of the standards, inch, foot, yard, rod, to one another. Other standards. Problems. E— Value. (i) Measurement of amounts of money by means of standards, as cent, 5-cent piece, 10-cent piece, 25-cent piece, etc. Comparison. Problems. (ii) Comparison of values of different coins. Prob- lems. (iii) " Making change." Problems. F— Time. (i) Measurement of time taken for the pacing of distances, the oscillations of a pendulum, etc. Comparison. Problems. A weight suspended by a thread will be found very serviceable for this work, as the time of each oscillation may be varied to suit the circum- stances, by shortening or lengthening the distance from the point of sus- pension to the centre of gravity of the weight. A clock or watch will be used to indicate the standard units of measurement, as second, minute, etc. UNIAKlU CULLtiit Uf ^UUCAII0I» IS ' J? HINTS ON TEACHING ARITHMETIC. (ii) Relation of second, minute, hour, day, etc. Comparison. Problems. Lessons on the sun dial and face of the clock will be found profitable. Factors. « A — Factors of numbers — From 1 to 144. B — Common factors of two or more numbers. Multiples. A — Easy multiples of numbers. B — Common multiples of two or more numbers. Fractions. A — Exercises on division into equal parts. B — Comparison of the whole and one of the parts, taking the part as the standard. C — Comparison of the whole and one of the parts, taking the whole as the standard. D — Easy fractional relations, as halves, fourths, eighths, thirds, sixths, ninths. i This work should not be begun until the pupil is able to form a pro- per conception of the word times; that is, as implying the ratio of one quantity to another. It is to be noted that the pupil will use this term long before it conveys to him its full meaning. He may say 3 times 6 feet = 18 feet, Just as he would say 3 times 6 apples = 18 apples, almost at the commencement of the course. There is a long step between 6 feet X 3 =- IS feet, as meaning 6 feet + 6 feet + 6 feet = 18 feet, and 6 feet X 3 = IS feet, as implying that IS feet is a quantity 3 times as great as 6 feet. In the former case the unit may be regarded merely as one of the like things which make up the whole, while in the latter it must be regarded as a measured quantity. The teacher should satisfy himself that the pupil has actually made a comparison of the quantita- tive values of G feet and 18 feet. When the pupil has done this, how- ever, he has the thought in mind which finds one form of its expression by moans of a fractional number. COURSE OF STUDY. 1^ Expression. A — Notation and Numeration — 100 to 1000. B— Roman Notation— 20 to 100. Problems — Types. A — Counting by fives and tens. (Introductory.) (i) I bought 20 yards of cloth at one store and 30 at another. How many yards did I buy? How much more is there in one piece than in the other? If I sell lx)th pieces at $2 per yard, how much shall I get for them 1 I divide the money equally among 20 men, how much does each get 1 If there had been twice as many men how much woiild each have got 1 (ii) John sold 20 quarts of milk in the forenoon and 30 pints in the afternoon. How many pints did he sell altogether 1 How much more did he sell in the forenoon than in the afternoon 1 At 5c. a quart, how much money did he get 1 If John's money is made up of 5-cent pieces, how many has he 1 How much cloth can he buy with it at 25c. per yard 1 B — Counting by sixes and twelves. (Introductory.) (i) In 1 2 yards how many feet ? (ii) In 48 feet how many yards ? (iii) Measure the length of the table, using a foot rule. What is its length in inches ] (iv) In 48 yards how many 6 feet measures ? (v) What is the value of 18 lbs. of rice at Gc. 1 \ y How much butter at 12c. could be bought for the same money ? 54 HINTS ON TEACHING ARITHMETIC. lis 3 I, ! T 1' I C — Goricnil Exercises. (i) How many quarts in 3 gallons ? In 3 gallons and 3 quarts 1 Compare 3 gallons and 3 quarts. At 5c. per quart find the value of 3 quarts. Find the value of 3 gallons. Com- pare 15c. and 60c. with 5c. How many times is 15c. contained in 60c. 1 How much greater is 60c. than 15c. ? How many times greater? At 15c. per hour, how long would it take to earn 60c. 1 (ii) How many yards in 18 feet? In 19| feet? In 72 inches? (iii) I had 15c. and I got 45c. more. How mfiny pencils at 5c. each did I buy with one-half of my money? I spent the other half in buying oranges at 3 for lOc, how many did I get? I sell the oranges at 5c. each. How nmch do I get for them ? How much must I add to what I have in order to make it $1 ? How many oranges at 3 for 10c. can I buy for .^1 ? How many at 3 for 5c. ? Why are there twice as many oranges in the latter case ? In giving problems, the following points should be noted : 1. The problems should usually be related to the exercise preceding them. 2. They sho:ild be related to one another. It is generally profitable to give a series of problems bearing on one another and developing one main idea. 3. They should call forth the pupil's best effort. Problems or other exercises which are too easy are useless. Of course the opposite error must be guarded agamst also. COURSE OP STUDY. 55 4. They shouM be carefully graded according to degree of difficulty. In the foregoing, the primary object is to give the pupil a clear conception of a measured unit. This is done in two ways, (a) by presenting the unit so that attention may be directed to it as a measured quantity, (h) by requiring the pupil to mea- sure suitable aggregates by means of it. All measurements should be performed as accurately as possible. It is not the amount of work done that counts most for either the purposes of number or the formation of right habits, it is the character of the work. Whenever a measurement is made, it should become the basis of a series of questions, leading the pupil to make comparisons in so far as he is aVjle to do so. The pupil should be allowed to use one unit until he has a good working knowledge of it, before another is introduced. New units should be related to those already known. Sup- pose, for example, that the pupil has become familiar with 1 foot as a unit of measurement by using it in determining such lengths as 12 feet, 16 feet, etc. Let him measure off say 12 feet, place in his hand a yard stick, saying nothing about its length, and ask him to measure the distance of 12 feet with it. He finds that the yard measure is contained 4 times in 12 feet. Question him so as to lead him to compare the length of the new unit with that of the old one. This is not by any means the easiest way to get the pupil to see the relation between the units, but it may be the best way, provided he has gained sufficient power to make the necessary comparison. But if he has not the power, it would be a serious mistake to try to force him to a conclusion. The stimulus should be as great as the mental condition of the pupil will warrant, but no greater. f if 56 HINTS ON TEACHING AlllTIIMPlTIC. OuTiJNK OF Work for Tniiin Pekiod. Fundamental JIules. A — Addition, Subtraction, Multiplication and Division. Practice to secure accura(;y and rapidity. B— Multiplication Table. From 1 to 400. C — Factors and Multiples — oral exercises. Weights and Measures. , A — Tables as required for exercises. B — Transformation of denominate units. (i) Simple — one denomination. (a) Reduction descending ) . Relate to each other. (6) Reduction ascentling ) («) Problems — oral and written. (ii) Compound — two or more denominations. (o) Reduction descending (A) Reduction ascending (c) Problems — oral and written. Weights and measures iiarison of lengths t>f lines, etc. (r) Problems — oral and written. (ii) Suiface. (a) Area of rectangles — land measurement. (b) Comparison of are«is of rectangidar sur- faces. (c) Problems — oral and written. (iii) Volume. (tt) Measurement of rectangular solids. (b) Comparison of volumes of different solids. (c) Problems — oral and written. (iv) Practical applications. (a) Fencing, ditching, tree-planting, etc. Ill I 0S HINTS ON TRACHINO ARITlIMETIfJ. (h) Carpt^tiiig, pHixM'iiig, painting, phwtoring, otc. (c) MeaHureraent of land. (d) Measurement of lumber. (e) Measurement of cordwood, stone, stonework, brickwork. (/) Capacity of rectangular tanks, bins, etc. (g) Problems. In all of this practical work the pupil should find his own data by actual uieasuremeut as far as possible. F — General review exercises and problems. Measures. A — Integral factors as measuroft of quantities. (i) Denominate units (siini)l(>). Exercises, (ii) Abstract units.* Exercises. B — Common measures of two or more quantities, (i) Denominate units (simj)le). Exercises, (ii) Abstract units. Exercises. C — Prime factors. D — Greatest common measure of two or more quantities. (i) By means of resolving numbers into prime factors. (a) Exercises. (6) Easy problems. t (ii) By means of the formal process. ' (a) Exercises. (6) Easy problems. • See imge 7. COURSE OP STUDY. 59 E — Gonoral roviow exerdaes and proV)lomH. Multiples. • A — Quantities which can ho niea.surod exactly by other ({uantities. (i) Denominate units (simple). Exercises on divisors. (ii) Abstract units. Exercises on divisors. B — Common nmltiples of two or more quantities, (i) Denominate units (simple). Exercises, (ii) Abstract units. Exercises. C — Prime factoi's. Tleview. D — Least Common Multiple of two or more quantities. (i) By means of i-esolving numbers into prime fac- tors. Exercises. (ii) By ni(\'ins of formal process. See Hamblin Smitli's Arithmetic, p. 48. Exercises. Easy problems. E — General review exercises and problems. Fractions — Common. A — Exercises on the comparison of quantities. (Review.) B — Division of units into equal parts. (Review.) C — ProliminMty exercises — no formal process. ^ 'omparison of the fractional unit with tlie prime unit, as j j j is 4 times also is \ of i j j ; 12 inches rr 4 times 3 inches, also 3 inches = | of 12 inches; 5 feet 5 times 1 foot, also 1 foot = J of 5 feet, < 60 HINTS ON TEACH INO ARITHMETIC. It 1 t'. t: ;» Such complementary thoughts must be kept before the pupil in order thut thu true idea of a fraction may be developed. It is one thing to learn to manipulate fractions, but a quite different thing to think of the relations which fractions express. (ii) Comparison of different fractional units with a common prime unit and then with one another, as 6 inches = | of 12 inches = 3 inches x 2 =: f of 12 inches = 2 inches x 3 == g of 12 inches. Therefore, J = f = g. Elxercises. (iii) Comparison of multiples of the fractional unit with the p;'ime unit, as 12 inches = t>vice 3 inches x 2, also twice 3 Inches = ^ of 12 inches.* Therefore, twice ^ = h. Exercises. (iv) Comparison of equal parts of the fractional unit with the prime unit, as 12 inches =12 times J of 3 inches, also ^ of 3 inches = ^^ of 12 inches. Therefore, l^ of | = j^g- Exercises. The pupil should be given many exercises such as chose indicated above before the formal teaching of the subject begins. D — Formal study. ^ (i) Notation. I (ii) Changing from one form to another. (a) Mixed numbers to improper fractions. (b) Lnpropcr fractions to mixed numbers. (iii) Changing from one denomination to another. (a) Equivalent fractions having different denom inators. (b) Equivalent fractions in lowest terms. (c) Equivalent fractions having a common denominator. Comparison. *It is here nssumed, of course, that 3 inches = J of 12 inches is well known. Tnu CMmes under (i). COURSE OF STUDY. 61 Tnu (iv) Addition, subtraction, multiplication and di- vision. (v) Complex fractions. Easy exercises. (vi) Practical problems. E — General review exercises and problems. Fraciions — Decimal. A — Notation. Relation to the Arabic (decimal) system. Reading decimals. Exercises. B — Changing from decimal to vulgar fractions and ince versa. Exercises. C — Addition, subtraction, nmltiplication and division of decimals. D — Practical exercises. Problems. E — General review exercises and problems. Fractions — Percentage. A — A particular case of fractions. ^ (i) Meaning of the term. Illustrations. (ii) Relation of percentage to decimal and common fractions. (iii) Exercises and problems. B — General application — Exercises and Problems. C — Particular applict^tions. (i) Trade discount. ♦ (ii) Insurance, taxes, etc. (iii) Commission. (iv) Profit and loss. C2 HINTS ON TEACHING ARITHMETIC. y < (v) Simple interest (vi) Bank discount, (vii) Partial payments, (viii) Compound interest, (ix) Stocks, (x) Exchange. D — General review exercises and problems. Proportion. (i) Simple, Exercises and problems, (ii) Compound. Exercises and Problems. Involution. Evolution. (i) By inspection. (ii) Formal process, (iii) Application — Easy exercises. General Review. 16. Plan of Teaching Nos. 1 to 20.* , Two. (a) 2=1 + 1; 2-1-1=0; 1x2=2; 2-^1=2. Three. P 3=1 + 1 + 1; 3-1-1-1=0; 3=1x3; 3+1=3. Four. (c) (i) Counting — forwards and backwards ; 4^=1x4; 4 + 1=4. (ii) Forming groups of two. (iii) 4=2 + 2; 4-2-2=0; 4=2x2; 4+2=2. * In onler to save H\vwii mich combinations iw 3 f 4, r>+2, 7 — 2, 11 — 6, et«., are hero oniittcti. TheHe are to be fully dealt witii lus explained elsewhere. It will be understood, of uounte, that concrete uniUi are to be dealt with at the beginning. See page 18. PLAN OP TEACHING. 63 I, are here lith ut the Five. (d) (i) Counting — forwards and backwards. (ii) 4+1; 2 + 2 + 1. Six. (e) (i) Counting — forwards and backwards ; 6=1x6; 6+1 = 6. (ii) Forming groups of three. (iii) 6=3 + 3; 6-3-3=0; 6=3x2; 6+3=2. (iv) Review (c) iii. (v) 6=2 + 2+2, 6-2-2-2=0; 6=2x3; 6+2=3. Seven, if) (i) Counting — forwards and backwards. (ii) 6+1; 3+3+1. Eight. (g) (i) Counting — forwards and backwards. (ii) Forming groups of four, (iii) 8=4 + 4; 8-4-4=0; 8=4x2; 8+4=2. (iv) Review (c) iii and (e) v. (v) 8=2 + 2 + 2+2; 8-2-2-2-2=0; 8=2x4; 8+2=4. ^ine. (h) (i) Counting — forwards and backwards, (ii) Review (b) and (e) ii. (iii) 9=3 + 3+3; 9-3-3-3=0; 9=3x3; 9+3=a Ten. (i) (i) Counting — forwards and backwards. (ii) Forming groups oi five. ' (iii) 10—5 + 5; 10-5-5=0; 10rrr5x2; 10-t-6==2. (iv) Review (c) iii, (c) v and (j/) v. (v) 10--2 + 2 + 2 + 2 + 2; 10-2-2-2-2- 2-0; 10 2x5; 10+2=5. ;[ 1^9 I tr; 64 HINTS ON TEACHING AlUTUMETIO. ' Eleven. (j) (i) Counting — forwards and backwards, (ii) 10 + 1. (iii) 5+5 + 1. Twelve. (k) (i) Counting — forwards and backwards. (i1) Review (c) i. (iii) 12:^0 + 0: 12-0-6=0; 12=6x2; 12+6=2. (iv) Review (c) i and (y) ii. (v) 12=4 + 4 + 4; 12-4-4-4=0; 12=4x3; 12 + 4=3. (vi) Review (/>), (c) iii and (h) iii. (vii) 12=3 + 3 + 3 + :i; 12-3-3-3-3=0; 12=3x4; 12 + 3=4. (viii) Review {efore him. Beginning with counting (by ones) as tiie fundamental process in numbering, the pupil is led step by step to detine aggregates by means of numbered units, thus pro«<>om- mitting to memory facts that are thoroughly under- stood. One of the weaknesses of primary number 6 66 HINTS ON TEACHING ARITHMETIC. I-Sf'i i >: " teaching consists in drilling mechanically on facts which, in so far as the pupil is concerned, have never passed through the thought stage. The proper order is, first comprehending the idea, second, mastering the idea hy making use of it frequently, third, drilling on it to secure perfect familiarity with it — the mechanical phase.* To reverse this order is to put the cart before the horse. (iv) By the tinie 20 or 30 is reached the pupil will have acquired some grasp of the subject. He will then Ihj in a position to make a fuller analysis of the lower numbers. This work should be carried on in no haphazard manner, but according to a well designed plan, leading the pupil to connect each new idea with others already in his possession. For example, such exercises as these taken in the order given are comparatively useless for the purposes of teaching : 6-1-7, 5 + 4, 14 + 8, 9 + 11, 23-8, etc. The pupil cannot see any connection which one part has with another, therefore the exercise can have little value except as a test of what is known. How different is the effect when a lesson is arranged so as to develop one leading thought, as for example : 6+7, 17 + 6, 23-7, 16 + 7, 33-6, 37 + 6, 16 + 17, 23-17, 33-16, 43 - 17, 2(i + 27, 43 - 26, and 37 + 26. This exercise is valuable because by performing it (a) The pupil has been led to relate numbers to one another as 6 + 7, to 16 + 7, etc., and he has consequently gained power. (6) The combination 7 + 6 means more to him than it did before, for he has made a wider application of it. (c) He has made progress in addition and subtraction. *Stre.s8 should belaid on the first and s<*cond of these rather than on the third. Drill exercnaes, although quite iiecesHury, if carried beyond a proper limit have a dead- ening effect. fLAN OP TKACHINO. 6? hat is it did third, bdead- (fl) llo lias don^ at least something towards forming the habit or relating one thing to another. The numbers given above extend beyond the stated limit of 20 or '^0 in order to show how the primary lessons may be connected with those more advanced. The pupil should l)e encouraged to discover different units in numbering an aggregate. When 16 blocks, say, are presentwl for the first time it is best to simply ask how many there are. Altliough an order for the selection of units is laid down in the foregoing outline, it is nijt intended that such order should l)e forced upon the pupil. One-half of the value of the lesson will be lost if he is not allowed to determine units for himself. In fact very much of the difficulty of arithmetic consists in recognizing units as being applicable to the measiu'ement of quantities. (v) At the same time that the analysis of the lower numbers is being carried on there will be progress ma«le in connection with the higher, proceeding as before, from outline to detail. The l>upil will pass gradually from 20 to 100, counting by tens, then by fives. The relations established between numbers under 20 should be inunediately applied to those above it. Whenever the pupil learns that 7 -|- 3 = 10 he should be re- (juired to use this knowledge in so far as he can. He will have a better grasp of 7-\-'.] when he has related it thus, 17-J-3, 27 -f 3, etc., than he could possibly have had Ix^fore doing so. Every new idea must be used in order to be fully mastered. The exercises in multiplication and division indicated above should be based on correspontling ones in addition and subtraction. The latter processes will therefore preccnle the former. It is not the intention here to advocate a return to the old metluxl of teaching tlu^ fundamental rules, either inde- pendently of one another, or in consecutive order ; but it is 68 HINTS ON TEACHINO ARITHMETIC. the intention to advocate their presentation in accordance with the logical sequence of their development. The pupil should be able to add 3 pears, 3 pears and 3 pears readily before he is asked to perform even the additional operation of counting the three addends. If this is true as to the lower, what must be said with reference to the higher phase of multi- plication? The same principle applies in the case of sub- traction and division. The prevailing practice of teaching addition, subtraction, multiplication, division and fractions (?) simultaneously, from the very commencement of the course, is based on the fundamental error of assuming that the pro- cesses of thought which these involve are co-ordinate rather than correlative. The course here outlined is, for the most part, suitable for the average pupil of 7 years of age. Any work in number df)nq with pupils below this limit should be very elementary imleed, such as counting and forming small groups of objects, stick-laying, drawing, etc. No regular number wt)rk should be prescribed for pupils below the age referred to for two reasons : (i) They have not, generally speaking, attained to the matur- ity of thought necessary to enable them to deal with the subject intelligently or profitably. (ii) There is plenty of other school work which is well suited to them — reading, drawing, nature study, oral composi- tion. Little children delight in these, because they can be adapted to their mental condition. 17. Seat Work for Junior Glasses. A — Objects. (i) Arranging shoe-pegs, sticks, etc., into groups of two, three, four, five, etc. SEAT WORK FOR JUNIOR CLASSES. 69 (ii) Laying sticks or pegs so as to roproscnt t'orms placed on })lackboard, arranging IkjIow each form the number of objects used, thus : t AD HI OD H O^^^ III nil mil mill iiiiiii iiiiiiii iiiiiiiii inmiiii (iii) Arranging sticks, pegs, etc., by threes, fours, fives, etc., so as to represent as many different forms as possible, thus : (iv) Arranging sticks, pegs, etc., into groups of ones, twos, threes, fours, etc., numbering the groups in consecutive order, thus : I I I I I I II I I II 1 I I A A AA AA AAA AAA AAAA AAAA AAAAA AAAAA (v) Forming bundles of toothpicks, etc., by tens, and afterwards making up such numbers as 16, 26, 36, 46, etc., by adding, and also by taking away. 70 HINTS ON TKACHING ARITHMETIC. I* I' m m (vi) Illustrating by means of pegs, trjotlipicks, etc;., operations -indicated on blackboard, ilius ; (See No. iv.) 1+0 =? 1 + 1 =? 1+1+1 =? 1+1+1+1 =? 1+1+1+1+1-? 1x1 = ? 1x2 = ? 1x3=? 1x4 = ? 1x5 = ? 2+0 ^__ ( f 3+0 -if 2 + 2 »' 1 3+3 -r 2 + 2 + 2 t 1 3+3+3 =? 2+2+2+2 f f 3+3+3+3 =? 2+2+2+2+2 = 5 f 3+3+3+3+ 3 = ? 2x1 = ? 3x1 = ? 2x2=? 3x2 = ? 2x3=? 3x3=? 2x4 = ? 3x4=? 2x5 = ? 3x5=? (vii) Illustrating by means of blocks operations already performed on slate, thus : 5x4—4x5 6x4 — 3x8 5 X 4 = 10 X 2 nnnnn nnnnn nnnnn nnnnn DDD DDD DDD nan DDD DDD DDD nnnnn nnnnn nnnnn nnnnn (viii) Laying one-inch, two-inch, three-inch, etc., sticks end to end, so as to represent lines of given length as 6 inches, 1 foot, 1 foot 6 inches, etc. Comparison. (ix) Finding the measurement of sticks numbered 1, 2, 3, 4, etc., by means of one-inch, two-inch, etc., measures and writing results on slates. Thus stick No. 4 contains the two-inch measure 8 times, therefore it is 2 in. X 8 == 16 inches long. Comparison. SEAT WORK FOR JUNIOR CLASSES. 71 (x) Defcerniinint» tlio Icnj^ths of sticks l»y placing small measures end to end heside them. Thus stick No. 7 requires 6 ft»ur-irich measures to make up its whole l(ui<(th,it is therefore 4 in. X 6=24 inches long. Questions as, how many 8 inches contained in the length of stick No. 7 1 How often must a threcNinch measure be rep(5ated to make up its length, etc. Comparison. (xi) Laying sticks of measured length so as to en- close 2 sq. inches, .3 s«|. inches, 6 sq. inches, 1 sq. foot, etc. Corresponding oral or written statements. Comparison of one unit, 2 units, 3 units, etc., with the whole area enclosed. The unit of surface must be made clear to the pupil. (xii) Laying sticks of measured length so as to enclose a stated area, giving it different forms. (xiii) Forming cubes and other rectangular solids of given size by means of cubical blocks. Com- parison. In all of the preceding either the idea of counting or mea- suring should be prominent. B — Representation of objects. (i) Representing on slates, pegs, sticks, blocks, etc., by twos, threes, fours, etc., as : D DD DDO DD DD DDD 72 HINTS OX TEACHING ARITHMETIC. (ii) Copyin;^ from lilacklxMinl dniwiu'^s, ilhist rating' o{M;niti<)iis, tlie pupil (l(>U'i'iiiiiiiii<{ tin; li^'lil- hund side of tlie equation, as . AA-^AA=? AAA+AAA=? TTTT-^TT=? HHHH-HH=? i (iii) Using figures in addition to that given in No. (ii), as : nnn + iiiin = ? 3 + 3 = (iv) Illustrating exercises, problems, etc. (first per- formed with figures), by means of circles, • dots, squares, etc., as : 5 + 4=9 4+4+1=9 4x3=3x? • I =M00TS 9» • • • DOTS ! • • ! • • • • tl 5+5+4= 5+5+5+5+5= 6x4=4x? •:l I' 1: 1 (v) Measuring lines drawn on slate by means of sticks, etc. Comparison. flFAT WOIIK FOR .TtTNtOn OfiAflSKft. n X? X? [ns of (vi) |)r.iAviiii,' oil slut(? s(|U)in's and rcctan^losof ^ivcn livvn. CoiiipariHuii of arttiiH. (vii) KoproH(Mitin^ rectanfflos of given area in different fonuH, etc. Comparison. C — No objects.* (i) Addition. (a) Adding l)y ones, twos, threes, fours, etc., the results being expressed by figures, as : 2, 4, fi, 8, 10, 12, 14, K;, 18, 20, 22, 24, etc. 4, 8, 12, Ifi, 20, 24, 28, 32, 36, 40, 44, 48, etc. 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, etc. 16 ft., 25 ft., 35 ft., 45 ft., 55 ft, 65 ft, 75 ft, 85 ft, etc. 13 pts. , 33 pts. , 53 pts. , 73 pts. , 93 pts. , etc. 25 ds., 50 ds., 75 ds., 100 da., etc. (h) Adding nund)ers arranged in columns, as: 4 14 24 7 17 27 4 4 » 4 » 4 > 4 4 — , etc. 9 19 29 39 49 59 8 > 8 « 8 8 > 8 » 8 — , etc. 9 19 29 39 49 59 9 9 9 9 9 9 a » 8 8 » 8 8 » 8 — , etc. 9 19 19 27 39 49 9 19 19 19 19 19 8 8 18 18 18 18 8 8 18 18 16 17 __ __ __. ftf/> ,,,,,, V\Aj. * Standunl uiiiUs of iiieaiiurc'iuent, utt ft., .yd., )>t., wk., etc;., should Ite kept proiui- nently before the pupiL >■■. 74 HINTS ON TEACHING ARlTHJTETIC, (c) Adding nura))ers connected by signs : f i 2+2=? 4+4=] 8 + 8 = ? etc. i 2+3=? 3+4=? 8 + 9 = ? etc. !l: G+4 = ? 16+4 = ? 2vS+4 = ? 10+9+1 =? 20+ 9+ 1 = ? 30+ 9+ 1 M 7+7+8 =? 7-f- 7+ 8 + = ? 7+ 7+ 7+ 7 + =? 7 + 8 + 9 + 10=? 10 + 11-12+13 = ? 14 + 15 + 16 + 17 + 18-? {d) Problems based on preceding exercises. (ii) Subtraction. (a) Subtracting })y ones, twos, threes, fours etc., from a given initial number, as: 12, 10, 8, 6, 4, 2. 30, 28, 26, 24, 22, 20, 18, 16, 14, 12, etc. 100, 96, 92, 88, 84, 80, 76, 72, 68, 64, etc. 150, 140, 130, 120, 110, 100, 90, 80, 70, 60, etc. 165 ft., 160 ft., 155 ft., 150 ft., 145 ft., 140 ft., etc. ^ (b) Finding the diflterence between numbers arranged in columns, as : » 18 28 38 48 58 6 6 6 6 6 6 16 25. 35 45 55 65 7 7 7 17 17 27 — "~"i m 35 25 35 45 • # » 18 18 18 172 172 172 172 172 4 14 24 34 44, etc. , etc. , etc. , etc. ___^d£^^ SRAT WORK FOR JUXTOR OLASffES. 75 (fi) Perff)nning operations indicated by signs, as: 9- 6 = ? 19- 6 = ? 29- 6 = ? 39- 6 = ? etc. 13-7 = ? 23-7 = ? 33-17 = ? 43-17 = ? etc. 75 = 15 + ? 75 = 25 + ? 75 = 45 + ? 75 = G5 + ? 40 30 = ? 34=? 32 = ? 20 = ? 24 = ? 22 = ? 10=? 14 = ? 12 = ? 125- 5 = ? 15 = ? 35 = ? 06 = ? 75 = ? 105 = ? 115 = ? 125 = ? . ((l) Problems based on preceding exercises. (iii) Aiidition and Subtraction. (a) Adding to and subtracting from a given numlmr, by twos, threes, fours, fives, etc., as: 50 90 55, ()0, 05, 70, 75, 80, 85, 90, 95, 100. 45, 40, 35, 30, 25, 20, 15, 10, 5, 0. 99, 108, 117, 12G, 135, 144, 153, 162, 171, 180. 81, 72, 63, 54, 45, 36, 27, 18, 9, 0. When pupils are sutticiently advanced they should be required to account for difierences, as, 99-81, 108 - 72, 117 - 63, etc. (b) Performi ng operations iixdicatef ; (2) to familiarise the pupil with the signs of operation. 76 HINTS OS TEACHING ARITHMETIC The former of ihoRe is too often lost Hij^ht of. A proper time-limit should be set for the performance of all such uxerciscH. (c) Problems involving addition and subtraction, (iv) Multiplication. (a) Making tables, as : 2x 1= 2 2x 2= 4 2x 3= 6 2x 4= 8 2x 5 = 10 2x 6=12 2x 7 = 14 2x 8 = 16 2x 9 = 18 2x10 = 20 3x1= 3 3x2= () 3x3= 9 3x4 = 12 3x5 = 15 3x6 = 18 4x1= 4 4x2= 8 4x3 = 12 4x4 = 16 4x5 = 20 5x1= 5 5x2=10 5x3=15 5x4 = 20 (/>) Multiplying numbers, as : 2 2 2 3 2 4 2 5 2 6 2 7 2 2 8 9 6 6 6 6 6 6 6 2 3 4 6 2 10 4 1 4 4 4 4 4 4 4 4 2 3 4 5 6 7 8 9 10 3 3 3 3 3 3 3 3 3 2 3 4 5 6 7 8 9 10 6 8 9 10 6x1= 6 6x2 = 12 6x3=18 C c 10 10 10 10 10 10 10 10 10 234^^6789 10 SEAT WOIIK FOR JUNIOR CLASSES. 77 imit Bhoukl 16 16 16 16 16 16 16 16 16 23456789 10 20 20 20 20 20 20 20 20 20 23466789 10 26 26 26 26 26 26 26 2() 26 23456789 10 666666 6 (> 6 20 30 40 50 (W 70 80 90 100 In such exercises the true idea of multiplication should be kept in mind, viz., the ripetition of the multiplicand so many times. For that reason one nn'^ipiicand (unit) should be kept before the pupil for a time. The exercit»> .. . be varied by changing the mu/dplvr. Tlie pupil should be rt. * >.>•<'■ to compare products in so far as he is able, and ac- count for their relation to one another. When the viultiplicaml is changed the new one should bear to the old a relation which the pupil can appreciate. For example, 2 might be followed by ^ or ti, 7 by 1 4 or 21, etc. (c) Performing operations indicated by signs, as : 5x4-? 4x8=2x? 1 = ? 2 = ? 3 = ? 4 = ? 25x 5 = ? 6 = ? 7 = ? 8 = ? 9 = ? {(i) Problems based on preceding exercises. 4x3 = ? 3x4 = -l 4x 8x3=? X 6 3x8= ^ 6 X ? 8 X 3 = ? 3 = ? 5 = ? 6 = ? 6 = ? 6 = ? 4x 8 = ? 8x 8 = ? 9--? 9 = ? 10 = ? 10 = ? 11 = ? 11 = ? Hi 0. 78 HINTS OS TEACHING ARITHMETIC. (v) Division (a) Making tables as 60- -10= 6 1 = = 72 GO- - 6 = 10 2- = 36 60- - 5 = 12 3 = 24 00 - 4 = 16 *?o • 4- = 18 60- - 3 = 20 <«i^- iy. = 12 60- - 2 = 30 8- = 9 60- - 1 = 60 9- = 8 * 12 = = 6 (b) Dividing nunil)ci> I, as : • 2)12 2)16 2)20 2)24 2)32 4)12 4)16 4)20 4)24 4)32 2)48 2)72 2)1)6 2)144 2)i88 4)48 4)72 4)96 4)144 4)288 6)48 6)72 6;96 6)114 6)288 8)48 8)72 8)96 8U44 8)288 2)108 3)108 4)108 6)108 12)108 (c) Problems based on preceding exercises. (vi) Multiplica.tion and Divisioc. (i) Exercises as follows : 12 ft. X 3= 18 ft. X? 12 ft. X 4 = 4 ft. X? 12 ft. x 6=3 it. x ? 12x6 = 144-r? 12x6-r3 = 12x? 12-i-6x8 = ? x 2. 100x2-=-50=16x2-f? 100-^2x8=? x2x4. (6) Problems involving multiplication and di- vision. (vii) Iteview exercises, involving the four hiutple rules. Problems. SEAT WORK FOn JUNIOR CLASSES. 79 Seat work may l)e excecMlina^ly profitable, or it may be just the opposito. This will (h'peiid on the circumstanoes umlrr which it is carried oti. The following are the principal con- ditions for successful work : (i) The exercises should Ix^ properly graded. (ii) They should be related to one another, so that what is taught U>-day may be iiscul to-morrow. (iii) Each exercise should, as a rule, develop e remembered, is that which grows out of the study of a subject for its own sake. Cards containing written exercises will ])e found convenient — especially wIumi there is lack of blackbo;ird space. They may be |)repared by the teacher alter school hours, thus saving time during the day. Kach c;ird should emphasize one main thought, /.fter the cards are prepare«i they should be num- bered, indexed, and filed, so as to l>e easily n'tVii'.'d to. ProV)lems should be related to oth»M- exercises. A few types are here j)resented. They may suggest one ov two Uiiefui lines of work. 1. John has S apples anenciKs, lost 2 an«• 44 88 10 to 110 55 f 55 55 55 55 G6 55 ti6 16 55 88 15 55 .88 1(»5 1(;5 418 (iv) Two column addition. Add the numbers 15, 65, 47, 84, thus : 15, 75, 80, 120, 127, 207, 211. This is an excellent exercise for senior pupils. e sum as 18. Ex- different . ^ 6, 7, i5, 47, iellent PRACTICAL SUGGESTIONS. 83 (v) Testing results. ' 28+ 20+ :)0+ :u 32+ 3:{+ 34+ 35 3(>+ 37+ :W+ 39 40+ 41+ 42+ 43 44+ 45+ 40+ 47 48+ 4S)+ 60+ 51 228 + 234 + 240 + 240 118 134 160 1(>6 182 198 948 342 986 735 429 834 927 4253 20(>3 2190 4263 (vi) Prove by casting out tlio 9's. This is the inetluxl generally enjployi;d by accountants. Much practice should he given in adding columns. L) Casting out the 9's. It is not \v(;ll to spend much time in trying to g(^t the pupil to comprehend the principles underlying the formal process until after he has acijuired a knowledge of the relations of denominate units. The uae of the process must be thoroughly understood. Subtraction should be taught in its relation to addition. Multiplication. (i) Exercises leading up to formal process. 600x8 = 4800 60x8= 480 6x8= 48 666 8 48 480 4800 666 8 4800 480 48 666 8 6328 6328 6328 5328 PRACTICAL SUGGESTIONS. 85 nhy the •r better, 2=6.) siuaimler lie pupil pi'()C«'SS ktions of Di-oughly bion. |66 8 28 (ii) III tliu following, coiiqNiro the multiplicuid Hiid the (liH'ereut proerf()rming operations hut they should also give increased power to relate quantities. Oliarts will b« found valuable. ^ :\ \ V 4 6^ i '^ :^^ t<*/ t/u 86 Hints on teaching ARiTHMETrc. (c) {d) ^2.25 448 1800 900 900 $1008.00 ^11.28 75 5610 789« $840.00 ^896 $112 $1008 $1128 $ 282 $840 Short methods should not be dealt with until the pupil hcas compre- hended the ordinary proce.socs fuUj'. At the proper time, however, such a problem as this will furnish an excellent means of discipline. Find the cost of 1728 cords of wood at $1.()'2^ per cord. (Short method.) The pupil must, of course, devise the method himself. The importauce of the law of commutation \\-\\\ be readily se-.n in this connection. Division. (i) Exercises leading up to short division. 0-7-2= 3 00-=-2= 30 600 -r- 2 =300 3)60 3)006 3)078 V Find the greatest nurabrr of hundreds, tens, and ones, exactly divisible bj 4 in 80, 84, 85, 92, 95, 440, 560, 564, 507, 648, 650, etc. Questions and exercises. (ii) Exercises leading up to long division. (To be performed by inspection.) 18-M5 = ? 22-f-21 = ? 36h-15=' 25~21 = ? 45 -f 15 = 2 42-r-21 = ? tllACTICAL StGGEStlONS. 87 compre- ver, such 5. Find method. ) portauce a. [ ones, 4, 5G7, formed 50-M5 = ? 60-M5 = ? 6C-M5-? 90-M5-? 100^15 = ? J50-^J5=? ICO H- 15 = ? 484-21 = ? 50-f-21 = ? 84-h21 = ? 90-: 21 = ? 100-^21 = ? 210-:- 21 = ? 220^21 = ? 425-=-21 = ? Find the greatest number of tens exactly divisible by 25 in 250, 255, 422, 525, 7G0, 840, etc. Practical methods. 39r))G5923(l(;G 3<>G (a) 2G32 2373 25G3 237G 39G)G5923(166 2G32 2563 187 187 (b) Such exercises as these will be found useful : (o) Divide 1342 by 90 (factors). (^) " " 21 (factors). (d) " - 226. ' (e) " " 33J. i Such as the foregoing are suitable for pupils who have made considerable ad vancenient in the subject. 88 HINTS ON TEACHING ARITHMETIC. I M m (c) Divide 1434227 (1) by 999; (2) by 998. (1) (2) 1434.227 1434227 1436 434 1 2 662 1437 808 101 This method cannot be regarded as very practical. It is applicable only in the case of divisors a little less than a power of 10. Exercises in factoring and finding multiples should accom- pany work in multiplication and division. These exercises will greatly increase the pupil's knowledge of number. They may take such forms as tiie following : (a) Find by inspection three factors of 64, 72, 96, 108, 210, 450, etc. (b) Find three multiples of 15, 20, 25, 75, etc. (c) Divide 12x3x5 by 15; divide 12x32x64 by 3x8x 16 ; divide 49 x 50 x 39 by 13x15x21, etc. If the exercises are well chosen the pupil will soon discover that the work may be shortened by cancellation. In teaching the simple rules, care must be taken not to allow the exercises to degenerate into performing merely me- chanical operations. For example, the pupil must understand when he multiplies $10 by 12 he has either combined 12 addends of $10 each or he has found an amount 12 times as great as $10. The latter conception is the higher, in that it involves a comparison of the unit $10 with the amount found. The aim should be to present the simple rules so that the pupil will gain a true notion of number as expressing the ratio between the unit and the whole quantity. Until this stage of thought is reached, multiplication and division cannot )plicable accom- cercises They )8, 210, 3x8x iscover I not to ily me- [rstand led 12 lues as that it Ifound. it the lig the 111 this cannot PRACTICAL SrGGKSnONS. 89 moan any tiling more than sliort proviesses of addition and subtraction respectively — a meaning which carries with it but little power. Fractions.* The term fraction seems to convey different meanings. It is use 1 (i) To ex})ress a ratio between two quantities, as 5 feet is ^ of 6 feet. Here a direct comparison is made between the quanti- ties 5 feet and 6 feet, and f expresses the relation between them, 6 feet being taken as the standard of comparison. It is this mode of thought which is employed in the solution of the problem : If 6 pencils cost 12c., what will 5 pen^'ils cost J when the solution assumes either of the forms 6 pencils cost 12c., .•. 1 pencil costs ^ of 12c., .•. 5 pencils cost 5 times ^ of 12c.; or, 6 pencils cost 12q., .-, 5 pencils cost g of 12c. We see this the moment we answer the (juestion, why does 5 pencils cost f of 12c.? Because 5 pencils is f of 6 pencils. The first form of solution differs from the second only in one respect — an intermediate step is taken in order that the compari- son may be more easily made.. It is less dilHcult to establish a comparison between 1 and 6 and then between 5 and 1, than to establish it directly between 5 and 6. The point to be noticed is that in both cases the fractions express ratio. . (ii) To denote an unperformed process of division and multiplication. According to this idea 5 feet = | of 6 feet, * For a complete treatment of fractions see Psychology of Number, by McLellan and Dewey — an admirable book just published. 1 1 'I' ■4 ' ii-'v ;•■!! 90 HINTS ON TEACHING ARITHMETIC. means that if we divide 6 feet into six equal parts and take five of them, the result will be five feet. The mode of thought in this case corresponds to that of finding the value of ^ of 1 2c. in the preceding problem, after the ratio has heen determined and stated. To find the value of | of 12c., we divide 12c. into 6 equal parts and take 5 of them. This view does not eiiable one to use a fraction in the solution of a problem, but merely as a process in evaluating the result. It is clear that when the pupil can perform the operations of multiplication and division intelligently he has but little to learn in order to deal with fractions used in this sense, If asked to divide 12c. into 6 equal parts and take 5 of them he will immediately do so, provided that his attention has been carefully directed to definite units of measurement, this beii g the necessary condition for the development of the true idea of fraction. But, it must be remembered, the difficulty of a fraction does not so much consist in finding the equal parts — the initial step — as in relating the parts to the whole. The latter implies establishing a ratio between a unit, or a number of units, and a whole quantity taken as the standard of com- parison. (iii) To denote a kind of concrete unit.* Suppose one j'^ard to be divided into three equal parts, then we may give the name foot to each part. 1 part would be designated 1 foot, 2 parts 2 feet, etc. In the same way, if we allow our- selves to lose sight of the relational idea, we may call eacli part by the name third. One part would be called 1 -third, 2 parts, 2-thirds, etc. Similarly, if we divide any other quantity into three equal parts we may give each part the name third. What is supposed to be a fraction is not a fraction at all, but a kind of concrete unit which has no fixed value. Unfortunately, much valuable time is wasted in drilling * See The Public School Journil, vol. xiv., No. 10. PRACTICAL SUGOKSTIONS. 91 d take tiought of 12c. rmined 2c. into eiiable merely bions of ittle to ise, If hern he as l)eeii is bell g 5 idea of by of a ts— The umber f com- 16 yard pan y give ignated Iw our- lU each i -third, other trt the not a fixed irilling small children unable to number 10 objects on fractions of which they cannot possibly have any proper idea. It is exceedingly important for the teacher t6 distinguish from one another the mental processes corresponding to the different cases given above. It is easy to fall into the error of assuming that because a pupil can manipulate fractional forms, which mr.y mean to him, in reality, very little, he is able to make use of fractions in the solution of problems, or in other cases where they represent not merely arithmetical operations, but definite quantity relations. As there is much difference of opinion regarding the time at which the teaching of fractions should begin, it may be well to indicate the ground on which such difference rests. The real question at issue is, how does the idea of fraction grow- up in the mind ? Does it arise from merely nwrnhnrhig acts of attention occasioned by objects, sounds, events, ideas, etc. (tliere can scaicely be any doubt that the idea of number — the liow 'ma7i}j — may be developed by any means that will occasion successive acts of attention), or from numbeiing definite quan- titative units ] In other words, is the fundamental idea that of time only, or does it involve both space and time ? If num- bering units, which are not quantitatively defined, can give rise to the idea of fraction, why should it be considered im- portant in teaching fractions to deal with units which are quantitatively e(i[ual to one another 1 On the other hand, if both number and definite quantity are included, why should fractions be inti-oduced until some prOj^ress is made in number- ing measured units ? It is sometimes urged by those who favour teaching frac- tions from the beginning of the course in number that young children use the language of fractions intelligently in ordinary conversation. For example, a child may say that 50 cents is half a dollar. Certainly, there is nothing to prevent a five m HINTS ON TEACHING ARITHMETIC. year old boy from rememl)eriiig two names for one thing. Ho will also remember that a dime is one-tenth of a dollar or one- fifth of 50 cents if t(ild frecjuently enough. He may learn further that one-fifth is equal to two-tenths! H this is taught by means of objects he may even "catch the trick" of illus- trating it after a fashion. Surely it is perfectly plain that all of this may be done and much more without involving a true conception of a fractional unit as related to a prime unit. The teaching of fractions usually presents considerable diffi- culty. The cause of this may generally be traced back to one or more of the following : (i) Attempting to deal with fractions at too early a stage. (ii) Regarding fractions as indicating merely processes of multiplication and division. (iii) Losing sight of the prime unit from which the frac- tional unit derives its meaning. This is closely allied to the last. (iv) Introducing symbols before their meaning is fully de- veloped. (v) Showing the pupil how to perf mn operations with frac- tions instead of allowing him to discover rules and methods for himself. « Preliminary Work. In teaching fractions the first step is to develop a proper notion of what a fraction is. This may be done by dividing strips of paper, lines, etc., into equal parts, and by dealing with denominate units. Compaie a line 1 foot long with another 2 feet long ; a line 2 feet long with another 4 feet long. Compare 5c. and 25c., 10c. and lOOc, etc. It makes no difference whether the pupil answers that 25c. is 5 times 5c. or that 5c. is | of 2oc., provided that a comparison of the quantities has actually been made. The idea of fraction is implied in both, although expressed only by the latter. PRACTICAL SUOCIKSTIONS. with feet akes mes of tion Tho order of steps indicated by tlie foregoing may be more fully statrd thus: (i) Separating the whole into parts (roughly) and /n/m/jfr- iiKj the parts, (ii) Dividing the whole into equal parts and numbering the parts, (iii) Comparing one or more of the efjual parts with the whole ; also, the whole with one or more of the ((jual parts. When the third stage is reached whole numbers are found insutHcicnt to define all of the ratios which may be established ; hence the necessity for frac- tional numbers. It will be seen that the very first mental effort made in numbering a quantity involves the idea of fraction as existing, not actually, but polentuUly. As the mind develops it demands a greater degree of accuracy for the satisfaction of its wants. The unit of mea- surement is made definite, and the necessary condition for the realization of the idea which at first existed only in possibility is afforded. As soon as fractions are dealt with in relation to the whole subject of number, many of the difficulties which are now experienced in teaching tiiem will vanish. Work in fractions will not be prematurely forced upon the pupil, but it will be deferred until such time as the require- ments of mental growth necessitate the comparison of (iuantitiea on the basis of the well-defined unit. The second step is to give such exercises as will dcn^elop in the mind of the pupil the principles underlying all fractional work. These principles may be stated as follows : (i) Multiplying or dividing the numerator of a fraction by a number multiplies or divides the fraction by the number. (ii) Multiplying or dividing the denominator of a fraction by a number divides or multiplies the fraction by the number. (iii) Multiplying or dividing both terms of a fraction by a number does not alter the value of the fraction. The method of dealing with principle i. may be illustrated thus : Compare \ oi a, foot and | of a foot ; Jj of 80 and ^ of $0 ; ^ of $14 and ^ of $14 ; 5 of a pint and ^ of a pint, etc. Compare f and | ; | and | ; f and |, etc.* There should be no haste in getting the pupil to make a formal statement of the principle. The aim should ])e to give * At first the pupil should be required to supply concrete units where no unit is given. 94 HINTS ON TEACHING ARITHMETIC. him power to make use of fractions, not simply to teach him how to perform operations. Tlie power gained will be proportionate to the amount of well-directed, independent thinking which the pupil does in connection with exercises in comparing fractional quantities. Formal Study. The formal study of the subject may begin by giving the pupil practice in changing the form of a fraction without altering its value. No new principle is involved in this. To change fourths to twelfths is quite as easy as to change yards to feet, provided the relation between one-fourth and one- twelfth is as weir known as that between one yard and one foot. Here, again, we see the necessity for exercises in com- parison. The reduction of fractions may take several forms of which the following may be regarded as types : — (i) Find the number of fourths of a dollar in $1|. (ii) Find the number of dollars in 1 fourths of a dollar. (iii) Find the number of fourths of a dollar in $|%. (iv) Express $f§ in its lowest terms. ' (v) Express %\ and %% as fractions having a common de- nominator. The exercises should be performed chiefly by the method of inspection. Viewed from either the standpoint of discipline oi' practical utility, it is well to give ample practice in easy frac- tions before proceeding to those involving large numbers. The next step is to teach the application of the fundamental rules to fractions. In introducing addition and subtraction, exercises should be given as : Find the value of %\ -f ij, IJ + Ii? i V^^^ 4- I pint, %\ - $iV, i yd. - i yd., etc. PRACTICAL SUGGESTIONS. 95 teaeli all be Mulent ises in giving rithout is. To ! yards id one- ,nd one in coni- w liicli ollar. kon de- thod of lline or ly frac- [nental should pint The pupil should V)e requirechto point out the resemblance which adding and subtracting fractional units bears to cor- responding operations which ho has aln^ady performed with other units. He should frecjuently illustrate his work by means of objects. The representation of lines on the blackboard will be found both convenient and useful. Multiplication and division present to the pupil greater dilUculty than addition and subtraction. This is chietly owing to the fact that the former involve a two-fold ratio. (i) The ratio expressed by the fractional unit. (ii) The ratio implied in the operation to be performed. It is true that $^ x 3 may mean nothing more to the pupil than $1 -j- $^ + $1 = $|, which expresses directly but one ratio, viz. : that which the fractional unit $^ bears to the prime unit $1. But with oidy the thought of a single ratio in mind S;^ x | must be meaningless. As soon, however, as the pupil grasps the idea that in S| x 3 = $|, 3 expresses the ratio between $\ and $|, he is in a position to comprehend the meaning of $i X J - $A. In teaching multiplication and division of fractions much practice should be given in comparing quantities. Compare $J and $2, $J and $f, J gal. and 5 gals., J lb. and 2 J lbs, etc. By what must we multiply J?| to get $2 1 How is this expressed 1 What part of 5 gals, is ^ gal. ? Express this on blackboard. The steps in teaching multiplication of fractions may be indicated as follows : (i) Multiply a fraction by a whole number, as : $ j x 2, pt. X 4, ^ X 2, j§ X 5, etc. 96 HINTS ON TEACIIINO AIUTIIMETIC. (ii) Multiply a w1h»1(3 number by a fraction, as : $6 x ^, J?() X I, 8 gals. X i, 10 X g, etc. (iii) Multiply a fraction by a fraction^ as : SJ* x \, S'f x J, The steps in teaching division are as follows : (i) Divide a whole number by a fraction, as : $2 -r $h, ') gals, -r 'i gal., 5 -f 3, etc. (ii) Divide a fraction by a whole number, as : .M -^ ^, ^$1 -^ 1 6, 5 JL •)r) »>t,o (iii) Divide a fraction by a fraction, as : '^\ -f $|, $^ -r g, A full interpretation of such an expi-ession as $2-f8^ = 4, generally pi'esents difficulty to the beginner. What are the lucanings which it conveys 1 (The pupil's answer to this will depend altogether on his power of interpretation.) (a) $^ may be taken from $2 f.)ur times, or to express the same idea in other words, $2 contains 4 half-dollars. {b) $2 is made up of 4 equal parts of $^ each. (c) $2 is four times as great as These represent different stages in relating. In (a) the parts (units) are numbered^ in (6) the equality of the parts (units) is considered, and in (c) the ratio between the unit, $^, and the whole, $2, is stated. Both {a) and (6) precede (c) in the order of thought. With- out the idea of equal parts the comparison implied in (e expressed as 6 tens, 6 ones, 6 tenths, 6 hundredths, etc. Change to usual form as 66.66, etc. (ii) Oral exercises. (a) Read 3.5, 33.25, 423.625, etc. (6) Express by figures 3 tens, 4 ones, and 5 tenths, etc. (c) Find the value of .5 of $10. Of .6 of $50. Of .25 of ). Of .25 of 100 lbs., etc. (d) Compare .2 and .4, .2 and .6, .2 and .8, .4 and .8, etc. (iii) Reduction of decimal to vulgar fractions and vice versa. (a) Represent a line 5 feet long on the blackboard. Divide it into foot lengths and bisect each of them. Give the class such work as : Point out ^ of the line. Express this as a decimal. Point out .25 of it, .6 of it, etc. Express these as vulgar fractions. I (b) Divide the line into inches. How many inches make ^ of the line ? Express as a decimal, etc. Com- parison. A great vruiety of valuable exercises may be given in this way. Tlieir difficulty, too, may be increased to any degree by taking lines differing in length. In this work the idea of comparison should be made prominent. (c) Represent a line 2^ feet long. On it describe a square. Divide the square into 25 equal squares. Give work similar to the foregoing. Divide the square into equal parts in other ways and proceed as before. PRACTICAL SUGGESTIONS. 99 once.) .enths, etc. E .25 of and .8, ze versa. jkboard. if them, he line, it, .6 of js make Com- ses may I may be jring in should Icribe a squares, tde the Lceed as {d) Write oa blackboard such fractions as f , f, f, etc., and have pupils give their equivalents as decimals, etc. In case there should be doubt as to whether or not the pupil has thought out any relation which he has expressed, have him illustrate it objectively. Addition and subtraction of decimal fractions are eatdly understood. Multiplication and division usually present diffi- culty in one particular, viz., finding the position of the decimal point in the product or the quotient as the case may be. Such exercises as these will direct attention to the main thing to be taught. (a) Write on blackboard the expressions 5.5, 55.55, 12.335, 183.932, etc. Change the position of the decimal point. How is the value of the number affected, etc. % (b) Multiply 642.937 by 10; by 100; by 1000. Divide the same by 10, etc. Give a sufficient number of exercises to familiarize the pupil with multiplying and dividing by powers of 10. (c) Multiply 12.4 by 2, 20, 200, etc. Exercises. (d) Multiply 125 by 4, .4", .04, .004, 4.4, 4.44, etc. Com- pare products, and account for their relations to one another. (e) Multiply 12.5 by 4, .4, .04, .004, 4.4, etc. Exercises. (f) Divide 44 by 20, 2, .2, .02, .002. Compare, and account for relations of quotients. (g) Divide 4.4 by 2, 20, 200, 22, 2.2, .22, .022, etc. Com- parison of quotients. A large number of easy exercises related to one another should be given. The teacher must not depend entirely on 100 HINTS ON TKACHINO ARITHMKTIC. the textbook for these, because a textbook, no mfttter how much merit it may possess, cannot, in some cases at least, meet the wants of the individual. Tt is important that the pupil should at first read deci- mals so as to bring out their full meaning. For in stance, 72.3 15 should be read as seven tens, two ones, three tenths, four hundredths and five thousandths. After decimals are well known a briefer form of expression will be adopted, as seventy-two and three hundred and forty-five thousandths. A common error in the school-room is to read such an ex- pression as 7.125 as seven decimal one hundred and twenty- five. This must be guarded against. Percentage. It is often considered necessary to give a series of lessons leading up to percentage as if it were an entirely new division of arithmetic. Such is a waste of time, for the pupil who has a proper knowledge of fractions has really nothing new to learn except the meaning of the term; 3% means y§^ or .03, both of which are well known to the pupil. In fact no such phase of the subject would appear in our textbooks but for the fact that it is found convenient to adopt 100 as a standard on which to base certain arithmetical calculations. Percentage should therefore be taught in its relation to fractions, of which it forms a particular ease. The steps in teaching percentage may be taken as follows : (i) Meaning of the term, per cent. (a) Statement. (6) Easy exercises to develop meaning of term, as what is 3% of $100] Of $2001 Of $400^ Of 1000 bushels, etc. PRACTICAL SUGGESTIONS. m how least, deci- tance, enths, Is are ed, as lis. an ex- wenty- lessons ivision ho has ew to [or .03, lo such 'or the ,rd on intage which lows: what 1000 (ii) Relation to decimals. («) Express 5%, 10%, 15%, 20%, 25%, 50%, etc., as deci- nials. Problems. (b) Express .5, .55, .525, etc., as percentages. Problems. (iii) Relation to vulgar fractions. (a) Express 10%, 20%, 25%, 30%, 50%, etc., as vulgar fractions in their lowest terms. Problems. (6) Express yV. ^o, zuj h h h Ih etc., as percentages. Problems. The pupil should become thoroughly familiar with relations commonly used, as ^ = .5 = 50%. By means of oral exercises, such as find tlie value of I of $88, find . 125 of $648, find 33i% of $900, etc., the practical ad- vantage of what is liere stated may be shown. The fractions most im- portant in this respect are i, ^, J, i, J, J, ^V. uV» tsV. »"iil^- j/ |— ^1.1 . • . Rate per unit = . 1 . '. Rate per cent = 10. Of course it will be seen that these exercises are suitable only for advanced pupils, for they demand considerable thijuj^ht power. Square Root. During the early part of the course .there should be given in connection with the work on factors and multiples, exercises on squaring small numbers, also on finding the square root of such numbers as 25, 36, 49, 81, 121, 144, etc. Before the formal process is taught the pupil should have a knowledge of terms and symbols ; he should also know some- thing of the application of square root in connection with such problems as these : (i) The area of a square is 1 6 square feet ; find its side. (ii) The area of a rectangle whose length is twice its breadth is 288 square yards ; find its length and breadth. (iii) The length of a rectangular piece of ground is to its breadth as 5 : 3, its area is 2160 square rods; find its length and breadth. 106 HINTS ON TKACHING ARITHMETIC. Constructive exercises will make clear how the square of a numl)er is made up. These may be carried on as follows : Describe on the blackboard a square whose side is 24 iiiciies, thus : . 24 in. G E 80 sq. in. 20 in. a 400 Bq. in. © 16 sq. in. 4 in. K 80 sq. in. F B H Area of square AC = area of square EH 4-area of rect. AK / -j-area of rect. CK + area of square GF ' = 400 sq. in. + 80 sq. in. x 2-f- 16 sq. in. 4 = 576 sq. in. ' After the pupil has worked out a few problems of this kind his attention will be directed to the relation between the number of units of length and the number of units of surface. He will thus perceive that 24^ = 202+20x4x2+42. Great care must be taken here, or there may be confusion in the mind of the pupil with regard to the different kinds of units. PRACTICAL SUOOESTION8. 107 of a ows : clies, sq. in. kind the rface. Ijlreat the dts. Simple exercises requiring the application of what lias heen learnt will follciw, as : find the square of 15, 16, 18, 21, 2.'i, 25, 27, 31, 32, 40, 41, 42, etc. The next step is to present such numbers as can easily he separated into squares by inspection, as : 121 = 10-i + 10 X 1 X 2 + 1'^ /. |/121 = 11 144 = 102 4-10x2x2 + 22 .-.1/144 = 12 225 = 102 + 10x5x2 + 62 .-.1/226 = 15 400 = 202 .-.]/ 400 = 20 484 = 202 + 20x2x2 + 2- .-. i/ 484 = 22 The formal process should then be taught by applying it to numbers whicli have ali-eady htton dealt with by the method of inspection. 102 = 100 10x0x2 = 120 62= 36 256 102 20x6- 6- = 256(10+6 100 166 120 36 36 26 256(16 1 156 156 Review Exercises. Among all the words found in the teacher's vocabulary none is more important than the word Review. Reviewing a sub- ject does not mean merely repeating exercises previously performed. It includes far more than this, and serves a much iiigher purpose. When a particular section or division of a subject is first brought under consideration, attention must necessarily be confined to somewhat narrow limits, otherwise little progress could be made. The new, in order to be ap- perceived at all, must be related to the old, it is true, but we 108 IIFNTS ON TEACH INO ARITHMETIC. must remomb«;r tlwit tlu; points of contact will be few, and the bonds of union will be w(!ak. To pass on to the next division of the subject, and then to the next, and so forth in rapid suc- cession would, under such conditions, result in the ac(|uisition of a mass of uni'(;lated or imperfectly related knowledge which would be of almost no value to its possessor. Knowledge gives power only when it is well organized. The chief value of review work in arithmetic consists in givin,:^ tlu; pupil a connected view of the subject in so far as he has stadied it. Suppose, for example, that he has finished a series of exercises on percentage, he is now in a better position than ever before to comprehend the whole subject of fractions. He can see the advantage of having different fractional forms and, conseciuently, he can avail himself of this in their appli- cation. The pupil will take the greatest pleasure in such review lessons, for they are the means of representing to him old ideas in new relations. - Review exercises should consist chiefly of problems involving principles, and also facts which should be remembered, as tables, etc. Even in the regular daily exercises much time may be gained by requiring the pupil to recall incidentally such facts of arithmetic as are of practical importance in everyday life. and the division pid suc- luisition e which ie gives }ists in ir as he ished a )osition ictions. forms appli- n such to him ' / olving ed, as le may I facts y hfe.