Special Method in Arithmetic

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Preface

It is our purpose in this volume to make plain to teachers in elementary schools the purpose of teaching arithmetic, to outline fully a course of study based upon this controlling idea, and to discuss and illustrate the method of handling some of the chief topics.

) It is strictly a teacher's book, and not designed for jise by children.

It has been generally felt that arithmetic stands aloof from other studies, but the attempt is made here to show a very intimate and important relation to the other studies, one vital in fact to the proper plan and treatment of the whole subject.

This volume is one in a series of special methods, as follows: Reading of English Classics, Primary Reading and Story Work, Geography, History, Elementary Science, Language, and Manual Arts.

The course of study for the eight grades based on the plans worked out in these eight volumes is now in press and will appear in two volumes.

Contents

CHAFTBR PAGE

I. Brief Historical Review i

II. Aim and Scope of Arithmetic ' . • .16

Chapter I

The history of arithmetic as a subject of common school study and the widely varying opinions of its importance that have prevailed at different times and in different countries constitute an interesting chapter in education. It is only within comparatively recent times that arithmetic has been held of much importance. It had no recognition in the classical schools (like that of Sturm at Strassburg) established by the humanists during the Reformation. Nor among the early classical schools of England and America had it much standing. " It was toward the close of the eighteenth century that the modem treatment of elementary arithmetic began to show itself. In the Philanthropin at Dessau, an institution to which education owes not a little, we find in 1776 very little improvement upon the old plan of pretending to teach all of counting, then all of addition, then all of subtraction, and so on. But in the following year Christian Trapp began upon entirely new lines, and in 1780 he published his 'Versuch einer Padagogik,' in which he worked out quite a scheme of teaching young children how to add and subtract, objects being employed, and the effort being made to teach numbers rather than figures. This he followed by simple work in multiplication and division, and he worked out a systematic use of a box of blocks illustrating the relation of tens to units, a forerunner of the Tillich reckoning-chest mentioned later. It is here that we may say, with fair approximation to justice, the modern teaching of elementary arithmetic begins.

" After the adoption of the Arabic (Hindu) numerals in Europe, about 1500, arithmetic was taught for three hundred years in an extremely formal and abstract manner. It consisted mainly in figuring by rule, without effort to develop or explain the meaning of processes." (Professor David Eugene Smith, "The Teaching of Elementary Mathematics.")

In the first quarter of the nineteenth century Pestalozzi gave the strong impulse that led to the adoption of arithmetic as an important study in the schools. His own school was famous for its success in number, and the methods he employed were followed and developed by his disciples, till finally Grube gave them a simple and systematic form which has been adopted largely in the schools of both Europe and America. '

Pestalozzi gave a strong emphasis to perception^ to the use of objects in developing the number sense. His successors worked out elaborate and systematic plans for illustrating number. The number ideas were developed before the figures were presented. Pestalozzi also gave great prominence to oral worky and in this connection to the vigorous mental gymnastics of arithmetic.

Number work was begun with children entering school in the first grade, and was continued through the course.

It would be interesting to trace the development of number teaching as worked out by Pestalozzi's disciples, by Tillich and Tiirk, Kranckes and Grube, because they show the extremes to which good ideas can be carried, and the reactions that naturally follow; for example, excessive emphasis of illustrative work, of mental gymnastic, of thoroughness, of logical order and mechanism.

These developments of method and of the course of study from Pestalozzi to Grube have had a powerful influence in shaping the present course of study in arithmetic, both in Germany and in the United States. An examination of the course of study worked out by Dr. Rein in his "Eight School Years" will show that he has adopted the Grube plan in a modified form, and has introduced into the method of treatment certain leading ideas of the Herbartian pedagogy, as follows: —

The old idea of beginning number work with observation is modified by Rein to the idea of starting out from things, and first of all not from artificial things such as cubes, blocks, abacus, splints, and other number devices, but from those familiar things and persons which children have known in their games at home, in nature study, and in the stories they have heard.

"The first number work should spring from that familiar group of objects already known to the children in story work and in nature study. . . . These objects and especially those common to the home are the very ones which press strongly for numerical expression and are the best basis for interest in number." When, through these familiar objects, the interest in number has been once awakened, it can be carried over successfully to the splints, blocks, and number machines and systematic schoolroom devices. "When gradually the attention has been turned to the formal relations of number, the objects fall into the background, the number machine (abacus or blocks) assumes the observational phase or instruction, until even this support is unnecessary, and only the pure number remains. Thus we pass over from object to symbol, from symbol to the number idea, and in this manner safely transfer the interest from objects to the region of pure number." (Rein, Pickel, and Scheller ; " First School Year.")

In building up the number idea in closest relation to home objects, to nature study, literature, and story, it is manifest that Rein has applied the doctrine of correlation vigorously to number work. ^ In the second place, in harmony with Herbartian pedagogy. Rein has reduced arithmetic to a relatively small number of large, and as far as possible, interesting, method units. The number six, for example, with all its possible combinations in the four processes, is one unit of thought. Its treatment should be based upon familiar experience, as, for example, the notion that the child in the first year of school is six years old. The single combinations, as 4 + 2 = 6, are not separately memorized at first but worked out in connected series (addition series, subtraction series, etc.). "Apart from the greatest possible objectivity in early number work, the chief emphasis is placed upon the working out of close and compact number series. The children should be induced and accustomed to build up these series without much questioning 'from the teacher. When once the series is firmly established, the following drill exercises in irregular order are in place." (Rein.) Here again it is clear that the working out of large units of thought, by building up series, by drills, comparisons, and applications, is based upon the formal steps of instruction.

Turning our attention to the history of arithmetic as a school subject in the United States, we observe first that for many years it has held a commanding place in the common schools. This has been due to two prevailing convictions: first, the notion that it was of great practical value to the common man ; and second, the vigorous mental training believed to belong to the processes and solutions of arithmetic.

\ It is only within the last fifteen or twenty years, since 1885, that these standard articles of the arithmetical creed have been seriously questioned. A closer examination has shown that the parts of arithmetic which are of direct service in common life constitute but a small proportion of the exercises in our arithmetics.

" For the ordinary purposes of non-technical daily life we need little of pure arithmetic beyond (i) counting, the knowledge of numbers, and their representation to billions (the English thousand millions); (2) addition and multiplication of integers, or decimal fractions with not more than three decimal places, and of simple common fractions;

(3) subtraction of integers and decimal fractions; and (4) a little division. Of applied arithmetic we need to know (i) a few tables of denominate numbers; (2) the simpler problems in reduction of such numbers, as from pounds to ounces; (3) a slight amount concerning addition and multiplication of such numbers, as from pounds to ounces;

(4) some simple numerical geometry, including the mensuration of rectangles and parallelepipeds; and

(5) enough of percentage to compute a commercial discount and the simple interest on a note.

" What, then, should be expected of a child in the way of the utilities of arithmetic? (i) A good working knowledge of the fundamental processes set forth above; (2) accuracy and restsonable rapidity, subjects which will be discussed later in this work; and (3) a knowledge of the ordinary problems of daily life. Were arithmetic taught for utilities alone, all this could be accomplished in about a third of the time now given to the subject." (Professor Smith, "The Teaching of Elementary Mathematics.")

This opinion seems now to prevail among thoughtful teachers, that only a small part of the old arithmetic had direct practical value. 4 The doctrine of mental discipline, of logical exercise through arithmetic, being grounded in a once prevailing psychology, is still held by many. But in the last few years the changes in psychological theory and in the whole course of study have been so great as to shake even this stronghold of the old arithmetic.

The doctrine of formal discipline has been largely undermined by the leading schools of modem psychology, both in America and in Europe. The old idea of separate faculties of the mind developed by different studies, for example the logical faculty by arithmetic, has now little recognition among psychologists. The doctrine of apperception has become in the main a substitute for the old notion of formal mental discipline.

As a consequence we are no longer willing to spend a large amount of time in drilling upon any subject further than it supplies us with ideas that will be useful to us in all our future studies and life. We do not study subjects for mere discipline, regardless of the positive^ ideas with which they equip the mind against future emergencies.

To exclude merely disciplinary topics from arithmetic is to shut out a large part of the difficult subjects of the old text-books.

A second influence^ of equal importance, has demanded a curtailment of all doubtful topics in arithmetic. The common school course has expanded itself so vigorously of late into new fields that the demand has become imperative for a radical reduction and simplification of our whole curriculum. It would be preposterous to allow arithmetic to absorb so much school time as formerly, when such subjects as American history, nature study, choice literature, manual arts, physilO SPECIAL METHOD IN ARITHMETIC

cal training, music, and English are demanding, and properly, a full share of the child's time.

In the last few years there has been an elimination from the books of a large number of the obsolete and over difficult topics in arithmetic for grammar grades, such as partial payments, arbitration of exchange, and compound proportion. Just at the present time there is a strong demand for a still greater reduction, for dropping out much of common fractions, troy weight, true discount, partnership, cube root, equation of payments, and several others. There is an equally strong demand for an bmission of difficult problems and greater concentration upon the simple elements of arithmetic so as to secure complete mastery of elementary processes. Arithmetic no longer stands by itself as an isolated instrument of training. The fundamental question is how can arithmetic work in cooperation with all the studies of the school to build up a well-organized and coherent body of knowledge such as a child needs.

It would not be difficult to show that all the leading ideas worked out by Pestalozzi and his successors have found more or less adoption in American schools. Our primary number work in graded

BRIEF HISTORICAL REVIEW II

schools has been strongly modelled on the work of Pestalozzi and Grube, But in this country there has been of late a strong criticism of the Grube method by Dr. Dewey and others. Dr. Dewey bases the development of the number idea upon measurement, and measurement implies activity, the adjustment of means to end. The number one is not a fixed thing, but a standard unit with which to measure some larger, as yet undefined, whole.

"These two conceptions — {a) the origin of quantitative ideas in the process of valuation (measuring), and {b) the dependence of valuation upon the adjustment of means to end {i.e. ultimately upon activity) are the beginning of all conceptions of quantity and number, and the sound basis of all dealing with them." (McLellan and Dewey, "Psychology of Number," p. 41.)

1 This appeal to activity in measuring by means of the standard units, we believe to be a sounder basis for the number concept than mere observation as practised by Pestalozzi. ;

The limitation of the first year's arithmetic to the number space from i to 10, according to Grube, is also strongly criticised on several grounds: (i) that it does not correspond to the natural range of a child's experience and interests ; (2) that it sets up a standard of thoroughness wholly premature; and (3) that it involves a mechanical routine without proper motive or interest.

Notwithstanding this criticism, the first year's regular work in number (in our course, the second grade) includes most of the number combinations from I to 10, also the additions and subtractions from 10 to 20, and some exercises in still larger numbers, according to the natural scope of the children's knowledge and interests.

In this country, for several years, efforts have been made by teachers of various schools and tendencies to secure an intimate connection between arithmetic and other studies, to use arithmetic as a positive means of interpreting clearly many parts of a child's growing knowledge and experience. In this way arithmetic contributes directly to a better understanding by a child of his whole envu-onment.

Even such a brief account as we have just given of the historical phases of number work suggests that a careful study of the various movements in number teaching is an excellent training in method. Historically all the different extreme conceptions of number work are strikingly exhibited by enthusiastic advocates. There is also a steady movement to a broader and more satisfactory view of number in its nature and purpose in education. "The history of the subject gives us a point of view from which we can see with clearer vision the relative importance of the various subjects, what is absolute in the science and what the future is likely to demand. ... So one who considers the historical development of arithmetic and its teaching will see how enormous has been the waste of time and energy, how useless has been much of the journey, and how certain chapters have crept in when they were important, and remained long after they became relatively useless." (Professor Smith, "The Teaching of Elementary Mathematics," p. 43.)

At the present time, with the light thrown upon our task by the history of method, we ought to gather up the controlling ideas that must organize for us our course of study in arithmetic.

1. Number work must spring in the lower grades from the activity of children in measuring and constructing with common standard units.

2. The clarification of number concepts can be \ greatly aided by the use of suitable, more or less artificial, illustrative materials, as the cubes ~ and blocks, the abacus and the splints for showing the decimal scale, etc.

3. Number work has its roots also in the consideration of those familiar objects and experiences which a child has encountered in his home, in reading and story, and in his other school studies.

4. The main units of study in the advance movement should be vague , quantitive wholes, requiring to be estimated and measured, thus giving motive

\ to the work.

5. As we move into the work of intermediate and grammar grades, many topics from geography, manual arts, history, and science should be worked out quantitively, so as to give the number interpretation to otherwise vague and undefined topics. This implies that arithmetic can throw a great deal of light upon many important topics in other studies and find its chief utility, not in its immediate business applications, but in the broad illumination it throws over the whole field of knowledge and experience.

6. The whole trend of thought in the more recent discussion of arithmetic is toward greater thoroughness and mastery of elementary processes. This implies abundant oral work, vigorous and strong

BRIEF HISTORICAL REVIEW IS

mental drills, self-reliant thought and expression. In other words, the arithmetic must be strong and virile, toning up the mind to concentrated effort and to the formation of the best mental habits.

7. The application of the doctrine of the formal steps to arithmetic by Ziller and Rein, — by which the subject-matter of number is broken up into large and valuable units of study, which are worked out inductively from measurement and construction with objects to abstract number formula and process, — this use of the inductive-deductive method brings arithmetic under the operation of the same principles as are known to prevail in the other studies.

In this respect the formal steps are a means of organizing, into a consistent plan, the various principles of method which have been worked out at successive periods historically. This general method plan may be then rationally applied to those topics which in the sifting-out process of educational history remain as substantial elements of the arithmetical course.

CHAPTER II Aim and Scope of Arithmetic

The chief aim of arithmetic is the mastery of the world on the quantitive side through number concepts. This means the ability to estimate quantitively in numerical terms the varied objects and forces in the physical world as related to man.

To gain this mastery k child must have a varied number experience, especially in measuring the things of the objective world. He must measure, analyze, compare, and express the results in number language. He must become familiar with the standard measuring units (conventionalized and natural) which are in common use among people. Such are the yardstick, the day, hour, and year, the pound and ton, the dozen and score, the square foot and square mile, the gallon and bushel, the scale of degrees on the thermometer, the degree of longitude, the acre, the horse-power, the dollar. The use of any standard unit is made more complex by its relation to smaller and larger units of the same

i6 scale, for which it must be often exchanged (as foot, with inch and yard). These measurements call for number expression and a great variety of number operations. As a consequence, we have a whole system of mathematical language with which the pupil must become quite familiar. It has taken thousands of years to work out a concise and satisfactory number language, a set of symbols of number and of number operations, simple and comprehensive, which now constitute an inheritance invaluable to every child. * In measuring and comparing objects, the successive processes of addition, subtraction, multiplication, and division spring up in a natural order, and these in turn are followed by factoring, fractions and decimals, percentage, and involution, a whole series of processes which must become familiar to any one who wishes to express himself in mathematical language, or to understand others when they so express themselves.

Alongside these measurements and processes is a large number of positive number facts, which each child must fully memorize in order to make any progress in the mastery of the processes themselves.

While number itself is an abstaiactifin, a mental act

T

_ by whichi we .put, niimhfir IHfias into thin^, the presence of objects is still necessary in order to put the mind into an attitude for thinking number. In this sense we may say that number concepts begin with the measurement of physical realities and develop toward abstract number and numerical processes, but that in the application of number we constantly return to physical things, and are ever constructing and estimating our material surroundings numerically.

On the one side we may say that the purpose of number work is to put a child in possession of the machinery of calculation ; on the other side it is to give him a better mastery of the world through a clear (mathematical) insight into the varied physical objects and activities. The whole world, from one point of view, can be definitely interpreted and appreciated by mathematical measurements and estimates. Arithmetic in the common school should give a child this point of view, the ability to see and estimate things with a mathematical eye. Whatever facts and processes are necessary to secure this kind of an eye and brain, the school should enable him to master.

The thing we aim at, therefore, is a completely practical and accurate mastery of our material sur-

AIM AND SCOPE OF ARITHMETIC IQ

roundings from the narrow point of view of number. It is not mathematical processes and discipline for their own sake. It is these as necessary instruments for achieving a definite result, the power to understand and make use of the world mathematically, the complete familiarity with the common standards as instruments applied definitely to all kinds of magnitudes. Moreover, the system of things that a child has occasion to know and interpret is the natural world as man has modified it; it is the world as seen from the standpoint of human conventions and institutions.

In this way it becomes evident that arithmetic stretches across all subjects, and includes a phase of all topics in the school course. It may sometimes seem to us that arithmetic is a wholly distinct branch of knowledge, and in its purely scientific and systematic form it is peculiarly abstract. It is not mathematics as a distinct and separate science that we have most to do with in the elementary school. It is rather the mathematical phase of every subject in the school course. Just as language, though a separate study, is present vitally in every study, so arithmetic, though distinct, is omnipresent in all subjects. If number, like language, is not present in every study, there is serious weakness and defect. For number is a point of view, a perspective, into which every subject must fall, else the student is blind and ignorant on that side.

This aim of arithmetic, as a point of view to be gained by which we can more clearly interpret and subjugate all experience and all kinds of knowledge in every subject of study, will help us to solve a number of controverted points, e,g. What topics, once common and even now common, should be omitted from our text-books ? How thorough shall be the mastery of processes ? What kind of mental discipline shall arithmetic give? What kind of applied problems, and how difficult, are needed } What is the relation of arithmetic to other studies, and to a child's outside experiences ? A clearly defined aim, such as we have proposed, should give a clear and positive answer to such questions.

Take the first question. What topics should be omitted from our text-books? Those whose facts and processes are not necessary as a general means of interpreting the child's world, e,g, they cannot use profitably the Roman or Chinese money standards, nor even the metric system in any complete form. All topics that are not broadly universal in their application can be left to the technical specialist. Surveyors' measure, for example, is narrowly technical and special. It is not a mode of measurement in common use among any except professional surveyors. Omit also topics which are becoming obsolete because better ideas and processes are taking their place.

"On the basis of social custom, our brief standard, the following things — very often taught — may well be omitted: —

"Apothecaries' weight.

" Troy weight.

" Examples in longitude and time, except the very simplest, involving the 15® unit, since our standard time makes others unnecessary.

"The furlong in linear measure.

"The rood in square measure.

"The dram and the quarter in avoirdupois weight.

" The surveyor's table.

" Table on folding paper.

"All problems in reduction, ascending and descending, involving more than two steps.

" The G. C. D. as a separate topic, but not practice in detecting divisibility by 2, 3, 5, and 10.

" All work with L. C. M., except of such very common denominators as those just mentioned.

"Complex and compound fractions as separate topics.

" Compound proportion.

" Percentage as a separate topic, with its cases.

" True discount.

" Most problems in compound interest, and all in annual interest.

" Problems in partial payments, except those of a very simple kind.

"The same for commission and brokerage; for example, all problems involving fractions of shares.

"Profit and loss as a special topic.

"Equation of payments — made unnecessary by improved banking facilities.

" Partnership — made unnecessary, in the old sense, by stock companies.

"Cube root.

" All algebra, except such simple use of the equation as is directly helpful in arithmetic.

" In addition to all of these, arithmetic may be omitted as a separate study throughout the first year of school, on the ground that there is no need of it, if the number incidentally called for in other work is properly attended to." (F. M. McMurry, "What Omissions are Advisable in the Present Course of Study, and what should be the Basis for the Same," address before National Department of Superintendence, 1904.)

The omission of the elaborate treatment of common fractions (which would save us at least a halfyear's work) is proposed on the ground that the decimal fraction is far shorter, easier, and simpler. In common usage and in scientific work the decimal has already taken the place of the more difficult forms of the common fraction.

Another disputed point concerns the kind of discipline furnished by arithmetic. The old doctrine of discipline and thoroughness carried with it a large amount of pure drill and formal exercise. The idea of using number as measurement, as a means of interpreting better our surroundings, would limit number processes and drills to what is necessary to this controlling purpose. It would not spend time, for example, in writing numbers to ten periods, nor in memorizing and drilling on the squares of numbers from 13 to 25 and above.

Those phases of number study which are of value to every child, which help him to understand better and quicker the varied objects of his experience, should be included in the number course. Those arithmetical processes which are least cumbersome and are most effective as ready means for necessary calculation, as the four fundamental processes, the decimal scale, and simple factoring, should be mastered. Those arithmetical processes, like the manipulation of complex and compound fractions, which are mere mental gymnastics, having lost their place in practical service, must be omitted. In this way we shall get a great reduction of time and mental strain and shall save energy for indispensable work in other fields.

Our controlling aim also suggests a much closer adjustment of arithmetic to the leading topics and investigations of other studies than has been usual in our manuals. Our text-books have used many so-called applied problems, but they have not made arithmetic, in any general way, a regular means of investigating and interpreting large topics in geography, history, and elementary science. This is now definitely proposed, so that the units of study in grammar grade arithmetic are typical occupations (as the profit and loss side of gardening or of managing a grocery store), administrative problems of taxation and revenue (as the city finances), public enterprises and geographical problems (such as the Panama Canal). In short, we are prepared to turn arithmetical calculation directly into the main streams of other studies, to take up the large problems of history and industry and to work them out arithmetically. This does not mean that arithmetic loses itself in these studies or becomes absorbed into them so as to disappear as a distinct study. It means rather that in the grammar school course of study arithmetic should focus upon the same large and important units as the other studies. Such is the arithmetical treatment of irrigation in the arid states of the West, or the cotton production, sale, and shipment of the Southern states. These topics require an illumination from arithmetic which they can nowhere else get. To pass these topics through the alembic of mathematical calculation is to distil from them a new and stronger meaning. It is to get a point of view from which to estimate all practical human affairs more accurately and more simply. If arithmetic does not reach forward to this result, it does not clarify the general field of knowledge; it is abortive. It consists of a complex machinery for which we have not found the use.