Southern Branch of the University of California Los Angeles Form L 1 This book is DUE on the last date stamped help JAR 2 9 1329 JAN 24 tg MAY-lg 1933 3 1991 Form L-9-15jn-8,'26 The Teaching of Mathematics The Teaching of MATHEMATICS BY RAYMOND E. MANCHESTER, A.M. PROFESSOR OF MATHEMATICS, STATE NORMAL SCHOOL OSHKOSH, Wis. SYRACUSE, N. Y.< C. W. BARDEEN, PUBLISHER 1913 COPYRIGHT, 1913, BY RAYMOND E. MANCHESTER OA\\ M 3 Co CONTENTS PAGE A period of reconstruction 9 Meaning of education 15 Values of mathematics 18 Daily use 23 Three groups of facts 25 Habit of logical thought 30 Generalizing power 33 Power to grasp situations 35 Individual thinking 37 Reconstruction of arithmetic 39 Materials of mathematics 41 Arithmetic by grades 44 Algebra 53 Geometry 54 Vocational departments 56 Methods and Modes 58 Analysis and synthesis 61 Induction and deduction 63 The Heuristic method 64 (7) 8 CONTENTS The laboratory method 65 The Socratic method . .-. 66 The Montessori method 67 Interest essential 69 Lecture and recitation modes 71 Summary 72 Mathematics and Education We are in a state of transition. The ideals of civilization are changing. Never in the history of mankind has the condition of unrest been so uni- versal. It is a dominant force. Whether or not the transition be from a lower to a higher plane depends much upon the teacher. One need not be a philosopher to appreciate this. Even those who think, work, and live by the day are capable of understanding the situation. During the span of the last one hundred years the human race has brought to a conclusion movements three thousand years in the forming. When one pauses long enough to reflect, truly marvelous are the strides mankind has taken. The whole method and mode of living have changed. With steam, oil, electricity, steel, cement, and light and sound waves man has elevated himself to a place only given the gods in times re- mote. Not only is he despotic ruler of the earth, but he reaches out his long, powerful, scientific arms to grasp the secrets of the universe. He has ex- tended his vision in two directions, to those regions his eye was not created to gaze upon. With the telescope he looks into God's own workshop and sees the stars and planets in the making; with the microscope he searches out the infinitesimal parts (9) 10 MATHEMATICS AND EDUCATION of His universe. His analysis is carried even unto himself, so that his own body, mind and soul are known in their parts and relationships. He even finds an analysis of the analysis. Man is uprooting all that was considered constant ; upheaving the very foundations of the old regime; making over his politics, his ethics, his morals, yes even his religion. Is it any wonder that he is dis- satisfied with his educational system? Is it any wonder that all those things, which appeared sane to his forbears, seem to his frenzied mind insane? Is it strange that the world is filled with promoters, agitators, reformers, and purists? Is it to be won- dered at that sophistry finds ideal conditions for germination ? By no means is educational unrest out of place. It is the necessary product of the situation. There could be no other result. Educational unrest, though grating to some, is nevertheless the logical thing. To close our eyes to it is folly, and is equiva- lent to dying with the old system. Because of this prevailing state of unrest two questions have shaped themselves: the one, what shall we offer the child who is destined by social law to serve as a worker, and, the other, by what method shall the material of this instruction be presented? In answer to these questions two great movements stand out distinctly, the one toward vocational education, the other toward informality. With reference to the first movement, educators at last agree that public education should face the 96% who are to RECONSTRUCTION 11 fill the producing class, the so-called masses. With regard to the second movement, new methods are found to be more efficient than the old, because men and women at last have discovered that the schoolroom contains red-blooded children. Perhaps no subject is so directly in the flux of the instability of things as mathematics. For cen- turies this subject has held the unenviable reputa- tion of being tortuous; yet man, because of his un- ending need of the subject, has endured silently. It has been one of the necessary evils of life, yet, did man but know it, wonderfully full of beauty in its symmetry and exactness. Indispensible to the man who builds, to the man who barters, to the man who meditates, it has ever been shrouded with a mantle of mysticism. One might say with truth that they who have the greatest need of the find- ings of mathematical investigation have the smallest opportunity to acquire a knowledge of them. The fundamental principles of mathematics could be written upon a calling card. From these simple facts a vast amount of detail has been accumulated through almost endless deductions. Under the spell of genius there has been constructed a most wonderful edifice upon this foundation, a struct- ure similar in many ways to a marvelous mansion with exquisite detail beautiful to look upon, but next to useless to the man who is to have a home. In both cases a wonderful foundation supports a structure of luxury, when by careful construction, a useful as well as an ornamental structure might have been built. 12 MATHEMATICS AND EDUCATION The magnificent is highly in order when the needs of mankind have been satisfied. With the 96% asking for recognition it is only proper that the first thought should be toward utility. So in the reconstruction of mathematics the utilitarian aspect is first in order, a mathematics for the builder and the barterer. To present such material in an in- teresting manner is an approximation to our ideal. To humanize mathematics is the problem. The solution does not lie merely in changing the order of presentation or in shortening the course by re- moving certain subjects, as many reformers firmly believe, judging by the many and varied text-books on the market. The teacher must take the re- sponsibility of the task and discover the secret of life. To do this, informality is absolutely necessary. To speak of the need of thorough training before an attempt is made to teach mathematics would seem unnecessary were it not for the fact that in- competence is the rule rather than the exception. Only in recent years under the guidance of general pedagogy have members of the teaching profession come to the realization that the difficulty of the subject calls for special preparation. The great number of under-prepared teachers in the profes- sion has caused a false philosophy to grow up ; even in our modern renaissance the bug-bear of practi- cability has caused an addition to the confusion. It has come to be recognized as a truth that there is an art of teaching a science. Teachers are be- ginning to appreciate the fact that mere acquain- STUDY OF NUMBER 13 tance with the science is not a sufficient preparation for the teaching of it. While this unrest has found most pronounced ex- pression in the new methods of teaching the lin- guistic subjects, there have been many, more or less successful, attempts to remedy existing evils in the presentation of the subjects relating to the development of the number concept. Especially with regard to elementary arithmetic is this true. Even yet, however, aptness in "riggers" is deemed sufficient qualification for the teacher in mathematics in many schools. Thanks, however, to the develop- ment of the psychological and pedagogical view- points, this custom is rapidly falling into disrepute. In apprehending the objective world, the indi- vidual is quite as appreciative of the number rela- tion as any other. It is a fundamental truth that he is conscious of his world in number as well as in time and space. Whether this relation be co- ordinate or subordinate need not greatly concern us except in philosophic discussion. The mere fact of its existence makes necessary a reaction on our part. In our educative system it is essential that attention be given to the study of the number phenomena, and that methods be devised for the study which shall approximate the best. Through- out all the centuries man has studied number, in- venting machinery in the form of symbol systems for his experiments. And, if we are to judge fairly the rise of civilization, it is quite as dependent upon the mechanical arts as upon the growth of language. 14 MATHEMATICS AND EDUCATION All the mechanical arts are dependent upon the basic facts of number. Yes, even the machinery of language is built upon the supposition that these relations are valid. The prevailing weakness of the mathematics teacher has led to two results, the appreciation of the fact that the subject is one of the hardest to teach, and the rise of the question of revising the course as offered. The pedagogy of mathematics is receiving world- wide consideration. In the older countries this is especially true, owing, no doubt, to the superior treatment of the science of general pedagogy. In America, although the problem is not so nearly solved, it is equally as vital and should call forth the best talent available. The findings of the many societies, the discussions in scientific publications, the founding of departments of educa- tion in our universities, and the reform of our nor- mal schools have been the dominant factors in the revolution in America. It is to be hoped that the mathematics teacher of the future may be so pre- pared that he or she can overcome the temptation to teach by rote and to follow time worn customs. The prevailing unrest is evidence of thought. Per- haps the movement toward the practical will over- carry in all probability it will but we may be thankful that there is movement. The subject is as thoroughly alive as any in the curriculum if the chains of bondage are loosened. Let us pray that the readjustment be rapid. MEANING OF EDUCATION 15 To justify a place for the study of mathematics as a course of study in our educational system, it is first necessary to have a definite idea of the mean- ing of education. We must have a solid base for that which becomes the superstructure. This is not difficult, inasmuch as the basic principles are so simple as to be accepted by all. A simple analysis is to consider society held in- tact through three relationships, (a) the relation- ship of the individual to the group, (6) the relation- ship of the group to the individual, (c) the relation- ships of the individual and of the group to nature. The relationships in turn might be subdivided into other relationships, but in this brief discussion the three mentioned are sufficient. It being evident that individuality has no meaning apart from the group, that the group has no meaning apart from the individual, and that nature has no meaning except in relation to the individual action and group action, we may be safe in considering education to have meaning only when the relationships exist. Education, whether it refers to the individual or to the group, has for its aim the perfection of rela- tionship. Education for the individual must have for its object the individual's better reaction to the group and to nature; education for the group, must have for its object the groupbetter reaction to the individual and to nature. Thus it is, that attempts to state specifically what education is are so futile, there being such a multi- tude of minor relationships and reactions involved 16 MATHEMATICS AND EDUCATION in the three larger relationships. Certain it is, that education is a process that must constantly vary. Keeping these facts in mind, it is not difficult for one to appreciate the fact that education is intimate- ly connected with the conscious aim of the indi- vidual and of society Broadly, education is at- tained only after a life has been lived through, only after the individual has drawn from experience in reaction of every sort; but for the teacher, education refers to that elementary beginning found in the school. Education is the acquisition of power to react toward the approximation of certain ideals, and of power to organize such reactions. The as- pects of education are many and varied, as for example, the biological, the psychological, the physiological, the ethical, the historical, and the philosophic, but back of all is the fundamental principle that education becomes meaningful only with regard to action and reaction. It follows that education does not have the same meaning for all. Group ideals call for differing reactions. For those following trade life, quite different reactions are called for than for those fol- lowing professional life. School training must be carried on with due appreciation of this fact. The materials selected and the method adopted in pre- sentation should be such as the group ideal may demand. There is, of course, a certain constant of educa- tion based upon the common characteristics of all men. This constant is well understood. It is the POWER TO SERVE 17 training of proper reactions to organized society, as such, with reference to the home, the state, and the church. Always, the point of contact is in the life of the individual. Defined in these terms edu- cation is power to react. Education is power to serve. To become educated is to attain such an approximation to the highest self as to be able to serve well. Self-perfection, whether it be in the individual life or in the group life, is the most ef- fective method of giving service. The individual at best is a unit in the group, and as such, serves best as an approximately perfect unit; while, on the other hand, the group is a totality of units having individual unity determined by the strength of its parts. Surely then, if power is education, the materials of education must be those best suited to produce power, and the methods of education must be those best adapted to the turning of these materials into power. In the selection of subjects for study, those richest in power-producing material must be chosen, while those most barren must be left out. If the subject of mathematics is to stand such a test, it must show that its material is that which can be used in the production of such a power, and that a method of teaching is possible by which this power can be generated. The successful teacher of math- ematics must, (a) become thoroughly convinced of the value of mathematics to the process of educa- tion, (6) make selection of material best adapted to the best development of the educational process, 18 MATHEMATICS AND EDUCATION (c) devise a best method to present the selected material to the student. Values of Mathematics The values of mathematics to the individual and to society should be determined by test of action and reaction. It should give to the individual, greater ability to react toward society to his better- ment; it should give society, greater opportunity to offer the individual conditions for a more perfect life and also greater efficiency in reaction to nature. Mathematics is valuable to the individual in offer- ing those methods and facts which aid him in the conduct of daily duties, and in helping him to develop power to react to social forms, institutions, and ideals. Mathematics is valuable to society in of- fering material for a closer knitting together of institutional life, for the creation of ideals, and for the engineering successes which turn the infinite powers of nature to the services of mankind. With language, number holds the center of the stage in educational inquiry. To recognize only the fact value of the subject is quite as narrow as to recognize only the other value. Thus, to be an efficient teacher of mathematics, one must acquire a certain balance. If one were to think the whole of any situation through, it would be found that the facts of mathematics are involved in some way. - All nature's laws are interpreted in number. Many of the more common phenomena are seldom thought of in number relation, owing to the general knowledge of the elements of the subject, but all the deeper VALUES OF MATHEMATICS 19 discussion is absolutely dependent upon number relation. ' Such sciences as astronomy and land measurement are almost entirely mathematical. The commonness of most of the relations of weight and measures is responsible for the lack of appre- ciation of the mathematical connections. In all sciences mathematics is indirectly connected. Also, a great majority of the occupations, into which students are likely to go, require mathemati- cal knowledge for successful participation. The great rise of industrial occupations generates a particular need of training. Common business intercourse calls at least for the elements, while so-called big-business is utterly dependent upon the validity of number laws. Not only has the subject a great fact value to society, but it typifies a method of reasoning, accept- ed by society as one of the most effective. Many consider this value far greater, directly and indi- rectly, to the student. All life problems must undergo analysis before a solution can be obtained. Situations must be understood in the relationship of their parts. To be capable of handling a situa- tion is power, regardless of occupation or social posi- tion. The study of mathematics is one of the most effective means to such- an end. In considering the values of mathematics, one is led into consideration of the larger discussions upon education. The awakening of the public has been stupendous during the past few years. In every state, city and town, the awakening is taking 20 MATHEMATICS AND EDUCATION place. Teacher and parent alike are striving to reconstruct a worn-out system, and despite many mistakes are accomplishing their end. One can imagine that some huge giant is sending chains of bondage flying, after fifty years of stupor. The producing class of this group of 80,000,000 people is becoming suddenly appreciative of the fact that it has been paying for something it has not received, and, with the spirit of new- world liberty fresh in its blood, is striking for its rights. And it will get them. The public, no doubt, has good grounds for de- manding a change in the educational system. In fact, it has been admitted by school men themselves that the system is not efficient. There has been, and is, educational waste. The public recognizes it in the inability of school graduates to cope with workaday problems. The public feels that the average graduate is not a sufficient reward for the time, energy, and expense necessary for his produc- tion. In reaction, the public has made severe criticism, verbally and in action. Through the public press, countless articles have been presented, showing faults and offering suggestions. In action, much money has been spent for expert service and for the construction of excellent school buildings equipped with the most modern apparatus. If we are to judge by attempted reforms, the public has arrived at three conclusions through its investiga- tion: 1st, that too much time is spent on theory and method, and too little time devoted to facts, FACT VALUES OVERESTIMATED 21 evidenced by the use of correspondence schools, technical schools, short course professional schools, and by the introduction of fact-courses into the public schools ; 2nd, that the courses of study should face the masses more directly, evidenced by night schools, vocational departments, summer schools, and by the cutting out of so-called culture courses; 3rd, that the physical education of the child is equal- ly as important as the mental education, evidenced by the rapid increase in the number of gymnasia in schoolhouses and of teachers of physical educa- tion. The first two conclusions vitally affect the teacher of mathematics. He must, or rather she must there being more women engaged in teaching than men thoroughly appreciate the need of change and be able to make changes for the best. With regard to the first conclusion, there is no doubt that the fact values of mathematics are over estimated and therefore over-stressed. The public, in attempt- ing to remedy one evil, is very likely to impose another. The facts of mathematics necessary to everyday existence are only the most elementary ones. Merely to cram these into the students' heads, together with many others necessary to fill out a course, is as futile toward the giving of an education as cramming the dictionary would be. Observation without reflection is barren. Never- theless there is an excellent lesson to be drawn from the demand for facts 22 MATHEMATICS AND EDUCATION The schools for the past seventy-five years have been preparing students for colleges and universi- ties. The colleges and universities have dictated the policy. Before the rise of the public schools to their present importance, this system was satis- factory because few students attended school regu- larly who did not aspire to a college education. Conditions have changed, however, without corre- sponding change in the course of study. Now 96% of the students in the public schools do not reach the college. Yet they prepare themselves for it. The public very justly is demanding a change. In mathematics it demands a change and, without the opportunity for due reflection, decides that facts must be emphasized. Very good, but the value of mathematics is not alone in the facts. It is valuable as well in offering a method of thought. This is appreciated by the teacher and causes his opposition to the public demand. The public de- mand is in fact just and must be recognized. The teacher must give and take. A system of mathe- matics teaching must be established, giving the facts demanded by the public and also giving men- tal development. Such a system will be discussed later. The valuable facts of mathematics may be grouped in three classes: 1st, those simple facts of use in the living out of average daily life, which are acquired in the study of elementary mathematics; 2nd, those facts of contingent value; and 3rd, those special facts which can be applied in engineering, in the DAILY USE OF MATHEMATICS 23 arts, in business, or in research. While it is true that no subject outside of language has such direct application to daily life, yet the importance to most persons is indirect. But even so, there is used in daily life only the elements of any subject. How often does the man of the street use his knowledge of history, of botany, of Latin, of chemistry, of French ? All people must converse and all people must know enough of mathematics to carry on ordinary business. Beyond this the fact value of mathematics affects the majority of people indi- rectly. Teachers should recognize this perfectly and drill eternally on elementary processes, but not to the exclusion of the methods of reasoning. The greatest value of the subject from the stand- point of fact is perhaps not realized by the great majority of people. The average citizen living out his daily routine of duties seldom, if ever, takes time to think how vitally his life is controlled by the laws of number. From the time he starts work in the morning until he rests at night he is constant- ly making use of the practical application of mathe- matics. Means of communication and transporta- tion have been perfected until we are truly citizens of the world, getting world news daily, and eating food raised many hundreds of miles away. Great bridges make it possible to cross waters, wonderful buildings make concentration of business possible, ocean lines connect the old and new worlds. Man has become cosmopolitan in the superlative. And what a wonderful force mathematics has been, the LOS ANGELES vlAL cCHOQt 24 MATHEMATICS AND EDUCATION foundation upon which this vast system has been built. From the historical viewpoint, the fact values of mathematics have always held an important place in the development of civilization. The ear- liest beginnings were with the utilitarian aspect uppermost. The Egyptians found a study of the elements of geometry of great value to them in determining the boundaries of their lands after the Nile inundations. The trading nations, notably the Phoenicians, needed the elements of arithmetic to carry on successful trading. In the wars, ap- plication was constantly made in war engines, as for example in the siege of Syracuse. In China attention was given mainly to the acquiring of a knowledge of the simple facts needed in daily life. Among those nations stressing religious life, only the practical values were studied. Even to the peoples of Greece and Rome, except in the philo- sophic schools, the usable facts of mathematics were the only ones of great value. Not only were the beginnings of mathematics strongly utilitarian, but throughout the growth of civilization the prac- tical aspect has been strongly uppermost in the minds of men. Most people in the world of organized society have little need to do more than work out simple relationships. But in all the myriad affairs of life the phenomena of number have some indirect bear- ing. So a knowledge of facts has its place, but no mere knowledge of facts can ever show us the goal FACTS ONLY MEANS 25 of education. Facts are only means. The old system knew the goal but not the means to reach it. Our system is rich in means but often misses the goal. Our teachers lose sight of the ideal in the study of facts and methods. The aim of present day education is to effect such a compromise as will maintain the lofty aims of the old-time school- master and overcome his trials by the use of care- fully devised methods of teaching. The first school- masters with their few students could give the per- sonal touch and get results without elaborate method, but with the growth of schools, lack of time made it necessary merely to hear recitations, and so re- sults became less satisfactory. To meet this sit- uation, the graded school was evolved, but here again numbers forced the teacher merely to hear recitations and to center all interest upon himself and his method. Today we are back to the in- formality of the beginning, attempting to increase the efficiency of the method by the application of modern pedagogy. We hope for success and there is reason to feel that we shall attain it. We have the method and the ideal. Can we get the two into our system by introducing the personal touch between teacher and student? (1) The facts to be grouped under the first class are those of elementary arithmetic, addition, sub- traction, multiplication, simple division, simple interest, proportion and mensuration; of elemen- atry geometry, simple construction, method of proof, and formulas for areas and volumes; and, 26 MATHEMATICS AND EDUCATION although not commonly taught in the grades below the high school, the elements of algebra, the use of the equation, and letter symbolism; the elements of trigonometry, computation by the use of angu- lar functions. These facts are of utmost practical value to everyone. Provision should be made to present this work as early as the mental strength of students will warrant. Inasmuch as students leave school rapidly from sixth grade, I should teach a student as much as possible by the end of that period, even at the sacrifice of some of the mental training material if necessary. The first task of the teacher is always to produce the highest quality of citizenship in the allotted time. The production of linguists or mathematicians is a secondary con- sideration. By the close of the sixth year in school, the student should have in his possession all the simple, daily usable facts of mathematics. He should also have acquired the elements of spelling, geography, grammar, United States history, civics, and the ability to read well and to write a good hand. All in addition to this should be rated second from a standpoint of value. Personal tastes of the teacher are not to bias the method and mode of teaching; in fact one's taste should be to raise the standard of mass efficiency to the highest during the first six years. Time and time over, the ideal swings back to the one expressed by the dictum, reading, writing, and arithmetic, simply because the public has a deep seated feeling that the largest number of people in any community has greater FUNDAMENTALS FIRST 27 need for the elements than for cultural development. The following newspaper item may be read with interest concerning this point: Cut Out School Frills The Three R's Not to be Lost Sight of in Chicago \ Chicago, 111., Dec. 20. Sweeping reform of the curriculum of the grammar grades of the city schools was advocated yesterday at the school management committee's meeting. If the board adopts a report which is to be presented, it will mean complete elimination from the schools of every- thing that tends to interfere with the fundamental principles of education. Sewing classes, reed and rattan work and similar courses, which now occupy the children's hands and minds, will be done away with if a motion made by Trustee John Guerin is carried. Mrs. Ella Flagg Young, superintendent of schools, was openly displeased at the action of the committee in appoint- ing a subcommittee to report on the matter. She made no secret of her feelings, and freely criticised some of the com- mittee members. "I have been working toward this end myself throughout my administration," she exclaimed. "Every time I start to do anything a committee is appointed to take the work out of my hands. It makes one wonder if a superintendent can accomplish anything." The motion made by Dr. Guerin that "everything that tends to interfere with the teaching of reading, writing, arithmetic, spelling, geography, physiology, grammar and United States history be eliminated from the school system" was seconded by Julius Smeitanka. Every member of the committee expressed himself as being unqualifiedly in favoJ of taking "mollycoddle trivialities" from the schools. From the attitude of the members there was apparently no question that the elimination would take place. Even those appointed on the sub-committee, while asserting that they were barred from speaking for publication because they were members of a committee, did not hesitate to de- clare themselves individually for the reform. (2) The second group of facts has contingent and cultural value. It embraces those facts under- lying the greater institutions of our civilization, MATHEMATICS AND EDUCATION as the railroad, the ocean liner, the sky-scraper, the bridge, the telephone, the telegraph, the auto- mobile, the wireless, the airship, automatic machin- ery, the elements of astronomy, etc. etc; the facts of the historical development of mathematics; the facts of the common trades and vocations with mathematical connections; and the facts of busi- ness intercourse. It promotes a student's general j aptitude for life, to know something of almost any I line of work he may enter. He has a fund of power which can be used in a great variety of ways. The student seldom knows exactly what his life work is to be; in fact ninety percent, of those who choose a line of work while young undergo change of mind. It is advantageous for the student to have a variety of material presented, so that he may discover his aptitude for one particular line. This group of facts also promotes the culture of the student more than any other group. It provides informational value upon many lines. It gives a more intimate relation to nature and her phenomena, thus open- ing a wider and better world to the student. It i encourages the appreciation of the triumphs of the j human mind. It gives him a more kindly feeling toward the well-related and beautiful. It guaran- tees a stock of thoughts tending to make points of contact between him and passing acquaintance, Hhus furnishing material for conversation. In short, this group of facts aids him to acquire self perfec- tion along general lines, to react to better advantage to society and to nature. From society's stand- THE THREE GROUPS OF FACTS 29 point there is increased opportunity opened to each individual. The facts falling under this heading are all those general facts ccvering the manual arts, the natural sciences, the growth of the science of mathematics, the business world, and the mechani- cal arts. They are those facts which enter into all thorough discussion of the science, and which the student acquires without special attention being given to their projection. They are the facts the students are bound to get, providing the teacher carries on his work carefully. (3) The third group of facts consists of those involved in the great engineering successes, such as electrical engineering; in the promotion of large business transactions; and in the development of the science for its own sake. While it is true that 96% of the students in school do not reach the university and need no special training to pursue their chosen vocation, the remaining 4% do get into the universities and, strange as it may seem, this 4% rules the world. It is necessary therefore to keep the door of opportunity open to that par- of this group who will fill the high positions of the engineering, business, and scholarly world. Since this group of facts is presented only in the higher educational institutions, the ordinary teacher is not especially interested, except is so far as prepa- ration is concerned. On this point little need be said. If the bright student is allowed to advance and finds a helpful and suggesting friend in the teacher, he asks nothing more. LOS ANGELES STATE NORMAL SCHOOL 30 MATHEMATICS AND EDUCATION Mathematics typifies a method of thought. It stands for a clear, direct development to a valid conclusion from accepted premises. Whether the method be synthetical or analytical, inductive or deductive, the argument is exact and sure of its end: so sure in fact, that no matter what doubts one has as to the certainty of other things, there is never a doubt that two plus two equals four. So wonderfully potent is it as a type of thought that men swear by it. In argument they put their faith in numerical statistics, confident that their hearers will accept as valid that which is based on number relation. As direct results of the acquirement of this method of thought several values come to the students. They are, (1) the habit and ability to draw conclusions from given data, (2) the appre- ciation of the necessity of a conclusion, (3) the ap- preciation of a logical development back of every v conclusion, (4) the acquisition of an effective method in other scientific subjects, and (5) the ability to generalize conceptions. Indirectly the develop- ment of such a type of thought gives power in (1) grasping situations, (2) use of symbolic language, (3) appreciation of the great advantage in validity, (4) habits of self study, (5) in development of men- . tal strength, (6) in developing habits of quickness, \ neatness, orderly arrangement, reflection, steadi- ness, and certainty. These various values, mentioned under the group of direct results, are usually passed off as falling under the culture category and there the argument HABIT OF LOGICAL THOUGHT 31 ends. As a matter of pertinent fact these values are fully as practical and utilitarian as the fact values. Perhaps more so, for the facts can be taken direct from reference books if the demand is suf- ficient to make it necessary to have them, while the habit of logical thought can only be developed. One of the greatest assets a person can have in any line of work is ability to draw quick and certain conclusions from given data. In every occupation situations constantly form themselves to be solved by the man or woman who has the faculty of quick thinking and quick acting. The world seems to have endless opportunities for just this sort of per- son, and although the slow ones invariably attribute the advancement of their co-workers to luck, the real reason is in their superior ability to arrange the data into details and fundamental principles and then draw a quick conclusion. Nearly all large business transactions are made upon newly created conditions, with the advantage always with the one who thinks the most rapidly. To be able to see the conclusions involved in changing conditions is in itself power. Not alone is there power in ability to draw a conclusion, but likewise in the appreciation that a conclusion is necessary. Whatever the conditions of a situation are or whatever the particular phase of social activity the data has reference to, there must be a necessary conclusion. It is not always appreciated by the ordinary thinker, that a given set of data necessitates a conclusion. Thus it is 32 MATHEMATICS AND EDUCATION that so many men and women, though excellent workers, fail to succeed when given responsibility. They have power to handle extensive detail but no power of organization, owing to lack of ability to grasp the significance of similar characteristics. In our particular civilization there is a strong feeling toward compilation of statistics. This is a direct example of the appreciation of the necessity of a conclusion. Statistics are compiled to make pos- sible the drawing of a conclusion. Were it not for the fact that a feeling exists that a conclusion is implicitly expressed, there would be no value in statistics. Young men in business life who have confidence in the necessity of a conclusion are con- stantly on the alert to discover one. They study the detail of their department and, it is needless to say, they succeed. For one to be trained to feel the necessity of a conclusion is to make for efficiency. It is a practical value and it is a cultural value as well. Then too, the conclusion may be given. It is power to know that back of it is a logical develop- ment. It is the first step toward analysis. In the activities of everyday life one is continually asking why the particular conclusion is valid. The only test is to consider the development leading to the conclusion and test each step for validity. To know that the conclusion has a logical development is evidently essential. Mathematics is most valuable in giving students their first insight into the method of inference and approximation. With no claim that the training GENERALIZING POWER 33 can be carried over directly (which offends many educators), it is safe to say that one finds in mathe- matics a reasoning to conclusions which is so simple and certain that it may be used with profit as a beginning for such training. Every event or fact, regardless of the outward appearance of contin- gency, has a development which one trained in mathematics readily understands. There are an- tecedent events and .facts, perhaps not evident, but nevertheless existent, which lead to the conse- quent. It is surprising to note the simplicity of an analysis once there is method and appreciation of necessary antedecents. Mathematics, owing to its certainty, gives perhaps the best elementary train- ing in such mental activity. In other sciences a method is absolutely necessary to develop working laws; since work in them rarely starts before the eighth or ninth grades, the method may be develop- ed through the study of mathematics. / One of the characteristics of a strong mind is the /generalizing power, organization of details and ar- rangement under heads and sub-heads. The great- est movement in the world today is toward organiza- tion. It is in the air. Trusts organize under the leadership of men of genius until they control the whole of the social group. In every occupation the tendency is to organize. The results of such organization may be good or bad, but if education is to follow the conscious aim of society there must be that in the materials of education and the pro- cesses of education which tends to develop such 34 MATHEMATICS AND EDUCATION power. Is it too much to claim that the study of mathematics is one of the best means to such an end, that is when mathematics is taught as a liv- ing science rather than a dead one, and when stu- dents are taught to evolve their own mathematics? Unfortunately there are those among us who fail to distinguish between formalism and business method. Fortunately they are in the minority. Otherwise the work of education could not have reached its present state of efficiency. In educa- tion, as in business, detail must hold a subordinate place and systematic handling of this detail makes for economy in time and energy. Modern busi- ness has appreciated the importance of system to the extent of constructing an office that approxi- mates a machine in its exactness and quickness. If he whose raw material is iron and wood finds it advantageous to give its greater attention to funda- mentals, how much more advantageous is it to him whose raw material is the growing mind of the com- munity, to put his greater attention on the funda- mentals?' A business man not only decreases ex- 5 Dense and time by systematic methods, but he im- Droves the quality of business done. So it is with the teacher. It has been said with truth, that a student on one end of a log and a teacher upon the other constitutes a school. Unfortunately, how- ever, modern school systems are not so ideal. There are forty students for one teacher in the modern school. Thus system in handling is made essential. Not alone in the administrative department is econ- POWER TO GRASP SITUATIONS 35 omy of time and energy desirous but in every school- room as well. Each class must have its individual unity as well as its subordinate place in the larger system. It should have its individual organization. The study of any group of facts touching daily life, whether classified as history, language, mathe- matics, or what not, is educative. There are de- velopments which the student gets regardless of the particular content. Among these may be men- tioned the power to grasp situations, the power of yusing symbolic language, the power of testing for \validity, the power of self-study, and the power /developing habits of quickness, neatness, orderly | arrangement, reflection, steadiness, and certainty. It is not claimed that only through a study of mathe- matics can these developments be attained, but it is claimed that the study of mathematics is one of the most effective means to such ends. Extensive discussion upon this point is not necessary. These developments are admitted by all thinking people, just as they are admitted by mathematicians for all other subjects. They are values the student gets from systematic study along any line.] < .4 To study mathematics for its own sake is equiva- lent to studying mathematics for the various values mentioned, plus mental pleasure. We study art, music, literature, language, political economy, in fact all subjects, partly for their value in applica- tion and partly because of the pleasure they afford. The pleasure value is always a large one unless we desire to make of ourselves mere machines. Man 36 MATHEMATICS AND EDUCATION does not live merely to eat and sleep. His educa- tion should not be merely the means of acquiring food, clothing, and shelter. Essential as these things are, a human life has a much deeper meaning than this. The full measure of human happiness comes with a love for the beautiful and ideal in all things. That education then is not complete, which does not touch upon the ideal. In mathe- matics, as in all else, there is a beauty and an ideal condition. It does give mental pleasure to have the power to appreciate this. Is it then in vain, that the love of a perfect demonstration is encour- aged? Not unless we limit our educational ideal to cover the crudities of life. To live is not a diffi- cult thing in this world; to live joyfully is always the difficult thing to do. There is a value in study- ing mathematics for its own sake just as there is a value in studying any other subject for its own sake. The only point to remember is that the utilitarian value must be first. If education for the group means such organization as will give greatest opportunity for the individual to realize his ends, then the individual, as a factor of the group, must become aware of the necessity of the individual's appreciation of what is meant by the word welfare. There must be in the educa- tion of the individual, that training which promotes his desire to think and do what is best for the group life. He needs to feel that that action which alone seems to promote his own well being, is not entirely what it seems; for indirectly be becomes a hindrance INDIVIDUAL THINKING 37 to better organized life, not only through his own negative action but through the example which prompts others to do likewise. The social-welfare side of education must combat not only the ten- dencies of individuals toward selfish action, but also age-long customs inaugurated by primeval group life. This phase of education, though well enough understood, is extremely difficult for the teacher to stress. It is not something which can be taught directly but rather must be presented in an incidental manner through constant suggestion and example. It is difficult to maintain an approxi- mation to the ideal among adults who have the responsibilities of citizenship upon them. How much more difficult for care-free children to find an in- terest in things of this sort ! In fact, there is more or less loose thinking con- cerning this point on the part of teachers. The fact that teachers are underpaid and overworked does not help them to have full conviction of their professed faith. They find it is often almost neces- sary to sacrifice the ideal for individual betterment. It is merely proof of the general statement that in- dividual welfare is most prominent in the individual mind. In this connection, the encouragement of indi- vidual thinking on the part of the student is of great value. Every person is endowed with different abilities. To get the best reaction, individuality is essential. Only through the group life can man find his highest development, and the highest par- 38 MATHEMATICS AND EDUCATION ticipation in group life is possible only for him who emphasizes his own uniqueness. The modern ethi- cal conception is, that ethical laws exist for man. The center of attention is not the form but the substance. The student must realize the power of originality in thought and action. In a system of free education, to which the entire community contributes, training in citizenship is not only ex- pected but demanded. Those qualities making for a better and closer relationship between the in- dividual and the community, stand out prominent- ly as qualities to be desired in the student. It is an expression of the very essence of democracy. 1 Education must have as a part of its aim the common good. Modern society is in a stage of conscious development. So well understood is this that education is denned as a process of adjust- ment. Mere knowledge in its narrow sense is, then, not sufficient. There must be training, covering points of social activity and also of leisure, for social welfare is as much concerned with individuals out of action as with those in action. But the greatest value, perhaps, is derived from the definition of mathematics as a quantitative ap- preciation of the world in which we live. Denned in this way mathematics becomes a point of view, and has its application in all things. Not only is it possible to find special application in the various problems of science, history, language, etc., but also to study these branches of knowledge mathe- matically. It is evident that such a definition holds MATHEMATICS AS A POINT OF VIEW 39 true only when one supposes that the world in all its aspects can be thought of quantitatively. Such supposition is probably universally held. Also it supposes that all situations must be judged from one point of view at a time, the quantitative point of view being one. Just as things are considered from the artistic point of view, the utilitarian point of view, or the ethical point, so can they be con- sidered from a quantitative point of view. If mathematics, arithmetic in particular, is de- nned in this way, its great value includes the various ones mentioned, all of which function directly to- ward the perfection of the general value. All sub- jects may be considered as points of view. His- tory, for instance, judges the events with reference to events which have taken place, events which are contemporary, and events which may take place in the future, it judges an event with regard to its place in a sequence. Mathematics may in this way become vitally connected with all school ac- tivity. This value need not detract from the values of the subject as a distinct science. With the emphasis placed upon the idea of mathe- matics as a point of view there is given a basis for reconstruction. Only those processes having most common application should be stressed, and all processes having the character of mental gymnas- tics should be eliminated. The following list of omissions are suggested by McMurry, "apothe- caries' weight, troy weight; examples in longitude and time except the very simplest, involving the 40 MATHEMATICS AND EDUCATION 15 unit, since our standard time makes others un- necessary; 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 denomi- nators as those just mentioned; complex and com- pound fractions as separate topics; compound pro- portion; 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 pay- ments 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 this, arithme- tic 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." With such omissions made there would be ample time for full consideration of all subjects quantita- tively, that is mathematically. Even with a list of omissions less complete, more time could be de- MATERIALS OF MATHEMATICS 41 voted to the subjects which directly concern us in daily life. When one considers that education aims not so much toward the acquisition of a body of facts as toward power to interpret these facts in the light of serviceableness, mathematics becomes a functioning process, as do all other subjects. Materials of Mathematics From the preceding discussion the power for successful reaction of the individual calls for (1) a group of facts so basic as to be easily applied to all human'activity and so well mastered as to be available for all situations, (2) a group of facts which, although not used in every application, may be used in part of every application (contingent facts), (3) a group of facts used only in advanced scientific work, as for example, engineering, (4) a method of clear, and simple reasoning leading from given premises to a sure conclusion, (5) an appre- ciation of the necessity of a conclusion where a se- quence of events is given, (6) ability to analyze a situation and see the sequence relation between parts, (7) generalization and organization of de- tails, and (8) habits conducive to clear thinking, such as those of quickness, exactness, neatness, steadiness, etc. The materials of mathematics must be selected carefully, not only to obtain that which is best suited to production of power for reaction, but also to give during certain time limits that material best suited to the varying strength of the pupil. Also, the material must be selected with respect 42 MATHEMATICS AND EDUCATION for the fact that students start dropping out of school rapidly by the end of the sixth year. Unless the teacher finds an interest in these points there is educational waste and general inefficiency. The materials of mathematical education are (1) those of arithmetic, and (2) those of the advanced courses, including algebra, geometry, trigonometry, analytic geometry, and calculus. Because of the fact that all students study arithmetic, while only a rela- tively small number study the advanced courses, and also because of the fact that arithmetic is stud- ied from six to eight times as long as any one of the other courses, it is evidently our duty to devote the greater part of our discussion to it. With better citizenship given as the aim of education, arithmetic must stand or fall upon its merits with reference to this thing. Age long customs cannot be defended upon mere sentimental grounds. Un- less arithmetic contributes proportionately to the time and energy expended upon it, there must be change either (1) in methods of teaching, (2) in materials used, or (3) in both. Education toward efficient citizenship must be through purposeful work. All other work becomes drudgery and drudgery is deadening. It is true, no doubt, that we get our number ideas from meas- urement, comparison, and relationship of things. The number idea is generated within the mind by constructive activity. This theory opposes the theory that number is something perceived, and makes number a creation of the mind. ARITHMETIC 43 Measurement of things is the basic operation or activity and the first to be performed in order of time. Out of the physical act of placing the hand upon objects and at the same time speaking words, (number names), or pointing to objects and speak- ing words, (number names), the first activity of a child, the unit idea must develop. Then com- parison of the unit idea with the whole idea, and finally the ideas of analysis, synthesis, and relation. Out of the mechanical activity of counting grows the idea of measurement, or application of the unit idea. Measurement is impossible without the unit, so in some manner the beginning idea of the unit is developed through mechanical counting. Just how, it is probably impossible to say. There is a development from the counting of mere separate entities, to the counting of separate entities having common characteristics. The next step is one of analysis of given wholes into evident entities. This is no doubt a mechanical operation in most part. Synthesis and organization is the next step, and finally comes relation. During the counting pro- cess the number names are learned. The second step, analysis, brings out the idea that the names refer to parts of larger things. It is through the process of counting objects with common character- istics and through this elementary analysis, that the idea of a unit, a standard for measurement, is first generated. The fundamental operations bring out synthesis and organization. The work involv- ing the relative size of quantities with regard to 44 MATHEMATICS AND EDUCATION the same unit, relative size of quantities with regard to different units, change of unit, etc., bring out the idea of relation. Following this outline the whole development of the number idea is worked out. The materials of this development must be selected with regard to these steps. It is quite impossible to give exact limits for the development for each grade, owing to differences in children and the differences in teachers, but in general the work may be blocked out as follows: Kindergarten All work should be incidental with counting and grouping, brought in as occasion arises. This is to depend entirely upon the judgment of the teacher and the age of children. But by all means the counting started at home should be continued, and continued toward unit appreciation. The word "half" may be used, and perhaps "quarter" under proper conditions. Paper folding, beads, blocks, etc. give material for such work. First grade Counting should be continued and application made incidentally to all possible things, as for in- stance, the number in the class, the pieces of mater- ial used, as paper, pencils, rulers, clock numerals, days in week, and number as it occurs in other subjects, etc., etc. Very little should be done to- ward developing the number work as a thing apart. Here again the judgment of the teacher must be the deciding factor. The advance made upon the ARITHMETIC BY GRADES 45 kindergarten should be to remove counting from the merely mechanical operation to a purposive operation. The work in grouping and in combina- tions should be carried on, with the idea of making the mathematics a quantitative point of view. Second grade The incidental work should be continued and the number space, from one to fifteen or twenty, dis- cussed according to strength of students and the ability of the teacher. In this work the counting may be made to embrace the fives and tens. As formerly, the application should be to common objects. In number conbinations the addition and sub- traction series should precede the multiplication series. With the multiplication series the begin- ning of the multiplication table may be started. In the application the use of standard units should be encouraged. Also there should be the one to one correspondence between name, number-picture, and the figures or symbol. The blocks, splints, clock face, (a study of the Roman symbols from one to twelve should be made to give the student the ability to read spaces), and units in linear, dry, and liquid measure are presented through the use of apparatus; all are useful not for their own sake but to help present the desired idea. Throughout the first two grades the oral work should predominate. It is not advisable that all written or seat work should be left to the later grades because education is attained through the 46 MATHEMATICS AND EDUCATION use of all the senses, but there can be no doubt of the fact that very much of this is impractical. Sup't Wright of Michigan goes so far as to say, "The en- tire time of each class period throughout the first four grades should be devoted to oral work, and no seat work should be given. Written work may be introduced in the fifth grade and continued on throughout the course. However, the greater part of the work up to the seventh grade should be oral. Give a daily oral drill in every grade. Seat work, unless carefully watched and a time limit put on the class, leads to inaccurate, slow, and untidy habits of work, and, except in advanced work, in- creases neither the mental power nor arithmetical skill." To summarize, the work of the first two grades should cover counting, reading, and writing, within the number space from one to one hundred. The number space from one to twenty may be studied through addition and subtraction tables. The multiplication table may be studied through the fives up to ten and in cases to twelve. Division may be introduced as the inverse opera- tion of multiplication. Fractional terms may be used, such as , $, i, and perhaps -5-. Students are usually familiar with the terms from home training, so it is desirable to keep this connection with the home work. The units of linear, liquid, and dry measure may be introduced in incidental ways, and if occasion arises the signs +, , X, and H-. ARITHMETIC BY GRADES 47 It is a debatable point, whether the work should go beyond this point, and much depends upon the strength of the children and upon the skill of the teacher. In some cases addition may be carried on to cover two and three place numbers with simple carrying, with subtraction as an inverse process. Also division may be introduced, provided there is no remainder. Counting may be carried on as far as students are able. Third grade All the work of the second grade should be re- viewed, and series formed for addition and subtrac- tion in the space from one to twenty. The counting should be reviewed to one hundred and also by 2's, 4's, 5's, 10's, etc. This is aimed to make the student familiar with the number names and symbols. Following the mere counting must come addition and subtraction series and an advance upon the multiplication table. Three to six place numbers may be added and subtracted. Drill work in the early grades is important, not for the mental discipline so much, as for the famil- iarity with elementary facts, needed to carry on satisfactory work within time limits. The decimal scale may be introduced and coins used for illustrative material. In this connection the reading and writing of large numbers, together with pointing off, is profitable. Compound numbers also may be introduced with measurements in pounds and ounces; quart and 48 MATHEMATICS AND EDUCATION peck; minute, hour, and day; square measure, etc. Throughout the year every possible application should be made to home duties, involving building, gardening, buying and selling, etc.; to other sub- jects of study, such as simple geography, natural science, story telling, etc.; to the childhood games, including score keeping, etc. During the work in the first three grades sense training is of greatest value. Courses of study are always at hand for the teacher, and in general should be followed. The teacher, however, is the ruling factor always. Her judgment is of greatest value, for every school room is filled with a different group of students, and to her is left the decision as to how the material should be presented. In communi- ties where the foreign population is large the home preparation is often inferior. Here the teacher finds the course of study to to be followed accord- ing to the strength of the pupils. Summary : Addition through six to eight place numbers, together with drill on combinations. Subtraction of numbers involving simple borrow- ing. Multiplication table to tens or twelves, as strength of students warrants. Problems involving bor- rowing may be given. Division following the multiplication table, and short division when there are no remainders. * Introductory work in fractions limiting the work to unit fractions. ARITHMETIC BY GRADES 49 Denominate numbers involving the standard units and simple measurement. Fourth grade Not only review in addition and subtraction, but in multiplication and the multiplication table. Numbers of two and three places may be multi- plied together. The decimal scale and operations with decimals may be studied with profit. Com- pound numbers are studied with reference to stand- ard units of linear, liquid, square, and dry measure, and avoirdupois weight. Short division, and from this process, long division. In work on fractions, the use of names and sym- bols, reduction, and application, with reference to money systems and objects of daily use, should be stressed. Students should be taught care in statement of problems, use of words, and should be kept busy upon applications to home duties, and to other subjects of study. Summary : Addition may be extended to include numbers as high as ten places or more, if conditions warrant. Applications should be made through concrete ex- amples, with especial drill upon the common things of life, as for instance, the money system. Subtraction of numbers, involving borrowing two or three place numbers, may be used. Multiplication of selected numbers to three places, *and drill upon the multiplication table. When occasion arises introductory work on factoring may be carried on. STATE N 50 MATHEMATICS AND EDUCATION Division, as the inverse of multiplication, and work in short and long division. The work in fractions may be extended to cover addition and subtraction of like fractions, and ad- dition and subtraction of mixed numbers. Denominate numbers are to be used and the operations of addition and subtraction performed upon them. The Roman number system should be extended to enable students to count. Fifth grade ' During the first part of the year the stress should be on the operations of addition, subtraction, multi- plication, and division. Counting, together with the reading and writing of numbers, should be carried into the large numbers. In the Roman system the work should be carried to a point where students find no difficulty in reading and writing numbers. During the last part of the year, intro- ductory measurements and the solution of rather simple problems in getting areas and volumes may be taken up. Summarized, the work covers material as follows: 1. Counting, reading, and writing, both Arabic and Roman numbers. 2. Four fundamental operations. (Very im- portant.) 3. Fractions. Application of four fundamental operations. Rigid reductions. Operations involving fractions and integers are taken up. ARITHMETIC BY GRADES 51 4. Factoring with reference to prime factors, and simple common denominators and common multiples. 5. Decimal fractions. Relationship between the common and deci- mal fraction. Application of fundamental operations. 6. Areas and volumes. (Like and unlike units of measurement.) 7. Denominate numbers. (Four fundamental operations.) Sixth grade The work in this grade takes on a practical turn, and measurement and applications are stressed. Review of fractions, denominate numbers, and decimals should be given early in the year. Factoring with work on Highest Common Factor and Lowest Common Multiple may be continued. In work on measurement some educators prefer to spend an entire semester. While this is perhaps not best, it is true that much time can be profitably spent on such work. Care should be exercised constantly through this period to keep students speaking correctly. With the giving of more concrete problems the necessity for correct speaking is made a more important part of the work. Especially should care be exer- cised in the use of new words incident to the develop- ment of the subject. Seventh grade The dominant thing during this year's work is the varied applications of mathematics to the af- 52 MATHEMATICS AND EDUCATION fairs of life. Two phases stand out, the mechanical application and the business application. Since business has an interest for all students to a greater or less degree, while mechanical applications direct- ly affect only a part of the student body, much time should be spent on the basic principles of business processes. This work should be preceded by a careful review of common and decimal fractions, denominate numbers, and the completion of meas- urement, etc. As a large topic, percentage is the point of con- centration. The principles of percentage should be emphasized, and later, the applications, as in- terest, business method (including profit and loss), insurance, taxes, and commission and discount. There should also be introduced in this grade elementary work in the solution of problems using a symbol for the unknown number, and the equa- tion. This will give the students an added interest in the harder problems and open up a method of solution at once more effective and more interest- ing. To the work on measurements may be added ele- mentary work in geometric proofs. The work must aim primarily toward a method of proof and the acquisition of simple facts. Eighth grade The work of this grade should be a complete re- view of all that has preceded, with special review upon all usable facts. It is at the end of this year that many students leave school or, what is equiva- ALGEBRA 53 lent, the first part of the ninth grade. Students should have covered the principles of the subject by this time so that concrete problems may be given covering every part of the subject. During this year the mathematics should have two purposes; (1) to give all useful facts to the students in review, (2) to lead gradually into alge- bra and geometry. This last purpose is really very important, owing to the fact that many students ,et discouraged during the first year of the high school course, due to the absolute change in subject matter and methods of teaching. There has been too much of a chasm between the graded school and the high school, and too much made of the need of a changed method of instruction. There has been a tendency to force students out of school at the end of eight years who could easily remain for two extra years. The development should be as gradual and natural as that from the seventh to the eighth grades. Algebra The problem of algebra in the high school is one that has been greatly discussed during the past few years. The movement has been toward the making of algebra practical. Text -books are on the market stressing practical application more than all else. This movement is in the right direc- tion but there must be more than practical applica- tion to justify a place in the high school course for algebra. As a practical science alone, that is, practical in the sense of applying to the small affairs 54 MATHEMATICS AND EDUCATION of life, it is not worth a year's time. Judged upon this ground educators have good reason to contem- plate the removal of the course from the curriculum. Algebra, however, has values far and above this sort of a practical one. It is absolutely essential to all who contemplate work in advanced science or in higher mathematics. As a training in general- ization it is unequaled. Either of these values alone would justify the retention of the subject. In the seventh or eighth grades the use of the literal symbol should be introduced as a symbol for the unknown number in the problem, thus mak- ing the equation a useful tool in the solution of the problem. This will not only add interest and tend toward greater efficiency but will open the student's mind to the next step in the development of number conception, that of generalization. To follow this introduction with a course in algebra presented as a generalization of arithmetic solves the problem of spanning the gap at present existing between arithme- tic and algebra. Always there must be a vital connection with the processes of arithmetic. " i Geometry The values of geometry are, (1) the body of facts relating to measurement of plane and solid figures and, (2) the training in methods of proof. The selection of theorems to be proved should be made upon this basis, that the number of theorems proved is not so essential as the careful discussion of a few basic ones. GEOMETRY 55 There is too much of text-book teaching, the memorizing of proved theorems, and too little of original thinking. Merely to reflect what is found in the book gives very little development and is a course in memory training rather than a course in reasoning. The value of the science to the pupil is in the stimulation of the habit of carrying an argument from accepted truths to a logical con- clusion. Except in the case of a few theorems the valuable thing is not the conclusion but rather the development of the power to reach it. Teachers would do well to use the text only as a reference book and spend nearly all the time on original problems, taking care that the sequence of the theorem is from the simple to the difficult. The text with model proofs is of course necessary during the elementary work, and the basic theorems must be proved. By careful suggestion upon the part of the teacher, the theorems presented in the text may be proved by the student as original theorems. Constant quibbling over minor points and insistance upon stereotyped methods have a tendency to deaden the interest. Let the study of geometry aim toward the life of the individual. Being able to reproduce a num- ber of theorems as given in a text is of little moment, but clear, consistent, reasoning ability is valuable to every man and woman. The average citizen very, very seldom has occasion to resort to the knowledge of geometric theorems, but a dozen times each day he needs to solve some situation. Start 56 MATHEMATICS AND EDUCATION the student thinking and the usable facts will be remembered without effort. Vocational departments The movements in vocational work are (1) general training of all children in simple construction and in use of tools, (2) the teaching of the trades, (3) the continuation schools. With respect to the first movement, all of the larger schools and many of the smaller ones offer some work in the manual arts. While as yet the work is often crude and inefficient, the ideas of manual training and domestic science are pretty generally accepted. The problem is not one of introduction into the schools so much as one of providing equipment and teachers. This will work itself out rapidly. Such work demands the ele- ments of mathematics and offers great opportunity for teachers in the way of applied problems. The trade-schools are usually conducted as such, and are not connected with the public schools. However those who enter the trade schools usually get their elementary work in the public schools and must depend upon the training received in them for their number conception. In the work of the trade school the mathematics usually embodies the facts and principles involved in particular trades, as for instance, carpentry, book-binding, salesmanship, machine work, etc. Each particular trade has its own body of mathematical truths but all depend on the basic principles of number. Here again the teacher may find excellent opportunity VOCATIONAL DEPARTMENTS 57 for application, by taking in turn problems arising in the various trades. The continuation-school, although well established in Germany, is a new idea for American cities. The students upon leaving school to enter the var- ious lines of vocational work are kept at school a few hours each week. They then receive training in the particular trade in which they are interested. It is very apparent to these students that education means dollars and cents to them and that the work they had disliked before now becomes of the great- est interest to them. The active teacher will at- tempt to lay the foundations of number before all students with a direct reference to the working life of the adult. For instance a problem in percentage means much more to a student when the student is determining the amount of money he or she might receive at a rate of 2% on the day's sales. Point out that clerks often get part of their salaries in this way and for a moment allow the student to pretend to be a salesman. Covering the materials of mathematics there can be no rigid rule to follow grade by grade, owing to the great difference in students and teachers. The large idea to keep in mind is that each and every student in school is to become a citizen of the com- munity. He or she must find an occupation and take a part in the community life. It may be to fill the high place or the low place in society, but, nevertheless, to work out an existence, together 58 MATHEMATICS AND EDUCATION with a certain degree of happiness and contentment. It is beyond the teacher's power to decide what is to be the exact station of each child, but this much he or she does know, that, whatever it be, the future man or woman will bless those who contributed toward the making of a well balanced man or woman. Teachers are, indeed, guardians of youth, and very, very much is upon their shoulders. Only the best of our young men and women should have the privi- lege of caring for those who are to be, in total, our future state. Teaching is never a task, but rather a wonderful opportunity. Hold to your ideals, teachers, and elevate your profession. Methods and Modes In determining the method and the mode of teaching mathematics the teacher must have one large fact constantly in view, namely, that more than 80% of students in school will not enter the high school. This means that over four-fifths of the students in school must get all the school train- ing in the grades below the ninth. It follows that if the teacher plans the mathematics as a prepara- tion of higher subjects the whole effort is in vain for over four-fifths of the students, except for the values they get incidentally. If the higher branches are worth sacrificing four- fifths of the student body for, would it not be better to reconstruct in such a way that the energy of preparation is saved, and to give in its place the elementary parts of the advanced branches? Alge- bra and geometry are subject to unending criticism METHODS AND MODES 59 because of their impracticability. The truth of the matter is that there is very much of value to all students in algebra and geometry, and a great deal that is not valuable to all students in the courses in arithmetic as offered. Courses in trigonometry, analytic geometry, and calculus also have valuable facts for students to know. Further, these advanced courses offer as good training for the mental faculties as excessive arithmetic, and are as easily grasped in their elementary forms by young students as arith- metic. There is only one condition when mutual exclusiveness holds good, that being when all stu- dents complete the entire course. Frankly, then, would it not be better for us to readjust so that all students may have the oppor- tunity to get an elementary knowledge of the sub- jects at present kept from them ? They would lose nothing in mental training. They would gain in the knowledge of facts. About all the students would lose is some of the hatred they now have for eternal drill upon uninteresting parts of arithmetic. The parts of algebra of value to younger students are literal symbolism, the equation (simple in one and two unknown quantities), and an elementary knowledge of the fundamental operations. Logar- ithms are very valuable for approximating. From geometry one should use methods of proof, con- struction and estimates of areas and volumes. From trigonometry, methods of solution of triangles should be studied. From analytic geometry and calculus, graphical representation, and methods of 60 MATHEMATICS AND EDUCATION approximation can be considered with profit. All this material can be worked into the mathematics of the sixth, seventh and eighth grades with no loss whatever to the child, as far as mental training goes, and with great profit as to acquisition of fact values. Perhaps there is truth in the statement that prospective teachers of mathematics would do well to forget all that they have heard of method and mode, thus giving room for so-called common sense. In many cases there can be no doubt of its truth. The moment the prospective teacher loses the con- nection between the method or the mode and its purpose, is the moment forgetting becomes a virtue. There is but one problem always and forever before the teacher, to interest the pupil, to stimulate a desire for more knowledge upon the subject at hand, and to choose and present such material as will carry the student to the goal. There can not be many methods, and these methods cannot be mutually exclusive. There are different methods which, though shading one into the other, exist distinctly. These are the synthetic, the analytic, the deductive, the inductive, the Socratic, the Heuristic, and the laboratory methods. While the teacher never finds it possible to employ these methods individually in practice, owing to his personal temperament and the unique reaction of his own student group, yet there is advantage for him to appreciate the classi- fication and the advantages of each particular method. It will aid him in the construction of his working machinery, the machinery of the recitation ANALYSIS AND SYNTHESIS 61 the machinery of his mode of presentation. As one whose aim is to aid others to acquire knowledge of a subject, he saves time and energy by and through his application of method in the selection and ar- rangement of material. Each method has its ad- vantages in the adaptation to special situations. The synthetic method is usually thought of in connection with the analytic method. It is a mat- ter of common belief that the one puts together while the other tears apart. While this is not strictly true the basic principle is valid. The syn- thetic method arrives at the desired conclusion by application of known truths. It is, therefore, de- ductive in its arrangements, working from the known to the unknown. The analytic method breaks the larger unsolved problems into its parts, thus pro- ducing more simple problems which are more easily solved. It is, therefore, inductive in its arrange- ments, working from the particular to the general. It does not follow, however, that the synthetic method is identical with the deductive method, or that the analytic method is identical with the in- ductive method. There is a shading between the two in each case. It follows that the synthetic method is the one for the elegant proof and the analytic method is the one for the workman searching for facts in the problem. In the class room the synthetic method is not the best for daily use, owing to the questions which arise in the student's mind as to why the particular operations were indulged in rather than 62 MATHEMATICS AND EDUCATION others. There is proof with no explanation other than that it serves the purpose at hand. The ana- lytic method is the one to stimulate the student to attack the problem, to tear it into its parts, to find the truth or the known and thus to solve the prob- lem. Once solved, the synthetic method may be applied with profit and pleasure, for now the stu- dent appreciates the beauty and advantage of the clear and direct line of reasoning leading from the given or admitted part of the statement to the de- sired conclusion. A very common example of syn- thetic method is to be found in the proof of the geometric theorem in which the proof proceeds from axioms and definitions to a logical conclusion. This method of presenting geometry is giving way to one in which the student is encouraged to search out the proof unaided. The analytic method is that employed in much of algebra, for instance, in the solution of an equation. Nearly all theorems are proved by analytic methods originally by the mathematicians; therefore it is advisable to encour- age students to work out their own solutions rather than to present the finished work for them to mem- orize. One fault of text book teaching is that the student is forced to use the synthetic method, this because the synthetic method is that which gives the text its finish and certainty, and is used by authors for that reason. There is a happy medium to which teachers should attempt to approximate. This medium would be a method by which the class-room work would fol- INDUCTION AND DEDUCTION 63 low the analytical, and the final statements would be reconstructed and preserved in the synthetical. Absolute hobby-riding and hero worship as to method is one thing teachers must ward against, Too often a teacher, because of momentary success, adopts one method exclusively, or because of the signal success of a brilliant teacher of his acquain- tance, he adopts a certain method, hoping for like success. Each teacher as well as each class pos- sesses an individuality, and methods should be de- vised to meet the requirements of the hour. The commonness of the words, deductive and in- ductive, would seem to make comment unnecessary. However, contrary to common notions, mathematics in the constructive stage is as inductive as deduc- tive, although common belief is that mathematics is the deductive science. No doubt the widespread influence of Descartes has had much to do with the prevalent idea. The finished form is deductive, but the working method is quite as inductive as deductive. The use of the inductive method is growing, just as the use of the analytic method is growing. It is to be hoped that thinking teachers will aid the advance of induction without losing the appreciation of the value of the deductive method for the finished proof. First allow the student to become interested in particular problems. The solution of these will generate a desire to know the general rule and its proof. From this, in turn, the advance to the particular is welcomed. An example might be cited in the teaching of addition. First 64 MATHEMATICS AND EDUCATION create interest by particular examples which can be visualized. From the special cases the rule is gradually learned. The rule is then applied to the particular. Thus the student goes over the ground in its entirety, developing the rule and applying it. The introduction of induction into mathematics in no wise detracts from the characteristic of exact- ness which is one of the subject's attractions. Rather, it eases the path by which the student comes to the final statements. It injects the feel- ing of the discoverer into the student, so developing the feeling of ownership which is so essential to the state of satisfaction. The Heuristic method is the method of encourag- ing the student to discover. The teacher must see to it that the student does not waste time and energy upon false leads, beyond the point of ap- preciation of their falsity. The method is perhaps the most valuable for the teacher to adopt if the teacher uses it in the proper way. As an absolute dictum it is as unfruitful of results as any other one used absolutely. There is as little result in asking students to discover things without sug- gestion as to ask students to drill eternally upon prescribed rules and formulas. Used with the proper discretion it is the common sense method for excellence. It has its advantages and disadvan- tages, its merits and demerits, but on the whole the spirit of the method is good. It stimulates effort if suggestion is properly given. It prepares the way for original thinking, provided the teacher THE LABORATORY METHOD 65 makes suggestions enough to keep the students interested. This method has been tried and has not proved successful in the pure form. Too much time is necessary for one thing. Not only does the class recitation move slowly, but much outside study is necessary, that is, to complete courses that are at present required. Again, there is a great tendency for the unscrupulous teacher to indulge in relaxation of effort, hiding behind the method which asks the student to work without help. This is deadening for the student. The student is as helpless without suggestion as he would be in a chemical laboratory. This point becomes espec- ially strong when one considers that the average intelligence of a public school class is low and the young students have not reached the age of reason. Still again, to refer a student to outside references is equivalent to allowing him to take facts directly from outside sources. This, except for the gain in prolonged search, is the same as to take facts from the text in use. But despite the disadvan- tages, the spirit of the method is good. Used care- fully it is effective. The laboratory method is comparatively new, though the principle is much the same as the one which prompted Pestalozzi to start his great re- form. Pestalozzi re-established the method of us- ing objects in teaching arithmetic after a three century discard, dating from the introduction of Hindu symbolism. The laboratory method is a new reform, encouraging the use of all available 66 MATHEMATICS AND EDUCATION material in the teaching of mathematics, supple- mented by the cutting out of a great part of what is now taught. More ground is covered in less time with stress placed upon acquisition of facts. The method is built upon the idea that since interest is the essential thing in teaching and since facts are the essential thing in life, these facts should be given to the student by contact with their applica- tion. This method encourages correlation of mathe- matics with the natural sciences. It suggests the teaching of the same group by the same teacher, to give greater opportunity to the teacher for ap- plication. Such a method would require a large number of axioms and accepted proofs. It would mean that intuition would be used as an aid to reason, for nothing that was evident would be proved. As in the case of the Heuristic method, there are disadvantages to be noted. Interest for the child is not identical with interest for the adult. Often the child can be as easily interested in success- ful abstract operations as in manipulation of mechani- cal objects. After all, people enjoy doing those things they are able to do well. A boy is as inter- ested in turning a cartwheel as he is in laying brick. It is not the operation but the doing that creates interest. The interest is not something intrinsic. It is created by the teacher. The Socratic method is well known, the question and answer method. It is destructive and requires excessive time. Both these facts are reasons why it is not a good method to employ entirely. 67 Teachers often become enthusiastic over what some man has presented as a model method, and by following it absolutely get into many troubles. Methods often read well which do not work well. There are many things a teacher must constantly bear in mind. The various methods are in pure form the constructions of men biased in their opin- ions. The laboratory method, for instance, in its pure form is as impractical as its opposite. The Socratic method is an example of a once famous one now in the discard, for there was but one Socra- tes and he will never be reborn. All methods have their merits and demerits. Some methods, owing to peculiar conditions, are more efficient than others. On the whole the best advice is that you each de- velop your own. To do this it is needless to say that you will find aid and inspiration in acquiring a knowledge of what other teachers believe and what other people do. At the present time the Montessori method is claiming more attention than any other, but it is doubtful whether such a method will be adopted in the public schools of this country for some time to come. However, approximations to it can be made with profit, especially with regard to the creation of interest as a means of solving the dis- cipline problem to replace the old method of main- taining discipline of immobility. The Montessori school has a setting and is conducted under con- ditions much more favorable than is to be found in the elementary schools of the United States, but 68 MATHEMATICS AND EDUCATION the principles upon which the method is built are so basic that every teacher can apply them. In brief the method is one encouraging the spontan- eous activity of the child both physically and men- tally, under direction. The school life is made as nearly like the ideal home life as is possible. Play is made purposeful without detracting from its attractiveness. Much of the success of the school, in fact most of it, is due to the very remarkable woman, Dr. Montessori. Her ideas are not new with her; in fact they have been the ideas of educa- tors for many years and have been worked upon in many cases successfully; but it has remained for this woman to produce the most striking suc- cesses. The American schools are at present conduct- ing elementary work along the same lines and successfully. In our system it is quite impossible to produce the striking setting Dr. Montessori has, but it is possible to make application of the under- lying principles in our elementary work. Dr. Mon- tessori follows Froebel in the theory of activity but has perfected the practical system to demon- strate the truth of her belief. She is radical but has successes to back her assertions. Her system is imperfect, as she admits, but it is the greatest step in the direction of informality yet taken. The work in numeration covers simple counting, the learning of number names, the meaning of zero, the fundamental operations, and simple frac- tional ideas. The striking part of the method is INTEREST INDISPENSABLE 69 the appeal to the child's interest. The mode is one of arranging the games so as to give the child- ren the feeling that accuracy is the all essential thing. For instance, one of the first games is to hand the children slips of paper upon which a num- ber is written, zero included, and then each of them places the slip of paper upon the desk, passes to a table and collects a number of objects equal to the number written upon the slip. The teacher then verifies the count. In this game the child drawing the number zero remains quietly at the seat, not taking objects from the table. The teacher empha- sizes the importance of taking the exact number of objects. The teaching of the operation is car- ried on in a similar way with different games, but advantage is always taken of the child's natural activity rather than repression of it. Interest is essential. It does not follow that one must become the slave of the student's idle dreaming. Systematic handling of the machinery of the recitation is essential to economy of time and energy. It does not follow that the class room work must become purely formal and mechanical. Charts and mechanical devices often aid the student to see certain relationships. It does not follow that the class periods must be one continual round of manipulation of charts, maps, blocks, etc. Prob- lems involving the facts of farming stimulate the student as do problems in steam and electricity. It does not follow that students are never interested in the abstract equation. Lincoln became presi- 70 MATHEMATICS AND EDUCATION dent with an educational foundation based upon the three R's. It does not follow that we must not consider other subjects. The pendulum swings from extreme to extreme with teachers who become intoxicated first with one man's theory then another's. The process is 2500 years old, yet the school-man's problem is as great as it was when Plato lectured in Athens. But there has been progress. Perhaps it has been in spite of the school-man. Perhaps the invention of printing, the telephone, the telegraph, the steam and gas engines, and the modern newspaper, have done it. Be that as it may, the teacher of mathe- matics is to realize first of all that mere aping the work of a genius will never lead to success. You must awake to the fact that you are a responsible person about to become adviser to a group of young men and women who crave some knowledge of number relation. Your tastes and hobbies may not appeal to them, so do not attempt to force your ideas into their young minds. Use the brain God gave you and have an ideal of success for your goal. If you are unfortunate enough to be without brains or ideals, seek another occupation. The method has to do with the selction and ar- rangement of material; the mode has to do with the presentation of it. It follows that in great part the mode follows directly the method. In general, however, the modes can be roughly grouped under two heads the lecture and the recitation. The lecture mode is that in which the teacher presents LECTURE AND RECITATION MODES 71 all material, the student taking notes upon the pre- sentation; the recitation mode is that which requires the presentation to be made by the student, the teacher merely acting as quiz master. In the ex- aggerated form the recitation mode becomes an examination mode, the teacher not even offering suggestions. About the same criticism falls upon the various modes that falls upon the various meth- ods. Each has its advantages and disadvantages and each has a place in certain situations. Proba- bly no one mode is advisable in the pure form. The mode of presentation should be such a com- bination of the Heuristic, laboratory, individual, and class recitation, as seems best in the particular school. Students must discover their own truths as far as possible, preferably by the actual contact with their applications. Also, students must have opportunity to advance as rapidly as their ability warrants. Lastly, there must be class recitation, for one large value of school life is the community life, the appreciation of the individual's relation to the group. Recitation in the presence of others is invaluable as a training in a life problem social contact. Above all else, teachers must realize that in- terest does not reside in an object or a method except through the teacher's suggestion. The teach- er must create interest. Once created, interest solves many problems, among them, discipline, lack of attention, and aimlessness. Get the student's interest and almost anything can be accomplished. 72 MATHEMATICS AND EDUCATION The particular mode must vary for every class and every teacher, because both classes and teachers have their personalities. No universal formula can be written to cover all cases. A few things are essential. (1) Each student should keep a note book in which to keep a permanent reocrd of work done. (2) Each student should have pencil, ruler, and compass in order constantly to be ready to visu- alize the conditions of a problem. (3) The class room should be as complete a laboratory as possible, in order that students may see the application of facts learned. (4) Individual work should be ex- hibited to stimulate rivalry. (5) Slow students should have careful attention until the teacher dis- covers the reason for slowness, then corrective teaching. (6) Good students should have every door open to them for advancement. Summary Mathematics, that oldest of sciences, is up for discussion. From the earliest times to the most recent, the body of science coming under the title mathematics has grown and enlarged until in its many branches and ever growing detail it dominates all knowledge. Years of expansion under the spell of men of genius have been followed by years of quiet, only again to be followed by greater expan- sion under other men of genius until now, after many hundreds of years, with a legion of great minds sacrificed to the cause, the science stands impreg- nable. For those who have not the opportunity to devote years of preparation to the larger and SUMMARY 73 deeper study of its far-reaching expansion, it stands as a wonderful, mysterious secret of nature. For those whose minds have travelled into those remote regions, hardly explored, there is a charm as stimu- lating as that in the investigation of the astronomer into the almost infinite regions of space. Part after part has unfolded until from the original trunk many branches have grown. Now in its last successes another phase presents itself, that of presenting the fundamental principles to growing minds in a way to stimulate more expansion and greater successes on the one hand, and on the other in a way to give all men the advantage of the facts discovered. The question of mathematics is everywhere in the air, and unfortunately some have misinterpreted the growing unrest. It is not so much a criticism of the science, as such,- as it is a demand for more knowledge of it. Years of careless teaching have resulted in breeding discontent. It is now for the teacher to satisfy the demand by developing a bet- ter method and a more effective mode. The question of teaching mathematics is not a new one, by any means. Mathematics teaching has been before educators as long as teaching has, but never before has the whole teaching problem been so prominent as today. Free schools and a great increase in the number of students, together with the great advance in civilization, has developed many new problems. In the last analysis the situa- tion is reduced to one in which the efficiency of the teacher is the controlling thing. Methods and modes 74 MATHEMATICS AND EDUCATION are aids to him who can apply them to advantage. To all others new ways to do things are confusing. To be successful the teacher must have a back- ground more staple than a mere method. The ele- ments of this background are, (1) an appreciation of the aim and scope of education built up not alone upon the passing ideals of the present civilization, but also upon the developing ideal of human per- fection, (2) a clear and unbiased judgment covering the values of mathematics to the student, (3) ability to choose the materials of education with regard to these values, (4) a method of arrangement of material, and (5) a mode of presentation. The ideal of success is one which stands before all who now teach and who expect to teach, and to approxi- mate this ideal is the innermost aim of each and every one. It is the ideal preached to all, by those whose business is that of teacher-making, and it is the ideal as well, for all who aspire to higher things in any work. To attain it becomes a ruling passion. Together with a background, the teacher must possess strength of will to work out the problem before him or her regardless of obstacles. Stewart Edward White tells a story of a man in the far north who received orders to find and take an offender of the law of the north. He knew what hardship and suffering it meant to pursue one as crafty as himself through the wilderness, but he had no fault to find or questions to ask. The next year he re- turned. He had found his man and completed his task. What the obstacles were, and what the suf- SUMMARY 75 fering was, no one knew, but he had completed the work he had attempted and had attained success. You teachers may also attain success if you possess the power of character to carry you through every situation. In modern language, be alive, be keen, be forceful, be resourceful, be open to suggestion, be ready to profit by the experience of those who have succeeded. Above all, be honest with your- self and with the world. Teaching, the oldest and most respected of professions, has a future for you, if you search for it, as great as it has offered those who have made the civilization which we enjoy. 000933 315 LOS ANGELES STATE NORMAL SCHOOL