
Mathematics as a Universal and Permanent Element in Educationby William Betz  1937 No one who is genuinely interested in improving the teaching of secondary mathematics will deny that in many places the subject has been poorly organized and stupidly taught. The same is true, however, of the other great fields of knowledge, and the solution is not to throw the subject out entirely but to reorganize it and raise the standards for those who wish to teach it. This discussion should help to clarify our thinking and furnish us with some different if not new ideas for further action. Whether all readers of THE RECORD agree with every statement in the following discussion is not of great concern. With mathematics being dropped from the course of study in some schools and many people seriously questioning its importance in general education, it is time that someone should try to state its case. Mr. Betz is well qualified, not only from the standpoint of training and scholarship, to speak concerning the place of mathematics in secondary education, but also because of the intellectual honesty that he has always exhibited and the rich classroom experience which gives to anything he says a quality that even those who may disagree with him are compelled to respect. No one who is genuinely interested in improving the teaching of secondary mathematics will deny that in many places the subject has been poorly organized and stupidly taught. The same is true, however, of the other great fields of knowledge, and the solution is not to throw the subject out entirely but to reorganize it and raise the standards for those who wish to teach it. Mr. Betz's discussion should help to clarify our thinking and furnish us with some different if not new ideas for further action. — W. D. REEVE, Professor of Mathematics, Teachers College. THE present worldwide crisis in human affairs has profoundly affected both the outlook and the spirit of education. In a period of change and transformation unparalleled in history, the school is once again facing the unavoidable and vastly difficult task of examining critically its basic philosophy, its ultimate and immediate objectives, its materials of instruction, its organization, and its modes of procedure. It is only too obvious that with reference to each of these aspects of the educational process there exist at present fundamental and vexing differences of opinion. The writer has discussed these divergent points of view at some length elsewhere.1 In American schools most of the current educational difficulties may be traced not only to the disturbed world outlook but more particularly to the following specific factors: (1) a confusion of objectives due to aimlessness, caused largely by a pragmatic opportunism; (2) the doctrines of "progressive" education, with their emphasis on immediate experience, individual interests, and "felt needs," and their disregard of race experience and sequential learning; (3) a policy of continuous and incoherent curriculum revision based on momentary interests, "social reconstruction," and superficial "orientation," to the exclusion of continuity and foundational training in essential lines of work; (4) the unsolved problem of mass education, with the resulting attempts at "adaptation" to individual needs and interests, all of which attempts have been unsuccessful because they have ignored basic causes and problems, have rejected standards, and have preferred an inconsequential tinkering with surface adjustments; (5) a wrong psychology of learning involving a mechanistic conception of the mind; and (6) a narrowly professional and culturally inadequate training of elementary and secondary teachers. At present, those educators who subscribe either to the policy of "social reconstruction" or to the creed of the childcentered school represent very powerful pressure groups. When carried to their logical conclusion, these programs lead to planless schools in which the subjective whims of the pupils dominate the scene. Under the impact of these tendencies the curriculum in some secondary schools has become a collection of odds and ends for which no one can have respect. More than three hundred subjects now constitute the collective educational menu of these schools. It has become accepted doctrine that, aside from a moderate contact with the social studies and some appreciational work in art, the pupil should be free to follow his own educational preferences. That this means following the line of least resistance in the vast majority of cases only seems to enhance the evident vogue of the doctrine. In this perplexing setting, mathematics has been receiving a peculiarly negative treatment. Because it is a cumulative subject and demands continuous, honest exertion over a period of years, it does not fit into the philosophy of planless and effortless education. Accordingly the decree has been issued that it must be eliminated as a prescribed element of general education. At best, it is to be tolerated, though in very reduced form, in vocational or technical courses. The prevailing propaganda against mathematics has, however, a number of other sources, among which the following are particularly significant: 1. It is true that mathematics is acknowledged as an indispensable tool of modern science and invention, but among the inventions for which technology must be held accountable are those monstrously efficient implements of war that are capable of causing wholesale slaughter and mutilation, and hence untold human suffering. Mathematics, so we are told, shares with the other sciences the responsibility for making such horrors possible. 2. Invention and technology, aided by mathematics, are also responsible for the industrial revolution and hence are accused of having caused the economic misery of our age. 3. Again, it is claimed that if science and education had given more attention to the social aspects of life, to mental and spiritual factors, to the promotion of social solidarity, we should not have witnessed the worldwide catastrophe that so nearly destroyed modern civilization. Hence the supreme need of the moment is said to be a greater emphasis on a socially unifying and healing type of education, and not on increased technical and scientific training. 4. Last, not least, the impression is prevalent that the traditional place of mathematics in the curriculum is due to the "exploded doctrine of mental discipline," and that, accordingly, the alleged educational values of mathematics are largely a myth. The impact of these and similar attitudes and convictions has brought about a concerted movement to curtail mathematical instruction in every possible way, down to the barest rudiments. It will be shown in the following pages that this agitation against mathematics is not based upon facts, and that, on the contrary, we may confidently expect a return to a more rational appraisal. Those who are informed need not be told that mathematics does not require a "defense." Nor should any correction of the false assertions about the alleged futility of mathematical instruction be branded as the outcry of "vested interests." The oldest of all the sciences is too firmly rooted in the very nature of things to justify any worries about its future. It is, however, a matter of considerable importance to protect the rising generation against the pauperized education that will result from an uninformed crusade against basic elements of education such as mathematics. For that reason, the time has come for presenting, however briefly, a confession of faith that shall be in harmony with facts instead of opinions. In particular, an attempt will be made to answer the following crucial question: Precisely what is the real or potential contribution of mathematics that warrants its mandatory inclusion in any system of general education? Since this discussion is not intended for mathematical specialists, and is concerned primarily with elementary mathematical instruction, reference will be made only to such everyday mathematical contacts as would seem to be within the reach of every educated person living in this age of science and of rapidly shifting horizons. THE UNIVERSAL BACKGROUNDS OF MATHEMATICS Mathematics Anchored in Fundamental Human Needs. Mathematics is the oldest of all the sciences. Authoritative investigations show that the story of mathematics goes back to the dawn of human history. We now know that mathematics is the joint contribution of many lands and the common heritage of all mankind. No one person alone invented mathematics. Its origin was as inevitable and spontaneous as was the instinct of selfpreservation. For at all times and in practically all places human beings were confronted with the necessity of thinking constantly of how to obtain food and clothing and suitable shelter. These everpresent needs caused even the most primitive people to be concerned with questions such as the following: 1. How many people must be fed, clothed, and housed? 2. How much food and clothing have we on hand? 3. How long will our supplies last? Evidently, such questions could be answered only by counting and measuring. These two processes, the outgrowth of hunger, cold, and want, became the universal, basic impulses toward the creation of the imposing structure of mathematics. For counting led to a knowledge of number, while measurement produced a familiarity with ideas of form and size. To this day, counting and measurement, number and form, constitute the foundation of the mathematical edifice. Number as a Great Social Instrument. The significance of our decimal number system as a universal vehicle of human progress can hardly be overstated. The numbers which we use so constantly in counting underlie the world's commercial computations. They are also the "tools of measurement," making possible the scales and measuring instruments of science and industry, such as rulers, speedometers, and gas meters. These numbers indicate the hours of the day, the days of the month, and the successive years of the calendar. They denominate the pages and chapters of a book, the houses on a street, the owners of automobiles, the telephones in our homes. The vast economy resulting from the use of tables is due solely to the allpervading science of number. We find our way across trackless water and air only by a system of latitude and longitude and by steering devices resting ultimately on the scientific use of number. All statistical and economic research and all graphic representations have a numerical foundation. Hence number is one of mankind's great unifiers. The numerals represent humanity's one universal language. Professor Charles H. Judd says, It is literally true in modern life that an individual cannot maintain himself socially or physically unless he can share in the general social use of number. . . . He who is master of the number system has a way of thinking that the race has worked out with infinite labor. He will never again fall back into the confused and inexact ways of viewing the world which are characteristic of his childhood. He has grown intellectually by assimilating the number system. Precision shows the contrast between primitive and civilized.2 Form—Universal Bond of Civilization. The second great basic "unifier" contributed by mathematics is the study of form. For we are living in a world, as even the ancients realized, which rests forever on the dual foundation of number and form. In the first place, nature is our permanent and universal museum of form. In all zones and climates, it has furnished the prototypes that man uses in endless variety in all his applied arts. The constant recurrence of certain natural objects not only suggested and made possible the basic vocabularies of mankind; it also served to emphasize the endless repetition of important type forms and ideas. Fruits, trees, flowers, animals, shells, crystals, and countless natural phenomena were the visible frames of reference that caused man to become conscious of forms and patterns. The sky looked like a hemisphere. The rainbow was a huge circular arch. The disk of the sun and of the full moon resembled circles. Raindrops were seen to fall in parallel lines. Primitive peoples were driven to a knowledge of this potential reservoir of form not merely by curiosity, but by sheer necessity. They all had to have food, clothing, shelter, weapons, tools, and simple household implements. All these fundamental needs led to the gradual discovery and perfection of such practical arts as building, farming, weaving, the making of pottery and baskets, and the like. Each of these arts and their related activities necessitated an everincreasing knowledge of shape, size, and position, and so prepared the way for our present science of geometry. This common and unavoidable background of the practical arts is responsible to this day for the universal "alphabet of form" that Is known and appreciated in all parts of the world. Thus, the force of gravitation caused builders everywhere to depend on the vertical and horizontal positions of the main structural units. This emphasized the inevitable use of the right angle, nature's own dominant angle, and hence of the rectangle. The rectangle has thus become the universal and basic figure of civilization. In the same inescapable way, the circle, the triangle, the cylinder, and the rectangular solid became members of that group of twenty or more forms which constitute the framework of every applied art. Hence a knowledge of these forms, involving their construction, their measurement, and their properties has always been, and will continue to be, a unifying bond of the highest order in the educational systems of all civilized nations. In the western world, at least, we constantly move in an environment that stresses the same universal type forms, especially in our architecture, in the products of industry, and in all our practical arts. Whether we live in America or in Europe, the daily routine of life displays the same alphabet of form in endless variation. Basic Mathematical Ideas and Their Universal Background. Throughout the long ages of recorded human history, the constant use of number and form and their resulting interaction created as byproducts an evergrowing recognition and conscious application of fundamental mathematical ideas and principles. Crude comparisons, characterized by such terms as many, much, greater, smaller, more, less, led eventually to increasingly precise methods of counting and measuring. This involved the important notions of quantity and of correspondence. For whenever we count, we establish a onetoone correspondence between the objects of the group to be counted and the successive numerals of our number system. Again, when we measure, we establish a correspondence between a given size or quantity and a point on a given scale. The ideas of measured quantity and of precise correspondence are basic in all scientific research. This fact has been well stated by Professor A. N. Whitehead: Through and through, the world is infected with quantity. To talk sense is to talk in quantities. It is no use saying that the nation is large—How large? It is no use saying that radium is scarce—How scarce? You cannot evade quantity. You may fly to poetry and to music, and quantity and number will face you in your rhythms and your octaves. Elegant intellects which despise the theory of quantity are but half developed.3 Gradually, there also emerged a real understanding of such cardinal ideas as those of equality, similarity, congruence, and symmetry. All these have a natural background. Thus, the leaves of a given plant or tree all have the same shape. They are similar. The idea of similarity underlies all sciences of classification. When we deal with objects that are exactly or nearly alike, such as the grains of an ear of corn, we have an illustration of congruence. Again, the petals of a flower, the wings of a butterfly, the reflection of a landscape in a body of water, suggest the omnipresence of balance or symmetry. The corresponding parts of congruent or symmetric figures are equal. The four cardinal ideas mentioned above have countless applications in modern life. The mass production characteristic of modern industry involves the constant repetition of definite patterns or models. Hence it is based on congruence. Whenever the same article is manufactured in various sizes, we have an application of similarity. Again, symmetry and equality are illustrated, at almost every turn, in the home, in the street, in shops and offices. A modern automobile is a symphony of these basic mathematical ideas. In the same way, it would be a simple matter to prove the universality of such concepts as direction, ratio, proportion, variation, functionality, direct and indirect measurement, and the like. Summary. Even a moderate acquaintance with the origin and the spirit of mathematics serves to bring out the fact that this great science has a universal background. It is anchored in the very nature of things. It is as indestructible as the force of gravitation and as permanent as hunger, being coextensive with nature, science, and technology. Hence the language of mathematics is part of the language of humanity. It is understood everywhere because the world is incurably mathematical. MATHEMATICS AS THE UNIVERSAL SERVANT OF MANKIND In the first place, it is absolutely true that mathematics is a universally available and indispensable servant of all mankind. In cooperation with other sciences and technology, it builds our cities, our roads, bridges, tunnels, factories, and industrial plants. It constructs our engines, automobiles, steamships, and airplanes. It makes possible the countless inventions and conveniences that have served to make modern life less arduous, such as laborsaving devices in the home, in the shop, and on the farm. It has given us the radio, the movies, and the electrified home, and it is about to perfect television. If ever our understanding of economics and our social planning are to rise above the rule of thumb stage, and above the level of unintelligent selfinterest, it will be done with the aid of mathematical principles. As shown below, practically all our sciences have a mathematical core. Should not education everywhere give a place of honor to so farreaching and significant an enterprise as the study of mathematics? In the language of a distinguished scholar, it cannot be denied that Mathematics underlies presentday civilization in much the same farreaching manner as sunshine underlies all forms of life, and we unconsciously share the benefits conferred by the mathematical achievements of the race just as we unconsciously enjoy the blessings of the sunshine.4 Secondly, the doctrine of the transfer of training, based on the careful research work of men like Lashley and Grata, makes those views seem incredibly obsolete which still ignore the impressive evidence now available as to the significance and the possibility of general training.5 We shall therefore briefly discuss the universal cultural bearings of mathematics, often doubted but never disproved. MATHEMATICS AS A UNIVERSAL MODE OF THINKING Let us proceed to examine those essential habits of thought which mathematics contributes and which should be incorporated in the educational program of every citizen of the modern world. Even elementary mathematics involves a continuous emphasis on at least three basic types of thinking which may be described as relational thinking, postulational thinking, and symbolic thinking. We shall now consider briefly their allpervasive character and their place in an uptodate mathematical curriculum. It should be observed at the very outset that the mathematician really uses the terms relation, dependence, transformation, and function as synonyms.6 What, then, is their general significance? First of all, these ideas are among the central concepts and tools of all science. To a large extent, modern civilization is their product. No one has stated this connection with greater clearness than Professor C. J. Keyser of Columbia University: Every one knows that one of the outstanding facts of our world is the great fact of Change. The world of events, whether great or small, mental or physical, is a flowing stream. Transformation, slow or swift, visible or invisible, is perpetual on every hand. But events are interdependent so that change in one thing or place or time produces changes in other things and places and times. With the processes of change every human being must deal constantly or perish. The processes of change are not haphazard or chaotic, they are lawful. To deal with them successfully, which is a major concern of man, it is necessary to know their laws. To discover the laws of change is the aim of science. In this enterprise of science the ideal prototype is mathematics for mathematics consists mainly in the study of functions and the study of functions is the study of the ways in which changes in one or more things produce changes in others.7 As to the allinclusive character of relations, the same writer says, Each thing in the world has named or unnamed relations to everything else. Relations are infinite in number and in kind. To be is to be related. It is evident that the understanding of relations is a major concern of all men and women. Are relations a concern of mathematics? They are so much its concern that mathematics is sometimes defined to be the science of relations.8 An equally broad statement, made by Professor Keyser, is the following: It seems that everything, great or small, is related to everything else. Can you think of something that has no relations at all? If you can, the thing has to you the relation of being thinkable by you, which means that, if you can, you cannot. Can you think of two things absolutely unrelated to each other? If you can, the two things are diverse, else they would not be two, but diversity is a relation, a very important one; and so it appears again that, if you can, you cannot. Whether we work or play or rest or sleep or dream or wake, our lives are immersed in a sea of relations. A few months ago Einstein was reported in The New York Times as having said: "I had the experience to find out by reason the meaning of relation. It was, for me, wholly different from any other thing in life. It seemed to me a revelation of the Highest Author, and I will never forget it." I know not if the report be true but I am sure it ought to be. When we are studying Science, said Henri Poincare, we are studying not things, but their relations. The universe appears to be an infinite network of relations. Being, said Lotze, consists of relations. To be is to be related. Not to be related is not to be.9 Another exponent of these important considerations, Professor A. N. Whitehead, the distinguished Harvard mathematician and philosopher, reminds us that science is "the thought organization of experience." This organization is made possible only by the conviction that there is such a thing as an "order of nature," and by observing the "interconnectedness of all events." "This possibility of disentangling the most complex evanescent circumstances into various examples of permanent laws is the controlling idea of modern thought."10 Now, one of the most obvious of natural phenomena is that of rhythm, of recurrence or periodicity. "The whole life of Nature is dominated by the existence of periodic events. . . . The presupposition of periodicity is indeed fundamental to our very conception of life."11 We are reminded of this fact every moment by the beats of the heart and the recurrence of breathing. "Our very natures have adapted themselves to such repetitions." In the same way "days recur, lunar phases recur, rotating bodies recur to their old positions." The study of heat, light, sound, magnetism and electricity is indissolubly connected with the phenomenon of waves of energy and hence of periodicity. When we tune in on a radio, we enjoy one of nature's dominant periodicities. Human speech and music are based on the rhythmic organization of nature. "On every side, we are met by recurrence. Apart from recurrence, knowledge would be impossible; for nothing could be referred to our past experience. Also, apart from some regularity of recurrence, measurement would be impossible. In our experience, as we gain the idea of exactness, recurrence is fundamental."12 Mathematics is the scientist's tool for stating the laws of recurrence in exact form. It is this fact which makes mathematics indispensable in the study of countless material, mental, and social phenomena. Wherever there is such a thing as "order in change," mathematics appears on the scene. Hence it is, as Klein expressed it, that the world is becoming "functionallyminded." "Industry and commerce, economics and politics are becoming saturated with functional ideas, so much so that there is an increasing demand for men with an expert knowledge of 'functional economics.' "13 In the mathematical classroom, the pupil encounters at every turn the study of relation, of dependence, and of functionality. He meets these ideas in the formula, in the equation, In every graph he draws, in the geometric constructions and relationships he investigates, in every case of proportion and variation, in all his tables, and in every problem given him to solve. Surely, it is Impossible to mention a more universal or a more liberalizing factor in all education than that represented by relational thinking. What relational thinking has come to mean in the study of the natural sciences, postulational thinking signifies in all cases of orderly or consecutive mental reactions. Curiously enough, although the advent of modern science dates back only to the period of Galileo, the conscious realization of the possibility and the actuality of "ordered thinking" was the priceless contribution of Greek philosophers and mathematicians. For many centuries the world's great model of "straight thinking" was Euclid's Elements, a treatise on geometry which for two thousand years remained without a serious rival. It was the first and the most famous attempt in history at "autonomous" thinking. Unfortunately, not until our own day was its real significance understood. Not its geometric propositions, but its method o£ approach, made it a landmark for all time. For here we have, for the first time, the conscious and masterful application of deductive procedures. Induction and deduction are two of humanity's great tools in its eternal search for truth. Both are instruments of discovery and also instruments of verification. When we proceed deductively, we begin with a few undefined terms or concepts and a few statements of relations that are assumed or postulated. For thinking cannot be carried on in a vacuum. We must have a starting point. This is furnished by the basic concepts and assumptions placed at the beginning of every deductive investigation. We then proceed to find out, by logical processes accepted as sound, what conclusions follow from this given foundation. In other words, just as we have in the physical world the inexorable relation of cause and effect, so in the mental world we have logical consequences following from given premises. "If this is true, then that is also true," is a relationship that applies, under given conditions, as unerringly as does the law of gravitation. Moreover, it is so universal in its scope that it would be hard to mention a domain excluded from it. There are many misconceptions about the educational significance of postulational thinking, of logic, in everyday life. One of these is to the effect that most people think "straight" anyway and that special training in that field is unnecessary. It would seem that even a slight acquaintance with newspapers and with current events should correct such a naive faith in untutored intelligence. The comfortable thesis that all the training in correct reasoning which we need in everyday life is acquired incidentally in connection with our ordinary experiences is not shared by so profound a thinker as Professor Dewey: While it is not the business of education to prove every statement made, any more than to teach every possible item of information, it is its business to cultivate deepseated and effective habits of discriminating tested beliefs from mere assertions, guesses, and opinions; to develop a lively, sincere, and openminded preference for conclusions that are properly grounded, and to ingrain into the individual's working habits methods of inquiry and reasoning appropriate to the various problems that present themselves. No matter how much an individual knows as a matter of hearsay and information, if he has not attitudes and habits of this sort, he is not intellectully educated. He lacks the rudiments of mental discipline. And since these habits are not a gift of nature (no matter how strong the aptitude for acquiring them); since, moreover, the casual circumstances of the natural and social environment are not enough to compel their acquisition, the main office of education is to supply conditions that make for their cultivation. The formation of these habits is the Training of Mind.14 Again, the farreaching effect of sound habits of thinking is pointed out by Dewey as follows: To prove a thing means primarily to try, to test it. ... Not until a thing has been tried—"tried out," in colloquial language—do we know its true worth. Till then it may be pretense, a bluff. But the thing that has come out victorious in a test or trial of strength carries its credentials with it; it is approved, because it has been proved. Its value is clearly evinced, shown, i.e., demonstrated. So it is with inferences. . . . What is important is that every inference shall be a tested inference; or (since often this is not possible) that we shall discriminate between beliefs that rest upon tested evidence and those that do not, and shall be accordingly on our guard as to the kind and degree of assent yielded.15 All scientific applications, and especially all attempts at prediction, are based on deduction. "If this concept, or this hypothesis, be sound, then such and such consequences must follow." By using this procedure, Adams and Leverrier located the planet Neptune which previously had not been seen by any human eye. Similarly, Lowell predicted the discovery of the planet Pluto, fifteen years before it was actually observed. Mathematics, especially in the form of geometry, represents the most nearly perfect system of postulational thinking which has been evolved by the human mind. It is also "the most original creation of the human spirit." When correctly taught, it is a great emancipator of mankind, for its interest is forever in the creative and correct use of clear concepts and foundational principles. We are therefore obliged to endorse Professor Whitehead's appraisal of the basic thinking which is characteristic of mathematics: Logic, properly used, does not shackle thought. It gives freedom, and above all, boldness. Illogical thought hesitates to draw conclusions, because it never knows either what it means, or what it assumes, or how far it trusts its own assumptions, or what will be the effect of any modification of assumptions. . . . Neither logic without observation, nor observation without logic, can move one step in the formation of science. We may conceive humanity as engaged in an internecine conflict between youth and age. Youth is not defined by years but by the creative impulse to make something. The aged are those who, before all things, desire not to make a mistake. Logic is the olive branch from the old to the young, the wand which in the hands of youth has the magic property of creating science.16 When a child has learned to speak his first words, he has begun to share the boundless magic of symbols. A mere sound, made meaningful by social agreement, has placed a part of his environment at his beck and call. And as he extends his vocabulary, he finds that by thinking and speaking in terms of meaningful symbols, he escapes increasingly from the tyranny of things and becomes a social being. Hence the invention of symbols, as Professor Dewey points out, was the greatest single event in human history.17 Without symbols, he says, "no intellectual advance is possible; with them, there is no limit set to intellectual development except inherent stupidity." And when the art of writing was added, man became, in the forceful language of Count Korzybski, a "timebinding animal." For he was able to record and hence to transmit his experience to future generations. From that time on civilization became cumulative. It is this fact which has made the symbolic art of reading one of the basic tools of humanity, for it is the dramatic recital of the unending adventure of human ideas in their onward march toward the future. Through his contact with the arts of speaking and reading the child's mind is placed in possession of the experience of the race. In due time he acquires the meaning of the particular word symbols and written symbols which humanity has created for purposes of counting. The written number symbols introduce the child to the vast domain of mathematical shorthand, humanity's universal shorthand. Perfected after thousands of years of continuous endeavor, this symbolism enables the modern child to perform feats of computation that would have seemed like miracles even to the giant minds of ancient Greece. Throughout the ages there has been at work a still more potent and comprehensive impulse in the direction of mathematical shorthand. It may be traced to the recurrent quantitative problems of everyday life and to the recurrent phenomena of nature. Thus, in countless situations it is necessary to find the area of a rectangle. After many vain attempts to solve this difficult problem, the use of square units led to the rule we now employ so commonly. In the course of time, all standard problems were solved by certain corresponding rules. Eventually, the suggestion presented itself to shorten these rules by means of symbols, precisely as number names had been replaced by symbols. Such a shortened mathematical rule is called a formula. Again, nature's rhythmic processes become a source of power, of control, only when they can be stated in a concise, objective manner. Once more, this is accomplished by the symbolism of the formula. Algebra has contributed the machinery for expressing in a universal code the solution of life's recurrent quantitative problems, as well as the cosmic relationships that reveal the secrets of nature. Hence this code represents the alphabet of science and industry. The formulas used by the engineer and the scientist are the cumulative result of agelong investigation and effort. They represent a treasure house of information, a key to knowledge, an armory of priceless value, shared by all mankind. Is all this a fantastic dream or an undeniable fact? Certainly the latter. Our clearest thinkers are substantiating the old Pythagorean doctrine, once dimly understood, that "number lies at the base of the real world." It is not an accident that during the nineteenth century pure mathematics made almost as much progress as during all the preceding centuries of human history. The forward march of science has coincided with the advance of mathematical thinking. "Galileo produced formulae, Huyghens produced formulae, Newton produced formulae," says Whitehead, and modern science is built on the formulae refined by men like Maxwell, Planck, Einstein, and de Broglie. MATHEMATICS AS A HUMANIZING ELEMENT IN EDUCATION It should hardly be necessary to remind educators that the very heart of the educational process is in thinking. It was thinking that distinguished man from the beasts and that lifted him to higher and ever higher planes. And correct thinking will continue to be our only hope in the direction of establishing "the good life" on this earth and in bringing about international understanding and good will. "The power to think," says Professor Bode,18 "is the educational kingdom of heaven; if we seek it persistently, other things will be added unto us." Again, Professor Dewey has reminded us that No experience having a meaning is possible without some element of thought. . . . Thinking is the method of intelligent learning, of learning that employs and rewards mind. . . . Processes of instruction are unified in the degree in which they center in the production of good habits of thinking. While we may speak, without error, of the method of thought, the important thing is that thinking is the method of an educative experience. The essentials of method are therefore identical with the essentials of reflection.19 Now, mathematics—especially the study of geometry—is the world's most potent laboratory of thinking, and thinking of a type that can be fruitfully applied in countless directions. For back of our industries, back of our inventions and our social institutions, there stands—as the originating and organizing force that keeps all things moving—the trained mind of the thinker. Why is it, however, that mathematical thinking is said to be capable of the extensive transfer to other domains that is so confidently claimed for it? The secret lies in the method of mathematics—the method of carefully selected and clearly enunciated postulates, of sharply and completely defined concepts, and of painstaking deductions or demonstrations.20 The constant necessity, in mathematics, of deriving the ultimate validity of an argument from its fundamental assumptions, the challenging opportunity of testing the correctness of every step, and the cumulative performance of building up an organically interwoven system of truth—these and the related aspects of mathematical "reasoning" constitute the distinctive glory of a subject which has fascinated a legion of enthusiastic admirers throughout the ages.21 Hence it is not an exaggeration, but literally true, that "every major concern among the intellectual concerns of man is a concern of mathematics." Mathematics is universally applicable precisely because of its intellectual and abstract character. There can be no dispute about the statement that 2 + 2 = 4. It holds true through time and eternity and in all manner of circumstances. It is customary to speak of "cold" figures and of "cool" logic. If this means that facts as such are independent of sentiment or personal whim, such phrases contain an undeniable truth. The propositions of mathematics are objective. They are of the form, "if p, then q" But the whole universe is built on relations. Obviously, then, the attribute of "coldness" is not an inherent quality of truth or of natural phenomena. Is a sunny landscape "cold"? To science it is a complex assemblage of molecular aggregates or vibrations, totally devoid of "color," "beauty," "purpose," and the like. To the artist and the poet, it is a source of rapture and of endless inspiration. Just so, mathematics, as a vast system of ideas, principles, and processes, may be viewed from the standpoint of emotionless, critical analysis, or from that of the artist who beholds with delight a finished masterpiece. MATHEMATICS AND INTELLIGENT CITIZENSHIP Upon the recent occasion of the opening, in Rockefeller Center, of the New York Museum of Science and Industry, Dr. Robert A. Millikan, speaking by radio from Pasadena, California, discussed eloquently the supreme importance of cultivating "a rational attitude of mind throughout the whole population." And on another occasion, also in New York City, Professor David Eugene Smith of Teachers College spoke with unexcelled eloquence on the theme "Mathematics in the Training for Citizenship."22 Is it necessary to point out that today our conception of "citizenship" should be extended to include "world citizenship"? At an everincreasing pace, the nations of the world are being driven to choose between cooperation and mutual understanding or inevitable decay or destruction. How does mathematics contribute toward the cultivation of a really social outlook in the arena of nations? First, by helping to implant in the youth of all lands a spirit of "rectitude." In mathematics, a statement is either right or wrong. Only correct processes will lead to a desired result. An incorrect answer will not "check." One cannot deceive, cheat, or lie in mathematics and "get away with it." Errors show up eventually with unfailing certainty. Is it necessary to point out the close analogy between mathematics and both personal and national integrity? Second, mathematics is no respecter of persons! It is democratic. There is no royal road to mathematical learning. All must travel the same straight and narrow path to master its fundamental structure. Truth is binding in all cases. Just so, the moral law is not to be infringed upon by the whim of the individual or of any nation. The Golden Rule is a universal precept. Third, mathematical principles, once established on an accepted basis, are eternal. With a brave confidence rising above the level of mere optimism, we may likewise expect the great ethical standards and ideals of mankind to prevail, finally overcoming all forms of lawlessness and blind egotism, in favor of genuine cooperation and enlightened social behavior. In the interest of improved international relations, the teaching of mathematics can and should make definite contributions even directly. Increasingly, good teachers and textbooks of mathematics are displaying in graphic form all manner of significant economic, industrial, social, and educational data. In the story of the world's supplies of food and raw materials is told the story of the essential interdependence of the leading nations. Such items as international security and the deplorable armament races point out the incredible drain on the financial lifeblood of the world. The study of taxation, insurance, welfare and relief expenditures, of hospitalization, budgets, pensions, savings, and the like, when handled even in elementary classes by wellinformed teachers, may become powerful vehicles for the creation of that social solidarity which has been the dream of humanity's finest spirits. MATHEMATICS AND ART We have referred to the fact that one of the basic phenomena of the world is rhythm, regularity, recurrence. This regularity applies not merely to events, but also to their visible manifestation in the realm of form. To this must be added the controlling force of gravitation which compels molecules as well as the giant suns to follow fixed paths. Hence we find in nature an allpervading principle of order, a definiteness in the structure of all material things, a symmetry that often extends to minute details: Nature's attention to a very small detail is well known in the case of the crystal, for if we examine the octahedron we find it a figure bounded by eight equilateral triangular planes meeting one another at 12 edges at an angle of 109° 8’, not, be it noted, 108°, 109°, or 110°. The symmetry and beauty of the various forms of snowcrystals are perfect, and the number of different designs that can be wrought on the basis of the sixrayed figure is surprising indeed. "To the production of the exquisite pattern of a crystal there go many more minutely nice arrangements than to the construction of a watch." We can understand the natural philosopher, Sir David Brewster, being so impressed with this precision that he would sometimes exclaim in his laboratory, "Oh, God, how marvelous are Thy works!" When we turn to the realm of the living, we find so much that shows design, end in view, aim to be achieved, order, method and system that the real difficulty is to decide what to speak of first.23 The regularity, rhythm, and symmetry of the universe we experience as beauty. It is the basis of all art. Now art has both a subjective and an objective aspect. Those who have no ears to hear cannot appreciate the most haunting music that may fill with rapture differently attuned souls; but the objective fact is that such music exists. Likewise, beauty of form is the objective basis of all the representative arts, such as painting, design, and sculpture. Increasingly, even the practical arts are aiming to reflect not mere utility but also beauty. The relation of mathematics to art is a very close one. When primitive man invented his first designs, nearly always geometric in form, he was at the threshold of art. Such designs, often identical in every respect, may be found in all lands and climates. At first, these designs may have been merely proprietary identification signs, very much like the trademarks now used in industry. That they also served an important purpose in connection with tribal rituals or protective ceremonies can hardly be doubted. This idea is still reflected in our various national or ecclesiastical emblems, the distinctive badges or insignia of private or public organizations, and the like. But patterns were used, above all, for purely decorative purposes, appearing eventually in all the practical arts, and in everincreasing variety. These decorative designs evidently correspond to a hunger of the human soul for beauty. At present, the young pupil in mathematics is initiated into this domain even at the junior high school level. In his later geometry work, he also studies and applies the principles which are used in scale drawing, in architectural plans, and in other constructive enterprises. He learns to appreciate the idea of perspective and of balance or symmetry. Under skillful guidance he will thus become acquainted with the universal alphabet of form and rhythm, and eventually he may even be induced to read such inspiring treatises as Hambridge's Dynamic Symmetry, or Birkhoff's Aesthetic Measure, or Bragdon's Frozen Fountain, or D'Arcy Thompson's Growth and Form, or Cook's The Curves of Life. Here is a reservoir of power and inspiration that should be the common possession of all mankind. Should not the schoolroom emphasize this common aesthetic background, thus adding another great element to the unifying contributions of mathematics? Whether our starting point is a purely practical motive or an interest on a loftier plane, there can be no doubt about the ultimate goal. Modern science began with measurement. Measures are great teachers. They teach truth. By modern methods, scientists measure accurately how things happen in nature and in experiment. Such measures build civilization. They are the numbers which rule the world of enterprise, the unseen frame of all achievement. The engineer puts measures to work in skyscraper, bridge, and other structures. Measures flow through his pencil to scale drawings. By means of measures the engineer builds his dream of beauty. When the cathedral stands finished, strong and beautiful, we forget the measures, but they remain forever the strength and beauty of the cathedral. The blue print speaks the language of measurement. In study, shop, and laboratory the world of "Tomorrow" is being traced in paper dreams set to measured scales. Measurements are thus shaping the pattern of the "Wonderlands of Tomorrow."24 SUMMARY AND CONCLUSIONS We have tried to present a brief account of the universal and permanent aspects of mathematical instruction. We have examined the service values of mathematics, its cultural values, and its humanizing bearings. We have seen how mathematics may assist the cause of international understanding and cooperation. The question now arises whether the type of mathematical teaching which is prevalent in our schools is in harmony with this picture. The answer, alas, must be negative. But the same thing can be said of every other human enterprise. Is our governmental machinery functioning perfectly? Are our courts always a source of pride? Or our churches? Is our home life ideal? Are we proud of our divorce statistics? The school, too, is far from being a perfect embodiment of our educational aspirations. This very condition, however, constitutes a challenge. Our teachers are not properly trained, our textbooks are still geared in terms of manipulative, mechanical processes, and our examinations completely ignore cultural considerations. We are moving in a vicious circle which can be broken only by heroic effort. Mechanistic examinations based on mechanistic books and on mechanistic teaching—that is the unfortunate rule rather than the exception. We must reverse the situation. Inspired teaching will lead to better books and to more adequate examinations. Above all, we must get rid of our archaic curriculum which lacks coherence and continuity, and substitute for it an integrated program in harmony with the suggestions of experienced teachers and of leading mathematical thinkers. In the preceding pages we have tried to offer evidence in favor of the potential breadth and the wide applicability of mathematical training and thinking. There are gratifying indications that this point of view will be recognized more universally in the near future. As Professor Harold Hotelling has forcibly expressed it. Mathematics, to my way of thinking, is the most general of all subjects. Everything else is more special than mathematics. There is nothing that has a richer profusion of applications, there is nothing that travels over the whole domain of human knowledge as does mathematics. There is no surer key to unlock all sorts of doors than mathematics. . . . The remarkable thing is that whereas other methods of research are appropriate to special domains, to special kinds of investigations, mathematics—the same, identical mathematics—is applicable to this enormous profusion of different kinds of investigation and is adapted to bringing to light truths of the utmost diversity.25 If this position is accepted as essentially sound, there seems to be no escape from the same writer's conclusion that "the educated man or woman of the coming generation cannot neglect to study mathematics."26 1 Betz, W. "The Reorganization of Secondary Education," The Eleventh Yearbook of the National Council of Teachers of Mathematics. Bureau of Publications, Teachers College, Columbia University, New York, N. Y., 1936. 2 Judd, C. H. The Psychology of Social Institutions, p. 104. The Macmillan Company, New York, N. Y., 1927; also, "The Fallacy of Treating School Subjects as 'Tool Subjects,' " in The Third Yearbook of the National Council of Teachers of Mathematics, p. 10. Bureau of Publications, Teachers College, Columbia University, New York, N. Y., 1928. 3 Whitehead, A. N. The Alms of Education and Other Essays, p. n. The Macmillan Company, New York, N. Y., 1929. 4 Slaught, H. E. "Mathematics and Sunshine." The Mathematics Teacher, Vol. 21, p. 246, May, 1928. 5 See, for example, Lashley, K. S., Brain Mechanisms and Intelligence. University of Chicago Press, Chicago, 1929; Grata, P. T., "Transfer of Training and Educational PseudoScience," Educational Administration and Supervision, Vol. 21, pp. 241264, April, 1935; also The Mathematics Teacher, Vol. 28, pp. 265289, May, 1935. Recently, Grata has completely restated the problem of transfer in a paper of farreaching importance, entitled "Transfer of Training and Reconstruction of Experience," The Mathematics Teacher, Vol. 30, pp. 99109, March, 1937. Attention may also be called to Dr. W. A. Brownell's summarizing monograph on the "Theoretical Aspects of Learning and Transfer of Training" in Review of Educational Research, Vol. 6, pp. 281290, June, 1936. Among his concluding sentences the following is particularly significant: "The possibility and the desirability of transfer cannot be questioned." Of interest, also, is W. L. Uhl's article entitled "Timidity About the Transfer of Training," in the JuniorSenior High School Clearing House, Vol. 7, pp. 493494, April, 1933. 6 Keyser, C. J. "Three Great Synonyms: Relation, Transformation, Function." Scripta Mathematica, Vol. 3, pp. 301316, October, 1935. It should be noted that "functional" thinking, in mathematics, is not identical with thinking that "works," or that is "practically applied." The mathematical meaning of the word "function" corresponds closely to its use in a sentence like the following: "His good disposition seems to be a function of [i.e., is dependent upon] his health." 7 Keyser, C. J. Mole Philosophy and Other Essays, pp. 9394. E. P. Dutton and Company, New York, N. Y., 1927. 8 Ibid., pp. 9495. 9 Keyser, C. J. "Three Great Synonyms: Relation, Transformation, Function." Scripta Mathematlca, Vol. 3, p. 306, October, 1935. 10 Whitehead, A. N. An Introduction to Mathematics, p. 11. Henry Holt and Company, New York, N. Y., 1911. 11 Ibid, pp. 164165. 12 Whitehead, A. N. Science and the Modern World, p. 47. The Maemillan Company, New York, N. Y., 1935. 13 Hamley, H. H. "Relational and Functional Thinking in Mathematics." The Ninth Yearbook of the National Council of Teachers of Mathematics, p. 6. Bureau of Publications, Teachers College, Columbia University, New York, N. Y., 1934. 14 Dewey, John. How We Think, pp. 27 ff. D. C. Heath and Company, New York, 1910. 15 Dewey, op. cit., p. 27. 16 Whitehead, A. N. The Alms of Education and Other Essays, pp. 178 ff. The Macmillan Company, New York, N. Y., 1929. 17 Dewey, John. The Quest for Certainty, p. 151. Minton, Balch and Company, New York, N. Y., 1929. 18 Bode, B. H. Conflicting Psychologies of Learning, p. 274. D. C. Heath and Company, New York, N. Y., 1929. 19 Dewey, John. Democracy and Education, Chapters XIXII. The Macmillan Company, New York, N. Y., 1917. 20 Keyser, C. J. "Humanistic Bearings of Mathematics," The Sixth Yearbook of "the National Council of Teachers of Mathematics, p. 50. Bureau of Publications, Teachers College, Columbia University, New York, N. Y., 1931. 21 Betz, William. "The Transfer of Training, with Particular Reference to Geometry," Fifth Yearbook of the National Council of Teachers of Mathematics, p. 193. Bureau of Publications, Teachers College, Columbia University, New York, N. Y., 1930. 22 Smith, David Eugene. "Mathematics in the Training for Citizenship." Teachers College Record, Vol. 18, pp. 211225, May, 1917. 23 FraserHarris, D. F. "Unity and Intelligence in Nature," The Great Design (Frances Mason, Editor), p. 266. Longmans, Green and Co., New York, N. Y., 1935. 24 Quoted from a radio address by Mr. Henry D. Hubbard of the National Bureau of Standards in Washington. 25 Hotelling, Harold, "Some Little Known Applications of Mathematics," The Mathematics Teacher, April, 1936, p. 158. 26 Ibid., p. 169.


