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CHAPTER 3: The Origin of the Modern Mathematics Movement.

"Experience, however, shows that for the majority of the cultured, even of scientists, mathematics remains the science of the incomprehensible".

Alfred Pringsheim

There was general agreement in the early 1950s and even before that date that the teaching of mathematics had been unsuccessful. Student grades in mathematics were far lower than in other subjects. Student dislike and even dread of rnathematics were widespread. Educated adults retained almost nothing of the rnatheinatics they were taught and could not perform sirnple operations with fractions. In fact, these people did not hesitate to say that they got nothing out of their mathematics courses. When this country entered World War 11, the military discovered quickly that the men were deficient in mathematics and that they had to institute special courses to bring up the level of proficiency.

Thõugh there are many factors that determine the outcome of any teaching activity, the groups that undertook reform focused on curriculum and argued that if this component were improved the teaching of mathematics would be successful.

In 1952 the University of Illinois Committee on School Matheinatics headed by Professor Max Beberman began fashioning a new, or modem, mathematics curriculum.

By 1960 the curriculum (at that time directed solely toward the high schools) was used on an experimental basis. Subsequently, the Committee undertook to provide an elementary school curriculum and gradually extended the teaching of both the elementary and high school subject matter to additional geographical areas. The experimental texts, in photo-offset form, were eventually published as commercial texts.

In 1955 The College Entrance Examination Board, whose function is to prepare college entrance examinations which meet the requirements of many colleges, decided to take up the problem of the high school mathe-matics curriculum and compose what it considered to be the proper one. It set up its own Cominission on Mathematics. In 1959 the Commission îssued its report, Program for College Preparatory Mathematics, and adjoined severa1 appendices which contained samples of recommended subject matter. It did not produce texts. During the years 1955 to 1959 and for severa1 years thereafter, the members of the Commission toured the country and campaigned for the kind of curriculum it proposed in its Program.

In the fall of 1957 the Russians launched their first Sputnik. This event convinced our government and country that we must be behind the Russians in matheinatics and science and had the effect of loosening the purse strings of governmental agencies and foundations. It may be coincidence but at this tiine many oher groups decided to go into the business of producing a new curriculum.

The American Mathematical Society, the organization concerned with research, decided in 1958 that its talents shou1d be applied to the fashioning of a high school curriculum, and it set up a new group, The School Matheniatics Study Group, headed by Professor Edward G. Beg1e, then at Yale University, to undertake the task. This group began its work by writing curricula for the junior and senior high schools and then extended its program to include the elementary school arithmetic curriculum.

The Nationa1 Council of Teachers of Mathematics set up its own curriculum committee, The Secondary School Curriculum Conimittee, which came out with its recommendations in an article in the May 1959 issue of The Mathematics Teacher. Many other groups, such as the Ball State Project, the University of Maryland Mathematics Project, the Minnesota School Science and Mathçmatics Center, and the Greater Cleveland Mathematics Program, were soon formed and began their work.

Individual high school and college teachers commenced in the late 1950s to write their own texts along the lines already foreshadowed or explicitly recommended by the curriculum groups. By the early 1960s a spate of such books had appeared, and many more have continued to appear since that time.

Rather surprisingly, the many groups and independent textbook writers all headed in about the same direction. Hence they have all, fairly enough, been described by the term "modem mathematics" (or "new mathematics").

The origin of the term modern mathematics is relevant. Even before the members of the Commission on Mathematics had determined just what they were going to recommend, they gave addresses to large groups of teachers. Their main message was that mathematics education had failed because the traditional curriculum offered antiquated mathematics, by which they meant mathematics created before 1700. Implicit in this con-tention was the assumption that young people were aware õf this fact and therefore refused to learn the material. Wou1d you, argued these educators, go to a lawyer or a physician whose knowledge of his profession was limited to what was known before 1700? Though these speakers were presumably informed in mathematics they ignored completely the fact that mathematics is a cumulative development and that it is practicaliy impossible to learn the newer creations if one does not know the older ones. Nevertheless, the Commission contended that we must drop the traditional subject matter in favor of such newer fields as abstract algebra, topology, symbolic logic, set theory, and Boolean algebra. The slogan of reform became "modern mathematics".

As it turned out, the reform offered as much a new approach to the traditiona1 curriculum as it did new contents, and some groups emphasized this fact. Hence the term modern mathematics is not really an appropriate description of the new curricula. However, perhaps because the propaganda value of the word modern was too useful to drop - a 1970 automobìle is c1early more desirable than a 1969 model - the terms modem mathematics or new mathematics have been retained.

While the modern or new mathematics curriculum as it stands today was being fashioned by the groups already mentioned, new groups appeared on the scene and began to recommend more radica1 reform. For example, an intemationaI group meeting at Royaumont, France, in 1959 urged the abandonment of virtually all the familiar courses in high school mathmatics, including Euclidean geometry. The conference declared these subjects to be outdated by electronics, relativity, computers and the soaring importance of abstract mathematics as the basìs of modern science. The new subjects were to be logic, structure, and the unity of mathematics as a whole and were to be taught in a new language This conference did not resu1t in the formation of another curriculum group, but it encouraged still furtber departures from the traditional curriculum.

Of the newer groups which have proposed more radical reforms we shall mention two. In the summer of 1963 a group of mathematicians assembled for The Cambridge Conference on School Mathematics. (Its report, Goals for School Mathematics, appeared as a publication by the Houghton Mifflin Company.) This group recommended the inclusion -by the end of grade twelve, the fourth year of high school - of many additional advanced topics drawn from the theory of numbers, abstract algebra, linear algebra, n-dimensional geometry, projective geometry, tensors, topology, differential equations, and of course, the calculus. On page 7 the report asserts,

"The subject matter which we are proposing can be roughly described by saying that a student who has worked through tbe fuIl thirteen years of mathematics in grades K to 12 (kindergarten through the fourth year of high school) should have a level of training comparable to 3 years of a top-level college training today".

The justification for advocating such a program when the already existing curriculum groups had barely begun to try out their programs or were still fashioning theiri was given in the Foreword by Francis Keppel, who was then United States Comnaíssioner of Education. He observed that recent curriculum changes are essentially different from those attempted in the past aad that the reforms have been eminently successful for the most part (How Dr. Keppel knew this in 1963 when most new mathematics curricula had hardly been tried is not clear.), so much so that

"it has sometimes been difficult to distinguish their shortcomings. Yet the shortcomings are there, and they are by no means insignificant. It cau be argued, in fact, that the deficiencies of the present reform movement are grave enough to threaten the expressed goals of the movements themselves".

Keppel then noted that the changes recommended by the Cambridge group were intended to represent the subject as the scholar saw the discipline, and that the students were assumed to be able to learn far more than they had been expected to in the past. The limitations of the teacher were noted too!

"Most curriculum reforms, practically enough, have chosen to limit their ambitions in the light of these realities. They have tended to create such new courses as existìng teachers, after enjoying the benefits of brief retraining, can competently handle. They have done so fully aware that they are thus setting an upper limit, and an upper limit that is uncomfortably close"

Keppel then continued:

"If the matter were to end there, the result might well be disastrous. New curricula would be frozen into the educational system that would come to possess, in time, all the deficiencies of curricula that are now being swept away. And in all likelihood, the present enthusiasm for curriculum reform will have long since been spent; the 'new' curricula might remain in the system until, like the old, they become not only inadequate but in fact intolerable. Given the relative conservatism of the educational system, aad the tendency of the scholar to retreat to his own direct concerns, the lag may well be at least as long as it has been during the first half of this century.

The present report is a bold step toward meeting this problem. It is characterized by a comlete impatience with the present capacities of the educational systein. It is not oniy that most teachers will be completely incapable of teaching much of the mathematics set forth in the curricula proposed here; most teachers would be hard put to comprehend it. No brief period of retraining will suffice. Even the first grade curriculum embodies notions with which the average teacher is totally unfamiliar.

None the less [sic], these are the curricula toward which the schools should be aimìng. . . ."

The second of the newer groups joining the movement to revi~e curricula, the Secondary School Mathematics Curriculum Improvement Study, was organized in. 1965 by Professor Howard Fehr of Columbia University. Its goal is to reconstruct secondary school mathematics from a globa1 point of view. It seeks to eliminate the barriers separating the severa1 branches of mathematics and to unify the subject through its general concepts, sets, operations, mappings, relations and structure. (We shall discuss these concepts later.) Professor Fehr's contention is that his organization of the subject matter will permit the introduction into the high school curriculum of much that has been considered collegiate mathematics. The work of the Cambridge group and of the Curriculum Improvement Study has proceeded slowly and their effect on the schools is not widespread as yet. Hence our account and evaluation of the modern mathematics movement wiIl concentrate on the curriculum efforts of the preceding groups, some of which are still at work on one aspect or another of the school programs.

The curricula which have been formulated by these several organizations are the product of group efforts in which research mathematicians, college and high school teachers and even representatives of industry have collaborated. On the face of it, such collaboration would seem to be a wise procedure. However, attempts to achieve a meeting of minds often result in compromises that are not satisfactory to anyone or which vitiate the thrust of. the effort. The point may be illustrated by the story that the famous dancer Isadora Duncan offered herself in marriage to Bernard Shaw and perhaps somewhat facetiously said, '. . . and think of the child who would have your brains and my looks. "Yes", said Shaw, but what if the child shou1d have your brains and my looks'

When one seeks to determine what changes these curricula offer, why these changes are desirable, and what reasoning or evidence can be proffered to support the desirabilîty of these changes, one is faced with a problem of considerable magnitude. It is true that in its 1959 report the Commission on Mathematics of the College Entrance Examination Board did describe the contents it recomniended. However, except for stressing that modern society requires a totally new mathematics the Commission did not defend the contents it proposed. Moreover, the various curriculum groups that did write texts not only extended the reform to the elementary school grades but did not necessarily follow the Cominissions recomendations. One wou1d have expected that each group wou1d have declared its own position and have presented its case for including or excluding particular topics and for adopting its own approach. No such documents have been issued. This is aIl the more true of the many texts published by individual authors which proclaim themselves to be modem in character. Hence we are left to infer for ourselves what the modem mathematics curriculum is and why it is presumably superior to the traditional curricultun. Could the absence of explanatory and justifying material be interpreted to mean that the advocates of modem mathematics are not too clear themselves on where they have headed, or are they fearful that explicit statements of the features and purported merits of their materials will not bear scrutiny? In any event, to determine the nature and qualities of the modern mathematics curriculum, one must examine the texts and listen to the speeches made by various proponents. At the moment, pending a fuller discussion, let us note that there are two main features of the new curriculum: a new approach to the traditiona1 mathematics, and new contents.

Since we intend to evaluate the new mathematics, it is necessary to consider on what basis one should judge it. One cou1d use as a criterion, Is the mathematics correct? The answer is yes, but the criterion is useless. Correctness does not guarantee that the students will take to the material, that they can absorb it, or that this particular mathematics is what should be taught.

Will it develop mathematicians? Even if it were the idea1 curriculum for the training of mathematicians one could not be content. The new mathematics is taught to elementary and high school students who will ultimately enter into the full variety of professions, businesses, technical jobs, and trades, or become primarily wives and mothers. Of the elementary school children, not one in a thousand will be a mathematician; and of the academic high school students, not one in a hundred will be a mathematician. Clearly then, a curriculum that might be ideal for the training of mathematicians would still not be right for these levels of education.

The contents should contribute to the goals of elementary and high school education and should be accessible to young people. The approach to the material should make the content inviting and aid comprehen. sion as far as possible. In particular the new mathematics should remedy at least some of the defects of the traditional curriculum. Unfortunately, in the field of education, unlike mathematics proper, one cannot give an ironclad proof that a particular principle or topic is right or wrong. But there are arguments which do enable us to decide.

Though a dozen or more groups have fashioned new curricula and by now many series of new mathatics texts are on the market, we have already noted that they all adopted about the same approach and contain about the same material. This uniformity has resulted in part from imitation. It is also a consequence of the emphasis and direction which mathematicians are favoring in current research and which we shall discuss at greater length later. Hence, though not every statement we shall make about the new mathematics applies necessarily to any one curriculum, it is fair to treat them as a single movement characterized by common features and content.

We intend to consider carefully the nature of the new mathematics prograxn and to discuss its merits and demerits. Before doing so we should like to inject a somewhat different but nevertheless relevant criticism. Reform of mathematics education was called for, but there is a serious question as to whether curriculum was the weakest component and should have been tackled first. It would, I believe, be generally conceded that the policy of universal education pursued in the United States is highly commendable, but our country was not and still is not prepared to carry on such a .program. Certainly we do not have enough qualified teachers; therefore the education in many parts of this country is woefully weak. Were more good teachers available they wou1d have been able long ago, by acting in concert, to remedy the defects of the traditional curriculum. Since the teacher is at least as important as the curriculum, the money, time and energy devoted to curriculum reform might well have been devoted to the improvement of teachers. It is true that in 1958 the National Science Foundation inaugurated and has maintaìned various institutes for the education of teachers. These institutes should have been used to improve the mathematìcal backgrounds of elementary and high school teachers so that they cou1d form more independent judgments of what is important in mathematics. Unfortunately, they have been used largely to teach teachers how to teach mathematics of unproven worth.

Whether or not curriculum reform shou1d have received priority, the historical fact must be faced that the new curriculum is at hand and is being widely used. Let us therefore attempt to evaluate it.

Professor Morris Kline

I am very grateful for the kind permission of Professor Kline's widow, Mrs Helen Kline for this book to be reproduced.

© Helen M. Kline & Mark Alder 2000

Version: 22nd March 2001

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