CHAPTER 9 - The Testimony of Tests.
"And lo! Ben Adhem's name led all the rest."
Despite all of the seemingly valid crticisms of both the traditional and the modern mathematics curricula, one feels that there should be some more objective criteria for determining whether a given program is more suitable or better than another. One would think that the framers of the modern mathematics curriculum would have experinented with many groups of children and teachers and thus produced some evidence in favor of their programs before urging them upon the country. The sad fact is that most of the groups undertook almost no experimental work. The sole significant exception was the University of Illinois Committee on School Mathematics headed by Professor Max Beberman. And even this group never offered any evidence for the superiority of its curriculum.
Beberman started in 1952 and for the first few years experimented with classes in the Uuiversity of Illinois high school. He a1so trained teachers to teach the new material in other schools and was prepared to spend years in testing his material. But by 1955 the Commnission on Mathematics entered the picture and began to proclaim what it advocated in its final report of 1959. In 1958 The School Mathematics Study Group (SMSG) was organized. During the summer of 1958 fourteen experimentaI topics for grades seven and eight were written. These were tried by about one hundred teachers in twelve centers during the school year 1958-1959. Some revisions in these units were made during the summer of 1959. By this time the SMSG writing groups had completed texts for all of the grades seven through twelve. These were used experimentally during the 1959-1960 school year and revised duñng the summer of 1960. Without further ado the SMSG began to sell the curriculum to the country. Seeing these competitors and others ready to market their products, Professor Beberman probably felt that his work would be lost in the shuffle and he began to campaign actively for his version of the modern mathematics curriculum. Thereafter, experimental work practically vanished and the rush to secure leadership took precedence over all other activity.
During the fall of 1960 the National Council of Teachers of Mathematics, which had put together its own curriculum through its Secondary School Curriculum Committee, conducted eight regional conferences in various parts of the United States. The purpose of these conferences was to give school administrators and mathematics supervisors information that would enable them to institute one of these "new and improved" mathematics programs. In other words, the millennium had arrived and the country was invited to rejoice and partake in it. The pamphlet The Revolution in School Mathematics, published in 1961, reported what was discussed during the 1960 conferences and states in its Preface, "We are now in a position to make a concerted effort toward rapid improvement of school mathematics. The general pattern is clear and the necessary materials of instruction are at hand. . . . We regard each of the new programs as a sample of an improved curriculum in mathematics which deserves the consideration of those interested in devising better mathematlcs programs." Thus by 1960 - after efforts had been devoted almost entirely to writing subject matter, while testing was practically ignored - the new currictila were being sold to the country.
Professor Edwin E. Moise, who participated in the SMSG writing effort and gave talks in favor of that modern mathernatics curriculum, unwittingly testified to the lack of experimentation. In his article in the booklet Five Views of the New Math (published by the Council for Basic Education) Moise said, "One thing was obvious, however, as soon as the books were written, and before they were tried: The improvement in intellectual content was so great that they surely would produce either an educational improvement or a collapse of classroom morale."
What kind of objective evidence should one look for to establish the superiority of any curriculum? One thinks almost immediately of tests. But one must be wary of the significance of mathematics tests. Presumably the mathematics courses have taught students to think through problems, to discover results for themselves, and to acquire insight into the concepts and proofs they have learned. Actually tests do not and to a large extent cannot measure such values. The usual tests require that a student answer a fair number of questions in a limited amount of time. If a question really called for thinking tbrough a new type of problem or discovering a new result it would require so much time that average students would not be able to do well in the time allowed. Even if the student made intelligent and signifìcant attempts to answer such questions his failure to reach a positive result would probably mean that he would receive little or no credit.
Hence, mathematics tests usually and almost perforce call for handing back infòrmation that has been learned and is merely being reproduced. The ability to memorize is the major faculty that is actually tested. While the acquisition of information is one objective of mathematics education, it is not supposed to be the sole or most important goal. Though most teachers would deny that they are testing memorization, their behavior belies their words. In Chapter 2 we noted that teachers do not permit their students to use their books during tests. But if the tests call for thinking on the part of the student, what vitiation of the tests wou1d result from the use of books?
Let us assume, however, that to secure objective evidence rather than the opinions of teachers we must resort to tests, and let us further assume that the tests do give evidence of learning. We must still be wary of what the results mean, particu1arly in the case of the modern mathematics movement.
In most schools the modern mathematics curriculum is offered to the better or best students. In fact many of the proponents of modern mathematics now state explicitly that this program is intended for the college-bound and college-capable students. Clearly these students will do better than the average.
It is also true that a great many teachers of modern mathematics courses are motivated to do a better job. There are several reasons. It is the more able and more enterprising who want to try new material. Some receive extra pay or other benefits for experimenting or teaching other teachers how to teach modern inathematics. Still others have been convinced by the literature advocating modern mathematics that it is a superior curriculum and respond to the challenge of presenting it. Finally, many teachers participated in fashioning one or another version of the modem curriculum and are determined to show that it is superior. When in addition these teachers tell the students that they are specially chosen and that they are participating in a noble experiment, the students usually respond to such compliments and put forth extra efforts. Certainly students taught under any of the preceding conditions will do better. Incidentally, the effect achieved by making students believe that they are the key people in an important experiment is known as the Hawthorne effect.
Tests have been given to small groups by teachers and curriculum designers who are proponents of modern mathematics. The evaluations usually claim tbat students were able to learn the new material and yet do as well in the techniques of traditional mathematics as students who are taught oniy the latter. This class of evaluations is suspect. Beyond the reasons previously cited for questioning the results of tests are other factors. The tests are not standardized; hence it is very easy for the teacher to bias the questions so that the knowledge of the traditional mathematics that is called for is minimal. Moreover, the results of any test must be interpreted. Let us suppose that the questions testing traditional material were ones used to test one hundred thousand students whose average grade was sixty per cent. The group being tested by the individual teacher might make an average grade of seventy per cent on the traditional material. Has this group done better? Not necessarily. This small group may be among the better students and might have averaged seventy per cent in the test given to the one hundred thousand students. Moreover, if this group were taught only traditional mathematics it might have attained an average grade of ninety per cent on the traditional material. Could not the genera1 inteligence of this small group be tested so that one would know how it compares with that of the one hundred thousand students? Any atempt to do this raises the entire issue of the reliability of intelligence tests. We know very little about what intelligence is and how to test for it.
The picture of what is accomplished in modern mathematics courses is confused by another factor. A numher of texts and courses based on them contain primarily traditional mateñal which is dressed up (contaminated?) with a smattering of modern mathematics. Chapters on modern mathematics topics are interspersed with chapters on traditional mathematics with no integration of the two approaches. Incidentally many of these hybrid texts (one could use a more apt word to describe the progeny of this unholy wedlock of traditional and modern mathematics) are the output of hypocritical authors who evidently wish to capitalize on both markets, modern and traditional. As a matter of fact, some of the strongest advocates of the new mathematics have written such texts. Other texts begin with a chapter on set theory, tben turn to traditional mathematics and thereafter never refer to set theory or any other topic of modern mathematics. Still other texts go beyond the introductory chapter on set theory only to the extent of using modern mathematics terminology in the body of the text, but these books still teach primarily traditional mathematics. Even totally traditional texts are now titled modern, new or contemporary.
When the teachers of courses based on such texts are asked whether they are teaching modern mathematics they will usually reply affirmatively. They are under pressure from chairmen, principals and superintendents to be up-to-date and since this means modern mathematics, they profess to be teaching it. If their students do well in tests based on these courses, the impression given is that students can and do learn modern mathematics, when in fact they are being taught and tested on the traditional mathematics.
One might be inelined to believe that these traditional texts souped up wìth a bit of modern mathematics could not be in wide use and so could not affect the evaluation of modern mathematics. Actually, such texts are the most popular ones because they cater to the tradi-tionally oñented teacher who wishes or is obliged to claim that he is teaching the new mathematics. He can pretend to do so by actually presenting the minimal amount contained in the texts, or he can omit this material wìthout aflecting the treatment of the traditional subject matter because the two classes of material are not integrated.
One international test known as the International Study of Achievement in Mathematics may be worth mentioning because it was conducted in 1964, by which time many of the United States entries had had some modern mathematics. The test actually consisted of several tests given to students of different age levels. At all levels the United States ranked low and this was especially true of the thirteen-year-old group, which ranked tenth. Japan, incidentally, ranked first at all the age levels and its students had only traditional mathematics. Despite the poor showing of our students, some proponents of modern mathematics tried to interpret the statistics as showing that those students who took modern mathematics dìd better than those who took only traditional mathematics. But the evidence was meager and dubious. Though the results might seem to favor the teaching of traditíonal mathematics even this conclusion may be erroneous. The countries differ vastly in their emphasis on mathematics. For example, the British high school student who takes up mathematics practically specializes in the subject. Moreover, the poorer students never get to an academic high schooI; they are eliminated by tests given at the age of eleven and are sent to vocational schools. In the Soviet Union and Japan the young boys and girls must do exceedingly well to gain admission to any college and they work hard to excel. Students in the United States are not under such pressure.
What one wishes to test, to the extent that tests are significant, are normal studnts taught under normal conditions, and for these we have no results. In fact no large sca1e testing of the quality of tbe modern mathematics program has been undertaken. At present the amount of effort devoted to assessing properly the claims of the proponents of modern mathematics is negligible in comparison with the claims. The superor understanding which tbe modern mathematics approach is snpposed to provide has not been demonstrated by tests or by any other objective measures.
The education of students in the modern mathematics curriculum has been in vogue long enough so that students have entered college with this background. Do these students perform better because they received a possibly superior education? It is almost impossible to answer this question. No large-scale tests of these students have been made. Moreover, it is difficult to segregate tbose who have had modern mathematics because, -as we have already indicated, many courses pretending to be modern are really either mixtures of traditional and modern or include just a smattering of modern mathematics. An informal consensus of college teachers is that students are now weaker in technique than those of ten or more years ago. But this fact, if it is a fact, does not necessarily point to defects in the modern mathematics curricu1um. The pressure on young people to secure a college education has brought many more students to college who are not as well prepared in all subjects and who are less motivated. Also, mathematics education is being speeded up at a time when slowing down would seem wiser. High school students used to take a full course in synthetic plane geometry. Under the modern mathematics program part of this course is devoted to a modicum of analytic and solid geometry. Previously, either in the last year of high school or first year of college, students took advanced algebra and solid geometry. This material has been practically eliminated. Further, students used to take a full semester of analytic geometry before taking the calculus. The analytics course, beyond introducing the major idea of relating equations and curves, a1so enabled students to improve their algebra, geometry and trigonometry. For the last ten years analytic geometry has been submerged in the calculus and very little of it is taught. Thus the student enters calculus with far less preparation than he used to have and is weaker as a consequence.
There are, however, other indications, apart from tests, that all is not well with modern mathematics. In a speech given at a Symposium sponsored by the Thomas Alva Edison Foundation and held in Pittsburgh in November 1960, Professor Beberman confessed that he was wrong in putting rigor into geometry. He even exclaimed that he could not understand how he could have made that mistake.
At this same Symposium, Professor Edward G. Begle, the director of the largest and most influential curriculum group, the School Mathematics Study Group, said at the very outset of his speech, "In our work on curriculum we did not consider the pedagogy."
Somewhat later in a speech given at a University Symposium on Mathematics, held at Ohio State University on November 16, 1962, Professor Beberman cast further doubts on the wisdom of his program, then ten years old. "I think in some cases we have tried to answer questions that children never raise and to resolve doubts they never had, but in effect we have answered our own questions and resolved our own doubts as adults and teachers, but these were not the doubts and questions of the children."
From this he progressed wíth commendable honesty to outright criticism. In another speech given on December 30, 1964; at the 1964 Christmas meeting- of the National Council of Teachers of Mathematics held in Montreal, Dr. Beberman confessed that "we're in danger of raising a generation of kids who can't do computational arithmetic." He. admitted that the new curriculum had failed to relate mathematics to the real world and that pedagogical principles had been ignored. Because excessive emphasis was being placed on esoteric branches of mathematics at the expense of fundamentals, and becanse of he hasty introduction of the new mathematics in the elementary schools, Professor Beberman feared that "a major national scandal may be in the making."
It is clear from these and many other indications that Professor Beberman had not been satisfied with the work of his group, and in the mid-winter of 1971-72 he went to England to study some new experimental curricula being fashioned there. Unfortunately Professor Beberman died shortly after arriving in England. There is some question as to what the Illinois group will do without his dynamic leadership.
The clearest evidence that the modern curriculum as expounded in the early 1960s is not satisfactory is found in a number of other statements by Professor Begle. He has admitted in numerous speeches that the SMSG curriculum has minimized the acquisition of skills and has faìled to present the relationship of mathematics to allied subjects. He then announced in an open letter (which appeared in The Mathematics Teacher for April 1966 and in Science for February 1966) the íntention to devise a totally new curriculum for grades seven to twelve. In this letter he states tbat the SMSG Advisory Board "feels that longer-range planning and experimentation is necessary and should be started now. This must be done to prevent the present materials from becoming frozen into a new orthodox pattern that would require another upheaval a few years from now."
But if the work from 1958 to 1966 was really good, why should there be any concern about the curriculum becoming frozen or about the necessity for a new up-heaval? The nearest thing to an answer one can find in Professor Begles letter is that the new curriculum "will be responsive to the rapidly developing needs for mathematics in our society."
A committee was appointed in 1966 to make plans for the writing of the new curriculum. By 1972 the revision concerned with the junior high school was completed.
A special version for low achievers and a tenth-grade course which would make the transition to the older SMSG eleventh- and twelfth-grade mateñals were also fashioned. In the SMSG Newsletter of February 1 972, Professor Begle described these programs. Though the content of the new junior high curriculum varies somewhat from the previous material, the chief features are not essentially different. For example," . . . structure is still definitely one of the unifying themes." Moreover, the elementary-school curriculum and the older material for grades ten, eleven and twelve were left untouched.
The prograin projected in 1966 will not be completed. It is known in professional circles that neither Professor Begle nor his sponsors have been satisfied with the partial revision. The financial support has been withdrawn and SMSG is being disbanded.
Beyond the evidence just presented and the cñticisms made in the preceding chapters, the arguments against the new mathematics are supported by the judgments of men who, we have every reason to believe, are impartial. Numerous critical articles have appeared in such professional journals as The Mathematics Teacher. There is no doubt that many more teachers, who would like to express their disapproval of modern mathematics, fear to do because it might incur the displeasure of their chairmen, principals or superintendents.
One expression of criticism should be noted. A few college professors (this author was one of them) got together, drafted a protest against the movement, and asked mathematicians if they wished to endorse it. It would have been possible to get hundreds of signatures, but since the objective was just to show significant opposition, it was decided that about seventy-five active, mature mathematicians should be solicited. The memorandum, entitled "On the Mathematics Curñculum of the High School", was published in The Mathematics Teacher of March 1962 and in the American Mathematical Monthly of March 1962. It warrants perusal and is reproduced here.
ON THE MATHEMATICS CURRICULM OF THE HIGH SCHOOI.
The following memorandum was composed by severa1 of the undersigned and sent to 75 mathematicians in the United States and Canada. No attempt was made to amass a large number of signatures by canvassing the entire mathematical community. Rather, the objective was to obtain a modest number from men with mathematical competence, background, and experience and from various geographical locations. A few of the undersigned, whose support is indeed welcomed, volunteered their names when they learned about the memorandum from a colleague.
The mathematicians of this country now have a more favorable climate in which to develop and gain acceptance of improvements in mathematics education. Indeed a number of groups have recognized the opportunity and are working hard and with the best of intentions to utilize it.
It would, however, be a tragedy if the curriculum reform should be misdirected and the golden opportunity wasted. There are, unfortunately, factors and forces in the current scene which may lead us astray. Mathematicians, reacting to the dominance of education by professional educators who may have stressed pedagogy at the expense of content, may now stress content at the expense of pedagogy and be equally ineffective. Mathematicians may unconsciously assume that all young people should like what present day mathematicians like or that the only students worth cultivating are those who might become professional mathematicìans. The need to learn much more mathematics today than in the past may cause us to seek shortcuts which, however, could do more harm than good.
In view of the possible pitfalls it may be helpful to formulate what appear to us to be fundamental principles and practical guidelines.
1.For whom. The mathematics curriculum of the high school should provide for the needs of all students: it should contribute to the cultural background of the general student and offer professional preparation to the future users of mathematics, that is, engineers and scientists, taking into account both the physical sciences which are the basis of our technological civilization, and the socia1 sciences which may need progressively more mathematics in the future. While providing for the other students the curriculum can also offer the most essential materials to the future mathematicians. Yet to offer such subjects to all students as could interest only the small minority of prospective mathematicians is wasteful and amounts to ignoring the needs of the scientific conununity and of society as a whole.
2.Knowing is doing. In mathematics, knowledge of any value is never possession of information, but "know-how". To know mathematics means to be able to do mathematics:to use mathematical language with some fluency, to do problems, to criticize arguments, to find proofs and, what may be the most important activity, to recognize a mathematical concept in, or to extract it from, a given concrete situation.-Therefore, to introduce new concepts without a sufficient background of concrete facts, to introduce. unifying concepts where there is no experience to unify, or to harp on the introduced concepts without concrete applications which would challenge the students, is worse than useless: premature formalization may lead to sterility; premature introduction of abstractions meets resistance especially from critica1 minds who, before accepting an abstraction, wish to know why it is relevant and how it cou1d be used.
3. Mathematics and science. In its cultural significance as well as in its practical use, mathematics is linked to the other sciences and the other sciences are linked to mathematics, which is their language and their essential instrument. Mathematics separated from the other sciences loses one of its most important sources of interest and motivation.
4. Inductive approach and formal proofs. Mathematical thinking is not just deductive reasoning; it does not consist merely in formal proofs. The menta1 processes which sug-gest what to prove and how to prove it are as much a part of mathematica1 thinking as the proof that eventually results from them. Extracting the appropriate concept from a concrete situation, generalizing from observed cases, inductive arguments, arguments by analogy, and intuitive grounds for an emerging conjecture are mathematical modes of thinking. Indeed, without some experience with such informal thought processes the student cannot understand the true role of formal, rigorous proof which was so well described by Hadamard: "The object of mathematical rigor is to sanction and legimize the conquests of intuition, and there never was any other object for it."
There are several levels of rigor. The student should learn to appreciate, to find and to criticize proofs on the level corresponding to his experience and background. If pushed prematurely to a too formal level he may get discouraged and disgusted. Moreover the feeling for rigor can be much better learned from examples wherein the proof settles genuine difficulties than from hair-splitting or endless harping on tivialities.
5. Genetic method. "It is of great advantage to the student of any subject to read the original memoirs on that subject, for science is always most completely assimilated when it is in the ascent state", wrote James Clerk Maxwell. There were some inspired teachers, such as Ernst Mach, who in order to explain an idea referred to its genesis and retraced the historical formation of the idea. This may suggest a general principle: The best way to guide the mental development of the individual is to let him retrace the mental development of the race -retrace its great lines, of course, and not the thousand errors of detail.
This genetic principle may safeguard us from a common confusion: If A is logically prior to B in a certain system, B may still justifiably precede A in teaching, especially if B has preceded Á in history. On the whole, we may expect greater success by following suggestions from the genetic principle than frorn the purely forrnal approach tomathematics.
6. Traditional mathematics. The teaching of mathematics in the elementary and secondary schools lags far behind present day requirements and highly needs essential; improvement: we emphatically subscribe to this almost universally accepted opinion. Yet the often heard assertion that the subject matter taught in the secondary schools is obsolete should be closely scrutinized and should not be taken simply at face value. Elementary algebra, plane and solid geometry, trigonometry, analytic geometry and the calculus are still fundamental, as they were fifty or a hundred years ago: future users of mathematics must learn all these subjects whether they are preparing to become mathematicians, physical scientists, social scientists or engineers, and all these subjects can offer cultural values to the general students. The traditional high school curriculum comprises all these subjects, except calculus, to some extent; to drop any one of them would be disastrous.
What is bad in the present high schoo1 curriculum is not so much the subject matter preseted as the isolation of mathematics from other domains of knowledge and inquiry, especially from the physical sciences, and the isolation of the various subjects offered from each other; even the techniques and theorems within the sarne subject appear as isolated, disconnected tricks to the student, who is left in the dark about the origin and the purpose of the manipulations and facts that he is supposed to learn by rote. And so, unfortunatel1y, it often happens that the material offered appears as useless and boring, except, perhaps, to the few prospective mathematicians who may persist despite the curriculum.
7. Modern mathematics. In view of the. lack of connection between the various parts of the present curricula, the groups working on new curricula may be well advised in seeking to introduce unifying general concepts. We think, too, that judicious use of sets and of the language and concepts of abstract algebra may bring more coherence and unity into the high school curriculum. Yet, the Spirit of modern mathematics cannot be taught by merely repeating its terminology. Consistently with our principles, we wish that the introduction of new terms and concepts should be preceded by sufficient concrete preparation and followed by genuine, challenging application and not by thin and pointless material: one must motivate and apply a new concept if one wishes to convince an intelligent youngster that the concept warrants attention.
We cannot enter here into detailed analysis of the proposed new curricula, but we cannot leave unsaid that, in judging them on the basis of the guidelines stated above (Sections 1-5), we find points with which we cannot agree.
Of course, not all mathematicians have the same taste. Mathematics has many aspects. It can be regarded as an instrument to understand the world around us: mathematics presumably possessed this value for Archimedes and Newton. Mathematics can also be regarded as a game with arbitrary rules where the principal consideration is to stick to the rules of the game: some such view may be considered suitable for certain problems of foundations. There are several other aspects of mathematics, and a professional mathematician may favor any one. Yet when it cornes to teaching, the choice is not a mere matter of taste. We may expect that an intelligent youngster would want to explore the world around him, but we cannot expect him to learn arbitrary rules: why just these and not others?
At any 1rate, we fervently wish much success to the workers on the new curricula. We wish especially that the new curricula should reflect more the connection between mathematics and science and carefully heed the distinction between matters logically prior and matters which should have prioity in teaching. Only in this way can we hope that the basic values of mathematics, its real meaning, purpose, and usefulness will be made accessible to all students, including of course, the prospective mathematicians. The recently expressed "widespread concern about a trend to excessive emphasis on abstraction in the teaching of mathematics to engineers" * points in the same direction.
* First Summer Study Group in Theoretical and Applied Mechanics Curricula, Boulder, Colorado, June 1961.
Lars V. Ahlfors, Harvard University
Hirsh Cohen, IBM
Herman Goldstine, International Business Machines Corp.
Deane Montgomery, Institute for Advanced Study
A. H. Taub, University of Illinois
© Helen M. Kline & Mark Alder 2000
Version 22nd March 2001