Chapter 1: Max

Chapter 1: Max

Max Beberman is generally regarded as the father of the New Math, his teaching and his curriculum project having achieved nationwide fame long before the 1957 Sputnik raised mathematics education to the level of a national priority. He was born in 1925, a Jew from Brooklyn: Stuyvesant High School and a BA in mathematics from CCNY in 1944, Army Signal Corps until 1946, upon discharge staying in Alaska, where he had served as a meteorologist in the Army, to teach math and science in the schools of Nome until 1948. He then returned (married, now) to Rutgers for an MA and, finally, Columbia University Teachers College for a PhD in mathematics education in 1953. His thesis director at Columbia was Howard Fehr, famous then and later as an authority on the teaching of school mathematics, and a man who directed the PhD theses of many future professors of mathematics education.

But as was common in the days when PhD degrees were fewer than they became later, Beberman was already an Instructor in education at the University of Illinois well before his degree work at Columbia was done. He arrived at Illinois in 1950 and with few interruptions spent his entire career there, dying at the age of 45 in early 1971. His early death was predictable, for in 1966, already seriously ill, he had gone to the Mayo clinic for a heart valve replacement; it was the failure of this valve that caused his death five years later, years during which he did not slacken in either his work or bon vivant style of life.

Max Beberman was an enormous worker, and the light in his campus office in the middle of the night was a beacon. People returning from late parties would sometimes drive past just for the pleasant reassurance that Max was still up there. As early as 1955 his friend and frequent correspondent on mathematics education, Bruce Meserve, wrote him a serious letter[1] begging him to let up on his terrific activity lest he work his way into an early grave. (Meserve was on the mathematics faculty at Illinois when Beberman got there, and was an early collaborator, but left Illinois in 1954.) The fact that Max's girth increased continuously during his lifetime, approaching sphericity, was also of no help to his health.

Beberman's principal work at Illinois was split: He was a professor of education at the University and a teacher in the University High School. His reputation was immediate; it was said that he could teach mathematics to a stone wall. In his classroom there were no back seats, for he engaged the entire class with what he was getting across. It was more than a Socratic dialogue, and he called it "discovery learning". The idea -- or at least the phrase -- has been institutionalized since Beberman's time, and is often turned into something of a fraud, and a cover for the failure to impart any real information or make any real intellectual demands on students; but for Beberman it was real and it was seen to work, though within limits.

Not all his teaching was in the discovery style; he would lecture like other professors when that seemed called for, and even hector students into saying things his way. To Beberman, no matter how the process of "doing mathematics" was begun -- or discovered -- by a student, it had to eventuate in mathematically accurate speech. By the end of his career thousands of people had seen him in action, for his curriculum project (UICSM) produced a number of 16 millimeter films of his classroom performance. In addition, his own personal appearances in schoolrooms around the country were frequent and legendary.

Yet being a fine teacher alone will not make anyone famous, or cause his work to be remembered beyond his time. There have doubtless been other teachers as good as Beberman, but he happened along at a propitious time, and in a propitious environment. 1950 was near the end of a dismal era in American public education in mathematics and science. The flood of war veterans that had engulfed the universities on "GI Bill" dollars from 1945 to 1950 was now subsiding, and these unusually mature students were being replaced by high school graduates of a more customary age, school-educated no worse in mathematics than their predecessors, but without the discipline and understanding that a few years of military service affords, or the seriousness about knowledge that a few more years of anything tends to bring. The new, postwar wave of college students without these advantages were not the same thing.

The professors of mathematics in the universities, by 1950, were therefore suddenly shocked by the wave of ignorance that seemed to have swept over their new freshmen. At Illinois it was William Everitt, Dean of the College of Engineering, who took an initiative to see if anything could be done to improve high school mathematics education, at least in Illinois, and at least for students intending to study engineering.[2]

At his initiative a faculty committee, including two professors in each of mathematics, engineering and education, was appointed to study what was lacking and issue a report to be disseminated among the high schools of the State of Illinois. Thus, Mathematical Needs of Prospective Students in the College of Engineering of the University of Illinois came to be published as University of Illinois Bulletin, vol 49, No 18 in October 1951, and given wide circulation.[3]

The committee writing this report not only brought three points of view to the problem, but conducted surveys of students and professors in engineering at the University, with some attention to the statistical validity of the results there, as well as to their own predispositions. Their 18 page report was printed and sent to high schools all over the State of Illinois Three pages were devoted to a simple listing of topics, 97 of them labeled "indispensable". Some of these, marked with an asterisk, were considered more advanced than the others, and while it was recognized that not all high schools might offer them, or students study them, and could be excused for not including them, all were warned that these topics would then have to be studied remedially once the student entered the engineering program, under the traditional rubric, "College algebra and trigonometry." Another 13 items called "not so fundamental", were recommended for superior students or those who had time for them in high school.

Typical of the "indispensable" items were

8. Scale drawing

9. Concept of an approximate number, precision of a measurement, significant digits, and rounding

17. Common special products, i.e., a(a+b), (a+b)(a-b),

18. Factoring such expressions as a² – b²,

56. Mensuration of plane figures

60. Concept of locus

68. Cylinders, cones, spheres; Concept of a definition, a postulate, and a theorem

69. Deductive proof

74. Solution of right triangles

while some of the indispensable items marked for "College algebra and trigonometry" were

38. Change of base of logarithms

39. Solution of exponential and logarithmic equations

44. Geometric progressions, both finite and infinite.

77. Definition of the trigonometric functions of any angle

More interesting today, perhaps, is the following (complete) list of topics classified as desirable but not indispensable, to be studied "if there is time available or by high-ability students whose rate of learning warrants supplementary work."

1. Extraction of square roots by means of the algorithm

2. Slide rule

3. Binomial theorem with fractional and negative exponents

4. Permutations

5. Combinations

6. Probability

7. The inverse, converse, and contra-positive of a statement

8. Polyhedral angles

9. Line values of trigonometric functions

10. Formulas for tangents of the half angle

11. Multiplication and division of complex numbers in polar form

12. De Moivre's theorem

13. Exponential form of a complex number

Even for the year 1951 one might wonder why the listing is so detailed. Items 11-13 are part of a package that can hardly be partitioned, after all. It is amusing that the archaic algorithm for taking a square root, a device that ceased to have practical value centuries earlier and whose intellectual underpinnings by 1950 were no longer understood by anyone but antiquarians, should have even received mention, but in fact the high school textbooks of 1950, many of them later editions of books thirty years old, did mention such things, and "college preparatory" students were still being put through it as "good for the mind." And whatever a "line value" of a trigonometric function might be, the committee didn't want the schools to waste time on it at the expense of (say) the "concept of locus" or any other item mentioned earlier as indispensable. The purpose of the document was not to make efficient mathematical sense, but to make it unmistakably clear to high school teachers across the state, teachers familiar with the kind of textbook available at the time, which sections of these books it would be safe to omit and which it was necessary to include. The committee's list had to use the jargon of the time to serve such a practical purpose. For example, the more sensible mathematical language that could have combined the three "topics" concerned with DeMoivre's theorem under one heading might well have failed to carry the proper message to a schoolteacher uncertain about complex numbers altogether.

Despite the appearance in this listing of such logical ideas as "contrapositive" and "axiom" (and the mysterious "inverse" of a proposition, which turns out to be the same as the converse, but expressed in contrapositive form), the context makes it plain that these notions were to be part of geometry, not algebra, and that nobody then intended the sort of axiomatic algebra that later became a hallmark of "the new math". The nearest thing to mathematical reasoning contemplated by the algebra entries was perhaps suggested by the item

47. Concept of equality including the symbol, and the postulates of equality

But this appears between the mention of geometric sequences and, after a similar item involving the word "inequality",

48. Use of the protractor

49. Use of the compass and straight edge

betraying the then popular confusion between equality as Euclid construed it, and posed axioms for, and equality meaning identity, as the word is used in algebra in discussing the solution of equations, in which context there simply are no postulates, after all. (cf. the later discussion of Minnick's methods book[4])

The opening of the report to Dean Everitt deplored the inadequacies of mathematical preparation of engineering freshmen, vigorously recommending more mathematics. It quoted from a paper by W.C. Krathwohl in the Journal of Engineering Education, vol 27 (1938), "Nobody can say what mathematics an engineer does not need", and "In particular, [the engineer's training] should emphasize general mathematical principles and methods of analysis rather than dexterity in few specified fields." And the remainder of the report was devoted to administrative comments on the University's admissions policies and so on. Despite Krathwohl's words, and the presence of the two young mathematicians, Assistant Professors Bruce Meserve and William A. Ferguson, on the committee writing this report, there was no great originality in this document. More mathematics for engineers, and earlier, but much the same sort of mathematics the schools were used to, and no mention at all of what a non-engineer might want with mathematics. Yet this report of limited purpose was destined to segué into a revolution.

Soon after the appearance of the report, the Principal of the University High School took steps to implement its recommendations locally:

102 University High School

December 18, 1951

Mr. Daniel S. Babb

Dr. Bruce Meserve

Mr. Max Beberman

Dr. R.E. Pingry

Gentlemen:

I wish to ask your help in planning for revision of our University High School mathematics offering.

The recent publication, "Mathematical Needs of Prospective Students of the College of Engineering of the University of Illinois," faces high schools with a number of problems.

1. The topics listed as essential (pages 12 to 14) are not accompanied by descriptions of the degree of competence required. While this was not within the scope of the committee who developed the publication, it seems essential to the organization of appropriate high school courses. Your work on this problem will undoubtedly be related to that of another interdepartmental committee to be concerned with evaluation of the mathematical competence of prospective University students.

2. The publication suggests that the competences to be required may be learned in three years of mathematics if the learning experiences are properly organized and taught, rather than in the four years of conventional mathematics courses mentioned in the bulletin.

3. All of you, I am sure, believe that mathematics should be a part of the general education of high school pupils as well as preparation for such specialized fields as engineering. This belief raises, for all but the largest high schools, the question of how to organize the mathematics offering so that both sets of needs are met.

4. The largest problem of all may be that of selecting and arranging the mathematical experiences of high school pupils to achieve the most effective learning.

I request that you serve as a committee (1) to study the problems involved and propose revisions in the mathematical program of our High School and (2) that after a new program has been approved by our staff, you continue your study of it as it goes into effect. I am asking Mr. Beberman to serve as chairman. Your committee can be of great help, not only to our school, but to others in Illinois as well.

Beberman, though only a year at Illinois, and as yet without the dignity of a PhD, was a member of a faculty committee that was formed to systematize the effort in 1951; indeed he thereupon immediately became the first director of what was called, modestly, the University of Illinois Committee on School Mathematics, and whose acronym, UICSM (initially UICSSM, for "Secondary School Mathematics"), is today remembered more as the name of a curriculum, and a system of teaching, than as the name of a group of people.

UICSM early, but not immediately, received financing from the Carnegie Foundation, and in later years, when it had grown much larger, from the U.S. Office of Education (OE), under its Cooperative Research Program, and even the National Science Foundation (NSF) for particular studies associated with its work (and of its junior offshoot, the University of Illinois Arithmetic Project, headed by David Page, also of the school of education). Beberman, already an experienced teacher of children, was now also a professor teaching future teachers to follow his methods if possible.

The quality of teaching in the schools was not the only problem UICSM had to face, for it was immediately apparent that the curriculum was even more fundamental than classroom procedure. Any Dean of engineering could see at a glance that what was being printed commercially for the high schools in 1950 was not what was needed, and any mathematician could also see that it was usually ignorant and often downright wrong, mischievous in its misrepresentation of the nature of mathematics even at the most elementary levels. UICSM therefore began with two threads of activity: training teachers and writing textbooks.

Now, Beberman was not a mathematician. Almost nobody associated with school mathematics in 1950 was a mathematician, and the textbooks showed it. Beberman's first friend at Illinois had been Bruce Meserve, a mathematician and later an author of some fine textbooks; but Meserve left Illinois in 1953 and Beberman's principal associate in UICSM for purposes of writing texts was Herbert Vaughan, a professor of mathematics whose specialty was logic. Some commentators on the failures of the New Math, in later years, attributed that failure to this single fact, that Vaughan persuaded Beberman that the proper study of mathematics at the high school level begins with some set theory and formal logic. Without a clear idea of how a list of axioms gives rise to the particular structures of mathematics -- algebra and trigonometry as well as geometry -- high school students intending to go to college would be limited, so the argument went, to a repertoire of handy rules rather than a mathematical education one could build on. This emphasis was later to be taken up by the most prominent of the other 'new math' projects, most notably SMSG.

That Beberman took Vaughan's advice can be seen in the textbooks they wrote together for the UICSM courses, initially simply produced by mimeograph for trial and ultimately published commercially [e.g., High School Mathematics, Course 1, 1964, and Course II, 1965, Boston, D.C. Heath Co.] Beberman's personal correspondence with both friends and critics often amounted to discussions of the language to be used, to make mathematics both logically correct and understandable to high school students.

A less than kindly account of the UICSM devotion to logical niceties has it that Vaughan, even before the establishment of UICSM, already had one or more book manuscripts in his files, rejected by commercial publishers, which he dusted off for UICSM when the opportunity arose. He might have had them, since certainly before the arrival of Beberman there would have been no publisher willing to take a chance on such a project; but there is also no doubt, as can be seen from Beberman's correspondence with mathematicians all over the country, that Max himself was a genuine co-author of what ultimately appeared, and not just the spectacular classroom performer of another man's scripts.

Furthermore, to say that Vaughan was "at fault" for the logical excesses of the Beberman program implies two questionable assumptions: that there was such fault, and that Vaughn was responsible. As to fault: Had Beberman asked almost any other relatively young mathematician of the time to serve as Vaughan did, as co-author and mathematical guide, he would likely have got much the same advice, for any mathematician looking at the usual school textbooks of the time would have noticed first off that they were lacking in logic and structure, and that indeed they were often downright foolish. While it does not follow that this mathematician would have prescribed much theory, or first-order functional calculus, with the technical notations and language used by professional logicians, something needed to be prescribed, and it is hard to see how the prescribers could avoid the tension between logical structure and pedagogy that characterized the programs known generically "the New Math", except by abandoning the call to reason -- as indeed happened later on in the reaction of the 1970s.

In point of fact, Vaughan was not initially a member of UICSSM, which consisted of a professor of education (Beberman), a mathematician (Bruce Meserve, a geometer, not a logician), a professor of electrical engineering (Daniel S. Babb), and a practicing high school teacher (R. E. Pingry). The initial committee was formally appointed on December 18, 1951 by Charles M. Allen, Principal of the University High School (affectionately called "Uni" by the students and their parents) of the University of Illinois at Urbana, the high school which was to serve as the initial laboratory for what became UICSM. Though Meserve left the University in 1953, the initial work on the syllabus was done by Meserve and Beberman:

For example, A Grade Placement Chart (tentative) was drawn up by Beberman on February 26, 1953[5] . Meserve had been granted a half-time "release" the preceding December by Professor Cairns, his chairman in the mathematics department, to work with Beberman, and this document was one of the early results of that collaboration, and not the work of Vaughn at all; it is a typescript signed by Beberman alone, and Vaughn was not to become part of the committee or its works until 1955, after Meserve left the University and the project.

(Actually, in consequence of some administrative dispute the details of which have probably died with the participants, both Beberman and Meserve left the University of Illinois in 1953, but Beberman, to the dismay of wife and children, returned to Illinois the following year from Florida, where he had already bought a house which then had to be sold again. Of course, his return was attended by a promotion and a raise. Meserve never returned, but remained a close friend of Beberman's to the end. This sort of thing is common in academe.)

Beberman's own draft placement chart (1953) names all the topics that should occur in Grades 9-12, as follows: Algebra and Arithmetic, Geometry, Trigonometry, Logic, and Statistics; and then, devoting a large double-size page to each of these topics, specifies what subheadings should occur in each of the four high school years. The page devoted to logic reads as follows:

NINTH YEAR

Generalizations

Definitions, conventions

Postulates

Theorems, corollaries

Undefined terms

Hypotheses

Inductive reasoning

Deductive proof

If-then statements

Converses

TENTH YEAR

As in ninth year

Indirect proof: reductio ad absurdum

Contrapositive

method of elimination

Inverses

Circular reasoning

Analysis and synthesis

Fallacies

Multiconverses

Necessary and sufficient conditions

Functional dependence

Class inclusion

ELEVENTH YEAR

Functional dependence

Implication

TWELFTH YEAR

Functional dependence

Implication

Mathematical induction

Postulational thinking

This document is clearly tentative, and clearly Beberman's, containing some ideas that a mathematician might not have put into those words (especially 'functional dependence', which by 1950 was an old-fashioned word, that Beberman might have been concerned with in his own study for a doctorate at Columbia Teachers College. It is the phrase used to designate "function theory" in the MAA 1923 Report[6], that is, it refers to the idea of a function being a relation of dependence between independent and dependent variable, and does not, in Beberman's listing, mean the rather specialized notion of the that name that occurs in multidimensional calculus.

Bruce Meserve, at about the same time, also published an article[7] outlining the topics a high school college preparatory program should contain, 67 of them, with reference to the report, among others, of the NCTM Commission on Post War Plans[8] , another document that would be deemed archaic by the "new math " standards of a very few years later.

The main problem, that of the tension between mathematical imperatives generated by the advance of mathematics since the 19th Century, and pedagogical imperatives, or what seemed to be imperatives in the "progressive era" that followed, could not help emerging during the 1950s, since all mathematicians in the first fifty years of the 20th century had learned to write in a certain style, in which axioms and close logical reasoning were predominant, and where no statement goes unproved, while the school teachers were still being taught a la Minnick. It seemed natural to most mathematicians that one should organize school texts tightly, making no statement that was less than complete and less than fully supported by logical reasoning from previous assumptions. As is visible in the talks given in some of the math education conferences of the 1960s, many mathematicians were even more fervent than Beberman and Vaughan in demanding a full-scale Bourbaki approach to school mathematics.

While Beberman (and Meserve, for that matter) had been conservative in the initial formulations of what freshmen in engineering needed to know, the 1955 team of Vaughan and Beberman were riding the spirit of the times, not yet realizing how difficult it would be in practice to assure high school students the benefits, such as they might be, of the rigorous approach. Nor could Beberman initially realize that this problem would arise at all in mathematics for the earlier grades, right down to Kindergartens. UICSM began, after all, as a University High School program, with an audience of college-bound students who even had to compete for entrance to the program. But while Beberman's own UICSM remained a college-preparatory high school math program until the end, "new math" reformers in the lower grades were constructing imitations, textbooks and examinations in which novel words such as "set" and "Venn diagram" appeared as newly necessary preludes to arithmetic, to bedevil the elementary school teachers and the parents of the children they were teaching.

Teaching modern mathematical discourse to children is not easy, and perhaps impossible on a mass scale even for the best teachers with the most clearly written textbooks. The fact that Beberman himself was so great a teacher probably obscured the necessary lesson for a few years, but those few years were crucial, since with the 1957 launching of Sputnik and the NSF creation of SMSG in 1958 the same taste for logical excesses was implemented on a truly enormous, and irrevocable, scale, before the lessons of the Beberman experiment were visible in contexts more ordinary than the University of Illinois laboratory school.

To begin with, Vaughan and Beberman didn't begin by writing "textbooks" at all. They wrote chapters and sections, mimeographed and given immediate trials by Beberman and other teachers in the University High School, and then subjected to immediate revision in light of classroom experience. This became the pattern in the later curriculum projects of the "new math" era as well, especially the School Mathematics Study Group as led by Edward Begle at Yale and Stanford, beginning in 1958 -- well after UICSM was established. As time went on, UICSM enlisted other high schools to do the same as at the Illinois base, but only after their teachers had come to Illinois for special summer instruction in the use of these materials. In later years "Summer Institutes" were held outside of Illinois especially for teachers of UICSM materials, and schools all over the country applied for permission to participate.

Beberman personally conducted correspondence with all these schools and the participating teachers, many of whom were known to him from his incessant travels around the country. His travels were initially intended to recruit schools and teachers for UICSM, in fact, but Max the showman never passed up a chance to secure public acceptance of his ideas via local newspaper reports -- with photographers -- of his visits. From beginning to end, Beberman did not permit use of UICSM materials (in classes affiliated with his project) whose teachers did not receive such training, and from beginning to end these teachers submitted systematic reports to UICSM on their success or failure, as they saw it, along with suggestions for improvement. To the newspapers it was a travelling show, worth at least a column and a photograph, and sometimes an editorial commending or condemning his approach, but to UICSM Beberman's travels were more than propaganda; they were for the education of the educators who watched him work, and who would later be training to imitate him.

By 1957 the UICSM materials were ready for a tryout on a broad basis for the first time, when, according to Willoughby [9], 12 pilot schools cooperated in a program involving forty mathematics teachers and about 1700 pupils. Thereafter, the number of experimental schools and classes increased rapidly. By 1958 the UICSM curricular materials, which now included lengthy teachers' guides as well as school texts, were made available to a wider public, that did not formally participate in the program or have the advantage of the kind of training Beberman insisted on for his experimental use. The results cannot be known, of course, but one can infer from the growing failure of the uses of these and allied (generally bowdlerized) materials over the next ten years that the initial intention, to keep the program strictly experimental and controlled, was wiser than could be maintained during the post-Sputnik clamor for "new math". Among other things, the experimental attitude had generated much of the enthusiasm UICSM teachers felt for the project, while the sudden wide introduction of "new math" materials into unprepared classrooms inspired fear more than enthusiasm at the grassroots level.

In due course, then, since much of the commercial material published during the 1950s were poor imitations or bowdlerizations of what Beberman and Vaughan intended, there were also produced overtly commercial versions of most of the tested and seasoned UICSM material: Beberman, M. and Vaughan, H.E. High School Mathematics, Course 1 (Boston: D.C. Heath, 1964) and Course 2 (Plane Geometry with appendices on Logic and Solid Geometry 1965). As these were commercial texts anyone could now use; but while the UICSM materials were produced only after long experience in class use by thousands of students, other textbooks of the 1960s were less carefully done and less correct. By that time the whole country was enthusiastic about "the new math", and every publisher had to have something that could be described this way.

The books by Beberman and Vaughan had their faults, but they were not the worst of the genre. The commercial publishing houses had a wealth of models from which authors able to write more attractively for ignorant audiences could take their material. SMSG, the post-Sputnik NSF-financed School Mathematics Study Group headed by Edward Begle, a much larger effort (which will be described below), including all grades K-12, was also such a source; but while UICSM and SMSG were at least mathematically correct many other authors simply didn't understand the purpose of this sort of writing, and offered embarrassing simulacra of logical discourse, advertised, to be sure, as "New Math." Furthermore, most teachers of "new math" were untrained for the task, let alone coached by Beberman.

Beberman himself never stopped learning mathematics as he went along, or at least elementary mathematics from a modern point of view. His trial drafts were sent to many mathematicians around the country, even to Morris Kline, the New York University mathematician who was the New Math's most articulate and frequent critic, for comment and criticism. In innumerable letters Beberman discussed details of nomenclature, the best axiom system to use for geometry and the best sequence of theorems and exercises, whether this or that topic should precede or follow that one or this, and so on. These epistolary debates were learned and serious, and Beberman never showed a closed mind to criticism, so much so that his colleagues sometimes wondered if he were straying from what they took (from his earlier attitudes) as the True Path.

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The hallmark of the New Math, Beberman's to begin with, but all the others to follow, was logical language. A typical conundrum in high school mathematics is the question of how one proves a trigonometric identity, that is, why the procedures by which one solves equations are suddenly prohibited in "the proving of an identity." Somewhere early in his career a student learns some rules of the form, "You can do the same thing to both sides of an equation." Most often this kind of statement is referred back to Euclid's "If equals be added to equals, the results are equal." This is true enough if the things in question are known to be equal, but if algebraic equations are to be solved, or identities proved, the objects being manipulated in this way are not yet known to be equalities, despite the presence of an "equals" sign in the middle of each line, and the manipulations are problematic in a way hardly mentioned in the traditional textbooks.

The experience of generations has shown that children taught "to solve equations" this way, taught without attention to the logical standing of their materials, do indeed develop reflexes rather than understanding. The more recent invention of the hand calculator, even the graphing calculator, has done nothing to change things. Give the dutiful child "x²-y²" as an exercise and he will write "(x-y)(x+y) ans." Show him "(x-y)(x+y)" and he will (unless he makes a mistake!) write "x²-y² ans." -- all without having been asked a question at all. Before 1950 in American schools, and to a distressing degree today as well, school mathematics was simply not construed as a collection of statements written in English, whether questions, answers, or theorems, and the feeble attempts to put order in the cook-books of high school algebra even since the ill-fated experiments of the 1960s are still an embarrassment.

In 1951 Beberman was one of the pioneers in deciding it could be otherwise. Why do freshmen come to college apparently believing that (A+B)²=A²+B²? Obviously because they have learned the wrong reflex. How can they be persuaded of the correct formula, that the answer is A²+2AB+B²? There are several ways to go about it.

A skilled traditional teacher of 1940 would first try numerical examples: If A is 3 and B is 4, then A²+B² is 25, while (A+B)² is 49. Clearly the badly remembered formula is wrong. Then he goes on, this traditional teacher, perhaps after other numerical examples illustrating the correct and incorrect formulas, to show the well-known diagram of a square of side length A+B, partitioned to show four smaller pieces, one a square of side A, one a square of side B, and two rectangles of sides A and B.

This diagram was obviously known in Babylonia four thousand years ago, and still has the power of conviction, but it teaches only that one lesson. Furthermore, from the point of view of a logician this diagram is irrelevant, since the correct formula concerns numbers, while the areas of the diagram are an application dependent on the properties of Euclidean space and a number of conventions by which we apply numbers to the study of that space. The real reason, goes the New Math argument, is that by the right-distributive law of multiplication, (A+B)(A+B) = A(A+B)+B(A+B), following which, by applying the law on the other side, we obtain (AA+AB)+(BA+BB). Further dickering with the associative law of addition, and application of the commutative law (AB=BA), produces the desired answer.

A complete rendition of this argument in the two-column format used for Euclidean proofs of the same era would occupy a full page of a textbook, yet one should not be too hasty in laughing at this example of logical overkill, since this particular formula is not the only point of the exercise. In fact, the distributive and associative laws must be understood by the 9th grade student of elementary algebra for many other problems of importance. Some of them are quite important even in daily life, some for future developments in mathematics that at least some of the University of Illinois High School students would be studying in a few years' time, whether for physics or medicine or finance, and some for the mere filling-in of the cultural background that all education, whether mathematical or literary, should provide.

But should one belabor the "obvious"? The distributive law is in fact understood by the average adult citizen, as may be illustrated by the laughter that invariably follows anyone's telling this allegedly true story:

A certain drug store was advertising "15% OFF -- ALL ITEMS!", and the narrator, having chosen her purchases, took her place in line at the check-out desk while the clerk was serving those in front. The clerk entered each price into the cash register, then calculated 15% of that price (as a negative entry in the cash register, i.e. as a subtraction from the total) before entering the next item. Each customer waited while this tedious string of entries and 15% subtractions went on, and it took a long time indeed, before each customer's grand (discounted) total emerged. When the narrator came to the front she suggested to the clerk that it would save a lot of time if she entered all the items as marked, summed them, and only at the end took 15% from the total. "Oh, I thought of that," said the clerk, "But it doesn't always work."

Now if failure to grasp so simple a principle evokes laughter, perhaps it is unnecessary to teach it in the schools? Apparently not, as Dean Everett of Illinois observed of his 1950 freshmen, who were unable to expand a power of a binomial or graph an elementary function. How abstract and how extensive the necessary instruction should be, on the other hand, is a question that was settled one way by Vaughan and Beberman, and in other ways by other reformers of the time; and the details of the matter are a center of serious disagreement in the community of mathematics educators to this very day. The last time there was general agreement, in the de facto national high school curriculum of (say) 1940, the policy was merely to ignore logical structure altogether, and to introduce rules as necessary, offering no pretence of justification. Indeed, it was not even understood even by teachers that there was a question. Geometry had axioms and proofs, and algebra was simply a different ball game. Everybody knows xy=yx, so why go on about it? This perception clearly had to change, and in the 1950s it changed most drastically indeed.

One letter written by Max Beberman in 1957 can illustrate the seriousness with which he took the necessity of precise language, in this case nomenclature itself rather than logic. In answer to what appeared to be a request that he criticize a chapter named Basic Mathematics in some proposed encyclopedia called Our Wonderful World, Beberman wrote a long set of comments on things he counted misapprehended or wrong in the draft, citing them by page number. It is clear from what he wrote what the original had said. Here are some of his comments:

(4) This is quite bad. Try this: The value of π can never be expressed exactly by a decimal numeral.

p.53 (1) You can write a numeral but you can't write a number, just as you can write a person's name but you can't write a person...

p.56-57 This section should be thrown out... One set of symbols can never be equal to another set of symbols, unless the symbols are identical (in appearance). There is no such thing as two equal numbers.

The first of these comments is clearly in answer to a statement such is commonly seen in school math books, saying something like "The number π can never be exactly known." This is of course untrue, since π indeed is exactly a certain ratio of two geometric lengths, and the misstatement Beberman is criticizing is not merely a trivial matter of English usage. School children learning such false statements do come to think there is something inexact about the idea of π, just as later they come to think the derivative of a function at a point is also some sort of approximation, making an impenetrable mystery of mathematics altogether. Add to this the mystique of "variables" (denoted x, y, etc.) being numbers that "actually vary", unlike those denoted a, b, etc., which do not, though they might change from problem to problem, and you have the beginning of a catalogue of horrors that every college calculus teacher is well acquainted with.

What can be a number that "actually varies"? Beberman and Vaughan sought to lay such nonsense to rest. In summer institutes they taught teachers to talk sense, in preparation for demanding sense of the high school children they would in their own turn be coaching. One exam question in one UICSM summer institute for teachers asks for the insertion of single quotes that make the following sentence meaningful and correct:

It is impossible to add 8 to 5 but it is easy to add 5 to 8.

The intended answer was

It is impossible to add “8” to “5” but easy to add 5 to 8

since “8” and “5” (printed in quotation marks) denote numerals, not numbers, and addition is something one does to numbers.

The distinction between number and numeral is far from the only subtle point insisted on by Beberman's project and the Beberman-Vaughan textbooks, though it is probably the one that attracted the most derision as the "new math" worked its way towards its own extinction. The language of sets was also new at this level, and while very little of what mathematicians call the "theory of sets" -- a profound study not at all suited to children -- is needed by daily mathematics, some of its more obvious nomenclature is important for understanding what is meant, for example, by "solving an equation".

That is, "Solve 3x-5=7" means "Describe the set of all numbers such that thrice that number diminished by five will produce 7." Without the idea of a "solution set", of which x is the name of a typical member, students in the past had regarded "Solve 3x-5=7" as a mere prescription for symbol juggling according to pointless rules. Such a student will be quite stymied if the problem is changed to "Solve 3x-5 = 1 - (6-3x)", which when treated in the same way produces the curious result "0 = 0", rather than what the traditionally drilled student would recognize as an answer at all. The correct answer here is that every number replacing x in the second equation yields a true statement of equality; but one would not think to investigate that possibility until “Solve 3x-5 = 1 - (6-3x)" is translated into a real question demanding an answer. In the high school classrooms of 1950 that sentence was thought to be understandable as it stood, but in truth “solve” is a word needing a good bit of definition. Beberman saw that without a language in which to ask mathematical questions there are no questions, only exam rituals.

These distinctions, second nature to logicians, were new to school mathematics and most of their teachers in the 1950s, but they seemed essential to Max Beberman if the imprecision of the actual algebra and geometry of the typical high school course were to be avoided, and it was the imprecision in the previous way of doing things that he saw as the great stumbling block to later learning of even the practical applications of mathematics, in particular the calculus the Dean of engineering wanted his freshmen to learn at Illinois.

Just the same, not everyone shared Beberman's point of view, not even every mathematician.

Unfinished, May 6, 2004

[1] UICSM Archive at Urbana, IL, correspondence of Beberman with Brucr Meserve

[3] Meserve, Bruce E., The University of Illinois list of mathematical competencies. The School Review 61 (1953) 85-92

[4]John Harrison Minnick, "Teaching Mathematics in the Secondary Schools" (New

York, Prentice-Hall, 1939).)

[6] MAA, National Committee on Math Requirements, Report: The Reorganization of Mathematics in Secondary Education. (Published by "MAA, Inc." 1923)

[7] The School Review LXI #2, Feb, 1953, p85-92

[8] NCTM Commission on Post-War Plans, Frist Report, Mathematics Teacher 38 (1945), 195-221, entitled Improvement of Mathematics in Grades K-14); also its associated Guidance Report, Mathematics Teacher 40 (1947), 315ff.

[9] Willoughby, Stephen S., Contemporary teaching of secondary school mathematics, NY, John Wiley 1967