Whatever Happened to the New Math?

School math textbooks sixty years ago were not written by mathematicians. The typical author was the chairman of a school science department somewhere, in a district large enough to make writing a textbook remunerative even if nobody else in the country used it. That his mathematics was ignorant was unnoticed by an ignorant public and cadre of teachers, and that his prose was abominable was perhaps admired, so strong was the general (mistaken) belief that mathematics is not written in prose.

Teachers, mainly trained in schools of education, knew little about mathematics to begin with; many habitually ignored anything demanding in their textbooks and took refuge in teaching the algorithms they had themselves learned as children. Textbook publishers wouldn't dare print a book containing something its predecessors did not contain, because no school would buy it. And which real mathematician would spend his time writing a school textbook that nobody would use?

Euclid's Elements, for example, history's greatest textbook of reason, had been bowdlerized, reduced or supplanted by things called more practical, but really esteemed for being easier to teach: interest rates, surveyors' triangles, and rigid algebraic rituals for the college-bound. Anyone with half a mind could recite them, but neither teacher nor student wasted a minute on their meaning or utility. Worse, each generation's authors added a bit of new misunderstanding to what might have been right in earlier editions.

Sputnik gave us a chance to break this gridlock. The 1945 atom bomb had already given physical scientists and mathematicians a prestige without precedent; now the Russian success of 1957 added fear, which paid better. The year 1958 therefore kicked off the largest and best financed single reform effort ever seen in mathematics education, the School Mathematics Study Group (SMSG), upon which the National Science Foundation (NSF) spent millions of dollars over its twelve-year lifetime.

Edward Begle, a professor of mathematics at Yale University, was chosen to head the new organization, and gave up topology for this new and unfamiliar calling. The existing professional educational bureaucracy, later called "the PEB" by William Duren, a reforming mathematician of the time, was thus suddenly outflanked by a new party. That is, the teachers' colleges, the National Council of Teachers of Mathematics, and all the State and Federal departments of education and nurture, who though loosely organized did still govern all teaching below the college level, were compelled for the time being to follow our lead.

What Begle saw in the schools could not be cured by a friendly environment, good lighting or deep pedagogical insight, so long as the textbooks, and the mathematical conceptions of thousands of teachers, amounted to a pack of lies. To put first things first, he assembled several separate teams of mathematicians to write exemplary textbooks, eventually covering all grades 1-12 and a bit more, that would be free of the ignorance, ambiguity, opacity, irrelevance and tedium of the traditional curriculum. He included practicing schoolteachers in each writing team, hoping (vainly as it turned out) to keep his textbooks within the realm of the classroom possible; but the mathematicians drove the effort. SMSG invited all commercial publishers to study, copy, or plagiarize these texts, which SMSG placed in the public domain as models, freely.

Simultaneously, SMSG established hundreds of Institutes, i.e., special college courses for existing teachers, some in the summers and some on Saturdays, to which eventually thousands (paid by the NSF) came to study the new material, to practice its pedagogy under the eyes of SMSG authors and master teachers, and then to carry the books back into the world for classroom testing on a nation-wide scale. The writing groups would reassemble summer after summer, study the reports from the field, and revise the texts and the teacher's guides for the next set of Institutes and experimental classes.

Almost half the nation's high school teachers of mathematics attended at least one such Institute during the 12 year life of SMSG; but an equivalent seeding was impossible for elementary school teachers, who outnumbered the high school math teachers ten to one. While there were some Institutes for elementary school teachers, these were mainly for experimentation. The SMSG books themselves achieved unexpectedly wide circulation, and were indeed, as Begle had urged, enthusiastically if often ignorantly imitated, even (or especially) at the more elementary levels. And the research literature produced in the colleges of education, and the journals of classroom practice written and read by teachers, were for the entire decade of the sixties dominated by obeisance to the SMSG program.

The result, after twelve years, was total failure. By any reasonable measure, and measures were taken, school mathematics was worse off in 1975 than it had been in 1955. The idiocies of the older curriculum had in most places been removed, but often to be replaced with new ones. Tom Lehrer's 1965 song New Math, lampooning the pretentious language used to justify an inability to calculate, had the mathematical community itself laughing at the follies committed in the name of promoting a better understanding of mathematics.

To take an example, the language of the "theory of sets" has been basic among mathematicians for a hundred years, and can ease enormously the path to much that people find perplexing in school. Anyone should be able to learn enough about sets and this vocabulary in a very few hours to permit him in consequence to understand an honestly presented course of high school mathematics including all the traditional material and more; his savings in time will have exceeded those few hours a hundredfold, and in understanding immeasurably. SMSG introduced set theory into its first books, which as it happened were for the high school level. Later books, written for grade-school years, also introduced the subject of sets, hoping later to make use of it when revised high school books were written. It therefore turned out that for a time -- all the time SMSG had, alas, in its short career – a chapter on sets appeared at the opening of every year's textbook, unfortunately making it appear as if sets were the be-all and end-all of Newmath. This redundancy was copied into the commercial texts of the time as well, and teachers leaped on it to the neglect of more prosaic matters, like getting a correct answer in arithmetic.

Easy as it looked, teachers didn't always get the notion of "set" straight themselves, and could teach the most egregious confusions as truth. One textbook lesson plan suggested that the teacher, as an example, distinguish the subset "boys" from the subset "girls" (in the set "this class") by asking the boys to stand, and then the girls to stand, and so on; one teacher I heard about then asked "the set of boys" to stand up. But while boys, being human, can stand, sets cannot. So fine a distinction may be meaningless to a third-grade teacher, or to anyone who has never made real use of it; but if exactly that distinction is not made plain, and into a habit of mind and speech, the notion of set is valueless in later mathematical reasoning.

On the other hand, SMSG and its imitators were also guilty of some pointless pedantry, ridiculous even if logically correct: "Write the numeral that names the number solving 3x -7 = 8," for example. That's not even English. If you actually ask a mathematician to write down his phone number he will cheerfully hand you a numeral without a moment's hesitation or apology. He can make the distinction, sure, but he only does it when it counts.

Just the other day I heard an aging academic say that Marxism hasn't failed, because it hasn't been tried -- not an original trope, for we have heard the same of Christianity for ages. Had SMSG really been tried? The mass of American teachers -- and children -- were not in the end exposed to, let alone taught, what the SMSG mathematicians prescribed. But to plead thus is only to evade responsibility.

Oliver Wendell Holmes once wrote that the American Constitution is an experiment, "as all life is an experiment." Experimental philosophers like Holmes understand that reality is not to be pushed around, neither by nine old men nor by a prestigious bunch of mathematical geniuses with a pipeline to the U.S. Treasury. Their prestige was unchallenged, their genius without peer, and their pipeline of pure gold; but the realities overwhelmed them. The cadre of teachers already out there had preexisting interests and capabilities, the public patience was shorter than experiments that could lose a generation of children, and the educational experts, the PEB, was gathering its strength for the political battle that finally turned the pipeline back in their direction.

Towards the end, Begle wrote, "I see little hope for any further substantial improvements in mathematics education until we turn mathematics education into an experimental science, until we abandon our reliance on philosophical discussion based on dubious assumptions, and instead follow a carefully constructed pattern of observation and speculation, the pattern so successfully employed by the physical and natural scientists." Begle himself died a disappointed man six years later, though he had continued after SMSG to work brilliantly towards a proper study of mathematics education. His disappointment was for the future more than for SMSG, because he foresaw correctly that PEB-sponsored research in education would not follow his sensible, if unexciting, prescription.

Meanwhile, the PEB, having taken back the schools, resumed educating its future leaders with exactly the "philosophic discussion based on dubious assumptions" Begle had warned of. It was the education of teachers that Begle had come to see as the truly intractable problem. SMSG, for all its faults, could solve the problem of choosing, pacing and stating an excellent curriculum; another ten years' experimentation would surely have removed what Duren called its “excessive enthusiasm for logical language," for example. But the SMSG Institutes had been hopelessly inadequate to the training of teachers, and the PEB is perforce in charge of the next generations. There is no market in sight for even a perfect SMSG curriculum.

The textbooks today are again not written by mathematicians, and indeed show no sign of SMSG influence whatever. They have eliminated the "set theory" they were all decorated with in 1975, and they are quite silent about numerals; so much is to the good. On the other hand, they contain even less mathematics than they did in 1955, except that at the college-preparatory upper levels some of them, intended for superior students and teachers, are a great deal better. The books for grades 1-8 come packaged for teachers with mountainous "Teachers' Guides," in which the mathematics is swamped into insignificance by the instructions on engaging the attention and improving the self-esteem of the students.

The general mathematical literacy, not notably improved by SMSG, has continued its decline under PEB management as well. Developmental psychology, not mathematics, informs the seminar rooms of the schools and the teachers colleges, while at the higher levels the research journals of the PEB are filled with what almost every mathematician today would condemn as being at best a waste of time.

Perhaps an example is in order:

From the anthology, Perspectives on Research in Effective Mathematics Teaching, Vol 1, published by the National Council of Teachers of Mathematics, Reston, VA 1988, the chapter, Interaction, Construction, and Knowledge: Alternative Perspectives in Mathematics Education, by Heinrich Bauersfeld (p.41ff) contains these insights:

Understanding Theories

As opposed to the context-bound ascription of meaning in everyday language use, scientific theories are presumed to rest upon the strict use of their technical terms. Researchers often pick the labels (the words, the "signifiers")for their key categories following a contiguity relation between the concept (the "signified") they have in mind and one specific of the many facets of meaning ascribed to the word in everyday use. Functioning as both help and hindrance, this facilitates initial understanding and access to the theory, but also gives rise to the illusion of an easy metabasis for criticism in both directions, from the theory directed outwards as well as against the theory from outside; whereas serious criticism of a theory would at least require an adequate understanding of the network character of its technical terms (see note 10). The construction of a metatheory capable of executing the critical comparison of competing theories will fail due to the impossibility of an uniting metaperspective and because of the (related) nonexistence of a universal language. How to proceed, then?

Good question.

Ralph A. Raimi

22 October 1995; slightly amended 12 April 2005