1. Mathematics in American Schools before 1950
Before about 1950, American textbooks in school mathematics were not written by mathematicians, indeed, they were not written even with minimal participation by mathematicians. By "school mathematics" I mean what is taught under the headings of arithmetic, algebra, geometry, trigonometry and the like, from kindergarten through the 12th grade. The textbooks of 1950 were written mainly by teachers and supervisors quite remote from mathematical research and the research universities; their own education had generally included mathematics at the undergraduate level if that. More often this education had been at a teacher's college, with only a very few of the teachers, supervisors, curriculum consultants, textbook writers, or even professors of mathematics education having accomplished the equivalent of a bachelor's degree in real mathematics at a good American mid-century university.
The 1950 worlds of mathematics and of school mathematics were essentially disjoint, and to the general public and to the teaching profession in the schools this did not seem a strange phenomenon. For one thing, the general public was unaware that "mathematics" and "school mathematics" designated different things, or that people called mathematicians, usually professors in universities, had any interests different from those of school teachers of mathematics, though probably they were "more advanced", whatever that might mean. Perhaps they could add a column of figures faster than the average school teacher, or operate a Comptometer in a bank, keeping track of complicated accounting procedures not taught to children.
The disjointness was, however, apparent to the teaching profession, for school teachers had gone to college and sometimes taken a few classes from mathematicians. In 1930 the average American schoolteacher was from a two-year "Normal School", and unless destined for high school teaching studied no mathematics whatever beyond what she had herself learned as a child. By 1950 many more had been to college, most of the old "Normal Schools" by then having converted themselves into four-year colleges of education, sometimes becoming part of an even larger conglomerate called a University. Thus, though Normal School graduates were still among us, the younger teachers of 1950 knew the difference between a university professor of mathematics and a school teacher of mathematics. In particular, those who were certified for teaching mathematics in a high school had generally been subjected to at least some courses called "College Algebra and Trigonometry", "Analytic Geometry", and "Calculus", the sort of thing an engineering student usually accomplishes in the first two years.
But school math teachers didn't ordinarily have to teach such things themselves. Typically, the last year of the high school curriculum for those few students who pursued mathematics into that last year, was a slender course in solid geometry, mainly mensuration of common solids, and a tedious course in trigonometry, the latter half devoted to the use of logarithm tables, with interpolation exercises, while all the interesting theorems, such as the Law of Cosines, were but memorized formulas into which numbers were put when "solving triangles".
Nobody had to be much of a mathematician to teach such things, it seemed. An ambitious teacher with experience in the classroom, perhaps having obtained the birds-eye view of the school that came from his having been promoted to Supervisor, often got the idea that he could write a better textbook than the one his people were using, and did so, earning a little money thereby; but it would never have occurred to him to consult some university mathematician in the writing. What was taught up to the 9th grade was arithmetic such as any storekeeper would understand and use daily, or if not a storekeeper an accountant, a bank clerk or carpenter. The geometry of the 10th and maybe part of the 11th grade was taken in degraded form out of Euclid's Elements, written over two thousand years ago; and while the Elements consisted of 13 books of considerable sophistication, especially beginning with Book 5, the high school plane geometry of even the best high schools in 1940-1950 contented themselves with the simpler parts of Books 1 and 2, plus some selected material about circles and ratios, but avoiding as if they were not there the difficulties associated with the irrational, which begin in Euclid's Book 5. Some books explained this material better than others, but new ones were written by teachers, sometimes professors in the teachers colleges, whose knowledge of the subject came only from previous textbooks of the same depth, plus their own pedagogical experience.
"College preparatory" students in 1950 were never required to study any mathematics beyond that much geometry, while the algebra of the 9th grade consisted of some symbolic ritual understood by very few, and applied to "story problems" of a certain few types that rarely were remembered (except to be mocked) by any adult a year or two later. Stephen Leacock, a Canadian professor of economics but better known as a writer of humorous essays, captured the public appreciation of algebra quite correctly in his essay A, B, and C: The human element in mathematics., in which it was suggested that mathematics was woefully regardless of what could be really interesting in the stories of filling cisterns when it treated the participants as mere ciphers. Algebra itself, which A, B, and C had been invented to make interesting, couldn't possible interest a sentient being. Of course Leacock himself knew better, but he judged his audience correctly.
If, apart from the few theorems of Euclid, one were to ask the typical "college preparatory" graduate of a good urban high school whether he had learned anything at all in mathematics in school, beyond what he could not help knowing by experience with banks and dime stores, he would most likely have answered "Nothing." He might even add, pridefully, the answer one would also get from the man in the street and the society hostess, "Oh, I was never good at math."
As an example from the 9th grade of my own time (1937), consider this problem: Two men are filling a cistern with water, each using a hose of a different diameter. If the first man working alone could fill it, whatever a cistern is, in three hours, and the second, working alone, could fill it in five hours, how long will it take them to fill it if they worked simultaneously?
It is documented by many a survey that the only people aged 25 or more who could solve that problem in 1950 were those few who worked professionally as scientists of some sort, or as teachers of allied subjects. The man in the street could not solve it and the Sunday comics mocked it. In the 19th Century the daily newspapers had often featured special columns for mathematical puzzles of this sort, much as we see crossword puzzles today, but in the 20th these were much more rare than columns for contract bridge and astrology. In short, school algebra was worthless to the general public, even the college-educated public, of 1950. Following the 10th grade, then, students who did not go into "manual arts" or "commercial" curricula in the schools, but instead elected the "college-prep" curriculum leading to non-scientific programs in the colleges, were as ignorant of mathematics as if they had never spent a day at it in their schooling. They knew the addition and multiplication tables and could use them, but so could my own parents, who had immigrated from Poland in 1922 and had never been to a mathematics class in their lives.
In the 11th and 12th grades, in the larger high schools of 1950, there were classes called "advanced algebra", "solid geometry" and "trigonometry." Though this much mathematics was never a requirement for admission to a university, "college preparatory" students who intended science or engineering careers would take them, along with one-year courses (also optional) labeled "biology", "physics" and "chemistry"; but unless a student encountered a truly unusual teacher these science courses consisted mainly of the memorization of old routines and "cook-book" laboratory demonstrations. Some of the mathematics traditionally taught, such as trigonometry, were "practical" in that surveyors and navigators needed to know them, but most of them were pure ritual transcribed from one textbook to another with increasing irrelevance, though with increasingly sophisticated notation as the science itself progressed, so that its public comprehensibility declined, even among adults who had “taken” these courses back in high school, with the exception of those who in college studied the sciences. Even a serious user of trigonometry generally learned what he needed in his job apprenticeship, as a machinist, navigator or surveyor, and not in the high school “trig” course. Hence Stephen Leacock.
In high school algebra one learned to "simplify" formulas by factoring, removing parentheses, dividing polynomials and the like, a set of skills that when first developed during the 17th and 18th Centuries represented an advance -- for scientists who needed to use such techniques -- over the cumbersome language of two hundred years earlier. To be sure, scientists do use these techniques today, and did in 1950, but their schoolteachers -- and the authors of the textbooks that they had themselves learned by -- had by 1950 forgotten both the origins of these manipulations and their purpose. Most algebra was poetry in an unknown language; indeed, "algebra" was in the popular consciousness synonymous with the incomprehensible. ("He might have been talking in algebra, for all I know.") One might in school learn, temporarily, to write down acceptable formulas for ritual examinations, yet not understand a word of it.
Solid geometry from the standpoint of Euclid was also impossible, even for many of the best scholars of ancient times, and it had become totally forgotten in the European Middle Ages. The only purpose that 1950 had for a subject of that name was a small collection of formulas: the volume of a prism or sphere, and -- maybe! -- its surface area, for example, but in most solid geometry books for the schools these formulas were given without the least attempt at proof or intuitive justification. Worse, a student studying such a book would not even arrive at the notion that proofs were possible, or relevant. For those courageous students who got that far, formulas like (4/3)πr3 were accepted as knowledge on the same basis as the Three Causes of the Civil War (Sectionalism, Secession, and Slavery, as I recall). If one were to stand on a street corner in 1950, asking every passer-by what is the volume of a sphere of radius five inches, or its surface area, he would find that only engineers and scientists -- and some schoolteachers -- knew the answer; and a large number of those would know it only from memory or use, with no idea of why the formula is what it is.
Or, to try a simpler question from plane geometry: If a square block in a city measures 50 yards by 50 yards, say, then to go from one corner to the opposite corner via the sidewalks requires a walk of 100 yards. But as the crow flies, diagonally, what fraction of that distance is saved by the short-cut? To ask why this should be, of even someone who knows the numerical answer (about 30 percent), will almost never elicit a coherent reply. Trigonometry takes up such questions in greater generality, dealing with triangles that are not half-squares, or even right-angled at all, and in the high schools of 1950 it was certainly the most sophisticated mathematics taught. Yet even here ritual dominated, even in the manipulation of trigonometric identities until they "came out right". Much of the time in my own 1940 trigonometry class was spent interpolating in logarithm tables.
Not that the good old days were universally better. In the 19th Century hardly anyone went to high school at all, let alone college, and the "prep school" programs for the elite contained even less mathematics than the 1950 program I have been describing, except in that Euclid was often taken more seriously and deeply than later. But with the new century, the American high schools began to prosper and increase dramatically in size and seriousness. The immigrant population flooding the country from Italy and Eastern Europe since 1890 had practically recreated the public high school, and put some of its graduates into competition with those of the elite "prep schools", so that such things as mathematics and Latin suddenly required more teachers and newer books, suited to a more democratic audience. It cannot be said that the changes were intellectually successful, at least formally, since by any measure the textbooks and readers of 1950 were less challenging than those of 1900, but despite the inanities of the curriculum the more ambitious children of the rising classes managed to make a success of school by themselves, building also on the ambition of their immigrant parents, the Carnegie libraries, and their own churches, community centers and debating societies.
This apparent success of early 20th century American secondary education for the masses did not extend to mathematics, however, which is something hard to find at random on library shelves, and hard or impossible to understand with the ignorant textbooks of the time, taught by teachers even further removed from mathematics than the textbooks they used. What was left of the Euclid as taught in the better prep schools of the late 19th Century was progressively watered down in the 20th; while what remained of 19th Century algebra, already rigid and pointless, cherished as "mental discipline" more than for any other reason, became even more misunderstood than geometry under the influence of the general attack on the intellect led by the progressives at the Columbia Teachers College. In public education the "progressive" philosophy was dominant, and it had little use for mathematics in particular.
Progressive educational theorists who were leading the way in the democratization of the schools distrusted any teaching that did not fulfill a "felt need" of the student. And while the child of illiterate parents might very well, through even a "progressive" school education, discover a need for literature, history and politics, random excursions into the library seldom generated such a need for mathematics. Inevitably, then, the decline in demand for mathematics courses in the schools was accompanied by a decline in the mathematical education of future teachers, a process that fed upon itself as the century wore on.
Yet during this time a veritable revolution had been going on within mathematics itself, even in America. In 1900 there were hardly any American mathematicians of note, and anyone curious about recent developments went to Europe, generally Germany, to study, either for a PhD or a postdoctoral stint. The only American mathematicians whose names were known in Europe in the 19th Century (and late in the century at that) were Simon Newcomb and Josiah Willard Gibbs, and they were better known as astronomer and chemist, respectively, than as mathematicians. Newcomb did take an interest in school mathematics, but as a university professor at Johns Hopkins his interest was in the prep school math of his potential freshmen, and not what we would call the K-12 curriculum today. Gibbs was little interested in teaching of any sort, and was largely incomprehensible even to his few students at Yale
Though there had been a great growth in American mathematics research and application between 1900 and 1950, they lived in industry, in the universities, and in the military, but not in the schools. From time to time there were committees of mathematicians formed by the American Mathematical Society or the Mathematical Association of America, to study ways of getting better mathematics into the schools. A beginning was attempted by the University of Chicago mathematician E. H. Moore, on his 1902 retirement as President of the American Mathematical Society. His Presidential Address of that year was largely devoted to an appeal for a better school mathematics program. His words were mostly wise, though unduly deferential to the new theories of progressive education, but they were really widely heard only within the mathematical community, and his subsequent efforts to influence the curriculum and teaching of the secondary schools' mathematics came to nothing. In the teachers colleges other words were heard, to quite other and more "democratic", effect.
In 1923 the MAA, an organization more explicitly devoted to problems of education, albeit at the college level, than is the AMS, which mainly fosters research in mathematics itself, issued a report of its National Committee on Math Requirements that could hardly have been wiser, as it strongly advocated a high school program emphasizing the notion of "function", which would surely be the unifying theme of mathematics for the forseeable future (as indeed it was); but there was no mechanism for carrying such recommendations into practice. That report is probably more widely read today than it was in 1923, i.e. it has more historical interest now than it had practical interest then.
It was hoped at that time, by mathematicians interested in school mathematics, that the newly created NCTM (National Council of Teachers of Mathematics) would be a vehicle for the suggested reforms, for the NCTM had been formed in 1920 with the urging and assistance of the MAA, and in its earlier years the senior organization had some influence in NCTM; but hopes for a cooperation between the community of mathematicians and the world of school mathematics were destined to be dashed, despite the fitful efforts of both NCTM and MAA to cross the widening divide.
For in competition with the 1923 MAA Report there had appeared at nearly the same time (1920) a report of a committee that had been formed before the war by the NEA (National Education Association), a report called The Problem of Mathematics in Secondary Education. This book was nominally the work of a committee formed before the first World War but in fact was largely written by the famous Columbia University Teachers College professor William Heard Kilpatrick, and it showed it. Kilpatrick's vision of a pulsating, vibrant education -- progressive, student-driven rather than dictated by authority -- ecstatic, free of the dead hand of the past -- held sway over the next thirty years while the MAA recommendations of 1923 were swept out of sight. Kilpatrick's vision of "progressive education", of even wider compass than that of his more famous predecessor and contemporary, the philosopher John Dewey, found such favor with schoolteachers who found it easier to teach less mathematics than more, and be praised for doing so, that his classes at Columbia turned into an unprecedented source of tuition money for that college. Who, after all, would not value children's psyches above mere intellect? Kilpatrick could not lose, and his literally thousands of students came to populate the forty-eight states and all the ships at sea. So it went until the Second World War, when this merely intellectual -- maybe philosophical -- debate concerning the nature and materials of education, hitherto troubling nobody but an elite, surged into the politics of the nation.
The State of Math Education in 1945
The Second World War brought to the attention of the public the enormous importance of science as no previous event had ever done. America had always celebrated its inventors, to be sure, from Eli Whitney to Edison and the Wright Brothers, so that technology certainly held the public respect, even reverence. But Whitney and Edison had no need of mathematics; invention (as opposed to basic science) was done with materials one could see and touch, not with symbols on paper. From ancient Greece to America and Europe in 1900, technology and industry were not the concern of philosophers. In Europe the Seventeenth Century featured a scientific revolution (Galileo, Newton), the Eighteenth Century an industrial revolution (Watt's steam engine and the power loom), and the Nineteenth Century a technological revolution (railroads, telegraph), and while this is the way schoolchildren used to be taught the progress of the practical arts such a listing totally neglects the purely scientific and mathematical advances of the latter two centuries. Mathematics flourished underground, as it were, while steam engines, cotton gins, telegraphs and harvesters were being developed quite outside the schools and universities. Even in the early 20th Century the great technological advances celebrated in the schools were associated with the names of Ford, Edison and Eastman, men who had little use for mathematics. But another thread was making its way, even if the newspapers weren't paying much attention.
For even in the early 20th century science began to converge on technology, especially with new discoveries in physics. Radios, electric power generation, petroleum exploration and refinement, differential gears in automobiles and the design of airfoils, for example, required calculations of increasing sophistication, based often on new mathematics even then being introduced, such as the Steinmetz application of complex number algebra to the analysis of electric circuits, and the application of differential equations to the governing of petroleum distillation. Engineering students, who hadn't even existed as such before 1850, were by 1950 no longer learning only such "practical" skills as mechanical drawing of tool parts, and computations concerning the strength of structural materials, but were learning the electronics and chemistry that had been developed in their own time, often using mathematics that in the time of their fathers had been considered only a philosopher's concern, as mathematics had indeed been only a philosopher's concern in the years between Pythagoras and Gauss.
In 1945 the public was startled with the results of work secretly done during the war just past, most particularly the development of radar and nuclear bombs. These had been the difference between victory and defeat, after all. Other developments, too, had required mathematical and scientific knowledge far beyond even "engineering school" training, let alone common knowledge and a basement workshop: The decoding of cryptograms, the statistical analysis of the efficiency of manufacturing processes, and even the economic analysis of the probable effects of a new program of taxation or foreign aid.
For a time the public appeared to be heeding the challenge to know more of science and mathematics than it had in the past generations, but this was a fortunate accident of the war itself: Young men came back from military service with respect for technology and a desire to participate in the new world. Aided by the "G.I. Bill of Rights", a college-tuition and living allowance entitlement for war veterans re-entering civilian life, they returned from the devastated cities of Europe and Asia to refill the American universities in unprecedented numbers, eager to learn.
University professors of the period 1945-1950 were unanimous in their delight at the suddenly improved quality of their students (this was as true in the humanities as in the sciences, actually), which was in fact as much due to the greater average age of the student body as to any intellectual influence of military experience, and their anxiety to make up for the lost years of the war was also evident; but however the causes might be weighted there was no denying that scientists and scientifically educated graduates in medicine and engineering were coming out of the universities in very satisfactory numbers during the era of the GI Bill.
All this was in considerable contrast to the situation in 1940, say, when the American military was appalled at the scientific -- and general -- illiteracy of their new draftees, and often had to conduct elementary school classes for their own inductees if they wanted to be sure to have enough gunners and typists, for example. But with the ending of the war the country was no longer terribly worried about the general literacy, which a draft army needed so badly, and when it looked upon the results of its concentration (via the “G. I. Bill”) on college education, especially in the sciences, in 1948 it found only success. Progressive education had been bypassed by military experience and age.
By 1950 this delusion evaporated. The college freshman classes were no longer seeded with returning war veterans, but were once again the same group of mathematically untutored children they had been in 1940. Professors of mathematics suddenly found themselves facing uncomprehending classes, their rhetorical questions met with silence, often sullen, their homework assignments beyond the capability of their more childish audience. Something awful seemed to have happened to the school preparation of their new students.
One event, a minor event to begin with, might be pointed to as signaling the recognition of the new problem: It was the appointment, by the Dean of the University of Illinois College of Engineering, of a committee to investigate the actual knowledge of incoming freshmen, with an eye to the publication of a document prescribing in outline for the schools of the State of Illinois a mathematical curriculum that would make it possible for the Engineering school to do its job with the products of that curriculum. Most importantly, the University intended to establish that curriculum experimentally, in its own University High School, to refine it by experience and see that it was more than armchair theorizing.
To say that this was a "minor event" is no denigration, only a recognition that by 1950 universities all over the country were facing the same problem and arriving at the same solution: the appointment of a Committee. This is what faculties do, and their reports are usually intelligent and to the point. The difficulty is then not in their own programs, where they suggest (for example) certain content for their freshman chemistry or physics courses, implying a corresponding high school background that is not terribly hard to outline; the difficulty is in inducing the behemoth of a public education system, a public anarchy in most states -- at least, in 1950 -- to make the necessary changes.
Even the existence of a University High School as a laboratory for curriculum and pedagogy, is fairly common. Illinois is not alone in having one, for in the major state universities (Michigan, for example, as well as Illinois) these schools are perfect training grounds for the future teachers attending the University's college of education. And Columbia University's Horace Mann school has been famous for many years. Indeed Columbia, at the same time as Illinois, was itself developing a new high school curriculum, under the guidance of Howard Fehr. So were many others, but the step from laboratory to statewide policy had always been a long one, if it existed at all.
In the back country of Illinois, far from the 116th Street subway stop for Horace Mann, that particular, local, curriculum study committee, called modestly the University of Illinois Committee on School Mathematics (UICSM) eclipsed all the others. Its first product was a list of “competencies” a high school student of mathematics, intending scientific studies in college, should end with, and produced a model of singular influence. The “competencies”, mainly the work of Bruce Meserve and Max Beberman, were conservatively stated, though demanding, and could have been written almost anywhere, but Beberman, new to the University of Illinois (and a charismatic teacher) changed their flavor to include more rigor in mathematical reasoning than a mere list of competencies seemed to imply, and personally carried his news to other colleges, schools and to the press, with increasing, though not really enormous, success until 1958.
Then, SMSG, a new gorilla, appeared on the block in consequence of a second public shock (second after the war, The Bomb, and radar): the Russian Sputnik of 1957. That Russia seemed to be superior to the United States in rocket technology (which wasn't quite so, actually), and demonstrably had acquired nuclear technology, added to the national concern in such a way that the federal government went directly to the American Mathematical Society for advice and aid, and, on the advice of committees of mathematicians, began pouring money into the problem in quantities Beberman and his allies never dreamed of. While Beberman's fame as a personality remained enormous, the infusion of Federal money into science and mathematics education following Sputnik relegated the Illinois program to second place, after the new organization, the School Mathematics Study Group, initially based at Yale University under the direction of a Yale mathematician, Ed Begle, began to flex its muscles in 1958.
That SMSG would have been created without the earlier work of Beberman is undeniable; indeed, Beberman would probably have been an important participant in SMSG if he had not earlier created UICSM, which was taking more than all his time. But the flavor of the mathematics promulgated by SMSG owed much to Beberman, for all that Beberman, himself a teacher and not a mathematician, owed much to the mathematicians for his own ideas. Unlike aborted reforms of earlier years, the "new math" of the 1950s was the creation of a combination of mathematicians and school-level mathematics teachers, each group giving credibility to the enterprise that neither group alone might have been able to command. Yet by 1958 the mathematicians had assumed the lead.
Thus, for the only time in the history of public education in the United States, mathematicians, as distinguished from professors of mathematics education, and from school teachers of mathematics, and from professors of education, were given the opportunity, via enormous public expenditures, to influence the schools. For ten or fifteen years their efforts were spectacular -- even given a name ("The New Math") -- and then they were gone, back to their universities and their theorems, leaving the field, after about 1975, to the educators who had been there before, to the colleges of education, and to the professional educational bureaucracies of the States and school districts.
The new directors of school mathematics education were, after 1975, much the same as the old, who had had command of the schools and their curricula before 1958, and were for good reason called, “The Professional Education Bureaucracy” or “PEB” by William L. Duren, a mathematician and Dean at Virginia, and prominent in advocacy of the “new math” reforms of the 1960s. As time went on, the Federal government continued to take an increasing role in public education, mathematics included, but without again calling explicitly on the mathematicians as it had in 1958. The federal concerns after about 1970 were concentrated in other aspects of education: the racial divide, for example, the Head Start and school lunch programs, provision for the dyslexic and the misfit, the problems of violence and drugs. The curricular initiatives of the 1960s took second place to these other social problems, and while the education professionals, and the funds available for education, flourished and increased as never before, the mathematicians as a group were no longer among them, and it does not appear that they will ever again be as influential as in "New Math" days. In fact, in the popular consciousness to this day, their effort and the written product of their work has been discredited. "The New Math" is now remembered as a giant mistake even by people knowledgeable in the history of public education in America.
For example, Edwin R. Schweber, a high school physics teacher of more than usual competence, wrote a letter to the editor of the public policy monthly Commentary [Volume 99, February, 1995], in reply to an earlier article by Chester Finn [volume 98, October, 1994]. Finn, though not a scientist, was a former Assistant Secretary of Education, so that one should be fairly confident that this exchange of views, at least insofar as it concerned "The New Math" of then-recent history, reflected the current wisdom of the time among people knowledgeable in education.
The main subject of Finn's article and Schweber's objection are not to the point just now, and in fact Schweber and Finn were largely in agreement; but in passing Schweber wrote, "Admittedly, giving control of education to those whose primary orientation is to the disciplines being taught is only a necessary, not a sufficient, condition: the new math was foisted on the schools by mathematicians, not by educators..." Finn's reply included this: "...Even while recognizing that it was the mathematicians who brought us to the debacle of 'new math', he [Schweber] would still have us trust experts to make curricular decisions.”
That is, whatever differences Schweber and Finn had concerning the participation of experts in curricular decisions, they agreed that the new math had been “foisted on the schools by mathematicians” and that it had been a “debacle”.
Even within the profession of math education we hear echoes of the same simplified view of the history of the time. On an email list called "math-teach", explicitly devoted to discussions of mathematics curricula and teaching methods, a certain Dominic Rosa wrote (June 14, 1997), "The new math curricula and textbooks were an absurd response, by misguided mathematicians, to the launching of Sputnik in 1987." On another such list, called AMTE, another message declared, also en passant, "Mathematicians without educators led to the failure of the 'New Math' of the 60s."
Both statements are incorrect. 1. The “new math” was not “in response to the launching of Sputnik”, though that launching certainly generated unprecedented federal education spending. Even so, that spending did not diminish following “the death of the New Math”, though it was directed at mathematics educators rather than at mathematicians, and it has been increasing without pause up to the present.; and 2. Whatever “the new math” was, it was in fact created with the fullest possible participation of “educators”, as may be seen, for example, from William Wooton’s history of SMSG, but also from the fact that most commercial textbooks, including those most enthusiastically in support of the new programs, continued to be written by “educators” with rarely, apart from the SMSG experimental textbooks, any significant participation of mathematicians.
What was true about all this misperception of the phenomenon of “the new math” was that there had been a participation of mathematicians at all, and that this participation was publicly construed as a good thing. If the errors of the preceding era could be laid to the ignorance of the PEB and the failings of the schools of education, should not the addition of mathematicians to the company of those seeking to improve things rather have helped matters, than have been responsible, as came to be the public view, for the seeming disaster to follow? How could this happen?
The story of this book is designed to answer both parts of this question: What did happen, and what was seen to have happened, during the period of "The New Math" in the United States from about 1952 to about 1975. Both stories are complex, for different things happened in different parts of the forest, and what was seen to happen depended very often on the observer more than the sight. Moreover, there is a third story, which is why what did happen, such as it was, happened in the way it did. There is a fourth story, of course, which was of why the public saw what it thought it saw during this period. Indeed there is no end of stories here, for there were also political causes and political consequences of all these happenings and appearances.
We will begin with part of the third story, which is about what mathematics education in the schools actually looked like in 1950. By that year, even before Beberman at Illinois and long before Sputnik and the SMSG, there were already mathematicians, distressed at the ignorance of their college freshmen in the aftermath of the GI Bill, who were looking seriously at the teaching of mathematics in the schools. The attempted treatment prescribed by "The New Math" was, after all, dependent on what the mathematical community saw as the nature of the disease. What did they see, then?
Ralph A. Raimi
Revised 8 July 2006
 Leacock, Stephen, Literary Lapses, Montreal, Gazette Publishing Co. 1910, p 118-125
 Lynd, Albert, Quackery in the Public Schools. Boston, Little Brown 1953
 Cremin, Lawrence A., The Transformation of the School: Progressivism in American Education 1876-1957. NY, Knopf 1961; Ravitch, Diane, Left Back: A Century of Failed School Reforms, Simon & Schuster, 2000
 Newcomb, Simon, Reminiscences of an Astronomer, Houghton Mifflin 1903; Rukeyser, Muriel, Willard Gibbs, Ox Bow Press, 1942
 Moore, Eliakim Hastings, Presidential Address: Foundations of Mathematics, Science, March 13, 1903.
 The Reorganization of Mathematics in Secondary Education, published by "MAA, Inc." 1923; also Houghton Mifflin Co.]
 NEA Commission (Kilpatrick, Wm. Heard, editor), The Problem of Mathematics in Secondary Education, USBE Bulletin 1920, no.1. Washington, D.C.:GPO, 1920 (written by Committee on the Problem of Mathematics in Secondary Education, and described on pages 192ff of NCTM's 32nd Yearbook.)
 Meserve, Bruce E., The University of Illinois List of Mathematical Competencies, The school Review 61 (1953), p.85-93
 Henderson, Kenneth B. and Dickman, Kern, Minimum Mathematical Needs of Prospective Students in a College of Engineering, Mathematics Teacher XLV (1952), p.89-93
 Duren, Wm.L., the chapter, Mathematics in American Society 1888-1988, in Duren, Peter, et al (Editors), A century of Mathematics in America (Amer. Math. Soc. 1988), Part 2, pp 399-447.
 Wooton, William, SMSG: The Making of a Curriculum (Yale University Press, 1965, 182 pages).