__Introduction__

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__.[1], 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)πr^{3} 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 20^{th}
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.[2] In public education the "progressive"
philosophy was dominant[3],
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[4]

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[5] 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[6]__ 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[7]__. 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[8],
and produced a model[9] 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”[10] 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."[11]
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,[12]
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

[1]
Leacock, Stephen, __Literary
Lapses__, Montreal, Gazette Publishing Co. 1910, p 118-125

[2]
Lynd,
Albert, __Quackery
in the Public Schools__. Boston,
Little Brown 1953

[3] 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

[4] Newcomb,
Simon, __Reminiscences of an Astronomer__, Houghton Mifflin 1903; Rukeyser, Muriel, __Willard Gibbs__,
Ox Bow Press, 1942

[5] Moore,
Eliakim Hastings,__ __Presidential Address: __Foundations of Mathematics__, __Science__, March 13, 1903.

[6]
__The Reorganization of Mathematics in
Secondary Education__, published
by "MAA, Inc." 1923; also Houghton Mifflin Co.]

[7]
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.)

[8] Meserve, Bruce E., __The University of
Illinois List of Mathematical Competencies__, The school Review 61 (1953),
p.85-93

[9]
Henderson, Kenneth B. and
Dickman, Kern, __Minimum Mathematical Needs of Prospective Students in a
College of Engineering__, Mathematics Teacher XLV (1952), p.89-93

[10]
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.

[11]
See http://mathforum.org/epigone/math-teach,
archive for June, 1997, and the thread named "TIMSS, etc."

[12]
Wooton, William, __SMSG: The Making of a Curriculum__ (Yale University
Press, 1965, 182 pages).