**NYC HOLD Honest Open Logical Debate on math reform**

**Are our school’s math programs adequate?**

** Experimental
mathematics programs and their consequences**

**New York**** ****University**** ****Law**** ****School**

**New York City**

**June
6, 2001**

**opening remarks**

**A Brief Review of the History of American Math Education Orthodoxy
**

**How Did We Come to This? **

Ralph A. Raimi

Professor Emeritus of Mathematics

My acquaintance with the
uses of mathematics goes very far back, to long before I became a mathematician. After two years of college I entered the
United States Army Air Forces and became, after considerable training, a Radar
Maintenance Officer, that is, an officer in charge of the shops on military
airfields where airborne radar sets were repaired or replaced on
airplanes. I was an expert on the radars
used in 1944, and had a good appreciation of the elements of electrical engineering,
such as it was in the days before transistors or computers.

I returned to the

However, my purpose in this Introduction is
historical. We are faced with a terrible
failure of mathematics education in the public schools, and not only here in
District 2. We believe the professional
educators are on the wrong track in addressing the problem, and so does the
public. The first question is, how did
we get here? Is the problem new? If not -- and the problem is not new, by the
way -- how have people tried to fix things in the past, and why have they
failed? What can we do this time that
will be better?

1. Before World War II nobody worried much about
math in the schools. Basic arithmetic
was so you would understand your bank account and grocery bills, and if anyone
was good at that and wanted more he could go to high school and get some
algebra and geometry (not well done, generally, in the period 1850-1950), and
then college. If he wanted to be a
scientist, fine, just as if he wanted to be a violinist. Science was a branch of philosophy in my
childhood, and mathematics even more so.
Its practical value, if any, was mainly for the future to discover. The technological heroes of my childhood,
people like Edison and Ford, didn't much use mathematics, and while there were
some scientists around who did, nobody thought a need for much mathematics
would percolate down to the shop floor and the farm.

But the future came sooner than anyone would have
thought. With World War II the public became for the first time aware that
advances in technology, such as radar, atom bombs, operations research, cryptography
and rocketry, required mathematical knowledge such as the public schools of the
1940s never imagined could be of any practical use.

The military itself was shocked at the mathematical
ignorance of the average draftee, in 1940 when

After 1945, when news about radar, jet airplanes, nuclear
fission and so on became public knowledge, there was increasing public attention
to math in the schools, which was elementary in the extreme, hardly different
from what it had been in 1900 and badly taught besides. Most college graduates of 1950 had never
taken a real course in mathematics in their college career, and in fact not
beyond the 10th grade in school, even in the so-called "college prep"
programs. After all, nobody in 1930
needed much math, and most of the 1950 high school teachers had been trained
even earlier than 1930. You can't teach
what you don't know, after all. Our
school math textbooks in 1950 were designed for grocery clerks, carpenters and
installers of carpeting, in other words for what the educators considered
"practical" and "real life", but a far cry from what the
world outside was in fact demanding; and the teachers who used these primitive
texts, except some at the high school level, knew no more than what was in the
books. Before 1940 there were many
distinguished educational theorists, usually called advocates of
"progressive education", who thought mathematics was __bad__ for
children, and could turn them into unsociable geeks and poor baseball players.

But even at the high school level, for kids who survived
the advice they got and retained their interest, the math wasn't doing the
job. By 1950 the colleges of engineering
were feeling the pinch, and the Dean of the

Beberman created a high school program of a new sort,
one that was real mathematics and prepared students for college mathematics with
logic and practice both. He and his
staff trained Illinois teachers to use his materials by the hundreds during the
summers of the period 1955-1970 (he had financing from a private foundation,
and later from the Federal government), and he traveled in many states
recruiting schools and teachers to try out his materials, and learn to use
them. His work got so famous that it
reached the newspapers and magazines, and was called "The New Math."

Of course Beberman's math was controversial, and many
people, including some mathematicians, considered it too abstract, too full of
logic, and too pedantic in tone, to be useful for the general public. But the general public wasn't what Beberman
had in mind; he was preparing future engineers and scientists. He didn't intend his "theory of
sets" and his careful distinctions between "number" and
"numeral" for the general run, only for those who would one day
really need such fine distinctions. Just
the same, there were those who wanted to use Beberman's methods in earlier
grades, especially the book publishers who wanted to cash in on the sudden
popularity of "The New Math", and so they hired people to produce
what the market seemed to demand. Alas,
the __National Council of Teachers of Mathematics__ was also enchanted with
newmath during the 1960s, and despite the warnings of Beberman himself (among
others), promoted some of these novelties intended for the college-bound future
engineer and physicist. Many very
questionable books were written for smaller children, to be purchased and used
by school officials who didn't know the real from the phony. Parents of small children protested some of
these idiocies being perpetrated on their children -- Tom Lehrer even wrote a
song about them --but many school districts held out for years before the whole
thing ended in disaster.

It is doubtful that things would have gone this way for
very long without the shock of the Russian launching of the satellite Sputnik
in 1957. The public was already aware
that science was important, and certainly willing to improve mathematics
education, at least for future scientists and engineers; but Sputnik created a
public panic, or at least a panic in Congress.
President Eisenhower appointed a Science Advisor and Congress suddenly
started to pour money into the National Science Foundation and the national
Office of Education, demanding instant science and mathematics. Numerous projects something like Beberman's

From 1958 to 1972 the SMSG enlisted hundreds of
mathematicians, school teachers and "mathematics educators" (meaning
professors of education who specialized in math), to write exemplary textbooks,
enrichment materials, teachers' guides and so on, and to try them out in
thousands of schools, using teachers specially trained in federally financed
summer programs called Teachers Institutes.
The SMSG books were not commercial, though one could buy them, but the
hope of the project was that commercial publishers would recruit experts to
imitate them, improve them, put them on the market and in general to improve
the educational system we had rather than have the federal government take over
the schools and their curricula.

As long as SMSG stuck to high school material, as Max
Beberman had, and used high school math teachers, who did after all have
better mathematical understanding than elementary school teachers, who are not
specialists, the new programs had some value, and were measured as in some ways
superior to what went before (though not as much as one might have hoped
for). In addition, the National Science
Foundation financed many summer Institutes for high school teachers, taught by
university mathematicians; and this raised the teaching level in the high
schools. But the public enthusiasm ran into earlier grades too, and every
publisher of textbooks insisted on having a line of "New math" books
guaranteed to make Einsteins out of every kindergarten child. It was impossible to get qualified people
even to *write* reasonable books at this level, let alone the cadre of hundreds
of thousands of teachers to make sense of what they thought they were trying to
do when they imitated SMSG materials.
And the number of elementary school teachers, who are not after all
expected to be mathematics specialists, was totally beyond the reach of even
the most ambitious congressional appropriation for summer Institutes.

The imitations, at the elementary level, of genuine
modern mathematics were awful, and often couldn't be understood by teachers or
parents, let alone the children, however bright, mainly because they picked up
on the most trivial parts of the "new math" and converted them into a
meaningless catechism even worse than the ignorant stuff that had passed for
elementary math in the previous generation.
Those books and programs, with rare exceptions, simply __couldn't be
understood__; and the fraud called "The New Math" finally outraged
the public that by 1975 it was a term of derision, and "Back to
basics" was the new demand from the public.

Besides, by 1972 the missile gap had ended, the Russians
were no longer ten feet high, and the __called__ the new
math, to elementary schools, instead of the basic elements of arithmetic and
geometry, was becoming a national disaster.
He died young, by the way, in 1970, before the disaster he saw in the
making had fully taken place, and ten years before the baby was finally thrown
out with the bathwater.

Yet the word had got around, especially in professional
educationist circles, that it was pointy-headed mathematicians from the universities
who had foisted "the new math" on the country, instead of the
mathematical "basics" really needed by children before they could
progress, some of them anyhow, to more sophisticated things. In truth, that era, from about 1955 to 1972
when SMSG died, was about the only time in the history of math education in

During the postwar years, when the mathematicians were
honored, along with physicists and others, they could lead the way, but behind
it all there was, after all, a much larger mathematical __education__
establishment of teachers, supervisors, principals, commissioners,
school-board members, mayors, governors, Congressmen, NSF bureaucrats --- all
necessary for the functioning of an establishment as enormous as a national
school system. Not to mention schools of
education. For a few years this
establishment was eclipsed by the glamour of the scientists, the builders of
the atom bomb, of radar, of cryptography, of moon landings; but this couldn't
last. Scientists don't teach in the
schools, they don't supervise the school libraries and lunchrooms, they don't
bake cookies for the PTA, they don't drive school busses, they don't teach the
multiplication tables.

Above all, the educational establishment was in a
position to get the rules changed. At
first, it sympathized with the public outcry for "back to basics",
which wasn't hard to do once "the New Math" got a bad name, but
within the educational establishment the progressive educators soon found their
own voices again. They testified before Congressional committees, persuading
them to increase education spending but with side conditions that made sure
the money would be directed by educational experts and not the failed
mathematicians, who were sent home to their research and graduate students. Had
it not been for the trumpeted failure of the scientists and mathematicians, an
anti-intellectual clamor as loud as had been the praise of "newmath"
ten years earlier, the NCTM would never have had a cash-paying audience for its
educational theories such as it got from Congress beginning with its 1980
manifesto __A Call to Action__, and continuing with the famous 1989 __Standards__.

Progressivism had received a bad name many years before,
so the educational progressivists now called themselves
"constructivists", and "constructivism" became the winning
mantra of the new era. The 1989
Standards were not standards at all; they didn't say what should be taught, or
in what grades. The book gave a dreamy
picture of happy, cooperative classrooms with the teacher permitting the
children to discover "their own" mathematics, deeply felt, at their
own rate, this being the "constructivist" (or progressivist) theme
since the time of Rousseau 250 years ago.
One striking corollary of this dream (NCTM itself called it "a
vision") was that numerical computation was mindless and unnecessary so
long as children were happy in their work and calculators were now invented to
make it a useless skill. Any
mathematician could have told them that answers are not the only virtue of
arithmetic study, but nobody asked.
There are other, equally damaging, features of the new, approved
programs, yet who was to tell them no?
The mathematicians had been sent home, and the money from the National
Science Foundation intended for the improvement of schools was instead paid out
to the true, certified educators, in expensive projects to manufacture
textbooks without arithmetic, classes without desks, and teachers without
subject-matter knowledge. "We teach
__children__, not mathematics."
I'm not too sure about the first part of that sentiment, which I have
heard many times, but I am convinced of the truth of the last part.

The NCTM era of constructivist math has held sway since
about 1980, not everywhere of course, even as "The New Math" of the
1960 didn't hold sway everywhere, but the past twenty years has finally
generated a vocal opposition from people who know better. It has taken just about as long this time as
it took last time. Beginning about 1995,
and centered in California at first, the mathematicians -- on their own time,
and with their own money, I should say -- have begun to speak out against what
they finally see is happening in the schools.
The example of

I will leave it to my colleagues here to describe in some
detail the programs that have resulted from this reaction to the "New
Math" of the 1960s, and the brief, unorganized "back to basics"
of the late 1970s. (Actually, "back to basics" never did get very
far, for it was neither progressivism nor a fruitful way to put the
alternative. Thus it had no organized
support, or program of its own, and was soon supplanted by a determined
organization that did have such a program, the NCTM.) The constructivist philosophy of NCTM became
official in 1980, and its products got rolling in 1989, relentlessly pushed by
a propaganda machine, oiled by well-meaning though ill-advised federal money,
such as American education has never seen before. The 1990s have seen the