NYC HOLD Honest Open Logical Debate on math reform
Are our school’s math programs adequate?
Experimental mathematics programs and their consequences
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
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