FIVE LESSONS FROM THE NEW MATH ERA
Any opinions, findings, conclusions, or recommendations expressed in this paper are those of the author and do not necessarily reflect the views of the Center or the National Research Council. The paper has not been reviewed by the National Research Council.
October 4, 1957 was a Friday. On the evening news, we learned that the Soviet Union had put a satellite into earth orbit, thus beating the United States and scoring a huge propaganda victory. All weekend, people talked about what was happening and what it might mean. Fingers were pointed and excuses made. There hadn't really been a race. Everyone knew that the U.S. was about to send up its own satellite, but we were waiting to do it right. If we hadn't separated our scientific effort to build a satellite from our military effort to develop new weapons, we could have been first. Anyway, the Soviets must be lying; they didn't have the capability to put such a heavy object into orbit. But no one could avoid the conclusion that Sputnik's ascent had brought with it profound political, military, and scientific implications for Americans.
When Sputnik went up, I was a first-year teacher. I was completing my fourth week of teaching seventh-grade science and ninth-grade algebra at a junior high school in Berkeley, California. In my algebra class, I was giving a test covering the chapter on equations and their uses in Virgil Mallory's 1950 First Algebra. Equations were said to be statements about the equality of two expressions and apparently required the presence of what was called a literal number, which was also called an unknown when it was used by itself in an equation. Near the end of the course, furthermore, literal numbers would be termed variables, as distinct from constants, because their values changed. This confusion of language and ideas was widely accepted as a natural part of introductory algebra. The new math reforms would attempt, among other things, to change all that.
Most of the new math reformers believed that the school mathematics program had become so entangled in senseless jargon and was so out of step with mathematics as then taught in the university that it needed a complete overhaul. The language of sets, relations, and functions would provide not only a more coherent discourse in the mathematics classroom but also a more meaningful structure for learning. Students would be drawn to mathematics by seeing how it fit together and, in particular, how the great ideas of modern mathematics brought order into the chaotic curriculum of literal numbers that were not really numbers at all, a notation for angles that did not distinguish between an angle and its measure, and operations on numbers that included turning them over, bringing them down, canceling them, and moving their decimal points around
Having just completed a teacher education program at the University of California at Berkeley, I was vaguely aware of some of the changes already underway in secondary mathematics. Max Beberman was leading a secondary mathematics curriculum project at the University of Illinois, and the Commission on Mathematics of the College Entrance Examination Board was developing a statement on how college preparatory mathematics needed to be reformed. But I was not prepared for the flood of activity that was to come.
At Stanford University the following summer, I attended an institute for mathematics teachers sponsored by the General Electric Corporation that was a forerunner of the summer and academic year institutes for mathematics and science teachers that were about to be funded at a much higher level by the National Science Foundation. At the institute, I had a class from Morris Kline, who was to become the single most prominent critic of the new math reforms and who was already collecting applications of mathematics that might be used in the secondary curriculum to counter the direction the new math was taking. Several years afterward, I went back to Stanford full time and participated in an NSF-sponsored academic year institute featuring courses by George Polya, also a critic of the new math reforms. I would later assist Polya in subsequent institutes and courses. When Ed Begle and the School Mathematics Study Group (SMSG)the largest and best known of the new math projects in the U.S.moved from Yale to Stanford in 1961, I had the opportunity to see some of the reform effort up close. Begle became my major professor for doctoral studies, and Polya was on my dissertation committee. Consequently, I was able to work with two eminent scholars who, while respecting each other's views, could not have had more divergent ideas as to what changes were needed in school mathematics.
In the current climate of cynicism about education, it may be impossible for people who did not live through the heady times of the new math to appreciate how confident the reformers were that fundamental changes could be made in U.S. mathematics education, given the will and the resources. There was no shortage of enthusiasm, ideas, or money. Many educators felt that at last school mathematics was headed for a genuine reconstruction and revitalization.
Today, my students look back at the new math era as ancient history and cannot imagine what the fuss was about. It is impossible in this brief paper to draw up a balance sheet on what something as multidimensional and diffuse as the new math did or did not accomplish, but there are some implications to be drawn from that time. In the comments below, I offer five lessons from the new math era that may be useful for anyone looking toward future reform efforts. These lessons are not all that I learned from my experiences during the era, but they capture much of what has seemed important to what has seemed important to remember in the years since then.
Copyright 1997 by the National Academy of Sciences. All rights reserved.