Whatever Happened To New Math?

Its founding fathers are dead, its disciples scattered, its millions long spent. Yet countless Americans still carry the revolutionary message of new math in their memories, if not always close to their hearts. Now in their mid-thirties or forties, these “new math kids,” myself among them, were part of a learning crusade that in the 1950s and 1960s marched through schools across the nation. For many of us, new math was a disaster; for others, a godsend.

Before the results could even be measured, new math became a near religion, complete with its own high priests and heresies. Chief among the hierophants were the University of Illinois’s Max Beberman and Stanford’s Edward Begle. Together with mathematicians and educators at universities in New York, Indiana, Massachusetts, Minnesota, and Maryland, they took aim at the mindless rigidity of traditional mathematics. They argued that math could be exciting if it showed children the whys of problem solving rather than just the hows. Memorization and rote were wrong. Discovery, deduction, and limited drill were the best routes to arithmetical mastery.

In practice, this meant learning how different number systems worked, that the number 9 in the decimal, or base ten, system would be the number 100 in base three. It meant learning about the set, a grouping of things: a beach as a “set” of grains of sand, for example. It meant learning the difference between a number like 7 and its representation the numeral, which could be expressed many different ways—21 minus 14, 7 times 1, VII. It meant learning to draw rulerlike number lines and divide them into sections to discover fractional multiplication. It meant learning about frames—boxlike symbols used as substitutes for the x, y, z ’s of algebra. It meant learning a new language with terms like open sentence, complementation , and truth set . It meant, in essence, learning to discover the hidden patterns in mathematics before knowing what they were called and reasoning out solutions before knowing rules—all at an earlier age than had ever been attempted before.

Beberman also urged a conceptual overhaul of math education. Mathematics should be taught as a language, he said. And like language, it should be considered a liberal art, a key to clear thinking, and a logic for solving social as well as scientific problems.

No educational proposal, before or since, has won such wide and quick acceptance. PTAs, politicians, and textbook publishers stumbled over one another to endorse the new approach to what was probably the worst-taught subject in American schools. Many high school teachers also were ecstatic, even though new math required that they work harder, perhaps retrain, and drop the drill sergeant’s mask for that of the muse. When the Soviets launched Sputnik, in 1957, the small new-math experiment, previously confined to a few score schools, became a national obsession. Parents went to night school to learn the new approach. The press hailed the reformers as the guiding geniuses of the most important curriculum change since Pythagoras.

By the mid-1960s more than half the nation’s high schools had adopted some form of the new-math curriculum. The figure jumped to an estimated 85 percent of all schools, kindergarten through grade twelve, a decade later. As Robert Davis, founder of the elementary-school program known as the Madison Project and now a professor at Rutgers University, said at the time, “U.S. math literacy can no longer be a matter of God and heredity.”

Now, almost thirty years later, we are hearing the saune lament about our mathematical skills and the same call for educational mobilization. Why? Where did new math go? And what happened to all those schoolchildren ready to experience, as one writer called it, “the wonder of why”?

If they were, like me, high school freshmen in 1964, they may have developed a lifelong aversion to anything associated with new math. Or they may have gone on to become brilliant mathematicians. The differences in experience and outcomes were as varied as the nation’s geography and, most important, only as good as the teachers. But what new math became is not what it was intended to be. In fact, there never was just one new math.

The need for curriculum change was apparent to educators everywhere after World War II. Wartime experience had shown that many high school graduates were too illiterate in math to be trained in radar and navigation. Scientific discoveries and new technology made such illiteracy a threat to America’s future.

Yet everywhere would-be reformers looked, orthodoxy reigned. Population growth and a national shortage of mathematics teachers had forced everyone from coaches to homemaking instructors to the blackboard. Few distinguished themselves. Often inadequately trained, these teachers relied heavily on rules-and-rote textbooks that in many ways had changed little since colonial times. Mathematics was presented not as a human enterprise but as a static subject about which everything was known, including a handy bag of computational tricks—like “carrying” and “borrowing” numbers. (Borrow from where?)

It was not surprising, said the reformers, that students lost interest. By the early 1950s two-thirds of U.S. high school students were ending their math careers after their freshman year. And many of those who went on to “advanced math” were still not prepared for college course work.

The dean of the school of engineering at the University of Illinois, William Everitt, proposed a solution: he would find a young, gifted teacher at the university’s laboratory high school and give him the task of designing a high school math curriculum to help prepare future engineers for college. Had he not found Max Beberman, in 1951, what became the University of Illinois’s Committee on School Mathematics (UICSM) might have remained a prairie dream. But under the guidance of this brilliant, chain-smoking New Yorker, UICSM became a new-math totem.

It is a particularly nice irony that the twenty-five-year-old Beberman was put in charge of this four-member reform committee. A graduate of City College of New York at age nineteen, Beberman had originally been denied a teaching credential because his Bronx accent was so unintelligible. Yet here he was being asked to help shape a new high school curriculum. Beberman loved the role, the limelight, and teaching children. With characteristic sagacity he quickly realized that the dull and lifeless textbooks available would ruin his chances, so with the help of the committee and a logician named Herbert Vaughan, he set out to write his own.

Five years later UICSM had produced loose-leaf texts for all four years of high school and put those texts to experimental use in four locations, including University High on the Urbana-Champaign campus. These early tomes were remarkable for their clarity, their weaving together of algebra and geometry, and their clear, but tacit, criticism of the piece-meal textbooks they were meant to replace. On the opening page of the 1954 edition of the revolutionary High School Mathematics First Course , students were introduced to number lines. By the third page they were negotiating positive and negative numbers and confronting the concept of sets. By the end of Unit 9 they had covered everything from coordinate planes to tangent ratios. A summary of concepts, rules, and principles followed each unit. New math had been born. And as classroom reports from pilot schools flowed in and lessons were revised, the message seemed clear: Children were learning more, staying with math longer, and even saying they enjoyed it.

This was exciting news for the Commission on Mathematics of the College Entrance Examination Board (CEEB), which incorporated many of new math’s guiding tenets in its influential 1959 report “Program for College Preparatory Mathematics.” (The commission’s members included Beberman, Vaughan, and David Page, a UICSM teacher who would soon focus his reform efforts on the elementary school.) CEEB’s goal was both to stimulate change in the high school curriculum and to expound the spirit of reform groups, not only in Illinois but at five other colleges and universities. Missing from the commission’s specific recommendations, however, was any call to modernize teachers.

Beberman was keenly aware of the teacher’s role as gatekeeper for the movement. Beginning in 1958 he used National Science Foundation funds to organize four-week summer teaching institutes on the Illinois campus and elsewhere. Even before Sputnik triggered the funding that broadened the program to nineteen thousand schools, Beberman and Page flew around Illinois, teaching classes and listening to teacher complaints. Such feedback prompted them to add more drill to their courses in 1959. For UICSM, teacher education was as important as its 1,430-page teaching manual.

National fretting over the loss of America’s scientific superiority was unleashing other forces in 1957 and 1958. At Yale the School Mathematics Study Group (SMSG), led by Edward Begle, began gathering teams of schoolteachers and mathematicians for “writing sessions.” Over the next fourteen years, SMSG would produce textbooks for every grade from kindergarten up, including special texts for less able and gifted students. The same federal and foundation grants that fueled SMSG propelled perhaps a dozen other reform programs as well. Together with UICSM, they dissected the traditional math curriculum from all sides. Some, bowing to the recommendations of the Cambridge Conference on School Mathematics, urged a radical acceleration of math curriculum so that calculus could become a high school subject. Others were more in the mainstream. Some aimed at high schools, others the primary grades. Although the public would eventually lump them all together as new math, the work of the ebullient Beberman and the commanding Begle, both of whom appeared in almost every popular periodical of the day, stood apart. Mutual friends and friendly competitors, they became the fathers of new math as well.

Thanks to the diligence of the Dominican nuns in my hometown of Bakersfield, California, I was blissfully unaware of this grand intellectual commotion in junior high school. I was learning math the old way, just as I had learned the multiplication tables, long division, and fractions. Drill, drill, drill.

In some Catholic schools, though, “Illinois math” took root early and flourished. Each summer scores of nuns, most of whom had never had more than a high school geometry course, made a trip to Urbana to “get modern.” In all, several thousand of them went over a thirteen-year period; those who couldn’t go could subscribe to one of twenty training films paid for by the U.S. Office of Education. What did these teachers learn? Robert Kanske, a former summer-institute teacher and now the senior program officer for the National Academy of Sciences’ Mathematical Sciences Education Board, remembers that it was supposed to be the Beberman technique. “But in four or six weeks you couldn’t teach Max’s instructional genius. You spent all your time on the math itself.”

While remedial training was undoubtedly useful for the teachers, it denied them initiation in the greater mysteries of discovery learning and nonverbal awareness—Beberman’s twin pillars of pedagogy. Both stemmed from his faith in the mental agility of children to discover an answer. To illustrate, Beberman believed students could “discover” that the order in which numbers were multiplied did not affect the sum, if they could grasp such equations as:

 

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He believed students could “discover” that multiplying parts of an equation together yielded the same result as adding them separately, if they could perform such computations as:

 

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Then, when the same students encountered a sentence like

(73 x 87) + (27 x 87) = (73 + 27) x 87

they could assert that it was true because they’d see it as a logical consequence of their other discoveries. With this knowledge a student who later took algebra and confronted the algebraic expression 3 a + 5 a would know that it was equivalent to 8 a —and also would know that 3 a + 2 b is not 5 ab .

The goal was for teachers to guide younger students toward the concrete discovery of abstract mathematical principles by deduction. This was unlike traditional methods, which usually had the teacher present a rule—“You can’t add unlike quantities”—and a sample problem, solve the problem, and then drive the solution home with a number of practice exercises. This old approach taught students to consider problems as types that, once recognized, could be solved by applying a formula. If the problem didn’t conform, most students were lost.

Creativity and curiosity died in those old-fashioned workbooks, and Beberman knew it. That’s why he often began classes with a note from a mythical student who thought that 5 plus 7 equaled 57 and that 9 goes into 99 twice. Trying to explain why he was wrong opened the door to understanding the decimal system and its morphological link to our ten fingers. From there one could move to the principle of zero and to the number system used by Martians with seven fingers—and to the binary code of computers.

But as Peter Braunfeld, a mathematician and Beberman associate at the University of Illinois, says, “Max could teach math to anybody. He was a wizard.” To make every teacher into a Beberman was impossible, as even UICSM’s own summer institutes were beginning to show. And as the appeal of new math spread to the elementary grades, the sheer numbers of teachers involved—more than 1.2 million in 1965—made the upgrading a nightmare.

Begle and his SMSG colleagues at Stanford (the program had moved there from Yale in 1961) attacked from a different direction. Their dream was to produce teacher-friendly new-math courses from summer writing sessions. Michigan-born, but with a “New Englander’s conscience,” Begle was careful to mix mathematical theorists and public and private school teachers in these sessions.

The mass appeal of SMSG texts speaks for itself. Beginning with the New Mathematical Library series, in 1959, the number of SMSG texts sold jumped from 23,000 copies to 1.8 million in just three years. Grant money poured in from the National Science Foundation—more than five million dollars bv the mid-1960s.

An important feature of the SMSG effort was the classroom testing of its courses and a recognition that what worked for some students might not work for all. From Chicago to San Jose, from black neighborhoods to cow towns, hundreds of thousands of children became part of a vast national experiment, all driven by the need to yank the country out of its deep and dangerous math rut. I was one of those SMSG children, and I knew I was in trouble from day one.

Enrolled in public school in the ninth grade for the first time since kindergarten, I was willing to accept that this was “Protestant” math. Yet nothing the nuns had taught me—except maybe discipline in the face of the unknown—could have prepared me for my first encounter with sets. This was not math as I had experienced it. The freedom of it all scared me. Where were the rules?

My teacher kept explaining that we were learning why things were so. I just wanted to know the answers. When told that the answers and the rules would reveal themselves, I felt like the Apostle Thomas after the Resurrection—very doubtful. As we moved into number lines, positive and negative numbers, and graphing equations, I panicked. No one in my family could help. I was the youngest and the lone new-math kid. Only the patience of a wonderful tutor helped me survive.

The democratizing of new math ensured that problems like mine would be repeated, for while many parents around the country did take advantage of special training sessions (like a “space-age” closed-circuit television course for parents in Iowa, financed by the Ford Foundation), most had no such opportunity. They felt befuddled by their children’s homework and embarrassed when they couldn’t explain why 1 plus 1 didn’t always equal 2. Max Beberman might have been willing to answer telephone calls at home, but few others were. The new-math revolution that the “pied piper of mathematics” had helped create was, by the early 1960s, no longer small, confined, or in any single person’s control.

Nor were the textbooks. At the beginning of the decade, commercial textbook publishers sensed the change in math education—and the potential profits—and leaped to meet the demand. Even the UICSM textbooks, produced loose-leaf by the University of Illinois Press, went commercial in 1962.

The trick for the publishers was to figure out how radical to be. The SMSG sample texts offered a guide, but companies were free to do everything from calling the old math new to pasting new math on the old, blending the two, or joining the revolution completely. Some companies offered careful and sound efforts. Others churned out a confusing hodge-podge. Before the sixties were over, an estimated six hundred different math textbooks had gone into use around the country, from kindergarten on up.

In their haste to jump on the new-math bandwagon, school districts frequently forgot the expensive lesson that Beberman had learned: Teachers must be nurtured and retrained in new-math techniques. Many teachers balked. They didn’t understand new math or why they were supposed to teach a roundabout way to answers that rules and procedures produced instantly. And for every first-grade teacher who introduced Cuisenaire rods—colored blocks used for the tactile discovery of fractions and division—a frightened traditionalist refused to budge.

Beberman heard their distress and gamely spoke out on their behalf. He knew that if new math was taught badly because teachers were unprepared, and if drills were mistakenly abandoned as unnecessary, children would not learn basic computation. At the 1966 meeting of the National Council of Teachers of Mathematics, he condemned what was happening as an “abortion of the [new math] revolution” and suggested that a major national scandal was in the offing.

Critics, generally ignored until now, began to find their way into the same newspaper and magazine articles that had once been so effusive. Morris Kline, chairman of the mathematics department at New York University, complained the loudest and longest, charging that new math was hopelessly abstract, elitist, confusing, and impractical. (His 1974 book Why Johnny Can’t Add was considered by some to be new math’s coup de grâce .) The satirist Art Buchwald joined the fray with an essay titled “Why Parents Can’t Add.” Tom Lehrer wrote a song about new-math subtraction—a song Beberman good-naturedly previewed to make sure it was mathematically correct—with lines like “The important thing is to understand what you’re doing, not get the right answer.”

While no critic advocated a return to the old days, each of the barbs had just enough truth to wound. For the first time, new-math proponents faced a restless audience. Reassuringly, the CEEB made the new-math standards part of its testing program in 1965, while RCA began preparing an eight-album new-math record set.

I changed high schools that same year and took geometry the old way. Old or new, high school math seemed strenuous, not particularly practical. But I was in the minority. Before long practicality became the measure by which all new math was judged. Applications—balancing a checkbook, paying your taxes—were something parents understood, and teachers ignored them at their peril. As Kline wrote, “Math serves ends and purposes and should be applied to show what it can accomplish.”

Begle countered that the new-math approach was more practical than the endless repetition of word problems because it prepared children for the future, and “since we do not know the answers to tomorrow’s problems, we cannot teach them.” It was an esoteric defense that Beberman could do little to bolster. Hospitalized with a heart ailment that eventually required surgical replacement of his aortic valve, he sat out much of 1966, disillusioned.

More professional battles were to come. As evidence slowly mounted that college-bound students trained properly in the new approach performed at least as well on standardized tests as those taught the old way—and felt more confident than their peers in tackling complicated math problems—there was little time to celebrate. Popular sentiment was beginning to shift, and with it congressional enthusiasm for the financing of educational reform. Good teaching and good textbooks could not be separated from bad. And for every bright student with a thirst for math, there was one who had trouble figuring the charges on his paper route. New math got no credit for the enthusiasm and all the blame for the ignorance, even in those school districts where it was never seriously adopted. When Beberman died suddenly in 1971, at the age of forty-five, federal funding died with him. UICSM shut down soon after.

Begle labored on at Stanford, but by 1972 SMSG had completed its textbook work. Its influence on math curricula remained enormous. But the long effort had taken a physical toll. Racked by emphysema, Begle would fight on futilely for serious research into how to teach mathematics better—an admission perhaps that better math had not produced better teachers. He died in 1978 at the age of sixty-three.

New math did not disappear with either Beberman or Begle. Where it had been successful, it lingered on in the teaching techniques of individual instructors and in watered-down new-math textbooks, which are still evident in elementary and high schools today. Because it put such stock in creativity and the abstract, new math appealed most to the brighter, college-bound students, who, some have argued, probably would have done well anyway. Average and marginal students could suffer dearly at the hands of an uninspired teacher and a poor textbook, and frequently did. But that spirit-crushing combination could exist independently of new math too.

The villain, if there is one, might be the country’s penchant for the “quick fix.” Had Sputnik not flown, UICSM, SMSG, the Madison Project, and the other experimental programs might have evolved slowly and carefully into a national curriculum; as it was, they were shoved to center stage, lavishly financed, and told to perform a miracle overnight. They couldn’t, so the country passed on to the next educational fad (“back to basics”), labeled the previous one a failure, and blamed it for low test scores and a decline in skills.

Did this fall from grace tarnish the new-math legacy? Undoubtedly. New math became a pejorative term. And because it was difficult to know if trying to understand the structure of math made it any easier, most teachers deserted discovery learning without any pangs. Still, few would dispute that there now is a willingness to teach tougher concepts in the primary grades. The reordering of high school math—putting all geometry together in the tenth grade, for example—also seems to be a lasting change. So, too, does the continuing move of calculus from college to a high school senior course.

Did the failure of new math bury the notion of a national curriculum? Probably not. Conservative critics thought they had won the battle against a national curriculum when they helped cut funding for educational reform. But our growing reliance on standardized testing has inadvertently produced what Lynn Steen, a mathematics professor at Minnesota’s St. Olaf College, calls a “national math curriculum that no one has planned.”

Ironically, new math’s most lasting impact might be that of a cautionary tale, as today’s curriculum reformers begin again—this time from the teachers up, not from the universities down. Would Beberman and Begle have applauded today’s attempt to make math less abstract, more meaningful, and more egalitarian by creating a single core curriculum that appeals to all? And how would they have felt about letting children progress at their own rates within this new curriculum?

It’s impossible to know. What is certain is that both men loved mathematics. And knew its power. Both were willing to sacrifice everything to share and transmit that vision to a fickle world. Still, as the nation continues its endless search for solutions, I am haunted—and chastened—by Beberman’s words: “Math is as creative as music, painting or sculpture. The high school freshman will revel in it if we let him play with abstractions. But insisting that he pin numbers down is like asking him to catch a butterfly to explain the sheen on its wings—the magical glint of the sun rubs off on his fingers and the fluttering thing in his hands can never lift into the air again to renew his wonder.”

SOME NEW-MATH CONCEPTS