There are three ways to teach math to the young. The old way forced rote memorization of basics and then, for most, stopped the lessons, continuing them only for those who had the inclination or ability to advance. The “new” way was to “expose” every student from the beginning, no matter their age or inexperience, to the highest, most difficult mathematical concepts, so that all might know how wondrous and astonishing math is.

The modern way, which may soon be upon us, is to let students define what math is to them or their culture, to let them discuss their feelings about what math means, and to work toward the goal of equality, that happy state when all are satisfied with their level of (self-defined) mathematical understanding. Two new books—*The New Math: A Political History*, by Christopher J. Phillips, and *Critical Mathematics Education: Theory, Praxis, and Reality*, edited by Paul Ernest, Bharath Sriraman, and Nuala Ernest—bring these distinctions to the fore.

People were long happy with the old way and for the happiest of reasons. It worked. Nearly every child eligible to attend school could be made to learn, or at least to memorize, that 8 x 7 = 56 and that triangles encompassed half as many degrees as circles and what simple consequences flowed from these facts. Not every child could advance beyond these basics, but few thought that all should.

That attitude began to change mid-twentieth century, a time in which greater proportions of children were enrolling in all levels of schooling. Because of the Cold War and the impression that America was falling behind, the concern was that kids weren’t learning enough and that they needed to be better thinkers. “New Math” was the result.

In *The New Math: A Political History*, Carnegie Mellon University assistant professor of history Christopher J. Phillips tells of its rise and fall, centering the tale on the School Mathematics Study Group (SMSG), an entity created in earnest by government money during the Sputnik era: “Although originally funded to work on textbooks for the ‘college capable’ students in secondary schools, SMSG gradually expanded its operation, producing textbooks for every grade and type of student, including material for elementary schools, ‘culturally disadvantaged’ children,” etc. The ascension of New Math was thus partly due to routine mission creep found in well-funded bureaucracies.

At its onset, parents were more or less happy with the status quo. But education theorists and others were not. Students taught in the old way could cipher to the rule of three, but they didn’t know the why behind the how: “[O]ne generally accepted axiom was that math textbooks’ and teachers’ traditional reliance on memorization and regurgitation gave students a misleading sense of what mathematicians *do* and what mathematics was *about*.”

Yet is it really of interest what professional mathematicians do? Filling out grant requests, for instance? At any rate, what mathematics is *about* is something argued over by mathematicians themselves. This was true during the time of New Math, as was pointed out by Morris Kline in his much-discussed *Mathematics: The Loss of Certainty* (1982), and is true now as shown by such authors as James Franklin in (the highly readable and recommended) *An Aristotelian Realist Philosophy of Mathematics: Mathematics as the Science of Quantity and Structure* (2014). Mathematicians are not unanimous about whether mathematics is nominalist or realist at base.

Educationists said the New Math would promote “mental discipline.” Rote was disparaged. A group of French mathematicians calling themselves “Nicolas Bourbaki” charged ahead and achieved a sort of intellectual celebrity. They shouted slogans like, “Euclid must go!” Why? Not because there was anything wrong with Euclid, but because Euclid had little relevance to what *current* mathematicians were doing. But what current mathematicians were doing could not be taught to any but the most advanced graduate students.

At the time of New Math, mathematicians were still coming to terms with the limitations of proof as discovered by Kurt Gödel. The roots of mathematics contained fundamental, unresolvable uncertainties; and, anyway, all research mathematics was by definition contemplating unsolved problems. So it was thought important to show these uncertainties and unsolved problems to students as early as junior high. Those children were, as you would expect, bewildered, especially since their grasp of the basics had been weakened by design.

“The authors [of new textbooks],” Phillips writes, “intended to deemphasize the most familiar representation of *doing* mathematics—the written forms of elementary arithmetic. The essence of mathematics was reasoning, not ciphering.” Little wonder that students put through the New Math tradition would not learn to cipher well. “At no point did SMSG’s authors promise that their textbooks would enable students to calculate more accurately or quickly.” And they didn’t.

There were, of course, critics who dismissed New Math efforts as “ephemeral research fads.” In the rush to progress, these voices were ignored and the textbooks were “fixed.” But it must be remembered, as Phillips explains, that “the reform of textbooks at midcentury was not primarily about accuracy, and there was nothing factually wrong with the previous textbooks.”

SMSG theorists wanted to “‘liberate’ students from the ‘shackles of authority,’” replacing old dogma with their more free-form but Byzantine rules. “The effect of SMSG’s textbooks on actual students was different from that imagined by authors or critics, however,” Phillips explains. “Students were obstinate; books were soiled,” and teachers “sometimes only had a shaky hold on the material.”

Not surprisingly, the “theory” behind many of the changes originated with John Dewey, who, as you might suspect, disliked memorization and ciphering. There was also plenty of psychologizing, drawing from Jean Piaget and others, and the democratic impulse to Equality. “By positing that all elementary students should learn the same version of mathematics,” writes Phillips, “SMSG’s plans for mathematical training ultimately implied that higher-level, structural thought should not be limited to the more ‘capable’ students.”

Parents, not surprisingly, were seldom consulted, but then parents had long grown used to giving their children’s education over to government control. But many parents balked when they couldn’t make sense of their children’s assignments. No less a personage than Richard Feynman, who was “burdened” in his youth by rote and ciphering yet somehow still blossomed into an extraordinary physicist and mathematician, complained that the New Math textbooks “included new mathematical terms and precisely defined concepts without consideration of the goals or needs of the schools.”

To these complaints was added *Why Johnny Can’t Add* (1973), Morris Kline’s argument that student deficiencies, which had become glaringly obvious, were *caused* by the New Math. Public mood had also shifted: “By the 1970s,” Phillips writes, “it was not clear that academic knowledge might actually be beneficial for the rest of society.” We hadn’t yet reached the point of understanding that academic “knowledge” might be downright harmful.

Then came the night James M. Shackelford, a chemist and therefore a man familiar with math, opened “his daughter’s elementary math textbook only to find problems that he himself could not solve.” He asked his daughter and her friends “to calculate 8 x 9 and was shocked to find that none of these fourth graders could do so.” Incensed, Shackelford started the influential Back to Basics movement. New Math withered due to onslaughts on multiple fronts.

New Math theorists resisted the dying of their light and carped that drilling fostered “structured, often militaristic classrooms.” Yet all history demonstrates that a drilled military routinely crushes untrained irregulars. Having students memorize facts was “brainwashing” them with “authoritarian” thinking, theorists claimed. Does this “brainwashing” hold for learning to play the piano? If you do not drill or submit to authority in music, you will stink and produce sounds nobody wants to hear.

Phillips says, “Opponents of the new math won.” They did, but then came the soldiers of the Common Core, who discovered new ways to make learning math about reasoning and not calculation. For example, in October 2015 a widely shared Imgur photo showed a teacher had marked wrong a third grader’s answer of 15 to the problem 5 x 3, because the student had demonstrated it as 5 + 5 + 5, not 3 + 3 + 3 + 3 + 3, as the “correct” understanding demanded.1

Phillips cites a 2013 *Atlantic* article praising the Common Core’s mathematical standards by Jo Boaler, a professor at Stanford’s Graduate School of Education, who argued that “employers do not need people who can calculate correctly or fast, they need people who can reason about approaches, estimate and verify results.”2 Boaler, who is also CEO and cofounder of YouCubed, which provides math education resources for students, parents, and teachers, apparently didn’t ask herself how these interesting people will know if their estimates are correct if they can’t calculate.

New Math is dead and Common Core lingers, its fate uncertain, but the view among educationists that math is oppressive and unbending has only grown stronger, which is the theme of the essays in *Critical Mathematics Education: Theory, Praxis, and Reality*, edited by University of Exeter emeritus professor of philosophy of mathematics education Paul Ernest, University of Montana professor of mathematics Bharath Sriraman, and Nuala Ernest.

Contributors Keiko Yasukawa, lecturer at the University of Tokyo School of Education; Ole Ravn, associate professor of electrical engineering at the Technical University of Denmark; and Ole Skovsmose, emeritus professor of learning and philosophy at Aalborg University and well-known in the field of academic mathematics education, are disturbed at how inflexible math is when used to calculate their departments’ budgets. Math plays in “[in]equitable distribution of resources,” “the bottom line,” and not having enough funds to go to fun conferences. This leads them to wonder if mathematics itself is *bad* or *good*—and since other departments have more money, they conclude that it is more bad than good.

Whatever you do, don’t get Skovsmose started on airline ticketing pricing. He devotes an entire essay to why he is displeased with how airlines use math. The poor fellow must have gotten bumped from a flight at some point. My evidence for this is that he constructs a vivid, supposedly fictional math scenario in which there exists a “tape-recording [of] the passenger, who states she has a valid ticket for this particular departure, and [of] the airline assistant who informs her that the airplane, most unfortunately, has been overbooked.” It goes on like that at some length.

Airline executives who sell what they don’t have ought to be tarred and feathered in public, but it’s not clear whether they shouldn’t be taught to make their sums come out right. Skovsmose thinks that the “‘bumping situation’…serves as illustration of the vast range of situations where a mathematical undercurrent provides a structuration.” It’s not explained what “structurations” are, but they sound like something to be avoided.

With the reader softened by examples of math being used for suspicious purposes, we come to Ubiratan D’Ambrosio, emeritus professor of mathematics at the State University of Campinas in São Paulo, who thinks math education can bring about peace—inner peace, social peace, environmental peace, and military peace. Why? Because “[m]athematics is deeply involved,” not just in academic department budgets and airline ticket pricing, but in “relations among nations,” “economics,” and even “government/politics.” If more people knew more math, D’Ambrosio claims, we could “build a civilization that rejects inequity, arrogance, and bigotry.” This will be accomplished not by classical mathematics education, by teaching addition and multiplication tables, for example, but by focusing on “ethnomathematics,” a system whereby we allow each person and culture to define what math is.

Karen François, professor at the Centre for Logic and Philosophy of Science at Vrije Universiteit Brussel, goes as far as saying that ethnomathematics thus defined is a human “right.” Nothing else can bring about Equality. She says the old ways of doing math created a “meritocratic position” which “cannot fully guarantee equal chances.” Worse, the old ways positively created inequality in outcomes. What François wants is a way to teach mathematics that mandates equal results. Ethnomathematics can do this.

She’s right, of course. If young George insists that 2 + 2 = 5, who’s to say George is “wrong”? In George’s self-defined mathematics, today 2 + 2 *is* 5. Tomorrow it may be something else. Who are we to insist on what the plus sign means anyway? Although François doesn’t mention it, ethnomathematics will make grading tests easy.

The rest of the essays in *Critical Mathematics Education* are devoted to putting “Eurocentric” mathematics in its place, shaming it for the ill and evil uses to which it has been put. *Out* are the “criteria of validity, reliability, and objectivity as they are understood and applied in scientific paradigm research,” and *in* is intense awareness of “social class,” diversity, equality, and all the other standard desires of modern academia—chiefly more money for academics.

Children might not be able to count under the system of ethnomathematics, but counting only counts if the student wants it to—and at least parents will be able to help with homework.