by Tim Paulson
Published by University Of Chicago Press, 2014 | 224 pages
Discussions between mathematicians and laypeople about the importance of mathematics tend to be frustrating for all parties involved. On one side, you have the mathematician waxing enthusiastic about the subject’s more exotic claims: “Some infinities are bigger than others! Imaginary numbers can have real, un-imaginary applications! The fact that some statements cannot be proven is itself proven!” Etc. On the other side, you have the frequently – and, in fairness, understandably – bewildered, annoyed, and skeptical layperson, to whom the mathematician comes across as demanding admiration, obeisance, and perhaps even government funding for something that can easily seem a quirky if not downright arcane hobby. Frequently, even reliably, the layperson asserts, even boasts with pride, of their own ignorance of the subject. This, unsurprisingly, comes across as anti-intellectual and backwardly pompous to the mathematician. How did it come to pass that mathematics, one of the most revered subjects throughout the history of Western culture – a subject that Bertrand Russell once said “possesses not only truth, but supreme beauty… sublimely pure, and capable of a stern perfection such as only the greatest art can show” – came to be dismissed so cavalierly by such a huge percentage of the population?
Though there is of course no simple answer, the most obvious explanation is that it has something to do with how we teach math in our classrooms. The history of US mathematical pedagogy in the twentieth century touches upon many of the largest political and historical trends of the nation’s broader history: issues in the educational system at large, competing political ideologies and cultural attitudes, even contradictions embedded within mathematics itself. Over the past few years, we’ve seen an attempt to revamp the way math is taught with Common Core, an initiative that has spawned a tangled web of unexpected alliances in its ensuing national dialogue (pro Common Core: President Obama, Jeb Bush, Bill Gates. Anti: National Education Association, Donald Trump, Louis CK). When similar reform projects have been undertaken in the past, they’ve yielded similarly perplexing reactions and results.
Christopher J Phillips’s New Math is a political history of the New Math initiative – one of the major large-scale twentieth century attempts at US Mathematical curriculum reform. Beginning in the late 1950s, and led by the National Science Foundation financed School Mathematics Study Group (SMSG) think tank, the movement grew – as with Common Core – out of a growing consensus that US students were underperforming in mathematics and that something had to be done about it. In the late 50s, particularly in the wake of the USSR’s successful launch of Sputnik in 1958, the concern was that the John Dewey school of thought – with its emphasis on practical, hands-on education – was failing to inculcate our youth with the mental discipline required by the modern, technological world. New Math, in turn, would be a style of teaching focused on conceptual understanding and logical reasoning, as opposed to just brute computation and calculation.
To anybody familiar with the pedagogical challenges mathematics presents, this sounds like a reasonable goal. To start, it should make the subject more engaging. Just as importantly, it would introduce the public to mathematics as the art form those who practice it know it to be. This point is important, though rarely appreciated: math as it’s taught in the elementary school classroom, and consequently the mathematics that most Americans are familiar with, has almost nothing in common with math as it is practiced, and appreciated, by actual mathematicians. A fitting analogy: working through grammatical exercises is to reading a great novel what practicing arithmetic is to contemplating a beautiful mathematical proof. Properly understood, math is perceived to be incredibly profound; it describes the physical world, yes, but its fundamental concepts run deeper that that; they arrive at conclusions about logic, knowledge, the infinite — truly foundational truths about the nature of thought and being itself that transcend the four-dimensional space-time that we inhabit. Mathematics’ claims are often sublimely beautiful, and elsewhere can appear, in their abstraction and counter intuitiveness, strikingly psychedelic. Furthermore, it turns out, math can actually be a lot of fun. Math problems are accurately described as abstract puzzles, the feeling of carrying out a math proof is close to that of playing a game.
The sad reality, however – at least for those of us with time to care about such things – is that few people are aware to any degree that this is the case, believing instead that mathematicians are just people who add, subtract, multiply and divide really big numbers.
The New Math initiative to rectify this, then, seems like a noble undertaking. The actual result, however, often ended up looking far less reasonable. Many parents and math tutors across the country, for example, had the frustrating — or at the very least humbling — experience upon encountering, for the first time, passages like the following in a New Math text:
Let’s attempt to show how 2 x -3 = -6 follows axiomatically from principles we’re already familiar with. Well, remember the additive inverse? Well, that tells us that given the number 3, there has to be some number, let’s just call it -3, such that 3 + (-3) =0. We also know about the multiplicative property of equality right? So we can say that 2 x (3 +( -3)) = 2 x 0. And of course multiplication is distributive, so from that we can easily deduce that (2 x 3) + (2 x (-3)) = 2 x 0. A simple substitution gives us that 6 + 2 x (-3) = 2 x 0, the right-side of which, since zero is obviously the multiplicative identity element, is simply equal to zero. So 6+ 2 x (-3) =0, which is just another way of saying that 2 x (-3) is equal to the (unique!) additive inverse of 6, which we represent with the symbol -6. And so it follows that 2x-3= -6.
One can justifiably question the efficacy of a math pedagogy designed for children that’s inaccessible to adults; where even the most rudimentary mathematical operation like adding two numbers can’t be performed without reference to the base-ten system’s positional notation and the axiomatic properties of Peano arithmetic (“It’s so simple, so very simple, that only a child can do it” sings Tom Lehrer in his song “New Math.”) New Math skeptics inevitably and understandably started forming.
As its subtitle indicates, The New Math: A Political History’s primary concern is less with the design of the New Math curriculum, and more with the political debates surrounding the New Math’s reception. One of the more intriguing aspects of the book is how resistant much of the political discussion surrounding it is to being mapped onto contemporary American political divides. The rhetoric of the New Math advocates, to modern ears, gives off a progressive or even new-age sensibility, with its aversion to rote memorization and its emphasis on deep-rooted understanding. The reality, however, as The New Math makes clear, is that the New Math movement was decidedly conservative and nationalistic in nature. Recall that it was, fundamentally, a backlash against Dewey’s progressive education model. Moreover, it emerged at the peak of the Cold War; Sputnik was in orbit, and the overwhelming consensus (oddly enough) was that the best way for the US to compete geo-politically was to provide our students with the most rigorous mathematical education possible. Advocates argued that “formulas and equations have taken the place of spears and guns,” and used phrases like “the scientific manpower gap,” and the “Cold War of the classrooms” (this phrase, by the way, was evidently so popular at the time, and is used so often in this book, that it feels like a missed opportunity that it wasn’t selected as the book’s title. Just saying.)
It’s interesting to note (and, in fairness, mathematicians of the era did, in fact, point this out) that the type of math at the heart of the New Math movement — abstract, systemic — would not, in fact, have been the first choice of any country looking exclusively to cultivate the next generation of weapons manufacturers. And here is where The New Math gets really interesting; it turns out that the nationalistic, reactionary, anti-Soviet impetus behind the creation of the New Math was not, in fact, the creation of better engineers, but rather the creation of better citizens. Which is, if you think about it, is weird. There was a period in our country’s not very distant past during which math’s ability to inculcate logical and rational thinking was seen as the ultimate panacea for America’s struggles. “Traditional methods of reason,” it was argued, would protect us from “Soviet Style Conformity.” In what may be the most amusing example that Phillips provides, one critic insisted that teaching New Math would cure our youth of both communism and Mccarthyism. The New Math wasn’t about solving math problems; it was about solving all of society’s problems at once.
The contradictions here, in light of today’s political climate, are fun to poke at. One could certainly hope that those in politics proudly branding themselves as anti-intellectual will discover that their idols once averred that the way to further their nationalistic ends was through a system of universal education that emphasized careful, rational thought and conceptual understanding (it’s pretty to think so, anyway). The most interesting paradox that the New Math movement presents, however, may be the idea of imposing a curriculum to battle conformity. It’s a paradox that runs deeper than just Math and education and into the heart of American democracy itself. Phillips includes a beautiful quote from Robert Hutchins that sums this up better than I can:
“The only serious doubt about democracy is whether it is possible to combine the rule of the majority with that independence of character, thought, and conduct which the progress of any society requires.”
In order to properly function, Democracy requires its citizens to view themselves as simultaneously autonomous individuals and as parts of a larger collective. They ought to be rebellious free thinkers but also have an enlightened willingness to conform. Both attitudes are vital, and yet both attitudes are clearly at odds with each other. It’s kind of a problem, really. If you squint your eyes a bit and are the type of person to think such things, this problem may appear to map nicely onto the contradictions found within math itself: rigorous, deductive logic vs. creative exploration and invention. With a bit more mental contortion, you might even be able to convince yourself that teaching such Mathematical reasoning is the best solution.
To be sure, a healthy dose of skepticism is in order here. Yes, we’re used to hearing all sorts of irrationality in public discourse, but the truth is, these fallacies really have little to do with the sort of puzzles mathematicians stew over. It’s likely that even if the New Math had achieved its lofty goals of transforming the teaching of mathematics in America, our society’s basic ills would remain. But still, in this reviewer’s opinion at least, the New Mathers — albeit sometimes for off-kilter reasons — were onto something. We want our kids to be smart; we want our kids to have fun; we want our kids to be cultured, to appreciate the finer things in life. That mathematics is practical and powerful few, if any, would deny. That it is also profound, fascinating, fun, and beautiful is lesser known, but equally true. A pedagogical revolution that could manage to supplant Math’s perfunctory rote memorization and brute calculation with something more exciting, that would spark within students curiosity and the desire to glimpse mathematics’ most profound truths – would, necessarily, be bound to entertain, to astonish, and yes, to educate. QED.
Daniel Goldman is a computer programmer and former Math instructor who occasionally likes to write. He studied Mathematics and Philosophy at Wesleyan University, and currently lives in Brooklyn, New York. He still has trouble grasping the full implications of Gödel’s incompleteness theorems.