Teachers of mathematics and parents of children who have a difficult time learning mathematics have long sought ways to teach and encourage those who don’t appear to “get” math. The struggle is not only against the perceived difficulty of the subject matter but also against societal assumptions about what it means to be good at math. In particular, students who ask questions about why math is the way it is, rather than seeing math as a collection of known facts to be memorized, are told implicitly or explicitly that they are not “math people.” Many of them internalize this message, stop asking questions, and learn as little math as possible, avoiding careers that require the use of mathematics and sometimes passing erroneous (and harmful) messages about mathematics on to their children.
In Is Math Real? How Simple Questions Lead Us to Mathematics’ Deepest Truths, mathematician Eugenia Cheng argues that those labeled as “non-math people” because of the questions they ask are, in fact, more like research mathematicians than are those who accept concept after concept as “obvious.” Chapters in the book begin with the kinds of questions sometimes asked by those who are too-often scoffed at by the “math people” in their lives: Why does 1 + 1 = 2? Why does -(-1) = 1? Why isn’t 1 a prime number? Using such questions as starting points for an exploration into what math is, how it works, why we do it, and why it’s good, Cheng sets out to demonstrate that questions are foundational to math, that they can lead to deep meaning, and that being good at math doesn’t mean seeing it as a collection of obvious facts. Rather, asserts Cheng, “It’s all about the processes, not the answers” (p. 110) and “math doesn’t have clear right answers, but rather, different contexts in which different things can be true” (p. 14), views shared by mathematicians everywhere.
Along the way, Cheng does an excellent job of laying out how mathematicians think about and do math while at the same time explaining specific “basic” math concepts clearly and in a way that shouldn’t make anyone feel stupid. She uses analogies frequently and effectively: doing math, for example, can be like jumping in puddles, walking on fresh snow, or climbing a mountain (p. 107). Recognizing that “[our] intuition [is] flawed and allowing it to change direction … is a bit like when our unconscious biases about people are shown to be false … we can either cling to our preconceived ideas, or celebrate our capacity for honing our thinking” (p. 132). Tossing i (the square root of -1) into our set of numbers is “as if Lego came up with a new type of special brick and the first thing we want to do is explore all the things we can now do if we add an unlimited supply of this new brick to our old stash of Lego” (p. 156).
Some reviewers have complained about a “political” tone to the book. While it’s true that Cheng uses for analogies or illumination some topics currently viewed as political, these are not the focus of the book, and it should be easy enough for readers to skim past these illustrations if they so desire. Or, preferably and more in line with doing mathematics, read them, explore them, and decide whether or not they’re logically supported or simply opinions of the author. Usually Cheng is clear about her own assumptions. For example: “It’s not logical to think that vaccines give no protection against illness, but it is valid to decide you fear the side effects of the vaccine more than you fear the illness. It is a different response to risk from mine, but at least then we can discuss that difference” (p. 132). It does seem to me that the section in which Cheng contrasts modern math-based-on-development with ethnomathematics would fit better in a different article or book, but Cheng does provide a viewpoint I have not spent much time thinking about and inspires me to think in more depth about my own assumptions about the nature of mathematics and the extent to which those assumptions have been culturally determined.
I spent a fair amount of time, while reading the book, questioning its title. Cheng was not, it seemed, probing a question about the reality of math as much as she was taking the existence of math as an axiom—an assumed truth on which to build knowledge. The deeper question, as she notes in the introduction, is about the existence of a kind of math where you can ask questions, where those questions can lead to deep meaning, where being “good at math” doesn’t mean greeting everything you’re taught with “well that’s obvious,” where there’s not necessarily a direct application to daily life. Is that kind of math real? By the end of the book, it occurred to me that “Is math real?” opens the book just as “Why does 1 + 1 = 2?” opens the first chapter. We’re not actually looking for an answer to that question. Instead, we’re looking for the contexts in which math is real, in which its patterns make sense, in which the way we define objects is important, and in which being able to move between different points of view is essential. Cheng demonstrates that there are such contexts and that they are open to all who ask questions about math.