What Is “Doing Mathematics” Anyway?
It might seem a little odd to raise this question five chapters into a book on mathematics education. After all, doesn’t everyone know what doing math is? Well, people generally think they know, but when you ask them, as I have on many occasions, you find that many don’t really know, and of those that give a definite answer, those answers often differ from one another and in many cases differ from the answer typically given by professional mathematicians like me. So before I get into more specific details of how video games can be used to enhance mathematics education, we need to make sure we are all on the same page about what we are trying to achieve. How Much Math Is There to Learn?I have already made the case that mathematics education should be primarily geared toward doing, as opposed to knowing-that it’s about acquiring knowing how rather than knowing that. The amount of material to know is extremely small. Take any term-long mathematics course, from elementary school through the end of high school, and you can write on a single postcard all of the key facts covered. In fact, the same is true right up through the end of the freshman year at a university. I know that from first-hand experience. I was a high-school student and then a university mathematics major during the 1960s. In those days, every course culminated in a major performance exam, and those examinations had to be taken without any reference materials, which meant that the examinee had to know by heart all of the definitions, facts, and formulas. A typical exam question back then began by asking the student to write
down a particular definition, formula, or theorem, and then apply it to solve a given problem. By the time I was in the upper grades of high school, I had decided I wanted to become a professional mathematician, so I was motivated to do well in the math exams. One of the techniques I adopted in order to ace the next exam was to write all of the basic definitions and formulas on a postcard and, in the weeks leading up to the exam, study them to the point of memorization on the bus to and from school each day. And you know what, there was never a course for which I required more than one postcard. (All right, I admit I did have to write fairly small on one or two occasions.) Incidentally, by the time I was at university, you could purchase professionally printed “revision cards,” but I always preferred to create my own. In fact, there is a good educational argument for doing so. The act of constructing the cards, involving the choice of what to put in and what to leave out, leads to deeper understanding of the material.But mathematics is not really about facts. It’s a way of thinking. You don’t measure an individual’s mathematical achievement by how much mathematics he or she knows, you measure how much mathematics he or she can do. This means that mathematics, at the school level at least, is much more akin to sports than to classroom subjects such as history, geography, literature studies, or sci-ence, where simply memorizing facts can correlate with success. This is not to condone the fact that such subjects are all too often taught that way. But, as all students quickly discover, the reality is that you can get by pretty well in those subjects simply by rote memorization, even in science, while the same is not true for mathematics.So what kind of thinking constitutes “doing math” as opposed to other modes of thinking, such as doing physics or writing a novel? It’s actually quite hard-perhaps even impossible-to write down a good definition, one that captures all of mathematical thinking yet does not also apply to other disciplines. In my writings elsewhere, I have advocated the definition that mathematics is the “science of patterns.” That’s a good description as far as it goes, but it takes a considerable amount of explanation to make it at all precise. What kinds of patterns? What makes it a science? How is the science done? For the purposes of this book, however, it’s fairly easy to say what “doing math” amounts to, since we are focusing on the mathematics typically taught in elementary and middle schools. As I noted in Chapter 3, that mathematics includes basic number concepts, arithmetic, proportional reasoning, numerical estimation, elementary plane geometry, basic coordinate geometry, elementary school algebra, quantitative reasoning, basic probability and statistical thinking, logical thinking, algorithm use, problem formation (modeling), problem solving, and sound calculator use. Notice that, with the exception of the first item (basic number concepts), everything in the above list is something that we do. That means that as you read this book, you should think of mathematics not as things to know, but as things to do.