August 22, 2011
Mathematical learning (and math as a hobby)
by Rishidev Chaudhuri
It is an oddly well-kept secret that mathematical learning is a very active process, and almost always involves a struggle with ideas. To a large extent, this is due to the nature of mathematical intuition: grasping a mathematical idea involves seeing it from multiple angles, understanding why it's true in a broader context and understanding its connections with neighboring ideas. And so, when you sit down to read through a proof or the description of an idea, you rarely do just that. Instead, digestion more often involves settling down with a pen and a piece of paper and interrogating the concept in front of you: “What is this statement saying? Can I translate it into something else? Can I find a simpler case that will help me gain insight into this general context? What about this makes it true? What would be the consequences if this statement were false? What contradictions would I encounter if I tried to disprove it? How does this concept reflect those that have gone before? How do the various assumptions used to prove this statement factor in? Are all of them necessary? Are there other ways to frame this fact that seem fundamentally different?” And so on. And this interrogation often involves taking your pencil and paper on long digressions, slow rambling explorations of ideas that help clarify the one you're trying to understand.
Similarly, proving a mathematical statement or solving a problem is an unfolding of false sallies and blind alleys, of ideas that seem to work but fail in very particular ways, of realizing that you don't understand a problem or a concept as well as you thought. And again, these are not wasted. In almost every case, if someone were to just give you a proof or a solution and you didn't either try to come up with it first or actively interrogate it once you had it (which is almost the same thing), you'd learn that the statement was true, but learn very little about why it was true or what it meant for that statement to be true. And much of the learning in a math class happens not in the lectures but afterwards, in the time spent on problem sets (and, if you had a choice between attending the lectures and doing the problem sets, you should always pick the latter).
Unfortunately, most people make it through a high school mathematical education without being taught this. This has unfortunate consequences and makes mathematical learning exceedingly vulnerable to expectation and self-belief, so that it is often seen as something you either can or can't do, and many people see the struggle as a sign of a lack of ability rather than as an intrinsic part of the learning. There are certainly children who, for whatever accident of genetics, upbringing or attentional prowess start out by being quicker at math. But this seems swamped by differences in temperament and confidence, or by the effect that initial quickness has on confidence. How you engage with the setbacks of learning seems more important than how quick you are1.
This was strongly brought home to me when running math classes. There would inevitably be two groups of people who could take the same amount of time to solve or almost solve a problem but be quite differently convinced about their mathematical ability (which, over a semester, ends up being self-fulfilling). Some students, ten minutes into wrestling with a problem, would find progress difficult and take this as a sign that they were learning what didn't work, were spending time understanding the problem, were edging towards a solution, were exercising their reasoning ability and so on. Others would start off anxious and ten minutes in, at about the same stage of reasoning, would come to me convinced that they were never going to figure it out, and that they were dumb or not good at math2. And yet the two groups didn't seem to have wildly differing levels of intuition and for the second group reassurance that they were participating in the right process or helping them follow the path they were on, even if it was headed in the initially wrong direction, would often lead them to the same solution. Strangely, while some of the job of a math teacher seems to be to help with mathematical intuition, a large part of the job seems to be palliative, compensating for something that they should have been told or learned but hadn't: be patient with yourself.
One of the inevitable tragedies of specialization is that most people don't take classes in most areas after college or high school. For some this is compensated by an amateur interest in history, say, or philosophy. But for the variety of reasons I mentioned, the reasons that make students think that mathematics proficiency is an extreme example of a natural talent and that it is hopeless to do math without this essential ability, few people seem to maintain an amateur interest in mathematics or study mathematics recreationally.
If it isn't clear already, I think this is a huge pity, especially because it is often motivated by a false assessment of one's mathematical ability. And it is also a pity because most people stop doing math just at the point when the fun stuff starts, just when they've worked through most of the tedious arithmetic and are finally ready to embark on sweeping journeys of abstraction. It's like taking dance classes but never going dancing.
And, as almost anyone with a sustained interest in mathematics will tell you, math contains some of the loveliest conceptual and aesthetic pleasures available to us. It is the locus of some of the grandest and most elegant ideas we know, the site of struggles to explore the nature of infinity, to abstractly describe form and space and to reason about the nature of logic. There is a profound sculptural beauty to the edifices of the great mathematical theories. Mathematical thought is as much a part of our intellectual and aesthetic heritage as the arts and philosophy. It would be a shame to miss this grandeur if you didn't have to.
Mathematics is also a very pure example of the pleasures of intellectual play. Large branches of math emerge from someone writing down a few rules and seeing what they can construct within those rules, asking what manner of objects a set of rules gives rise to, what conceptual universe they call into being, and how the objects interact within that universe. It feels like frolicking in some fantastical Borgesian garden.
There are many other pleasures, of course. There is the pleasure of learning the mathematical language, with its conceptual precision and logical power, and the pleasure of translation, as you begin to see what is general and abstract in patterns in order to mathematize them. And there are the smaller material pleasures of doing mathematics, like the satisfying tactility of scribbling over sheets and sheets of paper as you explore an idea. And mathematical notation is charming, with the Greek letters and the multiple squiggles and the host of symbols, each with its own history. It is the same charm I imagine for alchemy or highly symbolic esoteric teachings.
Learning math and working through theorems or problems takes time, but so what? There is no hurry, and you'll be exploring some of the deepest ideas we have. And the pleasure of investigating an idea, exploring it from every angle and then the thrilling leap (or slow clamber) of finally having it reach your intuition is unparalleled.
So where to start? Conveniently, many mathematics textbooks have no prerequisites (at least in the formal sense, in that they define everything they need but a lack of experience might make the logical steps harder to follow). For recreational study, with an eye towards aesthetics, either real analysis (which studies the intricate structure of the number line) or abstract algebra (which attempts to abstract out and study the structure of relations like addition or multiplication) are good places to begin. The Wikipedia articles on both have links at the end to online textbooks, and enough universities put coursework up online that it's pretty easy to track down resources for study.
1This is all anecdotal, of course.
2Unsurprisingly, these groups tended to be somewhat gendered
Posted by Rishidev Chaudhuri at 01:00 AM | Permalink