{∅, {∅}, {∅, {∅}}} : Rage Against the Meat Grinder

As has been the case in the past, I find myself having to look for inspiration from what I have written in the past.  It is as though I am reading someone else.

Now I am trying to see if I might gently guide myself back to more fulfilling rituals.

It is difficult to commit to exhaustive studies and rigorous proofs when it all seems to have been done in vain.   So, rather than seeing myself as guiding Holden, I am going to allow myself to see if a previous version of my "self" might guide this current bundle of nerves back into math mode ...


From Mathematical learning (and math as a hobby)

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 are.

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 math. 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.