Understanding maths

Note: this post is a translation of an older post written in French: Compréhension mathématique. I wrote the original one when I was still in university, but it still rings true today – in many ways 🙂

After two heavyweight posts, Introduction to algorithmic complexity 1/2 and Introduction to algorithmic complexity 2/2, here’s a lighter and probably more “meta” post. Also probably more nonsense – it’s possible that, at the end of the article, you’ll either be convinced that I’m demanding a lot of myself, or that I’m completely ridiculous 🙂

I’m quite fascinated by the working of the human brain. Not by how it works – that, I don’t know much about – but by the fact that it works at all. The whole concept of being able to read and write, for instance, still amazes me. And I do spend a fair amount of time thinking about how I think, how to improve how I think, or how to optimize what I want to learn so that it matches my way of thinking. And in all of that thinking, I redefined for myself what I mean by “comprehension”.

My previous definition of comprehension

It often happens that I moan about the fact that I don’t understand things as fast as I used to; I’m wondering how much of that is the fact that I’m demanding more of myself. There was a time where my definition “understanding” was “what you’re saying looks logical, I can see the logical steps of what you’re doing at the blackboard, and I see roughly what you’re doing”. I also have some painful memories of episodes such as the following:

“Here, you should read this article.
− OK.
<a few hours later>
− There, done!
− Already?
− Well, yeah… I’m a fast reader…
− And you understood everything?
− Well… yeah…
− Including why <obscure but probably profound point of the article>?
<blank look, sigh and explanations> (not by me, the explanations).

I was utterly convinced to have understood, before it was proven to me that I missed a fair amount of things. Since then, I learnt a few things.

What I learnt about comprehension

The first thing I learnt, is that “vaguely understand” is not “comprehend”, or at least not at my (current) level of personal requirements. “Vaguely understanding” is the first step. It can also be the last step, if it’s on a topic for which I can/want to do with superficial knowledge. I probably gained a bit of modesty, and I probably say way more often that I only have a vague idea about some topics.

The second thing is that comprehension does take time. Today, I believe I need three to four reads of a research paper (on a topic I know) to have a “decent” comprehension of it. Below that, I have “a vague idea of what the paper means”.

The third thing, although it’s something I heard a lot at home, is that “repeating is at the core of good understanding”. It helps a lot to have at least been exposed to a notion before trying to really grasp it. The first exposure is a large hit in the face, the second one is slightly mellower, and at the third one you start to know where you are.

The fourth thing is that my brain seems to like it when I write stuff down. Let me sing an ode to blackboard and chalk. New technology gave us a lot of very fancy stuff, video-projectors, interactive whiteboards, and I’m even going to throw whiteboards and Vellada markers with it. I may seem reactionary, but I like nothing better than blackboard and chalk. (All right, the blackboard can be green.) It takes more time to write a proof on the blackboard than to run a Powerpoint with the proof on it (or Beamer slides, I’m not discriminating on my rejection of technology 😉 ) . So yeah, the class is longer, probably. But it gives time to follow. And it also gives time to take notes. Many of my classmates tell me they prefer to “listen than take notes” (especially since, for blackboard classes, there is usually an excellent set of typeset lecture notes). But writing helps me staying focused, and in the end to listen better. I also scribble a few more detailed points for things that may not be obvious when re-reading. Sometimes I leave jokes to future me – the other day, I found a “It’s a circle, ok?” next to a potato-shaped figure, it made me laugh a lot. Oh and, as for the fact that I also hate whiteboards: first, Velleda markers never work. Sometimes, there’s also a permanent marker hiding in the marker cup (and overwriting with an erasable marker to eventually erase it is a HUGE PAIN). And erasable marker is faster to erase than chalk. I learnt to enjoy the break that comes with “erasing the blackboard” – the usual method in the last classes I attended was to work in two passes, one with a wet thingy, and one with a scraper. I was very impressed the first time I saw that 😉 (yeah, I’m very impressionable) and, since then, I enjoy the minute or two that it takes to re-read what just happened. I like it. So, all in all: blackboard and chalk for the win.

How I apply those observations

With all of that, I learnt how to avoid the aforementioned cringy situations, and I got better at reading scientific papers. And takes more time than a couple of hours 😉

Generally, I first read it very fast to have an idea of the structure of the paper, what it says, how the main proof seems to work, and I try to see if there’s stuff that is going to annoy me. I have some ideas about what makes my life easier or not in a paper, and when it gets in the “not” territory, I grumble a bit, even though I suppose that these structures may not be the same for everyone. (There are also papers that are just a pain to read, to be honest). That “very fast” read is around half an hour to an hour for a ~10-page article.

The second read is “annotating”. I read in a little more detail, and I put questions everywhere. The questions are generally “why?” or “how?” on language structures such that “it follows that”, “obviously”, or “by applying What’s-his-name theorem”. It’s also pretty fast, because there is a lot of linguistic signals, and I’m still not trying to comprehend all the details, but to identify the ones that will probably require me to spend some time to comprehend them. I also take note of the points that “bother” me, that is to say the ones where I don’t feel comfortable. It’s a bit hard to explain, because it’s really a “gut feeling” that goes “mmmh, there, something’s not quite right. I don’t know what, but something’s not quite right”. And it’s literally a gut feeling! It may seem weird to link “comprehension” to “feelings”, but, as far as I’m concerned, I learnt, maybe paradoxically, to trust my feelings to evaluate my comprehension – or lack thereof.

The third read is the longer – that’s where I try to answer all the questions of the second read and to re-do the computations. And to convince myself that yeah, that IS a typo there, and not a mistake in my computation or reasoning. The fourth read and following are refinements of the third read for the questions that I couldn’t answer during the third one (but for which, maybe, things got clearer in the meantime).

I estimate that I have a decent understanding of a paper when I answered the very vast majority of the questions from the second read. (And I usually try to find someone more clever than me for the questions that are still open). Even there… I do know it’s not perfect.

The ultimate test is to prepare a presentation about the paper. Do as I say and not as I do – I typically do that by preparing projector slides. Because as a student/learner, I do prefer a blackboard class, but I also know that it’s a lot of work, and that doing a (good) blackboard presentation is very hard (and I’m not there yet). Once I have slides (which, usually, allow me to still find a few points that are not quite grasped yet), I try to present. And now we’re back to the “gut feeling”. If I stammer, if there are slides that make no sense, if the presentation is not smooth: there’s probably still stuff that requires some time.

When, finally, everything seems to be good, the feeling is a mix between relief and victory. I don’t know exactly what the comparison would be. Maybe the people who make domino shows. You spend an enormous amount of time placing your dominos next to one another, and I think that at the moment where the last one falls without the chain having been interrupted… that must be that kind of feeling.

Of course, I can’t do that with everything I read, it would take too much time. I don’t know if there’s a way to speed up the process, but I don’t think it’s possible, at least for me, in any significant way. I also need to let things “simmer”. And there’s a fair amount of strong hypotheses on the impact of sleep on learning and memory; I don’t know how much of that can be applied to my “math comprehension”, but I wouldn’t be surprised if the brain would take the opportunity of sleep to tidy and make the right connections.

Consequently, it’s sometimes quite frustrating to let things at a “partial comprehension” stage – whether it’s temporary or because of giving up – especially when you don’t know exactly what’s wrong. The “gut feeling” is there (and not only on the day before the exam 😉 ). Sometimes, I feel like giving up altogether – what’s the point of trying, and only half understanding things? But maybe “half” is better than “not at all”, when you know you still have half of the way to walk. And sometimes, when I get a bit more stubborn, everything just clicks. And that’s one of the best feelings of the world.

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