How I Rewired My Brain to Become Fluent in Math

He highlights one of the problems I had with mathematics when I reached University. I was in the top 2% of students in the first few months. I considered it a trivial subject. "Here's a class of problems, here's how you solve them." Easy peasy. Then we started integrating and differentiating forms of problems we weren't familiar with. All of a sudden, everything we were taught didn't seem to apply any more.

Our lecturer told us: "I can see that some of you are struggling and are confused. You want to know how you become skilled at solving these problems? Forget your social life. Solve every problem in that 900 page textbook of yours. When you've done that, come to me and I'll give you another 900 page text book and finish that as well."

In one fell swoop Mathematics had lost all its appeal. I switched to Applied Math and aced it.

On the other hand, my brother actually followed this advice. For an entire year he spent a few hours every night doing Calculus till he recognized every different form they could conceivably throw at him. I asked him how he knew how to solve some problems. He said he didn't. It was all just memorization and recognition.

> You only ever need it to be able to think faster about certain advanced topics

This (and pretty much only this - you don't need rote learning to become fluent in a language, as evidenced by the way people learn their first language). Rote learning is what in tech we call caching things.

I very much prefer calling it like that, because it immediately highlights some benefits and drawbacks of the method:

- you cache things for quicker access, therefore it enables you to think faster

- the biggest benefit comes from caching things that are most frequently accessed; therefore learning multiplication table makes sense, it significantly speeds up every math you do in your head explicitly

- rote learning basics of a given field can help you learn it faster as again, the basic facts you need will be readily available in cache

- this matters because a lot of thinking people do has timeouts (aka "power of will", "energy", "patience", "curiosity") - if figuring out some connections between pieces of knowledge takes too long, you will get frustrated and give up early

- rote learning of history dates usually makes little sense because you end up caching vertices of a graph, while what you really care about are edges and traversing

- there is only so much you can fit in cache without losing its effectiveness, so memorize what matters, not everything

- rote learning is caching things, not developing understanding

- spaced repetition is a hack to force your brain from removing a particular information from cache prematurely; after few iterations it will understand that this piece of information you keep repeating is important and should stay cached

Also note that various memorization techniques are basically very fancy names for data structures optimized for human brains. For instance, "mnemonic link system" (aka. "chain method") is basically a linked list, where you use stories to build cons cells. There are variants of this method that are essentially doubly-linked lists or skip lists, and then you have other methods that implement association lists ("memory palace"), hash tables, etc.

This is important because you can apply knowledge about those data structures to evaluating those memorization techniques. For example, the reason it's difficult to insert another item in the middle or at the end of the list memorized with "chain method" technique is that you have to first traverse linked link to the place of insertion and then re-cons it there (develop two additional stories). It suggests that a/ maybe try to add new elements to the beginning of the list, and b/ you shouldn't use the "chain method" for remembering things you'll need to have random access to.

I'd say the prime proof of true learning is to have taught someone else to put in to action what you have learnt.

The "put forth" in the above is not entirely clear to me but I imagine that fits in there as a sub-division.

I first understood symbolic derivation after implementing code to apply the rules we'd learned (then I derailed completely and turned the expression evaluator I'd written for it into the core of a compiler).

All through primary school and most of high school, I stayed well ahead of the curve in maths because I could figure most things out from what we'd already learned, and finding algorithms was easier than memorisation.

E.g. I remember being given the task to sum all numbers from 1 to 100, and figuring out the algorithm for summing any series from 1 to n in time to be able to be intolerably smug about how quickly I had the result (I must have been 14-15 at the time so it was nothing out of the ordinary - the teacher assigned the task explicitly to see if anyone would find shortcuts or the formula). I then proceeded to spend the rest of the lesson figuring out how to modify it for all odd numbers, even numbers and various other classes of numbers. Up until then, that was how I spent most of my lessons, and my teachers tended to encourage it.

I lost most interest in maths when the teaching turned to become more about memorisation, and I was not taught the skills needed to figure out the underlying rules to be able to compensate for not memorising. If I'd been more interested, and invested the time in finding suitable resources myself, I'm sure I could've kept up, but it was still "just school" to me, even though I enjoyed math lessons until then.

It went from being my favourite topic to being something I actively avoided because of this.

I can't understand how anyone would decide to lose their social life for anything. But then, perhaps I am simply not the right person to become a scholar. Sure, I can put an hour or two a day of personal time keeping up to date, but I wouldn't see myself going "alright, let's waste an entire year of personal time for non paid formations" ... If I really have to do some learning, I'm going to try to do it through projects which I am paid to do.