Open thread, January 25- February 1

NancyLebovitz

If it's worth saying, but not worth its own post (even in Discussion), then it goes here.

I'm going to do the unthinkable: start memorizing mathematical results instead of deriving them.

Okay, unthinkable is hyperbole. But I've noticed a tendency within myself to regard rote memorization of things to be unbecoming of a student of mathematics and physics. An example: I was recently going through a set of practice problems for a university entrance exam, and calculators were forbidden. One of the questions required a lot of trig, and half the time I spent solving the problem was just me trying to remember or re-derive simple things like the arcsin of 0.5 and so on. I knew how to do it, but since I only have a limited amount of working memory, actually doing it was very inefficient because it led to a lot of backtracking and fumbling. In the same sense, I know how to derive all of my multiplication tables, but doing it every time I need to multiply two numbers together is obviously wrong. I don't know how widespread this is, but at least in my school, memorization was something that was left to the lower-status, less able people who couldn't grasp why certain results were true. I had gone along with this idea without thinking about it critically.

So these are the things I'm going to add to my anki decks, with the obligatory rule that I'm only allowed to memorize results if I could theoretically re-derive them (or if the know-how needed to derive them is far beyond my current ability). These will include common trig results, derivatives and integrals of all basic functions, most physical formulae relating heat, motion, pressure and so on. I predict that the reduction in mental effort required on basic operations will rapidly compound to allow for much greater fluency with harder problems, though I can't think of a way to measure this. Also, recommendations for other things to memorize are welcome.

Also, relevant

In general there the core principle of spaced repetition that you don't put something into the system that you don't already understand.

When trying to memorize mathematical results make sure that you only add cards when you really have a mental understanding. Using Anki to avoid forgetting basic operations is great. If you however add a bunch of information that's complex, you will forget it and waste a lot of time.

2whales12y

Nice, and good luck! I'm glad to see that my post resonated with someone. For rhetorical purposes, I didn't temper my recommendations as much as I could have -- I still think building mental models through deliberate practice in solving difficult problems is at the core of physics education. I treat even "signpost" flashcards as opportunities to rehearse a web of connections rather than as the quiz "what's on the other side of this card?" If an angle-addition formula came up, I'd want to recall the easy derivation in terms of complex exponentials and visualize some specific cases on the unit circle, at least at first. I also use cards like that in addition to cards which are themselves mini-problems.

18Shmi12y

In my experience memorization often comes for free when you strive for fluency through repetition. You end up remembering the quadratic formula after solving a few hundred quadratic equations. Same with the trig identities. I probably still remember all the most common identities years out of school, owing to the thousands (no exaggeration) of trig problems I had to solve in high school and uni. And can derive the rest in under a minute. Memorization through solving problems gives you much more than anki decks, however: you end up remembering the roads, not just the signposts, so to speak, which is important for solving test problems quickly. You are right that "the reduction in mental effort required on basic operations will rapidly compound to allow for much greater fluency with harder problems", I am not sure that anki is the best way to achieve this reduction, though it is certainly worth a try.