Anyone who has been in a class with me in graduate school knows that I'm a huge fan of flashcards. Well, really of spaced repetition for learning. The basic idea, first introduced to me in this Wired Magazine article, is very simple: we retain information best when space our exposure to it so that the time between exposures increases each time. For example, you might first require exposure to the idea immediately after learning it, then an hour later, then a day later, then a week later, etc. Any other spacing would be suboptimal, and would require either more time learning the material than is necessary, or would result in less material retained. Steven Strogatz and some others did a neat analysis of this problem from an abstracted perspective.

This puts a lot of responsibility on the part of the learner to figure out a system for when to review a card. Fortunately, a lot of software is already out there that automates this process. Some open source options include Mnemosyne and Anki. I use Mental Case.

I first learned about this approach the summer between my sophomore and junior year1. I immediately put the system to use. My first flashcard in the system asked, "What are Maxwell's equations?" One of my more recent flash cards asks, "To show that the estimator from Maximum Likelihood Estimation is consistent (converges to the correct value of \(\theta\)), is the result that the negative mean log-likelihood converge (in probability or almost surely) to a function whose maximum is the true parameter value enough?" (The answer is no. You need the convergence to be uniform in \(\theta\).) This gives you a flavor of how my interests have changed over time.

I used (and continue to use) that system for studying since my junior year of college. I've created notes for every class I've taken since then, and have now started making notes as I go off learning material on my own. I have 3546 notes in my collection currently, though some of them are 'completed' (I'll never see that Maxwell's equations card again, and to be honest, don't remember them anymore). I can't prove that they've cut down on the number of hours I spent studying, but I do know that after I started using them, I no longer had extended study sessions before exams. I spread all of the studying out over the course of the semester, and then beyond. This also gives me an easily searchable repository of all the results I've ever learned. I've found myself going back to it more times than not when I have a result on the tip of my mind, but can't quite grasp it.

I recently read a post by Rhett Allain at Dot Physics where he flatly states that he's not a fan of flashcards. Of course, his conception of flash cards and mine couldn't be further apart. I use flashcards as an excuse to commit concepts to memory (recently, they've been the only reason I go back over notes that I've written), and as a crutch to keep those concepts on the top of my mind. Rhett has in mind rote memorization of facts just to get past an exam. I've been so far removed from that world that it's hard to imagine returning to it. That's one nice thing about being a (pseudo-)mathematician. Results follow naturally from the definitions we set up.

Here's my comment on Rhett's blog. Hopefully I come off less as an ass and more as someone offering constructive criticism.

In mathematics, at least, I've found flash cards to be an invaluable tool. (Well, spaced repetition software like Mental Case.) Large parts of mathematics begin with simple definitions, which we then build on to develop the theory. But if we don't even know the most basic definitions, how can we hope to scale to the heights of the theory?

At least in my process, definitions come first, and then understanding. If I can't state what some mathematical object is, then I have no hope of having any intuition about it. During the courses I've TAed as a graduate student, I've stressed this fact with my students. Especially in a definition-heavy course like linear algebra. They have to know what it means for a collection of vectors to form a basis (i.e. a basis is a linearly independent spanning set for some vector space) before they can begin to get the intuition for what that means (a basis gives you a way of labeling all points in a vector space).

An anecdote about Feynman related to the "couldn't you just look this up online?" point:

'Richard Feynman was fond of giving the following advice on how to be a genius. You have to keep a dozen of your favorite problems constantly present in your mind, although by and large they will lay in a dormant state. Every time you hear or read a new trick or a new result, test it against each of your twelve problems to see whether it helps. Every once in a while there will be a hit, and people will say, "How did he do it? He must be a genius!"'

You can't make a connection unless you have the basic facts behind the connection readily at hand.

Admittedly, I probably use flashcards very differently from what you have in mind with this post. The notes I write for myself are much more about the concepts than any particular mathematical expression (though I try to keep those fresh, too). And I actively think about each note when I review ("Why is the expression this way and not that?", "What would happen to this theorem if we tried to weaken this assumption?", etc.), rather than perform rote memorization.

I've had thoughts of posting my flashcards online at some point. But I think they may be so tailor made to my own use that they don't be useful to others.

  1. That was my 'summer of the chemist,' when I attended a REU at Texas A&M under the advisement of Dave Bergreiter. I studied solute effects on PNIPAM, poly(N-isopropylacrylamide). That was also the summer when I learned that, unlike my brother, I would never be a synthetic chemist. Probably all for the better.