Using Anki to learn mathematics

I suspect that the value of memorization is unfairly maligned in learning mathematics. It would be wrong to say that mathematics is "only about learning high level concepts, not about learning facts." Quite simply, every field of knowledge involves storing information in the brain. This is indisputable. Therefore, it seems rational to (upon picking up a book) decide whether one is intending to learn the material permanently, or allow it to slip away. The basic gist of my argument is that it costs many many hours to initially learn some difficult concept in mathematics (like how to solve a class of exercises with a new technique) and very little time, speaking in comparison, to occasionally practice the skill every few months or so to ensure that one is keeping it fresh.

1. When you begin learning a new subject, you are quickly overwhelmed by a mass of definitions that you must master quickly in order to make sense of the material. For example: the taxonomy of rings. PID, UFD, Euclidean domain, etc. When does one imply the other?

2. As time passes, if material is not used it quickly subsides away unless it is refreshed. Even high level concepts can decay unless they are retrieved and stored in long term memory. In fields which are not your primary area of study, the problem is worse - I cannot tell you what the Hahn Banach theorem says at all, although it is an absolutely basic theorem of functional analysis. Even if you are studying one single area with many interconnected concepts, approaching it broadly is difficult. If I was only interested in logic, would I be able to remember everything I've ever read in higher recursion theory, geometric stability theory, proof theory, descriptive set theory, categorical logic and type theory, and so on? Of course not.

3. If the reading material is formula-dense, you may be able to understand what the formula *says*, but this is different than storing it in the memory. For example I am aware of the Kunneth formula but I have some difficulty recalling the exact statement to mind or the necessary preconditions.

4. You may want to carry in your pocket some convenient computational examples that illustrate a concept; so you may want to memorize two or three handy applications of a concept. For a definition you may want to carry around the canonical examples of that definition. Consider the technique that uses Mayer-Vietoris to show that if a property of homology groups holds on sufficiently small open sets of a space, then by induction it holds on every finite union of those open sets. (If the space is compact, then this lets you prove results about the homology of the whole space.) It is a beautiful technique - it would be nice to be able to carry around the standard examples where it is used. This is pedagogically convenient and facilitates discussion.

5. A possible response is that "ah, well if you know the basic idea it can be accessed when it is needed." This simply misses the point. Every piece of information has a certain value to us; in an world where we had infinite memory capacity, we would store all information in the world in memory; and in a world where learning took an infinitesimal amount of time we would never need to learn at all except in the case where we needed it in the moment in question. The real world lies somewhere in between these two extremes. For example, math is heavily social (consider your frequent talks with mathematical colleagues) and many people communicate better in a physical setting, or via teleconferencing, etc.. Perhaps even the majority learn more from verbal discussion, gesturing, diagrams drawn on a chalkboard, etc. Then the more knowledge people collectively hold in their minds, the more efficient this social form of communication can be; because when A is communicating an idea to B, the less frequently B has to interrupt and say "Stop, you're going too fast, I don't know what these objects are, what is the definition of this again?" and the fewer times A has to stop and say "ah, I forget the details now - you can look them up in this source later on." I have had conversations numerous times where I struggled to communicate with my peers, as I forgot quite a bit of basic representation theory and Galois theory after our first year of algebra and I often found myself lost in conversations.

Some examples of things I once knew but have since forgotten, to my irritation:

1. Give an example of a usage of the Excision axiom to compute the homology group of a space.

2. State Krull's principal ideal theorem.

3. Hilbert's Nullstellensatz implies an equivalence of categories between finitely generated k-algebras (k an algebraically closed field) and affine varieties in k^n. What is (one) precise statement of Hilbert's Nullstellensatz and the proof of this implication?

4. Are there any conditions needed on a ring to apply Noether's normalization theorem, or does it hold for any commutative ring?

5. What is the precise definition of a weakening and contraction comonad over a fibration? How does do we define quantification and notions of equality with respect to a W&C comonad?

6. Girard's "coherence spaces" that he invented as semantics for System F - what are they again?

And so on.

Each of these questions small enough to be written on a flash card. The answer is perhaps not necessarily storable in a flash card but one could list a reference - the point is that the flash card stores a memory. The idea is, anyway, to review the flash cards once a month or once every several months, not once every ten minutes. You try and recall the details, write them down for yourself, retrieve the memory and in doing so strengthen it.

After some research I have found this one https://apps.ankiweb.net/

which is highly praised by a few friends of mine who have used it for medical school (one friend has amassed thirty thousand flash cards dedicated to different parts of the body) and learning new languages (esp. Asian languages, as the foreign alphabets are very challenging to Americans who have only ever encountered the Latin alphabet growing up ). The reviews have sold me on it and i'm going to start using this for the time being as my primary (permanent) note-taking source. Every time I come across an interesting factoid I don't want to lose track of, either in my research or in a book, I make a card for it. As the initial review time, I put a gap of ten days. I think it would be safe to bump it back to 14 or 20.

Here is a link to my LaTeX file of Anki questions that I have been maintaining. (Note that you can edit the header of the LaTeX mode in Anki to include all the styles and newcommands i've defined.)

http://www.math.wisc.edu/~nicodemus/AnkiQuestions.pdf
http://www.math.wisc.edu/~nicodemus/AnkiQuestions.tex