I think I would have done well to think more carefully about the benefits of implementing a review system, when you posted this comment. It may have been true that I was getting by with my current study setup, but also accumulating “math recall debt” over time.
I think I underestimated the importance of knowing lots of math facts. For example, I initially replied:
So, I’m not trying to memorize everything. Leafing through Linear Algebra Done Right, I don’t remember much about what self-adjointness means, or Jordan normal form, or whatever. However, I don’t think that really matters. If I need to use the extraneous stuff, I know it exists, and could just pick it back up.
This underestimates the cost of picking it back up, which is linear for each fact in the dependencies I forgot, versus just maintaining the dependencies over time.
For example, when I’m studying quantum mechanics, it seems crazy to say “who cares about self-adjointness”, but not that crazy to say “who cares about self-adjointness” if I’m just doing reinforcement learning theory.
I gestured at another useful habit, but one which only works on skills I use regularly:
I am, however, able to regenerate fundamental things I actually use / run into in later studies. I have a mental habit of making myself regenerate random claims in every proof I consider. If we’re relying on the commutativity of addition on the reals, I reflexively supply a proof of that property. I came up with: use the Cauchy sequence limit notion of reals, then rely on the commutativity of rationals under addition for each member of the sequence.
I couldn’t use this technique to help myself remember what a correlated equilibrium is, because I don’t use concepts which build on ‘correlated equilibria’ very frequently.
I understood correlated equilibria at one point, but I don’t recall anymore, and that makes me sad. Now I’ll go back and put that chain of insights into Anki; if I’d been doing this from the beginning, I wouldn’t have to do that.
For those reading this thread in the future, Alex has now adopted a more structured approach to reviewing the math he has learned.
I think I would have done well to think more carefully about the benefits of implementing a review system, when you posted this comment. It may have been true that I was getting by with my current study setup, but also accumulating “math recall debt” over time.
I think I underestimated the importance of knowing lots of math facts. For example, I initially replied:
This underestimates the cost of picking it back up, which is linear for each fact in the dependencies I forgot, versus just maintaining the dependencies over time.
For example, when I’m studying quantum mechanics, it seems crazy to say “who cares about self-adjointness”, but not that crazy to say “who cares about self-adjointness” if I’m just doing reinforcement learning theory.
I gestured at another useful habit, but one which only works on skills I use regularly:
I couldn’t use this technique to help myself remember what a correlated equilibrium is, because I don’t use concepts which build on ‘correlated equilibria’ very frequently.
I understood correlated equilibria at one point, but I don’t recall anymore, and that makes me sad. Now I’ll go back and put that chain of insights into Anki; if I’d been doing this from the beginning, I wouldn’t have to do that.