Deliberate practice for learning proof-based math

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What would a deliberate practice for math look like? Specifically, while self-studying undergraduate-level and graduate-level proof-based math. (So I'm excluding earlier stages of learning math up through high school and non-proof-based calculus/linear algebra in college, contest math, and excluding research work. I'm interested in mathematical research, but that seems like a much more difficult problem to talk about. Why exclude contest math? mainly because the problems there seem boring to me.)

epistemic status stuff around deliberate practice: isn't ericsson's research suspect? why should we trust it? my answer is basically: we don't have to trust it, and it doesn't matter much whether his research is right or wrong. what matters to me is whether it's plausible enough to take seriously and try to learn something from it. and then once we apply it to our own learning, we can ask "did we get a bunch better at this skill than before using some of the ideas around deliberate practice?" In other words, we're using ericsson's research as a hypothesis to entertain, than as a conclusion or belief we maintain.

how is math a "highly developed field"?

both ericsson in Peak and cedric chin [1] claim that math is a field particularly suited to deliberate practice. but i just don't see how?? like, all the examples i've seen in the book are from tennis, memorizing digits, music, etc.

i looked through all of Peak and there was some discussion of mathematicians and their traits and stuff, but nowhere in the book did he talk about techniques for learning math or describe what kind of "deliberate practice" mathematicians do.

Some difficulties with applying deliberate practice in this setting search "What problems exist for practice in fields where no good training methods exist?"

"ill-defined sub skills" I think applies to math. What even are the separate skills in undergraduate math? ability to read a proof? ability to solve problems? those seem too broad as categories to me. maybe there's like "ability to solve a particular kind of problem". but textbook exercises don't come tagged with specific properties, so you can't really filter on these to improve your skill in particular ways.

undergrad proof problems are generally too long that you can't do drills like "get at least 95% correct on these problems" -- you can't run that many trials to practice.

can we break skills up using the images here at "Kathy Sierra’s Badass: Making Users Awesome" (see in particular the final image). so in a proof problem, i guess one thing that you could do (and that is commonly done) is to make the problem easier by relaxing some of the hypotheses, or proving it in a special case. this is a kind of thing that is already done though. another way to make it easier is to peek at the solution, and then try to solve it. that's also already done.

"lack of feedback" -- this one also applies when self-studying math. the only ways to get feedback are by looking up solutions or by posting to something like math SE. book/subfield-specific discord servers might finally change this, but it will be slow. This is related to Feynman technique fails when existing explanations are bad (when existing explanations are bad, you can't even use techniques like Feynman technique to generate pseudo-feedback).

but there are things like Anki i guess.

to some extent, as you get better at math, the better you get at generating your own feedback / using the fact that the math does or doesn't work as feedback. but that's still not the kind of feedback that deliberate practice is talking about, i think.

i think one can design e.g. multiple choice questions that will give good feedback. and you could even have a goal like "get 95% on this MCQ".

Parts of the definition of deliberate practice

let's look at the requirements:

  • purposeful practice ("a person tries very hard to push himself or herself to improve"[1]):
    • well-defined, specific goals[1] "well-defined goals (such as doing something 3 times in a row with no mistakes)": this, as explained above, is one of the difficulties. proofs are long and take a lot of time to do. you also can't just "regenerate" problems using random number generators the way you can with high school level problems. you can still have a well-defined goal like "solve the problems in this section" i guess, but that doesn't seem to be what this requirement is about? How can we judge a session of math study to be a success? maybe "I solved a problem" is good enough... but it sounds boring. This is more possible/meaningful with spaced proof review or regular Anki prompts, however. "The key thing is to take that general goal—get better—and turn it into something specific that you can work on with a realistic expectation of improvement."[1] So maybe one idea is to say something like "solve this problem while keeping a specific strategy in mind [e.g. don't forget to use induction if it's possible; try to generalize but also try to look at specific examples]"
    • focused:[1] "is focused (the person is intently interested in improving, rather than having their attention elsewhere)": i'm not really sure if this requirement is satisfied... like, i guess you're really focused on solving the problem in front of you, or really trying to understand what the textbook is saying. so maybe that counts? on the other hand, i don't think you are usually consciously aware of like "i'm trying to solve this problem so that i can improve in a specific skill"?
    • involves feedback:[1] as explained above, this is one of the difficulties.
    • getting out of one's comfort zone:[1] "involves getting out of one's comfort zone, practicing things on the edge of one's ability": i think this one is automatically satisfied. you don't solve problems that are obvious. you're always trying to solve problems that are new, that make you curious (i think one of the things curiosity tracks well is problems that are just at the edge of your ability, that seem "fun" because they are doable).
  • "informed by an understanding of how to do well" / "the presence of a theory of skill and practice guided by that theory": i'm pretty vague on what counts as a theory. like, i'm guessing "solve lots of problems and you'll eventually get good at math" isn't a theory (yikes, that sounds like naive practice!). Peak just talks about having access to experts, who supposedly can look at you and figure out what you need to improve on.

"This is the basic blueprint for getting better in any pursuit: get as close to deliberate practice as you can. If you’re in a field where deliberate practice is an option, you should take that option. If not, apply the principles of deliberate practice as much as possible. In practice this often boils down to purposeful practice with a few extra steps: first, identify the expert performers, then figure out what they do that makes them so good, then come up with training techniques that allow you to do it, too."[1] -- in a way, the method of "try to prove a theorem, and if you get stuck look in the book for a hint, and then figure out what insight or strategy you were missing and put it into anki" is exactly this.

another thought is that deliberate practice is often motivated as a way to prevent plateauing of skill; but in math, there is no worry that you are plateauing! as long as you're learning new math and solving problems you haven't solved before, and you're not forgetting too quickly (spaced repetition helps a lot with this), you can be sure that you're getting better all the time.


part of the thing to figure out here is why one wants to "get better at math"? what's the ultimate goal here? depending on this goal, i think the kinds of practice to do will be different.

unlike something like competitive chess or swimming, there isn't a single obvious goal one is trying to optimize for (although even in e.g. swimming you could optimize for goals other than getting as fast as possible). so the question has to be asked in math.

one of the things that seems different about how i learn math is that when i'm done learning something, it's natural to not try to push it and "get even better" at it by e.g. solving many problems. but maybe i should? my natural inclination is to instead go to something else i am curious about. i might come back to the first topic when something prompts me -- maybe i have a specific question, or i run into a specific difficulty.

another part of the difficulty is knowing what's an "expected" level of mastery, e.g. should one be able to just hear the name of any big theorem and be able to sit down and recite the proof? or is it ok if one struggles to prove it but manages to prove it eventually after like a few hours of work (taking cues from memory and also just one's general experience in solving problems)? The mathematics community has no clear standards for what a mathematician should know.

some goals for me:

  • i want to know why some things are the way they are. for this goal, it kind of depends on what you are satisfied with as an explanation.
  • ability to recognize when some piece of math is relevant to a situation and apply it.

See also


  1. 1.0 1.1 1.2 1.3 1.4 1.5 1.6 Peak: secrets from the new science of expertise. Anders Ericsson, Robert Pool.

External links

some links (that i didn't find very helpful, but this is all that i was able to find)

What links here