Deliberate practice for learning proof-based math - Revision history

Issa: /* Goals */

2021-10-25T18:37:24Z

‎Goals

Issa: /* Goals */

2021-10-25T18:27:26Z

‎Goals

Issa: /* how is math a "highly developed field"? */

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‎how is math a "highly developed field"?

Issa: /* Parts of the definition of deliberate practice */

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‎Parts of the definition of deliberate practice

Issa: /* Parts of the definition of deliberate practice */

2021-10-25T18:21:38Z

‎Parts of the definition of deliberate practice

Issa: /* Goals */

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‎Goals

Issa: /* Parts of the definition of deliberate practice */

2021-10-25T07:31:57Z

‎Parts of the definition of deliberate practice

Issa: /* Parts of the definition of deliberate practice */

2021-10-25T07:05:38Z

‎Parts of the definition of deliberate practice

Issa: /* Parts of the definition of deliberate practice */

2021-10-25T06:53:07Z

‎Parts of the definition of deliberate practice

Issa: /* Goals */

2021-10-25T06:00:38Z

‎Goals

← Older revision		Revision as of 18:37, 25 October 2021
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	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]].		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==		==See also==

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 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)?
+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]].
 ==See also==

← Older revision		Revision as of 18:26, 25 October 2021
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	both ericsson in Peak and cedric chin [https://commoncog.com/blog/the-problems-with-deliberate-practice/] 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.		both ericsson in Peak and cedric chin [https://commoncog.com/blog/the-problems-with-deliberate-practice/] 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==		==Some difficulties with applying deliberate practice in this setting==

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 * "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."<ref name=peak/> -- in a way, the method of "try to prove a theorem, and if you get stuck look in the book for a hint" is exactly this.
+"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."<ref name=peak/> -- 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.

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 * "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."<ref name=peak/>
+"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."<ref name=peak/> -- in a way, the method of "try to prove a theorem, and if you get stuck look in the book for a hint" 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.

← Older revision		Revision as of 18:18, 25 October 2021
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	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.		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)?

	==See also==		==See also==

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 let's look at the requirements:
-* purposeful practice:
+* purposeful practice ("a person tries very hard to push himself or herself to improve"<ref name=peak/>):
 ** '''well-defined, specific goals'''<ref name=peak>''Peak: secrets from the new science of expertise''. Anders Ericsson, Robert Pool.</ref> "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."<ref name=peak/> 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''':<ref name=peak/> "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"?

← Older revision		Revision as of 06:00, 25 October 2021
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	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.		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.

	==See also==		==See also==