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I have been reading a lot of books about deliberate practise recently like Angela Duckworth's "Grit", "Talent is Overrated" - Colvin and "Peak" - Ericsson and Pool. I want to apply these principles.

Can you suggest some concrete steps to implement the principles deliberate practise in my mathematical studies ?

As far as I understand, deliberate practise has two main principles :

  1. It is focussed - For example, if you're a tennis player, it isn't just to play the most games or to hit the most number of balls as fast as you can. Rather, it would be to specifically practise the motion of the forehand and focus on the wrists, elbows and shoulder to ensure that the form is as correct as it can be. If you're a cricketer, it isn't to just hit the most number of balls, but to ensure that the bat lift is straight, the weight transfer is right, head is steady and that the elbows, wrists and feet move as accurately as possible for the best possible shot. It is to put deliberate focus into those things and correct the form, rather than just focus on hitting the ball the most number of times. While doing mathematics, what can I choose as specific goals to improve ? In sports, it's easy to choose a small part of the form to improve. But, it's not so obvious what this goal should be in Mathematics.

  2. It has constant feedback - If you're a tennis player, you become conscious of which part of your swing is wrong exactly and decide if the ball doesn't have too much top spin. Having gotten that feedback, you specifically start tailoring that part of your swing consciously to increase the amount of top spin. If you're a writer, you don't just write more. You analyse your writing, and realise that your character psychology is fine but your description of events and settings isn't. So, you find some writings where those parts are excellent and specifically focus on improving that aspect of your writing. How do I analyze what mathematics I'm doing as good or bad ?

I want a set of concrete steps I can follow to improve my mathematical skills. I should note, that by mathematical skill, I mean ability to solve problems (Like the Olympiad type, which will later help me in research). I'm not asking for advice to increase my grade or marks. I'm interested in improving my mathematical ability.

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    $\begingroup$ Generally, "feedback" is what you get from a real live instructor! For work on Olympiad training, with feedback from others, you can try artofproblemsolving.com $\endgroup$
    – Gerald Edgar
    Commented Jun 2, 2016 at 14:23
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    $\begingroup$ You should be aware that Olympiad problem-solving is only very loosely related to genuine research in mathematics, and, in fact, has many attributes rather opposite. It's true that contest-math success has helped many people get their "foot in the door" for more advanced mathematics, but it's a rather different enterprise. If you're really thinking about contest-math, as opposed to research math, the appropriate optimization is significantly different, I claim. Can you clarify? $\endgroup$
    – paul garrett
    Commented Jun 5, 2016 at 21:43
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    $\begingroup$ I do not think it is a general fact that "most famous mathematicians have been good at contests". Being good at contests gets attention and publicity, which can be helpful, but the skills are significantly different. "Solving difficult problems" may or may not refer to contest problems, where usually there's a "trick", and very little genuinely advanced mathematical knowledge is relevant. But if you like contest problems, there's no harm in that. It's just that that is not a way to serious further mathematical education. $\endgroup$
    – paul garrett
    Commented Jun 6, 2016 at 17:13
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    $\begingroup$ Yes, it is possible to be a good mathematician despite being stumped by contest problems... which are designed to stump you, after all. There is not necessarily any meaning to the difficulties, or to the questions at all. They are just puzzles, mostly without meaning or significance beyond the contest situation. $\endgroup$
    – paul garrett
    Commented Jun 7, 2016 at 16:01
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    $\begingroup$ Mostly, no one can do Olympiad-level problems anyway. A few very quick, very clever kids can. Also, very quick, very clever kids can do many things. Still, such activity is only distantly related to mathematics as a scholarly and research activity. A different sport almost entirely, although very superficially similar. $\endgroup$
    – paul garrett
    Commented Jun 7, 2016 at 16:04

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Following up on (and repeating part of) a discussion-in-comments just below the original question:

As far as I can tell, contest-problem math is significantly different from genuine scholarly-study-and-research math. The goals, constraints, and sources are significantly different, although there is some overlap at the low end.

Practicing contest-math problem-solving is maybe not so good as a means to understand genuine mathematics. Even as generalized problem-solving, it is solving problems deliberately contrived by other clever people, designed to have specific difficulties overcome-able by things youngish people might know... with time being a significant, yet also artificial, constraint.

Yes, some people who are good at math contests are also good at other things, including genuine mathematics. But this misappraises causality to some degree, and confuses quickness with insight, potentially. It is true that being quick can be an advantage, but only if one persists proportionately longer. Quick and impatient, or quick and disinterested, or quick but lazy, and so on, are not productive combinations.

The original question in the title does not really seem to be the question. That is, the questioner expressed some disinterest in repetition. Yet the mathematical analogue of "muscle memory" is exactly what can afford speed-ups and "correct reflexes" in a variety of situations. Extensive rehearsals of "variations on a theme" seems to me the best way to assimilate new ideas... if there really are ideas rather than mere stunts.

So, yes, "deliberate practice" is plausibly at-least-partly synonymous with "rehearsal of variations on a theme"... until one can react reflexively in an appropriate fashion without much conscious effort. It does seem, in my experience, that one's subliminal capacity is much greater than one's explicit conscious capacity, but that it takes much "rehearsal" to assimilate things to a subliminal level.

Unfortunately, the exercises given in many of the standard upper-division and beginning graduate-level texts present an ocean of mix-and-match details, as though every possible logical combinations of hypotheses deserved equal consideration. This is bad training for sensibility, I think, in part because it does hark back to the contest-math side-channel information that there really is a human adversary who has contrived the problems to "challenge" us... not necessarily to edify us by bringing to our attention important issues that will help us subsequently.

Also, the contemporary style of mathematics seems to needlessly disparage heuristics, and exaggerate the importance of "proof", often elevating the supposed importance of proof far beyond any alleged significance or utility of the things being proven. To be able to allow oneself to optimistically skip over innocent-seeming intermediate steps to try to get an interesting conclusion is good practice! Also, after all, what's the point of worrying about a proof if we don't care about the conclusion? There are plenty of seemingly-trivial important things whose proofs are surprisingly subtle. (The intermediate value theorem from calculus, for example.) Given that people have finite capacity, it may be a bad allocation of resources to "practice" proving pointless or boring or artificial things.

So, in particular, I'd not advocate "drills" nor "doing all the exercises".

Rather, it seems to me that "understanding" mostly involves assimilation to a subliminal level, which usually occurs only after considerable rehearsal of the ideas. Not adversarial-style "exercises", but rehearsal. Something like the way one might rehearse lines for a play, or an explanation of a technical idea, or a piece of music for performance. As we know, mere "memorization" is a flimsy and inadequate version of assimilation of such things.

From the other end, exercising one's own critical judgement is another thing not really promoted in typical math coursework and textbooks, since there's typically a pretense of absolutism. Learning to look at arguments or computations "with eyes open", even if one wrote it oneself, is necessary. At the same time, one shouldn't sabotage a good heuristic by squelching it at the outset due to minor gaps (as opposed to literal errors). Nothing constructive about that.

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  • $\begingroup$ How do you recommend I spend my time to become a good mathematician ? $\endgroup$
    – Saikat
    Commented Jun 9, 2016 at 13:31
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    $\begingroup$ Trying to honestly understand things that are interesting to you, primarily. Secondarily, trying to see what (if anything) other math people find interesting in what they are doing... and form your own opinions... endlessly subject to revision, naturally. Thinking about things, but also looking at more and more sources to see whether they answer your question, or help you to answer your own. Subject to endless reconsideration as more information is acquired. Not at all a linear, systematic, algorithmic process. $\endgroup$
    – paul garrett
    Commented Jun 9, 2016 at 13:38
  • $\begingroup$ I'm interested in recreational mathematics, problem solving, puzzles, discrete mathematics, combinatorics and number theory. However, I don't know analysis yet. I'm in a computer Science engineering course right now in the bachelors state and I'd like to end up as a mathematician. I'll work really hard and try to improve. Since, I'm mostly self studying, I tend to skip steps in proofs and don't know when I should have written more steps, in particular, I struggle with writing down ideas in formal language using quantifies and sets. $\endgroup$
    – Saikat
    Commented Jun 9, 2016 at 13:47
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    $\begingroup$ Recreational math and puzzles are typically very low-prereq, and are a different genre than genuine math. Olympiad and contest math are more difficult, but still very low prereq. "Discrete math", "combinatorics", and "number theory" are very ambiguous: these can misleadingly refer to essentially elementary puzzles, and sometimes elementary number theory is hard to distinguish from elementary combinatorics, all low-prereq. But, in fact, discrete math and number theory are two very different things, and are also the subject of very sophisticated contemporary research. [cont'd] $\endgroup$
    – paul garrett
    Commented Jun 9, 2016 at 14:26
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    $\begingroup$ [cont'd] Learning to write coherent math is a task in itself, and probably is best done by imitation, by reading coherent writing of many others. Although much of this skill is exactly imitation, at the same time it can be refined by pointed self-examination: if what you write were written by someone else, would it seem reasonable? Not "could it be interpreted as being reasonable", but is_it_literally_reasonable? Until you are much more practiced, it is probably best to err on the side of too many details than not enough... especially if it's not clear to you what could be omitted. $\endgroup$
    – paul garrett
    Commented Jun 9, 2016 at 14:29
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I haven't heard of deliberate practice before but what you describe sounds pretty much like what we do in math courses at the university where I am (and also at most others in Germany):

We give homework that is focused on specific topics, notions and techniques. There are problems to train proof by induction, there are exercises to grasp the notion of continuity, there are problem sets on linear maps,... Basically any homework I give is focused on a special thing and I usually note the topic of the specific exercise in the title.

We provide weekly feedback for the exercise. The feedback is written down on the homework sheet and also communicated during the exercise classes.

There may be something that is different from the technique you described, namely an issue with time. Performing a formal proof by induction takes some minutes even when you really know what to do and can take much longer if you have to figure out how it works. Most students don't have to time to do many exercises on the same topic.

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  • $\begingroup$ I don't want to do many on the same topic. I'm not talking about exercise drills. I'm more interested in problems where you don't know exactly what to do rather than simple, procedural problems of applying a formula or a standard technique. $\endgroup$
    – Saikat
    Commented Jun 6, 2016 at 3:00
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You also need a goal to work towards (what is the "end game"?). You can be (1) focused and (2) receive quality feedback, but you need to create a clear objective regarding your studies. For example, what is the research that you would like to be involved in (as you mentioned in the OP)? It should be a specific thing, like "research in AI focusing on assistance of people with disabilities". That is, it should be measurable and clearly stated. Then, work backwards from your goal and apply deliberate practice to achieve it.

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If you want to be better at Olympiad work old Olympiad problems. This is the most efficient (deliberate practice) advice. If you lack the facility for some of the things in those problems, than study the courses needed to have that facility. Note that becoming super good at Olympiad problems is not the same as mastering courses which is not the same as facility for research.

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    $\begingroup$ You should use only one account. Please don't use a new account for each action that you do. $\endgroup$
    – JRN
    Commented Jun 11, 2017 at 13:37

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