How do I assimilate mathematical concept?

Question

Already knowing that the famous quotation "there is no royal road to mathematics", I believe that the most efficient and best way to learn mathematics is to make it intuitive to oneself, at least to me.

If I can't give any intuitive meaning to given statement, then I can't do anything. Likewise, if I can "see" how it is going on, then I can prove or disprove something easily.

So this is why I want to know the way to assimilate mathematical concept.

I've tried to prove any theorem on book I've been reading without peeking up the book, and solve exercises as many as possible time permits.

But no such methods immediately give me an intuition... only time permits me to obtain it...

Recently, I've read two opinion related to this topic on stackexchange, Paul Garrett said that one need to "rehearse" the idea(though I couldn't understand what "rehearse" means really) and Yuval Filmus said that if one wants to assimilate the idea, one needs to reflect on what one's learnt and ask oneself interesting question and try to answer on one's own.

Somehow I guess they were talking about seemingly similar method. what is it? any advice will be appreciated.

Answer 1

I agree that insight sometimes takes time, but don't dismiss the benefit of exercises. Sometimes, the insight monster will lurk in the background in your subconscious. If you do exercises, it may come out and give you the revelations later, not directly from the drill, but somehow aided by the time/exposure. If you solely chase insight and don't build your computational muscles, you probably will be inefficient at gaining mastery of either.

I suggest to read some of the writings of Richard Feynman. He was definitely in favor of insight (not mindless drill). But he also had huge value for sort of building a repertoire of tricks. Also, he felt that certain sorts of manipulations should become automatic and easy from practice (calculus and even algebra). Having these down cold ("bing, bing, bing") makes it easier to look at physics (perhaps for you higher math), without getting confused by errors. In addition, at one point he felt weak in classical E&M (and needed it for something). So he went and got a standard textbook and worked every single problem.

Bottom line: I would be worried by the spending four years and not describing your plan more strategically and then within that "as much exercises as time permits". I'd be very worried you are spinning your wheels and wasting a productive time of your life. These years are golden. Be savvy with how you spend them.

Answer 2

Expanding on the point already mentioned by guest's answer, I think you should stop focussing on building intuition. Instead, focus your time on solving the exercises even if, by the time you solve them, you don't feel like you've gained much in terms of intuition. Just solve the problem and move on.

This might seem very counterintuitive, but the point is that intuition is not an easy thing to build (at least, not for most people, for most topics). You don't, for example, magically obtain a strong intuition for how finite groups behave immediately after being introduced to the definition of groups. Things just don't work that way. Instead, intuition is slowly developed after attempting some (occasionally large) number of computational exercises, where you really work with whatever object you're studying "hands on" and get a feel for how they behave. This active process of manipulation is completely different from just reading proofs in your textbook, and cannot in my opinion be replaced.

A very practical way to go about doing this is by working through all the proofs you encounter in your textbook (or alternatively, the proofs presented by your professor in class). For textbooks, before reading through the proofs for each result, you can (and should) attempt to first understand what the result means, by for example writing down a few examples, or if that's not possible by rewriting on scrap paper the statement of the result and trying to gain some intuitive understanding for what the result claims. Then, you might attempt to come up with the proof yourself without looking at the proof presented in the book: this is often surprisingly easy once you've obtained a clear intuition for what the result is saying.

Of course, this is not always possible in practice. And there are some textbooks where the learning curve is so steep that it is practically impossible to come up with proofs of any of the claimed results on your own without reference to the provided argument (especially since, perhaps, a key idea that is used in the argument has not been introduced yet). However, at least the first step of understanding results---even if it means slowing down and taking an hour for a single page in the book---will be very instructive and lead to a significant increase in your learning quality and effectiveness.

In conclusion: stop thinking about building intuition. Actually do it by working with the objects you're studying, either through the (possibly computational or "tedious") exercises that are provided, or by working through and understand the results presented in your text/class.