Training to actively learn, think and use mathematics studied and construct everything on ones own

Question

Someone said this:

"Memorizing the entire Swedish dictionary will never make you mastering Swedish. Same to Maths, the only way is to actively think & use basics you absorbed so far to practice in advanced environment and explore fast. Learning by doing will make you speaking maths fluently, flexibly and intelligently. Babies are the best learner, they neither play with single words too much before trying to use them to speak, nor waiting too long to absorb new words to expend the zone."

And our professor said

"one should be able to use and prove main theorem of Calculus/Analysis/Linear Algebra/Algebra/Probability/Topology even without thinking like breathing every second. Especially one want to do research on mathematics or even applied mathematical aspect of some natural science"

I think it's a good idea, even though Euclid said "there is no Royal Road to geometry", but there're still good and bad paths to go I think.

But when I try to implement this when reading textbook, it seems even though trying to do the exercise after each chapter especially the ones that just train to put values into the formula or simply using concepts, no matter how many exercises to do with these, it doesn't help a lot to understand things. well, the opposite trial, by using textbook not too seriously, just like a manual to PC game, a little faster "scanning" to get the main idea and def/theorem, then try to self-motivately construct the def/theorem by myself and try to prove theorems on my own even having the big "picture" in mind to construct instead of trying to recall the details, only refer to the book when proceed next or when get stuck too long for getting main idea. When this is doing , it seems the brain works more actively, and the def/theorem is absorbed, like get the hammer controlled on hand instead of only read the manual of hammer.

The question is firstly to hear some comment on it especially the two quotations on the top.

And for the "practice makes perfect", will there be some selection principle of "good practice" to more actively learn certain mathematical textbook ? Since simple value-in-formula-use or basic concepts problems really doesn't help too much. Or say, should one put time doing more proof problems skipping some too simple questions even though the formula is newly learned. Because try to prove on own, it seems more using something actively, but value-in-formula-use, no matter how many problems to do, it seems the tool still not belonging to me.

Answer 1

One has to be practical, and consider the limitations of time available, and time one wishes to spend.

My own experience was that getting me going in research was mainly through writing and rewriting to make things clear, to me. Also explaining to others what I was doing and why was very helpful.

If you write something out five times, you may see how to improve it a little; and 2 months later another little; and so on. After five rewrites, it might become original! The point is that this method is a way of getting the rusty wheels of the brain turning over.

The composer Ravel said that you should copy (and he was very original, as we all know). If you have some originality, then that might come out as you copy. If not, never mind! In any case, it might take three copies to see a way of changing it.

It is said that Newton was an inveterate copier.

I also learnt from my supervisor, Michael Barratt. He showed me draft papers which started with a title, then author, then Introduction, explaining what the intention and results of the paper were. As you write the paper, it may turns out that the Introduction becomes inappropriate, so needs rewriting, and then the paper needs rewriting, the title may need changing, etc., etc. But I like the idea that the paper starts with an intention. It may be to solve a known problem, or to develop some mathematics which intuition tells you ought to exist, and that if it existed, it should be useful.

Another method is to write the work of A (person or area) in terms of the language of B (person or area); instead of going for "mountain peaks", you try to "fill in the valleys". This is useful work for a beginner, since you start by doing a kind of translation; through this you are bound to learn a lot, but also you have a guide to keep you going.

An example of area B is category theory.

Hope that helps.

Later: I'll add another point. Jose Montesinos told me he tells his research students: do what you find easiest! We all have to find some way of finding out how to get into developing the mathematics which is most appropriate to ourselves, without being too much distracted by the "should" and "ought" with which we are bombarded. What seems easy to oneself may not seem so eaay to someone else - we are all different!

Answer 2

Regarding the second quotation, I suspect what your professor meant was that elementary theorems should be completely mastered if you expect to work on more advanced material. If you expect to do research in analysis you probably ought be able to state and prove the Mean Value Theorem or define a compact space without looking these things up. If you try to move too fast through the basics, you'll find you're learning at a more and more superficial level as things get more challenging.

Regarding your practice of reading maths books "actively", this is indeed what we all do. You may have had experience of schoolbooks that only offer mechanical "drill" problems, but university-level texts will increasingly ask you to work out proofs and theorems on your own in the exercises, often developing important material there. You should always have pencil and paper in hand: as soon as you see a theorem try to prove it, as soon as you see a definition start listing examples and non-examples etc. I would skip routine exercises in a book if I was confident I could do them, although it's surprising how often problems that look routine turn out to be tricky!

Caveat: I'm not a professional mathematician, just an autodidact who's been at it a while.