Learning modulo sticking points, or "fluid study" in mathematics

Question

In graduate school I once tried to rapidly learn mathematics by working through a text and collecting (and precisely remembering) where I had been stuck, and what parts of arguments or computations I did not follow. After writing precisely down where I was stuck, I immediately moved on. Of course, I then immediately got in trouble with supervisors who spotted the holes in my understanding. I have, from time to time, wondered whether sticking with this approach longer may have paid off.

It seems that in this way, one can accumulate a lot of facts and intuitions about a lot of things. It may be argued that the sticking points might actually eventually be dissolved by frequent enough encounter with the same difficulty in a diverse variety of circumstances.

This is, perhaps, expressed in the extreme viewpoint (probably not seriously) stated by Halmos that one should "never read the proof of a theorem". The sentiment here being that one should instead think about what the theorem can do and eventually aim to prove the theorem for oneself. One envisions a mathematician spending time to collect theorems like mushrooms and using them to "cook his own mathematics" rather than poring over other people's proofs.

My question is: Is there evidence that some mathematicians effectively learn in the above fluid way described in my first paragraph? In particular, is it possible to develop deep insights by moving "laterally" and accumulating knowledge, cataloging where one is stuck as one moves along, but immediately moving on to further understand the landscape rather than remaining stuck for any significant period of time?

I am asking this question as a "devil's advocate" position to the one I actually espouse, as I have expressed in my answer here. I personally believe that depth is sacrificed by an approach like the above, and that it is too tempting to move on to something easier in the above strategy to develop anything substantial. On the other hand, such rapid lateral movement might better have a chance to uncover connections between different areas, and is likely to be much more fun than cracking one's skull on details. More interestingly, perhaps, does anyone teach this from this standpoint...especially to undergraduates? Undergraduates might enjoy a more pedestrian approach, and if there is evidence that significant gain can be acheived by treading more lightly, this might affect one's approach to undergraduate mathematics education.

If this is subjective and argumentative...or too metacognitive for the forum, I'll happily delete it.

Answer 1

I'll just address the issue of teaching and detail. I do think there is something to be gained in providing a big picture on a given topic. For the students who care, I hope to present them with some sort of roadmap to math beyond the course. By the nature of the discussion, there is no way they will be able to struggle through the details implicit within the big picture. For example, I would mention the Jordan form in linear algebra despite the fact I do not work out the gory details. Or, in calculus III, I will spend a little time at the end to alert them to the existence of these wonderful objects called differential forms. Are they ready to prove the generalized Stokes' Theorem on $p$-chains or whatever? No. Should I avoid it because of that? I say no. Whatever class I teach, I always look for opportunities to give brief pointers to avenues of future study.

That said, the bulk of my teaching is detail-oriented. I don't really grade the big-picture part. Furthermore, I tend to show more proofs than might be typical because I do believe that the details matter. Especially when you get to the point of generalization. How can you generalize an argument you never saw?

Unfortunately, I have seen students who seek a big-picture actively discouraged by certain professors. The desire to know more than you are currently able to prove is healthy in my view. So, this discouraging posture is an enigma in my view. I see a student who wants to know math. Good. Whatever skill they fall short in we should take the posture of encouraging them to attain that skill so they can own the new math rather than collect it as trivia. I think it's really important to give students a forum to express their excitement about math, we should never dismiss their interests. However, we should help them see what they need to unwrap before they can compress it down as Thurston says in your answer.