self-reliance and psychology in learning math

Question

It's hard to be more descriptive with the title without making it excessively long.

What I'm wondering is this: what is the "right attitude" or "right psychology" when confronted with a mathematical question that one is not able to solve or get right (within a reasonable amount of time)? Maybe this sounds like a silly question, but I've struggled with it for a long time.

I'm an independent-minded person. I get more satisfaction from work when I am able to do it by myself. And mathematics is a huge body of knowledge that requires the student to depend on everybody else's work. Obviously that's the case with pretty much every discipline; there's always a certain amount that we always have to learn from others. But what makes it more difficult in math is that there is a culture or expectation of self-reliance. "The proof is left to the reader" is pretty much a meme. Math educators frequently and deliberately omit information. Also math is highly intuitive. It's conceivable that a person could get pretty far working alone.

I've had students who get frustrated at themselves for not being able to solve a problem. On one hand, I feel like this attitude isn't healthy. The frustration will just break down their confidence. On the other hand, I completely understand it and feel the same way sometimes. What advice would you offer those kinds of students?

Answer 1

I don't think telling people how they should feel is generally helpful.

I am sure that there are lots of people who are "successful" mathematicians who have all kinds of different emotional relationships with the subject.

You might imagine one who is primarily motivated by fame and the approval of peers, another who gets really angry when they face a problem they cannot solve and attacks the problem relentlessly until it is vanquished, another who finds great joy in tinkering with puzzles and sharing their insights with others, another who is primarily motivated by supporting the community of mathematicians and would always network while solving a problem, etc.

Some of these perspectives might be healthier than others in terms of long term emotional well-being, but I am not sure any one of them is "correct" from the perspective of "success" in our academic and social systems.

Answer 2

This is an extreme example, but perhaps shows that there is no "right attitude," but rather there is a range of approaches depending on your cognitive skills and inclinations.

Richard Feynman was renowned for teaching himself algebra, trigonometry, and calculus at a young age, and then of course following with a brilliant career. Krass says in his book¹:

Feynman needed to fully understand every problem he encountered by starting from scratch, solving it in his own way and often in several different different ways.

Steve Hsu writes that "it was often easier for [Feynman] to invent his own solution than to read through someone else’s lengthy paper." And Hsu discusses the rumor(?) that Feynman never understood the conventional formulation of QED even after it was established that his approach was equivalent to Schwinger's.

Feynman was extreme in his need (and ability) to re-derive everything. Most are more comfortable grasping what others have accomplished and building upon that. There is a tradeoff between depth of understanding (if you derive everything from scatch) and the breadth of what mathematics you learn (if you build upon results you accept because you believe with effort you could derive them).

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¹Krauss, Lawrence M. Quantum Man: Richard Feynman's Life in Science (Great Discoveries). WW Norton & Company, 2011. Quote from p.7.

Answer 3

Life is finite and time has a value to it. You should use the most efficient manner of learning possible. This means to utilize more traditional read/practice/test strategies using the most pedagogically efficient texts. Occasionally, you can deepen your understanding by challenging yourself with derivations and the like. But don't make it your main approach. Life is short.