Exploration and Skill Development

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Different people have different goals with learning math. Some want to learn math to apply it to some other discipline of interest: ML/AI, rocket engineering, finance, bioinformatics, whatever. Others want to learn math to apply it to math itself as the end goal (like pure mathematicians).

Whatever your interest is, there is value in exploring it. The exploration can always help with motivation, provided it’s something you are actually interested in.

But at the same time, the exploration doesn’t really move the needle on developing/consolidating foundational skills until you’re at a high enough level where your explorations are actually exercising the foundational skills.

Say you are really interested in classical Euclidean geometry. How much value is there in working through challenge problems out of a book on that?

Say you don’t know your geometry and proof fundamentals. You don’t know how to prove triangle congruence and you find proof by contradiction confusing. In that case, the challenge problems aren’t going to do much for you, even if they’re fun to think about. But if you have the foundational skills necessary to grapple with challenge problems, then yeah, they’ll move the needle for you.

All to say, I would agree that there can be skill value in explorations if, during those explorations, you’re actively producing (not just passively consuming), and the things that you’re producing are turning out successful (even if they’re small successes).

On the other hand, some examples of exploration without value to skill development:

• Someone doesn't know much about complex numbers, series convergence, etc. They watch a video on the Riemann Hypothesis and are amused by all the pretty visualizations of the complex plane.
• Someone doesn't know much combinatorics. They pick up a flavorful book with a bunch of "how many ways can you ___" puzzles that seems really interesting and they work through problems but they never get any problems right and always just end up looking at the solution, which they don't really understand.

In general, if you get your daily dose of foundational skills training in, there’s no issue with poking around exploring areas you find interesting. For people who are into that, that can do wonders for motivation.

It’s just that sometimes people get confused and fall into failure modes like

• treating exploratory poking around as a substitute for foundational skills training, or
• attempting to produce research without the foundational skills in place.

(I’ve written a bit about failure modes in the talent development process; the failure modes above are #2 and #3 here.)

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