Key Insight: If you were the kid in math class who could usually just skip to the end of the chapter and start working out homework problems immediately, but then you lost that ability, it's still worth trying that quickly and failing before trying to read the chapter first, because you are more likely than average to have a problem-solving bent to your personality that makes this approach both more motivating and more enjoyable. It may even be faster than "front-loading" the knowledge, since now you have a map of what to work for.

Specifically, do not feel like this is somehow "cheating" or "not being rigorous enough", because it is a perfectly valid way to go about things.


It's been said, in various guises, that mathematics is something you do, not something you read.

I generally agree with this advice, but my agreement might just be due to a lack of familiarity. I've only had a few upper level proofs-based math courses, and they've had a "problem-solving" timbre to them (combinatorics, graph theory, differential equations).

Still, I always feel like I have a much better grasp of how to actually work with the theorems and definitions once I've run through some calculations and proofs of my own, especially proofs chosen by the authors to highlight one specific insight at a time. (For calculations: This goes double when I have correct answers to check against.)

In fact, recently I've noticed that even with topics I feel a lot of intrinsic, theoretical motivation to learn more about, I always get a boost in motivation after skipping to the problems first and trying (usually failing) them after a minute or two of effort.

For some reason, this gives me a visceral, intellectual hunger to figure out just what on Earth these egghead authors are yammering about earlier in the chapter that I couldn't figure out on my own. What is it! I want to know now!

From there I usually piece together a "working knowledge" of just enough about the definitions and theorems to actually solve a few problems, and then, finally, I go back and read the chapter, and suddenly details I feel fairly confident I would not have paid attention to are seen in a much more stark and important light. (For a small, recent example of such a detail: That infinite unions, but not infinite intersections, are allowed in the definition of a topology on a set.)

That all might sound obvious. That's because it is obvious.

The title, however, is not Towards but "Against Not Reading Math Books Problems-First", because not everyone is going to get the most out of reading math the way I do - I'm more of a problem solver than an abstract theory builder.

I fear there are some people, who would benefit from reading more in this style, but don't out of a fear of lack of rigor or some strange sense of scrupulosity. I know that's why I hesitated to approach it in this style. What if I miss something?

It turns out that that's not a huge worry after all. You can't remember everything, and you can only focus on one thing at a time, after all! Where your mind's eye turns in the seascape of ideas is important - you're looking for white whales, not white foam. Are you going to trust your own intuition to "know" the most important things to focus on, or are you willing to let the textbook author take the wheel and subtle guide you with practice problems? There's no shame, and often a lot of wisdom, in the latter. :)


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I think this generalizes to

  • Try the easy way
  • If you don't get what you want from that, try a more time-consuming/difficult way
  • Focus on what you want and what you get, not on what you think you should be able to get with more work/rigor.

I'd argue that last part can be pretty hard for some people to get used to, but yeah, that's the gist of it!