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I am studying a 400/500 level measure theory math book on my own. Right now, when I read it I try to read the proposition then the following proof. And then try to do the exercises on my own. I wonder if I should change it to:- 1. read the proposition 2. try to prove it on my own. 3. If I could not then proceed to read the proof from the book. 4. Do the exercises.

Or should I stick with everything above minus step 2. It can save me some time but I wonder if that is what good mathematics students do?

Any feedback from your experience would be appreciated.

Best,

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    $\begingroup$ There's no "should" about this. You want a mix, depending on your learning style and what works for you. Consider working with several books at once. Seeing multiple versions of theorems and proofs might help you triangulate on concepts. $\endgroup$ – Ethan Bolker May 10 at 20:49
  • $\begingroup$ What book are you using? Your success on step 2 will vary a lot, I think, depending on the book. $\endgroup$ – kimchi lover May 10 at 20:50
  • $\begingroup$ Thanks for the replies. Am using Robert Ash Probability and Measure Theory. Is there any more intuitive complement book that you would suggest? $\endgroup$ – Kevin May 10 at 20:52
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    $\begingroup$ I think there are two different modes of learning math, and both are vital but they are somewhat at odds with each other. Mode 1: Trying to grok the proofs and generally understand the material as deeply as possible. This is a slow process. Mode 2: Learning in a "big picture first" style. You look at a map of the world before you commit yourself to learning your way around a particular city. You get a high level, coarse picture of how a subject fits together before learning proofs in detail. Balancing these two modes is a difficult question of calibration / hyperparameter tuning. $\endgroup$ – littleO May 11 at 5:42
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    $\begingroup$ I think it is good practice to think about a proposition a bit before reading the proof. You don't always have to prove it yourself, but putting yourself in the problem-solving mindset very much aids in an intuitive understanding of the proof when you do read it. Think about how your attempt at a proof might start. $\endgroup$ – Jair Taylor May 11 at 7:33
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With regards to learning from proofs provided in a textbook, it may be very beneficial to

  • (step 1) mentally try to guess how a proof is going to substantiate its claim
  • (step 2) read the textbook-proof
  • (step 3) write a personalized proof while having access to the textbook-proof emphasizing clarity
  • (step 4) write a personalized proof while having access to the textbook-proof emphasizing brevity
  • (step 5) write a formal proof without having access to the textbook-proof.

Here "formal" means "able to be understood by other mathematicians---most importantly your instructor."

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