Writing mathematics one is only slightly familiar with

Question

Reading Paul Halmos' book I want to be a mathematician, he emphasizes early his personal experience with writing mathematics - using long, clear sentences, claiming that it is more beautiful, engages the reader to a greater degree and allows the writer to express his ideas more consistently. This is something that I agree with, and writing mathematics in a structured way has never been more easy, given the accessibility of $\LaTeX$.

However, I am not sure how useful this would be for me, as an undergraduate. I fear that I will end up wasting my time - that my writings will be nothing but thin summaries of the text. If this is the case, could it still be (personally) fruitful? After reading a (sub)chapter and gaining experience with the topic through problem-solving, is one capable of writing an independent text on the topic?

I am aware that this question has no clear answer, thus fails to meet the norms on math.SE. I hope that some free discussion will be tolerated, as I feel that it is a very interesting topic that many undergraduates (or even graduates, not that I know too many) struggle with.

Answer 1

I've often suggested to students (who want to improve their comprehension of the mathematics they are learning) to "rewrite the book" in their "own words", so to speak. That doesn't require that you eventually publish your rewrites, but it gives you the opportunities to

• answer, in writing, your very own questions: the ones that pop up while first reading a chunk of the material,

• meticulously work through the text's proofs after reading them carefully, to test how well you understand each of them, and/or to help "fill in the blanks" that help you to understand a given proof.

• work out the definitions and theorems you are learning in a manner that makes sense to you or in a manner you think would be helpful in teaching or tutoring a classmate,

• and ultimately, to consolidate and reinforce your understanding of the material.

I've actually done this with class notes, too, and the truth is, I rarely had to go back to reread my rewrites: the very act of re-writing the notes, in my own voice, helped me "own" the material.