How to Read a Math Paper
Every year, after the PRIMES program begins, I send a letter to our students about how to read a math paper. The students in my group are juniors just starting their research. They are often required to read advanced math papers—frequently the first research papers they have ever encountered. This year, I decided to post my letter online, in case it might be helpful to other aspiring mathematicians.
Dear PRIMES and PRIMES-USA students,
Reading math papers can be very difficult and overwhelming. I remember trying to understand every single word of my first research paper and getting stuck on the first paragraph for a long time. That was a mistake. I regret that no one ever taught me how to read math papers. As the joke goes, “There are only two kinds of math books: those you cannot read beyond the first page, and those you cannot read beyond the first sentence.”
Math papers are not stories. They are not meant to be read linearly from beginning to end. Depending on your goal, you read different parts in different ways. Here are some examples.
Goal: Decide whether to read the paper.
Read: The abstract and parts of the introduction.
Goal: See what was accomplished.
Read: The introduction, or locate and read the main theorems.
Goal: Learn a method that might be useful.
Read: Find the relevant method and focus only on that section.
Goal: Get a general idea of the topic.
Read: First understand the structure of the paper. Then try to grasp the main statements at a high level.
Goal: Master the topic.
Read: Read the paper several times, going deeper with each iteration. Try not to get stuck on a sentence; you might understand it on another try. Here is a potential list of objectives for each iteration: you can adjust them and change their order according to your needs.
- First read: understand the structure and the big picture.
- Second read: understand the definitions and main notions.
- Third read: understand the main statements and look at small examples.
- Fourth read: understand the ideas behind the proofs.
- Fifth read: go deeper and start reading the referenced papers.
- Sixth read: try to reproduce the proofs.
Goal: Check for acknowledgments.
Read: The acknowledgments and citations.
The main rule is to keep your goal in mind while reading a paper. If you do not have a specific goal, ask your mentor to suggest exercises or questions to guide your reading. Try not to feel discouraged if you don’t understand everything: the joke at the beginning of this essay implies that everyone has trouble understanding math papers.
Tanya
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