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Advice for Students

Mar 8, 2023

This is a work in progress. 2023/03/08

Students often ask me for advice on how to improve in mathematics. I find myself recommending the same strategies across a broad range of courses. These strategies may or may not work for you. Try them out in whatever order suits you best. If any of these things help, or you have a strategy that you think should be on here, please let me know.

  1. Be Kind to Yourself
  2. Time Block
  3. Watch Your Self Talk
  4. Read Slowly
    1. Read with a Pencil and Paper
  5. Identify The First Tricky Thing
  6. Teach Other People (or Rubber Duckies)
  7. Make Summaries or Mind-Maps
  8. Learn to Ask Precise Questions
  9. Keep Seeking Help
  10. General Life Advice
    1. Meditate
    2. Exercise
    3. Appreciate Those You Love

Be Kind to Yourself

Studying mathematics is hard. Being a student of mathematics is even harder. It is so easy to get bummed out and think that you will never make any progress. We all make errors in mathematics. However, errors lead to learning and progress. From time to time, you should acknowledge that you’re doing an awesome thing by learning mathematics. Be kind to yourself, forgive your own errors, and acknowledge that hard things are difficult.

Read Slowly

When you’re reading the textbook, or sources online, read slowly. There is a great quote from Bill Thurston, which talks about the rate at which we need to read mathematics.

I was really amazed by my first encounters with serious mathematics textbooks. I was very interested and impressed by the quality of the reasoning, but it was quite hard to stay alert and focused. After a few experiences of reading a few pages only to discover that I really had no idea what Iā€™d just read, I learned to drink lots of coffee, slow way down, and accept that I needed to read these books at 1/10th or 1/50th standard reading speed, pay attention to every single word and backtrack to look up all the obscure numbers of equations and theorems in order to follow the arguments.

When you’re reading material in mathematics, expect it to take a long time. This is common. You’re not a slow reader. Mathematics is just very very compact and requires careful attention to detail.

Read with a Pencil and Paper

To help stay active while reading, keep a pencil and paper handy. Paul Halmos has some good advice here. I think it is particularly important to work with physical paper and pencil.

Reading with pencil and paper on the side is very much better - it is a big
step in the right direction.  The very best way to read a book, however, with,
to be sure, pencil and paper on the side, is to keep the pencil busy on the
paper and throw the book away.

Time Block

Watch Your Self Talk

Identify The First Tricky Thing

Teach Other People (or Rubber Duckies)

Make Summaries or Mind-Maps

Learn to Ask Precise Questions

Keep Seeking Help

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