How To Study As A Mathematics Major

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1. mikevmMar 2018
   Ok, I'll take a crack at this:

   Up to high-school level:

   1. Precalculus: Precalculus: A Prelude to Calculus - Axler

   2. Calculus: The Calculus Tutoring Book - Ash.

   College:

   3. Preparation for Collegel-level maths:

   3a. General prep for high level maths: How to Study as a Mathematics Major - Alcock

   3b. Proof writing: How to Prove It - A Structured Approach - Velleman OR Book of Proof (2nd ed) - Hammack (it's free!)

   4. Mathematical Analysis:

   4a. Good prep for Analysis: How to Think About Analysis - Alcock

   4b. Understanding Analysis (2nd ed) - Abbott OR Yet Another Introduction to Analysis - Bryant (has full solutions) OR The How and Why of One Variable Calculus - Sasane OR Mathematical Analysis - A Straightforward Approach (2nd ed) - Binmore (has full solutions)

   5. Discrete Mathematics (a combination of set theory, combinatorics, a bit of discrete probability and graph theory): Discrete Mathematics - Chetwynd, Diggle

   6. Linear Algebra: Linear Algebra - A Modern Introduction (4th ed) - Poole

   7. Probability: Introduction to Probability - Blitzstein, Hwang + online course https://projects.iq.harvard.edu/stat110

   8. Statistics: (for Bayesian) Statistical Rethinking - A Bayesian Course with Examples in R and Stan - McElreath + online course https://www.youtube.com/playlist?list=PLDcUM9US4XdM9_N6XUUFr...

   Usually you'll be doing courses on #4, #5, and #6 simultaneously.

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2. gtaniFeb 2018
   Some books on proof, theorems/axioms, set theory, epsilon/delta/continuity/limits/differentiability, natural/rationals/reals/countability etc before heading into your first proof based LA or analysis sequence:

   - Kevin Houston "How to Think Like a Mathematician"

   - Keith Devlin "Intro Mathematical Thinking"

   - "How to Study as a Mathematics Major" Lara Alcock (for some reason, she/Oxford Press has 2 books with seemingly identical content under Math Major and Math Degree titles)

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3. jyunwaiMay 2024
   A lack of necessary prior knowledge is often a major reason for struggling with a problem. As a relatable example, suppose you're taking an exam. There's a problem near the end that you don't know how to solve—and the reason is that you haven't studied that topic enough. No matter how smart (as in, fast at learning) you are, you need to practice with similar or related problems to solve that issue.

   But suppose you're outside of an exam environment and have time to look up the relevant material. I've known a PhD candidate in a non-mathematics field who had to find a mathematical solution to a certain research problem. That person is smart but still needed a few months to learn the mathematical fundamentals to understand and solve the problem. In contrast, someone with a math background could have solved this far more quickly. But that person would have taken at least some months to get up to speed on the research literature for the non-mathematics part of the research problem, in order to properly understand its constraints and bigger-picture significance.

   Lara Alcock's book "How to Study as a Mathematics Major" touches upon this topic more directly. She encourages readers not to be too intimidated if other students in a course seem really smart: much of the time, the reason is not due to an innate difference in smartness, but rather prior exposure by other students to concepts in the course. Students who seem to find the material effortless often have already studied many of the topics in another course or could even be retaking the course after a previous attempt.

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