Looking for a book for mathematical thinking

Question

High school student here just wanting to be better at maths and to know how to approach and solve problems and also how to think like a mathematician. Please recommend a book to me it would help a lot.

Answer 1

I strongly recommend Polya's Induction and Analogy in Mathematics.

https://press.princeton.edu/books/paperback/9780691025094/mathematics-and-plausible-reasoning-volume-1

You can read the preface here.

You might be able to find the full text on line, but the real book is worth having.

Answer 2

Mathematical thinking is nothing you can just get inside a book. Maybe there are some of those books, but it really depends on the topic you are interested in. Knowing how to approach and solve problems is a task that needs years to get better and you will never be perfect at it.

You could start with a book about (Linear) Algebra or Analysis. However, I would suggest, that you maybe take a look at a lecture for beginners from a renowned university (e.g. there are free lectures from MIT, Princeton, ...) and try to figure out what's happening there.

Starting directly with a big book about Analysis can be very hard and frustrating.

I visited one lecture on Analysis 1 at university while I was in high school. That gave me great insights and really helped a lot in understanding mathematical thinking. As I started the university full time, I already got a great advantage over the other students. I also bought a book about Analysis, but that didn't really help me.

If you really want a book, I would suggest taking a book with a lot of examples and exercises so that you can apply what you learn

Answer 3

When I was a senior in high-school (1964/65) I had the same question, the same desire to learn more of mathematics and how mathematicians think. My math teacher gave me a book that fulfilled my quest and I still have that book today. At the start, a lot of the book was beyond me but I would often refer to it again and again as I made my way through college as a physics major.

The book is "What is Mathematics" by Richard Courant. I think there may be another author today or maybe even then but I am too lazy to go look. I know it is available via Amazon because I bought a copy for each of my high-school aged grandkids a couple of years ago.

It is not a "how to think like a mathematician" book but it is about mathematics and you learn how mathematicians think about these math subjects by reading the book. And, of course, reading many other books too.

Answer 4

Both Courant and Polya's books suggested above are excellent books you should read anyway. Another such book is Gowers' The Princeton Companion to Mathematics available here. However, I fear they might not be sufficient. I'd read these books before starting college and yet they didn't really prepare me for high school calculus being so vastly different from Calculus from Apostol's book.

The book that actually taught me what pure mathematicians care about, and convinced me that it was worth studying, was Tao's Analysis I. If you're going to study math, you will do real analysis (probably pretty early into the course), and Tao's book will make life a lot easier. The MAA sells the book in the US at this site, and I'm sure softcopies are not hard to come by online.