How can I improve creative thinking in Math?

Question

I'm a fourteen year old in the USA, who is currently in 8th grade, about to graduate in about 1 month. Currently I'm learning algebra. I have enjoyed math and recently I realized that math is not all about plugging in numbers. But I now, realized that math is a creative art, about finding different ways to solve a problem, and I want to find a way to think like how a mathematician thinks, but it's been hard. I have searched for resources like Polya's How to Solve It, and the Art of Problem Solving books, and even this site but it hasn't been sticking to my head. I ask this question specifically because I want to learn physics better and generally have a better grasp at math.

Answer 1

I had been doing competitive math when I was your age. Back then I could solve about 2/3 of all the competitive problems I was given (I called them the "easy (competitive) problems") but the others were simply totally out of my reach. I could spend 2-3 days thinking on a "hard (competitive) problem" without much progress. Then in a lower group were the problems in my textbook - these were "uninteresting" to me, I could solve them all.

So 3 groups in total: uninteresting (just exercises), easy competitive, hard competitive.

Of course for my classmates who were not very interested in math
(but still perfect students), even the "easy (competitive) problems" were not easy.

So I felt like there was a sharp division, either I could solve a problem in 1-2 hours or so, or I could think on it indefinitely and not solve it. So I felt like I could only solve the "easy (competitive) problems" which made me feel stupid (despite understanding the note in italic above).

So... two advices:

1. Read some theory too (or have someone teach you that theory, even better). That helps a lot and arms you with more powerful "weapons".

2. Think hard on hard problems, try 1-2-3 approaches, whatever you can invent yourself. Then if you still have no ideas how to solve some problems, or if you have ideas but cannot quite finalize them, read the problem solutions and make sure you understand every statement in the solution, every detail in the derivations.

If you do these two things long enough (say 2-3 years), you will notice that your definitions of "hard problems" and "easy problems" have shifted in the direction you want.

And I could argue that means your "creativity" has increased.

A few books to check (in fact any good book on competitive math would help):

https://www.amazon.com/USSR-Olympiad-Problem-Book-Mathematics/dp/0486277097/

https://www.amazon.com/Challenging-Mathematical-Problems-Elementary-Solutions/dp/0486655369/

https://www.amazon.com/Challenging-Mathematical-Problems-Elementary-Solutions/dp/0486655377/

Answer 2

One of the most important tools of a mathematician is having seen a wide variety of mathematics. Search for some recreational math and other accessible math topics online. Back in my day, I read random articles on Mathworld (https://mathworld.wolfram.com/cgi-bin/random.cgi). Even though most of the pages didn't make sense at the time it helped me familiarize myself with terminology and get a sense of what the field as a whole looked like.

Answer 3

I encourage you to twin Mathematics and Computer Science.

Programming with high level languages such as Mathematica will allow you, in particular by producing appealing graphical results, to stimulate your creative spirit, will help you to build conjectures, to ask you new questions.

I insist on the graphical representations. An example among many, I just wrote 20 minutes ago this answer dealing with spirals and complex numbers and geometrical series, with graphics that have taken me 5 minutes to produce. The graphical image of the spiraling convergence/divergence of a geometrical series complements the algebraic knowledge I have of it.