# How can I improve creative thinking in Math?

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.

• Use the three steps method: 1. practice; 2. practice; 3. practice.
– user65203
Apr 27 at 20:37
• @YvesDaoust In my opinion, it works better if you switch steps $2$ and $3$. Apr 27 at 20:38
• @TheSilverDoe: yep, that's even better.
– user65203
Apr 27 at 20:39
• The creative process usually comes with experience, the experience of solving tons of problems. What is usually perceived as a stroke of genius is really the accumulated know-how of solving problems. Apr 27 at 20:43

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).

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/Challenging-Mathematical-Problems-Elementary-Solutions/dp/0486655369/

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

• +1 for the comment at the end. "you will notice that your definitions of 'hard problems' and 'easy problems' has shifted in the direction you want" is very, very accurate. Apr 27 at 20:51
• @EdwardEvans Well, thanks, I am just saying it from personal experience. We all have our limits and they are different for the different people (the limits of our minds, and of our thought processes), but if we push ourselves, then the dividing line that I am talking about is shifting in our favor. I think everyone would agree. Apr 27 at 21:10
• @peter.petrov When you talked about learning about theory, could you be more specific? Do you have any book recommendations for studying theory, or will the books on competitive math help me go in the path I wanted to go? May 10 at 22:38
• @Plushiewaivee Go through the competitive math books. If you cannot solve a problem and you cannot understand the solution then it means you lack some theory. You can then lookup that theory and learn (on demand). May 12 at 13:14

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.

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.

• Programming is not computer science - it's programming.
– Nij
Apr 28 at 4:43
• @Nij Except that they're not. Programming, computer science, coding, software engineering - they're all different words for the same fundamental thing: writing instructions to get computers to do the things you want them to do. Apr 28 at 7:17
• No, programming is writing the instructions for the computer. Computer science is the study of whether instructions even can be written, which kinds of instruction lists are faster/use less memory/can be written using which other set of totally different instructions, and more.
– Nij
Apr 28 at 7:26
• @Nij Consider that English is not my mother language. I thank you for the explanations you have given. Maybe I should have said "using in particular CAS (Computer Algebra Systems) like Mathematica but as well other numericaly oriented software" Apr 28 at 7:42
• @Nij An afterthought: The word "informatics" which encompasses all theses different domains that is now dominant in many languages (informatique in French, informatik in Germany, informatica in Italy...) looks to be rarely employed in English. Am I right ? Apr 28 at 12:28