How to Practice Mathematics as an Art

A couple years ago, I was truly fascinated by 3Blue1Brown's videos. It was actually his video on an unexpected result that came from colliding blocks that really captured me in the art of mathematics. I couldn't believe that two simple blocks would produce $\pi = 3.1415926535897...$

As Grant explains in his video, the result comes from an observation that putting the mass and the velocity of the blocks onto an axis results in a circle. I won't go into the details here, but this is what I am talking about: trying, trying, and finding some observation (maybe more) that allows the problem to be solved. Now, I will list some resources for the discussion, and as such, this post will be some sort of review.

3Blue1Brown

Grant has been quite influential in mathematics education on YouTube and in classes, as most of his videos are shown around the world in classrooms. The biggest reason for it is how Grant explains and teaches: he uses animation and visualisation very effectively, which allows people to actually understand what he is saying, instead of it being "theoretical" and just "on paper."

In addition to his one-off videos, I would recommend his Essence of Calculus and Essence of Linear Algebra. Especially if you take a course around calculus, his videos around it are phenomenal, as they actually capture the ideas and observations that we make about calculus and the "why."

The Art and Craft of Problem Solving by Paul Zeitz

Mathematics is the art of problem solving—there is no way around it—and this book by Paul Zeitz is a perfect introduction to the art. Paul Zeitz prefaces his book by looking into the difference between "problems" and "exercises," where the latter is how mathematics is taught in schools. Zeitz continues with problem solving techniques, including trial-and-error and wishful thinking. Although Zeitz writes this book for students aspiring to start mathematics Olympiads, I think it's applicable to everyone. I, too, looked into the book from an informatics olympiad perspective. Furthermore, the book has chapters on different areas of mathematics, such as number theory, algebra, calculus, and geometry, and how to solve specific problems in them, with many problems attached at the end of the chapters.

I also want to get your attention to Paul Zeitz's foreword to the 2011 Math Survey. It is directed at to-be-mathematicians—about getting lost, getting obsessed, and getting together, which I also mentioned in my first post of the trilogy.

I do think that these two are good steps into practicing mathematics as art, but I also want to mention some other resources that I recommend.

• "How To Solve It" by G. Polya: It is similar to the book by Zeitz in the sense that it talks about problem solving.
• "Proofs" by Jay Cummings: A great textbook introduction to mathematical proofs.
• "Book of Proof" by Richard Hammack
• "My Best Mathematical and Logic Puzzles" by Martin Gardner: Great problems and puzzles that don't need higher mathematics.

And that's about it. While these resources may be of help to many, the most important thing is to just solve some problems and have fun.

Made and written by Mehmet Kutay Bozkurt. All rights reserved.