I wrote this article in 2013 at the request of Parul Mittal for her site on education, Rivokids.
In my last article, I talked about what draws mathematicians to their subject: it’s not its usefulness, though of course math is incredibly useful. Rather, we do it for the best of all possible reasons: because it’s fun and we enjoy it. We actually like thinking about mind-bendingly abstract, incredibly difficult puzzles for years at a time.
Now that may be fine for us professionals. But why should you do it? Once you are numerically competent and quite comfortable with all the math that you require in daily life (and perhaps, if you are an adult, all that you require for the technical aspects of your profession) why should you go beyond that?
By math here, by the way, I don’t mean more school math. I don’t want you to learn more algorithms and to do more drills! I want you to contemplate mind-expanding abstract ideas and gems of logical reasoning. Presented in this way, I think it doesn’t need much justification to pursue. But since I asked the question – why should you do math — let me answer it.
There are many possible answers. Here’s one: math contains a repository of some of the most beautiful and powerful ideas that humans have come up with. You would want to learn about them for the same reasons you might want to read the world’s great literature, or listen to a musical masterpiece. There are masterpieces in math too! We gain access to art or music through our visual and auditory faculties; to gain access and to appreciate a piece of mathematics, we use the mind. We engage, specifically, the faculty of reason. In this sense, a piece of mathematics is closest to literature – you have to read it, work through it. Some masterworks are too technical for a layperson to fully appreciate (which is also true, by the way, of the arts), but there is much that is entirely accessible.
Here’s another reason to do math: if you cultivate the skills required for mathematical thinking, you are really equipping yourself for problem solving in all kinds of contexts. A clear understanding of what is assumed and what is not. A rigorous analysis of arguments. The ability to communicate a complex argument to others. When you engage with a serious mathematical problem, it’s not going to yield immediately. You will encounter dead-end after dead-end (all of which is a necessary process), and then you will be forced to innovate and be creative. You develop the ability to think about a difficult problem in a sustained way — not for years, but perhaps a week? And is this not wonderful preparation for life? After all, the problems you encounter in school are designed to yield within a few minutes, but no serious problem we encounter in our personal or working lives will need less than a week of thought, or yield on the first attempt.
So how can you do math – where are you to go to get this exposure? You do need a good guide — you can’t get into it just by standing in front of, let’s say, a theorem, and looking at it, or by lying on your bed with your eyes shut listening to a cd of math, assuming you can find one. Math, as they say, is not a spectator sport – it takes active involvement on the part of the audience (i.e., you). Several mathematicians have written wonderful books for lay audiences, any of which would be a good place to start. Keith Devlin, Steven Strogatz and Paul Lockhart are all good choices. (If you look for their books on Amazon.com, you can usually take advantage of the Look Inside feature to read a few pages.) Keith Devlin also offers a free online course on mathematical thinking, available to anyone in the world – you may want to check it out on Coursera.
The course and these books I’ve mentioned are all for an audience of older teens and adults. If you are looking for mathematical activities for younger children, here’s a resource for parents and teachers of K-12 students: www.mathpickle.com. It’s so named because Gordon Hamilton, whose site this is, wants you to “put your students in a pickle” by giving them something difficult and meaningful to figure out. He is also a great proponent of board games that cultivate logical thinking and analysis. Games and puzzles are of course a great way to develop problem solving skills, and the best thing about them is that you don’t have to make a big project out of mathematical thinking. You can simply do it for the best possible reason: because it’s fun.