What is mathematics?

I am a mathematician and math teacher and I’ve often reflected on the question: What is mathematics? You would think I have a clear and ready answer to that question, but I don’t really. If you asked me that question I would likely give some esoteric answer, the same answer that would answer What is poetry? or What is music?, namely, something along the artful lines: that which is pleasing and uplifting to the human spirit. And then I would add some begrudging nod to the practical utilitarian side of mathematics, commenting on the computation and algorithm — the reckoning taught some 500 years ago and still experienced by most in school today.

How has human context shaped mathematics?

My context for learning and doing mathematics is, by-and-large, simply for the intellectual joy of it: puzzling out the mystery of numbers and relations, and enjoying the “buzz” of finding unexpected connections. As Gabrielle Emanuel shows in this July 2016 article, Why We Learn Math Lessons That Date Back 500 Years, the context for learning mathematics has been clear and direct in the recent past.

Is the context for learning mathematics still just as clear today?

After all, surely we each learn best if we are motivated by some personal question, need, or yearning.

I doubt that there is some single, overarching, good-for-all, motivation for studying the entire K-16+ mathematics curriculum. But I do know that mathematics curriculum is chock-full of moments to motivate, intrigue, or make you go “huh?”, no matter your personal style and taste.

Using your questions to motivate your learning

  • You want to impress your friends by doing three-digit multiplications in your head? Sure! Go ahead. What then is an efficient and easy-to-remember algorithm for doing this?
  • I personally have no interest in computing 273 time 992 (and if you insist I do it, I would pull out my smart-phone). But I do find it philosophically curious that “273 groups of 992” should give the same number of objects as “992 groups of 273.” Why should that be so?
  • And why is multiplying a number by itself called “squaring”? That’s a geometry word!
  • The ancient Greeks, with their advanced geometry and number work, failed to discover algebra. Why? That seems odd.
  • Actually, in algebra, is it even meaningful to write x² + x (that is, x squared plus x). Isn’t that an area plus a length?

The story of mathematics

Mathematics is a fundamentally human endeavor — created by humans, for humans, and is replete with humanness. All the practices and quirky jargon of mathematics have a human context and a human story.

And we humans enjoy stories. We are each intrigued and puzzled by different aspects of a story. So let’s teach mathematics with the honest context of human story in mind too.