Last week, Dan Meyer invited the folks at Mathalicious to opine on the meaning of the phrase “real-world,” not as it applies to MTV shows (though that would make for a great conversation), but as it applies to questions asked of students in a math classroom. This week, we responded, continuing what I believe to be an important and interesting discussion about the nature of what we mean when we demand that mathematics be made more “real” for our students.
Most of my thoughts on the subject are encapsulated in the Mathalicious response. (Both articles come highly recommended, and what I say below may not make much sense if you haven’t read them first.) The conversation got me thinking, though, and so I’d like to offer my own personal aside/addendum.
When I began writing in this corner of the internet in the summer of 2008, my goal was simply to talk about mathematical ideas in a way that was accessible for a general audience (and in particular, an audience that didn’t necessarily think of itself as mathematically inclined). I believed that talking about math through the lens of popular culture would help with this, because, in my experience, most people (a) prefer pop culture to math, and (b) don’t believe that the intersection between the two is all that rich. I probably haven’t always been successful in hitting the sweet spot in the intersection between pop culture and mathematics, but those successes have brought with them a not-insignificant sense of pride.
I’ve measured that success over the last several years, in part, by how accessible I’ve made some bit of mathematics to someone who wouldn’t have otherwise been interested. (I realize this is perhaps one of the least useful metrics ever conceived; I don’t even know what the proper units of measurement would be.) To put it another way, my goal has been to make math more “real” to people both in and out of the classroom. When I say “make math real,” I mean something like “make the beauty of mathematics more transparent,” though of course this simply replaces one squishy word (“real”) with another one (“beauty”). I guess what it boils down to is trying to imbue people with some of the joy for mathematics that I feel (on most days, anyway).
The world external to mathematics, if done properly, is frequently a resource when working towards this goal, since people have a variety of interests, not all of which are (obviously) mathematical in nature. This is why the work I do, both here and at Mathalicious, so frequently begins with something that is about the world at large, rather than about prime numbers or the Riemann Hypothesis (both of which are also things that I love dearly). Having said that, most of this work, for me, is in service of having great conversations around interesting problems, and strictly speaking there’s no requirement that the world external to mathematics make an appearance. Indeed, I don’t believe such externality is necessary in order for a task to be “real.” Beauty doesn’t discriminate when it comes to the origin of interesting questions. Sometimes, great mathematical questions are inspired by something external to mathematics (the search for these questions occupies a nontrivial amount of my time during the day). Sometimes, great mathematical questions are inspired by mathematics alone.
And sometimes, mathematics inspires “real-world” applications unbeknownst to the original authors. Elliptic curves were studied for hundreds of years before possible applications to cryptography were discovered by Neal Koblitz and Victor Miller in 1985. It’s conjectured that the zeros of the Riemann zeta function satisfy a property which was first discovered in the context of the quantum mechanics. In other words, there’s precedent for deep, pure mathematics to have unexpected connections to the “real” world around us. Or, to put it another way, the intersection between the external and internal worlds of mathematics is likely larger than any of us anticipate.
This is why, for me, keeping a task “real” in the sense I’ve described above is more important than keeping it “real” in the sense of drawing from something external to mathematics. A “fake” task, in my view, is one which doesn’t fit properly in either the internal or the external world (e.g. version C in the problem described by Dan). To the extent that I have no idea how deeply the internal and external worlds of mathematics intertwine, I can only do what I think is best: get equally psyched for both. And on days when I get other people psyched as well, all feels right the world. (Or worlds. Depending on your interpretation.)