I wrote a blog post a few days ago and I didn’t say what I meant to say. Here’s a second try.
I stumbled across a Twitter conversation today in which someone was arguing that we should teach complex numbers to 3rd graders. I won’t link to it, I’m not trying to shame anyone. Finding ways to explore “higher math” is an instinct I’ve had many times in my career, and that I’ve seen many other ambitious math teachers have. We have some idea of what the “rich” or “beautiful” math is, and we want to get to it faster! I think lots and lots of math teachers have had the instinct, “this math is so incredible, I want my students to see it!” Here’s the thing. The full sentence is actually, “I find this math incredible, I want my students to see it.” Humans (me, you, everyone else) are irreparably self-centered, and we have a hard time seeing things from someone else’s perspective.
Here’s what I meant to say in my blog post but didn’t: There is incredible richness and complexity in regular old school math that I think math teachers often take for granted. I wrote about these five equations:
I think many math teachers would agree that the best way to solve equation number one is to multiply both sides by 3. Some might appeal to seeing the equation as one-third of x is equal to 12, which one could argue is a different way of saying the same thing. But equation number two! That’s a zinger. You can solve it by multiplying by 3 also — two strategies could be seeing that it’s equivalent to the first equation, or knowing that 3 is the reciprocal of one-third and one can multiply by the reciprocal to make one. But in 7th grade, where we expect students to solve equations like this, the emphasis is on inverse operations. And that can really take students for a ride! The operation here is multiplication by a fraction, so inverse operations means dividing by that fraction. Again, if you have a good understanding of reciprocals this isn’t too hard — but lots of 7th graders don’t! It’s not intuitive for most 7th graders that dividing by one-third is the same as multiplying by 3, and lots of 7th graders would tie themselves in knots trying to get there. There’s a ton of math here, and I’ve only gotten through two “one-step equations.” (If you’d like an interesting intellectual exercise, pick your favorite method for each of the next three equations, then try to “break” it by finding a new equation for which your method is either very inefficient or doesn’t work. Also, for a gripping tour of the underlying math, Ben Blum-Smith’s blog post is a must-read.)
My point isn’t to argue for one method or another or to get in a back-and-forth about equation solving. My point is that there is way more complexity in solving these equations than one might think. Your typical math teacher could solve any of these in seconds without putting a pencil to paper. Lots of folks would argue that this math is “simple” or “rote,” but I’d argue it’s anything but.
Rather than searching for new, sexy math for students to learn — math that adults find beautiful or worthwhile — let’s spend more time plumbing the depths of school mathematics. What I described above is something most middle school math class sprint past without thinking twice, always in a rush to get somewhere else. There’s a lot there, and those of us with too much mathematical knowledge and experience are likely to overlook it.