Or, a Dispatch from the Trenches of the “Math Wars”
Dear Benedict Carey,
I very much enjoyed your book How We Learn. It blends the vast and varied harvest of research on learning into something light, flavorful, and nutritious. A psych-berry smoothie, if you will. It’s a lovely summer read for a math teacher like me.
But I’m also a blogger—which is to say, a cave-dwelling troll, forever grumping and griping. And so I’d like to dive into your chapter on practice (“Being Mixed Up: Interleaving as an Aid to Comprehension”). In it, you purport to remain impartial in “the math wars,” but it’s my view that you come down distinctly on one side.
It’s towards the end of the chapter that you hit the culture war in mathematics. On the one hand (you explain), there are top-down progressives, who urge children “to think independently rather than practice procedures by rote.” On the other, you’ve got bottom-up conservatives, who put their faith in “the old ways, in using drills as building blocks.” All fair enough.
Then, however, you land on a rockier claim:
This clash over math was (and is) about philosophies, and in math of all subjects it is results that matter, not theories.
Maybe so. But it’s far from obvious which results matter. That depends on what philosophy you subscribe to.
For example, one teacher might care a lot about standard textbook questions like these:
Another might scowl and spit upon those, preferring more open-ended questions:
Others might embrace all these questions. Or despise them all.
The disagreement isn’t just about the most effective path to excellence. It’s about what “excellence” even means.
At one extreme, you can view math as a fixed and specific body of knowledge, a pre-existing library of tools, techniques, and numerical skills. Or, at the other extreme, you can view mathematics as a frame of mind, more about habits of thought and broad problem-solving approaches. Like most teachers, I fall somewhere in the complicated middle. I have my own views about which tasks are meaningful, and which are rubbish—which are math, and which are mush.
In the end, it’s silly to say that “results” matter and “theories” don’t. Our choice of the former is inextricable from the latter.
Next, you dive into a study (promising that it consists of “real math” and implying that it ought to transcend the debate between mathematical progressives and conservatives). In the experiment, kids learn rules for computing the numbers of faces, edges, corners, and angles that a prism has, given the number of sides in its base:
Unfortunately, this is a task that many progressives would reject outright.
Here, students must recognize a code word (face, edge, corner, or angle) and match it with an arithmetic operation (+ 2, x 3, x 2, or x 6). To me, this represents a narrow and overemphasized aspect of mathematics: signal-triggered computation. You characterize it as high-level (“We are not only discriminating between the locks to be cracked; we are connecting each lock with the right key”) but I suspect I’m not alone in considering it rather drab and rote, not to mention irrelevant to the primary purpose of mathematics education.
What’s the alternative?
Well, first off, mathematics is about framing questions. So what is this “prism” thing we’re discussing—not to mention “faces” and “edges”? What definitions are we using? What properties are we focusing on? How are these 3D objects similar to (and different from) 2D objects? Are there borderline cases that are hard to categorize? For example, which of the following shapes would count as “prisms”? (And are there any that aren’t prisms, but still follow the rules?)
Now, as for those rules: sure, it’s nice to use them, but can we explain why they’re true? Could we have uncovered them for ourselves? Can we test the boundaries of their applicability, rooting out exceptions, or proving to our satisfaction that no exceptions exist?
These are the mathematical activities I care most about.
I don’t mean to hate on the study. Yes, I find the specific task pretty silly, but I buy the essential conclusion that interleaved practice (where you mix all four types of questions together) is more effective than blocked practice (where you practice each type in isolation). Insofar as my teaching is conservative and skill-focused (which it sometimes is—certain skills, I believe, demand automaticity), that’s useful guidance.
But I worry about your presenting this as a consensus vision of “real” math. Many of us believe that math is about far more than memorizing and applying procedures, just as we believe that a song consists of more than its bass line. It’s a shame to see someone trying to transcend the debate over the aims of the math curriculum, but then focusing entirely on how to accomplish the aims championed by just one “side.” It’s if someone said, “I won’t take a position on whether aliens exist. Now, here’s a method for contacting extraterrestrials that’s 39% more effective.”
I should be clear: This matter occupies only a few pages of what is, to be clear, a fun and practical book, from the story of Winston Churchill failing his classics exam to your delicious images of the creative process (“For me, new thoughts seem to float to the surface only when fully cooked, one or two at a time, like dumplings in a simmering pot”). So I thank you for the enjoyable book you’ve written.
In any case, I hope I’ve been able to offer a more detailed (if not entirely novel) perspective from the muddy trenches of these ongoing “math wars.”