*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.

Framing questions.

Reasoning.

Contrasting.

Extending.

Explaining.

Understanding.

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.”

Best,

Ben Orlin

Nice post.

I love how you are able to coherently describe the conflict that most math teachers feel. “These are the things I care about, but sometimes doing these things requires automaticity.” That’s the battle right there summed up in one paragraph. If I ignore rote memorization and automaticity in the wrong areas, will it impede my students’ collective ability to reach toward my real goals? What is the proper balance?

Outstanding thoughts.

Thanks, I’m glad that dilemma came through. I find it easy to get carried away with seething critiques of skill-oriented instruction, but the fact is, I want my students to develop skills. It’s no easy task to pinpoint which skills matter (solving simultaneous linear equations, yes; solving cubic equations, no; solving quadratic equations by means other than the formula, I’m not sure). But at least it frames the problem: as you say, practice the specific skills that are fruitful for problem-solving, and ditch the ones that are more “for their own sake.”

The bit about the dumplings reminds me of the famous comparison of the Sun’s surface in detail to “rice grains floating in a bowl of soup”; this despite the well-known fact that rice grains

sinkin a bowl of soap (though convection may stir them up while the soup is being boiled in a pot). But if they did float, they’d look like the surface of the Sun.Mmm, the Chinese place near my old apartment did a good sizzling rice soup. They should call it Sun Surface Soup. I’d have eaten even more.

I love good imagery, and somehow it’s even more fun when the imagery clashes with actual reality.

Even if one’s memory is atrocious it’s no good if one cannot remember what it is that one has forgotten, or never successfully learned in the first place. I know of the existence of a formula for sin(a) + sin(b), but I have to start from sin(p+q) and cos(p+q) and reconstruct it, EVERY TIME !

That’s half the fun of all those trig formulas, isn’t it? 🙂

Teaching trig has gotten me remembering more of those formulas, but I still haven’t memorized the ones turning sums to products (or products to sums). As you say, it’s a fun little exercise to derive them. Or takes 30 seconds to look them up.

What is the best “how to learn” book you’ve seen? Maybe this is it?

From your description of the math bit, it seems to miss an essential starting question: what do you want to learn? Presumably, there are categories of answers which lead to strongly different recommendations. For example:

– learn to ride a bicycle (or other skill): i) break the skill into key component sub-skills, ii) find activities that let you practice those subskills in isolation, iii) find activities that let you practice combinations of subskills, iv) some roadmap for when/how to deliberately practice those activities

– learn the capitals of all countries in Asia: i) a bunch of mnemonic techniques, ii) a schedule and workout regime for implementing those techniques

– learn how to discover new mathematical truths: i)??? ii)??? iii)?????????

– learn how to teach someone mathematics: …….You know, I can’t think of another “how to learn” book that I’ve read. This one had a really fun intro, describing Carey’s own experiences as a cram-and-forget student, and that roped me in when I was browsing in a Waterstone’s.

Carey focuses mostly on exploring a selection of the psychological research, and using it to develop practical advice for students. So I think he’s happy to bracket the question “What should I learn?” For most students, the answer is pretty simple: whatever the teacher wants.

But you’re right that “learning” is a broad word, capturing all kinds of different intellectual experiences. And I’ve certainly found it harder, in almost every way, to learn how to teach than I found it to learn any of the things I was taught in school!

I think Dan Willingham’s book “Why Don’t Students Like School?” is a good read. He makes some of the same assumptions you’ve mentioned above, but I found it useful.

My big take-away from his book was “memories are the residue of what we repeatedly think about”.

Eg. It’s not the task that matters, it’s the thinking and how often someone thinks about something.

In this book, Willingham seems agnostic on the whole idea of interpretations of thinking (or what some people call misconceptions) though which I see as a significant flaw in the new wave of cognitive psychology of which he is part.

G. Polya’s

How To Solve Itis a classic. You can find it on line quickly enough.Being a devil’s advocate – what make your 6 “mathematical activities” mathematical? They don’t seem to be relegated to this discipline of mathematics in any way. Rather, they seem to be the human condition, part of which involves solving the problems presented by our experiential reality, and part of which satisfies our need to communicate with others. If these are your 6 human activities that you consider to be what shapes this quality of being human we have deemed mathematical, than I suggest you focus on teaching children, not mathematics (and you have me as a huge cheerleader). Which brings up a need to have an answer to the question, what do you feel is your responsibility to teach what I will call the discipline of mathematics, or maybe stated as the content of maths?

/*Well, first off, mathematics is about framing questions.*/

well, since you evidently know what all the fuss is about.

what’s all the fuss about?

bye, now.