Understanding Math, and the Zombie Problem

I have been mulling this piece on the evils of explanations for a while. There’s many ways to approach this issue, and I highly recommend the extended discussion at Dan Meyer’s blog, as it captures experience-based teachers (mostly reform biased) with the traditionalists, who are primarily not teachers.

What struck me suddenly, as I was engaged in commenting, was the Atlantic’s clever juxtaposition.

All the buzz, all the sturm und drang about Common Core and overprocessed math has involved elementary school. The cute show your thinking pictures are from 8 year olds and first graders. Louis CK breaks our hearts with his third grader’s pain. The image in the Atlantic article has cute little pudgy second grade arms—with just the suggestion of race, maybe black, maybe Hispanic, probably male—writing a whole paragraph on math. The evocative image evokes protective feelings, outrage over the iniquities of modern math instruction, as a probably male student desperately struggles to obey meaningless demands from a probably female teacher who probably doesn’t understand math beyond an elementary level anyway. Hence another underprivileged child’s potential crushed, early and permanently, by the white matriarchal power structure unwilling to acknowledge its limitations.

And who could disagree? Arithmetic has, as John Derbyshire notes, “the peculiar characteristic that it easy to state problems in it that are ferociously difficult to solve.” Why force children to explain place value or the division algorithm? Let them get fluency first. Garelick and Beals (henceforth referred to as G&B) cite various studies finding that elementary school students gain competence by focusing on procedure first, conceptual understanding at some later point.

There’s just one problem. While the Atlantic’s framing targets elementary school, and the essay’s evidence base is entirely from elementary school, G&B’s focus is on middle school.

Percentages. Proportions. Historically, the bane of middle school math. Exhibit C on high school math teachers list of “things our students should know but don’t” (after negatives and fractions), and an oft-tested topic, both conceptually and procedurally, in college placement.

G&B make no bones about their focus. They aren’t the ones who chose the image. They start off with a middle school example, and speak of middle school students who “just want to do the math”.

But again, there’s that authoritatively cited research (linked in blue here):

gbquoteresearch

Again, all cites to research on elementary school math. The researched students are at most fifth graders; the topics never move above arithmetic facts. G&B even make it clear that the claim of “procedure without understanding is rare” is limited to elementary school math, and in the comments, Garelick discusses the limitations of a child’s brain, acknowledging that explanations become more important in adolescence—aka, middle school, algebra, and beyond.

G&B aren’t arguing for 8 year olds to multiply integers in happy, ignorant fluency, but for 14 year olds to calculate percentages and simply “show their work”. And in the event, which they deem unlikely, that students are just going through the motions, that’s okay because “doing a procedure devoid of any understanding of what is being done is actually hard to accomplish with elementary math.” Oh. Wait.

Once you get past the Atlantic bait and switch and discuss the issue at the appropriate age level, everything about the article seems odd.

First, Beals and Garelick would–or should, at least–be delighted with math instruction in 8th grade and beyond. Reform math doesn’t get very far in high school. Not only do most high school teachers reject reform math, most research shows that the bulk of advanced math teachers have proven impervious to all efforts to move beyond “lecture and assign a problem set”. Most math teachers at the high school level accept a worked problem as evidence of understanding, even when it’s not. I’m not as familiar with middle school algebra and geometry teachers, but since NCLB required middle school teachers to be subject-certified, it’s more likely they profile like high school teachers.

G&B don’t even begin to make the case that “explaining math” dominates at the middle school level. They gave an anecdote suggesting that 10% of the week’s math instruction was spent on 2-3 problems, “explaining thinking”.

This is the basis for an interesting discussion. Is it worth spending 10% of the time that would, presumably, otherwise be spent on procedural fluency on making kids jump through hoops to add meaningless detail to correctly worked problems? And then some people would say well, hang on, how about meaningful detail? Or how about other methods of assessing for understanding? For example, how about asking students why they can’t just increase $160 by 20% to get the original coat price? And if 10% is too much time, how about 5%? How about just a few test questions?

But G&B present the case as utterly beyond question, because research and besides, Aspergers. And you know, ELL. We shouldn’t make sure they understand what’s going on, provided they they know the procedures! Isn’t that enough?

Except, as noted, the research they use is for younger kids. None of their research supports their assertion that procedural fluency leads to conceptual understanding for algebra and beyond. We don’t really know.

However, to the extent we do know, most of the research available in algebra suggests exactly the opposite–that students benefit from “sense-making”, conceptual approaches (which is not the same as discovery) as opposed to entirely procedural based instruction. But researching algebra instruction is far more difficult than evaluating the pedagogy of arithmetic operations—and forget about any research done beyond the algebra level. So G&B didn’t provide adequate basis for making their claims about the relative value of procedural vs conceptual fluency, and it’s doubtful the basis exists.

I’ll get to the rest in a minute, but let’s take a pause there. Imagine how different the article would be if G&B had acknowledged that, while elementary school research supports fact fluency over sense-making (and fact fluency seems to be helpful in advanced math), the research and practice at algebra and beyond is less well established. What if they’d argued for their preferences, as opposed to research-based practices, and made an effort to build a case for procedural fluency over comprehension in advanced math? It would have led to a much richer conversation, with everyone acknowledging the strengths and weaknesses of different strategies and choices.

Someday, I’d like to see that conversation take place. Not with G&B, though, since I’m not even sure they understand the big hole in their case. They aren’t experienced enough.

Then there’s the zombie quote, where Garelick and Beals most tellingly display their inexperience:

Yes, Virginia, there are “math zombies”.

In high school, math zombies are very common, particularly in schools with a diverse range of students and thus abilities. Experienced teachers commenting at Dan Meyer’s blog or the Atlantic article all confirm their existence. This piece is long enough without going into anecdotal proof of zombies. One can infer zombie existence by the ever-growing complaints of college math professors about students with strong math transcripts but limited math knowledge.

I’ve seen zombies in tutoring through calculus, in my own teaching through pre-calc. In lower level classes, I’ve stopped some zombies dead in their tracks, often devastating them and angering their parents. The zombies, obviously, are the younger students in my classes, since I don’t teach honors courses. Most of the zombies in my school don’t go through my courses.

Whether math zombies are a problem rather depends on one’s point of view.

There are many math teachers who agree with G&B, who rip through the material, explaining it both procedurally and conceptually but focus on procedural competence. They assign difficult math problems in class with lots of homework. Their tests are difficult but predictable. They value students who wrote the didactic contract with Dolores Umbridge’s nasty pen, etching it into their skin. They diligently memorize the cues and procedures, and obediently regurgitate the procedures, aping understanding without having a clue. There is no dawning moment of conceptual understanding. The students don’t care in the slightest. They are there for the A and, to varying degrees, play Clever Hans for math teachers interested only in correctly worked procedures and right answers. Left as an open issue is the degree to which zombies are also cheating (and if they cheat are they zombies? is also a question left for another day). For now, assume I’m referring to kids who simply go through the motions, stuffing procedures into episodic memory with nothing making it to semantic, all to be forgotten as soon as the test is over.

Math zombies enable our absurd national math expectations. Twenty or thirty years ago, top tier kids had less incentive to fake it through advanced math. But as AP Calculus or die drove our national policy (thanks, Jay Mathews!) and students were driven to start advanced math earlier each year, zombies were rewarded for rather frightening behavior.

G&B and those who operate from the presumption that math can easily be mastered by memorizing procedures, who believe that teachers who slow down or limit coverage are enablers, don’t see math zombies as a problem. They’re the solution. You can see this in G&B’s devotion and constant appeal to the test scores of China, Singapore, and Korea, the ur-Zombies and still the sublime practitioners of the art, if it is to be called that.

For those of us who disagree, zombies create two related problems. First, their behavior encourages math teachers and policy makers to raise expectations, increase covered material, accelerate instruction pace. They allow schools to pretend that half their students or more are capable of advanced, college level math in high school while simultaneously getting As in many other difficult topics. They lead to BC Calculus pass rates of 50% or more (because yes, the AP Calc tests reward zombie math). Arguably, they have created a distortion in our sense of what “college math” should be, by pretending that “college math” is easily doable by most high school students willing to put in some time.

But the related problem is even more of an issue, because the more math teachers and policies reward zombies, the more smart, intellectually curious non-zombies bow out of the game, decide they’ll go to a state school or community college. Which means zombie kids just aren’t numbered among the “smart” kids, they become the smart kids. They define what smart kids “are capable of”, because no one comes along later to measure what they’ve…well, not forgotten, but never really learned to start with. So people think it really is possible to take 10-12 AP courses and understand the material (as opposed to get a 5 on the AP), and that defines what they expect from all top rank students. Meanwhile, those kids–and I know many–are neither intellectually curious nor even “intelligent” as we’d define it.

The Garelick/Beals piece is just a symptom of this mindset, not a cause. They don’t even know enough to realize that most high school math is taught just the way they like it. They’d understand this better if they were teachers, but neither of them has spent any significant time in the classroom, despite their bio claims. Both have significant academic knowledge in related areas–Garelick in elementary math pedagogy, which he studied as a hobby, Beals as a language expert for Asperger’s—which someone at the Atlantic confused with relevant experience.

Such is the nature of discourse in education policy that some people will think I’m rebutting G&B. No. I don’t even disagree with them on everything. The push for elementary school explanation is misguided and wasteful. Many math teachers reward words, not valid explanations; that’s why I use multiple answer math tests to assess conceptual knowledge. I also would love–yea, love–to see my kids willing to work to acquire greater procedural fluency.

But G&B go far beyond their actual expertise and ultimately, their piece is just a sad reminder of how easy it is to be treated as an “expert” by major publications simply by having the right contacts and backers. Nice work if you can get it.

And the “zombie” allusion, further developed by Brett Gilland, is a keeper.

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About educationrealist


25 responses to “Understanding Math, and the Zombie Problem

  • Hubbard

    Out of curiosity, could you give us some examples of how you trip up the math zombies? I’m not a teacher, but I see the equivalent in my line of work. Tests like the one from Grant Wiggins or are you able to do so with just a few questions?

    • educationrealist

      I just gave a test that was on quadratics mostly, and on factoring. So one question was a rational equation with 4 different factoring cases. Another question asked what resulted when you multiplied two lines together and gave five possibilities (3 quadratic, 2 linear), still another gave a quadratic in vertex form and asked questions about the zeros, the y-intercept, and the vertex. Finally, a question gave a bunch of quadratic graphs and asked questions about a, b, c, h, k, and factors.

      I have a zombie in the class, and he got the first one perfectly. Had no idea how to do some of the other questions, and did many of them the long way round.

      Meanwhile, some weaker students struggled a bit on the factoring, but got the question about the product of two lines completely right. Understood what h,k, were. So they got slightly more right than the zombie, despite struggling more in class and also getting much lower grades from other teachers. In my class, the zombie has a C+.

  • Calvin Hobbes

    Insanely great post! Bravo!

    Feynman wrote (or talked and somebody else wrote it down) about zombie physics in Brazil. It’s at the end here:

    http://southerncrossreview.org/81/feynman-brazil.html

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  • Frozen

    Sorry, I’m lost here in your many many words. So after all that rant, in the end you say “The push for elementary school explanation is misguided and wasteful.” So you agree with the article?

    • educationrealist

      My piece was 2000 words. The Garelick & Beals piece was 2252 words.

      Apparently, you have difficulty with both counting and reading comprehension. This would explain why you think it witty to insult my piece for length when the piece you approve of is 10% longer. It would also explain why you misunderstood both the G&B article and my own.

      The B&G article did NOT assert that elementary school explanation is wasteful. Its sole example was from middle school. And then it failed to make the case. You apparently didn’t understand this. Then you also didn’t understand my article, which said exactly that.

      So in very short words, so you can understand: The G&B article didn’t mean what you think it meant. Neither did mine. Not much point in my discussing your question until you figure out what one or the other article meant.

  • cthulhu

    If I understand your position, I think you’re saying that from your experience, understanding and proficiency can build on one another, but it is not a given that proficiency is the sufficient condition for understanding, and many (most?) students need some (lots of?) instruction specifically designed to elicit understanding.

    I really want to be cautious in generalizing from my experiences in both high school and college, because I may be atypical – I made it through fairly advanced analysis (including two semesters of PDEs, one of tensor calculus) and did well in those classes. Also, I realize this is anecdata. But I needed some good explanations of what was going on in the math, plus enough well-chosen problems to reinforce the explanations and develop proficiency. And I think this goes along with what Steven Pinker postulates in his books like “How The Mind Works” and “The Blank Slate” about how we co-opt various evolved brain modules for capabilities like abstract math for which the brain cannot have evolved.

    I wonder how far a math zombie (great phrase) can go though. Just thinking about it now, I’m having a hard time seeing how a zombie could do most calculus integration problems unless it’s screamingly obvious which technique should be used; that’s why integral calculus is generally considered to be harder than differential calculus – it’s much, much less cookbook. I don’t recall any of my college classmates who made it through, say, Calc 2 that seemed to be a math zombie.

    • educationrealist

      “but it is not a given that proficiency is the sufficient condition for understanding, and many (most?) students need some (lots of?) instruction specifically designed to elicit understanding. ”

      The first, yes. The second, I’m not sure what’s needed. I think pace, rather than “lots of instruction”.

  • DensityDuck

    You’ve actually been poking at this idea for a while now; there are stirrings of it in https://educationrealist.wordpress.com/2014/04/07/sats-competitive-advantage/

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  • Tempe

    Long live the math zombie. At least they can “do” maths rather than just talk about it.

    • educationrealist

      “Traditionalists” (as they like to style themselves) are incapable of grasping the fact that high school math exists, and that most high school math teachers aren’t constructivists.

      Besides, the whole point of the Garelick and Beals article was that math zombies didn’t exist. Now you’re all saying “OK, they exist. So what?” Big change.

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