Understanding Math and the Zombie Problem
Reprint. Regurgitating and forgetting is far too common among students with impressive math transcripts.
Original publication date: November 25, 20151 I’ve modified the opening.
My twitter followers might have noticed that I’ve lately (November 2024) been pretty cranky on the topic of acceleration. In fact, I oppose acceleration, which is not at all the same as tracking, and while the quote below isn’t the root cause of my opposition, it’s a good part of it.
This quote came from Barry Garelick and Katherine Beals “Explaining Your Math: Unnecessary at Best, Encumbering at Worst”, first published in November 2015:
Is it really the case that the non-linguistically inclined student who progresses through math with correct but unexplained answers—from multi-digit arithmetic through to multi-variable calculus—doesn’t understand the underlying math? Or that the mathematician with the Asperger’s personality, doing things headily but not orally, is advancing the frontiers of his field in a zombie-like stupor? (emphasis mine)
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. One can infer zombie existence by the numerous complaints of college math professors about students with strong math transcripts but limited math knowledge.2
I’ve seen zombies in tutoring through calculus, in my own teaching through pre-calc. In early math classes (algebra and geometry), I’ve stopped some zombies dead in their tracks, often devastating them and angering their parents. But I don’t teach honors courses, where the problem is rampant, so I don’t see most of them.3
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. (Others, like Barry Garlick, the G of the article, consider conceptual understanding more of a nice-to-have.) 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.
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, in my experience, 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, zombie behavior encourages math teachers and policy makers to raise expectations, increase covered material, accelerate instruction pace. The kids regurgitate, follow the procedures, and promptly forget everything after the test. The teachers know this, so they always schedule a week for review before midterms and finals (I do not do this). So the students are allowed to learn everything that they’ve forgotten again before the next test, and then forget it all.
Zombies enable 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.
Put another way, 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.
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. Garelick just finished ed school and taught a semester or so as a sub, finding it impossible to get a job.4 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 really misrepresent math instruction in America’s public schools, choosing instead to beat their own advocacy drums. 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.
(gleep. 9 years ago almost exactly) This piece was written shortly after the G&B article was published and I read a really interesting discussion at Dan Meyer’s old blog. As Dan observed, the exchange between Brett Gilland and Ze’ev Wurman is an epic exchange of different philosophies (although I dispute that Brett’s is “progressive”, as I agree with his take and I’m not progressive, but small point).
Anyway, my original article has two points. First, that Garelick and Beals (G&B) are using evidence gathered from elementary school classrooms to make general points about middle school and higher instruction, and this is just wrong, since the research on arithmetic has very different findings than the (much more limited) research on mathematics. I’m going to repost that part another day; those interested can reread it on the wordpress blog. The second point is due to a throwaway comment they made in the article about “math zombies”, and that’s what is presented here, with a few modifications.
One last note: this can be considered as a companion piece to my considerably more famous (in tiny circles!)
The Zombie problem is something that arises with bright students who fake it. The myth problem is one you see with low achievers. I mean, seriously, we’d all count ourselves lucky if low achievers were zombies.
To reiterate from the previous footnote: There’s a huge issue that I’ve written about often in which college students are entering public universities and community colleges with middle school math skills. The colleges know this and are actively engaged in fraud by eliminating remediation, etc. These students are not under discussion here. They are the Myth students of the link above.
Then there are the students who were admitted to competitive colleges with 4.5 GPAs, 1400+ SAT scores, and eight or more AP classes (Calc BC, Chem, US and World History, English Lang & Lit, etc) and their abilities don’t match their resumes in the slightest. These are the students I’m describing here. They have been around in large numbers for a while now, although it appears to have gotten much much worse since the pandemic, probably because a lot more of the kids are cheating. But leave aside the cheating and you still have the zombie problem.
I wrote about several cases of math zombies. All of them are really best suited for mathies, but if you just follow the conversation you can see the students involved are just going through the motions.
Tales from Zombieland, Calculus Edition Part 1 —Digs deep into calculus, which I don’t teach. This was one of the last years I tutored. Really good comments on both this and part 2.
Tales from Zombieland, Calculus Edition Part 2
Great Moments in Teaching: Browbeating Psychoanalysis—This is what a zombie looks like in class, or one of them.
Barry Garelick hadn’t worked full-time in a public school when he wrote that article. He did get hired at a middle school and worked for two years, then got non-re-elected. Ultimately got a job in a private Catholic school, according to his Linkedin profile. This wreaked havoc with his plan to write a book about his first year of teaching. His failure to find a job was almost certainly due to rampant age discrimination that used to be common in public schools (not since the pandemic). While I disagree with Barry on most issues, I’m very sympathetic and sorry he wasn’t able to work full time as he planned.