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?One of the sections of Barry Garelick and my recent piece on TheAtlantic.com that has turned out to be most controversial is this one. In follow-up posts/comments on two blogs (Dan Meyer's and Education Realist's), people have argued that such "math zombies" do exist, citing kids they've worked with who make it all the way through high school calculus and the AP exams without understanding the underlying math, and (Education Realist):
the ever-growing complaints of college math professors about students with strong math transcripts but limited math knowledge.I am closely associated with a number of college math professors, and am both familiar with and sympathetic to these complaints. Of particular concern are the many post-1960s high school texts featuring what some mathematicians have called “cookbook calculus." This is a curriculum that favors breadth over depth, and superficial applications of ready-made formulas over in-depth discussions of where these formulas come from and math problems that are conceptually challenging.
It is precisely for this reason that the careful reader will notice the phrase “multi-variable” modifying the word “calculus” in the above excerpt from our article. Multi-variable calculus is rarely taught in high school; readiness for multi-variable calculus implies what simply getting As in high school math doesn't imply: readiness of college-level math. To further clarify that we are talking about post-high school math skills, we extended the hypothetical zombie to someone operating on the frontiers of math.
OK, perhaps there are students who succeed in college-level math and even function quite well at the frontiers of math who are, nonetheless, zombies. When it comes to zombies, it gets philosophical: how can we know? But here’s another, more practical, question: who cares? As long as someone finishes a given math class ready for the next level of math, who cares whether they’re a partial or total math zombie?
Indeed, given the limitations of working memory, being a partial math zombie is probably a good thing. I'll go even further: it's probably a prerequisite to mathematical success--just as it is, mutatis mutandis, for success in everything from prose writing to piano performing.
As for ferreting out those math zombies who aren’t ready for the next level of math, I’m with Education Realist. As I wrote earlier here, you do this by assigning more conceptually challenging math problems—of the sort that simply can’t be solved if you lack the requisite depth of understanding. As Education Realist noted on DanMeyer’s blog, one can “ask test questions that ferret out zombies.”
Education Realist, though, isn’t OK with simply ferreting out those zombies who aren’t ready for the next level of math. For Education Realist, as we’ll see in my next post, the problem of math Zombiedom is much bigger and deeper than poor preparation for higher-level math.