In my last post, I categorized the critical comments on Barry and my Atlantic article into 7 categories. A similar, though generally more sophisticated set of critical comments appeared on Dan Meyer’s blog.
Again, there were those who took issue with our specific examples of expected explanations. They agreed that requiring such explanations isn't reasonable, that teachers should be flexible, and that explanations could be oral and informal. But they also argued that explanations in general are a good idea. And they are. But explanations are most effective and efficient when solicited in a teacher-centered discussion, or when used to help a student understand why he or she got a particular answer wrong.
There were those (including Dan) who conflated explaining answers and thought processes verbally and diagrammatically with doing math proofs (the latter is something we think there should actually be more of).
And there were those who conflated explaining answers and thought processes verbally and diagrammatically with showing work. Others seemed to think that the kind of work displays Barry and I were endorsing was work consisting only of mathematical symbols. But there are plenty of words that go into work-showing (and proofs), including reasons (“given"; “side-angle-side"; “without loss of generality”) and units (“miles per hour”; "liters of water loss per minute"; “pounds of salt per pounds of total mixture”).
There were those who cited student testimonials about how producing explanations enhanced learning. Many students would beg to differ, though not everyone wants to listen to them.
In the "communication skills necessary for math-related professions" category, there were those who specifically discussed how mathematicians themselves
use words in describing their discoveries all the time – and have for a long time. That’s why some doctorates in mathematics require a foreign language so that the candidate can read the mathematicians’ writings in the original language.The question, however, is whether requiring the kinds of verbal and diagrammatic explanations we critique in our article will help prepare future mathematicians to communicate with other mathematicians. The mathematicians I’ve talked to are skeptical. A related question: are mathematicians who learned math in pre-answer-explaining times deficient in their communication skills?
True, teachers, including mathematicians who teach other mathematicians what they have discovered, should be able to explain the math in question verbally and diagrammatically, as needed. But that doesn’t justify requiring K12 students to provide such explanations in their assignments. Teacher prep programs exist for a reason. And are teachers who learned K12 math in pre-answer-explaining times worse at teaching math concepts than their contemporary counterparts?
In addition, there were those (including Dan) who argued that a student with correct but unexplained answers, even my hypothetical student who “progresses through math with correct but unexplained answers—from multi-digit arithmetic through to multi-variable calculus”—still might not understand the underlying math. My response is twofold. One: this ultimately depends on what we mean by “understanding,” and matters only if the student’s understanding is insufficient preparation for the next level of math. Two: the problem would be remedied by assigning more conceptually challenging math problems--of the sort that simply can’t be solved if you lack the requisite depth of understanding. Such problems do exist: indeed, if they didn't, none of this would matter.
Perhaps the most-discussed argument was that of the supposed meta-cognitive benefits of explaining your answer (and of students listening to each other explain their answers), and, relatedly, of the purported need for teachers to understand what is going on in students' heads at a level of depth for which mere answers and work-showing are insufficient. One commenter simply writes “Metacognition! I can’t imagine anyone not seeing the value of that, but you never know."
In support of meta-cognition, some cited students who can crunch numbers or apply formulas but lack conceptual understanding. I’d say these students simply need practice with problems that require more conceptual understanding than mere “number crunching” and “formula applying” assignments do. In general, assigning more conceptually challenging math problems is a much better, and more efficient, way to help students develop conceptual understanding (and, as one commenter put it, to “ferret out” those “math zombies”) than bogging them down with verbal and diagrammatic explanations to problems that aren’t conceptually challenging.
One person cited kids who do problems incorrectly, but, when explaining them, realize their mistakes. That’s a benefit that can be achieved simply by soliciting explanations specifically for wrongly-answered problems for which an explanation is likely to lead to this sort of realization.
In short, my answer is more teacher-led discussion of underlying concepts, with teachers calling on students, as appropriate, to develop concepts and explanations (for which Japanese classrooms as described in this discussion are a good model); and more individualized practice with conceptually challenging math problems.
A comment I made to this this effect on Dan's blog led to a whole new thread of comments, which I will digest in my next post.