In the toughest math classes I took in college, it happened a couple of times that a certain classmate of mine would ask me to explain what was going on. He seemed to have more trouble understanding the material than I did, and my verbal explanations seemed to help him understand it better. So much better that he, now an accomplished engineer, went on to score higher than I did on all three class tests.

For many Reform Math acolytes, the ability to communicate your reasoning to others, and to talk about math more generally, is the apotheosis of mathematical understanding. It’s much higher level, supposedly, than “just” being able to get the right answer.

But does saying intelligent things about math necessarily mean that you can actually do math? One situation that often ends up suggesting otherwise is *tutoring*.

When you tutor someone one-on-one, at length, over a long enough period of time, it’s easy to think that you are comprehensively probing the breadth and depth of their understanding--simply by conversing with them broadly, and by asking the right questions and follow-up questions. Surely what your tutee says reflects what s/he knows. Surely it’s not possible for him or her to carry on articulately about something they don’t fully understand. And surely it’s not necessary to test their ability to do tasks independently in the more traditional, detached testing format of a written examination.

But some of the same things that make tutorials so great—their fluidity and flexibility, and the apparent close-up they provide into students’ understanding—are also their biggest downside. It’s not hard for tutors to accidentally provide more guidance than they intend to; to lead the tutee towards the correct answer; to fail to create situations, complete with awkward silences, in which tutees have to figure things out completely on their own.

Furthermore, when it comes to math in particular--symbolic, quantitative, and visual as it is--verbal discussion only captures so much. One can converse quite intelligently about limits, for example, without actually being able to actually find a limit, or about the properties of functions without being able to construct a formal proof of any of those properties.

Too often I’ve seen tutors grossly overestimate the ability of their more verbally articulate tutees to do the actual math—until they find independent testing turning out results much lower than they expected.

To put it in terms that are only *semi*-mathematical, clear verbal explanations are neither a necessary nor a sufficient condition for mathematical mastery. And it’s only the latter that correlates significantly with true mathematical understanding.

## Monday, October 13, 2014

### The Tutoring Fallacy: where clear explanations fall short

Labels:
explaining answers,
math,
Reform Math,
tests

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## 3 comments:

I find it very hard to agree with these statements. I have tutored Math semiprofessionally at many levels for almost two decades, and at every moment I was very keenly aware of what my kids do and don’t know. Tutoring is not at all just a peasant “intelligent” talk about the subjects – very far from it.

It depends on the tutor. I'm sure you're a great one!

Where I've seen problems is specifically with tutoring of older, accelerated kids in upper-level math courses, where the entire course is the tutoring itself. I should have been clearer about that.

This is something I've been working on a little. I think the problem for me is the lack of tests. As recent studies have shown, tests and studying for tests, is a very good way to learn. With tutoring, everything is always a bit low-key and low-pressure. Without a test, it's hard to make the student sit down with the materials and really absorb and come to terms with the math.

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