We have an interesting discussion going in the comments on The Problem with Manipulatives. I mentioned a vague memory of a quotation. Now I’ve found the source.

Originally published in 1970:

The continuing hullabaloo about the “new math” has given many a parent a false impression. What was formerly a dull way of teaching mathematics by rote, so goes the myth, has suddenly been replaced by a marvelous new technique that is achieving miraculous results throughout the nation’s public schools.

I wish it were true — even if only to the extent implied by entertainer (and math teacher) Tom Lehrer in his delightfully whimsical recording on “The New Math”:

“In the new approach, as you know, the important thing is to understand what you’re doing, rather than to get the right answer.”… Indeed, there is something to be said for the old math when taught by a poorly trained teacher. He can, at least, get across the fundamental rules of calculation without too much confusion. The same teacher trying to teach new math is apt to get across nothing at all…

— Martin Gardner

Foreword to Harold Jacobs’ Mathematics: A Human Endeavor

Unfortunately, I can’t embed the Tom Lehrer song Gardner mentioned, due to copyright restrictions, but here’s a link to YouTube:

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This is quite an interesting, and seemingly accurate conjecture…The quote “The same teacher trying to teach new math is apt to get across nothing at all” seems to pose a real problem with this switch to “new” mathematics, which I take as math taught with respect to relational understanding rather than instrumental understanding. Are we in a “bubble” era of mathematics teaching? It’s obviously not right to fire all instrumentally trained teachers, but how should we overcome this “problem”. Is Instrumental teaching wrong, or just different? What about a blend between the two? Schools with both types of instruction?

Never having taken an education course, I had to look up Relational and Instrumental Understanding. It’s an interesting distinction. I think the problem is not only with teachers, but also with many students and parents who are satisfied with (or even who demand) instrumental understanding.

A merely instrumental view of math almost guarantees that students will crash and burn out, usually in high school algebra or geometry. I frequently hear other parents say, “I hated geometry. I could never memorize all those theorems.” It makes me cringe.

How do we improve teaching? I think the easiest way to make a big difference would be to have elementary teachers specialize by subject. Let the teachers who hate math avoid it, and let the ones who enjoy math teach it. At least then we could avoid the tragedy of teachers passing on their own math phobia to students.