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 What was formerly a dull way of teaching mathematics by rote, so goes the myth, has suddenly been replaced by a marvelous new technique that is achieving miraculous results throughout the nation’s public schools.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:
Denise Gaskins' Let's Play Math 