I find this article to be incredibly insightful, even at 70 years old. It also contains many choice quotes.

"If we wanted to give our convenient inquiring friend, the Man from Mars, an idea of the best in human culture, we would hardly hand him a set of school texts and syllabi. Why should we do it with Johnnie and Susie?"

"Do they experiment and analyze and try out alternative procedures in some sort of planned sequence? The author of one important study finds that few of them seem to reason at all, and that reflective thought is not evoked. 'Instead, many of them appear to perform almost random calculations upon the numbers given. Where they do solve a problem correctly the response seems to be determined largely by habit.'"

Having known many mathematicians, I can confirm that they too solve problems this way ;)

I often wonder if the issue is that many educators feel there is an aspect of teaching that can be learned and mastered separate from the subject matter. In other words many think one can be a good math teacher without a passion for and a love of mathematics. The author's tone in discussing arithmetic conveys an attitude I often see with respect to one of the most powerful tools a child learns.

We see this in programming shops with the pointy-haired boss phenomenon. Management is taught as a science unto itself.

I'm reminded of a recent essay I read by the category theorist Eugena Cheng (http://cheng.staff.shef.ac.uk/illogic/illogic.html).

Imagine your children learning the "trivial" facts and algorithms of arithmetic from a teacher with this world view of what math is.

A follow up article:

http://www.theatlantic.com/issues/95dec/chilearn/muref.htm

It is worth noting that, up until the start of the second world war (i.e. right around the time this was written), Germany was the seat of intellectual power. As educated individuals fled for the last safe haven, which just happened to be America, they brought about a shift in that power. However, it wasn't until Sputnik and the scare of the commies being able to "drop nukes on our heads" that natural born American citizens started to really participate in the development of America's intellectual prowess.

It's also apt to note that many of the individuals educated in the knee-jerk response to sputnik are now reaching retirement, and, dare I say, it shows!