Conrad Wolfram on Teaching Mathematics in School
Loved this. Especially 3:20 to 4:50. His four step procedure is more or less the process of quantitative research, and Steps 1, 2 and 4 are really where human input is required.
I do think understanding calculation is important, but rather than teaching, say, long division in a rote fashion, getting students to write a program to do it would provide greater understanding.
As a kid, my first exposure to programming was turtle graphics. An excellent way for kids to learn programming and geometry at the same time.
(Source: http://video.ted.com/)
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