Is there a real difference in the way a 21st-century student learns math or science, as opposed to a 20th century student? Seriously.

Do you mean "Is there a difference in practice?" or "Should there even be a difference?"

For the first question, probably not. For the second, oh yeah. There is a qualitative difference between an abacus and modern computers. The difference in tools shifts the methods

and the frontiers of the subject matter for quantitative reasoning. The notion of data mining, for instance, is pretty new. Non-linear systems that can't be calculated can be simulated.

The basics are, of course, the same but new applications are unfolding all the time demanding a better and somewhat different foundation. Math and science need to be viewed as disciplines of inquiry, not a set of facts to memorize. The methods and techniques are more important than the results. I think that requires a shift in teaching.

Frankly, in my experience, not only are too many math and science teachers not doing this, but they don't even understand what I'm talking about. Nor are they familiar with research and discoveries in cognitive science on how we learn math and quantitative reasoning. It seems to me that cognitive science should be to teaching what biology is to medicine.