“Deeper conceptual understanding” in K-12 math comes with knowledge and practice to mastery, not with pointless struggle and reinventing of the wheel. Efficiency on paper is critical; the calculator tends to get in the way of learning. Each day, as I work through another chapter, I think, “Oh, yes. Right. I see that now.” Proper process is being reinforced for me; each time I cut a corner, I pay for it with an error. As I practice, I’m becoming faster, more efficient and more accurate. Recently I tweaked an algorithm to make it more efficient; this would not have come to me without skills and understanding.And this:
“Creativity springs unsolicited from a well prepared mind.”And this:
“Fundamental knowledge is the basis of creativity.”
– John Saxon, co-author of the Saxon Math textbook series
If proponents of fuzzy programs and constructivism had to use math in the “real world,” and were held accountable for the results, they would have to modify their views. In the “real world,” math is a tool, used to get a job done. What matters are clarity (understandable by others); efficiency (done relatively quickly); and accuracy (the result is correct). Math is a tool – like a hammer or drill. One doesn’t come to consensus on the philosophy of a drill; one learns to use the drill and then one uses it.You'll want to read the whole thing.