I'll completely agree with you that a student who can't multiply three by five doesn't understand mathematics. However, I'd argue that I'd rather have a studen

I'll completely agree with you that a student who can't multiply three by five doesn't understand mathematics. However, I'd argue that I'd rather have a student spend ten hours figuring out that 3*5 = 5+5+5 = 1+1+1+1+1+1+1+1+1+1+1+1+1+1+1 = 15 than to have a trained parrot that simply squawks the answer. The first student was slower, but it's the parrot that's mouthing a series of words without understanding. As for the rest, I disagree that I'm fighting that standard learning pattern. In nearly every subject, we start with a generality and then move into specifics. We learn about the Roman Empire before we learn about Cicero. We learn about teeth before we learn about incisors. We learn about protons before we learn about quarks. We learn about function calls before we learn the standard library. We learn about China before we learn about Beijing. Almost all learning is performed through iteratively refining a central abstraction with increasing detail.