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The argument isn't that we can teach mathematics without a firm intellectual foundation. The argument is that arithmetic is not that foundation. No one would

The argument isn't that we can teach mathematics without a firm intellectual foundation. The argument is that arithmetic is not that foundation. No one would argue that you should know the turn by turn direction to drive from the White House to the Liberty Bell and use that to generalize how to read a map. We don't force kids to memorize Harry Potter and then later expect them to generalize that our into literacy. History courses don't start with Supreme Court cases and wait until students generalize out the Constitution. By starting with the abstraction and moving toward the concrete, we give the information context and allow the students a good mental framework to build upon as they learn the details. Starting with the concrete and moving towards the abstract is needlessly confusing and robs the students of the firm intellectual foundation that they need to understand the subject.

jerf 15y ago
I'm not arguing arithmetic is that foundation; I'm arguing its an unremovable part of the foundation. There's more to it than that, but for all the well-documented foibles of true mathematicians when it comes to simple arithmetic, how much math can someone really be learning when they are staring blankly at 3 times 5? There's a lot more to programming than creating and calling functions, but I'm yet to see an interviewee struggle with how functions work in their putative "best" language but they're otherwise brilliant. You can't be progressing very far if you're burning that much mental effort on the very, very basics of the task at hand. By starting with abstraction and moving towards the concrete, you're completely fighting everything about how we experience life and learn. Nothing else works that way. The only abstractions you can teach that way are series of words that students can mouth without understanding, but have no true comprehension of.... gee, that sounds a bit familiar.
- rprospero 15y ago
  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.

jerf 15y ago

I'm not arguing arithmetic is that foundation; I'm arguing its an unremovable part of the foundation. There's more to it than that, but for all the well-documented foibles of true mathematicians when it comes to simple arithmetic, how much math can someone really be learning when they are staring blankly at 3 times 5? There's a lot more to programming than creating and calling functions, but I'm yet to see an interviewee struggle with how functions work in their putative "best" language but they're otherwise brilliant. You can't be progressing very far if you're burning that much mental effort on the very, very basics of the task at hand. By starting with abstraction and moving towards the concrete, you're completely fighting everything about how we experience life and learn. Nothing else works that way. The only abstractions you can teach that way are series of words that students can mouth without understanding, but have no true comprehension of.... gee, that sounds a bit familiar.

rprospero 15y ago
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.