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He explains that it was a "triumph of Greek mathematics" to prove that multiplying the top and bottom of a fraction is how you multiply two fractions. That is,

He explains that it was a "triumph of Greek mathematics" to prove that multiplying the top and bottom of a fraction is how you multiply two fractions. That is, we accept as true that you can multiply fractions this way, but we kind of just got lucky that it does work that way. Proving that it always works is non-trivial, but that is totally glossed over in math classes, doing a disservice to students. This is in context of dividing two fractions where kids are taught an arbitrary rule too early in their studies and it turns them off to math overall because it's become a class of memorizing arbitrary procedures instead of learning concepts. (That's all my summary of that segment with a little reading between the lines, not my personal view of it. Or it might be later, but have to consider it more.)

s-shellfish 8y ago
The problem with teaching math is you must give them the tools that were created last, so they have the tools to keep up with math. They then must use those tools to understand and prove what they first had to accept as true, lacking proof. Tend to think about constructive mathematics for this particular insight.
backpropaganda 8y ago
Dividing fractions is also easy. Did you mean summing fractions? Because, yeah, summing fractions is significantly harder than multiplying fractions, simply due to pure misfortune.
- mikekchar 8y ago
  I may be wrong, but I think the issue is: what does it mean to divide x by 1/2? It means to multiply it by 2. We can easily understand that dividing by 2 means to split something into 2 pieces. What does it mean to split something into 1/2 pieces? Why does dividing by 1/2 mean to multiply it by 2 and does it always work that way? Can you prove it? Summation of fractions is much easier to visualise, even if it is harder to calculate.
  - lagadu 8y ago
    Informally speaking, that can be visualised by asking "how many half pieces can you get if you have x pieces?" If I have two pieces I can break them down into 4 half pieces, which nets the same result as multiplying by two.
- alankay 8y ago
  This is easy to derive if you have a little algebra, but it is shoved at kids before they can derive it, so they don't understand it (which is anti-math). The dull ones go along with it, but the brighter ones often think they are stupid. I (and independently others) have worked out visual proofs of both fraction multiplication and division for younger kids. But I mostly think that "fractions as ratios" can be left until the child is older and has more chops for derivations. The Greeks thought about these as different ways to measure intervals, and they worked out the idea of "common measure" as a good solution to dealing with these problems.
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s-shellfish 8y ago

The problem with teaching math is you must give them the tools that were created last, so they have the tools to keep up with math. They then must use those tools to understand and prove what they first had to accept as true, lacking proof. Tend to think about constructive mathematics for this particular insight.

backpropaganda 8y ago

Dividing fractions is also easy. Did you mean summing fractions? Because, yeah, summing fractions is significantly harder than multiplying fractions, simply due to pure misfortune.

mikekchar 8y ago
I may be wrong, but I think the issue is: what does it mean to divide x by 1/2? It means to multiply it by 2. We can easily understand that dividing by 2 means to split something into 2 pieces. What does it mean to split something into 1/2 pieces? Why does dividing by 1/2 mean to multiply it by 2 and does it always work that way? Can you prove it? Summation of fractions is much easier to visualise, even if it is harder to calculate.
- lagadu 8y ago
  Informally speaking, that can be visualised by asking "how many half pieces can you get if you have x pieces?" If I have two pieces I can break them down into 4 half pieces, which nets the same result as multiplying by two.
alankay 8y ago
This is easy to derive if you have a little algebra, but it is shoved at kids before they can derive it, so they don't understand it (which is anti-math). The dull ones go along with it, but the brighter ones often think they are stupid. I (and independently others) have worked out visual proofs of both fraction multiplication and division for younger kids. But I mostly think that "fractions as ratios" can be left until the child is older and has more chops for derivations. The Greeks thought about these as different ways to measure intervals, and they worked out the idea of "common measure" as a good solution to dealing with these problems.

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