The American teacher looks at a problem they're going to use in a lesson and asks themselves, "how can I teach my kids to get the answer to this problem?" The Japanese teacher asks, "What's the mathematics they're supposed to learn from working on this problem? How can I get them to learn that mathematics?"
If you want better answers, ask better questions.
Maybe you, american teachers, are not the ones to blame, David. Maybe you can't spend enough time teaching maths because silly testing doesn't allow you. Maybe japanese teachers are more test-independent, I don't know, I just try to find an answer.
I don't know. We test more than I'd like, but I think Daro brings up a good point when he says we fill our minutes with methods and processes of answer getting rather than focusing on the math.
Maybe both educators should be saying, "Here is an interesting problem, lets see what mathematics results from investigating it?"
I liked his line about teachers inventing processes (e.g. FOIL, canceling), which we then have to teach and then distracts us from teaching real mathematics.
I think your question is much closer to the Japanese than American model. I don't know if we will ever fully remove ourselves from the "what are they supposed to learn" model. However, the mindset of the Japanese nicely dovetails with a more open-ended approach.
The process invention part really got me too. How many times have I heard, "we don't have time to [x]?" If we would eliminate the teaching of specific processes and in fact use [x] to allow students to invent their own processes, we'd actually come out ahead.
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