Wrong answers are part of the process too. Time and again in the Japanese classroom you’ll see, “Jen discovered an approach that doesn’t work. Jen, explain your discovery to the class.” Jen explains the discovery to the class. “Does everyone understand Jen’s discovery? Now let’s all figure out why it didn’t work. Jen, did you figure out why it didn’t work? Let’s figure it out.” It’s actually often easier to get to the math figuring out why an approach didn’t work than why an approach did work.
[N.B. Often times a student has correctly answered a different question, and asking "For what question would Jen's approach work very well?" is generative.]
Apostolos Doxiadis argues for more “paramathematicians”:
If our rationale for teaching a subject is circular – “you must learn it because it is useful, because it has uses, because it is useful, because you will need it later, because it is useful” – we won’t go a long way. A developing human being is many things, and chief among them a poet, an adventurer and a problem-solver. Give the poetry, the adventure and the problems, through stories, both small stories of environment and large stories of culture. Grip the heart – and the brain will follow.
As for the mathematicians themselves: don’t expect too much help. Most of them are too far removed in their ivory towers to take up such a challenge. And anyway, they are not competent. After all, they are just mathematicians – what we need is paramathematicians, like you…. It is you who can be the welding force, between mathematics and stories, in order to achieve the synthesis.
David Gessner built a shack for himself but left a gap between the door and the roof. He was rewarded when he didn’t patch that gap. Read about it and then imagine the contents of a blog post entitled, “Leave Some Gaps In Your Tasks.”
We may well ask of any item of information that is taught … whether it is worth knowing? I can only think of two good criteria and one middling one for deciding such an issue: whether the knowledge gives a sense of delight and whether it bestows the gift of intellectual travel beyond the information given, in the sense of containing within it the basis of generalization. The middling criterion is whether the knowledge is useful. It turns out, on the whole, … that useful knowledge looks after itself. So I would urge that we as school men let it do so and concentrate on the first two criteria. Delight and travel, then.