> when my main question was how the heck you stumble upon looking at the problem that way, vs all the failed ways
Well you stumble through half a dozen failed ways first, is the usual method. Eventually you gain "intuition" that lets you jump straight to the way that works and you can't explain exactly how you thought up the correct method, but it just feels so natural and anyway, you can explain in great detail why nothing else would work.
I'm teaching calculus this year so I sympathise with both sides. The problem is that the answer to "how did you know to do it that way?" is experience, as in all fields. But students rarely have the time to accumulate experience and intuition before finals, so there's frustration all around.
What I'm saying is I'm not looking forward to grading six hundred finals where many students will inevitably spend too much time on approaches that were obviously (to me) doomed from the start because they haven't stumbled enough to know it yet.
Well you stumble through half a dozen failed ways first, is the usual method. Eventually you gain "intuition" that lets you jump straight to the way that works and you can't explain exactly how you thought up the correct method, but it just feels so natural and anyway, you can explain in great detail why nothing else would work.
I'm teaching calculus this year so I sympathise with both sides. The problem is that the answer to "how did you know to do it that way?" is experience, as in all fields. But students rarely have the time to accumulate experience and intuition before finals, so there's frustration all around.
What I'm saying is I'm not looking forward to grading six hundred finals where many students will inevitably spend too much time on approaches that were obviously (to me) doomed from the start because they haven't stumbled enough to know it yet.