I teach calculus to students who have almost all taken calculus before. (Primarily first-year college students who took calculus in high school but didn't perform well enough to skip the course.)
For many topics, my students think they know the subject already, but all they actually know are a bunch of short-cuts or special cases. The problem is that when I try to illustrate the new ideas with simple problems, they fall back on their short-cuts rather than using what we've covered. Then when they get to the harder problems, they're completely lost, because the short-cuts don't work, but they haven't gotten any practice with the new material.
I can tell them "you have to solve this problem using this method", but that feeds into the lesson that math is an arbitrary pile of methods which we choose based on the professor's whim, and it causes them to resent the new method.
My question is what are good tactics to break through this: to either convince students that it's in their interests to rethink how they approach the simple cases, or even better, a way to make the parts they know useful rather than an obstacle.
(The original version of this question gave about a concrete example of a subject where this happens, but it attracted a lot of off-topic answers focusing on the example rather than the question, so I've removed it for clarity.)
( 4 months ago )
Sometimes it can help to take things to a ridiculous extreme, like finding the derivative of ln(sin(cos(ex2)))ln(sin(cos(ex2))). Tell them that the step by step process is like peeling off onion layers:
Take the derivative of the outside the function (the first layer), and leave the inside completely alone:
Now multiply that by the derivative of the second function (the second layer), and leave everything inside of that function completely alone: 1(sin(cos(ex2)))cos(cos(ex2))1(sin(cos(ex2)))cos(cos(ex2))
Now multiply by the derivative of the third `layer', and leave everything inside of that alone. Have students try it first, separately, then reveal the answer and see what they got: 1(sin(cos(ex2)))cos(cos(