[quote]This is why my brother (who is now a "rocket scientist"--aerospace engineering for the past 25 years) had to repeat calculus in high school. He didn't show his work on the problems (because he did them in his head and didn't have any work to show), and even though he always had the answers right, failed the class anyway. He refused to dumb it down to show work that he didn't need to do to get the answers, and failed. The teacher knew perfectly well that he knew more than the teacher did...
If the teacher knew that your brother knew more than s/he did, why did s/he fail him? Is the purpose of the class to learn calculus or to write stuff down the way teacher likes?
Richard Feynman talked about this idea in his essay
The Making of a Scientist. Some older kids were doing 2x +7 = 15 and he said "X is 4" and they said "You did it by arithmetic. You have to do it by algebra." The implication was that he didn't understand algebra.
So he taught himself algebra and argued that the point is to figure out what x is, not to follow a set of rules blindly. He also argued that the rules represented a way to get the answer if you don't understand what you're trying to do. Then he gave examples of people who could "do" calculus --- by using the rules --- but didn't truly understand it.
Food for thought.
Val
