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stoutfellowRubric: Problems intended to test students' skills should be nice, in the sense that proper application of the skills should lead to fairly simple results and, though the intermediate calculations may be complex, they should not be unnecessarily so. Students should complete such problems with a feeling of accomplishment, not exhaustion. (The combination is sometimes acceptable.)
Caveat: Nice problems are sparsely distributed. Thus, changing the parameters of a nice problem slightly is almost certain to transform it into a not-nice problem. Furthermore, beyond the most elementary levels, small computational errors are liable to have the same effect.
Note: This last is not necessarily a bad thing, as an astute student, seeing his/her computations going awry, may well realize that an error has been made.
Corollary: Problems intended to test students' skills absolutely must be carefully designed. Furthermore, when going over such a problem on the board, one must take special care to avoid computational errors.
All of which makes the following incident striking. On Wednesday, at the beginning of class, I went over some of the homework problems. In the course of one of them - finding the eigenvalues and eigenvectors of a 3x3 matrix - I made a computational error while calculating the characteristic equation. The resulting erroneous equation had small-integer solutions. It was not until I tried to find the corresponding eigenvectors that I realized I'd made a mistake. (The correct equation also had small-integer solutions.)
Very very unusual, in my experience.
Caveat: Nice problems are sparsely distributed. Thus, changing the parameters of a nice problem slightly is almost certain to transform it into a not-nice problem. Furthermore, beyond the most elementary levels, small computational errors are liable to have the same effect.
Note: This last is not necessarily a bad thing, as an astute student, seeing his/her computations going awry, may well realize that an error has been made.
Corollary: Problems intended to test students' skills absolutely must be carefully designed. Furthermore, when going over such a problem on the board, one must take special care to avoid computational errors.
All of which makes the following incident striking. On Wednesday, at the beginning of class, I went over some of the homework problems. In the course of one of them - finding the eigenvalues and eigenvectors of a 3x3 matrix - I made a computational error while calculating the characteristic equation. The resulting erroneous equation had small-integer solutions. It was not until I tried to find the corresponding eigenvectors that I realized I'd made a mistake. (The correct equation also had small-integer solutions.)
Very very unusual, in my experience.
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no subject
Date: 2007-07-21 02:05 am (UTC)no subject
Date: 2007-07-21 02:17 am (UTC)no subject
Date: 2007-07-23 11:57 pm (UTC)3 2 -3
-3 -4 9
-1 -2 5
In the determinant of xI-A, the first negative term (starting from the top left and moving down and left) is -(x-3)(-9)(2) or 18x-54. I accidentally wrote it as 9x-54. The correct characteristic polynomial was x(x-2)^2; I got x(x-5)(x+1) instead.