This semester, my Teaching Assistant assignment is being the grader for two courses. They are both upper-level undergraduate courses, having to do with math that's in my neck of the woods. The topics and assignments are pretty fun. One of them, Matrix Groups, is taught by my advisor. It's a tough course, with tough assignments. True to form, all of his assignments are really interesting and lead the exercise-ee to wonderful and useful mathematical topics.
But I want to write today about the other course, Differential Forms and Manifolds. We're working out of a
textbook by Stephen Weintraub. It's a nice book, although not as sophisticated as some others. The course is usually taught out of a really fantastic set of
notes written by my advisor, but someone somewhere decided that the course would use Weintraub's book this semester instead. This is fine. It's a nice enough book.
What is bothering me is that nearly all of the solutions are in the back of the book. These are not step-by-step solutions, just the final answers. But by allowing students access to the final answers, the textbook has short-circuited one of the grader's most important shortcuts.
This is only a 50% TA assignment for me, so even though there are fewer than 10 students in the class, there's no way I can read and think about every single word and symbol in the students' homeworks. But I usually don't have to. I can check certain key parts of their solutions, and I can see if their answers have the right "shape". If they were doing the correct calculations, if they have set up the correct integrals, the formulas should have a certain length and complexity, and in general a certain shape. Certain precise parts of the content may be incorrect, but they are probably only minor mistakes. And of course, the grader can check the students' final answers.
(This technique, of checking the "shape" of a student's solution, is also applicable to grading proofs as well as calculations, which is very convenient. When a math problem asks you to prove something, you usually already know what the final answer should be, so there's no sense in the grader checking that. Although, students do have a remarkable capacity for being stupid (or, if you prefer, for making silly mistakes). And of course, this included graduate students ...)
This is not high school math, or even beginning college math. The answers are not usually 1 or 5 or -2. More often, they are things like -95/3, or 3252 times Pi, or the square root of 5 divided by negative 2. Barring students copying from one another, you've got to figure that the answers are generally such weird numbers that there is simply no way that the student could have gotten the right answer without doing the problem correctly, unless there was some miraculous alignment of multiple mistakes.
But if the students already know what the final answer is, all bets are off! There is an exception, and this is in the wonderful circumstance that the answer in the back of the book is correct. Then the grader can go to town with the red ink.
I found this especially troubling in the last assignment, which dealt with the Generalized Stokes' Theorem. This marvelous theorem connects integrals over a manifold with integrals over its boundary. In a sense, it is simply an extension of the Fundamental Theorem of Calculus to much more complicated situations. I never actually learned the theorem in Multivariable Calculus. We didn't get that far. My first real exposure to it was when I took Differentiable Manifolds as a second-year grad student.
So in most of last week's exercises, the students were asked to verify Stokes' Theorem, for different objects. This amounted to computing an integral over a manifold, which was always a surface or solid body in three-dimensional space; then figuring out what the boundary of the manifold is, which would have been several curves or several surfaces; then computing some other integrals over the boundary of the manifold; and finally showing the these integrals gave you the same answer. This last part means showing that one number is equal to the sum of the others.
The main complication is that you actually want to add some of the numbers, and subtract the others. This has to do with orientations, which is a subtle and sneaky and confusing topic, and it's clear that many of the students have yet to come to grips with it.
Here's the problem. Usually there were only two numbers to put together, i.e. the boundary consisted of two pieces, and the only question is whether you add them, negate both of them and add them, negate one of them and add them, or negate the other and add them. But it's impossible to tell whether they really knew what they were doing, because they already knew what answer they were supposed to get, so they just did whatever gave the right answer!
If they only knew what answer they should get based on the other integral they calculated, that would be fine. Because then one calculation serves as a check for the other. But for all the integrals, they already knew what answer they should have been getting! The book told them! I had students who weren't able to do the problem, but who wrote down what the answer should be! As if that was an astonishing announcement!
I sense that I am ranting. I think I'll get back to work now.
