As a teacher, you have to take the incorrect results and the solutions that have been mangled and try to find any and all correct ideas in them and give partial credit where credit is due. Case in point: This problem from a calculus assignment handed in today (click to enlarge):

Here’s what I’ve had to work with so far:

- A student read the velocities as being in miles per hour instead of feet per second as they are given; then
*converted every distance in the problem into miles*, and then worked the problem. The data, coefficients on the various functions, and answers are totally off — but he’s doing all the right calculus.
- Lots of students used x^2 + y^2 = 600^2 to eliminate the variable in part (d), even though they should be using (2000-x)^2 instead. When carried through the calculus properly, this actually does yield the right answer, but the work looks completely different. Also, since the problem dictates what x has to be, it’s not entirely correct to set the formula in (d)/(ii) up with x^2. So some points need to come off for the setup but not for the answer. I look forward to explaining that one.
- A student set up the right total time function, but then applied the square root linearly — as in, sqrt(a^2 + b^2) = sqrt(a^2) + sqrt(b^2). Then he proceeded to take the derivative and get critical numbers correctly. So it’s a GIGO situation where correct methods are being applied to a seriously incorrect — and illegally simplified — objective function. How much partial credit do you give when the calculus is done correctly, but mainly because an illegal simplification nearly trivialized the derivative computations?

All of these require going in and finding exactly how far off from correct the work is. The work is often a lot closer to right than it looks; and each case is unique, complicated, and requires near-Solomonic judging abilities. I figure if this teaching gig doesn’t work out I can always go join the Supreme Court or be an NFL referee or something.

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That last one is particularly difficult to judge! I see the weak algebra skills constantly, and it makes it very hard to know what someone gets and what they don’t get.

I would tend to say that a student who makes an algebra mistake like that probably has some serious misunderstanding about other aspects of the subject, and would not give much partial credit. Then again you’re not teaching algebra in this course, and he does seem to have grasped the concept of using the derivative to maximize the function. But as you point out, the computations there have been rendered quite simple. So here’s how I read it: bad on the prerequisite algebra concepts, decent on the calculus concepts, indeterminate on the calculus mechanics. Send him back to Algebra 1? Give him half credit?

My god. Bring on the instant replays.

I practically crucify calculus students for mistakes like “sqrt(a+b) = sqrt(a) + sqrt(b).” That isn’t just a small arithmetical slip, it shows both a fundamental disrespect for the mathematics involved in the problem and it also shows that the student isn’t thinking carefully about what he/she is doing. They know better than that, and I tell them as much when they make such mistakes.

For the last one, I think it might help if you explain that you’re not so much deducting points for the algebraic error as refusing to grant credit for a trivialized version of the problem. I’ve had a physics professor take off a huge amount of credit for a simple sign error, simply because it trivialized the problem so far he no longer could tell if I understood the material (and in all honesty, my that-can’t-be-right alarm should’ve been going off when I realized I didn’t have to use anything I’d learned that month). At the time, I was annoyed about it, but it was absolutely the right thing to do. And I think that here, even if the particular error wouldn’t be an automatic

badgrade, the fact remains that it’s entirely his fault you can’t give him agoodgrade.