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.