How do you parametrize stupidity?

I have students submit an electronic homework problem every day in my Calc III class.   Today’s problem was “Find a vector valued function f(t) whose graph lies on the curve displaystyle x^2 + 4 y^2 = 16.

About 25 of the 60 responses I got back read exactly as follows:

displaystyle frac{x}{4} = mathrm{cosec}

displaystyle frac{y}{2} = cot

vec{f}(t) = 4 , mathrm{cosec} , vec{i} + 2 cot vec{j}

The probability of 25 students writing exactly the same derivation on a particular problem (especially one like this with infinitely many solutions) is small, and the probability that they would further divine this particular parametrization on simple inspection is smaller still.   But the probability of any of them also choosing on their own to write the function csc in the style of Leonard Euler as “mathrm{cosec}” is zero.

Clearly students are copying their answers from somewhere, which is bad enough… but vec{f}(t) = 4 , mathrm{cosec} , vec{i} + 2 cot vec{j}?   That doesn’t even make sense!   Even assuming they connect “cosec” with cosecant, cosecant and cotangent are functions… they require arguments before they have any meaning.

Actually, there were a few students who recognized that the lack of arguments was a problem, but apparently couldn’t quite figure out that vec{f}(t) was a function of t, and so they instead wrote

vec{f}(t) = 4 , mathrm{cosec}(vec{i}) + 2 cot (vec{j}),

which might be even worse.

I’m not sure what I’m more offended by from this: that so many students are cheating, or that they’re doing it in such a spectacularly dumbfuck manner.

Update

It turns out that a few students fessed up to the source of the common solution, which apparently comes from a website called Cramster, in which students pay a fee to access “complete” solutions to every problem in a given textbook.

I find this vaguely amusing: here I have students whose study skills are so poor they’ll pay money to plagiarize the online solutions of someone else who apparently has no better   mastery of the subject than themselves… but is making money of it nonetheless.

Capitalism and karma… who’da thunk they‘d go hand-in-hand?

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