### Find the errors!

#### by Willie Wong

I was tasked with grading the following exam question:

Using methods discussed in class this term, find the mean value over of the function .

The *conceptual* parts of the question are (based on the syllabus of the course)

- Connecting “mean value of a continuous function over an interval” with “integration”, an application of calculus to probability theory and statistics.
- Evaluating an integral by substitutions/change of variables.
- Familiarity with the trigonometric functions and their properties (periodicity, derivative relations, etc).

I was told to grade with an emphasis on the above, so I prepared a grading rubric such that the above three key ideas gave most of the points. Here’s an otherwise reasonable answer that unfortunately does not use the methods discussed in class and so would receive (close to) zero credit (luckily no students turned in an answer like this):

The function satisfies , i.e. it is odd. So the average . Since for every , we also have , the mean value of over that interval must be zero.

Here are some responses that can get quite a good number of points^{*} (at least more than the above answer) based on the grading rubric (I guess it means I wasn’t imaginative enough in coming up with possible student errors). (I took the liberty of combining some of the most awful bits from different answers; the vast majority of the students’ answers are not nearly that horrible^{**}, though only one student remembered that when changing variables one also needs to change the limits of integration.) Since most students who made any reasonable attempt on the question successfully wrote down the integral

(which is not to say no unreasonable attempts were made: just ask the poor bloke who decided that the Mean Value Theorem must play a role in this question), I will start from there. All mistakes below are *intentional* on my part. What amazed me most is how many students were able to get to the correct mean value…

**Response 1**

Integrate by substitution: we set then , so

.

Plugging back in we have

.

Since , we have that .

**Response 2**

Integrate by substitution: we set then . So

**Response 3** (this one receives fewer points because it didn’t integrate by substitution)

By the product rule of integration^{***}

We evaluate the first factor

so .

**Response 4** (getting even more ridiculous)

Since , by the rules of integration,

.

Since we have that .

^{*} A side effect of modern exam grading which emphasize fairness is the “no double jeopardy rule”. That is, if a mistake is made in one computation, and if the remainder of the computation is correct assuming the mistake, no more points should be deducted. Hence something like in response 2 would only get deducted some small amount of points for incorrectly computing and some small amount for writing down the wrong integration limits. The cynical side of me wonders if some students are not in fact exploiting this rule intentionally to great effect.

^{**} In fact, somewhere around 10 or 15 percent of the students did the problem perfectly (as long as we overlook the problem with limits of integration), with another 10 or 15 percent missing out on “perfect” only because they did not remember the values of and and didn’t want to risk guessing incorrectly.

^{***} The so called “product rule of integration” appeared uncharacteristically often during the midterm exam. Since clearly the lecturer would not have mentioned such nonsense, the fact that so many students “learned” this rule seems to indicate that when students study together, factoids can be easily mistaken for facts in a sort of crowd mentality. That this appears again on the final exam despite the lecturer having most likely clarified the issue in class maybe due to the “cheat sheet” system. For a lot of classes offered to non-majors, the instructors allow the students to prepare for themselves one or several sheets of paper, which can contain useful information and formulae, so long as the sheets are hand-written. The intention I assume is that the process of producing a cheat sheet would help with rote learning, since it involve looking up relevant information and copying it down by hand. The downside, however, is that once a mistake makes its way onto the cheat sheet, the student probably won’t remember to remove it even after the mistake was pointed out to him…

This exam question looks like a complaint magnet. Students who arrived at by whatever dubious means will feel entitled to a large chunk of the credit. If the correct answer were , that would not be an issue.

Also, the symmetry argument could be implemented as $, where by combining integrals on the left we get . The substitution $u=-x$ is a method discussed in class, after all.

“The exam question looks like a complaint magnet.” My thoughts too when I first laid eyes on it.

“Students who arrived … entitled to a large chunk of the credit.” Which is why we need to have some sort of grading rubric. If we grade consistently and objectively, the complaints have no leg to stand on.

“If the correct answer were…” Actually, most students who didn’t know how to do it never got a numerical answer. But we did have some joker guess (as in, without writing anything else) , which is not all unreasonable if you think about the domain of the function.

“Also, the symmetry … implemented…” That’s why we put in bold letter on the first page of the exam: unjustified statements will have points deducted. If a student have given the response as you wrote it, I can see sneaking in some points for the student technically satisfying the “use change of variables” part of the rubric. Furthermore, to arrive at zero, he/she will have to remember to change also the limits of integration, bettering his/her peers, and so I would likely be more sympathetic.

Unless students were provided with the rubric at the time of the exam (unlikely), they cannot be penalized for not following the rubric. They can’t read professor’s mind. My solution uses a method discussed in class (substitution), and is fully justified. I demand full credit for my solution.

Uh, you should read my comment again. Especially the sneaking some points in part.

The complaint magnet comment makes me think of when students first learn contour integrals in complex analysis. There are always a few who think writing zero for all of the contour integral questions will be enough to get them a pass…

Bloke? When did you become British? 🙂

If you’d recall, from 2009 to 2011…

“The cynical side of me wonders if some students are not in fact exploiting this rule intentionally to great effect.”

I don’t know whether it’s intentional or just an evolved behaviour, but I’m sure that a lot of my own students exploit this rule. In both exam and homework marking, I see a lot of “attempts” at questions that consist of throwing down every possible calculation that *might* be relevant: presumably these students have learned by experience that enough of their calculations will coincide with a point on the marking scheme to gather them a few marks. Apparently one can get quite far through the educational system, at least in Scotland, on this basis.

I suspect that this problem is an inevitable consequence of the very atomistic marking schemes that we’re encouraged to produce in mathematics. As mathematicians, we know that the core of a piece of mathematical work is not the individual lines of working but the logical connections between them (and, at a higher level, the choice and implementation of a suitable strategy), yet for some reason we allow ourselves to be coerced into producing marking schemes that cover everything except the logical connections and the suitable strategy… It’s as if my friends in the history department were to mark essays by giving one point for each occurrence of a date, one point for each occurrence of a trendy abstract noun and so on, and nothing for a properly evidenced historical argument. (This may of course be what happens in some institutions, but I’d prefer to think not.)

In any case, thank you for providing this wonderful “textbook” specimen of the problem, not to mention a moment’s sympathetic Schadenfreude!