## An Idea on Theory Exam Questions

In a discussion with a fellow student about grading, we came up with the following method which can make some Theory exam question with longer proofs easier to grade. I believe that, when used with caution, it can be a good tool to have.

Let me stress the word “caution”, since it is not a good idea to use this method on all problems on all exams.

The basic idea is to give the proof away, but in a scrambled fashion. Suppose the proof can be broken into \$k\$ “units of thoughts” (each unit is probably one or two statements). We will first append some \$n-k\$ junk units to the end of the proof, yielding a total of \$n\$ units. Then, we take a random permutation of the \$n\$ units and give the list to the students. The answer to such an exam question would then be an ordered sequence of numbers indicating the relevant units in the list.

There are many variations on this idea.

First we need to pick \$n\$. That clearly should depend on \$k\$. (Somewhere around \$1.5k\$ to \$2k\$ may be a good idea.)

Then we may decide to reveal the value of \$k\$ or not. (Revealing makes it a lot easier.)

Also, some of the junk units may actually be mathematically wrong but “reasonably sounding” in the context of the proof. We call these bombs. If the answer has a bomb, it automatically gets a zero.

And should we choose to add bombs, then maybe it is also an interesting twist to allow the students to pick out exactly all the bombs. For some classes, being able to pick out exactly all the bombs in a question should really qualify for a full credit for it. (Oh, Number Theory…)

Now that leaves the question of partial credits, which can be a highly subjective matter in the traditional “write-a-proof” format. But in this “pick-out-the-proof” format, it may become easier if the junk units are prepared carefully.

Finally, notice that the permutation needs not be the same for different students. This can be good for rooms with little elbow room. Grading is now a bit harder but should not be too bad.

1. I gave my comment on my blog:

http://rweba.livejournal.com/236005.html

2. And if the pieces are too confusing it might get tedious and be a burden on the good student who can actually solve the problem by themselves.

Robert, the way I would do such an exam question is to write down a proof sketch by myself first, then try to find the hidden proof in the list. (Also, scan through the list and kick away some statements that I think is irrelevant in the context of my proof sketch.)

My feeling is that for some questions, if the list is prepared with some caution, it should not be too much of a burden if the student can actually solve the problem by themselves.

3. OK, I think I agree.

The list would just have to be prepared with some care so as not to be too vague or confusing.

4. I’d love to see an example of this.

5. We will see. The semester is ending and there will be finals… the only thing is, doing this on a final exam may be risky since there an be unexpected outcomes and students may feel even more stressed dealing with a question using this new format. Must proceed with caution.

I will make another post showing the question if this really happens.