[Lowerbounds, Upperbounds]

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.