I remember that I read a blog post somewhere which tried to make a point on multiple choice questions for exams. I have the impression that mathematicians at German universities usually have the opinion that multiple choice questions are not an appropriate way to build an exam and to form grades. A frequent argument against multiple choice questions is, that it is important to check if the student has acquired techniques and is able to use them to solve problems; and this can certainly not be achieved by multiple choice questions. However, I think that there are certain points at which multiple choice questions could be the first choice even for this case.

In most exercise classes in the first courses in Analysis and Linear Algebra the following thing happens at least once: There is an exercise of the form “Prove that this and that holds.” or “Show that this object has that property.” Then some students hand in solutions in which there is a wrong argument hidden somewhere in the middle (while the general approach is ok). Then the teaching assistant tries to explain that the solution is not acceptable since the line of argument contains an error. And at this point the discussion often gets complicated: The discussion mixes aspects on the fact to be proven and the arguments used. For example, the teaching assistant tries to show that the argument is not valid by a counter example. However, this counterexample has the be considerably different from the fact itself (since this is true). The student respond that this is not a valid counterexample since it does not fulfill some assumption in the exercise. And there are several more problems which can occur during a discussion like that.

I think, that multiple choice questions are particularly well suited to put this problems into exercises. As an example, one may give the following questions:

Exercise 1 Which of the following arguments is not valid to show that the function ${f:[0,1]\rightarrow{\mathbb R}}$ defined by ${f(x) = x^2}$ is continuous?

1. For ${\epsilon=3/4}$ and ${\delta=1/2}$ it holds that ${|x-y|\leq \delta}$ implies that ${|f(x)-f(y)|\leq \epsilon}$.
2. For ${x_n\rightarrow 1}$ it holds that ${f(x_n) = x_n^2 \rightarrow 1}$.
3. The preimage of the open interval ${]a,b[}$ is the open interval ${]\sqrt{a},\sqrt{b}[}$.
4. For every ${\epsilon>0}$ one may set ${\delta=\epsilon/2}$ and then it holds that

$\displaystyle |x-y| < \delta \implies |f(x)-f(y)|<\epsilon.$

Exercise 2 Which of the following arguments is not valid to show that the unit step function ${f:{\mathbb R}\rightarrow{\mathbb R}}$ defined by

$\displaystyle f(x) = \begin{cases} 0 & \text{if }\ x\leq 0\\ 1 &\text{if }\ x> 0\\ \end{cases}$

is not continuous?

1. The preimage of the closed set ${\{1\}}$ is the open set ${]0,\infty[}$.
2. The preimage of the open set ${]-1/2,1/2[}$ is the closed set ${]-\infty,0]}$.
3. For ${x_n = 1/n}$ it holds that ${x_n\rightarrow 0}$ but ${f(x_n)\rightarrow 1\neq 0 = f(0)}$.

If this is given as a homework question which has to be handed in, one may add the sentence “Indicate why the argument is not valid and propose a correction if you think there is one.”

Although I did not test this kind of questions yet, I think that they could lead to interesting discussions is an exercise class since they somehow swaps to roles in the above mentioned situation.

While this is just one example which has come to my mind, I think that there are more situations in which multiple choice questions may be helpful. I still have to think more about situations in more advances courses\dots