In a traditional exam, there is a strong focus on facts and techniques. For instance, in a course on linear algebra, students are asked to diagonalize matrices and they have to check whether a given set forms a vector space, etc. Sometimes I think that a well-trained monkey should also do this well.

Recently I read a report on a different kind of task in an exam which had the following style:

Assume $f$ and $g$ to be real-valued functions satisfying $f(1)=2$, $f(2)=3,$ $f(3)=6$ and $g(2)=5$, $g(5)=3$, $g(6)=3$. Is this information sufficient to calculate $f\circ g$ of the value $5$? If the answer is "yes", then calculate the value, if not then argue why.

I found this task very interesting since it seems very easy for students who are familiar with the composition of functions and it seems nearly impossible for students who are not.

Question: Where can I find more non-standard tasks for (written) exams that try to focus on deeper understanding?

I would like to concentrate on undergraduate level.

My method for creating a concept-testing question is as follows. Start with the most open-ended question possible on the topic, then slowly refine it until it has a "correct answer." I've outlined an example for a conceptual derivative question below.

• Start with the generic question: "Tell me about the derivative," and recognize that the generic question is not very good because it still has many terrible yet technically-correct answers. For example, a student might answer "Tell me about the derivative" with the correct but unsatisfying

The derivative of a function is another function, like the derivative of x^2 is 2x and the derivative of 4 is 0.

• Refine the question to eliminate the possibility of empty answers. In my example, you are displeased because the student didn't mention what the derivative measures! So you might end up with the question "The derivative of a function is another function. Discuss what the new function represents." Recognize potential bad answers that are still technically correct. In this case, we have the correct but unsatisfying answer

The new function tells you how to get the derivative at each point. The derivative of x^2 is 2x and so if you want the derivative of the function at x=3, it would be 2*3 = 6.

• If you catch yourself just trying to get them to say a particular word, your question is actually a short-answer question in disguise. You were trying to write a conceptual question. In this case we catch ourselves hinting at the word "slope." Forget that. Put it in the question and ask why that is in there. Our new question would become "The derivative of a function is a new function that gives the slope of the original function. Explain why, by using the definition of the derivative." This question is getting a lot better but you aren't being clear enough about what you want to hear. So, for example, you might get the following answer from a really good student:

If you look at a line, like y=3x+2, then the derivative is [calculation of the derivative using the definition] which comes out to 3. This is the slope, so the derivative measures the slope.

• I am now realizing that what I really wanted was for the student to pick apart the definition of the derivative and tell me why it is what it is -- not show me that the derivative works. So I settle on "The definition of the derivative of f(x) at x = c is lim h->0 [f(c+h) - f(c)] / h. (a) Draw a sketch of f(x) = x^2. Label c, h, f(c), f(c+h), and f(c+h) - f(c) on the picture. (b) In one sentence and using part (a), point out how the derivative formula is really a slope formula."

I used essentially the last version of that question on my exam 1 this semester in freshman calculus. I was mostly pleased with the results.

• Outstanding and well-described example. Thank you for bringing in this attempt. – Anschewski Mar 29 '14 at 21:03
• @MichaelE2 I think it is important for students to realize that they should justify their definitions as well: definitions can be "wrong". It is really important that students know why we define the derivative the way we do, and Chris's question is great for testing that they do know that. – Steven Gubkin Apr 12 '14 at 13:04
• If you read his wording of part b again, I think you will find that your criticism does not really apply. He is asking (in my view) the student to show that the function of $h$ which is being taken a limit of (the limitand?) is in fact the slope of a particular line. The definition of slope of a line comes before the definition of the slope of the tangent line, and it is crucial that they understand that the definition of the derivative is as a limit of slopes of secant lines. – Steven Gubkin Apr 12 '14 at 19:23

I disagree with the "diagonalize a matrix" type exam questions. Such questions call for accurate, longish computation, not understanding. That I leave for homework. In an exam I'd ask why you'd diagonalize a matrix (e.g. explain how to do something using this, step by step, not do it). Set questions up so that computation mistakes don't invalidate the work.

Comming up with questions testing understanding, that the student can rapidly grasp, and can solve in a few minutes if they know the subject, is hard.

You can try problems of math olympiads and reduce them in difficulty or take them as inspiration.

Problem archive of IMO

For example, 43rd IMO (2002) Problem 5.

• I'd be interested in seeing one or two examples; would you be able to take one of those questions and show how you would make it easier? – Brian Rushton Mar 28 '14 at 14:46
• I have downvoted because I find it hard to believe IMO questions test for deeper understanding. Math olympiads in general do not strive for teaching conceptual clarity but for timeliness, techniques and having a huge question bank in your head. – Mark Fantini Mar 28 '14 at 15:55
• I stand by my comment. How does your example test for a deeper understanding of functions? – Mark Fantini Mar 29 '14 at 2:18
• I suspended judgment on this answer until I saw the example added, but now that it is added I have downvoted the answer. I don't think the IMO is actually a productive source of good conceptual questions. – Chris Cunningham Mar 29 '14 at 20:17
• @Fantini In course on linear algebra involving diagonalizing matrices, if a student is not able to do elementary calculations on fractions, you will fight for him to not leave him behind? He'll be very grateful to you, but don't ask the other students. – Toscho Mar 30 '14 at 17:53