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“To doubt everything, or, to believe everything, are two equally convenient solutions; both dispense with the necessity of reflection.” -Henri Poincare, Science and HypothesisDo your exams accurately represent what you value in your course? Only after many years of teaching did I begin to ask that question.
For instance, one of the goals for my upper division courses is for students to be able to articulate what mathematicians do. Another goal I have is for students to learn to generate their own questions for further investigation. Even though I might have seen a student exhibit such skills in the occasional conversation, the tools that I greatly valued were not showing up regularly in how I evaluated student progress.
Why Use Reflective Exam Questions
To give students opportunities to demonstrate these reflective skills, I began to assign reflection exercises as exam questions. There are certainly other ways to elicit such information--for instance, you could assign research papers or reflection journals--but I was interested in something that wouldn’t be additional work. Putting a question on an exam was a simple way to signal to students that I cared about their ability to process and reflect on what they were learning, in addition to the mathematical reasoning I expected them to demonstrate.
What I didn’t anticipate was the benefit reflective exam questions would have for me!
First of all, these questions made exams much more interesting to grade. (If you know me, you know that while I love teaching, I have never enjoyed grading. The monotony!) Now I say to myself: ‘if you grade all the other questions, then you get to read the reflections!’ Without reflective questions, the exams show very little of my students’ personalities. Having reflective questions helps me see the unique ways my students are thinking and feeling, and that gives me joy.
The second reason for adding reflective questions to my exams is that I often learn things from my student responses that help me become a better teacher. Sometimes students will explain an idea in a way that I had not considered. For instance, in reflecting about the importance of definitions in mathematics, one student described a definition as a choice of what conversation you are going to have with the material. That’s a metaphor that I now use in my own teaching!
Assigning Points to Reflective Exam Responses
My advertised grading system for such questions is simple: give me a thoughtful answer, and you’ll get full points. Less thoughtful responses get slightly fewer points, but students rarely fail to give thoughtful answers. That also makes my heart happy! Depending on the question, you may wish to give your students the question in advance, so they will have time to think of thoughtful answers and they can reflect on it as they study for their exam.
Below are five examples of questions I have used in the past, and some actual responses I have received.
- What three theorems did you most enjoy from the course, and why? Choose one theorem of moderate difficulty and reconstruct its proof.
I like this question, because the answers often surprise me. What I think is interesting is not always what they think is interesting.
One student responded: The moment in class where I was truly blown away was when we applied Van Kampen’s Theorem on the torus to derive its fundamental group...the simplicity of its application is a moment I will never forget.
- Formulate a research question related to the course material that you would like to answer. (You do not have to answer the question. Just ask a good question whose answer is unknown to you, and doesn’t have an obvious answer based on what you know from the course.)
One student responded: Is there a classification theorem for 3-manifolds? (This came after we had discussed the classification of surfaces.)
The main value of this question is that you signal to students that you value question-asking and conjecture-making. But students often rediscover questions of historical significance that lead to important conjectures or theorems. In such cases I have an opportunity to affirm the student’s intuition for asking a good question, as well as to answer it.
- Reflect on your overall experience in this class by describing an interesting idea that you learned, why it was interesting, and what it tells you about doing or creating mathematics.
One student responded: One interesting thing I learned from the class was the equivalence of open-cover compactness and subsequential limit compactness. Both of the definitions are quite abstract, but both end up being extremely significant in their consequences. I think this seeming disconnect between definition and consequence emphasizes the importance of definitions in mathematics. Definitions essentially frame the type of conversation you are going to have--some definitions that seem different produce conversations with similar results. Many definitions lead to conversations with results that are hard to predict.
What a thoughtful answer! I learned a new way to explain the importance of definitions from this response.
- How did the ideas of this course enlarge your sense of what it means to do mathematics?
One student responded: This class gave me a much better understanding of what it means to do mathematics than I had in the past. Most of our problem sets in other classes were applying theorems that we learned in class, and the problems were roughly of comparable difficulty. However, with this class, we did much of the learning on our own, through results that we proved. In addition, some of the problems were relatively straightforward, but there were several very challenging problems, where my group didn’t even have a clear idea where to start. This seems much more realistic to the life of a mathematician, where problems don’t present themselves in homogeneous sets.
From this response, I could see that my student was able to articulate what mathematicians do. Goal accomplished.
- I have emphasized the importance of struggling in mathematics: that it’s normal and part of the process of learning. Describe an instance, so far in this course, where you struggled with a problem or concept, and initially had the wrong idea, but then later realized your error. In this instance, in what ways was a struggle or mistake valuable to your eventual understanding?
This is one of the best reflective questions I have used. The prompt helps my students recall specific struggles that have helped them, and reinforces a theme I have emphasized in class. If you’d like to see a wonderful response to this question, you might enjoy my MAA FOCUS magazine article “The Value of Struggle.”
I’m sure you can think of other reflective questions that can advance your goals for your courses. Student reflections will help your students grow as learners and will help you grow as a teacher too.