Updated: Apr 28
The Covid-19 pandemic has upended our lives and our teaching, suddenly and without warning. Like everyone, I've been struggling with just keeping my head above water in the sudden shift. (I haven't even had time to blog, like I promised I would do more of this year). I have been mourning the loss of face-to-face interactions with students, as well as freedoms that I never realized I had. It has been jarring. We are all yearning for life as it was before, which--even if it wasn't perfect--was at least "normal" and familiar. But clearly life---and teaching---is going to be different from here on out.
But as James Tanton reminded us recently, we can shape what it becomes.
After weeks of forced "emergency teaching," I am now in more of a place of equilibrium, and it feels liberating now to actually *plan* some changes. Among them: what kind of questions I'll give on my final exam.
I speak often about how mathematical teaching often overemphasizes teaching specific facts or procedures, while underemphasizing all that goes into building mathematical explorers who have the habits of mind and confidence to solve problems they've never seen before.
In other words, we often overemphasize building skills rather than building virtues.
Virtues are aspects of character like: persistence, curiosity, imagination, a disposition toward beauty, creativity, strategization, and thinking for oneself. Any mathematician will tell you these are important qualities for a mathematician to have, and better yet, they are important for any other profession as well.
Both skills and virtues are important in mathematical problem-solving, but the latter is often shortchanged. It's usually only implicitly hinted at in teaching, if mentioned at all. One reason is that skills are easy to measure, but virtues are much harder to assess. This is why K-12 standardized exams focus on skills, and at least at the K-12 level, standardized exams unfortunately drive too much of what we teach. But if you talk with employers about what they are looking for in a mathematically-trained person, it is more often the virtues that they ask about. Can a student think? Are they creative? Curious? Do they have imagination? Moreover, the skills needed from math will change in the future, but the virtues needed from math will not.
Are these in the messages we explicitly send to students (and parents) about what a mathematical education is all about? Do we explicitly tell students: "One of my goals is to grow your mathematical persistence"? Or: "I want to give you a taste of sublime mathematical beauty." Or: "I am building in you habits of mind that will serve you well in tackling problems you've never seen before."
So even after writing a whole book about the way the proper practice of math can build virtue, and even after aspiring to teach math in this way, it dawned on me that these virtues have not appeared much in my student learning goals or the way I assess student learning. A standard college math exam usually looks like this:
Solve these N problems in this fixed amount of time.
Eek... that's what my exams usually look like. How do these exam questions show that I value the building of mathematical virtues? Of course, I always choose problems in which students will display persistence, creative thinking, and curiosity as they solve them, but do my students realize that these virtues are part of the goals I have for them if my exams just ask them to solve a bunch of problems? Why not have some questions on the exam where these virtues are explicitly called out?
So below are a few questions I am considering for a final exam. Notice that all of them are reflective questions that attempt to explicitly assess the development of virtues, like the ones I mentioned earlier: persistence, curiosity, imagination, a disposition toward beauty, creativity, strategization, and thinking for oneself. There are plenty more (developed in Mathematics for Human Flourishing)... but let's just consider these.
And they are all questions that can be used with untimed, open-book exams. During a pandemic, students are embedded at home and are not in environments where timed tests can be distraction-free. I don't think the pressure of timed exams will serve students well in these conditions. Why not try something different? And if you do have time restrictions, I encourage you to give out these questions beforehand. Students will appreciate the time to reflect and will give better answers.
Take one homework problem you have worked on this semester that you struggled to understand and solve, and explain how the struggle itself was valuable. In the context of this question, describe the struggle and how you overcame the struggle. You might also discuss whether struggling built aspects of character in you (e.g. endurance, self-confidence, competence to solve new problems) and how these virtues might benefit you in later ventures.
What mathematical ideas are you curious to know more about as a result of taking this class? Give one example of a question about the material that you'd like to explore further, and describe why this is an interesting question to you.
How has your mathematical imagination been enhanced as a result of taking this class? Give at least three examples.
Disposition toward beauty:
Consider one mathematical idea from the course that you have found beautiful, and explain why it is beautiful to you. Your answer should: (1) explain the idea in a way that could be understood by a classmate who has taken classes X and Y but has not yet taken this class and (2) address how this beauty is similar to or different from other kinds of beauty that human beings encounter.
Give one example of a mathematical idea from this class that you found creative, and explain what you find creative about it. For example, you can choose an instance of creativity you experienced in your own problem-solving, or something you witnessed in another person's definition or reasoning.
For any problems you cannot solve on this exam, suggest a strategy you might try to tackle the problem, and show what happened as a result. Describe any strategic gaps you were unable to bridge, and list 3 helpful insights that may help another person trying to tackle the problem. Doing so will earn you up to 1/2 credit on the problem.
Thinking for Oneself:
Choose one interesting problem from the text of medium difficulty that was not assigned. Describe why you find it interesting. Then either solve it, or find a solution online and work through it, using your own understanding to critique that solution and improve it. An alternative to this question might be: Write 10 true/false questions that illustrate a variety of ideas from this course that you might put on this exam if you were teaching the class. Give a key, explain the answers, then explain why you chose these particular questions and what you hope they will assess.
I teach mathematics at the college level, so you may have to adapt these questions to your own context and grade level. But these types of questions are appropriate at any grade level. You might be wondering now: how to grade such questions. I encourage you not to spend too much effort wrestling with that. The notes with each question suggest a natural rubric (e.g., 'describe 3 examples...') and I'd suggest, and announce, that you'll give full marks for very thoughtful answers where mathematics is correctly described.
You can see more examples of reflective questions that I've used before on this MAA blog. One thing I love about reflection questions is that they are much more fun to grade. And you'll see how your students think, which is one of the most important parts of teaching well.
Hang in there, everyone. I hope you are staying safe.
- updated 4/26 at 11pm to correct a few typos and fix some omissions.