Updated: Apr 28, 2020
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.