By Dan Teague, posted September 14, 2015 —

A few weeks ago, in preparation for the new school year, I took some time for my annual mid-August ritual. Each year it’s the same. Once I know my new teaching schedule, I think about my goals for each course and what I would like the residual for each course to be. The residual, of course, is what is left over, what sticks around, after the course has been completed. The residue is the knowledge, skills, and beliefs the students take with them, not just into the next course but throughout their lives. It includes the lasting impressions on their view of mathematics and on themselves as learners and users of mathematics. I think about what I want the answer to the residual question to be and, then, about how I should teach to make (or at least allow) that to happen.

A thought experiment is often helpful. Pick a class you are currently teaching. Suppose that, instead of having the end-of-course exam at the end of the school year in June, the exam for your course is to be given the following June—in June 2017. And your ability as a teacher is to be judged on the basis of your students’ performance a year after they leave your course. That is, suppose that what is being tested is the residual of your course. Would you teach the course in the same way you do now? Would you lecture more or less? Would you have the students engaged in more activities or use fewer activity-based lessons? Would you assign more repetitive homework or less homework that is more thoughtful? And if the way you would teach if the exam were given the following year is different from what you are currently doing, what does that mean?

Such thought experiments help me focus on keeping my courses active and the students engaged in thinking through and creating their own solution paths rather than only practicing techniques I’ve shown them. It encourages me to have more mathematical modeling in each course. Certainly, students cannot create all of what they need to know, but together we can spend significant time and intellectual energy on developing and applying the big ideas. We can aggressively seek to develop a healthy investigative attitude that values their creativity and ability to work together as important, even essential, steps in developing a robust mathematical residual.

As I go through this annual reverie, two quotes always spring to mind. They are the residuals of an article I read many years ago. The ideas are not mine, but they resonate strongly with me, even though the authors’ names are lost in memory. The first is a reminder that the students in our classes represent the hopes and dreams of their parents: “These are the best students their parents have. They are not keeping the good ones at home.” And the second is a comment about our students’ understanding of their relationship with us, their teachers: “No matter what you think about the intellectual ability of your students, you can be sure that they are smart enough to know whether you believe in them.”

So, as the new school year begins, I resolve again to teach mathematics to all my students as if their very futures depend on it. Because, in some fundamental ways, in tomorrow’s world, it does.

DAN TEAGUE, teague@ncssm.edu, teaches at the North Carolina School of Science and Mathematics in Durham. He is interested in mathematical modeling and finding problems that connect concepts from different areas of mathematics.