By Dan Teague, posted October 26, 2015 —

Common advice for new teachers is to be sure to do all the homework problems before you assign them. This is good advice. Much of what is possible in our classrooms comes from our reputation among students (and their parents). When students trust you, you have leverage and leeway in trying new things. A solid reputation allows for creativity in your teaching, which is often rewarded with creativity in student work. Everyone wins.

Building a reputation takes some time. The first requirement from parents and students is mathematical competence—knowing your stuff so that the inevitable errors at the board we all make are shrugged off as just that, instead of being viewed as a warning sign that “my teacher can’t do the problems either.”

So think carefully about your homework assignments and the example problems you use for class and make sure they move student understanding forward. Be judicious in the number of problems you assign. There is no good way for reasonable students to do twenty-five problems a night other than for them to close their minds, put their heads down, and grind. No one can do twenty-five problems thoughtfully, so choose rich problems that are not repetitive. As Jo Boaler noted in her NCTM presentation in April 2015, there is almost no brain activity when doing repetitive problems, each one like the previous one. Choose problems that make students’ brains spark.

“Do all the homework problems before you assign them” is good advice . . . for a while.

Wowing students with your ability to immediately solve every problem without making any false steps is nice and builds the reputation so fundamental to your success, but it gives students a very odd (and quite false) sense of what doing mathematics is about and how real mathematical problems are done. Students can easily come to believe that, to be good at mathematics, they must be able to do every problem without error and without thought. So, if they are like most students and have to work at it, they can come to believe they must not be very good. Moreover, they believe that mathematics is done by remembering how, and it is only a small step from there to believing that mathematics should be done by remembering how.

This means that they believe it is not possible for them to do something they haven’t been taught.

Students need to see their teachers figuring things out. You should be prepared to be unprepared. That is, to work a problem for the first time, thinking out loud as you go, so students see that working out a challenging problem often involves missteps. And corrections. And playing around with the problem. Students need to learn how to decide whether the approach they are trying is making progress and when to abandon it and begin again. Most important, they need to learn how to look carefully at their errors and use them to find a correct approach. All these skills are essential to creatively using mathematics to model and understand the world, and all need to be taught as a natural part of learning and doing mathematics.

Moreover, if we assign only problems we have done before, then we will assign only problems we can do, and our ability becomes a limitation for our students. How will they learn to do things we can’t? How will they become better than us? After all, if our students aren’t better than their teachers, then we are moving backward. One of my prime directives in teaching is to not let the limitations of my talents be limitations on theirs.

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.