I believe that one of the most important ways that teachers engage with high-performing students is by convincing them that effort matters. Bright students can miss this message in their early years, especially if they are not excited by the mathematics that they find comes easily to them. Since we obviously cannot demand the same level of struggling from all of our students, we must convince our high-performing students to take pride in their excellent work, even if only to see how excellent they can make it. That is why I believe I must deliberately carve out a place in my assessment system to reward diligence (and, alas, to penalize indolence). The most obvious way to do this is through homework.
I try to pitch my homework assignments toward the middle of the class, and I do expect everyone to do them. A high homework grade can be a great help to a diligent student who struggles on other assessments, while a low homework grade can deliver a clear message to a lazy student who tests well. It is very important, however, that students understand that the purpose of the homework is to give them something that is absolutely essential to their education (and not just in mathematics): experience in solving problems. While this is more important than ever in today’s educational environment, the message is ironically becoming more difficult for students to understand.
Armed with search engines, today’s students are much better equipped to find answers than to solve problems. Indeed, the answer (and perhaps even the complete solution) for almost any textbook exercise is probably available somewhere in cyberspace to a student who is motivated to look for it. I now spend two days at the beginning of the year explaining to my students as clearly as I can that it is the problem-solving journey on the way to the answer that will further their education, while the answer itself is worthless without the journey. In particular, I emphasize that each homework exercise must challenge them to do two things: (1) learn how to do the math, and (2) do the math. Then I try to make them appreciate the difference between those two tasks. When they are learning how to do the math, I urge them to use the book, their notes, and other resources, and I encourage them to collaborate with their friends, just so long as they are earnestly trying to learn how to do the problem. Once they have learned how to do the problem, however, they should write up the solution themselves. Collaboration is good; copying is bad. These are subtle distinctions for adolescents to make, to be sure, but they are growing up in a world where they are forced to confront them now or run the risk of wasting the most important years of their education engaged in the trivial pursuit of hollow, short-term goals.
I realize that some teachers prefer that their students not collaborate on homework at all, but I am a big believer in it, just so long as it is done right (at the learning stage, not the doing stage). In fact, I have found that such collaboration allows homework to stimulate better classroom discussions the following day. The students will have settled many of the simple issues among themselves, and the questions that reach me are more focused, refined by their earlier conversations. Even when they tell me “none of us could figure this one out,” I figure I am being handed a Teachable Moment!
One advantage of having dozens of homework grades recorded for each student is that you can afford a policy of “dropping” a few of each student’s lowest scores. I do this in recognition of the fact that even the best of students will have to make choices on some nights, and occasionally that might mean skipping my homework in order to (for example) polish a term paper. The down side of grading diligence is that some students will accumulate many more zero grades than you will drop, thus ensuring a low homework average. If they do well on other assessment, they might merely change a good overall grade into mediocre one, which I believe is entirely appropriate (and, I hope, instructive). If they do badly on other assessments, they will be burning their own safety net. In either case, they should be counseled about the consequences of their own choices.
At this point, experienced teachers will no doubt be wondering how I can possibly grade homework for all my students every night. This is an excellent question, and my reply will expose an aspect of my grading system that might not be replicable at other schools. I choose a random paper from each class and grade it with sufficient annotation that it can be used as a key; then I give all the papers to that student, who becomes the grader for that assignment. When the papers come back to me, I enter the grades and return them to the class. For the most part, students take this responsibility seriously and do a pretty good job; indeed, some students really get into it, adorning the papers with congratulatory comments and happy stickers when appropriate. Peer pressure is usually sufficient to get the student graders to get the papers back within a reasonable time, but I also let them know that my policy of dropping some low grades is dependent on the number of grades in my book. The policy is not perfect, but it does allow me to assign homework almost every night and hold my students accountable for doing it. In the end, I have the data I need to make diligence a significant factor in my assessment system. If we value diligence, we must communicate that to our students, and then we must assess it.
Next month I will offer some ideas on how to assess knowledge and cleverness, the usual hallmarks of the high-performing mathematics student, while keeping the rest of your students “in the game” until the end.