For more than 30 years, I have announced to each new class that mathematics is a participation sport, not a spectator sport. Before the mid-1980s, what I meant was that students were supposed to take my carefully chosen words of wisdom back to their dorm rooms and work hard on an assignment in order to understand the mathematics du jour. Only a few did that, of course. Most did not consider it important to participate -- other than on routine homework exercises -- until it was time to “study” (i.e., cram) for a test. And then their objective was not understanding but a grade.
My past behavior was analogous to a coach using the practice session to diagram plays on a blackboard -- so that players would not have to read the playbook -- and then sending them away to practice alone until game time. When it was time for the contest against the other team (Ignorance), they were expected not to communicate with each other, but to run their plays in isolation.
For most of my career, I had little sense of mathematics as a team sport, so perhaps my analogy for past classroom activity should be with tennis or golf instead of basketball or soccer. My practice time was devoted to lectures on the finer points of backhand or approach shots, with instructions to practice alone -- which students would mostly neglect to do -- until just before the tournament. Indeed, mathematics-as-participation-sport is both an individual and a team effort, but it doesn't make much sense to practice either type of activity alone. It makes even less sense to devote all the available coaching time to making students listen and take notes.
The metaphor of coaching is common among those thinking about and working on reform. We know what it takes: a mixture of short bursts of instruction and/or demonstration, longer practice sessions, motivational pep talks, the challenge of real competition, and assistance with in-game strategy. But we can't coach players who won't come out for the team, or who sulk instead of play, or who quit when they don't make the starting team. To keep our students engaged in the game we coach, we have to convince them that the pain and struggle are worth it -- that the game can be fun to play, that it is more fun if you work harder at it.
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Send comments to the author <das@math.duke.edu>Last modified: May 15, 1997