A critical mass problem

One of the things I like best about the way we teach math at Park is that the problems themselves serve as intrinsic motivation. Sure, not every kid is perfect about doing their homework or working as hard possible, but we’re far away from the situation where it’s the impending test that motivates a kid to do their work. Most kids are interested in the conversation that happens in class and almost can’t help but give thought to the problems before them.

Every now and then I have a class that thinks that the material is too easy, despite my feeling that most students in the class are not giving the material the thought it deserves, and sometimes even despite the fact that I know there are basic skills that most students have not mastered yet. This could happen in a geometry class, where it’s easy to trick yourself into thinking that an informal argument appealing to symmetry, say, is sufficient, when actually a proof is needed. Or, if the topic is algebra, a “which of the two quantities is bigger” question: to which savvy students often know that the answer is almost always, “they’re the same size,” even if they can’t provide the algebraic justification.

Often, it’s very smart students who have this view – they’re able to intuit their way to an answer for some problems without needing to go through the thought process that the person who wrote the problem intended. It’s great if they can do that, of course, but they may be missing a chance to generalize their method to future problems. That is, they may be missing the core content of the class. Even more importantly, in their eagerness to get the problem done, they’re robbing themselves of the opportunity to be a mathematician. If a problem seemed dumb… what do you suppose you were supposed to get out of the problem? What is its larger significance?

In these situations, if I give a test that I feel is reasonable given my expectations of the students, they don’t do very well.

Because we only give tests once a month or so, it takes too long to give students the feedback that they don’t understand everything they think they understand. Part of me has the impulse, then, to give them quizzes to hammer home the point. But this is not really what I want to do. For one thing, I don’t want students in my classes to feel that they constantly have to be completely on top of the skills and content in the course. Too often, we are in discovery mode, where we are debating the appropriateness of the very skills I’d be quizzing on. It takes time for the dust to settle. And perhaps more importantly… is a test or a quiz the only way to give feedback to a student about how they’re doing? Shouldn’t there be a way to give that feedback more naturally? In most of my classes, when the majority of the students understand the spirit of the class and the exploration, students will let each other know if their arguments are too vague. In the type of class I’m describing, where there isn’t this critical mass, it’s harder.

The way I have dealt with this issue in the past is to collect homework more often, either for a small grade or just for written feedback. Still, I’d like a way to send a message to these students that even the easiest problem contains a world of follow-up questions, generalizations, and connections to other topics. A message other than “teacher says,” of course.

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