11/12/2015 07:14

All that time on one problem

After working for an hour with a group of middle schoolers, I was recruited by a high school student to help with some calculus review. So we sat at a table and talked through limits and continuity on an artificial function. When we had covered all the main concepts and their application to the problem, the calculus student transformed into the driver who would take several of the middle school students back home.

Apparently the younger kids had been paying attention to our advanced math, for as the calculus book was being put away one of the waiting middle schoolers asked, incredulously, "You spent all that time on just one problem?"

"Yes," we replied, as if that sort of commitment were the most natural thing in the world. But we were not being fully informative about what we were doing.

We did use only the one problem as our guide for the entire time, but the conversation was more extensive than merely how to solve that one problem. For example, we didn't merely identify where the function is discontinuous; we also talked about what the mathematical concept of continuity means and how to test the continuity of a given function.

Our goal was to better understand the key concepts of limit and continuity. We used that specific problem as a device to facilitate that discussion. Our young interrogator was not imagining any goal broader than solving one particular problem. He saw math problems as a series of tasks to be performed; we saw them as a means to achieve greater mastery.

This difference in goals is a signal difference among players. In the actual reality game the more successful plays are those founded on broader, more integrative goals.