#### 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.

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