## Good Teachers Have More Than One Model

When students seem hesitant to ask questions in class or come to my office, I often say, “Please ask me questions.  Every question you ask makes me a better teacher and I want to be a better teacher.”

A student came by and asked me to look over her solution to this problem.

Her solution was overly complicated.  I explained that one could reason from the fact that the hour hand goes $2\pi$ radians in 12 hours to get a set up like this, ,noting that I had radians over radians and hours over hours.  Her original solution was correct if long and she didn’t want to rewrite it so I just told her to explain all her steps thoroughly.

A little bit later she knocked on my open door again.  She need to change rpm(revolutions per minute) to radians per second.  I built this edifice, ,showing her how the units canceled.  She asked several questions and concluded, “So you put the units where you want them and work from there?”  “Yes,” I said.  She seemed to have got it and left satisfied.

Later on it occurred to me that if she had asked the questions in the reverse order, she would have gotten a different explanation for the clock problem.  Since she would have seemed to understand dimensional analysis from the rpm problem, for the clock problem I would have reasoned with her that since we know there are $2\pi$ radians in 12 hours we can place elapsed time as shown here  and  get the answer through simple dimensional analysis.

With another student I might have reasoned in this different way.  Elapsed hours over twelve hours is the portion of $2\pi$ radians that the hour hand turned so  this expression gives the correct answer.

Students as they learn are building models in their heads.  We don’t know which one will “take” and there is no reason to assume that it will be our own personal favorite model.  Teachers need to collect as many different models as they can and not try to force a rote method on their students.  The more questions I get the better teacher I become.