One comment I received long ago – not on an end of course evaluation, but rather immediately after a particular lecture – changed the way I taught. It was in an intro calculus-based physics class. I knew the students had found that latest homework assignment a bit of a challenge, so I went through the problems – not just solving them to get answers, but carefully setting them up to show their similarity in approach and how to specialize each set up to correspond to that particular problem. My claim was that if you could set the problem up correctly, the rest was just manipulation to lead to an algebraic solution.
This particular student caught up with me as I left class and thanked me for going over all of that – then said, “I could follow every step – and it made so much sense. But what I want to know, is how did you know to make that step? I mean how do you make those decisions in setting up the problems?” I thought it was a very insightful question about problem solving.
Problem solving is just decision making. You want to ask yourself the right questions in order to be able to decide how to proceed with setting up the problem – formulating a “solution pathway”, in the words of one of my honors students. And that means deciding what the problem is about. That is, is it fundamentally a force problem or a motion problem? What principles apply – momentum conservation, energy conservation, both, neither? Are there special conditions we need to consider? And so on.
What that young woman asked led me to show my classes how to think about the problems in order to make the decisions needed to set up the solutions. It’s the difference between learning problem solutions and learning how to solve problems.