I think I found a clue about my bimodal distribution of grades.
Two weeks ago I watched a live video feed of the Escape from the Textbook conference. Paul Zeitz opened his presentation by pointing out that exercises and problems are different. That’s when the gears in my head starting grinding like a seized transmission, and I missed most of the rest of what he said.
He elaborates in his book The Art and Craft of Problem Solving. According to Zeitz, exercises are things you know how to approach. You might not know the answer right away, but you know what technique will work. When you are tackling a problem, you don’t know yet which technique to use — your task becomes investigating or maybe inventing techniques.
What I realized, uncomfortably, is that I tend to teach and assign exercises. Then I test problems.
Result: my tests were not differentiating between students who could use the techniques and students who couldn’t. They were differentiating between students who had pre-existing problem-solving skills and those who didn’t.
The students would complain that they needed practise beforehand with the same kinds of problems that would be on the test. I would complain that they hadn’t really understood the techniques, if they were memorizing how to apply them to a specific kind of problem. All of us were right, sort of. I had no idea that my students didn’t know how to do this.
As mortifying as it is to realize that I was blaming my students for a mistake I made, I can at least say that we used these situations to talk about learning and problem-solving. We developed the analogy of “the maze”, which is the twisty, unlit path between you and the solution. We talked about the difference between the kind of confusion you feel while inside the maze and the kind you feel when standing outside the maze before even having opened the heavy, scary-looking door. We shared techniques for “stepping inside the maze” — picking a strategy that might be helpful and following it as far as it leads, even if you can’t see the exit. (You definitely can’t see the exit from outside the door, so what’s to lose?) . We talked about keeping your eyes open, while following your strategy, for clues (you can’t see those from outside the maze either). This helped. Students tried to step inside the maze, and we at least had a vocabulary for talking about the uncertainty and fear they felt.
Now I realize it’s not enough to teach them the techniques of exercises. I must also teach them how to evaluate and maybe even invent techniques — so that they step inside the maze with a plan, rather than aimlessly. Even if the plan turns out not to work, it’s important to practise choosing one and checking afterward how you could have chosen better. My lessons no longer stop at “solve for the circuit’s time constant” or “solve for the circuit’s impedance”. They go on with “what characteristics tell you whether you should use the time constant or impedance” and “are there any circuits where neither one applicable? How can you recognize them? What could you do then?”
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I’m intrigued by this distinction between exercises and problems, especially because I so often use these words interchangeably. Your post makes me realize there is a real and powerful difference between the two, and that I need to be more deliberate about my language choice. I think there’s absolutely a place and time for exercises, particularly in building fluency, but we could probably make much more room for problems in our teaching– and then feel justified in assessing them (because we know we don’t want to assess exercises).
[…] to Level 4, and that’s an 80%. A Level 5 question is the name I give to questions that are not exercises but actually problems for most of the class. There are a number of ways to get a 5/5. All of them include both […]