A passing thought about problem-solving, while in the shower a few minutes ago:
Do students ever take the time to think about how they solve problems and the reasons — or lack thereof — that underlie the choices they make while solving one? Or is problem-solving, to them, just an unthinking act, to be completed but not considered?
What I have in mind are situations where a student does something in the course of a problem that has no basis, no reasonable connection to the context of the problem. Example: In a finance problem, students are given that the rent of an apartment goes up by 8% per year, and they are asked to find the year in which the rent will be double what it was originally; the student writes down the equation 0.08x = 100 and solves for x. I ask the student “Why did you do this?” and they have no idea why they did what they did; they just figure doing something would be better than doing nothing.
Which means that stopping and thinking about what would make logical sense as the next step in the solution is not an option for them; randomly generated algebra feels more like progress than this does. It looks as if they are getting done faster. And of course, they are brought up in a school culture where time-minimizing task completion (e.g. standardized testing) is the highest good. Going slowly is incredibly difficult for them.
What would happen to students’ conceptions about problem solving if we started asking them questions like this: “The rent on an apartment is increasing by 8% per year. You want to find out how long it is until the rent doubles. Give a thorough plan, detailed enough to be carried out by a classmate without doing additional work and justified at each point by logical connections to the problem and the mathematics you’ve learned, of how you would solve the problem.” In other words, ask them not to solve the problem but to explain how they would solve it. Or perhaps do this in an oral exam, or on videotape, and them have them watch themselves solve problems later. Would it change their minds?