In an earlier post I discussed the potential benefits of interleaving problems and questions in homework assignments.
Another interleaving-related idea came to me as I was reading a chapter from the National Research Council’s How People Learn: Brain, Mind, Experience, and School (2000). Chapter 2, “How Experts Differ from Novices” discusses the different ways that novices and experts approach problems. The chapter quotes a study by Paige and Simon (1966), in which experts and students were asked to solve algebra word problems like this one:
A board was sawed into two pieces. One piece was two-thirds as long as the whole board and was exceeded in length by the second piece by four feet. How long was the board before it was cut? (Paige and Simon, 1966, quoted in How People Learn, p. 41)
It turns out, of course, that this problem is impossible. Experts realize that quickly, while students try to figure out and apply the right procedure to get the answer without realizing the logical impossibility of it all.
What if students had to deal with impossible problems interleaved into their homework assignments, quizzes, and tests? Would we think that cruel? Or would it help prevent students from going on math-procedure autopilot while they work? What if they were told that one or two of the problems in their assignment were impossible? Would having them be on their guard and encouraging them to really read the problem help them or discourage them?
I think teachers could use impossible problems in a fun way that encourages students to work for a deeper understanding of what they’re solving. It might be all in the way things are framed. What do you think?