Expert Problem Solving

I've been reading Alan Shoenfeld's article Learning to Think Mathematically: Problem Solving, Metacognition, and Sense-Making in Mathematics, where he summarizes an interesting study about novice and expert problem-solvers.

Experts Don't Get Bogged Down

The study videotaped pairs of college students attempting novel problems and coded how much time they spent in different kinds of problem-solving activity. The following was typical of novice problem solvers:

"The students read the problem, quickly chose an approach to it, and pursued that approach. They kept working on it, despite clear evidence that they were not making progress, for the full twenty minutes allocated for the problem session. At the end of the twenty minutes they were asked how that approach would have helped them to solve the original problem. They couldn't say."

They did the same for pairs of math professors, and here is what they found:

From the article:

"The first thing to note is that the mathematician spent more than half of his allotted time trying to make sense of the problem. Rather than committing himself to any one particular direction, he did a significant amount of analyzing and (structured) exploring -- not spending time in unstructured exploration or moving into implementation until he was sure he was working in the right direction. Second, each of the small inverted triangles in Figure 4 represents an explicit comment on the state of his problem solution, for example "Hmm. I don't know exactly where to start here" (followed by two minutes of analyzing the problem) or "OK. All I need to be able to do is [a particular technique] and I'm done" (followed by the straightforward implementation of his problem solution) ....

... as he worked through the problem the mathematician generated enough potential wild goose chases to keep an army of problem solvers busy. But he didn't get deflected by them. By monitoring his solution with care -- pursuing interesting leads, and abandoning paths that didn't seem to bear fruit -- he managed to solve the problem, while the vast majority of students did not."

Teaching Self-Monitoring

A second study attempted to teach students this type of self-monitoring (though perhaps 'habituate' would be a better term). Shoenfeld had students spend approximately 1/3rd of their class time working on novel problems in groups of 3-4. His role was to be a roving consultant, and to press the students with the following 3 questions:

1. What (exactly) are you doing? (Can you describe it precisely?)
2. Why are you doing it? (How does it fit into the solution?)
3. How does it help you? (What will you do with the outcome when you obtain it?)

Initially, students can't answer his questions, because their way of working doesn't include thinking about them. Once they learn that he won't stop asking them, however, they begin to think about them ahead of time, before he asks. By the end of the semester, he reports, this type of thinking has become habitual.

Here is one activity graph for a pair of students who had completed this course.

He concludes:

"The point here is not that the students managed to solve the problem, for to a significant degree solving non-standard problems is a matter of luck and prior knowledge. The point is that, by virtue of good self-regulation, the students gave themselves the opportunity to solve the problem." [emphasis added]

This is like speed dating for math (if you'll indulge me the analogy for 2 paragraphs). Considering more candidate-strategies earlier gives you more chances of finding the right one for you. The difference is that speed dating prescribes a (short) fixed amount of time with each candidate. Problem solving would be more like speed dating if the speed-daters were left to decide for themselves how much time to spend with each person before moving to the next.

So how do you know when to move on? It's easy if you've run out of things to talk about, or if you don't know what to do next with your current strategy. But some people (and strategies) are seductive--it may take a long time to realize they're not taking you where you want to go. Savy and experienced daters, as well as mathematicians, learn to recognize signs of an interaction that seems promising or troubling. And this kind of subtle judgement is something that only comes with reflection on experience, I think.

What Now?

Thinking about this has suggested a few things I'm curious to try.

1). Brainstorm our options first. I might precede some problems with group- or class-wide brainstorming to get a short list of approaches that make sense to try.

2). Heuristics tell you what to try and what to look for. When I was teaching heuristics as skills, I focused on what each heuristic recommends that you try to do. Shoenfeld's questions make it obvious what this leaves out. Learning a problem-solving heuristic also means learning what to look for as you try, so you'll know whether the approach is working or not. It might also involve learning what you might do next with the kind of information you're likely to get. When first teaching students each heuristic through guided examples, I should make all three parts explicit--not just the first, as I had been doing.

3). Practice talking about all three parts. When a student was stuck, my first question was always "what could you try?" If they could give a coherent answer, I would often leave them to try. Instead, I should ask students all three questions: 1). What can you try? 2). What will tell you if it's working? (If you're looking for a pattern, can you quickly write a few types of pattern you might expect from prior experience?) 3). What then? (try to describe the pattern? connect its structure to the problem-statement? falsify it with test cases? represent it algebraically or graphically?).

Until students start to get a feel for how to answer the questions, this will probably have to be on a small-group or individual basis. Once students gain experience, I might stop all problem-solving work at 5 minutes, then at 10 minutes, then at 15, and have students write concise answers to these questions (no more than 1 minute each question).

4). Practice forced switching. Knowing how to make a mental bookmark of your current approach is immensely useful. It's required for high-level task-switching, but also to simply listen to and understand someone else's ideas. Being forced out of one's comfort zone is also sometimes required to experience the benefits of a new approach. Students might work for 5 minutes down one path, summarize their progress, then work down another path from scratch. This would be good practice for #3, and also let them know what it feels like to mentally start over, which is something that happens all the time in problem solving.