How To See It

“Our interpretation of Polya’s heuristics is that the strategies are intended to help problem solvers think about, reflect on, and interpret problem situations, more than they are intended to help them decide what to do when “stuck” during a problem attempt.” (From “Problem Solving and Modeling”)

This interpretation makes sense of Polya’s stated lack of interest in providing helpful, specific advice. I could buy it.

2 thoughts on “How To See It

  1. I’m going to try out a line of reasoning here, curious for your thoughts on it:

    A fundamental part of problem solving, in my mind, is that the problems being solved are “novel”, in that the specific context, topic, or question type is not one the student has seen before. We can break that down in lots of ways that are useful or not, but when thinking about problem solving strategies, if they exist, they must be able to be transferable, so that they work in multiple contexts and help students solve new problems.

    Let’s say for the purpose of my argument that the specific classroom structure a teacher is using is to pose a new problem to the class (here’s an example: ), observe students work, and debrief at the end (a Five Practices discussion would be another example).

    Let’s say students are engaged in the problem, and some subset of the students utilize a problem solving strategy that the teacher feels is useful for the rest of the class. Here’s where I think these heuristics can be particularly useful:

    Quick detour: I’ve been spending a bunch of time recently with Brown, Roedinger & McDaniel’s “Make It Stick”, and one piece they talk about in reference to learning is what they call “structure building” –

    “the act, as we encounter new material, of extracting the salient ideas and constructing a coherent mental framework out of them … High structure-builders learn new material better than low structure-builders. The latter have difficulty setting aside irrelevant or competing information, and as a result they tend to hang on to too many concepts to be condensed into a workable model (or overall structure) that can serve as a foundation for future learning” (2014, p. 153).

    According to this book, the research is inconclusive on whether structure-building can be taught, but there is some evidence that questioning can improve the performance of low structure-builders in making these connections.

    So back to my line of reasoning —

    I’m pretty skeptical of the idea of “teaching” a single problem solving strategy, in the sense of telling students that it is helpful to draw an auxiliary line, and then to expect that to make a meaningful impact on their ability to then solve those problems. Maybe I’m off-base there, but I see another use for Polya’s list of strategies — each strategy names a problem-solving schema, and this schema is useful in making those connections. Students don’t see the strategy for solving a problem as an isolated trick, but part of a larger structure of sense-making that they can use to solve future problems.

    There isn’t a ton to this — I’m really just proposing the potential of eliciting, naming, and connecting specific strategies, which ideally are connected and named across several lessons to build a structure that students are fluent in and able to apply in the future.

    I have a number of questions about that. What is the most useful course for students who are unsuccessful? Partner/group work? Structured discussion? Direct instruction? This also limits the transferability to a single strategy, and doesn’t teach problem-solving in general. This also says nothing of the background knowledge and preconditions for the problem-solving process to start.



    1. each strategy names a problem-solving schema, and this schema is useful in making those connections

      Dear Future People: this link to Polya’s The Pattern of Two Loci is likely to break, but it’s directly relevant to Dylan’s interpretation of Polya.

      I think this is getting close to how I see things, but I’m not sure.

      If I understand you, then I agree that the long-term trajectory of a math life is to organize more and more moves, strategies and techniques into meaningful categories that help us make sense of new situations. But I worry that the big-picture structures that Polya offers (if we’re going to read him as offering them) are still too abstract and general for nearly all k-12 students.

      Schemas can be general or specified just like concepts and strategies can. It’s true that set theory can describe all of mathematics, but that doesn’t mean that 3rd Graders should be expected to meaningfully see this sky-high conception of math. Similarly, Polya might be describing schemas that are true but still too general for our students.


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