There must be nearly as many definitions for problem solving as there are mathematicians and math teachers. I am not going to pick through each of them, but I do want to present my own, as I try to clarify what I mean by problem solving and what I want it to look like in my classroom. Here’s my stab at it.
A student engages in problem solving when a problem or situation requires that student to develop an approach by transferring prior knowledge to the context at hand.
This seems pretty simple, but I want to pick two fights here.
Puzzles Are Not (Necessarily) Problem Solving
First, I am going to exclude problems like this:
A farmer wants to cross a river and take with him a wolf, a goat, and a cabbage. There is a boat that can fit himself plus either the wolf, the goat, or the cabbage. If the wolf and the goat are alone on one shore, the wolf will eat the goat. If the goat and the cabbage are alone on the shore, the goat will eat the cabbage.
How can the farmer bring the wolf, the goat, and the cabbage across the river?
Maybe this breaks your heart. But thinking about problem solving in school mathematics, I don’t think it fits — I would label it a puzzle instead. Maybe there are connections that I am missing to the mathematics that my students are learning, but when I think about the insight necessary to solve this problem, I don’t see it as a principle that connects with the mathematics that K-12 students are asked to do. This is a difficult problem to solve, but I would argue that mathematical skill is neither necessary nor sufficient to solve it, and that the learning from solving the problem does not transfer to future mathematical skill. The solution does require a sort of cleverness that we always hope our students exhibit, but I don’t see a concrete connection between that and helping students become expert K-12 mathematicians.
I do think puzzles have value in giving students opportunities to be metacognitive or growth mindset oriented, or just to enjoy math class, and I think many puzzles do this well. I still don’t want to call them problem solving. This might seem like a silly distinction, but as I put together my ideas for how I want to teach problem solving, I see the cognitive activity that goes into solving this problem, and problems like it, as different from the daily problem solving I hope to see from my students.
Problem Solving Depends on the Student
My definition also deliberately includes any math problem that requires transfer for a student to solve. Solving the equation 5 – x = 3, or finding 15% of $24.00, could be problem solving for a certain student in a certain situation. I want to move away from the dichotomy between “problem solving” and “not problem solving”. I believe that the skills and mindsets of problem solvers are the ways that students should be learning and practicing math every day. I don’t want to single out any specific problem as problem solving because I think that the basic idea — considering what prior knowledge applies to a new problem — should just be what we do in math class, beginning with the more concrete and specific and moving toward the abstract and general as study on a topic progresses. We should be posing students with problems of varying difficulty, that require varying degrees of transfer, every day. A mindset of “now we’re going to do some problem solving” seems to imply that whatever it is we do to facilitate student problem solving does not apply to “normal” math. I think that it also leads teachers, in the name of “teaching problem solving” to give students problems that are too difficult too fast, or spend too much time on simple problems to “build the foundation” before throwing students into the deep end without the supports they need.
Problem Solving and Modeling
I think it’s worth mentioning that, while the degree of transfer that I’m singling in on is one major component in problem selection, there are others. One that comes to mind is the degree of mathematical modeling required — the degree to which the problem brings in the messiness of the real world, requires students to complete the modeling cycle, and put together a coherent model for application in a certain situation. This spectrum might look like this:
But the degree of transfer is what I am focusing on here, and if I want to put together a set of ideas ready to use in my classroom to teach problem solving, I am going to narrow my perspective to that axis — to what degree must the student transfer prior knowledge in order to solve the problem.
Summary, and Next Steps
I’m deliberately moving away from what is usually “problem solving” in conversations in the faculty room. I think that this conception of problem solving focuses too much on the cleverness of the problem and the novelty of the context, and too little on the way a student interprets the problem and the relationship between the problem and the mathematics being learned. I don’t think those problems I am labeling puzzles are useless, but I think they sideline “problem solving” to be “something different” rather than a cohesive part of math class, woven in with content instruction.
While this definition is useful for my purposes right now, I’m working to build from it a coherent idea of how I think problem solving can be taught, and what that can look like in my classroom. Check back to see where I end up.