Does the problem-solving literature have anything of value to guide instructional decision making? The answer is that, although the literature on problem-solving instruction presents ambiguous messages, five results stand out:

1. Students must solve many problems in order to improve their problem solving ability.

2. Problem solving ability develops slowly over a prolonged period of time.

3. Students must believe that their teacher thinks problem solving is important in order for them to benefit from instruction.

4. Most students benefit greatly from systematically planned problem-solving instruction.

5. Teaching students about problem-solving strategies and heuristics and phases of problem solving (e.g. Polya’s, 1945/1973, four-stage problem-solving model) does little to improve students’ ability to solve general mathematics problems.

Source: “From Problem Solving to Modeling,” Frank Lester and Paul Kehle, in *Beyond Constructivism*.

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I like this, but I’m a little fuzzy on what systematically planned problem-solving instruction means. Is there any more detail you could give from the research?

I’m also curious if the research has anything definitive to say on how facility with problem solving transfers from one context to another.

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This list is one that, in one form or another, Frank Lester has repeated in a couple of places. (See p.94 here http://files.eric.ed.gov/fulltext/ED314255.pdf or his piece in the Mathematical Enthusiast.) My sense is that he means that problem-solving instruction needs to be specific, targeted, explicit, and students need a chance to practice whatever domain-specific problem-solving move is on the agenda for the day.

This is contrast with the unsystematic approach that goes something like “well when I’m walking around and my kids are stuck I ask them to make a simpler problem” or whatever.

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Yeah I think our whole project is to be able to describe item 4 in detail. The clarification on what it isn’t is useful (since that sounds a lot like what I do) but more info on “domain specific problem solving moves” and how they get organized into an “agenda” is my personal grail at the moment.

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That’s a great project, and it’s close to mine. ! I’d argue that the existing lists of “habits of minds” or “Polya’s strategies” aren’t specific enough to be useful (per item 5), so this work becomes immensely difficult.

Also, I suspect that it’s impossible to separate this project from curricular development, since once we become aware of a promising problem solving move we will want to create curriculum that gives students an opportunity to use that move.

And then it gets further complicated, since if we want to help students develop their problem solving moves over the course of the year, we need to articulate what that progression might look like. There’s no “natural” way that student thinking will develop for a given topic (sadly, we can’t just have CGI for secondary school).

That leaves us with two options, I think:

1. Describe the process of thinking through these questions

2. Get our hands messy

Re 2, here’s what I think the bare minimum would be for getting our hands messy:

* For a given content area (e.g. trig) identify problems that kids of a given age can almost always make sense of (e.g. find the steeper ramp) without any instruction, but that can be most efficiently solved using important mathematical skills and concepts (e.g. use right triangle trig).

* Describe the development of student thinking and strategies for that class of problem. (e.g. first the compare visually, then they start looking at the measurements of the height, then the measurements of the height and width, then ratios, then any ratio of side lengths).

* Articulate teaching moves/feedback/hints that might help students progress from one strategy to the next.

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I actually wrote “our project” but then changed it because I didn’t want to presume. 🙂

And I agree with your outline. It does seem like there’s a vocabulary of connections that we (teachers and such) have internalized that we can make explicit. Can you find a right triangle to use in this problem? If there’s a circle, is your unknown related to a radius? If there’s a square root in your way, you can square both sides or rationalize. Etc. For each context, there’s a subset of often-helpful connections to make. A generic list of “try one of these!” is less helpful than “If you see … you might try …”. The latter sounds like anathema (totally procedural) but “If you see a circle, see if you can find a helpful radius” isn’t exactly turning students to a blind, unthinking algorithm.

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Oh wait! I did write “our project” !

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Ah, I thought you might have meant “our [department’s/school’s] project.” I’m with you now.

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Well that too. It’s a big “our”, there’s room for everybody.

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Numbers 4 and 5 appear to contradict each other, but I suspect there may be more to number 5. It wouldn’t surprise me that general and out-of-context study of problem-solving heuristics would do little to build problem-solving skills, but sharing strategies that the students used to solve a problem seems to be effective in increasing the use of strategies. I also find labeling those strategies, whether descriptively or just “Sally’s method” can help make them more visible and more likely for the students to use them again.

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Maybe we can take some of the difficulty out of the process by worrying less about planning ahead, and focus more on dramatically (by that I mean theatrically… with posters and blogs etc.) documenting the development of problem solving skills as it happens. Build the toolbox organically from whatever the students already have; just make sure to keep track of what the students have discovered and reference it vigorously. I think that half of developing problem-solving is simply that idea, that problem-solving skill builds with each problem you encounter, and that there is no one set of problem-solving skills: everyone’s own history contains their unique set. The point is to be aware of what you know and keep pushing yourself to learn more.

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