Metacognition… thoughts on teaching mathematical problem solving skills

Problem Solving: Where does the "F" go?

Problem Solving: Where does the “F” go? (Photo credit: Brian Metcalfe)

How do you go about teaching problem solving skills? How do you teach pupils the thinking skills necessary to solve mathematical problems in contexts that are unfamiliar to them?

These are difficult questions. I’ll declare upfront that I don’t have the answers to them fully. Nonetheless I have been researching and formulating possible approaches. This post shares with you where I am at with my current thinking and I hope will spark an interesting debate in the comments section where our readers can extend the ideas further.

I’d like to start by associating problem solving skills within the context of metacognition. Research evidence strongly concludes that the teaching of metacognitive skills has a significant impact on pupil progress. In a recent post about The Education Endowment Foundation I introduced readers to their work that shows which intervention strategies are effective on improving pupil progress. The teaching of metacognition and self-regulation has been found on average to improve pupil progress by +8 months. E.g. a pupil taught metacognitive strategies will progress by 8 months more than one not taught it over the period of one academic year. It’s clearly important, but how can it be done?

Let’s start by saying what it’s not. Metacognition has been a buzzword in the Thinking Skills initiative that many schools have participated in over the last decade. It has often been defined as thinking about thinking, which is a bit too vague to be useful. Lessons that try to develop pupils’ problem solving metacognitive skills are often in two forms. Either they are very example-led in the hope that pupils will make links in new problems based on their experience base of knowledge from seeing lots of problems solved, or they are open-ended where pupils are given many problems to solve and then those that solve them share their solution methods. The later case is really a more student-led version of the first type where we assume giving pupils many experiences and examples, that they will extract the common strategies and approaches for themselves. I think this is a big assumption and we need to be more explicit. In the same way that giving a pupil who can’t do column addition lots of questions to answer doesn’t move them forward if you don’t diagnose what their issue is, just throwing lots of open-ended maths problems at pupils won’t necessarily allow them to develop their own toolkit of problem solving strategies.

For some pupils we need to show them the toolkit and teach them how to use it one tool at a time.

A better definition of metacognition that I’ve have read is teaching one’s brain to control the thought processes it has for the purpose of directing it towards the management of their own learning. It’s like the conscious brain becomes a coach for itself. In the same way that we coach pupils by asking questions such as ‘what information is given in the problem?’ and ‘what areas of maths could we use to solve this?’, metacognition is when pupils ask these questions for themselves. This comes naturally to some, but not others.

I have a developing interest in Singapore Maths, the national curriculum and teaching methodologies used in Singapore. In their Ministry of Education document The Singapore Model Method for Learning Mathematics they define a set of Heuristics for Problem Solving that I think offer a starting point for categorising the various skills we use to solve problems:

Act it out

Use a diagram or model

Make a systematic list

Look for patterns

Work backwards

Use before/ after concept

Use guess and check

Make suppositions

Restate the problem in another way

Simplify part of the problem

Solve part of the problem

Thinking of a related problem

Use equations

Don’t these offer us a scaffolded structure for teaching problem solving skills? I think these heuristics act as a problem solving skillset that it would be possible to explicitly teach pupils. When showing them solutions to problems, couldn’t they examine this list and see which explicit strategies helped with the solutions? After seeing a range of different contextual problems where the key to their solutions was to make a systematic list, wouldn’t the pupils have a learned a valuable problem solving skill by looking in detail and focus at just one? Next time they could focus on problems that require using a guess and check approach and see the common strategy across different problems and contexts. Another lesson could focus on problems that require the drawing of an accurate diagram or model to solve.

Problem solving is a big skillset and something that needs breaking down into bite-size chunks. I think these heuristics offer a decent starting point. The next step will be to find problems that lend themselves to different heuristics so lessons can focus on just one skill at a time before they are then combined. The final step would be to get pupils to identify which strategies could be useful in solving problems. A flow chart could even be made based on the heuristics of things to ask themselves.

That just about represents the extent of my thoughts so far. I can see an opportunity for the teaching of metacognition to be more scaffolded and bite-sized for the pupils who need it that way, rather than some approaches that just label things as metacognition retrospectively when pupils have shown the skills. These thoughts need taking further. The heuristics need evaluating and possibly adding to. Problems that relate to particular heuristics need finding. There is much more work to do here and I hope you can contribute your thoughts and ideas in the comments section below!

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11 Responses

  1. Kathy says:

    I love your list. I’m going to put it next to my desk for encouragement throughout the year.
    The methods are not new, though. I’ve used books with chapters focusing on one method, such as guess and check. Watching someone use a method isn’t the same as learning it. In the same way, walking through a method isn’t the same as internalizing and understanding the method. I think teachers need to explain the method, like we might explain a joke that someone doesn’t get. Then, I think students need to reflect on the methods they have tried.
    The reflection part is something I’m trying to focus on more this year by using journaling. I usually explain methods, but this to often takes a long time before students are ready to hear and internalize. The test might come before they are ready. I’m hoping journaling speeds up the metacognition.
    Thanks for sharing your journey!

  2. Iris Aliaj says:

    William I really like your idea to explicitly teach problem solving skills in bite size chunks. I also really appreciate the emphasis on meta-cognitive skills and your specific definition of it.
    When I read your list, I thought it would be helpful for me to sit down with problems and practice metacognition myself, noticing what strategies I am using. In this way we can create the catalogue of problems that go with each skill.

    Kathy, I am very interested in your comment of using journalling to facilitate the development of meta-cogntive skills. I used journalling a lot last year for that very same goal and would love to talk with you more about it. I wonder how I can design journalling prompts and activities that thoughtfully and deliberately target certain kinds of thinking. So far for me it has been a lot of intuition work. How have you approached this process?

    • Sakayombo says:

      How do you help students in Secondary School understand the concepts of probability to enable them solve problems associated with probability using metacognitive strategies?

  3. Gil says:

    Very interesting article William. I am putting together a day for year 7 on problem solving strategies. I want it to be inspiring, dynamic and creative. I thought about revolving it around a Murder in the Maths Dept theme. Any bright ideas would be welcomed!

  4. Jim Doherty says:

    I have to assume that you are also familiar with Polya’s approach. I think his is a bit more compact. The idea of transferring these habits to students is SUCH a challenge.

  5. sepideh says:

    Dear William
    Thanks for your helpful notes, I want to know more about your method and results, so can i see your articles or more guide?

  6. Sakayombo says:

    Students in my classes have learning difficulties to understand probability. What mnemonic aids or diagrams can assist them understand the concepts and language of probability? How do I develop and use metacognitive strategies?

  7. oil wins says:

    Thanks for sharing your thoughts about memory function. Regards

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