The problem with problem solving (or, why I struggle to reset my clock)

When the clocks went forward in March and we arrived in British Summer Time, I made an abortive attempt to change the time on my car’s clock. I knew, from having eventually changed it six months ago, that this is a process entirely within my grasp and yet, after about 10 minutes of frustrated fumbling, I’d only succeeded in moving the time forward by 20 minutes. I gave up and resigned myself to having a clock that is 20 minutes fast for the foreseeable future. 

This has resulted in a few moments of confusion and panic over the past few weeks. Things came to head when I rushed to get my daughter to her music lesson last week. There’d been a fair bit of bitterness, recrimination and stress, only to find we had actually arrived ridiculously early. Sod it, I said, it’s time to get the manual out. Perhaps unwisely, considering how high feelings were running, my daughter scornfully pointed out that this was exactly what I’d eventually done six months ago. Quietly seething, I looked up the instructions and changed the clock to the correct time.

This is, I think, the problem with problem solving in a nutshell. The ability to solve problems is an evolutionary adaptation; very young children rapidly learn how to solve the various problems they encounter. They look about themselves to establish what means they have available and then apply these means to achieving their end, whatever that may be. In this way, they learn to roll over, fit blocks together, and get adults to do stuff for them. Most of the problems we solve are ones we need to solve again and again. As such, the most efficient way to solve problems is to remember the solution and apply what we did before. With sufficient repetition, we store the solutions to solved problems in long-term memory, which means when we face the same problem again we don’t even have to think about the solution. How efficient is that?

Problem-solving search is the name cognitive scientists have given to our biologically primary ability to solve problems. The two main strategies involved in problem-solving search are schema acquisition (remembering successful solutions, then recognising similarities between novel and previously solved problems) and means-end analysis (working backward form the goal until a workable solution is found). Most of the problems we face in life are ones we have already solved, or are close enough that we can generalise solutions from closely related problems. Schema acquisition is the key to expert performance within a domain, allowing experts to think about other, possibly more interesting things as they go about their business. But, when novices face a problem for which they don’t have a conveniently stored solution, they have to think.

In this paper, John Sweller explains why problem solving is an inefficient way to build up the schemas required for expert performance. The attention and processing required to engage in means-end analysis results in less capacity to store solutions in long-term memory in a way we can be easily accessed in the future. In order to build up useful schema, Sweller says “a problem solver must learn to recognize a problem state as belonging to a particular category of problem states that require particular moves.” This takes attention which, if it is being used to search for solutions, will not be available to recognise patterns.

Back to the clock in my car. Because I’ve successfully solved the problem of how to rest the clock, I’ve stored the solution in long-term memory, so why do I struggle to retrieve the solution every six months? This is the difference between familiarity and recall – although I’m familiar with the solution (that is to say, I recognise it when I look it up) I can’t bring it to mind when I want it. My car clock schema isn’t well enough developed for the precise solution to be stored, ready for recall. If I were to regularly practise changing the time then I’d quickly strengthen my schema and would, in due course, permanently store the solution. But I don’t; I only try to change the time twice a year. And because each attempt involves means-end analysis, all my working memory is involved in trying to solve the problem rather than trying to remember the solution.

This is exactly what happens in lessons. If students spend lesson time trying to solve problems – whether that’s writing an essay, choreographing a gymnastic routine, or using a computer program – they’ll be applying the only means a novice has available: means-end analysis. Even if they successfully solve the problem, the likelihood is they won’t remember the solution. But, if students spend lesson time having the patterns of problems explicitly modelled and then practise retrieving solutions, they’re likely to build up the schema necessary to solve future problems.

52 Responses to The problem with problem solving (or, why I struggle to reset my clock)

  1. Perfect analysis on the problem with problem solving. In maths this takes the form of rich tasks, extended problems, multi component questions, investigations and questions requiring just the schema described in the article. I’ve just taken a lower ability year 9 class with appalling prior behaviour on a systematic stepped algebra course, culminating on a chapter with problems in other contexts that require algebraic solutions. Each sub component of solving equations drilled in separate lessons, leading to procedural fluency in solving linear equations. Curiously, the chapter involving worded questions and diagrams went extremely badly on the first attempt. They needed explicit worked examples similar to the questions they were faced with. So, with clear didactic modelling on the board, they gained confidence and became increasingly independent in solving the problem solving questions. They had all the schema required to ‘do the math’ but couldn’t connect this to the problems at hand. Take away message; even problem solving needs explicit instruction, and filling a pupils mind with (domain specific) schema to draw upon supports better (domain specific) problem solving.
    The icing on the cake, the chapter on problem solving using algebra was sourced from a traditional GCSE (intermediate level) text book. This ONLY worked because of the traditional drilling of component skills as in good old fashioned math teaching. Even more impressive for me considering this is an inner city difficult year 9 class who are pretty impossible without me there, and because of their prevailing attitude to learning have learnt very little in year 7&8.

    Long winded, but cognitive science works. Traditional teaching works. Cognitive science validates, and improves traditional teaching.

    Great article.

  2. Of interest too is flexibility in learning. I think *some* problem solving in new contexts is important. Just as learning a procedure to solve a particular problem is important. The procedure to alter the clock is different from vehicle to vehicle, so only knowing your own vehicle won’t necessarily be helpful in new contexts.

  3. Great summary of why we should use traditional teaching methods

  4. Mario says:

    Why are the role of your emotions not part of the picture? Clearly, yours were getting in the way, probably affecting your working-memory. How successful can schema acquisition be with a negative frame of mind or lack of confidence? I can yell at my computer all day long and it won’t affect the way it functions. However, if I do the same with a child it certainly will have a negative impact. Sometimes, the way you portray the learning process seems a bit mechanical, computer-like, leaving out the emotional side of the learner.

    • David Didau says:

      Seriously? I just wrote the emotional stuff for colour. Focus on the facts dude.

      • Mario says:

        Are positive or negative feedback neutral in developing a schema?

        • David Didau says:

          If you infer from feedback that you’re unable to do something, then the schema you build consists of negative information, so PSS would turn up, “Oh, I can’t do this.” I wrote about it here

          • Mario says:

            What role do emotions play in the learning process? Is the acquisition of knowledge just a mechanical process where if you follow the right steps learning takes place?

          • David Didau says:

            Emotions heighten episodic memory. This can be counter productive when trying to recall semantic information.

          • Mario says:

            Sophie is surprised and delighted by the great applause she receives for the new way she plays a piece of music. The applause motivates her to continue learning and improving and, perhaps, even become a professional musician one day.

            The applause is an unexpected reward. This unexpected reward is associated with an increased release of the brain’s neurotransmitter dopamine in specific brain cells, stimulating learning and motivation.

            The three winners of the 2017 Brain Prize, English Peter Dayan, Irish Ray Dolan and German Wolfram Schultz, have identified how learning is linked with anticipation of reward, as in Sophie’s case, giving us fundamental knowledge about how we learn from our actions.


            “Mapping the connection between learning and reward is essential if we’re to understand human behaviour and how to improve treatment of brain disorders. With elegant experiments and mathematical models, the prizewinners have described how dopamine plays a crucial role in the motivation that drives learning,” says Professor Morten Kringelbach, himself a researcher of the hedonic brain at the universities of Oxford and Aarhus.


          • David Didau says:

            Interesting. This is classic Behaviourism. Reminds me of one my favourite psychology gags: that his experiments were concerned with pulling habits out of a rat.

          • Mario says:

            “Emotions heighten episodic memory. This can be counter productive when trying to recall semantic information.”

            Where can I find evidence about that? Thanks

          • David Didau says:

            Try googling it?

          • Mario says:

            If a student is nervous or insecure, don’t you you think is detrimental to the learning process, both on the acquisition and retrieval phase?

          • Mario says:

            Then, they do play a part in the learning process. Are you aware of any research regarding the influence of emotion on study techniques such retrieval, variation, interleaving, spacing, etc?

          • David Didau says:

            There’s loads of research which support the finding that memory is context dependent. The emotional context is just one of several contexts which we can tap into to help us retrieve memories.

          • Mario says:

            Thanks. I was thinking more along the lines of adding artificial pressure in either the acquisition or retrieval phase to see how it affects the learning. There’s evidence in motor skills that different types of learning are more robust than others under stress. For instance, too much focus on the self is detrimental for performance.


            OPTIMAL MOTOR LEARNING requires conditions that enhance learners’ expectations for future performance, provide learners with autonomy, and promote an external focus of attention.

  5. chestnut says:

    Is it just too obvious to say: why don’t men like using manuals? I am always the one in my house who changes the clocks on cars, cookers etc – If I don’t remember I use the manual. I have a husband and three sons who are all very good at maths but always try far too hard with the trial and error method – which is, as you say, a very poor method.

  6. The obvious answer here is ‘why don’t you Google it?’ e.g.
    21st Century Skills to the rescue.

    • David Didau says:

      In this case, googling it would be far more time consuming than using the manual. :/

      • Michael Pye says:

        Googling it would have brought up the manual.

        • Michael Pye says:

          Did you consider working in a group? One person could have found the manual, another could have read it, one person could have followed the instructions and finally a team leader could have reviewed the process and fed back ways to improve. Afterwards you could work together to create a poster showing you what to do next time.

          • Sounds like what happens – or used to happen – in schools, especially the poster. No-one learns anything and the kids who did most of the work have to share their marks with the ones who did nothing much.

        • David Didau says:

          Which would have been less useful than simply opening the manual, looking at the index and resetting the clock

          • Michael Pye says:

            No David, with my method they would have truly understood the process as they would be forced to discover it.

            (Quickly runs and hides)

  7. Jenny says:

    I don’t google it because I don’t always have a computer handy. I haven’t changed the clock in my car, I just remember it is an hour slow, but I do solve other problems. Trouble is next time they come up I don’t remember how I did it before and have to go through the whole trial and error thing again. But I can recall and discuss things I was traditionally taught 50 years ago, so I deeply suspect that problem solving won’t deepen knowledge, and knowledge (not just information, but processed information) is what we need and that knowledge only gets in our heads by being specifically taught, rehearsed and tested.

  8. Interleave, Davy! Interleave. Reset the clock twice. Then practice setting a couple of preset radio frequency buttons. Then fade, front and back. Then reset the clock. Then … as your typo suggests, rest the clock. It’s getting tired.


  9. Would it be correct to say that the error in contemporary “problem solving” lessons is generally that, regardless of what content is taught about problem solving itself (say Polya’s system of questions) the “problem solving” part boils down to nothing more than practice in means-end analysis, and simply practicing this skill has, after the initial stages, very little educational pay-off, because means-end analysis after lots of experience doesn’t look significantly different than it does after a small amount of successful experience?

  10. Michael Pye says:

    I struggle with the concept of means-end analysis in the context of CLT as I can’t yet bring myself to pay £120 for that damned book. (Any suggestion for cheaper yet still detailed explanations should be freely given). As I currently understand it constant practice of a complicated task would be frustratingly inefficient as the method leaves little working memory left to learn a reproducible method. Over time each solution would be different and hard for someone to recognize anything but the most obvious of patterns, misconceptions would likely also be as prevalent as accurate insights leaving at best an inefficient method and at worst abject frustration.

    Having just looked up a definition of means-end from sources (and contexts) other then Swellers CLT it seems the process is very useful in certain circumstances including AI’s and mathematical proofs even marketing as well as being our default approach to unfamiliar problems.. Here it is described as a iterative process where each successive alteration brings you one step closer to the end goal. The key seems to be accurate feedback that you are actually moving closer and not ending up in a dead-end or regressing.

    My understanding would then be that it is useful for novices in unfamiliar circumstances they are unlikely to repeat often and when getting more detailed information would be more effort. It is also useful for genuine experts who can’t simply access more detailed information because it is undiscovered or unavailable effectively becoming cutting edge research. (reversal effect I believe)

    As a critique I don’t quite understand why ME can’t be used alongside knowledge (even that it would seem to be greatly aided by it) and that in possession of clear tools (therefore reducing working memory) it pretty much describes most mathematical solutions which progress in clear steps. This does not seem to be what Sweller is saying. What have I misunderstood?

  11. Michael Pye says:

    I don’t have a good definitional of what Sweller is contrasting Means-end to making it hard to spot the differences. I am in need of a good dichotomy.

    • Grumpywearymathsteacher says:

      I don’t know what means-end analysis is either! I do know that if you never do any ‘problem-solving’ at all then when you present students with a ‘curve ball’, even if the ‘curve’ is incredibly mild, panic ensues. If they are only ever directly told exactly what to do then they learn that their naturally evolved problem-solving skills are not required in school and so they switch them off, along with any kind of common sense about whether the answer is sensible……

      • David Didau says:

        Means-end analysis is what you do whenever you come up against an unfamiliar problem. You learned how to do it before you could speak. What makes problem solving hard in maths isn’t a lack of problem solving ability, it’s a lack of mathematical knowledge. So you’re right, practising a range of problems through worked examples, learning to become familiar with their deep structures and building useful domain-specific knowledge is the way forward.

        • Michael Pye says:

          But why can’t means-end analysis use knowledge if it is iterative in nature?
          Is Swellers version different to the psychology definition above?

          I realise I am missing something as the phrase means-end is important to CLT I just can’t figure out why? The definition you just used is the one I already understand.

        • Grumpywearymathsteacher says:

          Er, that’s not what I meant…or what I said. (Were you replying to me, or have I misunderstood?) Students need to get comfortable attempting problems which don’t look like the worked examples they’ve done.

          • David Didau says:

            Why would you expect your students have to solve problems that didn’t share the same structures of any worked examples you’d shown them?

  12. Jenny says:

    I think teachers expect pupils to solve problems with new structures because they are convinced that discovering how to do them encourages learning. It may – but the knowledge may not be retained, perhaps.

  13. Michael Pye says:

    That would be the philosophical debate about values and beliefs. If I believe creativity is a result of unexpected connections between mastered knowledge then I would simple feed students a rich diet of clearly worked problems and leave to simmer. When you believe in creativity as a skill it is then logical to train it directly with a diet of unknown problems. It would simply be a matter of choice unless you believe research strongly indicates that one is superior. (Which I do).

    • Grumpywearymathsteacher says:

      I like your ‘unexpected connections’ description.
      About the worked examples/giving problems which are different in their ‘deep structure’: first of all in school level maths none of the structure is particularly deep! It’s the surface features that cause confusion, and to avoid what I believe is called ‘discrimination error’ we need to give students lots of problems which _look_ different. Of course they need to have the mastered knowledge already, but they also need to get used to looking for those connections.
      That giant of Maths blogging, Mr Reddy, has described how he has made ‘mash-up’ problems that require knowledge from 2 or more already-known topic areas to be applied in the each problem. This forces students to search for and connect the knowledge they need to use. Practice in these kinds of problems really does seem to make people better at solving them, and you can’t “worked example” them, like you can the sub-skills involved.

      • Michael pye says:

        Think I recognise your discrimination error but call it something else (because I little opportunity to discuss with anyone else). To me it came out of challenging common misconceptions, which led to focusing on explicitly pointing out differences using comparisons. This confused my colleagues who don’t specialise in maths (I teach Sen at college level). Why would I constantly force them to distinguish between time and money questions making them clearly articulate their rearranging. Why introduce improper fractions to force them to define a denominator correctly.
        Why discuss the debate over are squares rectangles. Why force them to think about division by getting them to divide by zero or one and worse of all why get them to multiple or divide a whole number by a fraction other then a halve or a third. Is this what you mean.

        You can use multiple choice worked examples to do the above , learnt that the hard way when trying to do recap sheets of multiple common issues.

  14. Ian Birch says:

    Hi David,

    Interesting readings and I appreciated hearing some of your talks I found.

    This thread made me think of my kids and their lego.
    When I was young almost all the pieces were the same, and once you learned how to put them together you could build almost anything. Ok everything was rather square but good enough for me. So step 1 I learn to put pieces together, step 2 I copy a couple of my elder brothers creations with som help from my dad and step 3 I start creating my own.

    Today my kids get lego and the pieces are all very different. They need a manual to put them together, I can hardly help them and once finished the figure falls apart as soon as they play with it and is almost impossible to put together again let alone build something different.

    Comparing this to learning (I hope) and setting your clock. Practising putting together the whole of Darth Vadars battle ship many times is futile. It would take ages and all I would learn is a set on nontransferable actions. In much less time I could train the basic actions of putting simple lego together that could lead to a more sustainable (maybe not as flashy) construction of Darth Vadars battleship and everyone elses battleships too.

    Learning the basics of “digital chronometer construction” would mean you not only could set your own clock but any other clock in any other car. Until they change the basic construction 🙂

    Unfortunately, at least here in Sweden, basics and the methods most suited for teaching them, are seen as outdated and unnecessary.


Constructive feedback is always appreciated

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