Solving problems and being creative

In the last couple of posts, I have fallen down a rabbit hole burrowed by the Education Endowment Foundation and others. I don’t think a knowledge-rich curriculum is something that can easily be tested and found to work or not to work, because it relies on the slow accumulation of knowledge across a variety of domains. It also brings into question what we are doing all of this for. If a child knows lots about the solar system but this does not aid her standardised reading performance this year then should we conclude that knowing about the solar system is useless? I don’t think so because I want students to know lots of stuff. In this sense, for me a knowledge-rich curriculum is more of an aim than a method.

Yet I do think that a knowledge-rich curriculum should help improve reading comprehension over the long term due to my understanding of how reading works. This is provided, of course, that students actually learn some knowledge as a result of the curriculum i.e. that the curriculum is effective in its own terms. And I also think that knowledge has a central role in all creative and problem solving endeavours.

What knowledge?

I am using a relatively loose definition of knowledge here. We can easily slip into high-minded discussions about the Western literary canon and whether it is representative of our diverse societies, but that’s only a small part of what I am referring to.

For instance, in the 2018 Northern Hemisphere Maths Methods VCE paper – bear with me – there was a question about discrete probability distributions that was set-out in an unusual way. Students needed to recognise that it was equivalent to the more standard version. This is something they could perhaps work out, creating new knowledge for themselves, or it could be something they learn via making a mistake and receiving feedback. In both of these cases, the knowledge will be more like a memory of an event – “remember that time when they asked that question where…” Alternatively, a teacher could explicitly teach this different presentation in class. This is the kind of tiny, grain-sized piece of knowledge that nobody gets excited about, but that can make all the difference.

And while I don’t believe in the existence of generic, trainable skills such as creativity or critical thinking, I do think that some kinds of knowledge can be effective across a number of different domains. Knowledge of logical fallacies, for instance, can help you evaluate lots of different types of argument based upon their form. However, I would contend that the more generic a strategy is, the less useful it is. A person may commit a logical fallacy and still be essentially correct about something. You need relevant domain knowledge to evaluate this correctness.

Similarly, back in the days when physics questions came without diagrams, the heuristic, ‘always draw a diagram’ had some utility for problem solving across the entire subject, from mechanics to electronics. Yet if you did not know the physics involved in a particular situation then drawing a diagram wouldn’t help much.

At the extreme end are heuristics such as ‘effort counts’ or ‘look at things from different perspectives’. You’ll only get marginal gains from applying them and then, only if you have something more substantial to build upon.

What does this knowledge do?

According to cognitive load theory, there are basically two ways to solve a problem. The first is to apply knowledge you already possess, either because you have generated that knowledge yourself or you have obtained it from others, or to randomly generate solution steps and test them to see if they move you closer to your goal.

This sounds as if experts and novices must operate in the same way when they are tackling novel problems, but it’s not quite like that. I would suggest that we ride the knowledge wave as close to the goal as possible before we resort to randomly generating and testing new steps. For relative experts, the wave of knowledge projects them far closer to the desired goal than relative novices and so they are much more likely to be successful. As Isaac Newton wrote, “If I have seen further it is by standing on ye sholders of Giants.”

Low knowledge individuals have to start guessing when they are still a long way from the goal.

Higher knowledge individuals begin the process of trial and error far closer to the goal.

Notice that we could mount an entirely equivalent argument about creativity, with the goal state in the case of creativity being a unique yet valuable product. Creativity is essentially a form of problem solving that precludes known solutions.

Novel problems do not stay novel for long. As successful solutions emerge and are communicated, they become subsumed into the knowledge base of individuals.

If this picture is accurate then there is little point in practising a ‘skill’ of solving novel problems or of being creative, because that is just practising the random generate-and-test stage. It is like practising rolling dice in order to become better at rolling sixes.

However, I don’t think this gives us the full picture. There is an emotional impact of wrestling with novel problems and students probably do need some training in order to learn how to cope with that and develop robust dispositions. Such training would have to be done well, however, because it could backfire. Similarly, a small amount of training in the use of intermediate level heuristics that apply across a range of situations is probably worthwhile.

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17 thoughts on “Solving problems and being creative

  1. All very true, if personal experience is anything to go by. I just wish the education sector could accept that being successfully educated is an amorphous condition/experience, as much emotive/subjective as anything else, and quite opportunistic in its application too. Unfortunately however, I see few signs that bean-counting is really coming to an end. I do wish I could persuade you to read my book – I think you would find a lit of inrerest in there. The Great Exception – why teaching is a profession like no other. Published by John Catt and available on Amazon or their own website.

  2. Succesful novel and creative problem solving is rarely a matter of random trial and error. Successfully finding novel approaches and solutions mostly relies on the generation of new analogies and metaphors. (Mary Hesse’s classic book on the role of models and analogies in science is a good place to start.) I’ve spent decades teaching university students how to identify, apply and test metaphors structuring social theory, and to practice generating, applying and testing their own analogies. Students attest to its utility in building their critical thinking skills and their capacity to devise ‘killer arguments’. In contrast decade teaching reasoning via classical approaches to logic and fallacies had little impact or transfer because it doesn’t reflect how we typically reason or solve problems (and it is of no relevance to the causal reasoning at the heart of science, whereas analogies are).

    1. Karey, Are you suggesting that all the philosophy 101 courses on logic and reasoning are a waste of time? And all the effort in math on concrete proofs is wasted? I pulled a few relevant books of my shelf and none mention metaphor. For example in Polya’s How to Solve It one of the classics for math folk. His dictionary of terms doesn’t include metaphor or anything like it.

      Your words here are not a great ad for your approach to convincing arguments.

      1. Polya was published in 1940s. No one was discussing metaphor in relation to scientific and mathematical reasoning then. Campbell in the 1920s was the first to take role of analogy seriously, but it didn’t receive systematic attention until Hesse and Kuhn examined analogy and models in 1960s, and Lakatos extended it to maths in 1970s Lakatos (Proofs & Refutations). Paul Churchland’s (1986) ‘Phase-space representation and coordinate transformation: A general paradigm for neural computation’ provides a model for the human brain as an analog device, mapping sensory and motor system, that emphasises reasoning in relation to action in physical space. Research like this, in cognitive science, has shown the common neural basis for analogy, models, maps, schemas and metaphors, and it is this element that is missing from the Stanford Encyclopedia of Philosophy entry on ‘Analogy and analogical reasoning’. The neural basis of Piaget’s concrete operational reasoning can be explained by this research.
        Lakoff’s work in cognitive linguistics draws these elements together (from the late 1970s) and explores the metaphoric foundations of formal logic (1987 Women, Fire and Dangerous Things), and mathematics (2000 Where does Mathematics Come From). There is little point searching for ‘metaphor’ in the index of philosophy of science or mathematics in works prior to 1980 at the earliest.

    2. The trial and error is selecting the metaphors/analogies and testing them for utility. More knowledge means more metaphors/analogies. Also arguing via the method you suggest is great for teaching or persuading but is insufficient for proof. Classical logic and fallacies are designed to help with that by identifying flawed reasoning. You are correct that most people don’t reason that way but it is a trainable skill, I am also aware that training people in logical reasoning can lead to improved rationalization and entrance flawed ideas.

      P.s Greg is “If I have seen further it is by standing on ye sholders of Giants.” the original quote (with the ye)?

      1. For the record, re: the quote, Newton is paraphrasing Bernard of Clairvaux, who wrote: “We are like dwarfs on the shoulders of giants. We see more things than the ancients and things more distant, but it is due neither to the sharpness of our sight nor the greatness of out stature. It is simply because they have lent us their own.”

    3. Studies of effective problem solvers demonstrate that they do not generate lots of possible solutions, they start by thoroughly understanding the problem and in the process of doing that the way to solve it is revealed. The result is that the problem is solved relatively quickly while the non experts are flailing around trying dead end non solutions.

  3. Greg,
    I think your diagrams are interesting. What I see from the opponents of KRC is characterizing it as something like the third diagram where the student is rote trained to recall any goal state using knowledge that has been poured into them filling the pail style.
    The alternative is always to place a problem outside their knowledge base. I think Barry Garelick has an interesting blog post https://traditionalmath.wordpress.com/2019/01/01/how-is-understanding-measured

    If you think about how an ill posed problem verses a well scaffolded problem looks with your diagram it is like moving the goal state vertically away from the closest point of the knowledge curve. You can create more struggle in two ways moving it further away horizontally where there is work but they have the best possible starting point or vertically where they are being pointed slightly or a long way in the wrong direction.
    Stan

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