Problem-solving is one of those twentieth century skills that schools are supposed to be teaching in order to prepare students for jobs that haven’t been invented yet. It is a key component of the ‘critical and creative thinking’ general capability of the Australian Curriculum. Yet problem-solving is a tricky concept. I certainly don’t believe it is a discrete skill like a golf-swing that can be developed through teaching and practice.
David Geary’s distinction between biologically primary and biologically secondary knowledge perhaps offers a useful way of thinking about problem-solving. General problem-solving skills are biologically primary because they have evolved alongside the human mind: we have always had to solve problems in a general sense. Tricot and Sweller suggest that mean-ends analysis is a good example of such a strategy. This is essentially the tactic of working backwards from the goal in order to assess your progress towards it. We all possess this strategy already and so there is no need to teach it to children.
Biologically secondary knowledge builds upon primary knowledge. Means-end analysis is at the root of maths problem-solving (unless we engineer its absence) but we then complement this by teaching a series of domain specific strategies. For instance, knowing how to factorise a quadratic is really useful if you need to solve a problem involving quadratic equations, yet it’s not much help in solving other kinds of maths problems, let alone the problem of a blocked drain.
One of the difficulties that maths teachers encounter is in enabling students to transfer problem solving procedures from one context to another. For instance, a student may be able to factorise a quadratic equation but may not realise that this is what is required in a particular problem. My approach to this is an explicit one where students begin work on problems that look very similar to each other but then gradually work through problems with varying contexts. At a later stage, different problem types need to be interleaved so that students may learn to spot the deep structure and therefore the type of solution required. All of this proceeds with lots of teacher guidance and demonstration and plenty of formative assessment to keep the trains on the track.
However, what if we could develop more general problem-solving strategies that could shortcut this process? These may be completely general skills like means-end analysis that can be applied to any problem or they may be intermediate strategies that apply to a broad range of problems within a particular domain. There is little evidence available that we can learn general problem-solving strategies but what of these more middling kinds of strategies?
For instance, ‘draw a diagram’ is a reasonably useful strategy to help solve a very broad class of physics problems. It might not help much in other kinds of problems but it has a general applicability in physics. Nevertheless, it is nowhere near as useful as a specific strategy for solving a particular problem. It works as a cognitive aid to enable us focus on the deep structure.
Some intermediate strategies have potential and they seem similar in nature to reading comprehension strategies. Research suggests that reading comprehension strategies produce a quick boost in reading performance but that they don’t really function as a skill in the sense that repeated practice of these strategies increases their effect. Perhaps ‘draw a diagram’ is a useful heuristic that operates in a similar way.
I can’t see any value in some intermediate strategies. For instance, ‘solve a simpler problem’ only works if you already know the deep structure of the problem because the two problems need to share structure for the simpler one to be useful. Yet if you already knew the deep structure then you would not need a more general problem-solving strategy.