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.

Drawing a diagram can help for a lot of mathematical problems as well, but the two main general strategies that I can think of for mathematics are:

1) Divide and conquer — Divide a problem into separate simpler problems that you can solve individually. For example, when factoring a polynomial, if you can write it as a product, you now have two simpler polynomials to factor.

2) Rewrite the problem as a simpler problem, or ideally as a problem you know how to solve. For example, in an equation involving rational functions you can cross multiply by the denominators, and you now have an equation involving polynomials.

The thing is, these general strategies are utterly useless without the specific skills needed to perform them. Not only do you need to be able to solve the resulting simpler problems (and recognise that they are in fact simpler) the act of dividing or rewriting also involves specific skills.

However, in my experience, one skill is almost always involved. That is algebraic manipulation and simplification. I would call it a general problem solving skill although I don’t think everybody would agree with me on that.

You can’t DO problem solving.

You can DO “find the roots of the quadratic’.

Here’s a link for you:

https://www.une.edu.au/about-une/academic-schools/bcss/news-and-events/psychology-community-activities/over-fifty-problem-solving-strategies-explained

I wonder though, if it’s possible to learn problem solving steps. Once upon a time schools taught logic and reasoning–being able to understand what is asked and follow a logical thought process to complete the task was an important part of that education.

The US Army FM 6-0 lays out a process: gather info and knowledge; identify the problem; develop criteria; generate possible solutions; analyze possible solutions; compare possible solutions; make and implement the decision. http://www.milsci.ucsb.edu/sites/secure.lsit.ucsb.edu.mili.d7/files/sitefiles/fm6_0.pdf

One could possiblly use these to write a history paper, for example.

We certainly spend a lot of time in history classes teaching research skills, which are a part of problem solving. I would say, though, that Prime Example Number One of how such skills are difficult to transfer is that many of my students are master video game players (requiring problem solving) yet are at sea when I ask them to use footnotes.

Worth reading

https://artofproblemsolving.com/articles/hard-problems

This covers the issues for experienced math problem solvers facing problems that stump them.

The assumption here is that there is a great deal of specialized knowledge but some problems are just really difficult until you get the right approach.

In math there are a lot of cases where a simplification is readily available. For example in combinatorial and number theory questions you can take smaller cases and get a feel for what is going on.

One point about the suggestions here is they don’t take a lot of teaching in the sense that reading the article covers the idea pretty well. The rest is practice so that using them is automatic.

The most useless problem solving strategy is “guess and check”.

It’s taught in NZ as a reasonable way to solve problems at younger years. Having internalised that they arrive at High School unprepared to solve problems properly, because they can always guess and check. Trying to teach proper linear solving to someone who is convinced that guess and check is an appropriate alternative is hugely frustrating.

Using guessing techniques actively destroys what you are trying to teach in Maths. It’s use as a “strategy” should never be taught by Maths teachers, ever.

(It’s not like people faced with an intractable problem in real life won’t use it anyway — it’s biologically primary to try things until one works.)

I use it only to show how time consuming it is and that algebra provides a more efficient way of solving a problem. After doing one or two guess and check problems, I’ll then give them a problem that has a fractional answer. This does not lend itself to guess and check because there are an infinity of choices of fractions vs the finite set of choices for integral answers.

The CPM series on algebra (used in the US) has students using guess and check in addition to the algebraic methods. They state in writing that it is a legitimate form of problem solving. Yes, and counting on one’s fingers is a legitimate form of addition/subtraction as well.