Written by Dr Colin Foster. Colin is a Reader in Mathematics Education and is interested in the teaching of problem solving in mathematics. He is the Director of the Loughborough University Mathematics Education Network (LUMEN), and the current focus of LUMEN is the design of a complete set of classroom resources for teaching mathematics to ages 11-14. There is a link to LUMEN at the end of this blogpost. The article is edited by Bethany Woollacott.

Introduction

People often say that the school mathematics curriculum should no longer be training children to be efficient calculators, doing fluent arithmetic and algebraic manipulation, because we now have computers that can do all of that kind of thing extremely quickly and accurately. Instead, the school mathematics curriculum should focus on teaching students to be confident problem solvers, ready to tackle unfamiliar (i.e. unseen) problems with creativity and ingenuity. Whether you agree with that or not, mathematics curricula across the world give ‘problem solving’ a high profile.

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“Instead, the school mathematics curriculum should focus on teaching students to be confident problem solvers, ready to tackle unfamiliar problems with creativity and ingenuity.“

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The phrase ‘problem solving’ is sometimes used just to mean answering familiar, routine exercises, which students have been trained to do, and where they merely copy the method that their teacher has shown them. But, in the mathematics education literature, ‘problem solving’ is rarely used in this way. Instead, it normally refers to “a task for which the solution method is not known in advance” (NCTM, 2000, p. 52). So, with problem solving, students have to scratch their heads and come up with an approach to something that they haven’t been shown exactly how to do.

Can problem solving be taught?

It isn’t obvious what it means to try to teach problem solving. If you teach someone a method for solving a particular kind of problem, then that class of problem is no longer ‘a problem’, because the student has a ready-made method for it. They can just turn the handle and get the answer. This is the normal way in which mathematics develops through history: someone clever comes up with a method for something for which there is a need, and then everyone else from then on can use that method and not have to reinvent the wheel. But amassing a larger and larger toolbox of methods isn’t the same as learning to be able to solve novel problems for yourself.

So, teachers are often exhorted to avoid this trap when focusing on problem solving, and instead let students struggle with problems without providing them with a solution method. However, this is often criticised as an inefficient teaching approach, and it could be demotivating for students to struggle for a long time without success. Can we really expect every student to reinvent methods that might have taken mathematicians hundreds of years to develop? But if the teacher helps by offering hints and suggestions then isn’t this taking away from the problem solving aspect?

Generic strategies

The most common approach that I see in schools is for teachers to focus on teaching generic problem-solving strategies, often with reference to George Pólya and his remarkable book ‘How to solve it’ (1957). Pólya’s book provides a long list of heuristics (i.e., general strategies) for solving problems, and many other people have also made lists of these kinds of things, which include strategies like ‘Draw a diagram’.

It is hard to argue with the value of these strategies, but, as Alan Schoenfeld discovered in his ground-breaking programme of research on problem solving, Pólya’s strategies “were broad and descriptive, rather than prescriptive, [and] novice problem solvers could hardly use them as guides to productive problem solving behavior” (Schoenfeld, 1987, p. 31). The strategies were true in the sense that that’s certainly what expert problem solvers actually did. But they were hard to apply in practice. A stuck student being advised ‘Draw a diagram’ is likely to respond, ‘What diagram?’. For this and other reasons, teaching generic strategies does not seem to improve students’ ability to solve mathematical problems.

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“A stuck student being advised ‘Draw a diagram’ is likely to respond, ‘What diagram?’“

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Specific tactics

In a recent article (Foster, 2023), I explored the literature on the teaching of problem solving in mathematics, looking for a better approach than teaching generic strategies. Those arguing against the teaching of generic strategies tend to be in favour of focusing solely on ‘content knowledge’. From a cognitive load theory perspective, the more relevant domain-specific knowledge you have in long-term memory, the more space that frees up in working memory to ‘problem solve’. However, in the article I consider examples of ‘easy but difficult problems’, that require very little in terms of what we normally think of as content knowledge, but which most people find very hard. A classic example is “Langley’s adventitious angles” shown in the diagram below: can you find x? (From Langley, 1922)

In my article, I argue that the missing ingredient that students are generally not explicitly taught, at least in the UK, is domain-specific problem-solving tactics. These are quite fine-grained (much smaller than ‘Draw a diagram’) and much narrower in scope. In this case – spoiler alert! – the relevant domain-specific tactic is ‘Draw in an auxiliary line’. This means a line which is not in the original figure, but which you add in yourself, and which creates new angles. Doing this, it may look as though you’ve made the problem more complicated, but in fact when you angle chase around your new line you find a solution relatively quickly (the article is linked at the bottom of this blogpost if you are interested). This tactic unlocks the problem quite dramatically, and it is not a one-off ‘trick’. ‘Draw in an auxiliary line’ is a tactic that will work across a range of geometry problems, including much simpler ones than this difficult problem.

In our current work at Loughborough on the design of the LUMEN Curriculum, we are thinking about how to prioritise a carefully-chosen list of high-leverage domain-specific tactics, and then how to teach these systematically and explicitly, through the use of a range of different problems. We want students to see which kinds of problems a particular tactic will unlock and which it won’t, and why. We hope that this will be a more equitable and reliable way to teach problem solving than leaving most students to struggle while a lucky few have a flash of inspiration.

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“We hope that this will be a more equitable and reliable way to teach problem solving than leaving most students to struggle while a lucky few have a flash of inspiration.“

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References

Foster, C. (2023). Problem solving in the mathematics curriculum: From domain-general strategies to domain-specific tactics. The Curriculum Journal. Advance online publication. https://doi.org/10.1002/curj.213

Langley, E. M. (1922). Problem 644. The Mathematical Gazette, 11(160), 173. https://doi.org/10.2307/3604747 

National Council of Teachers of Mathematics (NCTM) (2000). Principles and standards for school mathematics. NCTM.

Pólya, G. (1957). How to solve it: A new aspect of mathematical method (2nd ed). Princeton, NJ: Princeton University Press.

Schoenfeld, A. H. (1987). Confessions of an accidental theorist. For the Learning of Mathematics, 7(1), 30-38.