School mathematics curricula across the world give ‘problem solving’ a high profile, but what exactly does ‘problem solving’ mean? Can it be taught; and, if it can, how can this be done in an equitable way? These are the questions that I consider in my recent article in the Curriculum Journal (Foster, 2023).
‘Problem solving’ is sometimes used to mean students answering routine questions, like those seen in any standard school mathematics textbook. Students have been trained on these specific types of question, and they merely imitate the method that their teacher has shown them. But this is not how the term is used in the mathematics education literature. Instead, a ‘problem’ normally refers to ‘a task for which the solution method is not known in advance’ (NCTM, 2000, p. 52). When problem solving, students have to be creative and find an approach that they haven’t been shown how to do. Learning standard techniques for important classes of problems is a valuable part of learning mathematics, but this is not the same as learning to be able to solve novel problems for yourself.
Can problem solving be taught?
To learn to problem solve, teachers must avoid showing the students how to solve the problem, because that kills the problem solving and reduces the task to an exercise. However, this is sometimes taken to mean that students should simply be provided with problems and left to struggle, as any hints or suggestions given by the teacher would be deemed to undermine any problem solving. Some students may succeed with this approach, but it is likely to be those who are already advantaged. This does not seem to be an equitable approach that reliably and efficiently supports all students in becoming powerful problem solvers.
‘To learn to problem solve, teachers must avoid showing the students how to solve the problem, because that kills the problem solving and reduces the task to an exercise.’
The most common approach is for teachers to focus on teaching generic problem-solving strategies, such as ‘Draw a diagram’ (see Polya, 1957). For example, to make sense of connections between information provided in a problem, using a table or tree diagram might be helpful. Although successful problem solvers certainly use these strategies, Alan Schoenfeld (1985) discovered that they are too general to help a student who is stuck with a specific problem. Teaching generic strategies does not seem to improve students’ ability to solve mathematical problems.
Those arguing against the teaching of generic strategies tend to focus on ‘content knowledge’. From a cognitive load theory perspective, the more relevant domain-specific knowledge a student has in long-term memory, the more space that frees up in working memory to ‘problem solve’. The more theorems in geometry that a student knows, the more likely they are to be able to solve a geometry problem. However, although content knowledge is necessary, it is not sufficient.
In my Curriculum Journal 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 hard. Geometrical ‘puzzles’, for example, are difficult not because they depend on advanced geometrical knowledge, such as trigonometry; they require only basic facts, such as that the angle sum of a triangle is 180°. What is challenging is seeing how to make use of those simple facts – and generic strategies, such as ‘Draw a diagram,’ do not help with this challenge.
I argue that the missing ingredient is the systematic, explicit teaching of domain-specific problem-solving tactics. These are much finer-grained than ‘Draw a diagram’ and narrower in scope. A geometrical example is ‘Draw in an auxiliary line’, meaning a line which is not in the original figure, but which creates new angles that facilitate a solution. I show in the article how such tactics are commonly taught as part of teaching problem solving in Japan (Baldry et al., 2023).
At Loughborough University, we are designing a complete set of free resources for teaching mathematics 11–14 (the LUMEN Curriculum). We address problem solving through prioritising the teaching of a carefully chosen list of high-leverage domain-specific tactics. We want students to experience which kinds of problems a particular tactic will unlock and which it won’t, and why. We intend this to provide 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.
This blog post is based on the article ‘Problem solving in the mathematics curriculum: From domain-general strategies to domain-specific tactics’ by Colin Foster published in the Curriculum Journal.
Baldry, F., Mann, J., Horsman, R., Koiwa, D., & Foster, C. (2023). The use of carefully-planned board-work to support the productive discussion of multiple student responses in a Japanese problem-solving lesson. Journal of Mathematics Teacher Education, 26(2), 129–153. https://doi.org/10.1007/s10857-021-09511-6
Foster, C. (2023). Problem solving in the mathematics curriculum: From domain-general strategies to domain-specific tactics. Curriculum Journal. Advance online publication. https://doi.org/10.1002/curj.213
National Council of Teachers of Mathematics [NCTM]. (2000). Principles and standards for school mathematics.
Polya, G. (1957). How to solve it: A new aspect of mathematical method (2nd ed). Princeton University Press.
Schoenfeld, A. H. (1985). Mathematical problem solving. Academic Press.