There is no universal definition of what constitutes problem-solving in the maths classroom. As such, teachers often use broad definitions such as questions for which an appropriate method is not immediately obvious, or non-routine questions that force the learner beyond recall or rehearsal. The trouble is, what counts as non-routine varies dramatically depending on who is looking. A high-attaining learner might look at a simultaneous equations problem embedded in a geometry context and know straightaway what to do. Another learner, with less fluency or confidence, might find the same question difficult to interpret, or struggle to formulate an appropriate strategy.
This distinction brings us to a central tension: the label problem-solving should not be considered as a fixed property of the question, rather a comment on the experience of the learner.
Consider the questions:
“A train travels 120 km at an average speed of 80 km/h. How long does it take?”
“A train leaves station A at 09:00 travelling at 80 km/h. Another train leaves station B, 120 km away, at 09:15 heading toward A at 100 km/h. At what time do they meet?”
For most students, the first question is standard fare, but the second might be considered difficult. This is not, however, because the skills involved are so different, rather because the structure is less familiar.
This contrast highlights a deeper truth: what constitutes problem-solving is not inherent to the question itself, but arises from the learner’s familiarity with the structure and content. A question is only non-routine if it presents difficulty. It only becomes an act of problem-solving if the learner is unsure, hesitant, reflective. Once you are fluent enough, you stop solving problems and start executing procedures. This is why procedural fluency is a critical precondition for problem-solving. But it also muddies the waters, because the more fluent a learner becomes, the fewer things feel like problems, and there is no objective boundary.
Add to this the role of language. A mathematically competent pupil may still struggle with inference, layered clauses, or unfamiliar vocabulary. Problem-solving questions often introduce extraneous words or real-world contexts that amplify cognitive load.
Cultural capital also plays a role here. Pupils may find some problems more or less accessible depending on their familiarity with the context being used. In a recent session I hosted on embedding problem-solving at KS3/4, we discussed a question set in a Roman context. One delegate simply annotated the task with ‘cultural capital’ and they were absolutely right to do so. The context, though engaging for some, added an unnecessary barrier for others, making the problem harder not because of its mathematical demands, but because of the knowledge assumed.
This prompts a broader consideration: in light of the varied learner experiences shaped by fluency, language, and background knowledge, how should problem-solving be conceptualised within the curriculum and classroom practice?
If we accept that learners differ in fluency, background knowledge, and language confidence, then it follows that no single question can be definitively classified as a problem-solving question. A task that is routine for one learner may pose a significant challenge for another. This is not a reason to discard the concept entirely, but rather to use the term with greater precision, and perhaps with more humility. It remains important that teaching eventually ventures into what is commonly understood as problem-solving territory, as seen in some international curricula. Such teaching would include exploring heuristics, strategies, and general principles for approaching unfamiliar problems.
The first responsibility lies elsewhere, however. Before these more advanced ideas become meaningful, learners need secure fluency, coherent mental schemas, and the language to reason clearly. The challenge for teachers and curriculum designers, then, is not to search for silver-bullet problem-solving questions, but to recognise that a learner’s ability to engage is shaped by their level of fluency, the coherence of their mental schema, and their access to the language and cultural knowledge needed to decode a task.
Another blog post, coming soon.
George Bowman
Founder, Maths Advance
MATHS ADVANCE BLOG 
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