What makes a problem a problem in mathematics

From the puzzles in school textbooks to the unsolved questions that drive research, ‘problems’ lie at the heart of mathematics. But what exactly makes something a mathematical problem? 

Should we retire the term “problem”? 

The term has a long and varied history in mathematics education.  In different contexts, it can mean a routine exercise, an open-ended investigation or simply any mathematical task. Sometimes it refers to applying mathematics in real-world contexts; elsewhere, it’s linked to particular teaching approaches. In short, problem is a word that carries many meanings — some precise, others much looser. So, should we retire it altogether, or can it still serve a useful role in how we think about mathematics teaching and assessment? 

Word problems: familiar but limited 

At one end of the spectrum are “word problems”, those short narratives that describe a simplified mathematical situation. Learners must build a mental model, identify the mathematical structures and extract relevant quantitative information, determine the necessary calculation(s) and calculate. Word problems are a staple of classrooms and assessments and we know a lot about what makes them hard or accessible.  They can be useful for practising certain skills, and structured strategies can help students succeed. But many word problems can be solved by applying a familiar, rehearsed method. The challenge is often linguistic or procedural rather than conceptual.  

Genuine problems: perplexing and exploratory 

At the other end of the spectrum, we have “genuine”^[1] problem solving - situations for which the learner has no ready-made solution method. These are, by nature, “perplexing”^[2]. They demand reasoning, creativity, and persistence to explore an unfamiliar or non-routine situation. Such problems may have multiple valid approaches or even multiple solutions. Their educational value lies less in arriving at an answer and more in the process of exploring, conjecturing, and generalising. 

A landscape of mathematical problems 

Given these contrasting meanings, should we look for a new word altogether? Or might we reserve “problem” only for the most perplexing, open-ended tasks? Perhaps neither. There is a landscape of mathematical problems, each offering its own kind of value.  Not every task needs to be genuinely open-ended or unfamiliar to be worthwhile. What they share is a certain mathematical richness: opportunities to reason, connect ideas, and see patterns that goes beyond, and complements, routine procedural exercises.  

Focusing on what matters 

While debates about terminology are a distraction, there are, however, important questions worth our attention. What kinds of mathematical richness are most valuable at different stages of learning? How can such activities support - and be supported by - fluency and foundational knowledge? How do we design them to be both accessible to - and stretching for - all learners? What we shouldn’t waste energy on is debating terminology.   

Whether we call them problems, investigations, questions, challenges, or tasks, what matters is that learners encounter mathematics as something to be thought about, not just done. 

Professor Camilla Gilmore is a lead curriculum specialist on the Maths Horizons team.

Camilla's research aims to uncover the cognitive processes involved in mathematical thinking and the implications of this for mathematics education. She is Professor of Mathematical Cognition at Loughborough University and Director of the ESRC Centre for Early Mathematics Learning. She is a member of the British Society for Research into Learning Mathematics (BSRLM) and the Early Childhood Maths Group (ECMG).

^[1] Askew, M. (2020). Reasoning as a mathematical habit of mind. The Mathematical Gazette, 104(559), 1-11. 

^[2] Schoenfeld, A. H. 1992, ‘Learning to think mathematically: Problem solving, metacognition, and sense-making in mathematics’ in D. Grouws (Ed.) Handbook for Research on Mathematics Teaching and Learning New York: MacMillan (pp 334 – 370).