If you have spent any time in a mathematics classroom, you will know that the most interesting answers students give are often the incorrect ones. Teachers spend a lot of their planning time trying to anticipate errors and misconceptions so they can address them head-on. This requires scrutinising the mathematics, trying to view it through the eyes of a novice, and thinking about how prior learning might lead students towards sensible but incorrect statements. <\/p>\n\n\n\n
The ability to accurately anticipate the errors that are most likely to arise is a teacher super-power, but how good are mathematics teachers at this? And is it something that can be improved through deliberate practice? In our recent paper (Stannard and Foster, 2025), we explored how well secondary mathematics teachers can predict the prevalence of specific student errors, and whether this improved with repeated feedback. <\/p>\n\n\n\n
A crucial aspect of Pedagogical Content Knowledge<\/strong><\/p>\n\n\n\n For decades, educational researchers have drawn on Lee Shulman\u2019s (1986) concept of Pedagogical Content Knowledge<\/em>. This is the special blend of content knowledge and pedagogical knowledge that is unique to teachers. It is not just knowing mathematics, but knowing how to teach mathematics. <\/p>\n\n\n\n Within this framework sits a crucial sub-domain known as Knowledge of Content and Students<\/em>. This includes anticipating student conceptions and misconceptions. It is the awareness, for example, that when teaching the addition of fractions, a noticeable proportion of the class are likely to try adding the numerators and adding the denominators.<\/p>\n\n\n\n However, while most experienced teachers are aware of a variety of common misconceptions, predicting how an entire class are likely to distribute their answers across plausible errors is a demanding task, requiring a highly developed mental model of student thinking. <\/p>\n\n\n\n Investigating teachers’ predictive skills<\/strong><\/p>\n\n\n\n To investigate this, we worked with seven secondary school mathematics teachers. Over a total of 37 sessions, of around 20-30 min each, the teachers were presented with multiple-choice diagnostic questions from Craig Barton\u2019s free Diagnostic Questions website<\/a>, which hosts thousands of questions designed to expose student thinking. <\/p>\n\n\n\n Diagnostic questions do not contain haphazard distractors (incorrect options). Instead, every incorrect option is designed to reveal a specific identifiable error or misconception. For each question in the study, the teachers were asked to predict the percentage of students from a large dataset (also provided via the website) who would choose each of the four presented options. <\/p>\n\n\n This task requires the teacher to do far more than merely identify the correct mathematical answer. They must deconstruct each distractor and understand the particular reasoning that might lead a student to choose one option over another. In addition, they must estimate the prevalence of that specific error within the broader student population.<\/p>\n\n\n\n Teachers are highly skilled and improve rapidly<\/strong><\/p>\n\n\n\n We found that our teachers were very good at this and got much better remarkably quickly.<\/p>\n\n\n\n Over the course of the sessions, all participants demonstrated measurable improvement in the accuracy of their predictions. The most dramatic gains did not take months of gruelling practice or hours of theoretical lectures. Instead the sharpest increases in accuracy occurred over just the first three sessions. The vertical axis of the graph shows a measure of how inaccurate<\/em><\/strong> the teachers\u2019 predictions were. You can see this generally decreasing, particularly across the first three sessions. <\/p>\n\n\n\n The mean improvement across a single session had a Cohen\u2019s d<\/em> effect size of 0.606. This suggests that the simple act of repeatedly engaging in this specific predictive task, coupled with the immediate feedback of seeing the actual student data, rapidly hones a teacher’s diagnostic intuition. Teachers quickly became better calibrated as they had their predictions challenged by the data. <\/p>\n\n\n\n Implications<\/strong><\/p>\n\n\n\n It seems worth creating structured opportunities for teachers to repeatedly anticipate student errors and compare their predictions against empirical data. Teachers who can better anticipate what is going to happen in their classroom are likely to be better prepared to respond effectively. <\/p>\n\n\n\n A mathematics department could dedicate 10 min of their weekly meetings to looking at a diagnostic question, making individual predictions about the distribution of student answers, and then discussing the actual data. This could be a high-impact activity that could accelerate the development of teachers\u2019 predictive accuracy.<\/p>\n\n\n\n A teacher who accurately anticipates errors is far better equipped to design robust explanations. They can select appropriate examples that deliberately highlight and resolve anticipated misconceptions. They can pivot their teaching in the moment when a predicted error arises, because they have already thought through possible responses. <\/p>\n\n\n\n Mistakes are an inevitable and valuable part of learning mathematics. They provide a window into student thinking. Learning to better predict these mistakes might be a strategy for becoming a better mathematics teacher.\u00a0<\/p>\n\n\n\n About the authors<\/strong><\/p>\n\n\n\n Aidan Stannard is a secondary school mathematics teacher in Derbyshire. He is particularly interested in developing oracy and problem-solving skills within a mathematical context in his school. He originally undertook this research as part of his undergraduate degree at Loughborough University. <\/p>\n\n\n\n Colin Foster is a Professor of Mathematics Education at Loughborough University. His research focuses on the learning and teaching of mathematics in ways that support students\u2019 conceptual understanding. He is particularly interested in the design and use of rich tasks in the mathematics classroom, and in finding ways to enable students to develop the necessary fluency in mathematical processes to support them in solving mathematical problems. <\/p>\n\n\n\n References<\/strong><\/p>\n\n\n\n Shulman, L. S. (1986). Those who understand: Knowledge growth in teaching.\u202fEducational Researcher<\/em>,\u202f15<\/em>(2), 4-14. https:\/\/doi.org\/10.3102\/0013189X015002004<\/a> <\/p>\n\n\n\n![\"\"](\"https:\/\/blog.lboro.ac.uk\/cmc\/wp-content\/uploads\/sites\/54\/2026\/05\/image-1.png\")
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