I am a Reader in Mathematics Education in the Department of Mathematics Education at Loughborough University, UK. My research interests in mathematics education focus on the learning and teaching of mathematics in ways that support students’ conceptual understanding.
Colin Foster, Jeremy Hodgen and Dietmar Kuchemann exlore how the use of examples and counter-exam... more Colin Foster, Jeremy Hodgen and Dietmar Kuchemann exlore how the use of examples and counter-examples can support students developing understandings of definitions.
In this paper, we analyse a large, opportunistic dataset of responses (N = 219,826) to online, di... more In this paper, we analyse a large, opportunistic dataset of responses (N = 219,826) to online, diagnostic multiple-choice mathematics questions, provided by 6–16-year-old UK school mathematics students (N = 7302). For each response, students were invited to indicate on a 5-point Likert-type scale how confident they were that their response was correct. Using demographic data available from the online platform, we examine the relationships between confidence and facility (the proportion of questions correct), as well as gender, age and socioeconomic disadvantage. We found a positive correlation between student confidence and mean facility, higher confidence for boys than for girls and lower confidence for students classified as socioeconomically disadvantaged, even after accounting for facility. We found that confidence was lower for older students, and this was particularly marked across the primary to secondary school transition. An important feature of the online platform used is ...
Exploring even something as simple as a straight-line graph leads to various mathematical possibi... more Exploring even something as simple as a straight-line graph leads to various mathematical possibilities that students can uncover through their own questions.
Canadian Journal of Science, Mathematics and Technology Education, 2014
The frequent misinterpretation of the nature of confidence intervals by students has been well do... more The frequent misinterpretation of the nature of confidence intervals by students has been well documented. This article examines the problem as an aspect of the learning of mathematical definitions and considers the tension between parroting mathematically rigorous, but essentially uninternalized, statements on the one hand and expressing imperfect but developing understandings on the other. A small-scale study among schoolteachers sought comments on four definitions expressing differing understandings of confidence intervals, and these are examined and discussed. The article concludes that some student wordings could be regarded as less inaccurate than they might seem at first sight and presents a case for accepting a wider range of more intuitive understandings as a work in progress.RésuméLa fréquente mésinterprétation de la nature des intervalles de confiance de la part des étudiants est bien documentée. Cet article analyse la question en tant qu’aspect de l’apprentissage des définitions mathématiques, et considère la différence entre d’une part la répétition d’énoncés parfaitement rigoureux sur le plan mathématique, mais qui n’ont pas été intégrés, et d’autre part l’expression de concepts encore imparfaitement maîtrisés, mais qui dénotent une certaine compréhension. Une étude réalisée auprès d’un petit groupe d’enseignants a sollicité leurs commentaires au sujet de quatre définitions exprimant différents degrés de compréhension du concept d’intervalle de confiance, commentaires qui ont ensuite fait l’objet d’une analyse et d’une discussion. L’article conclut que certains énoncés des étudiants sont moins inexacts qu’ils ne pourraient sembler à priori, ce qui suggère qu’on peut accepter une plus vaste gamme d’énoncés intuitifs comme l’expression d’une ‘compréhension en devenir’.
Maths teacher Colin Foster finds a glass of fizzy orange can help students get to grips with the ... more Maths teacher Colin Foster finds a glass of fizzy orange can help students get to grips with the tricky topic of ratio.
International Journal of Mathematical Education in Science and Technology, 2013
ABSTRACT In a high-stakes assessment culture, it is clearly important that learners of mathematic... more ABSTRACT In a high-stakes assessment culture, it is clearly important that learners of mathematics develop the necessary fluency and confidence to perform well on the specific, narrowly defined techniques that will be tested. However, an overemphasis on the training of piecemeal mathematical skills at the expense of more independent engagement with richer, multifaceted tasks risks devaluing the subject and failing to give learners an authentic and enjoyable experience of being a mathematician. Thus, there is a pressing need for mathematical tasks which embed the practice of essential techniques within a richer, exploratory and investigative context. Such tasks can be justified to school management or to more traditional mathematics teachers as vital practice of important skills; at the same time, they give scope to progressive teachers who wish to work in more exploratory ways. This paper draws on the notion of a musical etude to develop ´ a powerful and versatile approach in which these apparently contradictory aspects of teaching mathematics can be harmoniously combined. I illustrate the tactic in three central areas of the high-school mathematics curriculum: plotting Cartesian coordinates, solving linear equations and performing enlargements. In each case, extensive practice of important procedures takes place alongside more thoughtful and mathematically creative activity.
Colin Foster, Jeremy Hodgen and Dietmar Kuchemann exlore how the use of examples and counter-exam... more Colin Foster, Jeremy Hodgen and Dietmar Kuchemann exlore how the use of examples and counter-examples can support students developing understandings of definitions.
In this paper, we analyse a large, opportunistic dataset of responses (N = 219,826) to online, di... more In this paper, we analyse a large, opportunistic dataset of responses (N = 219,826) to online, diagnostic multiple-choice mathematics questions, provided by 6–16-year-old UK school mathematics students (N = 7302). For each response, students were invited to indicate on a 5-point Likert-type scale how confident they were that their response was correct. Using demographic data available from the online platform, we examine the relationships between confidence and facility (the proportion of questions correct), as well as gender, age and socioeconomic disadvantage. We found a positive correlation between student confidence and mean facility, higher confidence for boys than for girls and lower confidence for students classified as socioeconomically disadvantaged, even after accounting for facility. We found that confidence was lower for older students, and this was particularly marked across the primary to secondary school transition. An important feature of the online platform used is ...
Exploring even something as simple as a straight-line graph leads to various mathematical possibi... more Exploring even something as simple as a straight-line graph leads to various mathematical possibilities that students can uncover through their own questions.
Canadian Journal of Science, Mathematics and Technology Education, 2014
The frequent misinterpretation of the nature of confidence intervals by students has been well do... more The frequent misinterpretation of the nature of confidence intervals by students has been well documented. This article examines the problem as an aspect of the learning of mathematical definitions and considers the tension between parroting mathematically rigorous, but essentially uninternalized, statements on the one hand and expressing imperfect but developing understandings on the other. A small-scale study among schoolteachers sought comments on four definitions expressing differing understandings of confidence intervals, and these are examined and discussed. The article concludes that some student wordings could be regarded as less inaccurate than they might seem at first sight and presents a case for accepting a wider range of more intuitive understandings as a work in progress.RésuméLa fréquente mésinterprétation de la nature des intervalles de confiance de la part des étudiants est bien documentée. Cet article analyse la question en tant qu’aspect de l’apprentissage des définitions mathématiques, et considère la différence entre d’une part la répétition d’énoncés parfaitement rigoureux sur le plan mathématique, mais qui n’ont pas été intégrés, et d’autre part l’expression de concepts encore imparfaitement maîtrisés, mais qui dénotent une certaine compréhension. Une étude réalisée auprès d’un petit groupe d’enseignants a sollicité leurs commentaires au sujet de quatre définitions exprimant différents degrés de compréhension du concept d’intervalle de confiance, commentaires qui ont ensuite fait l’objet d’une analyse et d’une discussion. L’article conclut que certains énoncés des étudiants sont moins inexacts qu’ils ne pourraient sembler à priori, ce qui suggère qu’on peut accepter une plus vaste gamme d’énoncés intuitifs comme l’expression d’une ‘compréhension en devenir’.
Maths teacher Colin Foster finds a glass of fizzy orange can help students get to grips with the ... more Maths teacher Colin Foster finds a glass of fizzy orange can help students get to grips with the tricky topic of ratio.
International Journal of Mathematical Education in Science and Technology, 2013
ABSTRACT In a high-stakes assessment culture, it is clearly important that learners of mathematic... more ABSTRACT In a high-stakes assessment culture, it is clearly important that learners of mathematics develop the necessary fluency and confidence to perform well on the specific, narrowly defined techniques that will be tested. However, an overemphasis on the training of piecemeal mathematical skills at the expense of more independent engagement with richer, multifaceted tasks risks devaluing the subject and failing to give learners an authentic and enjoyable experience of being a mathematician. Thus, there is a pressing need for mathematical tasks which embed the practice of essential techniques within a richer, exploratory and investigative context. Such tasks can be justified to school management or to more traditional mathematics teachers as vital practice of important skills; at the same time, they give scope to progressive teachers who wish to work in more exploratory ways. This paper draws on the notion of a musical etude to develop ´ a powerful and versatile approach in which these apparently contradictory aspects of teaching mathematics can be harmoniously combined. I illustrate the tactic in three central areas of the high-school mathematics curriculum: plotting Cartesian coordinates, solving linear equations and performing enlargements. In each case, extensive practice of important procedures takes place alongside more thoughtful and mathematically creative activity.
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