Colin Foster

Mathematical études: embedding opportunities for developing procedural fluency within rich mathematical contexts

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 étude 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.

Resisting reductionism in mathematics pedagogy

Although breaking down a mathematical problem into smaller parts can often be an effective solution strategy, when the same reductionist approach is applied to mathematics pedagogy the effects are far from beneficial for students. Mathematics pedagogy in UK schools is gaining an increasingly reductionist flavour, as seen in an excessive focus on bite-sized learning objectives and a tendency for mathematics teachers to path-smooth their students’ learning. I argue that a reductionist mathematics pedagogy severely restricts students’ opportunities to engage in authentic mathematical thinking and deprives them of the enjoyment of solving richer, more worthwhile problems, which would forge connections across diverse areas of the subject. I attribute the rise of a reductionist mathematics pedagogy partly to an assessment-dominated curriculum and partly to a systemic de-professionalisation of teachers through a performative accountability culture in which they are constantly required to prove to non-specialist managers that they are effective. I argue that pedagogical reductionism in mathematics must be resisted in favour of a more holistic approach, in which students are able to bring a variety of mathematical knowledge and skills to bear on rich, challenging and non-routine mathematical tasks. Some principles for achieving this are outlined and some examples are given.

Confidence Trick: The Interpretation of Confidence Intervals

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’.

Teaching with tasks for effective mathematics learning

There has recently been increased interest in the research and development of task design within mathematics education. This follows the production over time of theoretically-informed materials and approaches such as those from Realistic Mathematics Education (De Lange, 1996), Connected Mathematics (Lappan, Fitzgerald, Feyl, Friel, & Phillips, 2009) and the Centre for Research in Mathematics Education at the University of Nottingham (also known as the Shell Centre) (Swan, 2006). In this book, Sullivan, Clarke and Clarke make an important contribution to the growing literature on mathematics tasks as they outline the findings from their three-year project Task Types in Mathematics Learning, in which they worked with grade 5 8 teachers from three varied clusters of primary and secondary schools in Victoria, Australia. They describe a wide variety of different mathematical tasks and their observed effects in the classroom, how teachers’ pedagogical practices were informed through the process, and insights from the students. They also report on a subset of their teachers who took responsibility for designing and teaching extended sequences of lessons. All of the authors’ research is ‘‘based on an assumption that choice of tasks and the associated pedagogies are key aspects of teaching and learning mathematics’’ (p. 14) and that ‘‘the nature of teaching and what students learn are defined largely by the tasks that form the basis of their actions’’ (p. 57). In line with other researchers in this area, they see tasks as fundamental to all mathematics classroom activity. The authors ‘‘use the term task to refer to information that serves as the prompt for student work’’ (p. 13), and the study focuses on three task types:

The definition of the scalar product: an analysis and critique of a classroom episode

In this paper, we present, analyse and critique an episode from a secondary school lesson involving an introduction to the definition of the scalar product. Although the teacher attempted to be explicit about the difference between a definition and a theorem, emphasizing that a definition was just an arbitrary assumption, a student rejected the teachers definition in favour of his own alternative. With reference to this particular case, we seek to explore some ways in which teachers can introduce mathematical definitions to students so as to support, rather than attempt to circumvent, their mathematical sense making. In this regard, we believe that it is important to develop learning opportunities for students which help them to gain some appreciation of important structural and historical reasons that underpin the definitional choices made.

Closed but provocative questions: curves enclosing unit area

This article describes a task leading to work on curve sketching, simultaneous equations and integration to find the area enclosed between two curves. An initial closed question is used to confront students with a provocative answer, which they then explore in a much more open-ended way.

The Significance of a Square-Root Rule

In this article, I comment on a rough square-root rule for statistical significance which turns out to be equivalent to testing whether an observation lies within one standard deviation of that expected.