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

Foster, Colin

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 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.

• Journal Article Type: Article
• Publication Date: Jan 1, 2013
• Deposit Date: Apr 15, 2014
• Publicly Available Date: Apr 15, 2014
• Journal: International Journal of Mathematical Education in Science and Technology
• Print ISSN: 0020-739X
• Electronic ISSN: 0020-739X
• Publisher: Taylor & Francis Open
• Peer Reviewed: Peer Reviewed
• Volume: 44
• Issue: 5
• DOI: https://doi.org/10.1080/0020739X.2013.770089
• Keywords: Cartesian coordinates, enlargement, fluency, learning for understanding, practice, solving equations, task design, traditional and progressive approaches to the teaching of mathematics
• Public URL: https://nottingham-repository.worktribe.com/output/1004181
• Publisher URL: http://www.tandfonline.com/doi/pdf/10.1080/0020739X.2013.770089