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

Foster, Colin

doi:10.1080/0020739x.2013.770089

Cited by 21 publications

(16 citation statements)

References 4 publications

(3 reference statements)

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“…Too often, 572 C. Foster developing mathematical fluency on particular technical skills is seen as a prerequisite for tackling more complex problems. Instead, rich problems might provide more motivating and meaningful contexts for developing the desired fluencies (Foster, 2013a). Ariely (2010) laments some aspects of the 'division of labour' prevalent in modern society:…”

Section: Micro-learning Objectives Agendamentioning

confidence: 99%

Resisting reductionism in mathematics pedagogy

Foster

2013

The Curriculum Journal

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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.

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Section: Micro-learning Objectives Agendamentioning

confidence: 99%

Resisting reductionism in mathematics pedagogy

Foster

2013

The Curriculum Journal

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“…In this context, it has been suggested that a mathematics task genre of etudes might be capable of addressing proceduralfluency at the sametime as offeringa richer experienceof learning mathematics (Foster, 2013b(Foster, , 2014. Etudes are mathematics tasks in which extensive practice of a well-defined mathematical procedure is embedded within a rich mathematical problem (Foster, 2013b).…”

Section: Introductionmentioning

confidence: 99%

“…Being secure with important mathematical procedures gives students increased power to tackle more complicated mathematics at a more conceptual level (Codding, Burns, & Lukito, 2011;Foster, 2013bFoster, , 2016, since automating skills frees up mental capacity for being creative (Lemov, Woolway, & Yezzi, 2012, p. 36). Devising ways to support the development of robust fluency with mathematical procedures is currently a focus of attention.…”

Section: Introductionmentioning

confidence: 99%

Developing mathematical fluency: comparing exercises and rich tasks

Foster

2017

Educ Stud Math

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Achieving fluency in important mathematical procedures is fundamental to students' mathematical development. The usual way to develop procedural fluency is to practise repetitive exercises, but is this the only effective way? This paper reports three quasiexperimental studies carried out in a total of 11 secondary schools involving altogether 528 students aged 12-15. In each study, parallel classes were taught the same mathematical procedure before one class undertook traditional exercises while the other worked on a Bmathematical etude^(Foster International Journal of Mathematical Education in Science and Technology, 44(5), 765-774, 2013b), designed to be a richer task involving extensive opportunities for practice of the relevant procedure. Bayesian t tests on the gain scores between pre-and post-tests in each study provided evidence of no difference between the two conditions. A Bayesian meta-analysis of the three studies gave a combined Bayes factor of 5.83, constituting Bsubstantial^evidence (Jeffreys, 1961) in favour of the null hypothesis that etudes and exercises were equally effective, relative to the alternative hypothesis that they were not. These data support the conclusion that the mathematical etudes trialled are comparable to traditional exercises in their effects on procedural fluency. This could make etudes a viable alternative to exercises, since they offer the possibility of richer, more creative problem-solving activity, with comparable effectiveness in developing procedural fluency.

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“…These curves intersect when 4x 2 -18x + 22 = -2x 2 + 12x -14; i.e., when 6x 2 -30x + 36 = 0. So x 2 -5x + 6 = 0, so (x -2)(x -3) = 0, so at (2, 2) and (3,4). The area between these curves is 1 ( Figure 1).…”

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confidence: 99%

“…Some closed questions can have answers that are so provocative that they invite further exploration or demand an explanation, leading to rich mathematical thinking [1]. I often like to offer students a closed starting point, which draws on routine procedures that they have learned, but which leads to a provocative result which motivates them to go on to pursue related mathematics in a more exploratory way [2,3].…”

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confidence: 99%