www.mathematicaletudes.com
“Colin Foster is designing etudes that develop mathematical fluencies with style and flair, not to mention an afterglow of insight.” Phil Daro, lead author of the mathematics Common Core State Standards, used by most states in the United States 
The Mathematical Etudes Project aims to find creative, imaginative and thoughtprovoking ways to help learners of mathematics develop their fluency in important mathematical procedures.
Procedural fluency involves knowing when and how to apply a procedure and being able to perform it “accurately, efficiently, and flexibly” (NCTM, 2014, p. 1). Fluency in important mathematical procedures is a critical goal within the learning of school mathematics, as security with fundamental procedures offers pupils increased power to explore more complicated mathematics at a conceptual level (Foster, 2013, 2014, 2015; Gardiner, 2014; NCTM, 2014). The new national curriculum for mathematics in England emphasises procedural fluency as the first stated aim (DfE, 2013).
But it is often assumed that the only way to get good at standard procedures is to drill and practise them ad nauseum using dry, uninspiring exercises.
The Mathematical Etudes Project aims to find practical classroom tasks which embed extensive practice of important mathematical procedures within more stimulating, rich problemsolving contexts (Foster, 2011, 2013, 2014, 2017a, 2017b). Recent research (Foster, 2017a) suggests that etudes are as good as exercises in terms of developing procedural fluency – and it seems likely that they have many other benefits in addition.
For more details see the papers listed below or scroll down for some example tasks.
Colin Foster
University of Nottingham
References
Foster, C. (2011). A picture is worth a thousand exercises. Mathematics Teaching, 224, 10–11. (Extra material)
Department for Education (DfE) (2013). Mathematics Programmes of Study: key stage 3, national curriculum in England. London: DfE.
Foster, C. (2013). Mathematical études: Embedding opportunities for developing procedural fluency within rich mathematical contexts. International Journal of Mathematical Education in Science and Technology, 44(5), 765–774.
Foster, C. (2014). Mathematical fluency without drill and practice. Mathematics Teaching, 240, 5–7.
Foster, C. (2016). Confidence and competence with mathematical procedures. Educational Studies in Mathematics, 91(2), 271–288.
Foster, C. (2017a). Developing mathematical fluency: Comparing exercises and rich tasks. Educational Studies in Mathematics. https://doi.org/10.1007/s106490179788x
Foster, C. (2017b). Mathematical etudes. NRICH article available at: https://nrich.maths.org/13206
Gardiner, A.D. (2014). Teaching mathematics at secondary level. The De Morgan Gazette, 6(1).
National Council of Teachers of Mathematics (NCTM) (2014). Procedural Fluency in Mathematics: a position of the National Council of Teachers of Mathematics. Reston, VA: NCTM.
Examples of Mathematical Etudes on Different Topics
Number · Adding fractions Foster, C. (2014). Sum fractions. Teach Secondary, 3(5), 48–49. (NRICH version: https://nrich.maths.org/13205)
· Addition and subtraction Foster, C. (2016). Sums of pairs. Symmetry Plus, 59(1), 14–16.
· Highest common factors Foster, C. (2012). HCF and LCM – Beyond procedures. Mathematics in School, 41(3), 30–32. Foster, C. (2012). The what factor? Teach Secondary, 1(4), 56–58.
· Multiplication of integers Foster, C. (2016). Making products. Teach Secondary, 5(5), 31–33. Foster, C. (2017). Surprise, surprise! Teach Secondary, 6(1), 42–44. 
Algebra · Graph sketching Foster, C. (2008, March 7). You’re having a graph. Times Educational Supplement – Magazine, pp. 58–59. Foster, C. (2011). A picture is worth a thousand exercises. Mathematics Teaching, 224, 10–11. (Extra material)
· Simplifying expressions Foster, C. (2016). The simple life. Teach Secondary, 5(2), 31–33. (Resource sheet pdf; NRICH version: https://nrich.maths.org/13207)
· Solving equations Foster, C. (2013). Connected quadratics. Teach Secondary, 2(1), 46–48. Foster, C. (2015). Expression polygons. Mathematics Teacher, 109(1), 62–65. (Resource sheet doc)
· Straightline graphs Foster, C. (2012). Straight to the point. Learning and Teaching Mathematics, 13, 6–10. 
Ratio, proportion and rates of change · Percentage change Foster, C. (2014). What’s the deal? Teach Secondary, 3(7), 34–35. (Resource sheet pdf) Foster, C. (2017). Pink paint. Teach Secondary, 6(4), 32. (Full lesson plan pdf)

Geometry and measures · Bearings Foster, C. (2017). Get your bearings. Teach Secondary, 6(8), 32–33. (Resource sheet pdf and solution sheet pdf)
· Calculating angles Foster, C. (2014). Angle chasing. Teach Secondary, 3(4), 40–41. (Resource sheets pdf) Foster, C. (2015). Clock watching. Teach Secondary, 4(8), 31–32. (Spreadsheet resource xls)
· Enlarging a shape Foster, C. (2012). Working without a safety net. The Australian Mathematics Teacher, 68(2), 25–29. Foster, C. (2013). Staying on the page. Teach Secondary, 3(1), 57–59. (Resource sheet pdf)
· Nets Foster, C. (2017). A fitting challenge. Teach Secondary, 6(6), 48–49. (Full lesson plan)
· Perpendicular bisectors Foster, C. (2015). Crossing lines. Teach Secondary, 4(3), 31–33.
· Rotations Foster, C. (2015). Repeated rotations. Teach Secondary, 4(1), 35–37. (Resource sheet pdf and Geogebra files ggb, ggb)

Probability and Statistics · Calculating the mean Foster, C. (2015). The meaning of the mean. Teach Secondary, 4(6), 37–39. (Resource sheet pdf)


© Colin Foster 2018