“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 USA


The Mathematical Etudes Project aims to find creative, imaginative and thought-provoking 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 problem-solving 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

Loughborough University




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 Technology44(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/s10649-017-9788-x

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

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.

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)


Straight-line 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)



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)



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.



Foster, C. (2015). Repeated rotations. Teach Secondary, 4(1), 35–37. (Resource sheet pdf and Geogebra files ggb, ggb)


Calculating the mean

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


© Colin Foster 2020