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, 2017,
2018). Recent research (Foster, 2018) 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.

The video below introduces the idea of mathematical etudes, and presents the findings from Foster (2018).

[The article for teachers that I refer to is here and the full research paper is here. The lesson plans for the tasks are here, here and here, and the worksheet for the enlargements task is here. The slides from the presentation are available here.]

For more details, please see the papers listed below, or scroll down for some example tasks.

**Colin Foster**

**Loughborough University**

**References**

Foster, C. (2011). A picture is worth a thousand exercises. *Mathematics
Teaching*, *224*, 1011. (Extra material)

Department for
Education (DfE) (2013). *Mathematics Programmes
of Study: key stage 3, national curriculum in England*. London: DfE.

Foster, C. (2013). Mathematical ιtudes:
Embedding opportunities for developing procedural fluency within rich
mathematical contexts.
*International Journal of Mathematical Education in Science and Technology*, *44*(5),
765774.

Foster, C. (2014). Mathematical fluency *without* drill and
practice.
*Mathematics Teaching*, *240*, 57.

Foster, C. (2016). Confidence and
competence with mathematical procedures. *Educational Studies in Mathematics*, *91*(2),
271288.

Foster, C. (2017). Mathematical etudes. NRICH article available at: https://nrich.maths.org/13206

Foster, C. (2018).
Developing
mathematical fluency: Comparing exercises and rich tasks. *Educational
Studies in Mathematics, 97*(2), 121141. https://doi.org/10.1007/s10649-017-9788-x

Foster, C. (2021, March 31). Developing fluency with procedures without using traditional exercises [Video]. YouTube. https://https://www.youtube.com/watch?v=OSrnbf9YyO0

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**

Foster, C. (2020). Number snakes. *Teach Secondary*,
*9*(8), 7071.

*Adding
fractions*

Foster, C. (2014). Sum fractions. *Teach Secondary*,
*3*(5), 4849. (NRICH version: https://nrich.maths.org/13205)

*Addition
and subtraction*

Foster, C. (2016). Sums of pairs. *Symmetry Plus*,
*59*(1), 1416.

*Highest
common factors*

Foster, C. (2012). HCF and LCM Beyond
procedures.
*Mathematics in School*, *41*(3), 3032.

Foster, C. (2012). The what factor? *Teach Secondary*,
*1*(4), 5658.

*Multiplication
of integers*

Foster, C. (2016). Making products. *Teach Secondary*,
*5*(5), 3133.

Foster, C. (2017). Surprise, surprise! *Teach Secondary*,
*6*(1), 4244.

*Percentages*

Foster, C. (2014). Whats the deal? *Teach Secondary*,
*3*(7), 3435. (Resource sheet pdf)

Foster, C. (2017). Pink paint. *Teach Secondary*,
*6*(4), 32. (Full lesson plan pdf)

Foster, C. (2019). Alternative vouchers. *Teach Secondary*,
*8*(8), 9091.

*Decimals*

Foster, C. (2018). Almost zero. *Teach Secondary*,
*7*(8), 8485.

Foster, C. (2008,
March 7). Youre having a
graph.
*Times Educational Supplement Magazine*, pp. 5859.

Foster, C. (2011). A picture is worth a thousand
exercises.
*Mathematics Teaching*, *224*, 1011. (Extra material)

*Simplifying
expressions*

Foster, C. (2016). The simple life. *Teach Secondary*,
*5*(2), 3133. (Resource sheet pdf; NRICH version: https://nrich.maths.org/13207)

Foster, C. (2018). Beat the calculator. *Teach Secondary*,
*7*(5), 8687.

*Solving
equations*

Foster, C. (2013). Connected quadratics. *Teach Secondary*,
*2*(1), 4648.

Foster, C. (2015). Expression polygons. *Mathematics
Teacher, 109*(1), 6265. (Resource sheet doc)

*Straight-line
graphs*

Foster, C. (2012). Straight to the
point.
*Learning and Teaching Mathematics*, *13*, 610.

*Ratio,
proportion and rates of change*

*Percentage
change*

Foster, C. (2014). Whats the deal? *Teach Secondary*,
*3*(7), 3435. (Resource sheet pdf)

Foster, C. (2017). Pink paint. *Teach Secondary*,
*6*(4), 32. (Full lesson plan pdf)

*Simultaneous equations*

Foster, C. (2019).Knowing the unknowns. *Teach Secondary*,
*8*(1), 8687.

*Bearings*

Foster, C. (2017). Get your bearings. *Teach Secondary*,
*6*(8), 3233. (Resource sheet pdf and solution sheet pdf)

*Calculating
angles*

Foster, C. (2014). Angle chasing. *Teach Secondary*,
*3*(4), 4041. (Resource sheets pdf)

Foster, C. (2015). Clock watching. *Teach Secondary*,
*4*(8), 3132. (Spreadsheet resource xls)

*Enlarging
a shape*

Foster, C. (2012). Working without a
safety net.
*The Australian Mathematics Teacher*, *68*(2), 2529.

Foster, C. (2013). Staying on the page. *Teach Secondary*,
*3*(1), 5759. (Resource sheet pdf)

*Nets*

Foster, C. (2017). A fitting challenge. *Teach Secondary*,
*6*(6), 4849. (Full lesson plan)

Foster, C. (2019). Spider on a cuboid. *Teach Secondary*,
*8*(6), 116117.

*Perpendicular
bisectors*

Foster, C. (2015). Crossing lines. *Teach Secondary*,
*4*(3), 3133.

*Rotations*

Foster, C. (2015). Repeated rotations. *Teach Secondary*,
*4*(1), 3537. (Resource sheet pdf and *Geogebra*
files ggb, ggb)

*Trigonometry*

Foster, C. (2018). Five triangles. *Teach Secondary*,
*7*(2), 3031.

Foster, C. (2015). The meaning of the
mean.
*Teach Secondary*, *4*(6), 3739. (Resource sheet pdf)

Foster, C. (2020). Statistical puzzler. *Teach Secondary*,
*9*(1), 8485.

© Colin Foster 2024