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, 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.
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. (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.
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.
Foster, C. (2019). Alternative vouchers. Teach Secondary, 8(8), 9091.
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. (2018). Beat the calculator. Teach Secondary, 7(5), 8687.
Foster, C. (2013). Connected quadratics. Teach Secondary, 2(1), 4648.
Foster, C. (2012). Straight to the point. Learning and Teaching Mathematics, 13, 610.
Ratio, proportion and rates of change
Foster, C. (2019).Knowing the unknowns. Teach Secondary, 8(1), 8687.
Enlarging a shape
Foster, C. (2012). Working without a safety net. The Australian Mathematics Teacher, 68(2), 2529.
Foster, C. (2019). Spider on a cuboid. Teach Secondary, 8(6), 116117.
Foster, C. (2015). Crossing lines. Teach Secondary, 4(3), 3133.
Foster, C. (2018). Five triangles. Teach Secondary, 7(2), 3031.
Foster, C. (2020). Statistical puzzler. Teach Secondary, 9(1), 8485.
© Colin Foster 2023