Hands up if you’ve never set your pupils a block of practice questions that are all similar but slightly different. You could describe this as a ‘tedious exercise’ – and maybe you are right. Sometimes the initial stages of learning a new technique is tedious. It’s part of the skill of a teacher to encourage and motivate pupils to get past a tedious stage before they can progress to doing something more interesting with their acquired skill – some sort of ‘lovely rich task’.
I’ve taken the two quotes: ‘lovely rich tasks’ and ‘tedious exercises’ from a group of publications by Dr Colin Foster, Reader in Mathematics Education in the Department of Mathematics Education at Loughborough University. He’s a fan of a type of practice task he calls an etude after a type of musical practice which involves embedding the practice activity in a rich task. You can read about edudes in maths here: https://nrich.maths.org/13206
I am uncomfortable when an education academic uses emotive language like this. Teachers and pupils deserve researched and reasoned arguments, not rhetoric. But Foster is doing both. He’s carried out a well-designed trial of the impact of using his etudes vs massed practice at the start of teaching a new topic. You can read his paper here.
I’ve summarised his three experiments here.
- Study 1: Expression Polygons
This study focused on solving linear equations and compared the effectiveness of traditional exercises to an etude named “Expression Polygons.” In this etude, students were presented with polygons where each side represented an equation to be solved. The results indicated no significant difference between the traditional exercises and the etude in terms of improving procedural fluency, with a Bayes factor close to 1, suggesting an inconclusive outcome. - Study 2: Devising Equations
In this study, the focus remained on solving linear equations, but a different etude named “Devising Equations” was used. Here, students were tasked with creating and solving their own equations to achieve whole-number solutions. The findings demonstrated that this etude was almost as effective as traditional exercises, with a Bayes factor of 5.92, providing substantial evidence that there was no significant difference in their effectiveness. - Study 3: Enlargements
The third study shifted the focus to geometry, specifically the task of enlarging shapes on a grid. The “Enlargements” etude required students to determine possible centers of enlargement to ensure that the enlarged shapes fit within a grid. Similar to the previous studies, the results showed no significant difference in procedural fluency development between the etude and traditional exercises, with a Bayes factor of 5.20, suggesting substantial evidence of equivalence.
Of the three studies, the first is inconclusive, but the second and third show no improvements of etudes over traditional practice exercises.
In short, he’s produced evidence that well-designed practise tasks which embed the practise in a context, or problem to solve is not worse than massed practice. Which is why, I suppose, it is okay that he uses emotive language… if one task is tedious and another isn’t and they are equally effective, then why wouldn’t you use the interesting one?
I still have some reservations. Cognitive Load Theory suggests that adding additional context at the very start of instruction leads to worse outcomes (see the redundancy effect). I would feel happier if etudes were introduced after pupils had got the hang of the simplest version before adding context. Then, by all means, practise in as many different contexts as you can.
Reading for Learning 