The nature of mathematics practice offered to students and the impact of practice-through-progress

Practice plays a critical role in the teaching and learning of mathematics. This thesis provides new insights into practising in mathematics, what teachers attend to when choosing practice tasks, and explores, for the first time, the impact of practice-through-progress. I take a pragmatic approach to research design guided by the research questions using four theoretical frameworks: attention, desirable difficulties, variation, and practice-through-progress.

In the first part of the thesis, I explored mathematics teachers’ views on practising by analysing questionnaires, teacher interviews, and the practice tasks teachers used. From this, I identified four key points regarding how teachers choose and use practice tasks. First, teachers tended to choose tasks that gradually increase in difficulty but often feel pressured and dissatisfied with their choices. Second, teachers often sourced practice tasks online so they were not necessarily part of a coherent journey considering beneficial strategies such as spacing, variation, and interleaving. Third, knowing the effective practice types is insufficient; translational research is needed to support teachers in understanding when, how, and why to use different forms of practice. Fourth, less choice and more support and guidance seemed effective when the shared material was sufficiently high-quality. Based on these points, I suggest that many mathematics teachers would benefit from high-quality curricular materials to support professional decisions about practising.

The second part of the thesis considers the impact of practice-through-progress: incorporating previously learned material when teaching new material. Drawing upon existing literature, I contribute a tighter definition of practice-through-progress, proposing three types of practice-through-progress: extrinsic, intrinsic, and hierarchical. Two pre-registered randomised controlled trials compared: (a) Alternating-interleaved practice, switching between fractions and area; (b) Blocked practice, fraction problems followed by area problems; (c) Combined practice-through-progress where all the shapes had fraction dimensions. Both experiments found that all three methods had similar effects on procedural fluency, even during initial learning, with practice-through-progress potentially seen as a desirable difficulty. Practice-through-progress shifted learners’ attention away from the underlying ideas towards unconscious competence. Subsequently, an expert interview suggested practice-through-progress might reduce forgetting, boost confidence and competence in problem-solving, and create a more connected view of mathematics for both learners and teachers. Practice-through-progress developed fluency just as well as traditional exercises and might have other benefits.

These findings provide an important first step in in exploring the impact of practice-through-progress. Future research might usefully explore the cumulative impact of practice-through-progress over time with the goal of supporting learners’ mathematical development.