Teaching using contextualised and decontextualised representations: examining the case of differential calculus through a comparative judgement technique

ABSTRACT

An ongoing debate concerns whether novel mathematical concepts are better learned using contextualised or decontextualised representations. A barrier to resolving this debate, and therefore to progress in the discipline, has been the paucity of validated methods of measuring students’ understanding of mathematical concepts. We developed an innovative and efficient method for measuring, in experimental settings, students’ understanding of any mathematical concept using comparative judgement. We demonstrate the method by applying it to the comparison of learning outcomes from two teaching conditions. Participants (260 15–16 year olds across six schools) were introduced to differential calculus using contextualised or decontextualised representations. We then assessed participants’ comparative conceptual understanding of derivatives. We found evidence that contextualised and decontextualised representations were equally effective at promoting student learning in this context. The assessment method yielded valid and reliable results, suggesting that it offers a robust and efficient approach for the problem of assessing conceptual understanding in experimental or other comparative settings.

KEYWORDS:

• Decontextualised representations
• calculus
• conceptual understanding
• contextualised representations
• randomised controlled trial

Acknowledgements

The research was funded by a Nuffield Foundation grant to IJ, CG and MI. CG is funded by a Royal Society Dorothy Hodgkin Research Fellowship, and MI was funded by a Royal Society Worshipful Company of Actuaries Research Fellowship. We are grateful to Rod Bond who designed and delivered both teaching interventions.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 Grades B to A* represent about the top third of GCSE mathematics candidates. They were the top three grades in GCSE mathematics on a scale running from G (lowest) to A* (highest) in addition to U (unclassified).

2 The split-halves technique produces an underestimate of the true inter-rater reliability because it effectively calculates the reliability for just half of the total judgements used to generate the CJ scores used in the analysis.

Additional information

Funding

The research was funded by a Nuffield Foundation grant to IJ, CG and MI. CG is funded by a Royal Society Dorothy Hodgkin Research Fellowship, and MI was funded by a Royal Society Worshipful Company of Actuaries Research Fellowship.