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.

What do mathematicians mean by proof? A comparative-judgement study of students’ and mathematicians’ views

“…Comparative judgement, as used here, allows us to take useful steps in this (Bisson et al, 2016(Bisson et al, , 2019Jones et al, 2019) and problem solving (Jones and Inglis, 2015). We have also demonstrated the value of comparative judgement in assessing proof comprehension, as captured by undergraduate students' ability to summarise given proofs Davies et al (2020).…”

“…In mathematics education, comparative judgement has been used to assess constructs including proof comprehension (Davies et al, 2020), conceptual understanding (Bisson et al, 2016(Bisson et al, , 2019Jones et al, 2019), problem solving (Jones and Inglis, 2015) and general reasoning in primary school (Hunter and Jones, 2018). In all cases, the construct is considered important but resists rubricbased assessment.…”

Comparative judgement, proof summaries and proof comprehension

“…First, the multiple-choice test itself did not achieve acceptable internal consistency. A similar problem hampered Bisson, Gilmore, Inglis, and Jones (2019), who applied CJ-based assessment in differential calculus: there, too, a standardised instrument performed sub-optimally outside the context in which it was developed. If the substantial work done in generating standardised instruments needs to be repeated in each context, that further motivates a need for alternative low-resource assessment methods such as CJ.…”

Concept maps as assessment for learning in university mathematics

“…However, a major limitation of this study is the lack of robust evidence to make generalisable claims about the utility of concept mapping as an assessment tool for conceptual understanding. Future research could focus on a detailed analysis of qualitative features of concept mapping that differentiate it from conventional assessments and by comparing it with the existing valid methods for assessing conceptual understanding, such as a comparative judgement (CJ) approach, which was successfully used for assessing conceptual understanding of some mathematical topics: derivatives, p-values and use of letters in algebra (Bisson et al, 2016(Bisson et al, , 2020Jones et al, 2019).…”