Abstract
This paper presents two studies of an intervention designed to help undergraduates comprehend mathematical proofs. The intervention used multimedia resources that presented proofs with audio commentary and visual animations designed to focus attention on logical relationships. In study 1, students studied an e-Proof or a standard written proof and their comprehension was assessed in both immediate and delayed tests; the groups performed similarly at immediate test, but the e-Proof group exhibited poorer retention. Study 2 accounted for this unexpected result by using eye-movement analyses to demonstrate that participants who studied an e-Proof exhibited less processing effort when not listening to the audio commentary. We suggest that the extra support offered by e-Proofs disrupts the processes by which students organise information, and thus restricts the extent to which their new understanding is integrated with existing knowledge. We discuss the implications of these results for evaluating teaching innovations and for supporting proof comprehension.
Highlights
Most students learn about proving at least in part via encounters with written proofs
E-Proofs belong to a class of interventions that provide reading support targeted to specific content. Other such interventions have been designed to support careful reading of textbook sections by combining direct instruction on what to read with questions to prompt reflection (Alcock, Brown, & Dunning, 2015; Shepherd, 2005), or to promote careful reading of single proofs by asking comprehension questions tailored to those proofs (Conradie & Frith, 2000; Cowen, 1991)
It is not obvious how to balance these issues: there can be a fine line between productive struggle and distressing failure. These results suggest to us that e-Proofs provide help that is too much or too directive, and that other approaches seem more promising for instructors wishing to support undergraduate proof comprehension
Summary
Most students learn about proving at least in part via encounters with written proofs. This is obvious: students are often expected to learn by watching and listening as a lecturer presents proofs at a board (Weber, 2004). Students in almost all instructional situations are expected to use books or online materials to support their learning. This is appropriate: we want educational systems to produce young people who can critically assess written arguments, and we want to develop the generation of mathematicians and scientists— those people will spend much of their working lives learning from complex written materials. We pursue this line of research, reporting two studies on the effects of an intervention designed to help undergraduate students read and understand mathematical proofs
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