We recently reported a study in which undergraduate students and research mathematicians were asked to read and validate purported proofs (Inglis & Alcock, 2012). In our eye-movement data, we found no evidence of the initial skimming strategy hypothesized by Weber (2008). Weber and Mejía-Ramos (2013) argued that this was due to a flawed analysis of eye-movement data and that a more fine-grained analysis led to the opposite conclusion. Here we demonstrate that this is not the case, and show that their analysis is based on an invalid assumption.
Matthew Inglis and Lara Alcock
Matthew Inglis and Lara Alcock
This article presents a comparison of the proof validation behavior of beginning undergraduate students and research-active mathematicians. Participants' eye movements were recorded as they validated purported proofs. The main findings are that (a) contrary to previous suggestions, mathematicians sometimes appear to disagree about the validity of even short purported proofs; (b) compared with mathematicians, undergraduate students spend proportionately more time focusing on “surface features” of arguments, suggesting that they attend less to logical structure; and (c) compared with undergraduates, mathematicians are more inclined to shift their attention back and forth between consecutive lines of purported proofs, suggesting that they devote more effort to inferring implicit warrants. Pedagogical implications of these results are discussed, taking into account students' apparent difficulties with proof validation and the importance of this activity in both schooland university-level mathematics education.
Mark Hodds, Lara Alcock and Matthew Inglis
In this article, the authors report 3 experiments demonstrating that a simple booklet containing self-explanation training, designed to focus students' attention on logical relationships within a mathematical proof, can significantly improve their proof comprehension.
Sven Trenholm, Lara Alcock and Carol Robinson
As part of a dramatic recent shift in tertiary education, many undergraduate students now learn mathematics via fully online courses. At present, the mathematics education research community knows very little about this shift, and in this report, we consider implications of an investigation into the instructor experience of fully online undergraduate mathematics courses. To compare instructor experiences of fully online and face-to-face teaching, we explored assessment schemes, feedback processes, and approaches to teaching using a survey and semistructured interviews. The main emergent theme was instructor concern about the loss of short-cycle face-to-face human interaction. We argue that this concern is serious but should be seen as an opportunity for education researchers to leverage knowledge about effective mathematics teaching to simultaneously alleviate instructors' difficulties and promote and study pedagogical development.
Lara Alcock, Paul Hernandez-Martinez, Arun Godwin Patel and David Sirl
In this article, we argue that although mathematics educators are concerned about social issues, minimal attention has been paid to student–student interactions outside the classroom. We discuss social network analysis as a methodology for studying such interactions in the context of an undergraduate course. We present results on the questions: Who studies with whom? What are students’ study habits, and are these systematically related to the habits of those with whom they interact? Do individual and collaborative study habits predict attainment? We discuss the implications of these findings for research on undergraduate learning and on social issues in mathematics education, suggesting that social network analysis may provide a bridge between mathematics education researchers who focus on cognitive and on social issues.