Assessing Proof Reading Comprehension Using Summaries

In this paper, we explore the role of mathematical proof summaries as a tool for capturing students’ reading comprehension of a given proof. We present an interview study based on mathematicians’ pairwise evaluations of student-produced summaries of a proof demonstrating the uncountability of the open unit interval. We present a thematic analysis, exploring features of mathematicians’ pairwise decision-making and their priorities in evaluating summaries. We argue that the students’ proof summaries shared several properties with traditional modes of proof-writing and were frequently evaluated against similar conventions. We consider the consequences for research and practice with proof comprehension and conclude that proof summaries have the potential to form the basis of a new approach to assessment in this area.

Comparative judgement, proof summaries and proof comprehension

Proof is central to mathematics and has drawn substantial attention from the mathematics education community. Yet, valid and reliable measures of proof comprehension remain rare. In this article, we present a study investigating proof comprehension via students’ summaries of a given proof. These summaries were evaluated by expert judges making pairwise comparisons, which were used to generate a score for each summary. This approach, known as comparative judgement, has been demonstrated to generate reliable and valid scores when assessing other mathematical constructs. Our findings suggest that comparative judgement can produce valid and reliable assessments of the quality of student-produced proof summaries. We also explored which features of students’ proof summaries were most valued by the expert judges, and found that high-scoring summaries referenced a proof’s logical structure and the mechanism by which it reached a contradiction.

Exploring students’ proof comprehension of the Cauchy Generalized Mean Value Theorem

Undergraduate students majoring in mathematics often face difficulties in comprehending mathematical proofs. Inspired by a number of studies related to students’ proof comprehension, and Mejia-Ramos et al.’s study in particular, a test was designed in relation to the proof comprehension of the Cauchy Generalized Mean Value Theorem (CGMVT). The test mainly focused on (a.) investigating students’ understanding of relations between the statements within the CGMVT proof and (b.) the relations between the CGMVT and other theorems. Thirty-five first-year university students voluntarily participated in this study. In addition, 10 of these students were subsequently interviewed to seek their opinion about the test. Test results indicated that most of the students lacked an understanding of the relations between the mathematical statements within the CGMVT proof, and the relations between the CGMVT and other theorems. The results of interviews showed that this type of assessment was new to students and helped them to improve their insights into mathematical proofs. The findings suggested such a test design could be used more frequently in assessments to aid instructors’ understanding of students’ proof comprehension and to teach students how mathematical proofs should be learned.

FLAWS IN PROOF CONSTRUCTIONS OF POSTGRADUATE MATHEMATICS EDUCATION STUDENT TEACHERS

Intending to improve the teaching and learning of the notion of mathematical proof this study seeks to uncover the kinds of flaws in postgraduate mathematics education student teachers. Twenty-three student teachers responded to a proof task involving the concepts of transposition and multiplication of matrices. Analytic induction strategy that drew ideas from the literature on evaluating students’ proof understanding and Yang and Lin’s model of proof comprehension applied to informants’ written responses to detect the kinds of flaws in postgraduates’ proof attempts. The study revealed that the use of empirical verifications was dominant and in situations. Whereby participants attempted to argue using arbitrary mathematical objects, the cases considered did not represent the most general case. Flawed conceptualizations uncovered by this study can contribute to efforts directed towards fostering strong subject content command among school mathematics teachers.