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,…

Teachers need ways to efficiently assess students' cognitive understanding. One promising approach involves easily adapted and administered item types that yield quantitative scores that can be interpreted in terms of whether or not students likely possess key understandings. This study illustrates an approach to analyzing response process…

The current study aims to explore the impact of the two-column format in writing simple mathematical arguments. That is to say, a structured method of presenting a mathematical proof or argument by using a tabular layout with two columns. The underlying goal of the research reported in this paper is to inform understanding of how to effectively…

Research using the Anthropological Theory of the Didactic suggests different models of how student learning may evolve in the progression of undergraduate mathematics coursework: from elementary courses in Calculus to more advanced courses in Analysis. An ideal model suggests that the theory-driven learning in the latter serves as a natural…

We discuss a practical method for assessing mathematical proof online. We examine the use of faded worked examples and reading comprehension questions to understand proof. By breaking down a given proof, we formulate a checklist that can be used to generate comprehension questions which can be assessed automatically online. We then provide some…

We argue that examples can do more than serve the purpose of illustrating the truth of an existential statement or disconfirming the truth of a universal statement. Our argument is relevant to the use of technology in classroom assessment. A central challenge of computer-assisted assessment is to develop ways of collecting rich and complex data…

Drawing upon his experience, more than 60 years ago, as a psychometric support person to a very special teacher of brain damaged children, the author of this article reflects on the productive use of educational assessments and data from them to educate - assessment in the service of learning. Findings from the Gordon Commission on the Future of…

Specifications grading is a version of mastery grading distinguished by giving students clear specifications that their work must meet, and grading most things pass/fail based on those specifications. Mastery grading systems can get quite elaborate, with hierarchies of objectives and various systems for rewriting and retesting. In this article I…

Multiplicative reasoning is essential for students' engagement with various mathematical concepts. Although the field's understanding of children's multiplicative concepts has grown over the past 30 years, relatively few studies have examined the development of multiplicative concepts with whole numbers, and even fewer have studied this phenomenon…

The fact that students have difficulties in constructing proofs is well documented. However, some of these difficulties may be lessened if instructors and students have access to a common evaluation framework. Operating in the theoretical tradition of heuristic inquiry, a proof error evaluation tool (PEET) is constructed that may be used by…

Tasks regularly completed by elementary teachers reveal the mathematical nature of their work. However, preservice teachers demonstrate a lack of depth of mathematical thought. This study investigated the criteria preservice teachers intuitively used to evaluate algorithms. The intent was to use that knowledge as a foundation for modeling…

This submission contains the Proceedings of the 2012 Annual Meeting of the Canadian Mathematics Education Study Group (CMESG), held at Laval University in Québec City, Québec. The CMESG is a group of mathematicians and mathematics educators who meet annually to discuss mathematics education issues at all levels of learning. The aims of the Study…

In this study we investigated relationships between convergent and divergent thinking with an emphasis on fluency, originality, flexibility and elaboration in the mathematical domain. A related purpose was to examine relationships between problem types in mathematical tasks. The math section of a performance-based assessment was used to assess 857…

This study was designed to identify the types of understandings, skills, and beliefs that affect pre-service teachers' evaluations of students' mathematical arguments in classroom contexts. Thirty-four pre-service teachers read a transcript of a third grade lesson in which the students were expected to prove a generalization. To investigate…

This article describes a sophomore transition-to-advanced-math course taught via a modified Moore method at Lebanon Valley College.