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Mathematics is the discipline that a significant majority of most incoming researchers in mathematics education have prior qualifications and experience in. Upon entry into the field of mathematics education research, these newcomers–often students on a postgraduate programme in mathematics education–need a broadened understanding on how to read, converse, write and conduct research in the largely...
This study proposes a framework for examining ways in which prospective teachers integrate mathematical knowledge acquired in advanced topics courses into explanatory knowledge for school teaching. Participants, all of whom had recently completed coursework in abstract algebra, were asked to explain concepts connected to the school mathematics curriculum, such as division by zero and even numbers...
This paper describes a small-scale research project based on workbooks designed to support independent study of proofs in a first course on abstract algebra. We discuss the lecturers’ aims in designing the workbooks, and set these against a background of research on students’ learning of group theory and on epistemological beliefs and study habits in higher education. We organise our analysis of student...
This article presents the results from a study of 535 early undergraduate students at six universities that was designed to describe their views of the meaning of proof and how these views relate to their attitudes and beliefs towards proof and their classroom experiences with learning proof. Results show that early undergraduate students have difficulty with mathematical proof. In particular, the...
Counting problems provide an accessible context for rich mathematical thinking, yet they can be surprisingly difficult for students. To foster conceptual understanding that is grounded in student thinking, we engaged a pair of undergraduate students in a ten-session teaching experiment. The students successfully reinvented four basic counting formulas, but their work revealed a number of unexpected...
In this paper we present a case study of an individual student who consistently used semantic reasoning to construct proofs in calculus but infrequently used semantic reasoning to produce proofs in linear algebra. We hypothesize that the differences in these reasoning styles can be partially attributed to this student’s familiarity with the content, the teaching styles of the professors who taught...
Introductory college calculus students in the United States engaged in an activity called Peer-Assisted Reflection (PAR). The core PAR activities required students to: attempt a problem, reflect on their work, conference with a peer, and revise and submit a final solution. Research was conducted within the design research paradigm, with PAR developed in a pilot study, tried fully in a Phase I intervention,...
The Calculus Concept Readiness (CCR) instrument assesses foundational understandings and reasoning abilities that have been documented to be essential for learning calculus. The CCR Taxonomy describes the understandings and reasoning abilities assessed by CCR. The CCR is a 25-item multiple-choice instrument that can be used as a placement test for entry into calculus and to assess the effectiveness...
We investigate the nature of Calculus I homework at five PhD-granting universities identified as having a relatively successful Calculus I program and compare features of homework at these universities to comparable universities that were not selected as having a successful program. Mixed method analyses point to three aspects of homework that arose as important: structure, content, and feedback....
In mathematics education research paradoxes of infinity have been used in the investigation of students’ conceptions of infinity. We analyze one such paradox – the Painter’s Paradox – and examine the struggles of a group of Calculus students in an attempt to resolve it. The Painter’s Paradox is based on the fact that Gabriel’s horn has infinite surface area and finite volume and the paradox emerges...
Combinatorial enumeration has a variety of important applications, but there is much evidence indicating that students struggle with solving counting problems. The roots of such difficulty, as well as ways to mitigate such difficulty, have not yet been thoroughly studied. In this paper, one particular aspect of students’ counting activity is explored – the use of the problem-solving heuristic of solving...
This paper discusses findings from a research study designed to investigate calculus instructors' perceptions of approximation as a central concept and possible unifying thread of the first-year calculus. The study also examines the role approximation plays in participants' self-reported instructional practices. A survey was administered to 279 first-year calculus instructors at higher education institutions...
I am very grateful to Keith Weber for thoughtful comments on an earlier draft of this review. This work was supported by a Royal Society Worshipful Company of Actuaries Research Fellowship.
A rich understanding of key ideas in linear algebra is fundamental to student success in undergraduate mathematics. Many of these fundamental concepts are connected through the notion of equivalence in the Invertible Matrix Theorem (IMT). The focus of this paper is the ways in which one student, Abraham, reasoned about solutions to Ax = 0 and Ax = b to draw connections between other concept statements...
Although feedback is a very important component of assessment in higher education, there is substantial evidence that students view traditional methods of feedback as deficient in a number of respects. In this paper we explore how students perceive generic feedback on a mathematics assignment provided via screencasts. Our study is based on a Differential Equations module taught to first and second...
In this paper, I identify five effective proof reading strategies that mathematics majors can use to comprehend proofs. This paper reports two studies. The first study is a qualitative study in which four successful mathematics majors were videotaped reading six proofs. These students used five proof reading strategies to foster comprehension: (i) trying to prove a theorem before reading its proof,...
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