Ann O’Shea

A study of creative reasoning opportunities in assessments in undergraduate calculus courses

ABSTRACT In this article we present the findings of a research study which investigated the opportunities for creative reasoning (CR) made available to first year undergraduate students in assessments. We compared three first year calculus courses across two Irish universities using Lithner’s framework. This framework sets apart imitative reasoning (IR) (analogous to rote learning and mimicry of algorithms) and CR (which includes plausible mathematically-founded arguments). We report on the differences between reasoning opportunities in specialist and non-specialist courses and compare our findings with those of other similar projects. In addition, we will provide a critical reflection on Lithner’s framework as a method for classifying tasks in calculus courses and suggest a modification.

Designing Mathematical Thinking Tasks

Abstract Research has shown that the types of tasks assigned to students affect their learning. Various authors have described desirable features of mathematical tasks or of the activity they initiate. Others have suggested task taxonomies that might be used in classifying mathematical tasks. Drawing on this literature, we propose a set of task types that are deemed appropriate for undergraduate students and which foster mathematical habits of mind. These are: generating examples; analyzing reasoning; evaluating mathematical statements; conjecturing and/or generalizing; visualizing; and using definitions. We give rationales for our choices and examples of each type of task suitable for use in an introductory calculus course.

Decision points in mathematics lectures

Abstract We report on a study of decision-making in mathematics lectures. The data comes from accounts of classroom incidents written over two academic years by three university mathematics lecturers who set out to reflect on, and improve, their practice by collectively engaging in a professional development project using Mason’s Discipline of Noticing. The accounts were analysed to locate decision points during lectures. We report on a classification of these decision points and give examples. We found that 72% of the decision points that occurred were student-focussed while the remainder focussed on the practice of teaching. The professional development implications of our study will be discussed.

Threshold Concepts and Undergraduate Mathematics Teaching

Abstract Traditionally, many undergraduate mathematics courses have been defined in terms of mathematical content and the techniques in which students should become proficient or theorems they should be able to prove. This can result in a reliance on shallow or rote learning by students, despite the fact that the main goal of a mathematics lecturer1 is usually to foster mathematical understanding in their students. In contrast, it is suggested that placing an emphasis on the threshold concepts involved in a course can enable teachers and students to focus on what is fundamental to the study and mastery of their subject.