In this article, we describe how we systematically included example generation exercises into our introductory Analysis course and explain our motivation for doing so. Feedback on these exercises, collected from 39 students via course evaluation questionnaires, is presented. We also provide examples, taken from one-to-one interviews with seven students, of how engaging with the exercise caused two students to expand their concept images of divergent sequence, and resulted in three students developing techniques for generating certain types of sequences. The merits of incorporating these exercises into an advanced mathematics course are discussed.
Motivation
Definitions of mathematical concepts play a pivotal role in advanced mathematics courses at university. For a given mathematical concept, students are expected to engage with its formal definition and to ultimately validate and produce proofs based on this definition. Undergraduates often meet this mathematical practice for the first time in an introductory analysis course, and since we both have taught such courses, we are aware of the difficulties that students encounter. As a result, we are always interested in activities that can be incorporated into our advanced mathematics courses to assist students in successfully engaging with, and using, definitions.
Fortunately, there has been quite a bit of research carried out on students’ understanding of, and reasoning with, definitions in advanced mathematics courses. Alcock and Simpson [1] have surveyed some of the literature and they highlight two problem areas that students have in relation to definitions. Firstly, a student may not be aware of the importance of definitions in advanced mathematics and the role they play in accurately defining concepts and producing proofs. Secondly, for a given concept, a student’s concept image may not be well-developed and may not coincide with the definition. Here the notion of concept image is as described by Vinner and Tall [2]:
“… the total cognitive structure that is associated with the concept, which includes all the mental pictures and associated properties. It is built up over the years through experiences of all kinds, changing as the individual meets new stimuli and matures” (p. 152).
In this article, we describe how we systematically included example generation exercises into our introductory Analysis course and explain our motivation for doing so. Feedback on these exercises, collected from 39 students via course evaluation questionnaires, is presented. We also provide examples, taken from one-to-one interviews with seven students, of how engaging with the exercise caused two students to expand their concept images of divergent sequence, and resulted in three students developing techniques for generating certain types of sequences. The merits of incorporating these exercises into an advanced mathematics course are discussed.
Motivation
Definitions of mathematical concepts play a pivotal role in advanced mathematics courses at university. For a given mathematical concept, students are expected to engage with its formal definition and to ultimately validate and produce proofs based on this definition. Undergraduates often meet this mathematical practice for the first time in an introductory analysis course, and since we both have taught such courses, we are aware of the difficulties that students encounter. As a result, we are always interested in activities that can be incorporated into our advanced mathematics courses to assist students in successfully engaging with, and using, definitions.
Fortunately, there has been quite a bit of research carried out on students’ understanding of, and reasoning with, definitions in advanced mathematics courses. Alcock and Simpson [1] have surveyed some of the literature and they highlight two problem areas that students have in relation to definitions. Firstly, a student may not be aware of the importance of definitions in advanced mathematics and the role they play in accurately defining concepts and producing proofs. Secondly, for a given concept, a student’s concept image may not be well-developed and may not coincide with the definition. Here the notion of concept image is as described by Vinner and Tall [2]:
“… the total cognitive structure that is associated with the concept, which includes all the mental pictures and associated properties. It is built up over the years through experiences of all kinds, changing as the individual meets new stimuli and matures” (p. 152).
