Operable Definitions in Advanced Mathematics: The Case of the Least Upper Bound

This paper studies the cognitive demands made on students encountering the systematic development of a formal theory for the first time. We focus on the meaning and usage of definitions and whether they are "operable" for the individual in the sense that the student can focus on the properties required to make appropriate logical deductions in proofs. By interviewing students at intervals as they attend a 20 week university lecture course in Analysis, we build a picture of the development of the notion of least upper bound in different individuals, from its first introduction to its use in more sophisticated notions such as the existence of the Riemann integral of a continuous function. We find that the struggle to make definitions operable can mean that some students meet concepts at a stage when the cognitive demands are too great for them to succeed, others never have operable definitions, relying only on earlier experiences and inoperable concept images, whilst occasionally a concept without an operable definition can be applied in a proof by using imagery that happens to give the necessary information required in the proof. Mathematicians have long "known" that students "need time" to come to terms with subtle defined concepts such as limit, completeness, and the role of proof. Many studies have highlighted cognitive difficulties in these areas (e.g. Tall & Vinner, 1981, Davis & Vinner, 1986, Williams, 1991, Tall, 1992). Some authors (e.g. Dubinsky et al, 1988) have focused on the role of quantifiers. Barnard (1995) revealed the subtle variety in students' interpretations of statements involving quantifiers and negation. Nardi (1996) followed the development of university students' mathematical thinking by observing and audio-taping their first year tutorials. This highlighted the tension between verbal/explanatory expression and formal proof, and tensions caused in proofs by quoting results of other theorems without proof. Vinner (1991) drew attention to two modes of use of definition - the everyday, and the technical mode required in formal reasoning. In this paper we report a longitudinal study of the individual developments of students in their first encounter with a formal mathematical theory, to see the growth of their use of definitions in building concepts and proving theorems.