Högskolan i Skövde

Focusing on the potentiality of students’ ways of treating a mathematicalmaterial this thesis aims to investigate how students use their conceptualunderstanding when working with mathematical tasks in calculus. Two casestudies were carried out to explore students’ understanding of thresholdconcepts. The first study, an interview study, explored engineering students’understanding of limit and integral. The second study, a problem solvingstudy, involved students within a mathematics programme, working on achallenging task including the concepts function and derivative, requiringproof by induction. Drawing on a theory of contextualisation data wereanalysed within a constructivist research framework following the principlesof intentional analysis. The results reveal that the students in themathematics programme expressed their understanding in a formal contextin which also intuitive ideas played an important role. They used intuitiveideas and formal reasoning in a dynamic interplay with several functions:to control intuitive ideas, to offer a new basis of reasoning, to reduce thecomplexity of the problem and to push the problem solving process forward.The engineering students expressed their conceptions in an algorithmiccontext, in which procedural knowledge was predominant and the operationsof the concepts were seen as defining features and a basis for understanding.However, faced with probing questions, the students appeared to shift to acontextualisation foregrounding ideas relating to conceptual dimensions ofcalculus. These contextual shifts display the transformative aspect ofthreshold concepts allowing the development of conceptions and students’awareness of ways of thinking and practising in mathematics.