Högskolan i Skövde

I studier av elevers begreppsuppfattningar fokuseras ofta elevers svårigheter och missuppfattningar, men det finns också forskare som menar att dessa ofullständiga eller felaktiga uppfattningar utgör en del av utvecklingsprocessen. Jag har i min forskning valt att lyfta fram den potential som finns i elevers agerande när de möter ett matematiskt material. Med det perspektiv jag har tagit vill jag visa hur elever använder de begreppsuppfattningar de har och hur de förmår utnyttja sina uppfattningar i ett matematiskt arbete. Genom att studera det sammanhang i vilket universitetsstudenter tolkar ett matematiskt material kan jag visa på en dynamik mellan olika aspekter av en begreppsuppfattning som aktualiseras under ett matematiskt arbete. Resultaten från två studier, en intervjustudie och en problemlösningsstudie, visar att dessa studenter, trots brister i sina begreppsuppfattningar, ändå förmår att utnyttja ett samspel mellan olika aspekter av matematiken på ett sätt som har stora likheter med en matematikers arbetssätt. I kapitlet presenteras de två studierna. Därefter presenteras de sammantagna slutsatserna från dessa studier. I det avslutande avsnittet diskuteras resultatens relevans för matematikundervisning på alla nivåer.

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.

Studiens syfte är att visa hur en växelverkan mellan intuitiva idéer och formella resonemang kan gestalta sig i en problemlösningsprocess. Studien visar att universitetsstudenter redan under sitt första år av matematikstudier förmår utnyttja en sådan växelverkan. En grupp studenter har arbetat med en analysuppgift som berör begreppen funktion och derivata samt inkluderar ett induktionsbevis. Studenterna utnyttjar i den kreativa processen intuitiva idéer och formella resonemang i ett dynamiskt samspel. Växlingarna har ett flertal funktioner: att kontrollera intuitiva uppfattningar, att skaffa nya utgångspunkter för problemlösningsprocessen, att ekonomisera resonemang och att driva arbetet vidare.

The aim of this study is to illuminate the interplay between intuitive ideas and formal justification. In a perspective where we focus on the students’ competences we describe the interaction between intuitive ideas and formal structures, as they appear especially in relation to concept definitions and formal proofs. A group of university students have been working on a task in calculus. The task includes the concepts function and derivative and requires of them to use a proof by induction. The discussion between the members of the group has been analyzed in accordance with the principles of intentional analysis, a method by which we regard the students’ activities as intentional. This method of analysis makes the process of interpretation visible. It also makes it possible to make explicit the competences of the students’ and the mathematical content of their actions. In the study examples are given of the interplay between the intuitive ideas and the formal structures and also of the function for this interplay. The students have rich concept images and access to intuitive ideas relevant to the concepts brought to the 4 fore by the task. During the group discussion all components of a complete proof are included in the students’ reasoning. The students create a proof by induction which matches the ordinary pattern for such a proof, but they do not themselves regard it as a proof fitting into the ordinary scheme of argumentation, as they remember it from text-books and teaching. The students put heavy demands upon the formalization of their ideas and these demands are sometimes hampering the problem solving process, but they also encourage the students to expand their search for a solution to the problem at hand.