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# Matematisk och pedagogisk kunskap – Lärarstudenters uppfattningar av begreppen funktion och variabel

Licentiate thesis
Authors Mikael Borke 2017-05-22 Kerstin Pettersson Institutionen för matematiska vetenskaper Göteborg 2017 Department of Mathematical Sciences, Algebra and Geometry sv hdl.handle.net/2077/52381 function, variable, student teacher, concept definition, concept image, conception, mathematical knowledge for teaching Educational Sciences

## Abstract

The concepts of function and variable are important in compulsory and upper secondary schools, and at the university. Teachers' mathematical knowledge, as well as their pedagogical knowledge, have an effect on their teaching. The aim of this thesis is to investigate student teachers’ concept images (Tall and Vinner, 1981) and mathematical knowledge for teaching (Ball et al., 2008) of the concepts of function and variable. Questionnaires and follow-up interviews were used to collect data. Six categories of student teachers’ explanations of the concept of function are identified: expression, dependence on a variable, rule, correspondence, machine, and relationship between variables. Most of the student teachers’ explanations of the concept of variable are that it is a quantity that can vary. Almost all the student teachers interpret the word "variable" as independent variable. They demonstrate concept images of the concept of a function, which includes constant functions, piecewise defined functions, and that a functional value should be uniquely determined. However, some of them display potential conflict factors in their concept images. The student teachers demonstrate knowledge of content and students of the concepts of function and variable, when they reason about students' difficulties with the concepts. They demonstrate specialized content knowledge of the function concept, when they choose appropriate representations of functions for different purposes. A student teacher demonstrate specialized content knowledge of the concept of variable, when suggesting that teachers should distinguish two aspects of the concept: the varying aspect of a variable, and variable in the sense of an unknown number in connection with equations which has a unique solution.

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