y måste bero av x – gymnasieelevers förståelse av det matematiska begreppet funktion

Abstract

Objective: The aim of the study is to describe pupils' understanding of the mathematical concept of function. How do pupils define the concept of function? What images of the concept of function evoke when they solve tasks, which involve identifying and constructing functions?

Theory: A student's thinking about a mathematical concept depends on more than just the formal definition of the concept; therefore Tall and Vinner introduce the term concept image to describe the role cognitive structures play when students learn about concepts. The cognitive structure includes all mental images, associated properties and processes that an individual associates with a given concept. According to Sfard, an individual's understanding of mathematical concepts may have different character: an operational conception, where a concept is conceived as a process and a structural conception, where the given concept is conceived as an object, that is, as a whole.

Method: 16 pupils at the Science Program at two different upper secondary schools in Sweden answered a questionnaire on the mathematical concept of function. In addition, five of the 16 pupils in the survey group were interviewed on their understanding of the concept of function. Tall and Vinner’s theory of concept definition and concept image and Sfards theory of operational and structural understanding of concepts, were used to analyse the data.

Results: Four categories of the pupils’ definitions of the concept of function were identified; Correspondence, Dependence Relation, Rule and the Graphical Representation. Every second pupil in the survey group mistakenly believes that the equation does not represent a function, with the justification that the value of y is independent of the value of the independent variable. A majority of the pupils in the survey group do not specify the condition that a function has to assign a unique value to every number in its domain. One consequence of this is that many pupils falsely believe that the equation of a circle represents a function, even though it does not meet the condition of a unique functional value. Some pupils in the survey group evoked a concept image of function involving one or several of the following aspects: The graph of a function has to be connected. Piecewise defined functions are rejected. A function must be represented by a single formula. Each of these images is a potential conflict factor, which is at variance with the formal definition of the concept of function.