Student teachers’ knowledge of students’ difficulties with the concept of function

Mikael Borke

doi:10.31129/lumat.9.1.1661

Abstract

An important part of the mathematics syllabuses at the secondary school level in most countries is the concept of function. However, secondary school students often experience difficulties with this concept. These difficulties are well-known in the research literature. The study applies the mathematical knowledge for teaching (MKT) framework, including the category knowledge of content and students (KCS). Teachers’ ability to anticipate students’ difficulties is one aspect of KCS. The aim of this study is to investigate secondary mathematics student teachers’ KCS regarding the concept of function. Ten mathematics student teachers participating in a Supplementary Teacher Education Program answered a questionnaire about fictive secondary school students’ various difficulties with the concept of function. Follow-up interviews were conducted with four of the respondents. Compared to the findings of previous research on students’ difficulties with the concept of function, the respondents in the study sometimes provide reasonable suggestions about the sources of students’ difficulties. Some of the respondents demonstrate an aspect of KCS when they suggest that students can reason that a function must be defined by one algebraic expression only, and that students only know about continuous functions. However, no respondent suggests that one source of students’ difficulties with a constant function with an implicit domain is the missing domain. In addition, some respondents take for granted that students can interpret the algebraic representation of a piecewise-defined function and translate it into a graph.

Highlights

The concept of function is an important part of mathematics (Freudenthal, 1983), and of mathematics syllabuses at the secondary school level in most countries (National Council of Teachers of Mathematics, 2017; Swedish National Agency for Education, 2012)
What knowledge of content and students (KCS) do the participating secondary mathematics student teachers demonstrate regarding the concept of function? In particular, suggestions about students’ difficulties in recognizing constant functions and piecewise-defined functions, difficulties regarding the one-valuedness property of a function, difficulties related to the various representations, and students use of prototype examples are considered
The suggestions from the respondents in the present study are consistent with the findings of Hatisaru and Erbas (2017), who propose that teachers demonstrate an aspect of KCS when they suggest that students may have difficulty in recognizing constant functions with an implicit domain

Summary

Introduction

The concept of function is an important part of mathematics (Freudenthal, 1983), and of mathematics syllabuses at the secondary school level in most countries (National Council of Teachers of Mathematics, 2017; Swedish National Agency for Education, 2012). Teachers’ ability to anticipate and resolve students’ errors and misconceptions regarding the concept of function, ability to interpret students’ incomplete reasoning, and to anticipate what tasks students will experience as difficult are aspects of knowledge of content and students (KCS) which in turn is part of MKT (Nyikahadzoyi, 2015). This knowledge influences the teacher’s decision on how to respond to students’ questions (Even & Tirosh, 1995). Two of the most frequently used Swedish mathematics textbooks for upper secondary school (Sönnerhed, 2021) define a function as a relationship between two variables that satisfy the onevaluedness property: If the relationship between two variables x and y is such that each x-value, according to some rule, gives a unique y-value, we say that y is a function of x. (Alfredsson, Bråting, Erixon, & Heikne, 2011, p. 288)

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