Academic journal article International Education Studies

Equations, Functions, Critical Aspects and Mathematical Communication

Academic journal article International Education Studies

Equations, Functions, Critical Aspects and Mathematical Communication

Article excerpt

Abstract

The purpose of this paper is to present the mechanism for effective communication when the mathematical objects of learning are equations and functions. The presentation is based on data collected while the same object of learning is presented in two classes, and it includes two teachers and 45 students. Among other things, the data consists of video-recordings of lessons and tests. In the analysis, concepts relating to variation theory have been used as analytical tools. The results show that effective communication occurs in the classroom if it has the critical aspects in students learning as its starting point. The communication in the classroom succeeds or not if the aspects of the content supposed to be treated is the same as or different from the aspects of the content of the teacher's representation, and if the aspects of the content of the teacher's representation are the same as or different from the aspects discerned by the students. The results also show that the students cannot make sense of the difference between the highest/lowest value of a quadratic function and the maximum/minimum point; the difference between a quadratic equation and function; the students also have difficulties in solving a quadratic equation if it appears in a new context. The argument of the functions is identified as critical aspect in this study.

Keywords: communication, equations, functions, teaching, learning, dimensions of variation

(ProQuest: ... denotes formulae omitted.)

1. Introduction

1.1 The Specific Problem

Mathematical knowledge is seen as an important requirement to develop society. Despite the increased interest in people with deeper mathematical knowledge, there is a constant stream of new articles which indicates that students have unsatisfactory knowledge in mathematics. Because equations and functions are often conveyed in symbols, oral and written communication about mathematical ideas is recognized as an important part of mathematics education. Students do not necessarily talk about mathematics naturally; teachers need to help them learn how to do so. The main questions in this paper are: How can theory help us to understand and support students' developing mathematical learning? It is possible to understand the mechanism for an effective communication which may lead to students' understanding pattern and structure, the logical analysis, and calculation with patterns and structures when working with algebra and functions during the classrooms lessons? My hypothesis is that through the communication that occurs in the classroom (e. g. listening, talking and writing), students are prompted to organize, re-organize and consolidate their mathematical understanding, as well as analyze, evaluate and build on the mathematical strategies of others.

The basic idea of the mathematical theory of communication, as developed by Claude Shannon:

The fundamental problem of communication is that of reproducing at one point either exactly or approximately a message that has been selected at another point. (Shannon, 1949, pp. 31)

The success or failure of communication is a matter of the relation between thought contents of speaker and hearer (Frege, 1918). Research on effective communication primarily focuses on a process-oriented approach where the focus is on the transfer of messages, coding and analysis (e.g., Nilsson & Waldemarson, 1990). In addition, there is a semiotic line where discussions take place about how messages interact with humans to create meaning (e.g., Morgan, 2006; O'Halloran, 2005), as well as a socio-cultural approach in which communication is defined as an activity which attempts to get an interlocutor (possibly oneself) to act or feel in a certain way (Sfard, 2002). Sfard (2002) found that communication is effective if communicative aims are met and if the discourse focus is clear. Sfard (2002) defined the concept discourse as a dynamic process denoting a specific act of communication, verbal or not, with others or with oneself, synchronic (e. …

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