Realistic mathematics education : Eliciting alternative mathematical conceptions of learners

This paper focuses on a Grade 8 mathematics intervention that has been developed and implemented in a local high school in South Africa. The theory of Realistic Mathematics Education (RME) has been applied in the design of the intervention, which aims to assist low attaining learners in a remedial programme that is run during school hours. Based on the premise that eliciting and addressing learners’ alternative conceptions in mathematics is beneficial in assisting them to improve their understanding, the paper seeks to explore the role that RME plays pertaining to this particular supposition. The paper presents and discusses examples of learners’ responses to contextual problems given to them during the course of the intervention. Introduction Literature that has been generated from studies done in a number of countries (including other developing countries such as Indonesia) has shown that the Realistic Mathematics Education (RME) theory is a promising direction to improve and enhance learners' understanding in mathematics (Armanto, 2002; Fauzan, 2002). To explore the potential of using RME for teaching remedial classes, a case study of twelve Grade 8 low attaining learners in mathematics is being conducted at a high school in South Africa. The theory of RME has been employed in the design and implementation of an intervention in an attempt to address perceived deficits in the learners' conceptual understanding of basic mathematical concepts. Within this framework, this paper seeks to the explore the following question: Can the theory of Realistic Mathematics Education (RME) play a role in eliciting the alternative mathematical conceptions of learners in order to assist them in overcoming these? Although a large percentage of learners at the school do take extra mathematics classes of some type after school, there remain a growing number of learners who continue to perform far below the expected level. A few years ago the school took the decision that certain learners in Grades 8 and 9 were achieving such low results in Mathematics and English that it would be in the best interests of these learners to allow them to attend additional lessons in these two subjects during school. In order to accommodate these lessons within the school day, and to reduce the workload of these learners, they are not required to take a third language, which is usually compulsory for all learners in Grades 8 and 9. The Mathematics and English teachers respectively of these learners give the remedial Mathematics and English lessons and each teacher decides what instructional methods, content and material to use for their own classes. There is therefore not any policy or specific methodology for these remedial lessons. The time is created on the timetable and certain learners are identified (by their poor academic performance) as needing the remedial assistance and the task of the teacher is to provide this. From the personal experience of the author, as well as validation from the teachers at the school, it appears that the remedial mathematics lessons have mostly been a repeat of the normal mathematics lessons rather than an intervention to diagnose and address the 1 The word remedial is used here as these classes are referred to as remedial classes by the school Realistic mathematics education 54 mathematical difficulties experienced by the learner. This paper describes and analyses a case study being carried out with two grade 8 remedial classes. Theoretical underpinnings of RME Realistic Mathematics Education has its roots in Hans Freudenthal's interpretation of mathematics as a human activity (Freudenthal, 1973; Gravemeijer, 1994). To this end, Freudenthal accentuated the actual activity of doing mathematics; an activity, which he envisaged should predominantly consist of organising or mathematising subject matter, taken from reality. Learners should therefore learn mathematics by mathematising subject matter from real contexts and their own mathematical activity rather than from the traditional view of presenting mathematics to them as a ready-made system with general applicability (Gravemeijer, 1994). These real situations can include contextual problems or mathematically authentic contexts for learners where they experience the problem presented as relevant and real. The verb "mathematising" or the noun thereof "mathematisation" implies activities in which one engages for the purposes of generality, certainty, exactness and brevity (Treffers, 1987; Gravemeijer, Cobb, Bowers & Whiteneack, as cited in Rasmussen & King, 2000). Through a process of progressive mathematisation, learners are given the opportunity to reinvent mathematical insights, knowledge and procedures. In doing so learners go through stages referred to in RME as horizontal and then vertical mathematisation (see Figure 1). Horizontal mathematisation is when learners use their informal strategies to describe and solve a contextual problem and vertical mathematisation occurs when the learners' informal strategies lead them to solve the problem using mathematical language or to find a suitable algorithm (Treffers, 1987; Gravemeijer, 1994). For example, in what we would typically refer to as a "word sum", the process of extracting the important information required and using an informal strategy such as trial and error to solve the problem, would be the horizontal mathematising. Translating the problem into mathematical language through using symbols and later progressing to selecting an algorithm such as an equation could be considered vertical mathematisation, as it involves working with the problem on different levels. (Adapted from Gravemeijer, 1994.) Figure 1: Horizontal mathematisation ( ); Vertical mathematisation ( ) The traditional formal and authoritarian approach to teaching mathematics that has dominated in South African classrooms for a number of years has not afforded learners many opportunities to make use of horizontal mathematisation. Mathematics lessons are often presented in such a way that the learners are introduced to the mathematical language relevant to a particular section of work and then shown a few examples of using the correct algorithms to solve problems pertaining to the topic before being given an exercise or worksheet to complete. The exercises or worksheets are usually intended to allow learners to put the algorithms they have been taught into practice and may even contain some contextual problems that require the use Contextual problem Mathematical language