Realistic Mathematics Education ( RME ) for teachers

Realistic Mathematics Education ( RME )

Introduction

This report describes and synthesizes literature that was surveyed on the topic realistic mathematics education (RME). This topic will be part of the literature review chapter of the Ph.D. research proposal. The title of the research is Computer Assisted Curriculum Analysis, Design and Evaluation for Mathematics Education in Indonesia (CASCADE-MEI). The main aim of this study is to develop a computer-support system which supports students teachers in Indonesia designing mathematics lesson plans based on the realistic approach.

Problem Definition

The literature review started with the following problem definition: ‘how to design mathematics lesson plans based on the realistic approach? This problem definition was split up into several smaller units of questions that focus on one aspect of the problem definition. It was hoped for that the survey would become more manageable. These smaller units have been formulated as questions. The answers to these questions constitute the sections that make up this report. The questions were:

What is realistic mathematics education?

What are the characteristics of realistic mathematics education?

How is the realistic approach related to the constructivist approach?

How to design mathematics lessons based on the realistic approach?

Search Method

The search began with several talks with the mentor and an expert on realistic mathematics education. From all of them some books were borrowed. They gave several hints in which directions to search such as the type of report and the recently year of publications. With the problem definition in mind a list of possible key words was made. Literature has been sought with the following key phrases: realistic mathematics education, assessment methods in mathematics education, lesson development, and constructivism in mathematics education. The last keyword is used in order to compare it to RME approach.Two ways of searching were used in this review. The first way of searching was through the internet with all of the above mentioned keywords. The goal of this search was to find recent publications such as journals and web sites.

By using some search engines such as yahoo (at address http://www.yahoo.com) and altavista (at address http://www.altavista.com ) the exact phrase of keywords were typed in advanced search part and the results appeared. They were web-sites and web-pages that contain on-line articles. The first and the most useful keyword is realistic mathematics education, because some of the results ( 14 web sites by Yahoo or 24 web sites by Alta vista are) deal with RME. The results from others key words are assessment in mathematics education( 86 web sites), lesson development (363 web sites) and constructivism in mathematics education (53 web sites).

However, most of the results were found only the list of references, instead of the articles. The most useful web sites that are selected that contains online articles and resources are:

Freudenthal Institute (address at http://www.fi.ruu.nl), contains some on-line articles about realistic mathematics education such as the remesa project and norma project.

NCTM organization (address http://www.nctm.org) that contains online journal and teaching standards of mathematics in USA, and

The Third International Mathematics and Science Study (TIMSS) is the largest and most ambitious international study of student achievement ever conducted. The address of TIMSS center is http://wwwcsteep.bc.edu/timss

· A number of on-line articles about the constructivist approach also available in a web-page t the following address:http://www.inform.umd.edu:8080/UMS+State/UMD-Projects/ MCTP/ W W W / Essays.html.

· A lesson development page http://www.eirc.org/integrate/tsld015.htm that contains steps in developing lesson plan and an example of lesson development software at http://www.deixissoftware.com/

Another way of searching was a search through libraries. Three different libraries were searched, SLO library in Enschede, TO (the automated online catalog of the library of the Faculty of Educational Science and Technology) University of Twente, and central library of the University of Twente. Some articles in the Journal of Research in Mathematics Education and the Journal of Educational Studies in Mathematics found in at SLO library in Enschede. In the TO library, by using on-line catalog, eight of books about realistic mathematics and also some journals were found. While in the central library of University of Twente, a dissertation about realistic mathematics was found. In addition, two books used during the realistic mathematics course were very useful.

2. What is Realistic Mathematics Education?

Realistic Mathematics Education (RME) is a teaching and learning theory in mathematics education that was first introduced and developed by the Freudenthal Institute in the Netherlands. This theory has been adopted by a large number of countries all over the world such as England, Germany, Denmark, Spain, Portugal, South Africa, Brazil, USA, Japan, and Malaysia (de Lange, 1996).The present form of RME is mostly determined by Freudenthal’s view on mathematics (Freudenthal, 1991).

Two of his important points of views are mathematics must be connected to reality and mathematics as human activity.

First, mathematics must be close to children and be relevant to every day life situations. However, the word ‘realistic’, refers not just to the connection with the real-world, but also refers to problem situations which real in students’ mind. For the problems to be presented to the students this means that the context can be a real-world but this is not always necessary.

De Lange (1996) stated that problem situations can also be seen as applications or modeling.Second, the idea of mathematics as a human activity is stressed. Mathematics education organized as a process of guided reinvention, where students can experience a similar process compared to the process by which mathematics was invented. The meaning of invention is steps in learning processes while the meaning of guided is the instructional environment of the learning process.

For example, the history of mathematics can be used as a source of inspiration for course design. Moreover, the reinvention principle can also be inspired by informal solution procedures. Informal strategies of students can often be interpreted as anticipating more formal procedures. In this case, the reinvention process uses concepts of mathematization as a guide.

Two types of mathematization which were formulated explicitly in an educational context by Treffers (1987) are horizontal and vertical mathematization.

In horizontal mathematization, the students come up with mathematical tools which can help to organize and solve a problem located in a real-life situation. The following activities are examples of horizontal mathematization: identifying or describing the specific mathematics in a general context, schematizing, formulating and visualizing a problem in different ways, discovering relations, discovering regularities, recognizing isomorphic aspect in different problems, transferring a real world problem to a mathematical problem, and transferring a real world problem to a known mathematical problem.

On the other hand, vertical mathematization is the process of reorganization within the mathematical system itself. The following activities are example of vertical mathematization: representing a relation in a formula, proving regularities, refining and adjusting models, using different models, combining and integrating models, formulating a mathematical model, and generalizing.

Freudenthal (1991) stated that “horizontal mathematization involves going from the world of life into the world of symbols, while vertical mathematization means moving within the world of symbols.” But he adds that the difference between these two types is not always clear cut.

Figure 1 illustrates the process of reinvention. It shows that both the horizontal and vertical mathematization take place in order to develop basic concepts of mathematics or formal mathematical language.