• He should provide for authentic versus academic contexts for learning;
• There should be provision for learner control; and
• The educator should use errors as a mechanism to provide feedback on learners’
understanding.
on the process of mathematisation. Fauzan et al (2002) concur that Realistic Mathematics Education stresses the idea that mathematics is a human activity and mathematics must be connected to reality, and that it must be real to the learner using real-world context as a source of concept development. According to Benson (2004) Realistic Mathematics Education involves putting mathematics into recognizable, real life, contexts to allow the pupils to engage with the mathematics and generate solutions in a variety of forms, which encourage discussion in a more informal atmosphere whilst moving towards a more formal solution. This model is based on the principle, that learners see meaning in their schoolwork when they connect information with their own experience.
Treffers and Beishuizen (2000) state that in addition to illustrating the applicability and relevance of mathematics in real-world situations, realistic mathematics involves taking realistic context situations as the starting point or as the source for learning mathematics. They point out that this important realistic mathematics education viewpoint does not mean adding a few application problems to your mathematics lessons, but rather a complete reversal of the teaching/learning process, where the emphasis is no longer on the teacher transmitting knowledge and concepts, but on the children finding mathematical patterns and structures in realistic situations, and becoming active participants in the teaching/learning process. At this point one can draw parallels with the constructivist theory. Treffers and Beishuizen stress that
‘realistic’ also relates to mathematical activities which are experientially real to a child.
According to Mudaly (2004) realistic mathematics educators place tremendous emphasis on the idea of making a mathematical idea real in the mind of the learner.
Similarly Zulkardi (2003) notes that mathematics must be close to children and be relevant to everyday life situations, and that ‘realistic’ refers not just to the connection with the real- world, but also refers to problem situations which are real in students’ minds.
Bottle (2005) also points out that ‘realistic’ does not just mean real-life situations. She explains that the term ‘realistic’ is taken from the Dutch word ‘zich REALISEren’ which can mean ‘to realise’ or ‘to imagine’. It therefore means that contexts that are realistic to children (although imaginary) are included. According to Bottle, this means that using stories can be useful in relating mathematical ideas to young children. Similarly games, television, videos and films can be considered as useful environments within which to develop mathematical learning, as all of these environments become real in a child’s imagination as they act them out.
Realistic Mathematics Education as a teaching and learning theory represents a significant departure from traditional ideas about teaching and learning mathematics. According to Meyer (2001) “perhaps the most obvious difference between an instructional sequence based on Realistic Mathematics Education and a more traditional sequence is their starting point. In explaining algebraic representations, most traditional algebra texts start with abstract expressions involving variables and rapidly move to formal equations and their manipulations.
After attaining some degree of facility in manipulating equations, the student of algebra is invited to apply these skills to context-based problems. Materials based on Realistic Mathematics Education, however, reverse the progression. Instead of starting in the abstract
realm and moving toward the concrete application, the mathematic starts in contexts and gradually progresses to formal symbolism. This shift allows students, for example, to engage in meaningful, pre-formal algebraic activity in earlier grades than they traditionally have.
Through a structured instructional sequence, students explore and rediscover significant mathematics that anticipates the more formal representations found in traditional algebra.
The usefulness of the Realistic Mathematics Education approach is evident in the success that the Netherlands achieved in the recent international comparisons. In 1995, 26 countries took part in the Third International Mathematics and Science Survey (TIMSS). Nine year old children in the Netherlands topped the scores from all the other European countries participating. It is interesting to note that in the Netherlands many children learn much of their mathematics through real-life problem-solving (Bottle, 2005).
The researcher views mathematical modelling as being very relevant within a Realistic Mathematics Education framework. Mathematical modelling engages learners in activity;
they are able to apply mathematics to reality and they are able to see the usefulness of mathematics.
In summary, Meyer (2001) and Widjaja (n.d) lists the following as characteristics of realistic mathematics education:
1. use real life contexts as a starting point for learning- the starting point should be real to students, allowing them to immediately become engaged in the situation;
2. use models as a bridge between abstract and real that help students learn mathematics at different levels of abstraction- at first the model is a model of a situation that is familiar to students. By a process of generalizing and formalizing, the model eventually becomes an entity on its own and is possible to be used as a model for mathematical reasoning;
3. use student’s own production or strategy as a result of doing mathematics. By making free production, students are guided to reflect on the path they themselves have taken in their learning process and, at the same time, to anticipate its continuation;
4. interaction is essential in learning mathematics, both between teacher and students, and between students and students. Explicit negotiation, intervention, discussion, cooperation, and evaluation are essential elements in a constructive learning process in which the student’s informal methods are used as a lever to attain the formal ones;
5. connection among strands, to other disciplines, and to meaningful problems in the real world. Learning strands cannot be dealt with as separate entities; instead an intertwining of learning strands is exploited in problem solving.
Vaid (2004) points out that Realistic Mathematics Education demands that teachers establish a link between children’s own world and the world of mathematical ideas. From a psychological perspective, this relates to Lave's theory of situated learning which argues that knowledge needs to be presented in an authentic context, unlike most classroom learning activities which involve knowledge that is abstract or out of context.
Eade (2004, as cited in Vaid, 2004) outlined the key aspects of Realistic Mathematics Education as follows:
• It is realistic – the term 'realistic' refers to situations which can be imagined by the learners. Realistic Mathematics Education initially presents knowledge within such a concrete context allowing pupils to develop informal strategies, but gradually through the process of guided 'mathematization', allows students to progress to more formal, abstract, standard strategies. These contexts are chosen to help students' mathematical development, not simply because they are interesting! The Realistic Mathematics Education problems, set in real world contexts, are presented, so that along with giving meaning and making mathematics more accessible to learners, they also illustrate the countless ways in which mathematics can be applied.
• It involves 'mathematization' - This can be split into two kinds.
o HORIZONTAL MATHEMATIZATION: This is when the students' discover mathematical tools which can help to organize and solve a problem located in a real-life situation.
o VERTICAL MATHEMATIZATION: This refers to the process of reorganisation within the mathematical system itself, for example refining and adjusting models or generalising to create more challenging mathematics and hence to a greater use of abstract strategies.
• It is procedural vs. algorithmic – Realistic Mathematics Education stresses understanding processes, rather than learning algorithms. Students 'discover' the mathematics for themselves, and so multiple solutions are encouraged and valued.
• It incorporates the use of effective models - The use of various models e.g. ratio tables and combination charts, provide a more visual process of doing mathematics.
• It encourages 'guided reinvention' - this implies beginning with the range of informal strategies provided by students, and building on these to promote the materialization of more sophisticated ways of symbolising and understanding.
• Due to students' directing the course of lessons, Realistic Mathematics Education requires a highly 'constructivist' approach to teaching, in which children are no longer seen as receivers of knowledge but the makers of it, and the role of the teacher is that of a facilitator.
• It promotes 'historical simulation' - Allowing the students to begin at the basics, using informal strategies and constructing the mathematics for themselves, simulates the discovery of the mathematics and allows them to appreciate the complexity of the mathematics.