Resume: Developing Realistic Mathematics Education Mediating between Concrete and Abstract

(Koeno Gravemeijer)

The approach in realistic mathematics education for mediating between concrete and abstract based on self developed models known as domain specific theory for realistic mathematics education. The characteristic of this approach is bottom-up, because the initiative is with the students. In addition, this approach is expected to guarantee insightful mathematical knowledge, and the realistic concept of generalizing is presented as bottom-up alternative for the top-down concept of transfer.  

Long Division with Manipulatives

Formal crystallized expert mathematical knowledge is taken as starting point for developing instructional activities in the mainstream information processing approach. Representational models and manipulatives are designed to create a concrete framework of reference in which the intended mathematical concepts are embodied. Because of everyday life is not pure enough for learning mathematics, it is though too complex and much distraction, that is why abstract mathematical knowledge and procedures are introduced and learned with manipulatives.

Research Findings

Doing analyses carefully from a constructivist point of view have resulted in a convincing explanation of what goes on in regular classroom and what goes wrong with conveying knowledge with the help of manipulatives. Moreover, shortcomings of manipulatives based learning sequence are exposed as lack of insight and problems with application and these problems stem ignoring the importance of informal knowledge and strategies.

New Approaches

There are three current approaches that recognize the existence of childrens’ informal strategies followed by RME approach. For instance, Cognitively Guided Instruction (CGI) had idea to help teachers constructing a referential framework used to guide their students in spontaneous learning process.

In addition, there is cognitive apprenticeship as a teaching model based on the assertion of the indexical character of knowledge and keywords in this approach are coaching, scaffolding and fading.

Moreover, socio-constructivist believed that all knowledge is self-constructed, thus mathematics education should acknowledge idiosyncratic construction and foster classroom atmosphere where mathematical meaning, interpretations and procedures are explicitly negotiated. Then, a suitable domain specific instruction theory in mathematics education can be found in the theory for realistic mathematics education.

Realistic Mathematics Education

Realistic mathematics education is rooted in Freudenthal’s interpretation of mathematics as an activity had characterized as an activity of solving problem, looking for problems and organizing a subject matter and the main activity is organizing or mathematizing. Then, mathematizing relates to level rising in a mathematical sense, the activity on one level is subjected to analysis on the next, the operational matter on one level becomes a subject matter on the next level, such as generality (looking for analogies, classifying, strusturing), certainty (using a systematic approach, elaborating and testing conjectures), exactness (modeling, symbolizing, definig) and brevity (symbolizing and schematizing).

In RME, mathematizing mainly involves generalizing and formalizing. Furthermore, there are two reasons for mathematizing as the central of mathematics teaching: firstly, mathematizing is not only the major activity of mathematicians, it also familiarizes the students with a mathematical approach to everyday life situations. Secondly, mathematizing relates a reinvention, the final stage is formalizing by way of axiomatizing. Mathematics education organized as a process of guided reinvention, when students can experience a similar process as the process by which mathematics was invented.

Developing Long Division

Contextual problem are used as starting point in the realistic approach, preferably problem allowed for variety of informal solution procedures. In other words, applied problem precede instruction on the algorithm.

Teaching-Learning Process

Furthermore, mathematical discourse is identified with conjecturing, justifying and challenging. RME places the students in quite a different position than traditional education approaches. Students have to be more self-reliant. They cannot turn to the teacher for validation of their answers or for the direction for a standard solution procedure.

Key Principles of Realistic Mathematics Education

There are three key principles of RME that can be seen as heuristics for instructional design. Firstly, guided reinvention, and progressive mathematizing. In this principle, students should be given the opportunity to experience a process similar to the process by which mathematics was invented. Secondly, didactical phenomenology. According to this principle, situationa where a given mathematical topic is applied are to be investigated for two reasons. Firstly, to reveal kind of applications that have to be anticipated in instruction, secondly, to consider their suitability as points of impact for a process of progressive mathematization. Then, the third principle is self-developed models played in bridging the gap between informal knowledge and formal mathematics. In RME, models are developed by the students themselves. At first, a model is situational model that is familiar to the students. Then, there is transition from model-of to model-for by process generalizing and formalizing the model becomes model for mathematical reasoning.

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