Realistic Mathematics Education (RME)

As explained in introductory part, innovations in learning and teaching algebra are really required. Therefore, in this study, a series of lessons is designed. The idea of Realistic Mathematics Education that makes the most of the applications of

mathematics concepts in human‟s daily activities is chosen as a design heuristic that underlies activities in the design. This idea of RME based teaching has been adapted by countries including Indonesia with the so-called Pendidikan Matematika Realistik Indonesia (PMRI). A number of efforts have been performed to introduce the implementation to education components and authorities in the countries, like involving several schools to be pilots of PMRI classes, conducting researches on PMRI teaching, teacher trainings, and studying contexts that might be applicable in Indonesian classrooms (for further readings, see Putri, Dolk, & Zulkardi, 2015;

Sembiring, 2010; Sembiring, Hoogland, & Dolk, 2010; Zulkardi, 2002).

The choice of RME as underlying of the proposed designs is mainly due to the three key principles of RME (mentioned in Gravemeijer, 1994), such as 1) guided reinvention, 2) didactical phenomenology, and 3) self-developed models.

Guided Reinvention

The idea of reinvention in RME is based on the view of mathematics as a process as well as a product of learning. The idea believes that students would learn better if they could discover the concept for themselves. Thus, students must be given opportunities to experience the process of how certain mathematical concepts were invented. This principle implies two components that should exist in mathematics teaching, such as, activities that lead into a concept in mathematics, and the mathematics concept itself.

To stimulate the process of reinvention, the teacher‟s role is crucial. Here, teachers should mainly act as a facilitator of learning. They scaffold their students‟

thinking process with questions. On the other side, the students also play a very important role. They are the main actors during the learning process. They do activities, explore the concepts, and try to reveal the mathematical ideas within the activities. In this phase, the students are required to be more self-reliant in doing their tasks.

Taking this principle into account, in facilitating students‟ learning on equations, balancing activities might be provided for students. During the process, the students can be given opportunities to explore algebraic concepts within the balance scale with minimal guidance. They will do experiments with the balance themselves, and represent what they find in their own representation.

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Universitas Sriwijaya Didactical Phenomenology

This concept is based on Freudenthal‟s (1999) belief that mathematics concepts, structures, and ideas were invented to organize and explain phenomena in the physical, social, or mental world. Connecting those applications with the learning process of young students is what we call didactical phenomenology. With this principle, students can recognize where the concept they are learning could be applied.

This principle recalls the need to present a context that allows students to show their range of mathematical ideas. It allows students to express their mathematical understanding by relating it to the contexts they are familiar with. Thus, it builds the students‟ common senses in the process of learning.

Based on the above explanations, there are two characteristics that situations should have to be considered a good context. First, the situation should contain an application of a mathematical concept, and second, it should support the process of mathematization. In other words, the context should contain mathematical concept, and be doable, and allow for reflection.

Self-Developed Models

Given the reinvention principle, the existence of a model is needed to facilitate a bottom-up learning process. The model would help students to bridge the contexts with mathematical concepts to achieve the learning goals. The model can be a scheme, description, ways of noting, or simply the students‟ understandings toward and uses of certain concepts to explain the more complex one.

In RME, it is very important that the students construct their own models. The students‟ initial model can be derived from their informal knowledge or strategies.

During the lessons, the students are expected to formalize their initial understanding and strategies to work with a certain concept. This is why enhancing students‟ prior knowledge is required. The development of students‟ modeling becomes a concern in an RME teaching. Gravemeijer (1994) explained four levels of emergent modeling used in RME teaching such as, situational, referential, general, and formal.

Situational

This level is where a general context is first introduced. Thus, the model developed is still context-specific. The students should rely on their informal knowledge or experience to understand the situation.

Referential

This stage is where the promoted models, concepts, procedures, and strategies are explored. Those mathematical ideas are still context-bound. Hence, the students are working with problems within the context.

General

This phase starts when the discussion about procedures, strategies, concepts, or models become the focus. The movement from the referential to general model happens due to generalizations and reflections toward activities in the referential phase.

Formal

This highest level is shown in the students‟ uses of formal strategies or procedures in solving any related mathematical problems. Hence, the mathematical knowledge they have gained can be used to solve any mathematical problems across contexts.

These levels of modeling could be illustrated in figure 2.1: