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Conclusion and Discussion

6.3. Recommendations

In this section, we give recommendations concerning realistic mathematics education in Indonesia and using structures in mathematics lesson. These recommendations are made based on the finding of this research.

6.3.1. Realistic Mathematics Education in Indonesia

Realistic mathematics education (RME) can contribute to developing a traditional teaching in Indonesia to a more progressive learning. In our research, RME has underpinned the classroom activities and we have seen how students learned better in such an environment. The use of context, in this case we used the candy packing context, has stimulated students to think of a way of solving problems. With a good context, mathematics is meaningful so that it makes sense for them. A context

109 also should allow students to mathematize by creatively inventing new strategies, in that way, students can construct their own understanding.

The emergence of models supports students’ transition from concrete situational problems to more formal and general mathematics. The model serves as the stepping stone for students. With a model of students relate the concrete objects to the model, and with a model for students can use the model to represent and support their thinking.

In RME classrooms, the contributions from the students are highly promoted.

Students learn to share and listen to each other’s idea through a discussion where strategies are discussed and compared to determine which ones are more sophisticated. In a discussion, students can learn from their peers and the collaborative development of knowledge among students can be made possible.

During the research, we found that the classroom we worked with was still struggling in establishing socio norms and socio-mathematical norms. Nevertheless, a good start has been made as this class has developed an open learning atmosphere where students are allowed to use their own strategy. In this classroom students have freedom to use different strategy, but they are not promoted to discuss and choose the best strategy. This condition does not support the mathematical development of low achiever students because they are still allowed to use counting and not stimulated to use other more sophisticated strategies. More efforts are still needed to continue the development of the socio-mathematical norms, where students are aware and have the ability to choose the best strategy. We realize that it takes time for students to develop such norm, but it is important to keep on nurturing the students with constructive learning attitudes.

6.3.2. Using structures in mathematics lesson

From this research we have contributed to a local instructional theory about classroom activities that support students learning of addition up to 20 by using structures. In this section, we recommend teachers, educators and public policy makers to employ structures in school mathematics.

This research has shown the use of structures in teaching students how to abbreviate addition strategies. We also think that structures can be used in many other mathematical domains such as geometry and statistics. By structuring, students are

110 stimulated to find patterns, make connections, group and decompose elements to find a better strategy of solving mathematical problems.

We have shown that structuring allows students to mathematize, that is by recognizing patterns and regularities and use the patterns for further thinking.

Structures in an egg box and math rack has supported students to construct double and decomposition to 10 strategy of solving addition up to 20 problems. Many researches have shown the advantages of structuring in other early mathematics domains. Van Nes (2007) used spatial structures for developing students’ number sense. Mulligan, et al (2004) showed that students’ ability to structure has a positive correlation with their mathematical achievement. Thus we recommend public policy maker to include structuring in school mathematics curriculum in Indonesia. The breaking and grouping, finding patterns in a collection is an important informal mathematical skills which haven’t got much attention in school mathematics. By structuring, students learn to look at a problem from different perspectives that allow them to discover a creative solving strategy. Nevertheless, in Indonesian school, students are not used to this creative way of thinking since they are always taught the standard procedure or the algorithm of solving problems.

6.3.3. Further studies

This research has found differences in students’ ability in recognizing and using structures. We could hypothesize some levels in the development of using structures for doing addition up to 20.

1. Non structural: Students who do not recognize structure and keep on rely on counting all strategy.

2. Informal structural: Students who can recognize structure of small group by perceptual subitizing, and then use it for conceptual subitizing, but can not do counting by grouping due to their inability of basic addition.

Therefore, at some point they still use counting all. For example: when solving 8 + 6, a student could show an eight in the upper bar of a math rack by moving 5 beads at once, and then 3 more beads at once. After that he showed six in the bottom bar by moving 5 beads at once and one more bead after that. But when determining the total number of beads, this student still used counting all strategy. At this level, students are able

111 recognize the structures but unable to employ the structure in their formal mathematical thinking.

3. Formal structural: Students who can recognize structure and use structure for their counting reasoning.

These findings have raised some new questions such as, how do students differ in their ability of structuring? What basic knowledge underpins students’ ability to structure? How to support students to structure better? Further research is needed to answer those questions.