Conclusion

In this part we would like to answer the proposed research questions. It should be noted that the following answer is the outcomes of this study. So, the outcomes are limited by the target group and participants as described in the Chapter III.

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We mean to know how the students can learn to understand the coordinate system. We would try to elaborate the answer to that question by answering the proposed sub questions, 1. What context that can be used to help the students to understand the coordinate system; and 2. How that context can support the students to learn and understand about the mathematical idea of coordinate system. In order to avoid redundant of information, we would like to address both questions in once.

The findings suggest that the “experientially real” context problems which are proposed in this study are able to support students in learning and understanding the big ideas of the coordinate systems. We can use the cinema problem to help the students understand how a system works. From this activity the students understand that the system used in the cinema locates the seat uniquely and precisely. They found out that it is impossible for a seat to have two different locations and vice versa. In addition, the students notice the role of rows and columns in a system. They can locate or tell the location of seat by considering the row and column. They also understand that locating a system only by considering one parameter (row or column) is not enough.

The next problem, airplane problem, gives chance to the students to make their own system. It also supports the students to understand the good system and the idea that each point should be uniquely located. The students come up with different system; numbering the seat from one to 148 and using the similar system as in the cinema (involving alphabet and number). From this activity the students senses that numbering the seat from 1 to 148 is not as effective as numbering the rows with alphabet and the columns with number. They know that using 1-148

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system consumes time and need effort to find the seat. So, besides learn about the idea of each point is unique, the students also make a judgment of the most effective and better system.

The students broaden their understanding about coordinate system by making a system to locate an object in the plane. “the sunken ship 1” context problem offer an opportunity for the students to experience another kind of system which is different from the system used in cinema and airplane. This context, again, makes the students understand of an idea that each point is unique.

Furthermore, here they show an improvement of the development level in locate a point. With the teacher’s guidance, the student’ thinking move away from locating a point by only consider one parameter to the sophisticated system. For instance, the system involves angle and the system considers two perpendicular parameters.

The students learn about the origin through the rice field problem. The modified drawing of the rice fields bring the students into the problematic situation. This problematic situation enhances them to think about an unique origin. The students sense that to locate an object precisely, they need to know in where the origin is. They found out that different origins (in a system) can produce different locations with the same clues/ coordinate. In addition, this context can bring the students to the more abstract level. Through this context, the students can see the grids, vertical and horizontal axis which is used to introduce the positive coordinate to the students. From this activity, the students made an agreement on how they should locate an object (horizontal, vertical).

Students’ understanding about the positive and halves coordinate system is challenged by giving them the context problem that force them to think about

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negative coordinate. The sunken ship 2 context offers that opportunity. Making the students locate the rescue team’s ships which are placed in the left and below the light house make the students think about the negative coordinate and so the halves coordinate.

In the end, the students are introduced to Cartesian coordinate system.

Here their knowledge of origin, idea of each point is unique, and the agreement that they made to locate the point help them to understand the Cartesian coordinate system and its rule. The last proposed context is an abstract context which can make the students aware of some common mistakes happen when locating or plotting a point in the Cartesian system. By asking the students to analyze and choose a right answer out of three options can make the students to keep aware of the agreement on locating point which had been made.

So, in general we can conclude that students learn to understanding the coordinate system from the concrete to the more formal level. They learn the mathematical ideas of the coordinate system in each activity through the context.

In other words, the real context and the challenging activities proposed in this study can enhance the students’ thinking and understanding about the coordinate systems. Moreover the familiar and meaningful context motivates the students to learn about the coordinate system. This is also supported by the post test result which shows improvement than the pre test. This implies that the learning activities which are designed in this study play an important role in supporting the students (in this study) to understand the mathematical ideas of the coordinate systems.

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Beside the contexts, the finding suggests that the teacher also plays an important role in supporting the students to understand the coordinate system.

Probing question asked by the teacher makes the students think through and critically about the mathematical contexts which have been addressed. Moreover, how much the teacher gives helps to the students affects the students’ thinking and learning.