DEVELOP STUDENTS’ GRAPH INTERPRETATION SKILLS OF LINE GRAPH: “A STUDY OF FIFTH GRADE STUDENTS USING KMS (KARTU MENUJU SEHAT) AS THE CONTEXT”

2. Theoretical Framework

Graphs in the curriculum In the Indonesian curriculum, the students learn about graphs for the first time is in the second semester of sixth grade in which they learn how to transform data into bar graphs, line graphs, or pie graphs. As an introduction to data, they are taught to collect and represent data in a table in the previous semester. The teaching and learning process is only focused on drilling students to construct and read different graphs. One of the goals that the students should achieve is reading what is clearly stated in the table and graph. This teaching and learning process is more structuralistic than realistic. Furthermore, there is no deep attention given to the students’ development on graph’s reading and interpreting skills.

Realistic Mathematics Education (RME) and Pendidikan Matematika Realistik Indonesia (PMRI)

In learning mathematics, Freudenthal [5] emphasized that students should be allowed to invent their own idea and use their own strategies. He also argued mathematics as a human activity. It indicates that instead of giving procedures, mathematics should be designed and taught based on the students’ learning process. Therefore, the present study has developed instructional activities on interpreting line graphs. The students could gain more insight about how to interpret graphs through experiencing a

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sequence of meaningful activities instead of only copying the way they have interpreted any graphs before. The process of designing instructional activities in this study was guided by five tenets of realistic mathematics education defined by Treffers in Bakker [6]. On the other hands, the instructional activities designed in this study were only a part of longer series of learning trajectories on constructing and interpreting graphs. In this study, by using a local context, KMS that is usually given in POSYANDU, PMRI was chosen as the approach to teach students to interpret line graphs .

3. Methodology

Research Approach

In terms of research approach, there were three phases in the present study. They were preparing for the experiment, experimenting in the classroom, and conducting the retrospective analysis. In the preparing for the experiment phase, the Hypothetical Learning Trajectory (HLT) was resulted. It consisted of a sequence of instructional activities complete with mathematical goals, conjectures of students’ thinking, actions, and strategies. In the next phase, experimenting in the classroom, the HLT has been applied in the teaching and learning process. Also, the entire data set collected from the teaching experiment has been analyzed. It consisted of the students’ written works and video recording during the lessons. The analysis process is mostly conducted by comparing the HLT with the students’ actual learning process. This kind of analysis is called as retrospective analysis.

Participants

The participants for this study were six fifth-grade students. These participants were chosen and rated by their classroom teacher. They were chosen as representative among all the students in their classroom by considering their average level in mathematics. Before conducting a lesson, there were a pre test and interview to investigate their initial point.

Instructional Activities

In designing activities for this study, the students’ learning line in interpreting line graphs has been considered as a crucial point. Therefore, the general hypothesized instructional activities have been designed. By providing two tasks, the students were asked about and facilitated to understand the line graph display (including data plotting) and then gain the meaning (both explicit and implicit meaning) of the line graph by writing a story.

Instrument

There were two tasks about line graphs used in this study. They were about weight and height growth and the students were asked to read and interpret them. For the detail, they are described below and illustrated in figure.

Task 1: The weight growth of a baby during 5 years old The students were given a line graph of baby’s weight growth during 5 years. The

label in the horizontal axis is age and the label in the vertical one is weight. There are 3 lines illustrating different speed of growths but the students were only asked to focus on the lowest line. From this line graph, the students were asked to write sentences as an illustration they can get and tell from the line graph display.

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Fig. 1. Task 1: a line graph of baby’s weight growth

Task 2: The height growth of a boy until 24 years old In this task, the students were given a line graph of a boy’s height growth up to 24 years old. They were asked to write a story based on what they can gain and tell.

Fig. 2. Task 2: a line graph of boy’s height growth

4. Results

Interpreting task 1

In interpreting the first task, Adel, one of the students, became the first who tried to compose a sentence based on the line graph. She wrote that the baby’s weight was 5 kg when it was 1 year old. Using Adel’s answer, a discussion has been held.

Re : Adel said that at one year old, the baby’s weight is 5 kg, Is it right? All

: Nodding

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Re : From where? B : There is a point

Re : If I ask this one (pointing at the intersection point of 2 years old and 7.5)? How old? R/S

: 2 years old Re

: The weight? A/B/C : 7.5 kg (S: 8 kg) Re

: How many kg in year 3, Raka? R

: 9 years old, ohhhhhhhh 9 kg Re

: Shenina, at year 4? S

: 10 (B: 11) I : 11

Re : Now, when he was born? All

: Silent for a moment C : 2 kg (B: at the very start of the line)

Note: A (Adel), B (Bali), C (Chia), I (Ihsan), R (Raka), S (Shenina), Re (Researcher)

From the transcription above, it is clear that Adel ’s idea has opened the discussion. It was like conjectured before that there will be a student who paid attention to the dots on the line (see fig.1). Her answer invited Bali to share his opinion by stressing that Adel’s answer could be seen from the dot on the line. All the students have been questioned one by one, with different questions, and some of them have answered correctly. It gave indication that they who answered correctly have recognized the dots on the line refer to the scale both in the horizontal and vertical axis. This kind of activity has brought the students to the discussion of plotting a dot if the horizontal axis shows 0. At this time the researcher stressed the students to not only show the dot but also the way to get it. There were three kinds of answers. For more detail, look at the figure bellow.

(a)

(b)

(c)

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Fig. 3. The students’ works in interpreting line graph of baby’s weight growth (a) Shenina’s work, (b) Ihsan’ work, (c) Raka’s work, (d) Bali’s work

(e) Adel’s work, and (f) Chia’s work

Almost all the students were able to plot a dot or read a dot. Some of them need a line to make it clear and the rest just only need to see the dots. However, only two students have plotted the dot correctly ((e) and (f)) to show the weight when the baby was born. The other did a mistake on determining the dot ((a)), the way to find it ((c) and (d)), or both ((b)). After comparing and discussing their answers, finally all the students have learned more on determining that dot and the way to get it on the line graph. It was Bali who could compose a sentence for illustrating the baby’s weight when it was born by saying ‘it should be found out by drawing a vertical line from 0 as high as 2 ’. In the end of this activity, although the students still could not write sentences as an illustration of the line graph, but they have learned much how to make sense or gain meaning from the line graph display. To examine the way they started to write a story, see their activity in the task two.

Interpreting task 2

In describing information from the line graph of the boy’s height growth up to 24 years old, the students were working in group of male and female. The first thing the students tried to recognize from the line graph display was the intervals. Although it was written 0, 30, 60, …, 150 in the vertical axis, but there was a mark between two intervals. Bali and Ihsan who worked in a group were asking the same question to the researcher, ‘Is it right that at the age of 4, the height is 95?’. They have drawn a vertical line from 4 until intersecting the line graph. From the intersection point they drew a horizontal line until intersect the vertical axis right on the mark between 90 and 120. Without thinking deeply, Bali thought that it was 95 rather than 105. This was a classic mistake. However, it was the male group also succeeding to comprehend that the intervals for the horizontal axis were four. They were able to predict and estimate the position of 1, 2, and 3 years. In the end of activities, both groups have composed sentences as an interpretation of the line graph. Here is the translation from the male group ‘When he was born, his weight is 80 kg cm, in age 4 his height is 105 cm, in age 8 his

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height is 135cm, in age 12 his height is 150cm, in age 16 his height is 167cm, in age 20 his height is 168cm, in age 24 his height is …’

And, here is the translation from the female group ‘The growth of the boy when he was born is 50cm, and in age 4 his height grows up to 105 cm and the more his age the higher his height, age 8: 130, age 12:155, age 16:, age 20:, age 24: ’

What can be concluded from the descriptions above is both groups have a little difficulty in making estimation of the appropriate scale. As expected, both groups also tend to make list the age and the correspond height and did not connect their description with the line graph display. However, the female groups have improved their sentence in which they said “the more his age the higher his height”. Although the description students wrote were not perfect, their process of thinking in interpreting line graphs can be captured.

Conclusion From this study it can be concluded that the context, KMS in POSYANDU, has helped the students much to gain the meaning of line graph display. The students’ activities in the classroom have also developed since the students spent most of the time for an effective group and class discussion. The students’ development in gaining the meaning of line graph has been recorded. By imagining the context, they have struggled with the line graph display in the first task. In this task they started to understand that the line in line graph consists of many dots and each dot refer to the scale both in the vertical and horizontal axis. Furthermore, they have tried and succeeded to show the baby’s weight when it was born and the way to get it (data plotting). When working with the second task, the students ’ ability in writing a story from the line graph was examined. From the way they wrote the story, it can be seen that they tend to write what is clearly stated in the line graph. They just listed the data they can get from the line graph by using the interval of the scale in x axis as the reference point. However, the female group showed notable progressive by not only listing the data but also a simple conclusion they can get from the line graph display. It was only the matter of time and the more they learn the more they will develop. Finally, if the activities in this study could develop the fifth-grade students’ graphing skills, then these activities should be more effective for the sixth grade students.

5. References

[1] Ates, S. & Stevens, J. T. Teaching line graphs to tenth grade students having different cognitive development levels by using two different instructional modules. Taylor & Francis Ltd. 2003.

[2] Ersoy, A. F.. The effects of calculator based laboratories (cbl) on graphical interpretation of kinematic concepts in physics at metu teacher candidates. a thesis submitted to the graduate school of natural and applied sciences of middle east technical university. 2004.

[3] Leinhardt, G., Zaslavsky, O., & Stein, M. K.,. Functions, graphs, and graphing: tasks, learning, and teaching. American Educational Research Association. 1990. [4] Parmar, S. R. & Signer, B.,. Sources of error in constructing and interpreting graphs: a study of fourth-and-fifth-grade students with LD. Journal of Learning

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disabilities. 2005. [5] Freudenthal, H. (1991). Revisiting Mathematics Education: China Lectures. Dordrecht: Kluwer Academic Publishers. Gravemeijer, K., & Cobb, P. (2006) Design research from the learning design perspective. Educational design research (pp. 17-51). London: Routledge.

[6] Bakker, A. (2004). Design Research in Statistics Education. On Symbolizing and Computer Tools. Amersfoort: Wilco Press.

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