The First Cycle of Developing Teaching Materials for Fractions in Grade Five Using Realistic Mathematics Education

There are three questions that will be answered in this study, namely (1) what are the contexts that can be used to introduce the meaning of multiplication of two fractions and to find the result of multiplying two fractions, (2) how to use these contexts to help students construct the understanding of the meaning of multiplication of two fractions and find the result of multiplying two fractions, and (3) what is the impact of the teaching-learning process that has been designed by researchers on the process of students' knowledge construction. Learning approach which was used in developing teaching materials about fractions is realistic mathematics approach. Lesson plan was created for fifth grade elementary school students. The type of research used is developmental research. According to Gravemeijer and Cobb (in Akker, Gravemeijer, McKeney, and Nieveen, 2006) there are three phases in development research, namely (1) preparation of the trial design, (2) the trial design, and (3) retrospective analysis. This paper presents the results of the first cycle of three cycles that have been planned.

According to Lamon (2001, in Ayunika, 2012, the development of understanding of the meaning of fractions in the teaching-learning process is a complex process because the concept of fraction has a number of interpretations, namely (1) fraction as a part of the whole, (2) fraction as the result of a measurement, (3) fraction as an operator, (4) fraction as a quotient, and (5) fraction as a ratio.
There are 3 questions that will be answered in this study, namely (1) what are the contexts that can be used to introduce the meaning of multiplication of two fractions and to find the result of multiplication of two fractions, (2) how to use these contexts to help students construct the understanding of the meaning of multiplication of two fractions and to find the result of multiplication of two fractions, and (3) what is the impact of the teaching-learning process designed by researchers on the process of students' knowledge construction.
According to Gravemeijer (1994), Realistic Mathematics Education (RME) is rooted in Freudenthal's view that mathematics as a human activity. If implemented, the basic philosophy of RME brings about a fundamental change in the process of teaching-learning mathematics in the classroom. The teacher in teaching and learning activities should no longer directly provide information to the students, but he/she provides a series of problems and activities that can be used by the students to build their understanding of mathematical concepts that leads to the formation of formal mathematical knowledge. In other words, in the RME approach, the teacher plays a role as a facilitator to their students. According to Widjaja, Fauzan, and Dolk (2009), to be able to act as a facilitator, the teacher must facilitate students' learning by using contextual problems, asking questions that guide students to develop their thinking processes, and leading class discussions in order to help the students in constructing their understanding of the mathematical concepts that are embedded in the contextual problems.

METHOD
In the first cycle, there were four students involved that came from the fifth grade of a private elementary school in Yogyakarta. The approach used by researchers to develop students' learning materials and teacher's guides in this study was realistic mathematics education approach. The development of the learning materials and the teacher's guide was to be conducted in three cycles.
Data analysis was done based on the data of video recording, taken during the teaching and learning process, and the student worksheets. The steps were undertaken in the first cycle following phases of the development research developed by Koeno Gravemeijer and Paul Cobb (in Akker, Gravemeijer, McKeney, and Nieveen, 2006).

The First Phase of the First Cycle
The objective of the teaching-learning process that used the learning materials developed by the first researcher was to facilitate the students so they (1) understand the meaning of multiplication of two fractions, and (2) are able to determine the result of multiplication of two fractions.
Before students experienced the learning process designed by the first researcher, the students had learned about fractions in fourth grade, involving (1) the meaning of fractions, (2) ordering fractions, (3) simplifying fractions, and (4) adding and subtracting fractions.

The Second Phase of the First Cycle
The contextual problems were explored and solved by students for four meetings, i. e.: The First Meeting a. The first problem Yesterday afternoon during school recess, the teacher saw two groups of children who were sharing bread. The first group consisted of two students who were sharing a piece of bread. The second group consisted of four students who were sharing two pieces of bread. Do you think that each student in the first and second groups got the same amount of bread?
b. The second problem Yesterday afternoon during the school break, the teacher also saw two groups of other children who were sharing bread. The first group consisted of two students who were sharing a piece of bread.
The second group consisted of three students who were sharing two pieces of bread. Do you think that each student in the first and second groups got the same amount of bread?
c. The third problem The third problem consisted of four questions. In each question, there were two groups of children who were sharing the bread. The number of the children and the amount of the bread of each group were different. Students were asked to choose whether they would be a member of the first or the second groups and the reason why they determined their choice.
First question: there were four children sharing two pieces of bread at the first group, while there were six children sharing two pieces of bread at the second group.
Second question: there were four children sharing two pieces of bread at the first group, while there were six children sharing three pieces of bread at the second group.
Third question: there were three children sharing two pieces of bread at the first group, while there were four children sharing three pieces of bread at the second group. The Third Meeting a. The first problem (inspired by the problems in the book titled "Young Mathematicians at Work: Constructing Fractions, Decimal, and Percents") Today fourth grade students of Mekarsari School will make observations at some objects of art and culture in Yogya.

The First group will visit
Kasongan, Bantul students bread The third group will visit the center of batik art. students bread students bread The fourth group will visit the silver products.
The Second group will visit Affandi's museum.

students bread
When students returned from the observation activity, the students began to argue that the bread that was distributed to each student in the group did not have the same amount, because there were some students who got more than other students. Did each student get the same amount of bread?
b. The second problem (inspired by the problems in the book titled "Young Mathematicians at Work: Constructing Fractions, Decimal, and Percents") Mrs. Niken gives the following questions to the students. A student, named Bulan, was of five students. The group received three pieces of bread. How much bread was obtained by Bulan? The pictures below were the students' answers. Do the answers produce equivalent fractions? Can you show it?
Titin 's answer Rudi's answer Susi's answer Andi's answer The fourth meeting was used for an evaluation activity. The following were the questions given to students in the evaluation process:

The First Question
Bu Vivi makes a pan cake. Bu Vivi will divide the cake to 8 neighbors, namely Bu Dina, Bu

The Second Question
Today, the 4th grade students of Karya Elementary School will make observations at some objects of art and culture in Yogya. The students are given some bread by the school, to be eaten during lunch time. Here is the place to visit, the number of students in every group, and the amount of bread in each group.
At the time of the distribution of the bread, the bread for group 1 is combined with the bread of group 3, while the bread of group 2 is combined with the bread of group 4. After that, each combination of groups is asked to share the bread for its members fairly. Does each student of that grade, who takes part in the visit, get the same amount. In other words, is the distribution of bread in that way fair? The first group will visit Kasongan, Bantul students bread The third group will visit the center of batik art.
students bread students bread The fourth group will visit the silver products.
The Second group will visit Affandi's museum.

students bread
The The students in the first group divided the bread for the first group into two equal parts. Then the students in the second group divided each of the first and the second pieces of bread for the second group into two equal parts. After that, they gave each part to each student in the second group. The conclusion is that students in the first and second group received the same portion, i.e.
half of bread.

The second problem:
Method 1: In the first group, the students divided the bread for the first group into two equal parts and each student obtained a half of the bread. In the second group, the students divided each of the first and second pieces of bread for the second group into three equal parts and each student obtained of the bread.
 The number of students who participated in the first meeting was four.
 The first problem was done individually.
 For the first problem, the four students answered in the same way.
 The second problem was done individually.
 For the second problem, there were two students who answered using method 1, and there were two students who answered using method 2.

Method 2:
In the first group, the students divided the bread for the first group into two equal parts and each student obtained half of the bread. In the second group, the students divided each of the first and second pieces of the bread for the second group into two equal parts.
Then every student in this group was given half of the bread. Then they cut the remaining bread into three equal parts. Students wrote that the bread slice was of = . So, the portion obtained by each member of the group was + = + = = . The students said that the portion obtained by each student in group 2 was more than that obtained in group 1 because each student in group 2 got extra bread from the rest of the half part of bread that was divided by three.

The third problem:
The first question: Students divided each pieces of bread for the first group into two parts, and for the second group into three parts.
Students chose group 1 because they wolud get more bread than each student in group 2.

Second question:
Students divided each bread for the first group into two parts, and for the second group into two parts. Students said that the portion for each student in group 1 was equal to the portion for each student in group 2.
The third question: because the part of group 1 more than group 2.

The fourth question:
Students divided bread for the first group into four parts.
Students divided each piece of bread for the second group into five parts. In the classroom, there were two groups of students were worked on this fourth question.
The first group's answer was they chose group 2, because each student in group 2 got more bread than each student in group 1. The second group's answer was they chose group 1, because each student in group 1 got more bread than each student in group 2. 2 The first problem: The students in the classroom said that the part obtained by each friend Mr. Hongki is . The reason given by students is because after Mr. Hongki divides the bread into two equal parts, Mr. Hongki divides each section into two equal parts. Thus, each Mr. Hongki's friend gets the same portion, namely part, although the form of cake obtained by each person is not same.

The second problem:
Method 1: The piece A of bread 1 is of = . The piece B of bread 2 and the piece C of bread 3 is of = × = . So, the biggest piece is piece A.

Method 2:
The piece A of bread 1 is of = or divided by 2 or of . The piece B of bread 2 is of = or that comes from divided by 3. The piece C of bread 3 is that comes from divided by 3 or of .
Students did not say which one was the biggest piece. 3

The first problem
For Titin's answer: Method 1: Students redrew the picture in the student  The third meeting was only attended by two students who also worksheet, and provided shading on the first piece of each piece of the bread. Then the students gave a check on the answer. In other words, the students said that Titin's answer was correct.
Method 2: Students redrew the picture in the student worksheet, and wrote down the names of students who received each part of the piece and each part is . Names written by the students for each piece of bread were Candra, Rudi, Adi, Budi, and Bulan. Then, the students shaded three pieces that belonged to Bulan, and concluded that the portion was obtained by Bulan is For Rudi's answer: Method 1: Students redrew the picture in the student worksheet, and shaded the parts acquired by Bulan, i.e.
the top piece of the first piece of bread, and the leftmost bottom of the third piece of bread. The students wrote that the small part obtained by Bulan is of = , and the whole portion that belonged to Bulan is + = + = . After that, the students made the cross sign, indicating the students stated that Rudi's answer is wrong.

Method 2:
The students redrew the picture in the student worksheet, and wrote down the amount of bread in each piece. Each of the big piece is , and each of the small pieces is . Students wrote the method to obtain a small piece is ∶ 5. Then the students concluded that the whole portion of bread obtained by Bulan is + = × × + = = .
Susi's answer: Andi's answer: Method 1: The students redrew the picture in the student worksheet, and put each of the numbers 1, 2, 3, 4, 5 three times consecutively, and shaded the parts acquired by Bulan,i. e. the top three pieces of the first bread. Then students wrote that the whole portion obtained by Bulan is .

Method 2:
The students redrew the picture in the student worksheet, and put each of the numbers 1, 2, 3, 4, 5 three times consecutively, and shaded parts acquired by Bulan, i. e. the bottom three pieces of the third bread Then the students wrote that the whole portion obtained by Bulan is .

CONCLUSIONS AND RECOMMENDATIONS
There are four things that can be inferred from the results of the exploration and responses of the students to the problems and evaluation given by the researchers: 1. From the first problem in the first meeting, the students built a model for a half to solve the problem. This model was also used by the students to solve the second and third problems. Due to the use of the model of a half in solving the second problem, the students could build an understanding of the meaning of of = .
2. From the students' understanding of the meaning of of = , the students could develop an understanding about ∶ 2 or of = for the second problem in the second meeting, of = for the third problem in the second meeting and the first problem in the third meeting, and of = × = and of = × = for the third question in the evaluation.
3. The students were able to build up an understanding about the multiplication of two fractions and how to find the result of multiplying two fractions.
4. To further strengthen the establishment of formal knowledge of how to multiply two fractions, students need more experiences through exploration activities and solve other problems in addition to the ones that are already given.