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THE TEACHERS’ EFFORTS TO ENCOURAGE THE STUDENTS’ STRATEGIES TO FIND THE SOLUTION OF FRACTION PROBLEM IN BANDA ACEH

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The Teachers’ Efforts to Encourage the Students’ Strategies to Find the Solution of Fraction Problem in Banda Aceh

Rahmah Johar and Marisa Afrina

Syiah Kuala University

Author Note Rahmah Johar and Marisa Afrina, Department of Mathematics Education, Teacher Training Education Faculty, Syiah Kuala University This research is part of my research that supported by Ministery of Education for Hibah Bersaing Grant BCHP: 258/H11/A.01/APBN‐P2T/2010 Correspondence concerning this article should be addressed to Rahmah Johar, Department of Mathematics Education, Teacher Training Education Faculty, Syiah Kuala University. Darussalam. Banda Aceh. Indonesia E-mail: [email protected]


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Abstract

One of the Realistic Mathematics Education (RME) characteristic is using students’ own construction. Teacher stimulate the student to be more active and creative for develop their ideas or strategies. The purpose of this study was to describe the students’ strategies and teachers’ efforts to encourage them. The subject is student fourth grade of the Islamic Elementary School (MIN) Suka Damai Banda Aceh, Indonesia. Data was collected via video and field note for record the interaction between student-student and student-teacher in classroom to find their own strategies for solving fraction problem. The results of the research are there are some students’ strategies to solve the problem, those are 1) trial and error, 2) using diagram, and 3) using pattern. Furthermore, teachers’ efforts to encourage the students’ strategies are a) giving some choices for the student to use concrete material, pictures, or calculation, b) giving some choices for the student to use any strategies as the student can, c) asking the question to stimulate the students’ thinking or clarification the students’ answer, and d) giving compliments for the students’ efforts. Key words: students’ strategies, teachers’ efforts


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The Teachers’ Efforts to Encourage the Students’ Strategies to Find the Solution of Fraction Problem in Banda Aceh

Mathematics learning objectives is not only to calculate, but also to develop

thinking skills, reasoning, problem solving, and communication. One of the approaches which can be used in teaching mathematics is realistic approach, known as Realistic Mathematics Education (RME). RME was introduced by Freudenthal in the Netherlands in 1968. According to Freudenthal as cited in Gravemeijer (1994), students should experience mathematics as a human activity. It is an activity of solving problems, looking for problems, and organizing a subject matter. In addition, Panhuizen (1996) suggest mathematics must be taught in the order in which the students themselves might be inventing it. Gravemeijer (2010) complements the RME tries to relate with the experiential reality of the students. About term, ‘real’ in ‘realistic’ has to be understood as real in the sense of being meaningful for the students. Therefore, the choice of realistic problems plays a very important role.

To develop RME in Indonesia, Indonesian mathematicians and mathematics educator was designed “Realistic Mathematics Education in Indonesia” or Pendidikan Matematika Realistik Indonesia (PMRI), known as Indonesian version of RME. Dutch consultants from the APS Netherland and Freudenthal Institute (FI) of Utrecht University have been involved to coaching the PMRI team. The team that consists of teacher educator and teacher started a pilot program of PMRI from 2001 until now. The PMRI team wants to change mathematics education in such a way that most children will be able to do and enjoy mathematics to develop their mathematics knowledge, skills, and strategies. Dolk (2010) suggested such changes start in classroom. Creating learning community in the classroom asks for another role of teachers and of students. One aspect of these new norms is related to the teacher’s expectations and beliefs about students’ mathematical thinking. Gravemeijer (2010) reminded RME further requires the teacher to play an active role in orchestrating productive whole-class discussions and in selecting framing mathematics issues - context problem - as topics discussion.

To implement RME in Indonesia, teacher educator and teacher in primary school collaborate to design the context problems for Indonesian Primary School students, it is use to develop the students’ thinking/reasoning in mathematics. Context problem in RME must be accessible, inviting, and worthwhile solving, also challenging and it must be obvious to the students why an answer to a give question is required. In short, problem must be meaningful. The emphasis on higher-order reasoning implies that the problem situations should be fairly unfamiliar to the students. In other words, problem solving in RME does not mean simply conducting a fixed procedure in set situation. Consequently, the problems can be solved in different ways (Panhuizen, 1996). This problem we known as non-routine problem because the mathematical procedures that children must use to solve them are not obvious (Reys, et.all, 2007), they don’t know how to solve “comfortably” with routine or familiar procedures (Schoenfeld, 1983).

The problem likes this one: suppose mother baked 1 cookie. If she has 2 children (or 4 or 6) and each child gets the same number of cookies, how many cookies does each


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child get? It is a routine problem for Indonesian Primary School students at fourth grade. Because they can be solved by applying a mathematical procedure in much the same way as it was learned about simple fraction for share topics. But, the problem 15 kg meat for 20 poor people is as a non-routine problem for them, therefore context problem in RME. The student didn’t know yet about the fixed procedure they can solve, so they want to develop their own strategies.

Student at grade 4 in primary school need to discuss with the other student to solve the context problem and teacher play the role to stimulate them. Teacher encourages the student to be more active and creative for develop their ideas or strategies. In general, Horn et.all (2005) proposed the strategies for creative instruction, those are 1) student-centered learning, 2) Use of multi-teaching aids assistance, 3) Class management strategies, 4) Connection between teaching contents and real life, and 5) Open questions and encouragement of creative thinking. These research objectives are describing the students’ strategies and teachers’ efforts to encourage the students’ strategies to solve the fraction problem at fourth grade of the Islamic Elementary School (MIN) Suka Damai Banda Aceh, Indonesia.

a. Context Problem in Realistic Mathematics Education

Context is one of the prominent characteristics of RME. Borasi defined context as a situation in which the problem is embedded (in Panhuizen, 1996). In other word, context means the situations to which they refer. In the learning process, context gives meaning to the content (Johnson, 2007). The context need not necessarily refer, however, to real life situations. The important point is that they can be organized mathematically and that students can place themselves within them. Panhuizen (1996) offered two requirements of problem in RME, problems must be meaningful and informative. In order for problem to be meaningful, the context problem must be obvious to the students why an answer to a give question is required. Meaningful also respect to subject matter, it cover the entire breadth and depth of the mathematical area. Another requirement is informative. It means the problem must be as clear as possible to the student and the students must have the opportunity to give their own answers in their own words. They can solve in different ways/strategies and different levels.

About problem itself, Lester (in Zawojewski and Lesh, 2003) defined a problem as a task which: (1) individual or group confronting it wants or needs to find the solution, (2) there is not a readily accessible procedure that guarantees or completely determines the solution, and (3) the individual or group must make and attempt to find a solution”. In this line teacher have to stimulate the students to find their own strategies to solve the problem. b. Teachers Efforts to Stimulate the Students’ Strategies to Solve a Context

Problem Problem solving-it means solving the non-routine problem or context problem- is a

primary goal of mathematics teaching and learning and is considered to be the essence of mathematics (NCTM, 2000). However, some students fail in solving problems typically


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defined as non-routine and teachers encounter difficulties in supporting the development of the students’ problem-solving competency (Kolovou, et al. 2009). The teacher has to play an active role in orchestrating productive whole-class discussions and in selecting framing mathematics issues - context problem - as topics discussion (Gravemeijer, 2010, Gravemeijer and Cobb, 2006). The teacher not only asks the student to ‘explain your strategy’ but also ‘show how you got your answer’ (Panhuizen, 1996).

In general, the initiator of “how to solve the problem” is the Hungarian-born mathematician George Polya. He, in general, proposed four-stage model of problem solving (Polya, 1973): understanding problem, devise a plan for solving it, carry out your plan, and look back to examine your solution. However, Bell, Swan, and Taylor (in Zawojewski and Lesh, 2003) explained a number of studies have found that these steps of general strategies is not particularly helpful for mathematically unsophisticated students. In addition, Reys (2007) said Polya’s model can be less than helpful if taken too literally. Moreover, the steps are not discrete, and it is not always necessary to perform every step. For instance, while trying to understand a problem, students may move into the planning stage without realizing they have done so. Or simply understanding a problem may enable students to see a solution without any planning. Children need specific strategies that will help them move through the steps in a productive way. Polya himself delineated many of these strategies, and many textbooks provide lists of the strategies geared to various grade levels.

Schoenfeld in his own work (in Zawojewski and Lesh, 2003) offered three metacognitive questions to help students reflect on their current understanding of a problem:

1. What (exactly) are you doing? (Can you describe it precisely?) 2. Why are you doing it? (how does it fit into the solution?) 3. How does it help you? (what will you do with the outcome when you obtain it? For additional guidance in helping the student to solve the problem, teacher

introduces or offers some strategies for the students. There are some strategies to solve the problem (Polya and Pasmep in Shadiq, 2004; Sobel, and Maletsky, 2004; Reys, 2003; Zawojewski and Lesh, 2003), are as follow: - Ignore the impossible thing

In this strategy after understanding the problem by formulating what is known and what was asked. When you find things that do not relate to what is known and what is asked should be ignored.

- Draw a picture It is one of the most common problem solving strategies taught in school today. When teaching this strategy, stress to children that there is no need to draw detailed pictures.

- Look for a pattern Recognizing, describing, extending, and generalizing patterns are important components of algebraic thinking (NCTM, 2000). In many early learning activities, children have to identify a pattern pictures or numbers. In problem solving, students look for patterns in more active ways- for example, by constructing a table, drawing of dots, etc.

- Look for a similar problem


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Children who know how to solve a given problem that is somewhat similar, even if the second problem is also somewhat more difficult. When given a problem that seems too hard, children can apply this strategy by setting the problem aside for a moment and solving a similar but simpler problem. Then can try using the same method to solve the original method. In this study, for example, we give 2 problems for 2 lessons. First problem is easier then the second one. First problem:

Mr. Ahmad has 15 kg of meats that will be given to 20 orphans. How much meat does each orphan get?” Second problem:

Mother has 25 kg of rice. Every day, she cooks it as kg. How many days the rice

will be finish?” Student looks for the similar of both of problems, for instance similar about sharing idea and the meaning of ¾.

- Guess and check (trial and error) For years, children have been discouraged from guessing. Of course, random guessing is not good problem solving, but guessing can be a useful strategy if the students incorporate what they know into their guess-that is, if their guesses are educated guesses rather than wild guesses. Teacher must help the students learn how to refine their guesses efficiently.

As we discussed, giving students problems that are just within their reach, that

challenge them to reach solutions, helps them make sense of mathematics. Teacher must think carefully about how you organize and manage classroom instruction to ensure that all students have worthwhile problem solving experience. Important considerations are how to manage time, classroom routine, and needs of individual students (Reys, 2003).

Base on the above statement, the teachers need to encourage their students’ strategies to solve the mathematics problem. Teacher encourages the student to be more active and creative for develop their ideas or strategies. Horn et.all (2005) proposed the strategies for creative instruction, as folow

1) Student-centred learning The role of teachers is as facilitator rather than lecturer, helping students with self reflection, group discussion, and group activities. Questions for group discussions and presentations are prepared. Students are given reedom to choose from what perspective they will solve the problem. Throughout the class, teachers act as a learning partner, inspirer, navigator and sharer, while students transform from passive listeners to observers, performers and co-learners.

2) Use of multi-teaching aids assistance The three subjects analyzed are good at using multi teaching aids to assist their instructions. They ingeniously used creative teaching aids, such as paper crusher, toy block, hammer, student writing script, power-point, computer and multimedia to excite students thinking, broaden the viewpoints and encourage further discussion.

3) Class management strategies


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The subjects showed sophisticated management, created friendly interactions and treated students with recognitions of their individualities and needs. They are more like friends to their students, speaking with gentle tones and body language. When students express ideas, they would not interrupt nor give judgments immediately. Instead, they gave guidance, more open questions, or conveyed their personal experiences as references.

4) Connection between teaching contents and real life Integrated Activities help students to develop the ability to express and realize them in daily life, find real life examples to evidence what they learn, and relate what they learn to life experiences. The subjects indicated creative ideas also come from real life; the key point is that teacher should be sensitive to feel, find, think and convert into instruction. 5) Open questions and encouragement of creative thinking Open questions for the students stimulate them to be students’ creative thinking

Method

The subjects for this study were 22 male students grade fourth of MIN Suka Damai

Banda Aceh, Indonesia. This school was as a partner school of PMRI since 2007. The students have learned about introduction of fraction, equivalence fraction, and division of whole number without left (for instance 20:5, 124:4). The students are divided in 5 groups of 4-5 peoples in heterogeneous ability to solve the fraction problem. We only focus on 2 groups of 5 groups, namely group A and group B.

Actually, working in group is familiar for the students but the students are not so independent, they prefer waiting for guidance teacher to discussing each other in group. Teacher in this study is a university student. She is my student and she is doing her research in the same title but a little bit different focus with this study. So, I am as her advisor also as observer in this research.

Data was collected via video recorder, camera digital, field note, and interview with teacher. The data are about interaction between student and student, student and teacher when they are solving the problem. We collect the strategies of the student in their paper and poster. Dialog between teacher and student is very important to know how the teachers’ efforts to encourage the students’ strategies to solve the fraction problem. There are 1 university student take a picture, 4 university students fill the field note, 1 student take a video, and I am as observer for whole activities in classroom. I held interview to both of teacher and observer to make sure the data were already collected.

There are two fraction problems for student, one problem for each lesson, as follow: First problem:

Mr. Ahmad has 15 kilogram of meats that will be given to 20 orphans. How much meat does each orphan get? Second problem:

Mother has 25 kg of rice. Every day, she cooks it as kg. How many days the rice will

be finish?” The first problem is about sharing problem. The student in grade 4 of the Islamic

Elementary School (MIN) didn’t know yet about the solution because they only know


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about the simpler one, for instance 1 cookie is divided to 2 children (or 4 or 6). They can elaborate their strategy to solve the first problem.

The second problem is about division whole number with fraction. Students in grade 4 didn’t know this concept yet from their school, but they can solve the problem by using addition and subtraction concept of fraction. The PMRI team of Indonesia, accompanied by Dutch consultant, gave this problem for students in some schools since last year.

Before the problem is given to student in grade 4, I asked the student of Primary Teacher Education College (PGSD) of Syiah Kuala University to solve the first problem as prediction how the strategies of student in grade 4 to solve that problem. At that time, the teacher will give the lesson in primary school grade 4 as observer. She is invited to learn how to stimulate the students’ to solve the problem and how many possibility the strategies will appear.

Results

The results of this study will students’ strategies and teachers’ efforts to encourage

the students’ strategies to solve the fraction problem at fourth grade of the Islamic Elementary School (MIN) Suka Damai Banda Aceh. The first is about the result of the first lesson and second one is about the result of the second lesson. a. The First Lesson

In beginning of lesson, teacher review about fraction concept by using teaching material as students recognize around them. Teacher shown 1 kilogram of sugar in one bag and this sugar will share to 5 small bags. Then teacher ask the student to think how many kilograms of sugar in each small bag? And students mentioned kilograms as the answer.

Then, the teacher gave a problem and asked the student to solve it in group. Teacher said there is a mathematics problem in the student worksheet and asked the student to read the problem and understanding it. The student can use any strategies that easy to them, teacher reminded. During the lesson, the teacher monitor 5 groups of students one by one to giving motivation, checking the group answer, asking the question, and clarifying the students’ strategies. While the teacher observes one group of students, some time the other group called the teacher to give comment for their answer. Next explanation it will describe the strategies of group A and group B of students and the teacher’ efforts to encourage students’ strategies as a follow. 1) Group A

There are 5 students in group A: Agus, Irfan, Haikal, Iman, and Juni. First, one of student in this group read a problem. But, around 2 minutes nobody talk something in group A, also in the other group. So that, in front of class, teacher asks the student to pay attention what the meaning of problem is, “15 kilograms of meat will be given to 20 orphans; does each orphan will have 1 kilogram?”. The student answers ‘no’. The teacher asks the student to discuss in group to solve the problem. The table below describes the students’ strategies and the teacher’ efforts to encourage the group A strategies.


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Table 1 Students’ Strategies and Teacher’s Efforts for Group A for the Fisrt Lesson

Students’ Strategies Teacher’s Efforts

Student in group A started to solve the problem with multiplication strategy Iman wrote multiplication 15 with 1 and 2, as a picture follow Iman : “How can 15 divided by 20?,

15 multiply by what number to get 20?,if 5 x 3 is 15, it’s still less than 20

Teacher ask the matacognition question for the student Teacher : What do you want to do to solve

the problem? Student : “It can use division…” Teacher : “How can it work?” “You may use many way, you may

make a drawing, or counting, it’s up to you”

Because of they didn’t succeed yet, they will change their strategy and they ask the teacher support Haikal : do we make a picture?, what

picture we will make, teacher?

Teacher gave motivation to students to do anything, and give some choice Teacher: you may imagine the meat as any

thing you want, it might be square, circle, or anything what you want”

Student makes a picture to represent of meat. Agus’ pictures:

(Irfan ask to Agus, how many parts of meat? Agus confuse about his picture) Haikal and Irfan draw a picture as below

Teacher clarifies the students answer Teacher : “How many pieces of meat?” Juni : “ The meat are 15 pieces” Teacher : “How many pieces are made if 15

is divided into 2 parts?” Imam : “It’s more than 20” Teacher : “How many more?” Imam : “5…” Teacher : “What 5??” Imam : “5 kilograms” Teacher : “How about the others?” Haikal : “I try to redivide” Teacher : “How many parts do you want to

divide?” Haikal : “4 parts…” Teacher : “Why do you divide into 4 parts?” Haikal : “To make it enough 20…” Haikal make a new representation as a


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picture

The students confuse how to say the result; 10 pieces of rectangle are divided by 2, and 5 pieces of rectangle are divided by 4. Irfan : “So, ½ is added by ¼, isn’t

it?”

Teacher gave a confirmation to the students’ answer. Teacher : “Yes, look back to your picture of meat, and which one is a

half and which one is ¼ ?”

Because students didn’t know yet about addition of fraction with different denominator, students asked how to write the answer down Haikal : “Is it like a picture before?

(fraction in shaded shape) Is it needed to shade picture?”

Teacher asked students to make a draw of fraction ½ and ¼. Teacher : “Yes…” Haikal : “I have to make ½ parts of meat, then added by ¼” Agus : and then, we have to collect

them together in the same picture, right?

Teacher : “Yes… Right...” Agus : like this?

Teacher : Yes. How many is this? (point to

shaded shape) Srudent : ¾ Teacher : ¾ what? Students : (thinking) Haikal : ¾ kg for each orphan Teacher : Give applause for your friend…”

Students in group A write their result down in a poster, as a picture follow.

Teacher’s efforts can encourage students to finding the strategy to solve the problem.


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2) Group B

There are 5 students in group B: Fahrizal, Ikhtiar, Furqan, Arsyad, and Risqan. The table below describes the students’ strategies and the teacher’ efforts to encourage the group B strategies. Table 2 Students’ Strategies and Teacher’s Efforts for Group B for the First Lesson

Students’ Strategies Teacher’s Effort

Students tried to understand the problem. Students made interaction among them in group. Fachrizal : (Asking about the way to

solve the problem with his friends in group)

Arsyad : (Reading a problem) 15 kilograms divide into 20 orphans. 15 : 20. mmmmhh 15 cannot be divided by 20

Teacher asks the student to understand the problem and ten thinking how to start it.

Some students change their strategy, using concrete material. Fachrizal : (Fold the one piece of paper) Arsyad : (Count the part of the folded

paper and measure one part using a ruler, he confuse). “Shape... 15 cannot be divided by 20“ (getting angry)

Teacher tried to give a simple problem, but in similar meaning. Teacher : “If you have 1 kg of sugar, you can share it to 5 small bags, aren’t you?? Arsyad : “Yes, I can.... but 15 can not be divided by 20 Teacher : “If 1 can be divided by 5, 15 also can be divided by 20, please try again”

Students tried to think hardly, but they still try to fold one piece of paper. The fold became bigger and bigger (because of 20 fold). They continue folding until they reach 20-fold.

Teacher gave a motivation to be an action as a students’ friend and asked question to open students’ mind. Teacher : “What did you fold? People or

Poster of group A for first lesson


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meats?” Arsyad : (recounting the part of folded

paper, each fold around 1 cm, until 20) ‘it can be’

They kept trying to fold a paper, but after folding a paper, they seem confuse, because they think it cannot work.

Teacher asked about the meaning of problem to the students Teacher : “Come on… keep trying first… 15 kilograms divide to 20 orphans” Arsyad : “It can’t”

Students still hard to think 15 kilograms of meat are divided to 20 orphans

Teacher changed the context into a cake Teacher : “How if 15 cakes divide to 20

children… can it work?” “I believe you can find the answer Risqan : 15 divided by 20 Arsyad : 20:15 Teacher : How to do that? can it work?”

Student could imagine if 15 is divided by 20 will get a result half and a half, but they still confuse to write down the answer. Furqan : “It will get half and a half…”

Teacher gave motivation to students to write down everything in their mind to find the solution Teacher : “Which one is a half? Which meat is?” Arsyad : “Can I write it down, to make it clear ?” Teacher : “Yes, you may write everything, every shape, like a circle or anything, it’s up to you”

Students’ get a new idea and tried to draw a picture, but still in confuse. Arsyad : (made a picture of rectangle and divided in 20) “It’s really unusual… How can be 15 divided by 20?” (Keep trying to divide with long division algorithm)

Teacher gave motivation on and on to make students keep spirit to solve the problem. Teacher : “Keep spirit…” “Let’s see… for example, If I have 2 mangoes, and I want to give them to 4 of my friends, how many part for

each friend get?” Risqan : “½ part” Teacher : “Yes you are right… and now, if you have 15 kilograms of meat, how many part of 20 orphans can have it?” Arsyad : “1 ½…..” Teacher : “Come on… keep trying…”


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Students became more enthusiastic to solve the problem when they used a picture. Arsya : “If divide into a half, we will get

more….”

Teacher kept giving motivation to students to make them conscious to solve the problem. Teacher : “How many?” Arsyad : “10…” Teacher : “How about the left? So, what should we do?” Arsyad : “Share again, doesn’t it? Teacher : “Yes, excellence…” Arsyad : (drawing 1 rectangular as

follow

Arsyad : “Yes, it’s ½ and ¼ Teacher : How much? Arsyad : (drawing)

Arsyad : 3/4

Students in group B felt very happy because finally they could find the right strategy to solve the problem. They rewrite their answer on poster as follow.

Teacher’s effort actually can encourage students to find strategy.

In the first lesson, the strategies of group A is drawing the picture and using the

pattern to solve the problem. They draw 15 shapes and 10 shapes of it are divided by 2 and the rest 5 shapes are divided by 4. In the beginning, they start to trial and error strategy with multiplication 15x1 and 15x2. To encourage the students, the teacher asking metacognitive question, giving motivation to students to do anything with making a picture or calculate, clarifying/confirmation the students’ answer, and give the reinforcement/compliment for student.

Poster of group B for first lesson


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The strategies of group B is the same with group A, drawing the picture and using the pattern to solve the problem. The shape is not the same with picture of group A. Group B draw one big rectangular and divide it to 15 parts, 10 of them are divided by 2 and the rest of 5 parts are divided by 4. Group B spent their time in the beginning to calculate 15:20 and 20:15 (they can’t do it because it should be decimal number), and trial and error to fold one piece of paper. The teacher knows the student didn’t clear about the problem and the student difficult to represent 15 kg of meats for 20 orphans. So that, the teacher asks some similar problem, ‘1 kg of sugar divided to 5 small bags’, 15 cakes divided to 20 friends, and ‘2 mangoes for 4 friends’. The teacher gives motivation for the students, those are ‘keep spirit’, ‘come on….. keep trying’, ‘I believe you can find the answer’.

Both of group A and group B didn’t know yet how much kg of meat if they have ½ kg of meat and ¼ kg of meat. Base on guidance from the teacher they using the idea of collecting ½ and ¼ in the same picture to find the result of ½ + ¼.

b. The Second Lesson Before teacher asked students to working in group, teacher provided a picture about

a rag-bag rice which content 25 kilograms. Then teacher read the information about “Mother has 25 kg of rice. Every day, she cooks it as kg . And then teacher

asked the question, “What do we want to know from this information?, write your answer down in the whiteboard. Agus wrote in the white board : “How many days mother cooks the rice?” Teacher : “Yes, good…. Who want to complete Agus’ sentence!” Arsyad : “How many days rice will be over? (wrote down in whiteboard) Teacher : “Is Arsyad’s expression right”? Student : “Yes…” Teacher : “Is ¾ equal to 1 kilogram? Student : “Not yet... It has to be added by ¼ to get 1 kg Teacher : “Yes, you are right... so, if mother cooks 1 kilogram of rice for 1 day, How

many days the rice will finish?” Student : “25 days...” (some students answered simultaneously) Teacher : “Good.... in 25 days, How about if mother cooks ¾ kilograms of rice in 1 day?” Student : “Of course, it will be more than 25 days….” Teacher : “Ok... so, this is our problem for today that you will be discussed in group. Any

question?” Student : (silent, it mean clear)

1) Group A There are 3 students in group A: Agus, Haikal, and Juni. Iman move to group B,

and Irfan is absent because of he is sick. The table below describes the students’ strategies and the teacher’ efforts to encourage the group A strategies. Table 3 Student’s strategy and Teacher’s Efforts in Group A for Second Lesson

Students’ Strategies Teacher’s Efforts

Student start discussing the meaning of the problem

Teacher asked student to explain what they know about the problem.


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Teacher: What is the information in this

problem? Agus : (Reading the problem) “mother has

25 kilograms of rice, she cooks it ¾ kilograms per day”

After understanding the problem, students tried to drawing one by one square till 25 squares, 1 square is representation of 1 kg of rice. Each square divided by 4, students shade 3 parts of four (it mean ¾ kg) Agus draw the picture as below ¾ ¾ ¾ ¾

Teacher gave response to a picture and asked the meaning of the picture.

Teacher : “What picture did you make, Agus? Would you like to describe it to me?”

Agus : “It is a picture of ¾ kilogram…”

Juni and Haikal help Agus to continue draw the picture and shade it. After they counting until 25 square, the rice is still left 1/4, in each square. The student shade again, 3 pieces of ¼ is become ¾, it can to 1 day. They continue to count and shade all of square, as picture below Haikal : “I will be shaded again and

again, but I got 26 days…” Agus : (continue recounting)

Teacher gave a direction to the student by asking some questions about picture that is made by students. Juni : “teacher……………we already

finish......” Teacher : “Really…. Let me see the

picture! Haikal : “This is the picture, but it still be left 1 over”


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The student skip 1 part of square (it mean ¼ kg of rice). The students said the ice will be finish after 33 days Haikal : (write 33 days)

Teacher kept giving questions to clarify students’ answer. Teacher : “Are there any left over?” Agus : “Yes, it still, but only 1 left” Haikal : “No, it’s only ¼ kilogram” Teacher : “Yes, very good... so what is the answer?” Student : “33 days plus ¼ kilogram” Teacher : “Yes, give applause....and now make a draw in poster paper” Student : “Ok…”

This is the strategy of students in group A

Teacher asked for students to present the students’ work.

2) Group B There are 3 students in group B: Fahrizal, Arsyad, and Iman (from group A).

Ikhtiar, Furqan, and Risqan are absent because one of them sick and the other join in the other school activity. The table below describes the students’ strategies and the teacher’ efforts to encourage the group B strategies.

Table 4 Students’ Strategies and Teacher’s Efforts in group B for Second Lesson

Students’ Strategies Teacher’s Efforts Some students cannot understand the problem, but one of the student give suggestion. Iman : “25 x ¾” Arsyad : “Not 25 x ¾ , but it is 25 :

¾ (Arsyad tried to drawing some of rectangular as folow).

Teacher aske the student to look back to examine the students’ answer Teacher : “Ready? Did you find the answer?” Arsyad : “100 days…..” (because there 100 parts in his picture) Teacher : “Wow… that’s too much… keep

trying… mother cooks ¾ kilogram of rice for 1 day, is it equal with 1 kilogram?”

Haikal representative of group A was

presenting their strategy on poster for

second lesson


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(Arsyad divided 1 big rectangular to 25 parts and each part divided by 4, he shade 1 part with blue color = ¼, so he got 100 parts) Facrizal : (Counting the rectangle

that had been made by Arsyad)

Student check the part in rectangular Arsyad : (recounting)

Teacher asks the students to recognize the representation that student made. Teacher : “How many? Student : “I got 25 days” Teacher : “Does it still have left over?” Arsyad : “Yes…” Teacher : “What should we do for them?” Arsyad : “We have to be divided again” Teacher : “Yes, right…”

The student confuse and they didn’t see the similarity of this problem with last problem Arsyad : “It’s ¾ , it cannot be

divide…

The teacher ask the student to compare this problem with last problem Teacher : “Of course it can… like what you did it yesterday” Iman : “Yesterday is easier” Arsyad : “Is it 50 days?” Teacher : “How can you get it?”

Arsyad suddently recounted and found the result. Arsyad : “33 days? But it’s plus 1

day”

Teacher asked again to students and gave compliment. Teacher : “How can you get it?” Arsyad : “33 days? But it still be left over as 1” Teacher : “So, what should we do for 1 Part (it means ¼ kg)?” Arsyad : “Is it divided by 4? Means ¼?” Teacher : “Yes Arsyad, good… you are right”

This is the strategy that is found by group B

Teacher asked for a student to write down the conclusion and present their result of a group in front of class.


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In the second lesson, the strategies of group A is drawing drawing one by one square till 25 squares, 1 square is representation of 1 kg of rice. Each square divided by 4, students shade 3 parts of four (it mean ¾ kg). Then they take 1 part from each square, so that they got 25 days + 8 days. The rest is 1 part (=1/4 kg). Total is 33 days, rest ¼ kg. The students in group A didn’t get some difficulty to look for the pattern. But they didn’t know ¼ kg rest for how many days. Teacher effort for group A is asking the meaning of the picture, clarifying/confirmation the students’ answer, and give the reinforcement/ compliment for student.

The strategies of group B is the same with group A, drawing the picture and using the pattern to solve the problem. As draw in last lesson, student in group B drawing big rectangular and divide it to 25 parts, each part is divided by 4. Total part is 100, they said for 100 days. Teacher ask the student to look back to examine the students’ answer. The student said “25 days”. And then the student put different color for each rest part in blue color (it mean ¼ kg). They got by them self the correct answer is 33 days and left ¼ kg. For students in group B the teachers’ effort are asks the students to recognize the representation that student made, compare this problem to last problem, clarifying/confirmation the students’ answer, and give the reinforcement/compliment for student.

Both of group A and group B knows about the rice will be finish for 33 days and there is ¼ kg left. They didn’t know for how many days ¼ kg of rice.

Discussion To solve the fraction problem, both of group of students start with calculation, for

instance 20:15, 15:20, 25 x ¾, they said ‘can 15 divided by 20?, 15 multiply by what number to get 20?, 5 x 3 is 15, it’s still less than 20’. It means, the student known mathematics problem is just calculate, using mathematics operation. They didn’t realize mathematics as a human activity that (Gravemeijer (1994). In this case, teacher plays the important role to help the student make sense of mathematics. Teacher ask the matacognition question for the student, for example ‘What do you want to do to solve the

Asryad representative of group B was

presenting their strategy on poster for

second lesson


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problem?’. This question help students reflect on their current understanding of a problem (Schoenfeld in Zawojewski and Lesh, 2003).

In the first lesson, group B spent their time in the beginning to trial and error to fold one piece of paper. The teacher knows the student didn’t clear about the problem and the student difficult to represent 15 kg of meats for 20 orphans. So that, the teacher asks some similar problem, ‘how about 1 kg of sugar divided to 5 small bags’, the student give the right answer ‘1/5’. How about ‘15 cakes divided to 20 friends, the student give answer as follow:

Risqan : 15 divided by 20 Arsyad : 20:15 Furqan : “It will get half and a half…” Teacher : “Which one is a half? Arsyad : “Can I write it down, to make it clear ?” Teacher : “Yes, you may write everything, every shape, like a circle or anything,

it’s up to you” Based on the dialog at above, teacher guidance the student to look for a similar

problem. It is one strategy of solving problem (Polya and Pasmep in Shadiq, 2004; Sobel, and Maletsky, 2004; Reys, 2003; Zawojewski and Lesh, 2003).

When the students are getting stuck, the teacher gives motivation for the students,

for instance ‘keep spirit’, ‘come on….. keep trying’, ‘I believe you can find the answer’. The teacher also offers to student do anything with making a picture or calculate. This ways will impact to student to get a new strategy.

While the primary school students in grade 4 are solving fraction problem, they need stimulate from the teacher to give them some support to star solving the problem. Some time they call the teacher to monitor their solution to get comment. When the student confuse, the teacher ask the students to understand the problem, recognize the representation that student made, compare this problem to last problem, and clarifying/confirmation the students’ answer. Teachers give motivation for the student in reinforcement/compliment ways.

Drawing a picture and looking the pattern is one of strategy can implement to solve the fraction problem for primary school students in grade 4, because they didn’t learn yet about the multiplication of fraction and division of fraction.

There is a little interaction between student-student in solving the problem, because sometime they ask teacher to monitor and give them comment, not discuss it with their friend in group. The effort of teacher to provoke interaction between student-student in solving the problem is not maximal, because sometimes the teacher did not ask them to discuss each other in group.

References


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Dolk, M., Widjaya, W., Zonneveld, E. & Fauzan, A. (2010). Examining Teachers’ Role in Relation to Their Beliefs and Expectations about Students’ Thinking in Design Research. In in Sembiring, R. K., Hoogland, K., & Dolk, M., (Eds), A Decade of PMRI in Indonesia, Bandung, Utrecht: APS International

Gravemeijer, K.P.E. (1994). Developing Realistic Mathematics Education. Technipress:

Culemborg, Netherland. Gravemeijer, K.P.E and Cobb, P. (2006). Design Research from a Learning Design

Perspective. In Dekker, van den, Gravemeijer, K., Mc Kenny, S., & Nieven, N. (Eds). Educational Design Research (pp. 17-51). London: Rontledge.

Gravemeijer, K.P.E (2010). Realistic Mathematics Education Theory as a Guideline for

Problem-Centered, Interactive Mathematics Education. In Sembiring, R. K., Hoogland, K., & Dolk, M., (Eds), A Decade of PMRI in Indonesia, Bandung, Utrecht: APS International

Horn, J-S., Hong, J-C., Lin, L-J.C., Chang, S-H., & Chu, H-C. (2005) Creative Teachers

and Creative Teaching Strategies. International Journal of Consumer Studies, 29(4), 352-358.

Johnson, Elaine. B. (2002). Contextual Teaching and Learning: What it is and Why It’s

Here to Stay. USA : Corwin Press. Kolovou, A., Van den Heuvel-Panhuizen, M., & Bakker, A. (2009) Non-Routine Problem

Solving Tasks in Primary School Mathematics Textbooks – A Needle in a Haystack. Mediterranean Journal for Research in Mathematics Education 8 (2), 31-69.

National Council of Teachers of Mathematics (2000). Principles and Standards for School

Mathematics. Reston, VA: Author Polya, G. (1973) How to Solve It. Princeton NJ: Princeton University Press. Reys, R.E., Lindquist, M.M., Lambdin, D. V., Smith, N. L. (2007) Helping Children

Learn Mathematics. USA: John Wiley $ Sons Inc.

Schoenfeld, A. H. (1983). The Wild, Wild, Wild, Wild, Wild World of Problem Solving (a Riview of Sorts). An International Journal of Mathematics 3 (3), 40-47

Shadiq, Fajar (2005), Pemecahan Masalah Dalam Pembelajaran Matematika, PPPG Matematika, Yogyakarta, Indonesia


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Van den Heuvel-Panhuizen, M. (1996). Assesment Realistic Mathematics Education. Utrecht, Netherland: Freudental Institute

Zawojewski, J. S. & Lesh, R. (2003) A Models and Modeling Perspective on Problem

Solving. In Laesh, R., & Doer, H. M. (Eds). Models and Modeling Perspectives on Mathematics Problem Solving, Learning, and Teaching (pp. 317-336). Mahwah: Lawrence Erlbaum Associates, Inc Publishes.

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