One of the important aims of teaching mathematics is to prepare children to meet the mathematical problem of everyday life. That is why we need to make link between the mathematics problem and their real-world. The students need appropriate context situation in order allows them to see the connection of school mathematics to situations and contexts that they found outside the classroom.

The real contextual situation should engage their interest that influences motivation of the children so that they can follow their own direction. In this way children can develop their own understanding toward the problem. The real context suggests that children familiar with the situation in the context and also they have experience about that.

The contextual situation should give mathematical conflict for the students. In that context students can learn through their effort, when they do not know the answer they have to think about that. Therefore, they can learn deeply.  When the students have to find how many marbles are obtain by each child if they have to fair share 3 chocolate for 4 children, they have to think deeply because they have to give reason how they divide the chocolate. It could be many different strategies to find the answer based on the level of understanding of each student. It could provoke cognitive conflict for them because they have to give the reason to convince their selves about their answer.

If the children interest with the problem by connecting the problem with their experience then they can be more understand about the problems. There are many chances that student success in mathematics because the problem can provoke cognitive conflict of the students.

A.  Introduction

Fraction is taught in school as an important subject to understand the next topic in mathematics.  Many students are familiar with fraction as a notation of number consisting of numerator and denominator. For the students as the learner, definition of basic concepts of fraction has to be frame with care because we find many cases related to fraction in our life.  But sometimes they have difficulties in understanding what the meaning of fraction is. They only use the procedural algorithm of fraction that they learn from school without understand what the meaning of fraction itself. When asked to add two fractions and get an integer answer, they added the numerators or the denominators of the two fractions.

To introduce the meaning of fraction we must also investigate the history of the fraction. Given that emphasize, this is our way to teach mathematics inspired by history (based on ICMI study). Therefore, in this case we focus on constructing the teaching and learning process in the class by introducing the students how human develop and learn about fraction. Introducing fraction with the history leads the students understand what the basic concept of fraction is.

Since the students do not recognize the meaning of fraction, it will be introduced through the context that is familiar with them and connect with the history of fraction. In this paper we will use the Egyptian system as starting point to understand the concept of fraction.

In this paper, we will implement the introducing of fraction inspired by The Egyptian System using the following trajectories:

B. History of Fraction

Fractional word actually comes from the Latin “fractio” which means break. To understand how the fraction has evolved into a form that we know today, we must step back further to historical time to find out the process of notation of fraction.The history of fraction passed through long time and long journey. There were several notation fractions evolved simultaneously with the finding of number systems in historical times.

1.  Egyptian numeral hieroglyphics

In 1800 BC, the Egyptians were writing fractions. Their number system is base 10, so they have separate symbols for 1, 10, 100, 1000, 10000, 100000, and 1000000. Writing system of ancient Egypt like in the following picture is called hieroglyphics:

The Egyptians wrote all the fraction using fractional units (fraction which has 1 as a numerator). They put a picture of the mouth (which means the part of) above the number. For example figure 3 represented the fifth.

They stated other fragments as the number of fractional units, but they are not allowed to repeat a fractional unit on this side. In other words, if we interpret to our number system today is like in the following:

In Egyptian systems, 2/3 or any unit fraction (fraction with one as the numerator, like 1/7) were expressed in a simple, straightforward way. 1/2 had a sign of its own ( ), as did 2/3 ( ). And the other unit fractions were just the symbol (meaning “part”), with the denominator expressed as an integer, under this symbol. I drew the “r” after the symbol, to show that it is pronounced “r”. 1/7 would be that symbol with seven, small vertical lines under it ( ).

The big disadvantage from Egypt system is that it was very difficult to calculate. To try to overcome this, the Egyptians made a lot of tables, so they can find the answers to these problems.

2. Ancient Rome

In Ancient Rome, fraction is only written using words to explain part of the whole. They are based on the unit of weight which was called “as” in which one “as” was made up of 12 uncia. Therefore, the fraction in that time was centered on twelfths. For example:

112 was called uncia (means 1/12)

124 was called semuncia (means 1/12)

1288 was called scripulum (means 1/288)

However, those words make it very difficult to perform calculations.

3. The Babylonians

Beside of that, there were the other people, The Babylonians, who came up with a more reasonable way to represent fractions. They did it before the Roman method. Their number system based 60. Babylonians extended only to enter their number fractions in sixtieths, as we did for the tenth, hundred etc. However, they do not have the kind of zero or a decimal point. This makes it very confusing because they can be interpreted in different ways. They also made a group of ten.

Here is an example:

Babylonian number 12 and 15

The Babylonians has also expressed the notation of fraction in their system. For example they expressed the fraction which is interpreted like in the following table:

Sixtieths X60 Unit Number

Although the Babylonians had a very sophisticated way of writing fractions, it contains the deficiency also. Around 311BC they designed Zero. It made number system become easier. However, they still didn’t recognize about a decimal point, so it was still difficult to distinguish the fraction and integers.

4. India

The format which we know today is directly derived from the work of Indian civilization. The success is the way to write fractions as numbers which have three main ideas:

i)     Each number has the symbol,

ii)   The number depends on the position of digit in whole number,

iii) Zero is no need to interpret and also to fill the place of the missing units

In a fraction of India was written very similar to what we do now, with one number (the numerator) and the other above the other (denominator), but without the line. For example:

5. Arab

It was the Arabs who add the line (and sometimes pulled horizontally, sometimes slash) which we now use to separate the numerator and denominator: 3/4.
So here we have the pieces as we now recognize. It’s amazing to think how much thought has been entered into the way we write.

C. The Implementation of Teaching and Learning

Because the characteristic in our paper (based on ICMI study) is about learning mathematical topics inspired by history, therefore we use the implementation of experiental mathematical activities. An experimental mathematical activity used in learning activity is a method activity. In this case, students are asked to solve the problems in contextual world, the problem about dividing chocolates equally. This problem is similar to the Egyptian system in which they divided food equally. We choose the Egyptian system as a method to introduce fraction concept. We make the lesson plan which will be implemented by teacher in grade 3 primary school. We use the basic concept of fraction implemented through the context. The teaching and learning process was started with a little explanation about The Egyptian System whereas in 3000 BC Egyptians have an interesting way to represent fractions. The teacher also shows to the students the number system of Egyptian System in historical times (like we mentioned on the previous page).

The Egyptian System has a notation for 1/2, 1/3, 1/4 and so on. This is called a unit fraction. In Egyptian System, it does not allow them to write 2/5 or 3/4 or 4/7 as we see today.
Instead, they can write each fraction as the sum of the fractional unit where all the fractional units in a different form.

For example,

3/4 = 1/2 + 1/4

6/7 = 1/2 + 1/3 + 1/42

After the students are introduced with The Egyptian System, the teacher asks students to involve in the mathematical activities such in the following:

1. The teacher brings 5 chocolates. She said that she will share those chocolates to 8 people. She asks the students to find the strategies in sharing 5 chocolates for 8 people equally without using calculator.

2. Our prediction is the students will add together and then divide it become 8 parts. Of course it will be difficult if they don’t have the precise measurement tools. Therefore, they have to find another way which is easier until they find the unit fraction such in the following:

• First of all they divide each chocolate become two equal parts (making half of each)
• When the students come up into the fifth chocolate, we hope the students do the same thing i.e dividing each part become two equal parts until they get 8 equal parts

4. From those activities, we hope the students can determine which one is the larger between 5 of 8 and 4 of 7.

Sometimes if we ask children in addition and subtraction problems such as 3+4 and 6-2, they do not know what they should do to answer because those are no meaning for them. But if we ask them problem based on their experience, they may know how to solve that. We can illustrate the problem in which familiar with them so that they can construct the idea into formal level and solve the problem in their own way. They model the situation and then think the strategies. Model means the representation on how one change thing to another form as a tool to solve the problem. To support children develop mathematical model they should have situation that intrigue them.

One of situation that familiar with children is bus context that has the potential to encourage children to model addition and subtraction. In real world passenger getting on and getting off the bus so teacher can ask children what is going on in the bus and then go into mathematical situation, how many passenger in the bus. Teacher can involve children by using bus game. One child can play role as the driver and he has to count how many passenger in the bus when the bus stopped in each bus stop.

While children engage in situation, they have experience the situation of adding and removing people. They realize when passengers get on it means they need to count on all the people on the bus; passenger on the bus and new entering passengers. For example when the bus stopped with two passengers in its then three passenger get on the bus, they may count on the passenger one by one or count it as two and three become five. And then they gradually come up with their own models of the situation. They can model the situation by using picture.

Commonly they draw the bus and the bus stop and number representing the number of people. At first children draw all objects close to real action. Slowly they understand that not all in the picture need to draw. Then they represent the situation which people get on and get off the bus by using boxes and arrows. The boxes represent the number of people and arrows represent the action of operation. By modeling the situation helps children grasp an understanding of the operation of addition and subtraction.

The developmental process of children by modeling the picture into boxes and arrow shows that they generalize the specific situation into many situations. Teacher can help children to generalize the situation by using number line. Number line is used as a tool to generalize all addition and subtraction problems. The arrow in model of situation can be replaced by a leap and the number in the boxes can be conceived in a line. Open number line is used to avoid confusion whether the intervals or the marks denoting the intervals are being counted on a number line. By using number line children move from real situation to formal level.

Involving children in situation is important to bring them in constructing model to solve the problem. When children can model the situation then teacher can use this model to bridge learning from informal level to more formal level, moving from situation to model of situation and then go to model for all situations. Especially in addition and subtraction, children may use their experience in bus later move into model that they have constructed then go to more formal level by using number line.

Setelah melalui proses yang panjang dan melalui beberapa tahap seleksi diputuskanlah 13 orang mahasiswa dari Universitas Sriwijaya (UNSRI) dan Universitas Negeri Surabaya (UNESA) untuk berangkat ke Belanda. Mereka akan melanjutkan program IMPoME di Utrecht University. Pada tanggal 5 Februari 2010, 7 mahasiswa UNSRI dan 6 mahasiswa UNESA dipertemukan di DIKTI sekaligus mengikuti pelepasan oleh pejabat DIKTI dan Direktur NESO.

Pada tanggal 6 Februari 2010 mereka disambut oleh pihak Freudenthal Institute dan diperkenalkan dengan lingkungan kampus, dosen-dosen dan staf-staf Freudental Institute. Kegiatan minggu pertama di Utrecht adalah mengikuti workshop bersama 5 orang mahasiswa Ph.d, yaitu Fenna van Nes tentang Spatial ability and number sense (young children), Aldine Aaten tentang Picture books and concept development in mathematics, Angeliki Kolovou tentang Problem solving in Primary school, Marjolijn Peltenburg tentang Mathematics for special needs children,  dan Mirte van Galen tentang children’s learning of arithmetic facts. Kemudian mengunjungi salah satu sekolah di Utrecht yaitu Paulusschool.

Di sekolah ini mereka melakukan observasi yang dipandu oleh Fenna van Nes. Berikut ini beberapa media pembelajaran matematika yang digunakan di sekolah ini.

There are many problems related to vision geometry in our daily life and I think it can bring children to find out how the world around them works and explore it extensively. We give them the use of vision lines with their thumb through the following experiment.

Stretch your right arm, put up your thumb and look with one eye. Let your thumb coincide with a fixed vertical line on the blackboard. Now look with your other eye. What happens with your thumb?

By making a schematic drawing from above, it become clear why the thumb shifts to the right when we switch eyes.

In this way, the children get acquainted with natural phenomenon, some basic geometric concept. They get a grasp on fundamental structuring principle, in this case the principles of projection. Also, they will consider that looking must be regarded as making a straight connection line from eye to the object. Another orientation aspect has to do with taking point of view. When they look at the thumb with their right eye then switch the eye, their thumb making a move to another side. This phenomenon show that a change point of view influences the direction in which you see something.  By introducing students with geometric phenomena and by letting them solve simple geometry problems, they can learn better understanding their surrounding world.

Exploring the concept of this part of geometry, the teachers are expected to organize a wide variety of geometric activities. Not all the children interested in the geometry subject. Therefore these differences within introducing geometry to the children will bring them interesting way to learn geometry. The children are curious, want to know how the world around them works. The teacher should take the advantage of that by making a challenge for students to learn geometry as a subject that closed to their everyday life. As I mentioned before geometry is part of human activities. We look all kind of shapes and phenomena in the surrounding of our activities. Perception in geometry contributes to the development of spatial insight that is important aspect of our intellectual capability.

The first day, Jaap talked about the learning trajectory of proportion and ratio for grade 4, 5 and 6. It is important that students recognize situations where proportions are used. Also, we discussed about the essential of the ratio table in calculating with proportions. Aad talked about geometry which is connected with visualizing space on paper. He gave us a problem about seeing things the intuitive way. We used the idea of vision line to explain the landscape problems such as why we will lose sight of high buildings if we approach those buildings. We also learned about the formal perspective with side view and top view.

The second day we continued our lesson about vision line with Aad. We focused on the way student think about visualizing space on paper. For didactical part, we discussed about article by Parmjit Singh, Understanding the Concepts of Proportion and Ratio Constructed by Two Grade Six Students. This article investigated the ways in which the students began to develop proportional thinking by compares how two students construct proportional reasoning. It is essential to determine what enables students to make a transition from solving ratio problem by iterating composite ratio units to using multiplication and division.

The third day, we did some activities with Aad to show perspective view by draw object with vision line and compare the original and the sketch. We also tried the classical method with side- and top- view by transfer the image screen to side- and top- view. With Jaap, we did some activities to differentiate proportion and ratio by using part of our body.

Although these activities were just three days, we have got many new experiences and new knowledge from those.

This observation report will tell about the process of teaching and learning using realistic mathematics approach at first grade at MIN 1 Palembang. This observation was conducted on Thursday, July 23^rd, 2009. It was the first observation at the beginning of the semester. Based on the curriculum, the teacher taught number subject to introduce number up to 10. The week before, the teacher and I discussed about the lesson plan and the context that we would use. And we decided to use number cards and candle in context birthday party.

At the beginning of the lesson, the teacher told about birthday party to make students interest. The teacher asked students to tell about their birthday party. One by one told about their birthday party and told about whatever they found in birthday party such as balloon, gift, cake, candle and clown. Students seem enthusiastic when they told about their birthday party. Then the teacher showed a birthday party picture and asked a student to tell about the picture in front of class and the student count anything that he or she found in the picture.

After count things in the picture, the teacher put some candles in front of class then asked students if they are 6 years old, how many candles have to put on the cake. Teacher asked students to go to the front of the class to put a number of candles needed. Then teacher together with students re-count the candles and then asked another student to do this activity. By doing this activity, students are expected to develop an understanding of the count.

Furthermore, the teacher gave a set of number cards, number 1 to 10, and a sheet of picture to pairs of students. Then they were asked to raise the suitable number card while the teacher appoints group of things on the picture. After have a game of number cards, students were asked to match a picture with the number by make a line to connect the picture and the number.

By using realistic approach, the students are expected to understand mathematics by doing it. These activities make students able to count and know the number by count things that they found in their activities.

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A rope is divided into twelve spaces by make a sign in each space of the rope. Then students arrange this rope to make many kinds of plane figures and doing various activities by using this rope. It is better if student work in pair in this activity.

Activities of students:

1. Make a square and determine the dimension.

This activity makes students think about characteristic of rectangle that it has sides with the same length.

2. Make various rectangles in various dimensions

This activity makes students think about how many possibilities to make rectangle by using twelve spaces of the rope with possibilities dimensions of rectangle.

3. Make various triangles

This activity makes student think about kinds of triangles made from twelve spaces of the rope and the possibility to make different triangle.

Finding Prize

There are many ways to make student interesting in mathematics lesson. One of them is make a problem close to their real life. Like in realistic mathematics, the problems are given close to their real life, so it can make students easy to understand the problems.

One of the problems is finding prize,

The first problem

Then the students answer the answer sheet

1. Without knowing prize each of stuff, determine which one is more expensive? Give your reason!
2. How many calculators can be bought by Rp.50.000? Give your reason!
3. How much the prize of one glasses? Give your reason!

The second problem

1. Which one is more expensive, an umbrella or hat? Explain!
2. How much the prize of three umbrellas? Three hats?
3. How much the prize of an umbrella and a hat?

The purpose of this activity is to build student’s understanding about elimination concept. Based on student’s answer seem that each student has their own strategy to solve this problem. Even sometimes the teacher does not think the alternative answer given by students. Students can express their own opinions and not glued on one-way solution. Various answers indicate that students free to give their answer and maintain it by give the reason, so their ability to communicate can be increased.

Classroom Observation Report

Mathematics classroom observations require all of the students of International Master Program in Mathematics Education to observe the elementary schools in Palembang. The Master Program’s students are divided into 4 groups to 4 different schools, there are SDN 98, SDN 117, MIN 1 and MIN 2 Palembang, so that one of is our group, which is going to MIN 1 Palembang.

School’s Profile

School Name : MIN 1 Palembang

Address : Jl. Jenderal Sudirman, Km. 4 Palembang-Indonesia

Phone Number : 0711-360115

Head Master : Dra. Rasunah A. Manan, MM

The Number of Teachers : 40 Teachers

Class Involving in PMRI : First grades and Second Grades

The first classroom observation was on May 7, 2009. It is in MIN 1 Palembang. Since 2004 MIN 1 Palembang has been joining with PMRI and involved in PMRI activity. In 2006, it has delegated 2 teachers to following training and workshop PMRI. They are Mrs. Tartilah and Mrs. Aida. This training held for 3 days in Bandung. Then in 2007 MIN 1 return to delegated teachers to following training and workshop PMRI in Yogyakarta. They are Mrs. Anisa and Mrs. Risma.

The aim of first observation is to introduce the new members of PMRI team in South Sumatera to the school and the teachers who will be invited to cooperate about PMRI Project. Beside that, observing how teaching and learning process in class, especially in first grades, second grades and third grades. However class observation couldn’t be done because of our late arrival in that time, it’s about at 11 AM, the classroom activity of first grades students has already finished and the other classes there were no mathematics classes in Thursday.

After introduce ourselves, the conversation was continued to schedule observation with Mrs. Tartilah the first grades teacher, Mrs. Annisa the second grades teacher and the last Mrs. Risma the third grades teacher. They said that from 11^th until 17^th May sixth grades following national examination and the school will have its examination for students between first grades to fifth grades, starting from 1^st of June 2009, so that there no more classrooms activity. We couldn’t do any classroom observation until the beginning of next semester. Because of the condition, so that’s better to have many discussions with the teachers to prepare for the materials and media for the new semester starting on July.

Discussion continued to mathematics teaching-learning process in class. Mrs. Tartilah told that the learning process in mathematics class had using of PMRI approach. But she got problems to find the suitable media that can be used and got hardship to develop new idea to make media. During this time, media that she uses such as straws, ice cream sticks in teaching addition and multiplication of number and anything surrounding students like whiteboard, windows and door in teaching geometry. Teachers expected that they have other media to teach mathematics concepts.

The implementation of PMRI in class had been making a good progress in students’ understanding. In the first of learning process some students shy to work in team with their friends and afraid to give their argument but finally they enjoy the activity in the class. She also told us about the text book that’s used. It’s from Ganeca Exact publisher. That’s not a problem if they used another book, but the most important things is the PMRI implementation.

In the end of our conversation with Mrs. Tartilah said that the school’s holidays will be begun on May 11, 2009 until May 21, 2009. Before left the school, the teacher said that they want to have a discussion on May 28, 2009 to discussing about our preparation for the next semester.