The program will be carried out for a period of 2 years and 2 months which is divided into 3 stages, namely 8 months in Indonesia (Unesa or Unsri), 1 year in the Netherlands (University of Utrecht), and 6 months for research and thesis writing in Indonesia.

The scholarship for the period of studying in Indonesia for the Lecturers or CTAB (not including teachers) will be provided by Ditjen Pendidikan Tinggi Depdiknas (BPPS) while the period in the Netherlands will be funded by StuNed.

The eligibility of candidates are as follows:
• a citizen of Indonesia;
• minimum education is S1 in the field of mathematics education or mathematics with a Cumulative Index of
Achievement or I.P.K. of at least 3.0;
• minimum 2 years work experience in the field of mathematics or has been
formally appointed by the rector as a lecturer in the institution as a CTAB;
• being ready to join and finish all classes during the period of scholarship;
• has TOEFL score (ITP TOEFL is preferable) at least 475;
• being in healthy condition;
• not in the process of studying abroad in the period of the last 2 years;
• on the 1st of December 2010, the candidate’s age is no more than 40 years old for men, and 45 years old for women.

Deadline: The last date for registration: 31th of December 2009.

Download the Form at the website : www,pmri.or.id and www.p4mri.net.

All of the completed documents should be sent to:
Prof. R.K. Sembiring, IP-PMRI, Gedung Basic Sciences Center A ITB, Jln. Ganesha 10 Bandung 40132. or by e-mail (zulkardi@unsri.ac.id or metricas@bdg.centrin.net.id)

The announcement will be posted on the website www.pmri.or.id and all participants who are qualified will be directly contacted. The commencement of classes will be in May 2010 which is initiated by an intensive course of English. In case the candidates cannot fulfill all of the specified requirements to go to the Netherlands, they may continue their regular S2 studies at Unesa or Unsri.

Contact Person (sms): Zulkard Unsri(+628127106777), Agung Lukito Unesa (+628165428611) and Martha IP-PMRI (+628164202261)

]]> http://zetrahp.wordpress.com/2009/10/20/international-master-program-on-mathematics-education-sholarship/feed/ 0 zetrahp http://zetrahp.wordpress.com/2009/09/14/150/ http://zetrahp.wordpress.com/2009/09/14/150/#comments Mon, 14 Sep 2009 15:45:51 +0000 Zetra http://zetrahp.wordpress.com/2009/09/14/150/ ]]> Two professors, Jaap den Hertog and Aad Godjjin from Utrecht University, The Netherlands, gave lectures to IMPoME students in Sriwijaya University at 7th to 9th September 2009. They taught about Didactic (Teaching-Learning Trajectory) and Geometry. The students studied from 08.00 am to 17.00 pm like students in Netherland because some of them will study there next year about realistic mathematics education. This class was not only for studying mathematics but also to selecting some candidates who will get experience in Frudental Institute.

In the first day, Jaap taught students about Fractions, Percentages, Decimals and Proportions (A Learning-Teaching Trajectory for Grade 4,5 and 6). The Students followed course very enthusiastically in spite of fasting month. They made interesting discussion and also student activity in group working. In the second day, They studied about Understanding the concepts of proportion and ratio constructed by two grade six students (Parmjith Singh). IMPoME students were asked to analyze two student answers in grade six. How Students could solve proportion problems by using their thinking. In the third day, Jaap taught about Proportional reasoning among 7th grades students with different curricular experiences.

In geometry part, Aad taught students about vision line which is a new lesson for them. In the first day, Aad asked students to analyze two photos in different view. It was interesting lesson because students could look something in different vision. In the second day, They studied about building with cubic bloks which looked in different view as side- and top-view. Students had to find the original building. In the last day, students studied about Perspective drawing: central projection from space to plane. They also did activity outdoor, and it was first activity done in Sriwijaya University. All students were very enjoyable and satisfied to study with Jaad and Aad. They hoped that they could meet them in Utrecht University in Netherland. See U later.

1. INTRODUCTION
New curricula and teaching strategy have been developed for the topics of middle school mathematics. The Connected Mathematics Project (CMP) is one of the new middle grades curriculum projects to develop a complete mathematics curriculum with teacher support materials for grades six, seven, and eight. This curriculum is structured to develop students’ knowledge and understanding of mathematics that is organized around interesting problem settings. Students solve problems and observe patterns and relationships. The CMP curriculum treats the major areas of mathematics like number, geometry, measurement, algebra, statistics, and probability. In the seventh grade, three units deal with the ideas of similarity, ratio, and proportions and their applications. The approach in those units is to encourage students to construct their own procedures for doing rational number computations, solving proportions, and applying those skills to applied problem solving. Students collaborate in work on the problems, sharing through mathematical reflections, discussion, and journal writing.

In a traditional curriculum, each arithmetic operation is taught with a focus on developing student proficiency in well-defined computation algorithms to ensure speed and accuracy of execution. The text material provides problems and the teacher demonstrate the solutions to sample of problems to the students. After that, students solve problems according to the given format.

There are three focuses in this research comparing between traditional and CMP curriculum:
a. How do the conceptual understandings, computational skills, and              problem solving strategies and success of CMP and traditional                    curriculum students compare?
b. Does the new CMP approach do successfully lead students to                        construct effective strategies for fraction, decimal, percent, ratio,            and proportional computation?
c. Do CMP students develop flexible and/or effective strategies for               solving contextual problems involving rational numbers and                       proportions?

2. PROPORTIONAL REASONING
A mathematician said that a proportion is a statement of the equality of two ratios. The students’ ability to reason proportionally develop through-out grades 5-8. Following ideas of Frudenthal (1978,1983), proportional reasoning problems can be described in three broad categories: a) comparing two part of a single whole; b) comparing magnitudes of different quantities with an interesting connection; and c) comparing magnitudes of two quantities. Three types of tasks are reported in the literature for assessing proportional reasoning: a) Missing value problems; b) Numerical comparison problems; c) Qualitative prediction and comparison problems. Freudenthal (1978,1983) has pointed out that missing value and comparison proportion problems can be solved by three distinguishable approaches related to: a) Internal ratio (within a magnitude); b) external ratio (between two magnitudes); c) refraining from computation until the result has been found formally.

3. PURPOSE OF THE STUDY
The goal of this study is to describe the character and effectiveness of proportional reasoning by students with different curricular experiences. The main purpose was to compare CMP and traditional curriculum students in American middle schools about how they proportional reasoning.

4. METHOD
The data was gathered during the first school years (1994-1995) in which CMP sample consisted of students from eight seventh grade classes and seven difference teachers in some areas in America. The control sample consisted of students from six seventh grade and six difference teachers also in America. The equivalence indication of the two samples was given by the standardized test results reported by evaluation team (CMP and Control). These results show that the Control student’s scores were slightly higher than the CMP students at the beginning of the year, and slightly lower at the end of the year. 187 students were in the CMP sample and 128 students in the Control sample.

Both samples were tested on variety of proportional reasoning problems and were distributed randomly in each participating class. Researchers interviewed about 25% students to gain more knowledge about their understanding. There are five rate of density problems that did in both samples about numerical comparison, missing value problem, integer number structural, fraction and structural number, and population density information. There are three major categories that were identified: Correct answer (correct answer, correct answer with correct support work, and correct answer with incorrect support work), Incorrect answer (incorrect answer, incorrect answer with partial understanding, and incorrect thinking), and no response.

5. RESULT AND DISCUSSION
Most of the students from both samples responded to the problems with support work. Analysis of the written support showed that the CMP students demonstrated much more proficiency in writing than Control students. CMP students were more frequently instructed to write and talk about the idea. Incorrect answer with partial understanding often occurred when the problem called for calculations followed by reasoning.

- Comparing students’ performance on rate problems 1 and 2
The first two rate problems (a numerical comparison and finding a missing value) required the same knowledge of rate, relating to amount and cost. The performance on the missing value problem was better than on the numerical comparison. The missing value problem may be easier because it consisted of two identical parts, each less complex than the numerical comparison problem. The CMP students performed well on the missing value problem because there is possibility of effects o their performance on the comparison problem. One’s evidence is the high percentage of CMP students using unit rate for the comparison. The CMP students performed significantly better than the Control students on the first two rate items.

• A mini case study
There were nine different strategies based on analysis of the problem task and students’ written test papers and individual interviews for solving the numerical comparison rate problem:
1. Comparing the ratio of two different variables using external ratio or a functional method.
2. Comparing ratios of the same variable using internal ratios or a scalar method.
3. Comparing the cost of the some quantity by finding common factor or common multiple quantities such as price per unit.
4. Comparing amounts for the some cost by finding common factor or common multiple costs such as unit per price.
5. Building up strategy.
6. Looking at ratios of differences between the same variables.
7. Responding to the numbers but not the context of a given problem
8. Relating to only one variable by ignoring part of the data in the problem.
9. Affective responses to numerical data and question.

- Comments on rate problem 2
Analysis of students’ responses to the second rate problem also showed use of a variety of strategies to solve the problem. There were still many students who manipulated numbers in ways that did not reflect the structure of the given information or the question. All students from both samples applied the same strategies for both parts of the problem. Very few were successful on one part and not on the other. The students’ thinking is essentially additive when they build up the smaller quantity to the larger one in equal increments.

- Comparing students’ performance on rate problems 3 and 4
Both of this numerical comparison’ problems require the same knowledge of rate relating to distance and time. The second problem was more difficult because it contains both decimal and fraction numbers for the time. Most of the incorrect answers for both samples stem from confusion between distance per unit of time and time per unit of distance. Teachers should be well-advised to use a variety of time and distance units. The CMP students performed better than Control students on both of the distance/time rate items. The CMP students used the unit rate strategies twice as often as the Control students. The superior performance of CMP students is a result of the problem solving approach presented by the CMP curriculum where students construct personal understanding of key mathematical structures. The CMP students are more familiar with this context due to the CMP problem based curriculum.

- Comments on rate problem 5
The study included one other rate problem about density of feral cat populations in two cities. In this problem, a numerical comparison is required, and integers are used in spite of larger density problem. The performance on the density problem for both sample decreased drastically. The causes are not familiar factor and the real activity large integer numbers used in the density problem.

6. CONCLUTION
The main purpose of this study was to compare proportional reasoning of seventh grade students with CMP curriculum and Control curriculum. This report has focused on numerical comparison and missing value proportion problem with several different contextual settings and number structure. These two samples were tested with a written exam and interviewed some students randomly. The results, The CMP students clearly outperformed the Control students on the collection of tasks and on each individual rate problem. Reforming curriculum students were capable of providing a good quality of written and oral explanations to their work. Both reform and traditional curriculum students have a long way to attain mastery of basic proportional reasoning strategy and skills.

The CMP students competed very successfully with the Control students and also demonstrated ability to develop a variety of strategies. The CMP students are encouraged to construct their own understanding and procedures for doing rational number computation, solving proportions, and applying those skills to applied authentic problem solving tasks. There are more factors that could affect CMP student performance. First, the CMP curriculum is connectivity between the units, an important factor for acquiring knowledge in the rational number domain. Second factor is the level of CMP teachers’ preparation and background versus the traditional teachers.

7. IMPLICATIONS FOR TEACHING
Researchers identified many demonstrated strengths and many difficulties encountered by students when they were dealing with proportional reasoning concepts and their applications. They also found evidence confirming conjectures about the effects of number structure and context familiarity on task difficulty. Both the complexity of numbers and the contexts should be more manipulated by curriculum materials and instruction. All teachers concern about teaching the complicated subject matter included under the umbrella of proportional reasoning. Freudenthal (1983) has recommended that learning process of ratio and proportionality must be steered in such a way that sources of insight are not clogged during the process. The varieties of strategies and ways students find to solve problems are encouraging. Students are capable of developing their own repertoire of sense-making tools to help them to produce creative solutions and explanations.

UNDERSTANDING THE CONCEPTS OF PROPORTION AND RATIO CONSTRUCTED BY TWO GRADE SIX STUDENTS

Parmjith Singh

BACKGROUND

There are many students having problems with the basic concepts of fractions, rates and proportion and with problems involving these concepts. Students have difficulty reasoning multiplicatively that uses in proportion problems. They often use additive reasoning in solving a problems where multiplicative reasoning is required. This research tried to find ways to help students reason proportionally. They have a problem to identify a litany of task variables that affect problem difficulty. It is important to analyze mathematical structures and children’s solution processes in developmental precursors to the knowledge needed to function competently in a domain.

MULTIPLICATIVE SCHEMES IN PROPORTIONAL REASONING

To determine what knowledge was critical for understanding ratio and proportion is the goal of this analysis. There are two main issues to be discussed about multiplicative schemes and the nature of students’ proportional reasoning. There are three researches studying about multiplicative structure. First, Vergnaud (1983) researched about building multiplicative structure as consisting in simple and more complex situation of proportion. The analysis of students work and teaching situation are his approach. Second, Confrey (1995) doing approach to multiplicative structure is based on the idea of splitting where it is an action of duplicating and constructing simultaneous splits of an original. Steffe (1988) has argued that the key to students’ meaningful dealings with multiplication is its ability to iterate abstract composite units.

The differences between Confrey’s and Steffe’s approaches are that Confrey observes multiplicative actions as independent of addition ideas, in other side, Steffe’s approach considers early multiplicative acts as making natural use of counting based mental structures. There are two significant related changes –in what the numbers are and in what the numbers are about- when children move from additive to multiplicative. Researcher believe that multiplicative reasoning is an entry point to the world of ratio and proportion. He would elaborate how students’ thinking of composite unit schemes can develop into proportional reasoning.

The research question:

How students in grade six understanding the concept of proportion and ratio and how they construct some methods to solve many different task?

RESEARCH METHODOLOGY

This study was conducted using a constructivist theory of learning that focuses on a child’s construction of proportion reasoning. The researcher monitors each student how he/she think and use the concept of ratio and proportion. He set up to explore how two students construct proportional reasoning and use a unique way of accomplishing some tasks.

Clinical interviews

The methodology adopted for this study was clinical interviews that have two main advantages. First, allowing to interventions where students were encouraged to elaborate on their statement and judgments (Opper, 1975). Second, this approach to gathering data provides for a continual interaction between inference and observation (Cobb, 1986b). The use of interviewing as a successful tool of research must be accompanied by appropriate learning tasks. A set of learning tasks developed by the researcher and adapted from the literature was used to analyze the ways learners construct proportional reasoning knowledge from the problem solving activity.

The students

The researcher did observation in the two beginning grade six girls (11 years) Alice and Karen. They were in different class but a common mathematics teacher. The teacher said that Alice was a very bright student and A students. In the other side, Karen was a hard working student and in the second top class of the school for sixth graders.

Content knowledge of Students

Schoolteacher said that Alice had had been taught the ratio type of questions in a section on the topic of money though no terminology of ratio or proportionality was used. From Alice’s work sheet, all the tasks used the unit method, finding the rate for one and multiplying to get the rate of money. Alice had studied this method in class while had not as her class was slower than Alice’s class.

ANALYSIS AND RESULTS

Interpretation of Karen’s activity

Karen used strategy with meaningful insights on multiplicative schemes in proportional reasoning. She constructed iterable ratio units to find the answer by coordinating these units such that one ratio was distributed over the next ratio. She was able to unitize the units in a composite and was able to deal meaningfully with composite units. She was able to take a ratio as a composite unit and maintain the ratio unit of its elements. The key foundation in Karen’s meaningful dealing with proportional reasoning was the ability to iterate composite units. Karen used relationship as a countable unit to find the answer a ratio unit method. There are two advantage that method, first to avoid any fractional or decimal computation, and secondly, to become a powerful method for solving all missing value tasks within the contexts that made sense to Karen.

Researcher believed that Karen’s method was a scheme according to Glasersfeld (1980) that consisted of three part: the experiential situation, the child’s specific activity or procedures, and the result that students expect. Karen was able to articulate the goals of her actions and give mathematical meaning to the procedures or reasoning she undertook and internalize her actions. She used her fingers on several occasions to re-present pattern. Finally, She was able to generalize her actions across similar proportionality task.

Interpretation of Alice’s activity Alice’s conceptualization in proportional reasoning is based on the unit method, a memorized procedure rather than a conceptual one. She used the unit method to get the answers and solve various problems. She could not describe her reasoning in a meaningful way and the procedure she used. She was not able to make sense of activity and think in terms of the composite ratio unit. Researcher believed that Alice’s procedural orientation influenced her action in dealing meaningfully with ratio and proportion.

DISCUSSION

Different levels of multiplication reasoning can be seen in the strategies utilized by these two students- Karen and Alice. Karen had constructed multiplication schemes in solving proportional reasoning while Alice used procedural method than conceptual one. Alice tried to memorize a procedure so that she could not construct and explain whether her strategies were or were not viable. Her thinking was based on additive reasoning rather than multiplicative one in which each sequences of counting acts into composite units. Karen unitized the composite unit to find a ratio unit and then iterated it to its referent point. She coordinated two number sequences and was able to preserve the relationship in the iteration. She was able to deal meaningfully with composite units. In the other hand, Alice was not able to think in terms of composite of ratio units and make decision about which unit strategy to use. Although she was successful to use that method to solve some tasks, she was not able to use it to solve a problem requiring qualitative proportional reasoning.

Researcher believed that there are varying degrees of sophistication between Karen’s and Alice’ solutions to the problems based on whether they formed composite ratio units and reinterpreted the problems in terms of those units. Karen’s method of iterating composite units seems to avoid the additive strategy that Alice used. The iteration scheme is based on a simple action scheme that can underlie the solution to proportionality. Unit ratio strategy (find the rate for one and multiplying to get the rate for many) may not help students develop multiplicative reasoning. Interpreting a situation of that unit may help to understand the nature of composite units. The unit method-a meaningful understanding-becomes essentially instrumental for the children to development of insight (Freudenthal, 1979) in multiplicative reasoning.

CONCLUTION One of the goals of this study was to investigate the ways in which the students began to develop proportional thinking. There are three essential components to use operation with composite units. First, one needs to explicitly conceptualize the iteration action of the composite ratio unit to make a sense of ratio problems. Second, one needs to have sufficient understanding of the meaning of multiplication and division. Third, one need to have sufficiently abstracted the iteration process. The unit method should not be taught to students until having a good grasp of the unit coordination schemes. Further research is need to elaborate how students’ composite uit schemes of multiplicative structure can develop into proportional thinking.

Core insights for proportion

Core insights
There are many forms of mathematical descriptions with fractions, percentages, decimals and proportions, and actually they have their own rules and procedures that students must do a lot of practices to remember them. The purpose of this article is to make a simple and narrower programme without mastering procedures but understanding the principles. Researchers refer to core insight for proportion by using contexts.

Proportions are everywhere

Students will find some problems about proportions like enlarging and reducing photos, prices comparisons, recipes, comparing probability, gears on mountain bike, shadows change, and graphs and diagrams. They already have some experience with proportions in the upper grades of primary school. This book, researchers explained general concept of proportion in grades 4-6. Proportion can be placed as fractions, percentages, and decimals. In the beginning of this chapter, researchers discuss the concept of direct proportions and then ratio table. After that, they move on the more problematic issues and finally will return to proportions. The end of this book tells about sketching out a learning-teaching trajectory.
Reasoning with proportions

Proportions
Proportions are based on the concept of ratios (direct proportionality). Students can use ratio table to show clearly because They can expand the number of columns to suit their needs. In mathematical terms, a directly proportional relationship between two units is called a “linear relationship” because it can be drawn as a straight line on graph and pass through the origin. In reality, there is often only an approximately direct relationship. The graph and the ratio table are not a description of a true condition, but in fact a mental model.

The ratio table
The ratio table is an important role in the curriculum on proportions in grades 4-6. It is an ideal aid for making handy calculations and gaining insight because the table invites students to write down intermediate steps. The ratio table is much clearer because every intermediate step has meaning. It can also be used for calculating with percentages and decimals. The ratio table is a mental model as well as a work sheet. The table helps students reason with proportions and is at the same time a handy calculations tool.
The ratio table gives a good overview. But is not in itself any better than a table or list in which the number are placed under each other. The advantage of the table is that all the numbers have their own place and that the unit of measurement must stay the same. There are good reasons for choosing a ratio table as the standard form of notation, but that does not mean that the ratio table should be introduced without any discussion.

Reinventing the ratio table
The reinventing comes down to two things: the choice of a systematic notation method and discovering for themselves how useful a ratio table is for doing calculations. The first, students must realize that a systematic notation method has great advantages. They often write down a list of items more automatically than they will choose a table form and discover what they need to write down for the list to be complete. The students need to explore independently which step they can make in the table and which will not work. They can learn some rules by reasoning from a context problem and by explaining to other students how they have reasoned. Teachers can help by making the mathematical steps in the table explicit.

Comparisons using proportions
Proportional comparisons are based on a mental experiment. The students have little trouble with the mental experiment underlying the conversion, because it is so logical to star from a directly proportional relationship between two problems. Many presuppositions are required to be able to make a proportional comparison. Adults are so used to thinking in proportions that they mostly do not realize that they are making an artificial mathematical transition. For children, the difference between absolute and relative comparisons is not nearly as self-evident. Teachers should focus sufficiently on the aspect of comparing absolute terms versus comparing in relative terms. They should encourage students to think a lot about the function of mathematical tools. This can be done via investigative activities and open questions without a pre-determined approach. Students should do too many activities in which the method of comparison is pre-determined.

Composite units
Composite units are different types of units and combining them leads to a new unit. In the history of mathematics it took a long time for such unit to become accepted. For students composite units are troublesome, for example, velocity is a difficult concept. The ratio table simplifies calculations with composite units since in such a table students can work with the separate units rather than with the composite unit directly. Composite units are special as in everyday life people often calculate with a standardized proportion. The advantage of such standardization is a direct comparison that people can make. How things are standardization is no more than a mutual agreement and different from one country to other areas.

Indirect proportion
Directly proportional relationships are very common, but students must still consider about not directly proportional relationships. For example, The cost of a mobile telephone cell already costs a few cents before the conversation can begin, in fact the cost of it is usually made up of a starting rate and a charge related to the length of the cell. Another example of proportional relationship is that between length and area when enlarging a photo. Students are often confused by making something twice as big does not make the area twice as big. The relationship between length and area is not linear,but quadratic: the area increases by the square of the length.

The global learning-teaching trajectory for proportions
Students have stated with a broad exploration of proportions before grade 3. This exploration primarily has a qualitative nature and covers situations dealing with dimensions and measuring. Students learn about simple visualization. There are four parts of learning-teaching trajectory about proportions.

Ratio Table
Ratio table is one of the part of the learning-teaching trajectory. After introducing it, students must develop their feelings for recognizing situation in which they can be useful. They must develop some strategies for making clever use of the ratio table. They can start by using more formal strategy into standard method. People can distinguish between situations where both numbers in the ratio refer to the same unit and situation where the units differ. The level of working with a ratio table usually does not create any extra complications to make composite unit to be easy to understand.

Percentages and decimals
Using the ratio table for percentages and decimals is a part of the learning-teaching trajectory. Percentages involve converting proportions to percentages as well as calculating with percentage. Proportions or ratios can be converted into percentages by calculating. The ratio table is well suited to global calculations and estimating the result using in iterative approach.

Proportion Comparisons
An important first step of proportion comparisons is to distinguish between absolute and relative comparisons. The next step is to extend into situations in which there is no strictly linear relationship, but where people act as if there is one.

Indirect proportions
There are some subjects that are not linear standing apart from others. To avoid student misunderstanding on those materials so teachers must pay attention to these relationships with a certain regularity.

I INTRODUCTION

This first classroom observation report will tell about studying-learning process from the beginning to the ending that I did in the third grade at 23th july 2009 in MIN 1 Palembang at 2.00 pm. It explain about process at the classroom, including student activity, teamwork, and problem solving. Finally, I observed the goals after studying like result and conclusion. To know more about studying-learning process I asked teacher before and after teaching process and students after studying-learning process.

II START OF THE LESSON

The teacher started the lesson by reviewing the last lesson to know that students still remembered or not. After that, she introduced what lesson will be studied today about number symbol and name. Next, she asked me (Zetra) to continue studying-learning process.

I started the lesson by telling and asking students about something that they found and used in their life for examples house number, body height, ice cream cost, tree height, car length and cow weight. Also, I explained on how to work on a problem both individually and in groups. I gave positive respond when the students had comments or questions. In addition, they look enjoyable the lesson so some of them started to give examples like their house number. One of them said that her house number is 2396 but I explain that they still studied number 100 to 999. I asked students to raise their hands when they wanted to give comments, questions, or answers. Only a few students did it because they were still shy.

From these results, it could be inferred that most students enjoyed studying mathematics by using PRMI. Some of them had many ideas and comments about their lesson.

III DURING THE LESSON During the lesson, I managed the classroom by guiding the students to give their reasons about picture that I put in white board (picture). they started to communicated their arguments and solutions. One student came to in front of class to explain that picture. She explain to their friends that the number under picture was number symbol, and number name was written by alphabet. For example in the picture 573 was symbol, and name was Five hundred and seventy-three.

I continued this process by putting some pictures about their live activities in the white board. Finally, I asked a volunteer to stand in front of class and showed their friends about a picture that put in the right of his T-shirt shoulder. All students said that it was district symbol, and district name was south sumatera. So, they understood difference from symbol to name of number.

After discussion process, I gave students worksheet (picture). There were four questions about number name and three questions about number symbol. Some students could finish this question quickly and perfectly.

IV END OF THE LESSON

I guided the students to make conclusion after the whole class discussion and answering students worksheet. Finally, I concluded the lesson based on the students’ contributions or solutions and gave them assignments.

V RESULT

Before the end of the studying-learning process, the students did some task, and more than 70% students could answer the question quickly and correctly. There are two examples of students’ answers. VII CONCLUTION From the results, It can be concluded that students understood and able to answer the questions in students’ worksheet. Some of them could develop their own example and distinguish between number symbol and name. From data on the list, it can be concluded that students followed well in studying-learning process using PRMI. However, some of them could not solve well that problems because they did not focus when their friends explained.

VIII INTERVIEW RESULT

From interviewing the teacher, it can be concluded that.

1. The students enjoyed studying mathematics by telling real contexts as usual with their activity in their life.

2. They understood quickly about the lesson if the teacher could explain interestedly and clearly.

3. They could solve task or homework quickly and easily. After studying-learning process,

I interviewed some students, and it can be concluded that.

1. The students like study mathematics because they always use it at home like playing monopoly game and buying food at store.

2. They enjoyed studying mathematics today because the teacher told about their life activities before studying.

3. They always got good score in mathematic projects because mathematics was easi to follow.

Silabus

RPP

MEMBACA DAN MENULISKAN BILANGAN (Materi Ajar kelas III)

The firstproblem is in figure 1.

Figure 1

The cost two glasses and one calculator are Rp 50,000.00, and one glasses and three calculators are Rp 50,000.00.

The Questions are :

1. Without calculating cost each item, Which one is more expensive?            Explain your reason

2. How many calculator can be bought by Rp 50,000,00? Why?

3. How much of a glasses? Give a reason.

Solution :

Almost all students can answer that question, and their solutions are different among one students and others.

This is one of their solution.

1. Glasses is more expensive than calculator because cost of a glasses            equal cost two calculators. That conclusion is made depend on figure      1.

2. One glasses and three calculators are Rp 50,000,00. Because cost of      a glasses equal cost two calculators, two calculators and three                    calculators are Rp 50,000,-. Someone can buy 5 calculators.

3. Cost of a glasses is Rp 20,000,- because cost of a glasses equal cost         two calculators.

The second problem is in figure 2.

Figure 2

The cost two umbrellas and one hat are Rp 80,000.00, and one umbrella and two hats are Rp 76,000.00.

The Questions are :

1. Which one is more expensive, a umbrella or a hat? Why?

2. How much of three umbrellas? How much of three hats? Explain your      reason.

3. How much of a umbrella and a hat? Give a reason.

Solution :

This is one of their solution.

1. Umbrella is more expensive than hat because buying two umbrellas      and one hat is more expensive than one umbrella and two hats.                  Differences price is Rp 4,000,00.

2. If Umbrella in the second equation change to hat in the first                         equation, Students get new equations cost of three umbrellas equal           Rp 84,000,00 and cost of three hats equal Rp 72,000,00.

3. Cost of a umbrella is Rp 28,000,00 because cost of three umbrellas         equal Rp 84,000,00, and it is divided three. Cost of a hat is Rp                    24,000,00 because cost of three hats equal Rp 72,000,00, and it is        also divided three.

Students usually use context to find solution of linier equation system. They can answer it by thinking logic and looking figure. They do not theory about linier equation system, but they answer a problem what they know in the figure.

There are some subjects which can be used roof as learning model like number theory, Triangle, Quadrilateral, Hexagon, and Symmetry.

Number theory

A teacher can use roof (figure 1) to explain number line, positive-negative number, addition-division number and comparison number. For example, Students can discuss how to calculate 3 + 2 or 4 – 6 base on figure 1. They must understand number line first, so they can also understand other subjects. A teacher is only hoped to guide his/her students find concept so they can use that to answer many question about number.

Figure 1

Triangle

There are three triangles can be made in roof. First, right triangle which sides 3,4, and 5 is in figure 2. Students are hoped to understand about Pythagorean theorem, which states in any right triangle, the square of the length of the hypotenuse equals the sum of the squares of the lengths of the two other sides. Second, equilateral triangle which is in figure 3 sides 4,4, and 4. all sides are the same length. An equilateral triangle is also a regular polygon with all angles equal to 60°. Students can find it because they make it by themselves. Last, isosceles triangle which sides 5,5, and 2 is in figure 4. two sides are equal in length. An isosceles triangle also has two equal angles: the angles opposite the two equal sides. A teacher only guides his/her students to reinvent triangle, and they must develop their ability to analyze triangle theory.

figure 2               figure 3             figure 4

Quadrilateral

Students can make many kinds of Quadrilateral such as square, rectangle, parallelogram, rhombus, and kite. First, Students make square which sides 4 in Figure 5.They can calculate that area by dividing it and perimeter by summing its sides. all four sides are of equal length (equilateral), and all four angles are right angles. the diagonals perpendicularly bisect each other, and are of equal length. Second, Students are able to make two different size rectangle. In Figure 6 is rectangle which sides 5 length-1 wide and 4 length-2 wide in figure 7. So, students can know that all four angles are right angles. Equal conditions are that opposite sides are parallel and of equal length, or that the diagonals bisect each other and are equal in length. There are two model of rectangle. They can compare figure 6 with figure 7 about those area and perimeter. Third, There are two parallelogram which are different area but same perimeter. They side 5,1 in figure 8 and 4,2 in figure 9. Students can compare that parallelogram and make conclusion that parallelogram is a quadrilateral with two sets of parallel sides. Equal conditions are that opposite sides are of equal length; that opposite angles are equal; or that the diagonals bisect each other. Next, Students can make rhombus that sides 3 in figure 10. It is a parallelogram in which adjacent sides are of unequal lengths and angles are oblique (not right angles). “A pushed-over rectangle. Finally, Students invent kite that is two adjacent sides are of equal length and the other two sides also of equal length. This implies that the angles between the two pairs of congruent sides are equal, and also implies that the diagonals are perpendicular.

Figure 5                    Figure 6                            Figure 7

Figure 8                             Figure 9                 Figure 10                Figure 11

Hexagon

Using that roof, students can make hexagon which sides 2 units, and they can also analyze its figure. Each angle is more than 900, and if they divide it, they can get 6 equilateral triangle. So, there is some relation between hexagon and triangle.

Figure 11

Figure 11

Symmetry

There are 2 kinds of symmetries that students can find from roof media by making letters. First, They can make some letters like in figure 12 – letter M, figure 13 – letter C, and figure 13 – letter U, and find definition about reflection symmetry. They have to know what mirror symmetry is and also know other figures which have that symmetry in their life. Second, Rotational symmetry which is an object that looks the same after a certain amount of rotation is in figure 14 – letter Z and figure 15 – letter S.

Figure 12                Figure 13          Figure 14

Figure 15    Figure 16

Roof is a media which can be use to explain some mathematics subjects such as Number theory, Triangle, Quadrilateral, Hexagon and Symmetry. First, Students study about natural number and number line from it. Second, They study about equilateral triangle, isosceles triangle, and right triangle and square, rectangle, parallelogram, rhombus, and kite. Next, Hexagon is one of figure which students can make and study. Finally, By making letters students can study about reflection symmetry and rotational symmetry. So, students are able study many things by using roof as media.

Snake and stair game is a game which is played by four students. Each student has a pawn, and he/she gets opportunity to shack cube. Cube has number 1 to 6. A student will put his/her pawn building on number in cub that he/she gets. Student’s cub which stand in stair will climb that stair but in snake will reduce, so they use mathematics process in this game.

Snake and Stair Game

Example :
1. A student’s pawn put in number 11 than he shack a cube and get number 5, so he move his pawn 5 step . Finally, the student put his pawn in number 46 because he got stair. How many extra point that student get ?

This is mathematics model about it :
11 + 5 + . . . = 46

2. A student’s pawn put in number 51 than he shack a cube and get number 3, so he move his pawn 3 step . Finally, the student put his pawn in number 26 because he got snake. How many extra point that student get ?

This is mathematics model about it :
51 + 3 + . . . = 26 or 51 + 3 – . . . = 26
If Teacher gives abstract problem, some students can not solve it. This illustration can help students find model and know how to solve many problems about it.