A Concrete Situation for Learning Decimals


The research has been done by Puri Pramudiani in 2008. Together with Zulkardi, Yusuf Hartono, and Barbara van Ameron, she wrote it in a Journal on Mathematics Education, volume 2, nomor 2, page 215-230.

The aim of this research is to investigate one situation that enables the students to learn decimals in a meaningful way. It is very important because decimal is one of mathematics domains that is often used in calculation. Unfortunately, mostly in Indonesia, decimal is taught only as another notation for fractions or percentages. Students assumed that decimal is only the number with point (comma) without knowing the meaning of it. Therefore, the learning is meaningless and students can not use it in the concrete situations.

Based on that condition, Puri used RME (Realictic Mathematics Education) in her research. It was chosen because RME underlies the design of context and activities. According Zulkardi and Ilma that was quoted by Puri, the context is a main point for students in developing mathematics. It should be meaningful and real for students’ mind. In this research, Puri used measurement activities as the context, especially the activities of measurement of weighing duku, weighing the body, weighing the rice, and the volume of beverages.

The subject of this research was 26 students and a teacher for grade 5 SDN 21 Palembang Indonesia, but only 7 students involved in the series of activities of pilot experiment. She chose small group because she expected  to be more focused to the adjustment of HLT (Hypothetical Learning Trajectory). HLT is a part of planning mathematics lesson which consist of the goals, the mathematical tasks, and the hypotheses about the process of students’ learning.

The phases of students’ learning in this research are :

  1. Knowing the existing of decimal form through contextual situation (weighing duku and body)
  2. Exploring the meaning of one-digit decimals through weighing rice
  3. Using number line as a model for placing the position of one-digit decimals (partician based ten)
  4. Exploring the meaning of two digit decimals through measuring the volume of beverages
  5. Using number line as a model for placing the position of two-digit decimals (partision based tenth of tenth).

Based on the result of post assessment and interview, 69% students shows a good ability in reading the scale, 85% students have a good knowledge of decimals, and 77% students could master the idea of density of decimals on the number line, and 92% students chose number line as the tool to placing the magnitude of decimals.

Before the activities of measurement, students thought that there were no other numbers between two consecutive whole numbers. However, after the activities of measurement, students realized that decimals exist in between two consecutive whole numbers and desimals were needed to measure the things precisely. Through the series of activities, the students could develop their idea into the density of numbers on the number line which bring them into the idea of partitioning base ten and tenth of tenth.

Puri conclude that the context and activities designed (weigh and volume measurement) can become the concrete situation for learning desimals. From this research, we can show that desimals can be taught in a meaningful way. Students could discover desimals by themselves and develop their ideas to come to the number line as a model for placing their magnitude.

From this research, we can learn many things, such as :

  1. Desimals can be realized in the studends’ mind
  2. Weight and volume measurement can be a context for learning of decimal that is meaningful for students.
  3. Students able to develop their idea if the learning is meaningful.
  4. Using RME in other material can help teacher to reach the goals of the learning.

Understanding of RME

Understanding of RME is one of the sub titles of the Prof. Zulkardi’s thesis that titled Developing A Learning Environment On  Realistic Mathematics Education For Indonesian Student  Teachers. This thesis was written in 2002

Prof. Zulkardi explains that the philosophy and characteristics of RME consist of views on what mathematics is, how pupils learn mathematics, and how mathematics should be taught. It is strongly influenced by Hans Freudenthal’s concept of mathematics as a human activity that means the students aren’t passive recipients of ready-made mathematics, but they should be guided to discover and reinvent mathematics by doing it themselves.

According to Van Hiele, that was quoted by Prof. Zulkardi, there are 3 levels in learning mathematics. First, students can manipulate the known characteristics of a pattern that is familiar to them. Second, students learn to manipulate the interrelatedness of the characteristics. Third, students start manipulating the intrinsic characteristics of relations. Based on that levels, RME starts from the first level. It is different with the traditional instruction that is inclined to start from second or third level. In order to start from the first level, RME starts from a meaningful contextual problem. Furthermore, by guided reinvention through progressive mathematization, students are guided didactically to progress efficiently from one level to another level of thinking.

Five tenets (characteristics) of RME such as :

1.      The use of context in phenomenological exploration

RME uses real problem and contextual situation as the starting point of learning. The phenomena by which mathematics concepts appear in reality should be the source of concept formation. The process of conceptual and applied mathematization according de Lange is showed below.

Figure 1. Conceptual and applied mathematization

Figure 1. Conceptual and applied mathematization

2.      The use of models or bridging by vertical instruments

Four levels of models that are developed by the students in RME according to Gravemeijer are described below.

Figure 2. Levels of models in RME

Figure 2. Levels of models in RME

3.      The use of pupils’ own creations and contributions

Students create concrete things and free productions, such as write an essay, do an experiment, collect data and draw conclutions, design exercises or  test for other students.

4.      The interactive character of the teaching process or interactivity

Interaction between students and students and teachers is very important. Students are engaged in explaining, justifying, agreeing and disagreeing, questioning alternatives and reflecting.

5.      The intertwining of various mathematics strands or units

In RME, the integration of mathematical strands or units that is often called the holistic approach is very important. An intertwining of learning strands is exploited in solving real life problems.

These characteristics can be used as a study guideline both in the process of adapting RME curriculum materials to the Indonesian context and in the process of pre-service training for student teachers in teacher education.

Three levels of construction that was developed by Streefland in realistic mathematics lesson materials are

1.      The classroom level

At this level, instructional activities are designed based on all the characteristics of RME. How all the characteristics of RME are pictured in a model for designing RME curriculum materials is showed below.

Figure 3. A model for designing RME curriculum materials

Figure 3. A model for designing RME curriculum materials

2.      The course level/instructional sequence

After the materials from the classroom levels were tried out and revised, students are expanded to other contents and contexts in order to develop the instructional sequence of that topic.

3.      The theoretical level

A theory in the form of a local theory for a specific area of learning is constructed, revised, and tested again during additional cyclic developments.

RME exemplary lesson materials refer to learner materials and teacher guides that are used as a learning trajectory for teacher and consist of content materials, learner and teacher activities, and assessment.

RME materials are related to real-life activities and uses contextual problems that should be appropriate for the goals of the particular mathematics topic. Three levels of goals in mathematics education that were characterized by De Lange are lower level, middle level, and higher-order level. While the goal of the traditional program were classified as lower level that are based on formula skills, simple algorithms and definitions, RME goal includes middle and higher level goal (reasoning skills, communication and the development of a critical attitude).

While the RME teacher in the classroom as a facilitator, an organizer, a guide, and an evaluator, the students work individually or in a group.

De Lange formulated 5 guiding principles of assessment in RME, such as

  1. The primary purpose of testing is to improve learning and teaching
  2. Methods of assessment should enable the pupils to demonstrate what they know rather than what they don’t know
  3. Assessment should be operationalize all of the goals of mathematics education
  4. The quality of mathematics assessment is not determined by its accessibility to objective scoring.
  5. The assessment tools should be practical

Assessment can be conducted in the classroom using strategies both formative and summative.

Based on that explanation, we can conclude that to apply realistic teaching and learning in the class, we have to refer to all of the RME characteristics.

PMRI in Aceh

buku a decade of PMRI

PMRI in Aceh is one of the sub titles in a book that titled  A Decade of PMRI in Indonesia. This book was written by Rahmah Johar in 2010.

According to Gravemeijer that was quoted by Rahmah, RME (Realistic Mathematics Education) is a theory of mathematics learning that uses reality as the starting point in the learning process and aims to support the students in reinventing mathematics in problem-centered interactive instruction. Since 1968, Netherland has been developing  that learning approach. And Indonesia has been adaptating it as PMRI (Pendidikan Matematika Realistik Indonesia) since 1998. In Nanggroe Aceh Darussalam, the local PMRI-team of University Syiah Kuala began socialization of PMRI in 2006, such as conducting workshop, doing research, writing student textbook, and coaching primary school teachers. There is a significant progress, even thought there is still challenges in the future.

Beside abstraction and formalization, difficulties in learning mathematics is caused by the style of teaching that is meaningless. While teacher is only explain the mathematics concept or procedures and give an example, the students have to listen, write down, and do the exercise. They quickly forget the content and can’t apply it in the daily activity. It isn’t equal with the realictic approach. In the realistic approach, student rebuild their mathematics idea, explore concepts by their creativity and initiatives. They are regularly organized to work in groups, so they can discuss and learn to respect different opinions and solutions. According to Sembiring, it is not only reform the mathematics education, but also teaching democratic culture. Beside that, it is suitable with KTSP. According to this curriculum, students have to be engaged in learning experiences, involving mental and physical process through interaction between students and students, students and teacher, students and their environment, as well as students and learning resources.

Hans Freudental (Gravemeijer, 2010) argued that students should be engaged in mathematics as a human activity that means an activity of solving problems, looking for problems, and organizing a subject matter. To implement RME, the teacher needs to design realistic problem to be discussed in the classroom and have to be a mediator and facilitator.

The socialization of PMRI in Aceh started in 2006. Cooperation was sought with the teacher educators of other Teacher Education Colleges in Aceh, the Institute for Quality Assurance of Aceh province, and local Department of Education of province. The goals of that socialization such as to inspire teachers and get them acquainted with that approach and to be willing and confident to implement it in their class. Now, there are 36 primary schools that have connected to the local PMRI center. From each school, 5 teachers, one from each grade 1 to 5 are involved in the socialization. In the first year, workshops focused on Grade 1 and 2, in the second year, it focused on Grade 3 and 4, and in the third year, it focused on Grade 4 and 5. The topics of these workshop are simulation of teaching, discussion about students’ thinking, analizing videos, and reflections and planning follow up activities.

The simulation is leaded by a teacher educator or a key teacher, a teacher who has gained ample experience in the implementation of PMRI and can share their experiences. The consultant from Netherland could also give comment and reflect. When discussed about students’ thinking,  teachers analized the students answer. There are many videos, such as measurements of length, measurements of weight, the net of cube, volume, area and perimeter, and integer number. Then, it would be followed by implementing of PMRI.

Not only in the development of RME, the concept of a hypothetical learning trajectory is used in socialization of PMRI in Aceh. The teachers have to design a learning route/learning trajectory around a well specified topic that is supported by the teacher educators of the local PMRI-center. The main focus of this reseach is the frequency of students’ interaction that is analysed by making sociograms. The categories of students’ interactions such as the student/teacher explains the question, the student asks question to student/teacher, the student responds to a question from student/teacher, the student answers the question from student/teacher, and students do irrelevant activity (for example: run away or day dreaming). After analizing data, it will be made conclution.

Writing material is very important in socialization of PMRI in Aceh. It supports analizing the curriculum, thinking about ways of changing the teaching strategies, and coaching the teacher in the school.

There are over 2000 schools in Aceh. Therefore, there are many strategic plan to implement PMRI there, such as institutionalize the local PMRI center, increase the intensity of the contacts among elements, increase the number of people who do design reseach on PMRI, continue and expand the cooperation with international teacher training institutes, and continue with designing and writing Islamic PMRI lesson materials.

Based on the experience of implementation of PMRI in Aceh, we able to learn that RME is a mathematics learning approach that is meaningfull so students are motivated to reinvent of mathematics idea by their creativity and inisiative. Teacher isn’t dominant in the class but they become the mediator and fasilitator. Interaction is very important, not only between student and teacher but also between student and student, and between student and learning resources. To implement PMRI, we should engage many elements of education to do workshop, writing material, and follow up.

Gundu and Benthik for Learning Measurement of Length


The title is Design Research On Mathematics Education : Indonesian Traditional Games As Preliminaries In Learning Measurement Of Length. This research has been done by Ariyadi Wijaya in 2008 and was written in Prosiding Konferensi Nasional Matematika XIV. The aim of this research is to develop an instructional activity to increase students’ understanding of basic concept of length measurement. It is very important because length measurement  is one of mathematics domains that is used in our daily activity and has been a part of human civilization since centuries ago. It is taught from kindergarten until grade 6 of primary school. This research applied RME (Realistic Mathematics Education)because the teaching and learning in RME should be related to reality through problem situation. Reality means it isn’t  always encountered in daily activity, but has to be experimentally real for students. While students are motivated to select their own strategies and explore their ideas, teacher is giving guidance for them. The reality that was chosen is traditional games, especially Gandu (playing with marbles where each player has to throw their marble from starting point to a pole. The winner is the player whose marble is closest to the pole) and Benthik (All players in each team have to hit a stick then measure the distance of the fallen stick. The winnner is the team which has the longer distance). Those games rich linear measurement concept, such as comparing, estimating, and measuring length.

This research used design research or development research that had 5 features, such as learning theories development (including designed instructional theory and used Indonesian Traditional Games as starting point of learning), intervationist nature (flexible), has a prosective and reflection component, cyclic character (invention and revision form on iterative process), and the theory  has to deal with the real work. Its phases are preliminary design, teaching experiment, and retrospective analysis. For linking the instructional theory with the concrete teaching experiment, it used HLT (Hypothetical Learning Trajectory) that has three components: the learning goal that defines the direction, the learning activities, and hypothetical learning process (prediction of how students thinking and understanding). The data in this research were written and audio visual data.

This research chosen the students of second grade of  SDN Percobaan 2 Yogyakarta as the subject. They were divided in 2 groups to play Traditional Games. Then, they had to measure length use blank ruler (ruler without number, for understanding of the covering space), normal ruler (ruler with number, to investigate how students measure the length of object), and broken ruler (ruler without zero point, to understand covering space and conception that any points can serve as starting point of measurement). This research was held during 3 weeks. The tryout of the HLT had been held in grade 1 and grade 2. The tryout in grade 1 was held to investigate pre knowledge of student, but the tryout in grade 2 was to investigate the achievement of the student in learning.

The result of this research consists of 2 majors, for grade 1 and grade 2. Student in grade 1 already perceived of the transitivity and indirect comparison ideas, but only imitated the appearance of ruler without getting sense of number. Most of them understood the concept of measurement as covering space, but didn’t perceive the concept of identical unit, one of the conceptual accomplishments of length measurement. Student in grade 2 understood the relation between size of unit and result measurement, but didn’t perceive the idea that measuring is covering space and the concept that any number can serve as zero point of measurement. The advantages of this result are Benthik and Gandu as one of Indonesian Traditional Games that can be used to motivate students to reinvent the identical unit of measurement and transfer the concept of covering space from non standard to standard measurement.

The conclution don’t directly connected to the Indonesian Traditional Games, but  it able to be taken as an important consideration in deciding the Indonesian Traditional Games as preliminaries of teaching and learning of measurement length.

From this research, we able to learn at least 4 things. First, the crusial of using traditional games in learning to teach the student how important of respecting of our local tradition. Second, keep heritage of tradition. Third, the important of adaptation of RME in local situation. The last, RME able to increase students’ understanding because it uses the meaningful situation that closes with the students.