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A Concrete Situation for Learning Decimals

Puri Pramudiani, Zulkardi, Yusuf Hartono, Barbara van Amerom

IndoMs. J.M.E
Vol.2 No.2 July 2011, pp. 215-230

Decimals is important domain in mathematics. Indonesian curriculum, nevertheless, leads most teachers to introduce decimals merely as another notation for fractions or percentages. Teaching and learning of decimal is conducted in a very formal way, in which decimal is directly converted from fraction which has denominator ten or one hundred without using of concrete situations to introduce its concept to students. As consequence, many students assumed that decimal is just the number containing point (comma) without knowing the meaning of it. At the same time, they might think that there were no other numbers between two concecutive whole numbers.
The researcher, therefore, conducted this study aiming at developing an instructional program that enables students to discover decimals and get insight about their magnitude through measurement activity as a meaningful way. RME underlies the design of context and activities used in this study.  
This research was done under design research methodology, conducted in six lessons at grade 5 SDN 21 Palembang by involving 26 students and a teacher as research subject. The learning activities in this study were generally designed to help students explore the notation and the meaning of decimals. There are four activities design by researcher in this study; those are:
Activity 1
Playing come closer game with an expectation that students are able to determine the numbers between the other numbers. Through this activity, it could be known that whether students have already perceived an idea about decimals based on their daily experiences or they have no idea at all about any number between two concecutive whole numbers.
Activity 2
Measuring the weight of the things (duku and body) using weight scale and digital weight scale. Through those activities, students were encouraged to do the precise and accurate measurement so they eventually could find decimals in weighing body activity, for instance, by observing the scales that when the needle pointed to the position between two consecutive numbers there should be comma numbers (decimals) in it, e.g between 35,6 appeared on the digital weight between 35 and 36.   
Activity 3
Measuring the weight of rice helped students to invent the meaning of one-digit decimals by finding that there were ten partitions containing one digit decimals between two concecutive whole numbers which eventually lead to the idea that decimals refer to the number of base ten, i.e: 0,2 is at the second stripe from ten stripe overall (two over ten).
Activity 4
Measuring the volume of beverages aimed at developing students’ acquisition for the idea of two-digit decimals. By using model of measuring cup, the students could perceive the idea that there are decimals between two concecutive whole numbers, and between two concecutive one-digit decimals there are other decimals, namely two-digit decimals.  Furthermore, they also found the idea that decimals refer to the partitioning base tenth of tenth by noticing the stripes provided by measuring cup.
To conclude, context and activities designed (weight and volume measurement) can become the concrete situation for learning decimals. It could provoke students’ thinking about decimal idea developing from informal level to pre-formal level containing insightful mathematical ideas. Or the other word, decimals can be taught in a meaningful way by applying RME approach, however it was known as abstract for most students.

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UNDERSTANDING RME


Mathematics as a branch of science that is structured and systematically organized is considered playing an immense role in optimizing the ability of human thinking. Therefore, mathematics learning in schools is expected to be a media of establishment thinking skills particularly for students. There are, nonetheless, some major problems in mathematics education in Indonesia, such as: quality of curriculum materials, teaching methods and assessment strategies, emerging a gap between the intended and the implemented curriculum. It is, therefore, necessary to design the integrated learning which is expected can reduce such gap, namely Realistic Mathematics Education or so called Pendidikan Matematika Realistik Indonesia (PMRI).
RME is a theory of teaching and learning in mathematics education that was initially developed in Netherland. It stressed the idea that mathematics is a human activity as Hans Freudenthal’s concept (Freudenthal, 1991). According to him, pupils should not be treated as passive recipients of ready-mathematics, but rather that education should guide the pupils towards using opportunities to discover and reinvent mathematics by doing themselves. There are five characteristics (tenets) of RME (de Lange, 1987; Gravemeijer, 1994):
·      The use of context in phenomenological exploration
In RME the instruction should not be started with the formal system, but its starting point should be experientally real to the students, allowing them to become immediately engaged in the contextual situation.
·      The use of models or bridging by vertical instruments
Model virtually refers to situational models and mathematical models that are developed by the pupils themselves. Level of models in RME (Gravemeijer, 1994) are;
o  Situational level, where domain-spesific and situational knowledge are used within the context of the situation.
o  Referential level or ‘model-of’, where models and strategies refer to the situation described in the problem;
o  General level or ‘model-for’, where a mathematical focus on strategies dominates over the reference to the context;
o  Formal level, where one works with conventional procedures and notations.
·      The use of pupils own creations and contributions
Pupils should be asked to be active and initiative by creating concrete things, reflecting on their learning process. They might be asked to do an experiment, collect data, draw conclusions, or write an essay.
·      The interactive character of the teaching process or interactivity
Pupils should be motivated to develop their confidence by making good interaction in instructional process. They are encouraged to discuss their own thinking rather than focusing on whether they have the right answer.
·      The intertwinning of various learning strands or units

Those five tenets of RME then should be represented in designing or redesigning curriculum materials using realistic approach. Streefland (1991) used three levels of construction in this case: (i) The classroom level. Based on the characteristics of RME, instructional activities are designed in this level. Materials used are started from meaningful context and then intertwinned with other strands or units. Tools such as symbols or diagrams are produced to support the materials. Then, making interactive class by encouraging students to interact with others and giving them assignment leading to free productions.  (ii) Course level. At this level, materials are expanded to other contents and context in order to develop the instructional sequence of the topic. (iii) The theoretical level. `All activities in two previous levels form the source of theoretical production for this level. In addition, a local theory is constructed, revised, and tested again during additional cyclic development.
RME Exemplary Lesson Materials refers to learner materials and teacher guides used as a learning trajectory for teachers in the RME classrooms. Generally they consist of three components, namely: (i) Content materials. RME material should be associated with the students’ environment so it is real experientally for students. (ii) Learner and teacher activities. In classroom, the RME teacher’s roles are as facilitator, organizer, guide and evaluator, whilts students should be active in constructing their own idea. (iii) Assesment. It can be conducted in the classroom using strategies both during the interaction process (formative), and products of their solutions (summative).

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A DECADE OF PMRI IN INDONESIA


Realistic Mathematics Education (RME) Theory as a Guideline for Problem-Centered, Interactive Mathematics Education
(Koeno Graveimeijer)
Over the last decades, there has been a huge alteration in mathematics education in Indonesia; instruction which has been widely believed as transmission of knowledge has turned into learning as the construction of knowledge. In line with this case, there has been a shift away from teacher-centered class toward problem-centered, interactive mathematics education. This shift surely will require a change in how instruction is conceptualized. RME so called as Pendidikan Matematika Realistik is a domain-specific instruction theory which can support teachers in the incarnate of a problem-oriented classroom condition.   
Turning into this such new classroom culture certainly requires a different didactical contract as well by adopting classroom social norms for students, such as the compulsion to construct their own idea, explain and justify their solution, understand other students’ reasoning, and ask the unclear explanation. It is realized, however, that implementing those new didactical contracts takes a significant amount of effort. It is not easy to encourage students construct their own idea, start sharing their thinking and no longer rely much on the teacher, since students are likely accustomed to being passive information-recipient in the learning process.
Reforming mathematics education virtually rests on two pillars: the ability of the teacher to create a problem-oriented classroom culture and then engage with students in interactive instruction as well as the design of instructional activities that allow for the reinvention of mathematics together with the ability of the teacher to support this reinvention process.
Relating to that case, RME is a domain-spesific theory that can offer guidelines for instruction aiming at supporting students in constructing, or reinventing mathematics in problem-centered interactive instruction. RME further requires the teacher to play an active role in orchestrating productive whole-class discussions and in selecting and framing mathematical issues as topics for discussion. RME then would refer to mathematics instruction based on practical problems in an everyday life context. The ‘real’ in ‘realistic’ has to be understood as real in the sense of being meaningful for the students. According to the RME theory, instructional starting points have to be experimentally real for the students.
In elaborating and applying RME approach, of special focus will be the role of concrete materials, context problems, and the cultivation of mathematical interest. To begin with, concrete materials can help students bridge the gap between their informal knowledge and the more formal abstract knowledge they need to acquire. Considering this case, RME alternative is to provide manipulatives to students as a means for scaffolding their own thinking. Manipulative can be a powerful means to support students in building upon their own ideas and scaffolding their thinking. It  allow abstract mathematical knowledge become more concrete and easier to understand for students.    
 As emphasized before, the instructional starting points in RME have to be experiantally real for students. Therefore, one of its focus is using context problem which has two goals; to offer the students a motive concerning the intention to create a situation in which it makes sense for the students. Students are supposed to be able to image themselves in the situation how the problem is cast.  Another objective is to offer students footholds for a solution strategy, in which students have to build upon their own thinking.
A further issue of corcern with context problem is that RME aims to eventually surpass the level of just finding solutions to practical problems. Rather, it also aims to teach students to start thinking mathematically. In RME, reinvention requires a combination of horizontal and vertical mathematization. Horizontal mathematization is mathematizing activity applied to a subject matter of reality whilst when it applies to a mathematical matter, it’s called as vertical mathematization. To solve the the practical problems, horizontal mathematization actually is sufficient to perform by students. Nevertheless, to invent new mathematics as well as enhance students’ mathematical skills and insights, students also have vertically mathematize their own mathematical activity. To help students in this process, teachers will have to cultivate the mathematical interests of students.


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Abbreviating Strategies of Addition and Subtraction up to 20 through Structures (Meiliasari, 2008)



The research written by Meiliasari was done under design research methodology, conducted at  SDN Percontohan Kompleks IKIP Jakarta, during the period of May to August 2008. In this study, researcher concerned about addition and subtraction up to 20, particularly for those in the low grade of primary school by applying Realistic Mathematics Education. Noticing to the evidences found, most of children very likely rely on using fingers to keep track of their counting when doing addition or subtraction at the beginning. They count all objects one by one that surely will spend a lot of time, for instance the sum of 3 + 4 = 1, 2, 3, 4, 5, 6, 7. This is assumed as the simple way to solve addition and subtraction problems. When encountering the larger numbers, nonetheless, such way is no longer effective.  
The researcher, therefore, conducted this study aiming at developing an instructional program that helped students to abbreviate strategies of addition and subtraction up to 20 through structures. Structures of numbers, either line or group and combination model as well, were used as a visualization to enhance students’ thinking process in constructing meaningful and flexible structures such as using ‘doubles’,’ splitting’, and ‘friends of 10’ which eventually will make students easier to solve the addition and subtraction problems.  
The research was divided into two parts; part 1 was conducted between May to June, whilst part 2 was running in July. Over those periods, five students who represented the high, middle, and low achiever groups were picked in the 1st part. Then, hypothetical learning trajectory will be tested in the learning experiment in the 2nd part. Lastly, the collected data through interview and video recording during teaching learning experiment were analyzed in the retrospective analysis phase.
The learning activities in this study were generally designed to help students count effectively. To begin with, the researcher gave students the candy packaging activity to develop their structure awareness. In this activity, they were asked to make several arrangements of candy packing –in fives, tens, or other group structures- with an expectation that they would be able to realize that structuring helps them do a faster counting.  In addition, working with concrete object (candy) as the characteristic of RME would make students construct the concept of material easily.
The next activity was providing the double song and completing the worksheet concerning about double structure. Then, the 3rd  activity was opened with flash card games forcing students to count quickly how many objects in the card. To do so, they should recognize the structure well instead of counting objects one by one. For example, when having the 8 card, some students might see 8 as 10 – 2, whilst other students use doubling structure 4 + 4 = 8. Through this activity, students could realize the importance of structure in shortening the counting process.
The following week, the activity was followed by developing students’ understanding of the ‘friends of 10’. For example, if there are only 7 candies in the box, to make it 10 how many more need to be put in? The result showed that friends of 10 strategy has allowed students to work faster and easier. Morerever, a trick so called ‘number pair’ in Indonesian language, pairing the numbers whose same first letter such as satu-sembilan, dua-delapan, and so on, worked well on students’ memorizing.
To conclude, a challenging and interesting instructional program will make students more motivated and get involved in the activity actively. It is important to make activity goes along naturally with the real context in students’ surrounding. As children mostly do in doing addition or subtraction, counting concrete objects using finger is simple and always works. When working in larger number, however, students should be encouraged to use flexible and meaningful strategies such as splitting, doubling, and friend of 10 to make their counting process easier and more effective.

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