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Several studies on learning mathematics for rural area's student indicate that students have difficulty in understanding the concept of division operation. Students are more likely to be introduced by the use of the formula without involving the concept itself and learning division separate the concrete situation of learning process. This underlies...
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... prospective teachers students at Surya College of Education Tangerang derived from Ambon, Serui, Yapen, and South Sorong, Papua, and also a teaching assistant who acted as a model teacher. This study consists of three steps done repeatedly until the discovery of a new theory that a revision of the theory of learning is tested. Overall, the stages that will be passed on this research can conclude in the diagram (Figure 2). The learning activities start from review the students' multiplication ability of 1-9 and then conduct a fair share balls to introduce the concept of division operation in concrete level as a partitive process (Figure 3). Furthermore, students learn the division process of 2 digits by 1 digit numbers using the relationship between multiplication and division. Lastly, students are introduced to division with remainder and how to solve the problems of division operation in Math GASING for various forms of division. After that students do mental arithmetic activity namely mencongak as one of evaluation process in this learning activities and exercise by using student evaluation sheet. As a result, students was able to master the division operation in Math GASING seen the results of the final evaluation. For more details, the researcher will discuss the results of the learning process of division operation in Math GASING, which is divided into three stages that are called preliminary design, teaching experiments, and retrospective analysis. At this stage, researcher is beginning to implement the idea of division operation in Math GASING by reviewing the literature, conducting observations in elementary school, and ends with designing hypothetical learning trajectory (HLT), as shown in Figure 1 above. A set of activities for learning division operation in Math GASING has been designed based learning trajectory and thinking process of students who hypothesized. The instruction set of activities has been divided into four activities that have been completed in 2 meetings, start from review material multiplication, introduction the concrete form of division operations as a partitive process using the ball activities, division process of 2 digits by 1 digit numbers using the relationship between multiplication and division, introduction the solving ways of division operation problems in Math GASING, division operation with remainder, a variety of fun activities that make students happy in the learning process, and ends with the evaluation process. In teaching experiment, researcher tests the learning activities have been designed in the preliminary design stage. There are the activity of multiplication operation material reviews from 1 to 9 conducted from an verbal question to conduct activities aimed to master multiplication operation from 1 to 9, introduction to the basic concepts of division using the ball activities to the students with the discussion moderated by teacher model, doing division operation in Math GASING with remainder from the front to back, and the last the mental arithmetic evaluation and also student worksheet evaluation written in the form of division operation issues as the formal problems or about the story problems based on the context of day-to-day problems. When the teacher models have started to see students do not get excited, then the teacher models provide educational games that make fun learning activities, because it is becoming one of characteristics in Math GASING learning process. Division process in Math GASING is different with division process in mathematics in general. As a result, all activities which have been designed can be used to answer the research question above. Learning trajectory which has been modeled in Figure 1 are the activities undertaken in this study to guide students mastered division operation, as a result of the partitive process. So that, researcher designed an activity fair share balls, so many students received ball is the result of division operation, while the process of dividing the ball equally to each student emerged as the partitive process until all the ball is evenly divisible by student who has been determined amount (the divisor). From this activity, teachers guide students toward the concept of division operation formal model without remainder. Students learn the division process of 1 digit by 1 digit and 2 digits by 1 digit numbers using the relationship between multiplication and division. For example, 12:4, the teacher to ask the student, “4 multiplied by how many are the result equal to 12?”. This is the quick way for division operation, so the children can answer the result is to ask them. For second activity, start from using the number of balls in a fair divisible and the amount of ball that still have remaining, despite being shared fairly. Furthermore, from this activity, teachers guide students toward the concept of division operation formal model which is the result of partitive process has remainder. In Figure 6, Jhon performs the division process starting from the front, which is 16 divided by 3 and the result is 5 remainder 1, then the remainder becomes tens with one digit behind as unit to be 12 divided by 3 and the result is 4. Because there is no remainder then 8 straight divided by 3 and the result is 2 remainder 2 after that 23 divided by 3 and the result is 7 remainder 2. All results are written sequentially starting from the front, so that the result from 16.283 divided by 3 is 5.427 remainder 2. The goal is the students are able to pass through the critical point that has been designed that is division with remainder. So that, they can do mencongak to solve all division problems given. Based on all the activities above, it can be seen that the students have gone through the process of activity based on experience, moving toward a more formal, the understanding of formal level from the critical point, and then reached into the formal level desired as the ultimate goal of this learning activities. In the design of this study, researcher used the learning steps of division operation in Math GASING, starting from the students’ ability to review multiplication operation from 1 to 9 and relationships between multiplication and division that can be used as the starting point of learning division operation. After that, the teacher models simulating the activities fair share sweets, to instill the concept of division operation, up to trace the division by remainder as a critical point of learning division operation in Math GASING. When the activity takes place, the dialogue is very good in the process of introducing the basic concepts of division operations. In the dialogue, it seems that students feel learning division operation in Math GASING looks so easy and so much fun. As a result, the learning process can guide students in understanding the basic concepts of division operations. It can also be seen from the student evaluation of learning division process given by the teacher to evaluate student understanding. As a result, students seemed to be able to apply basic concepts of division operation in solving each problem is given in terms of evaluation. Therefore, it can be seen that learning division operation in Math GASING can use to raise students' understanding of basic concepts integer division operations or in other words, the design of this study can be used as the starting point of learning operations division. Based on the result of this research and discussion that has been described above, researcher can conclude that the learning of division operation in Math GASING have a very important role as the starting point and improve students' motivation in learning division operation. In addition, the activities that have been designed in such way that student find the concept of division operation as the partitive of shared process, which is fundamental to the division operation in Math GASING. This process begins with the ball activities, then move into the process of how each student gets distributed balls fairly, ranging from division without remainder to division with remainder, and ends with the completion of division operation in Math GASING (formal). Lastly, each student can do mencongak for any given division problem and resolve many division questions very quickly and precisely where is both of this are one of assessment forms in Math GASING. In the other hands, this result is in line with Prahmana’s results (2013) which talk about learning division operation at urban students in Indonesia. Thus, researchers assume that the LIT can be applied to all students with any background in Indonesia. Suggestion that researcher can give is before doing teaching experiment, researchers were able to transfer the expected good learning design which has been made to the teacher model, so that, a fatal error not occurs while teaching experiment took place. It’s because the teacher model is the teacher who did the teaching process, not the researcher. In addition, Math GASING can also be developed into the design of other learning number operations. Lastly, from the HLT has been designed, researcher should be added more steps before entering into a formal level. It is still a big task for researcher to redesign HLT and try it again in the process of further research. Gravemeijer, K. (1994). Developing Realistic Mathematics Education . Utrecht: Technipress, Culemborg. Gravemeijer, K. (2004). Local Instructional Theories as Means of Support for Teacher in Reform Mathematics Education. Mathematical Thinking and Learning, 6(2) , 105-128. Lawrence Erlbaum Association, Inc. Akker, Gravemeijer, Gravemeijer, J.V.D., K., K. Gravemeijer, & (1994). Eerde, D.V. Developing K., (2009). McKenney, Design Realistic S., Research & Nieveen, Mathematics as a N. Means (2006 Education for ). Education Building . Utrecht: a Design Knowledge Technipress, Research Base . London: Culemborg. for Teaching Routledge in ...
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