Third PMRI Learning Design

Third PMRI Learning Design: “Concept of Angle Measurementby Using Pizza Model”

Achmad Dhany Fachrudin1

Sitti Busyrah Muchsin2, dan Ummy Salmah3

International Master Program on Mathematics Education (IMPoME 2012) email: dh4nyy@gmail.com, sittibusyrah@yahoo.co.id, ummysalmah@ymail.com

A. Introduction

Mathematics as a branch of science that is structured and organized in a systematic, this is conscious to have a role in optimizing the capabilities of human thinking. As stated Plato (Gredler, 1986) in his teaching which states that in order to develop the mind, we must learn mathematics. Awareness is also evident in mathematics education policy formulation in Indonesia. According to the Ministry of Education (2003), one goal of mathematics learning is how to think and reason in drawing conclusions. Therefore, learning mathematics is expected to be the correct vehicle in the formation of human thinking abilities.

The establishment of the ability to think through the mathematics is an absolute for every human being and particularly special for the pupils. But the learning process in schools today is more teacher-centered. Pupils tend to be passive and not given the space to discover mathematical concepts themselves.

Therefore, it is necessary to design an integrated learning close to the pupils and especially fun for pupils. Learning approach that is most appropriate is Realistic Mathematics Education Indonesia (RMEI). RMEI learning is an approach that uses contextual issues (contextual problem) as a first step. Based on the results of studies in several countries, Freudenthal (MKPBM Team 2001) revealed that RMEI very beneficial because (1) to make mathematics more interesting, relevant and meaningful, not too formal and also not too abstract, (2) to consider the level of pupil ability, (3) emphasize learning mathematics on „learning by doing‟, (4) to facilitate

the completion of mathematics without the use of algorithm standard, and (5) use the context as a starting point for mathematics learning.

In this paper we presented the results of research on learning design „Measuring Angle‟ in elementary school 179 Palembang by using RMEI approach uses context in this piece of pizza, where the context is very general and close to the pupils in the current era.

The goal of the learning process RMEI approach using context pizza and teaching aid pizza pieces made of carton, namely: (1) to determine whether the pizza and pizza models can help pupils to measure angles with a non-standard unit (2) to determine whether the model pizza can deliver pupils to understand the measurement of angles using a protractor with standard unit (degree).

B. Design Research

In designing learning angle measurement, observer/researcher team took notice about characteristics of design research. Akker and colleagues (2006) formulate a research design as a methodology that has the following five characteristics:

1. Interventionist nature: Design research is flexible due to the design of learning activities can be changed during the study to adjust the learning situation.

2. Process oriented: Design based lesson plans and tools or teaching aid that are used to assist learning.

3. Reflective component: After implementation design learning activities, conjecture analysis of each learning process as compared to the actual learning process. 4. Cyclic character: The process of evaluation and revision. The learning process is

actually used as the basis for revising the next activity.

5. Theory oriented: Design based theory test must relate to teaching experiment. Because this study design has been implemented, it can be explained that the research/observation that we as a observer team do this, the first research design characteristics, interventionist nature, we cannot apply because the learning process only takes place in one meeting, but the learning process is remain flexible and

adapted to the conditions/ circumstances of the learning process that took place at that time. Process oriented that we apply assisted with teaching aid such as model piece of pizza. For the fourth characteristic, cyclic character in the evaluation process we submit suggestions for further research, while our revised limit with only one meeting so we convey through the weakness of the design we made. This design is also in accordance with the theory oriented, which based the design theory and testing associated with teaching experiment.

Stage I: Preliminary Design

In the preliminary design stage we conducted a literature review on the topic that be adapted from Education Unit Level Curriculum and the school syllabus. The standard of competence is to use measurements of angle, length and weight in problem solving while the competency standard is determine the angle with standard unit and nonstandard unit. Further discussions between the team of observer/ researcher with the teacher of the class in the form of conditions of pupils in a group setting, and teaching purposes in the form of equipment and materials used in the learning and adjustment schedule. For pupils in a group setting, the teacher initially wanted the cooperative learning model type think pair share, but as the discussion continued on the use of teaching aid made of cardboard which if the type of think pair share is used it will need more props so we agreed that mathematics learning implemented are cooperative learning model type STAD (Pupil Team Achievement Division).

Some of learning activities designed to facilitate pupils‟ thinking heading angle measurement with standard unit. In detail, the learning activity angle measurement using pizza carton is described as follows:

a. Activity I: pupils are introduced to the context of the problems associated with measuring angles.

In this activity the teacher show the packaging pizza for a few pieces of pizza and gave some questions related to this typical Italian food. What is the shape slice of

pizza if put together? What is the shape of pizza which still intact? This activity aims to lead pupils to the concept of angle measurement.

b. Activity II: pupils measure angles using a model of pizza carton.

In this activity pupils measure angles with a piece of pizza cartons by coincide the angle of pizza carton at the vertex given in worksheet. The purpose of this activity is to bridge pupils‟ way of thinking in solving problems that involve the measurement of angles. It is a bridge to understanding how to measure angles with a protractor. Then assuming pupils know the angle of the circle equal to 360o, then by doing this activity pupils will find the angle of pizza carton.

c. Activity III: pupils measure angles measured at II activity using a protractor. In this activity pupils are proving the angles that have been measured by using pieces of pizza on the previous activity using a protractor. The expected goal that pupils be able to measure the exact angle that is able to find a way to measure angles with respect to the midpoint of the arc, the initial value that is zero degrees, the edge of the scale arc coincides with another angle legs.

d. Activity IV: pupils measure angles individually

In this activity pupils measure more complex angles such as 87o, 103o, and so forth in order to train pupils to be more careful in looking at and calculating the scale shown on the protractor.

Expected in the design process can be seen at the iceberg below.

Stage II: Teaching Experiment

Implementation of learning which refers to a design that has been created by teachers and the research team is conducted on November 14, 2011, in class 4D SDN 179 Palembang for a 2 hour lesson. As for being involved in this activity was a teacher, researchers and pupils of class 4D.

The learning process started with the teacher divides pupils into five groups. Furthermore, teacher explained briefly learning objectives and the everyday life‟s problems associated with the material that will be learned by the pupils. While the researchers helped the teacher by allotting the worksheet and tools media will be used by pupils, the teacher explains a brief description of the pupils‟ tasks. Tools media used are circular pizza toys that made from a paper which has been divided into 12 equal sectors.

Figure 2. contextual problem and the model used

At the first activity that was measuring angle by using a toy pizza, pupils still seem confused about how to put a pizza correctly to measure the angle in the worksheet. But after getting the guidance of teacher and researchers, all groups could measure angles with a pizza toy correctly. Then the teacher gave instruction to pupils to draw dotted lines to indicate one size pizza toy at an angle that is measured. Actually, based on a design that had been made, pupils do not need to make a dotted line as a marker of every unit of the pizza, but the teacher took an initiative to ask

pupils to give dotted lines to make pupils got more understanding one corner unit in question.

Figure 3. pupils measure the magnitude of the angle by using different methods On this activity, there are many methods used by pupils in measuring the angle. One of them is by arranging all the pizzas that can fulfill the angle measured, while the other group putting one pizza then gave the line, and then put a second pizza until they get some number pizzas that fulfill the angle measured.

After performing the measurement of angle by using a toy pizza, pupils continued the activities that are answering the conclusions in the worksheet. At first many groups seemed unable to conclude that the magnitude of one pizza‟s angle is 30°, but with the guidance of teacher and researchers, finally all groups could conclude that the magnitude one pizza is 30°.

Figure 4. Pupils’ conclusion

After that, the teacher asked each group representation to go to in front of the class to present the results of their work and discussions. One pupil read out the results of her work discussion while other pupils listen and match their results. Eventually, all groups had the same results. The teacher then gives explanations and together the pupils conclude about an activity that has done that the magnitude of each pizza‟s angle is 30.

Figure 5. Each groups presenting their work

Then the teacher asked the pupils to do activities and answer the questions on the second worksheet which is about measuring angle by using the standard measuring tool, while the researchers help by distributing the worksheets. At first the pupils still seemed difficult to understand the commands in worksheet. But, with the guidance of teachers assisted with the researchers they could understand the commands in worksheet. On this second activity of each group were asked to measure the angles of the first worksheet, but this time by using a protractor. Although some groups perform the measurement again using the bow as the

worksheet‟s command, researchers found a group that did not perform this measurement, they just write down the back of the angle they've got on worksheet-in the first event.

After all groups have completed all the activities and questions on the second worksheet, the teacher asked all the representatives of the group to present and match the answers respectively. There was a class discussion occurred on this presentation, because there are some differences of the each group‟s answer. The teacher guided the discussion and help pupils to conclude the discussions and activities that they have done. Then on the last activity, the teacher had the pupils to finish the exercise on the pupils‟ book individually.

Figure 7. Presentation of the second worksheet Figure 6. Pupils do the activity in second worksheet

Figure 8. Pupils do the exercise individually

On the measurement of angle by using a protractor in the second worksheet and pupils‟ book, many pupils still had difficulties in measuring the angle that is not a multiple of 10, like 86 and so on. Therefore, the researchers and teacher give some guidance so that pupils are able to measure gradually and correctly.

Stage III: Retrospective Analysis

In general the learning process took place properly and smoothly. Pupils seem enthusiastic following every activity in both groups and individuals provided by the teacher. Each group showed good cooperation and compact in doing each activity which are given. At the beginning of the study, when the teacher explained material by using the context of pizza, pupils seemed very enthusiastic especially when drawing up pizza boxes into one full circle (Figure 9a). Similarly, when pieces of pizza which includes 12 pieces distributed to each group and they were asked to compose pieces of pizza so that it forms a full circle (Figure 9b). All members of the group appear to be snatched away to compose pieces of pizza. Teacher and researchers keep guiding them to get together and not snatched away to organize them.

Figure 9a Figure 9b Pupils were arrange pieces of pizza in their groups

In the first activity on the first worksheet, each group worked cohesively. They were not having trouble measuring the angles given by using a piece of pizza. When putting together pieces of their pizza at an angle that will be measured, as it turned out pupils understood that piece of pizza should be put in the middle of the intersection of two lines that form that angle. The interesting discussions occurred when they were asked to specify a large angle for one piece of pizza (Activity 2).

Figure 10. Pupils seemed enthusiastic in measuring angle by using the pieces of pizza

Most pupils have already known that the angle of a pan of pizza is equal to the angle of circle, 3600. They are a bit of confusion when determining the angle. There were several ways that they used. There was a group that directly used a concept of division. They divided 3600 with 12 slices of pizza and concluded that the angle of each slice of pizza was 300. There is also a group that is trying to find the multiplication of 12 which the result is 360. But at the end, they realized also that the results of 30 that they can get by simply dividing 360 by 12. A pupil in the Group 5 suddenly replied spontaneously that the angle of one piece of pizza that is 300. When he was asked what the reason, the pupil replied that piece of pizza has the same number of digits on the clock and the angle formed between 2 numbers on a clock that is 300. But there was unexpected event when suddenly a pupil used protractor to measure the angle of a piece of pizza.

Figure 11. Pupil’s activity in determining the angle of a piece of pizza

The teacher then asked the pupils to save their protractor. When determining the angle, pupils were not so experienced difficulty, because they just multiplied the number of pieces of pizza with 300. Some of the ways that pupils answered showed that they have a different way of thinking to find the same answer.

On the second worksheet, the difficulties experienced by many pupils when deciding how many piece of pizza that they need to measure the angle of 1350 and 1200 on the number 4 and 5. For question number 4 that determining the angle of 1350 all groups responded correctly that they need 4 ½ pieces of pizza. For determining the number of pieces of pizza at an angle of 350, none of the group answered correctly (4 1/6 pieces). Pupils seemed confused when determining part of 1/6. The answer finally emerged from the pupils during the class discussion was taken over by the teacher.

Figure 12. Pupils’ activity in measuring 350

by using the pieces of pizza and protractor

At the end of the activity, when pupils were required to measure the angles by using a protractor, there were still some pupils who looked confused to put the midpoint of the Protractor. Similarly, while reading a scale printed on the Protractor, few pupils seemed confused to determine a scale where they were supposed to use. With the guidance of teachers and researchers, pupils were instructed to do how to measure exactly. But after finishing the lesson, indeed there was still a small percentage of pupils who were still confused and should receive additional training so that they are increasingly adept at measuring of angle by using a protractor.

Figure 13. Pupil’s mistake when measuring angle by using protractor

C. Conclusion

To teach the concept of angle, especially in measuring angle can be started with the context which is known by the pupils. One of them is the context of the pizza. Round-shaped pizza may be used to introduce the concept of angle. With pieces of pizza which has a great point 300, pupils can use it as a tool of measurement to determine a certain angle.

In general, the learning of measuring angle by using the context of the pizzas running successfully. The learning indicator could achieve well, however there are still many obstacles and problems occurred in the learning process. The pupils looked enthusiastic and worked attractively in their groups when measuring the angles by using a piece of pizza. They measured angle by using a protractor too. Although

pupils still seemed confused and mistaken when measuring angles using a protractor, pupils still looked enthusiastic when measuring the angles. Learning indicators agreed in advance has been achieved in this learning process.

Here this the iceberg of the learning process of measuring angle which was done by pupils in Grade 4 of SD Negeri 179 Palembang:

Figure 14. Iceberg of Learning of Measuring Angles

References

Depdiknas. 2003. Kurikulum Berbasis Kompetensi: Standar Kompetensi Mata Pelajaran Matematika. Depdiknas: Jakarta.

Gredler, M. E.B. 1991. Belajar dan Membelajarkan (Learning and Instruction Theory Into Practice). Terjemahan oleh Munandir. Jakarta: Rajawali.

TIM MKPBM. 2001. Strategi Pembelajaran Matematika Kontemporer. Bandung: Universitas Pendidikan Indonesia.

worksheet‟s command, researchers found a group that did not perform this measurement, they just write down the back of the angle they've got on worksheet-in the first event.

After all groups have completed all the activities and questions on the second worksheet, the teacher asked all the representatives of the group to present and match the answers respectively. There was a class discussion occurred on this presentation, because there are some differences of the each group‟s answer. The teacher guided the discussion and help pupils to conclude the discussions and activities that they have done. Then on the last activity, the teacher had the pupils to finish the exercise on the pupils‟ book individually.

Figure 7. Presentation of the second worksheet Figure 6. Pupils do the activity in second worksheet

Figure 8. Pupils do the exercise individually

On the measurement of angle by using a protractor in the second worksheet and pupils‟ book, many pupils still had difficulties in measuring the angle that is not a multiple of 10, like 86 and so on. Therefore, the researchers and teacher give some guidance so that pupils are able to measure gradually and correctly.

Stage III: Retrospective Analysis

In general the learning process took place properly and smoothly. Pupils seem enthusiastic following every activity in both groups and individuals provided by the teacher. Each group showed good cooperation and compact in doing each activity which are given. At the beginning of the study, when the teacher explained material by using the context of pizza, pupils seemed very enthusiastic especially when drawing up pizza boxes into one full circle (Figure 9a). Similarly, when pieces of pizza which includes 12 pieces distributed to each group and they were asked to compose pieces of pizza so that it forms a full circle (Figure 9b). All members of the group appear to be snatched away to compose pieces of pizza. Teacher and researchers keep guiding them to get together and not snatched away to organize them.

Figure 9a Figure 9b Pupils were arrange pieces of pizza in their groups

In the first activity on the first worksheet, each group worked cohesively. They were not having trouble measuring the angles given by using a piece of pizza. When putting together pieces of their pizza at an angle that will be measured, as it turned out pupils understood that piece of pizza should be put in the middle of the intersection of two lines that form that angle. The interesting discussions occurred when they were asked to specify a large angle for one piece of pizza (Activity 2).

Figure 10. Pupils seemed enthusiastic in measuring angle by using the pieces of pizza

Most pupils have already known that the angle of a pan of pizza is equal to the angle of circle, 3600. They are a bit of confusion when determining the angle. There were several ways that they used. There was a group that directly used a concept of division. They divided 3600 with 12 slices of pizza and concluded that the angle of each slice of pizza was 300. There is also a group that is trying to find the multiplication of 12 which the result is 360. But at the end, they realized also that the results of 30 that they can get by simply dividing 360 by 12. A pupil in the Group 5 suddenly replied spontaneously that the angle of one piece of pizza that is 300. When he was asked what the reason, the pupil replied that piece of pizza has the same number of digits on the clock and the angle formed between 2 numbers on a clock that is 300. But there was unexpected event when suddenly a pupil used protractor to measure the angle of a piece of pizza.

Figure 11. Pupil’s activity in determining the angle of a piece of pizza

The teacher then asked the pupils to save their protractor. When determining the angle, pupils were not so experienced difficulty, because they just multiplied the number of pieces of pizza with 300. Some of the ways that pupils answered showed that they have a different way of thinking to find the same answer.

On the second worksheet, the difficulties experienced by many pupils when deciding how many piece of pizza that they need to measure the angle of 1350 and 1200 on the number 4 and 5. For question number 4 that determining the angle of 1350 all groups responded correctly that they need 4 ½ pieces of pizza. For determining the number of pieces of pizza at an angle of 350, none of the group answered correctly (4 1/6 pieces). Pupils seemed confused when determining part of 1/6. The answer finally emerged from the pupils during the class discussion was taken over by the teacher.

Figure 12. Pupils’ activity in measuring 350

by using the pieces of pizza and protractor

At the end of the activity, when pupils were required to measure the angles by using a protractor, there were still some pupils who looked confused to put the midpoint of the Protractor. Similarly, while reading a scale printed on the Protractor, few pupils seemed confused to determine a scale where they were supposed to use. With the guidance of teachers and researchers, pupils were instructed to do how to measure exactly. But after finishing the lesson, indeed there was still a small percentage of pupils who were still confused and should receive additional training so that they are increasingly adept at measuring of angle by using a protractor.

Figure 13. Pupil’s mistake when measuring angle by using protractor C. Conclusion

To teach the concept of angle, especially in measuring angle can be started with the context which is known by the pupils. One of them is the context of the pizza. Round-shaped pizza may be used to introduce the concept of angle. With pieces of pizza which has a great point 300, pupils can use it as a tool of measurement to determine a certain angle.

In general, the learning of measuring angle by using the context of the pizzas running successfully. The learning indicator could achieve well, however there are still many obstacles and problems occurred in the learning process. The pupils looked enthusiastic and worked attractively in their groups when measuring the angles by using a piece of pizza. They measured angle by using a protractor too. Although

pupils still seemed confused and mistaken when measuring angles using a protractor, pupils still looked enthusiastic when measuring the angles. Learning indicators agreed in advance has been achieved in this learning process.

Here this the iceberg of the learning process of measuring angle which was done by pupils in Grade 4 of SD Negeri 179 Palembang:

Figure 14. Iceberg of Learning of Measuring Angles

References

Depdiknas. 2003. Kurikulum Berbasis Kompetensi: Standar Kompetensi Mata Pelajaran Matematika. Depdiknas: Jakarta.

Gredler, M. E.B. 1991. Belajar dan Membelajarkan (Learning and Instruction Theory Into Practice). Terjemahan oleh Munandir. Jakarta: Rajawali.

TIM MKPBM. 2001. Strategi Pembelajaran Matematika Kontemporer. Bandung: Universitas Pendidikan Indonesia.