Nikmatul Husna (firstname.lastname@example.org)
Sri rejeki (email@example.com)
According to Van Hiele, there are five levels of hierarchy in geometry learning (the yearbook NCTM). The five levels are: level 0 (visualization), level 1 (analysis), level 2 (abstraction), level 3 (deduction), and level 4 (rigor). The level of visualization is often referred to as the recognition rate. Students are familiar with the basic concepts of geometry involving simple plane figure as square, triangle, rectangle, parallelogram and others. At this level of analysis the students already understand the concept or informal geometry based on the analysis of parts and components. At the level of deduction, deductive reasoning students were beginning to develop, but is not well developed. At the level of rigor, students are able to work on a system of axioms.
As stated by Wubbels, Korthagen & Broekman in Yenni B. Widjaja & Andre Heck (2003), in the mechanistic approach, mathematics is seen as a system of rules and algorithms, doing mathematics is analogized with machine works which means verifying and applying the rules to problems that are similar to previous problems. The downside of this approach to learning is when students forget the formula area of a circle, then certainly they can not find the correct answer.
Based on the explanation above, students need a learning experience that can make them understand the concept of how to calculate the area of circles. Rote merely short-term memory and without a deep understanding in its concept, students will be difficult to solve various problems. Therefore, the observers along with classroom teachers design learning area of a circle through Pendidikan Matematika Realistik Indonesia (PMRI) approach that begins with the context of the covers of circular cake tins and circle of cardboard that has been cut into 16 sections to find the formula for the area of a circle by analogized the concept of calculating the area of plane figure that has been learned in the previous grade.
This learning activity held on Friday, November 9, 2012, in the sixth grade B at Elementary School 98. Students who are involved in the learning process amounts to 33 students comprising 17 men and 16 women. Learning activities are carried out in collaboration with the sixth grade math teacher Mrs. Kartini, S.Pd.
B. RESEARCH DESIGN
1. Preliminary Design
1.1 Curriculum Analysis
Before the observers make instructional design, observers analyze the first semester of sixth grade Elementary School. This material is located on the third competency standards that calculate the area of simple polygons, area of circles and the volume of triangular prisms. While the basic competence is calculating the area of circles. Before studying the material area of a circle, the students have learned the value of π and circumference of a circle.
1.2 Learning Design
Design learning activities conducted by the observers with a sixth grade mathematics teacher who is Mrs. Kartini, S.Pd. Learning begins with a real-world context of the covers of circular cake tins. This context is taken with the assumption that goods are often encountered by students in their daily lives. Learning activities continued to provide opportunities for students to develop their understanding in concept which will be found. Media used in the learning process is several circles that has been divided into 16 sections each.Learning activities are planned in a group discussion. The following are student activities and conjecture or students hypotesis.
First Activity: Arranging the 16 pieces into various shapes of plane figure
This activity is the first step in finding the formula of the area of circles that has a relationship with the previous material which is various forms of plane figure. These activities are in accordance with the characteristics of the intertwining of realistic mathematics education or linkage between concepts in mathematics.
Activity Description: Each group gets a cardboard circle. The given carton has been divided into 16 sections and students are asked to cut it. The cutting pieces of the circle have to be set up to be various forms of plane figure that has previously been studied.
Allegations of student’s thinking:
- Each group Each group will arrange the pieces of circle into a different plane as these following figure.
Figure 1. Various shape of plane figure from the cutting pieces of circle
- Students are only able to put together pieces of the circle into the form of a parallelogram, plane figure which is the most easily prepared like this following figure.
Figure 2. Parallelogram which is shaped from the cutting pieces of circle
- Students are not able to put together pieces of the circle into the form of an irregular plane figure.
Second Activity: Finding the formula of the area of plane figure which has been formed
In the second activity students can find the area of a circle formula based on broad plane figure preconceived. This activity can work well if students understand the formula for the area of plane figure.
Activity description: The teacher guides students in finding the area of a circle formula by asking the parts on a plane figure that is formed in the form of the base or height. Students are required to link the base and height as the parts of the circle.
Allegation of student’s thinking:
- Students assume that the base of the formed plane figure resemble a straight line which is part of the circumference of circle.
- Students understand that the height of the formed plane figure is the radius of the circle.
- In determining the length of the base, students only count part of it. Students do not link between how many parts with the circumference of a circle.
2. Teaching Experiment
In this activity, the teacher is Nikmatul Husna while the other group members, Sri Rejeki and the sixth grade teacher acted as observers. The learning process begins by providing students apperception by asking the previous material learned which is finding the value of π and circumference of a circle formula. After that, the teacher asks a plane figure with a formula that has been studied before. Students mention plane they have learned are: rectangle, square, parallelogram, trapezoid and triangle. When the teacher asked the students if the formula of broad flat wake up, students have difficulty in answering. Only a small percentage of students who still remember what the formula for the area flat wake. For the vast trapezoidal formula none of the students who still remember. Finally broad trapezoidal formula question can be answered by a student with a notebook open.
Apperception activity continued by providing contextual issue of the area of a circle. Contextual problem given is a mother’s desire to beautify the circular cake tin lid. Mother wants to cover the cake tin lid by providing wrapping paper on the top. What is the required minimum of wrapping paper to cover the lid Eid cookies. Students are asked to think about the issue separately and ask students to give their opinions on the issue. Some students gave their opinion that the problem is related to the area of a circle. Once the activity is completed apperception, teachers continue to give motivation to the students about the importance of studying the area of a circle and continued by stating learning objectives.
The process of learning activities given followed by an explanation by the teacher about the activities that will be carried out today to find the area of a circle formula. A circle is divided into several pieces which is usually referred to as pie. The pieces can be arranged into various forms of plane figure. The teacher asks the students what are the plane figure that can be formed from pieces. Students mention a wide range of plane figures namely: rectangle, parallelogram, triangle and trapezoid.
Teacher asks students to sit in groups according to divided group at the previous meeting. Each group received two half-circles made of cardboard with different colors, scissors, glue and paper flow chart. The purpose of providing two half circles with different colors just to make a good color combination in plane figure formed. In a given circle there have been lines that will help the students to divide the circle into 16 parts.
Each group was asked to cut the circle into 16 pieces. Then the students were asked to arrange the pieces of the circle into a plane figure as they want. Each group looks so enthusiastic in participating in this activity. They shared task and discuss how the shape panel to be formed.
Figure 3. Students work together to form plane figure from the cutting pieces of the circle
In the activities of forming a plane figure from the cutting pieces of circle, there are some groups who find it difficult. They are difficult to adjust the position of the circle pieces. There are several groups that directly attach to the circular piece of paper flow chart, but flat wake formed irregular. The group was forced to give up the pieces again. This activity requires creativity and cooperation among members of the group. Of the eight groups that exist. There are four kinds of flat wake is formed.
Figure 4. Plane figures which are formed by students
After each group finished form the plane figure from circular pieces, teachers provide guidance to students in finding the area of circle formula. Teachers begin by asking how the base of plane figure formed to students. At first the students associate with the confusion in the circle. The teacher explains to the students that in any part of the base plane figure formed a part of the circumference of a circle. Teacher asks students to calculate how many pieces of the circle on any part of the formed plane figure, after it compared to how much all the pieces of the circle. The value obtained is associated with the circumference of a circle formula. After that, the teacher goes on to provide questions about height or width of the formed plane figure. Teachers draw a line from the edge of the end piece to the pieces in the plane figureof the circle formed. Seeing the line, students can understand that the height or width of the formed plane figure is the radius of the circle. Once students understand the relationship between plane figure and its circular form, students are asked to find the area of circles formula based on the formula of broad plane figure. Teachers along with other observers around the groups to provide assistance to students who are experiencing difficulties. After all the groups finished in finding the formula circumference of a circle, the next activity is displaying the work of students on the board and present it.
During the learning activity, there were students who used their mobile phone. As the student was observed by the observers several times and they still use it for learning, observers take their mobile phone until the lesson is completed. The mobile phones is placed on the teacher’s desk and at the end of the lesson, the student was only allowed to take it.
Not all the planned activities can be accomplished. To discuss the activities of the wider contextual circle previously provided can not be implemented because of the class period has expired.
3. Retrospective Analysis
After implementing learning design that has been designed, observers and sixth grade math teacher doing the reflection. Overall the teaching and learning process is progressing well. Students actively pursue all activities in learning. Fellow group members established good cooperation because they have to share in the task of completing the work presented.
In apperception activity, the teacher gives the students questions about the formula for the area of plane figure they have learned before. Many students have forgotten the formula for the area. Not even a single student who remember the formula for the area trapezoid. Should sixth grade students are not having problems with the formula for the area of plane figure. Since the basic competencies prior to this lesson is to calculate the area in terms of a lot which is a combination of two simple plane figure. Based on these basic competency, students should have frequent use of the formula for calculating extensive broad polygons.
In the activity composed pieces wake cycle into various plane figure, students look so enthusiastic in messing ngatik position cuts the circle. All members of the group are actively involved in arranging it. Of the eight groups that exist, there are four kinds of flat wake that is formed is a rectangle, two kinds of parallelograms and triangles. But no one group that can form a trapezoid. Trapezoid is one-up that can be formed from flat pieces of circles.
In the event, look for a formula area of a circle based on a formula that forms broad plane figure, students have high doubts in determining which is the radius of the circle. Because previous teachers provide guidance to students by taking the rectangle which form a single wide radius circle, the triangle formed group also considered that the height is the fruit of your fingers. Though the height of the triangle they form is four fingers. After the teacher gives understanding back to the group, eventually they can write the correct step in finding the area of a circle formula.
Each of the planned activities can not be all done. To discuss the activities of the wider contextual circle previously provided can not be done. This is because teachers do not control the use of time well and are impaired in learning. One of these disorders is the finding of a few students who use their mobile phone in learning activities. When students are asked to give their mobile phone objections and time consuming enough until finally they would give up
In realistic mathematics education recognized the iceberg that describe how the students’ understanding of mathematical concepts from the real thing to the formal stage, where students understand abstract mathematical symbols.
Figure 5. Iceberg in finding the area of circles formula
Based on the observations of the two observers, the area of a circle with a learning context circular cake tin lid, the students can perform the activity well though there are some students who do not participate actively in group work because students are not accustomed to working in learning. The results of student work also showed positive effects in which students can arrange the pieces of the circle into three distinct wake the flat rectangle, triangle and two different parallelograms.
However, not all the planned activities can be accomplished. To discuss the activities of the wider contextual circle previously provided can not be implemented. This is because teachers do not control the use of time well and there is an interruption in the learning