By : Margaretha Madha Melissa / 09301244013
Mathematics Education’09
A. Introduction
Mathematics in Junior High School have its function to encourage the students to think logically, analytically, systematically, critically, creatively and be able to collaborate with others. Those competencies are needed for the students in order that they can get, access, and employ information to preserve their live. The aims of teaching learning of mathematics in Junior High School are: to understand the concepts of mathematics, explain the relationships among the concepts and apply the concepts to solve the problems accurately and efficiently; to develop thinking skills, learn patterns, and characteristics of mathematics, manipulate them in order to generalize, to proof and to explain ideas and mathematics propositions; to develop problems solving skills which covers understanding the problems, outlining mathematical models, solving them and estimating the outcomes; to communicate mathematics ideas using symbols, tables, diagrams and other media; to develop appreciations of the uses of mathematics in daily life, curiosity, consideration, and willingness to learn mathematics as well as tough and self-confidence. Therefore, teacher should teach students with realistic mathematics. It will make teaching learning process more interest, easy to understand, and students may relate or make connection between mathematics and daily life.
B. Content
Many students assume that mathematics is difficult subject, abstract, study about numerical, memorize many formula, not interesting, and not relate to our daily life. So, teacher has to change students’ assumption. Now days, there is new teaching model, realistic mathematics. In realistic mathematics, mathematics must be close to children and be relevant to every life situations. However, the word ‘realistic’, refers not just to the connection with the real-world, but also refers to problem situations which real in students’ mind. There is mathematization process in realistic mathematics. There are two types of mathematization, horizontal and vertical mathematization. In horizontal mathematization, the students come up with mathematical tools which can help to organize and solve a problem located in a real-life situation, transferring a real world problem to a mathematical problem. On the other hand, vertical mathematization is the process of reorganization within the mathematical system itself.
Now, we will study about circle using realistic mathematics through iceberg approach. What is iceberg approach? Iceberg approach is an approach that illustrate like ice mountain that represent some steps from daily life problem/real world problem to be mathematical problem. In the iceberg approach, there are some steps such as: concrete mathematics, concrete model of mathematics, formal model of mathematics, and formal mathematics. To understanding about concrete mathematics, concrete model of mathematics, formal model of mathematics, and formal mathematics, we will review a textbook “Mathematics for Junior High School Year VII” by Marsigit. This book is based on School Based Curriculum (SBC) 2006. This book contains 7 chapters for year VII in Junior High School. We will focus on chapter 6 about a circle, then we will identify where is the concrete mathematics, concrete model of mathematics, formal model of mathematics, and formal mathematics.
1. Concrete mathematics
Concrete mathematics is real world problem that apply in a figure about things in our daily life then followed by statements that support the figure and leads to the mathematical topic. Concrete mathematics in the topics circle can we look at page 171 in Mathematics for Junior High School Year VII (Marsigit, 2009). In that page, we can see the figure of wheel. Then, beside the figure, there are statements like “Wheel was used initially as a spinner while making porcelain. Then, it was used to move a cart. Starting at this, the used of wheel was spread out all over the world. Currently, round shaped like wheel can be found in everything, like machine inside your watch, and tyres of your bicycle. Trucks and trains are also used wheel. The radii of wheels are also various from just several millimeters up to tens of maters. Although it can be use for many things, they apply the same principles. What are the principles? You may guest!” The figure of wheel as spinner while making porcelain and wheel in train leads students to the topics circle. Students will more easy to understand about circle by look at the figure.
1. Concrete model of mathematics
Concrete model of mathematics is specific figure in our daily-life that show a subtopic from a big topic. Concrete model of mathematics will help students to imagine and easy to understand the problem relate to the subtopic. We can see concrete model of mathematics on page 211 171 in Mathematics for Junior High School Year VII (Marsigit, 2009). Here, the subtopic is the minimal length of belt connecting two circles. The figure describes a chain that is twisted on two gears. Then, it is followed by question, “Have you ever thought to calculate the minimal length of the chain needed in order to move the gears?” The question useful to make student curious and try to find how to calculate the minimal length of the chain needed in order to move the gears.
1. Formal model of mathematics
Formal model of mathematics is specific mathematical figure that show a subtopic from big topic. We can see formal model of mathematics for the minimal length of belt connecting two circles on page 213 in Mathematics for Junior High School Year VII (Marsigit, 2009). The figure in formal model of mathematics not about things in our daily life, but figure in mathematical. In this case, the figure of the minimal length of belt connecting two circles is two circle with different radii that connecting by external tangent alliances. In the figure, radii for big circle is assume as R, radii for small circle is assume as r, the external tangent alliances is assume is show as 
, the circumference angle in big circle is assume as a, and the central angle in small circle is assume as (3600 - a). In that figure we also can see the characteristic of external tangent line, always perpendicular to a radius of the circle that has end point on the point of tangency.
, the circumference angle in big circle is assume as a, and the central angle in small circle is assume as (3600 - a). In that figure we also can see the characteristic of external tangent line, always perpendicular to a radius of the circle that has end point on the point of tangency. 1. Formal mathematics
Formal mathematics is mathematical writing, using symbol, number, variable, constant, operation, and so on. We can see formal mathematics about the minimal length of belt connecting two circles on page 213 in Mathematics for Junior High School Year VII (Marsigit, 2009). In that case, formal mathematics is shown by formula to find the minimal length of belt connecting two circles based on formal model of mathematics (figure 6.32).
A. Conclusion
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In realistic mathematics, mathematics must be close to children and be relevant to every life situations. We may use realistic mathematics through iceberg approach to teach our students. In iceberg approach, there are four steps such as: concrete mathematics, concrete model of mathematics, formal model of mathematics, and formal mathematics.
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