REALISTIC MATHEMATICS THROUGH ICEBERG APPROACH OF LEARNING CIRCLE IN JUNIOR HIGH SCHOOL

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.  

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

Formal mathematics

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.

USING ENGLISH FOR MATHEMATICS EDUCATION

A.      INTRODUCTION

As a human, we need to communicate with other people. To communicate with other people, we need languages. There are so many languages in the world. Indonesia itself also has many local languages. The most popular language is English. The English language, which takes its name from its origins as the native tongue of the people of England, is today the most widely spoken language in the world. Because English is so widely spoken, it has often been referred to as a "world language". It is spoken as a first language by a majority of the several nations, including the United Kingdom, the United States, Canada, Australia, Ireland and New Zealand. Now days, in Indonesia also try to use English in education and school beside bahasa Indonesia as mother language and besides local languages. Some school in Indonesia use English to deliver teaching learning process, including teaching learning of mathematics. 

B.      USE ENGLISH FOR MATHEMATICS EDUCATION

English is an international language, so we all try to learn English, practice English, write in English, speak English, read English article, communicate in English, and listening English from native speaker. Not only learn the English well, we also have to know about England culture. Because using English or learn English without knowing the culture is not recommended. If we live in Yogyakarta or Bali as a tourist destination, we need English to communicate with tourist. Salesman in Malioboro street also can communicate with tourist using English. If we want to go overseas, we also need English to communicate with other people. There are many books and references write in English. So, we as educate person have to communicate in English. English is very important for us, because after we learn English we may know the world well.  Communication is the activity of conveying information. Communication has been derived from the Latin word "communis", meaning to share. Communication requires a sender, a message, and an intended recipient, although the receiver need not be present or aware of the sender's intent to communicate at the time of communication; thus communication can occur across vast distances in time and space. Communication requires that the communicating parties share an area of communicative commonality. The communication process is complete once the receiver has understood the message of the sender.    There are four kinds of communication, such as  Nonverbal communication, Oral communication, Written communication and its historical development, barriers to effective human communication. Therefore, English is important for our life, English is our personal need, English is use to communicate with other people around the world.

English is a personal need, so we also using English in the school, include teaching learning of mathematics. There are the factor affect the teaching learning process, such as: educational facilities, media/teaching media, teaching aid, teaching methods, psychological aspect, context of teaching learning (culture), curriculum that can be explained into syllabi, lesson plan, student worksheet, teaching learning resource, and assessment, paradigm that formed by theories and theories that formed by philosophy, technology. All elements above always move and affect the teaching learning process. There are many ways to develop teaching learning of mathematics. First of all, we have to make our students more active in the class, because student able to develop their own creativity. I think that there must be an interaction between student and student, student and teacher in the class. we use English to make interaction with students, so English is very useful. If there are interaction between student and student, student and teacher, so teaching learning of mathematics change from teacher centered to student centered. The other benefits are student become critical thinker, able to connect mathematics in our daily life, and more creative. The other ways to develop teaching learning of mathematics are developing and implementing lesson plan, apperception, various method of teaching, various media/teaching aid, various interaction, the scheme for achieving student competencies (cognitive scheme), small group discussion, student’s reflection/presentation, student work sheet, and assessment.

If we are talking about teaching learning of mathematics, so it is relate with curriculum development. From year to year, Indonesia always try to improve the curriculum and try to find the best curriculum which can be used for all school in Indonesia and can improve students understanding and students achievement that imply in national examination. Now, Indonesia use SBC (school based curriculum). In SBC, every school can develop syllabi, lesson plan, student worksheet, teaching learning resource, teaching material, and assessment itself. Here, teacher has opportunity to make an innovative syllabi, lesson plan, student worksheet, teaching learning resource, teaching material, and assessment his/herself or collaborate with other teacher.

After curriculum development, there is activity to build the theory. Teachers or lecturers also often have research about teaching learning, about science, about student activities, try to develop the theory or create new theory. Because, the characteristic of professional teacher is who continually improves the quality of teaching and learning. There are many kinds of research, such as qualitative research, quantitative research, small research, class action research, experiment, lesson study, and so on. Qualitative research is a method of inquiry employed in many different academic disciplines, traditionally in the social sciences, but also in market research and further contexts.  Quantitative research refers to the systematic empirical investigation of social phenomena via statistical, mathematical or computational techniques. Then, the characteristics of class action research are research in a certain place, no need control class/comparison class, certain topics, certain class, there are phases, there are improvement/reparation in every phase, and the result can’t be generalized. On the other hand, experiment can be generalized. The phase in class action research are reflection, planning, action, observe, reflection again. Then, lesson study is action to increase teacher capability to arrange learning process in the class by research and make reflection systematically about learning process. The aim of the lesson study is to increase teacher’s attendance about what students need for their study. The point of lesson study is to make relationship or cooperation between teacher and teacher or teacher and lecture to observe, understand, lesson, and make reflection about learning process in the class. There are some steps to create lesson study, such as: preparation, do it, reflection and correction 1, reflection and correction 2.

After we can build the theory, the high level is reflective the paradigm and philosophy. We know that paradigm is formed by theories and theories that formed by philosophy, technology. Then, philosophical research is conducted by field experts within the boundaries of a specific field of study or profession, the best qualified individual in any field of study to use an intellectual analysis, in order to clarify definitions, identify ethics, or make a value judgment concerning an issue in their field of study. So, after we can build the theory, we cannot just stay in that step, but we must try to develop our ability by producing scientific publication. We may publish our achievement not only with our friends, but show to all people in the world using English as international language and using smart technology.

Then, after we study about the world characteristic, compare the traditional and innovative way of mathematics teaching, quantitative and qualitative approaches, use English for mathematics education, we know that “I’m the world, I have everything, mathematics is myself, English is myself, teaching learning of mathematics is myself.”

C.      CONCLUSION

English is an international language. English is important for our life, English is use to communicate with other people around the world. There are some steps to use English for mathematics education. First of all, English is our personal need to communicate with other people or communicate mathematics itself. Because, English is a personal need, so it is include teaching learning of mathematics. If we are talking about teaching learning of mathematics, so it is relate with curriculum development such as develop syllabi, lesson plan, student worksheet, teaching learning resource, teaching material, and assessment itself. After curriculum development, there is activity to build the theory, develop the theory or create new theory. Then, after we can build the theory, the high level is reflective the paradigm and philosophy by producing scientific publication.

THE COMPARISON OF QUANTITATIVE AND QUALITATIVE APPROACH

A.      INTRODUCTION

In education world, we often hear or see or do a research. Teachers or lecturers also often have research about teaching learning, about science, or about student activities. Because, the characteristic of professional teacher is who continually improves the quality of teaching and learning. There are two kinds of research such experiment and class action research. The characteristics of class action research are research in a certain place, no need control class/comparison class, certain topics, certain class, there are phases, there are improvement/reparation in every phase, and the result can’t be generalized. On the other hand, experiment can be generalized. Research can be done by two methods/approaches such as quantitative approach and qualitative approach.      

B.      THE COMPARISON OF QUANTITATIVE AND QUALITATIVE APPROACH

In a research there are two kinds of methods/approaches such as quantitative approach and qualitative approach. Besides that, there are also quantitative and qualitative data. We need data to do a research. Then, we will examine the differences between qualitative and quantitative data. Quantitative data usually deals with numbers. The characteristic of quantitative data is data which can be measured. Example: length, height, area, volume, weight, speed, time, temperature, humidity, sound levels, cost, members, ages, etc. On the other hand, qualitative data usually deals with description. The characteristic of qualitative data is data can be observed but not measured, such as colors, textures, smells, tastes, appearance, beauty, etc. For example if we want to observe an oil painting, we get quantitative data like the length of oil painting is 20 inch x 14 inch, with frame 3 inch, the weight of oil painting is 8 pounds, surface area of oil painting is 280 square inch, and the cost is Rp 500.000,00. We get qualitative data such as the colors are red, green, blue, yellow, orange, purple; the frame is brown, texture shows brush strokes of oil paint, peaceful scene of the country, smells old and musty. Other example, if we want to observe teaching learning of mathematics in class VII, we get quantitative data such there are 32 students, 20 girls, 12 boys, 70% students get score more than 7 in mathematics, average mathematics scores of students is 8,5. We get qualitative data like many students enjoy learn mathematics, there are interesting discussion among students, facilities in the class is complete, there are good environment, etc.

Before we compare quantitative and qualitative approach, we should know the meaning of quantitative and qualitative research. Qualitative research is a method of inquiry employed in many different academic disciplines, traditionally in the social sciences, but also in market research and further contexts. Qualitative researchers aim to gather an in-depth understanding of human behavior and the reasons that govern such behavior. The qualitative method investigates the why and how of decision making, not just what, where, and when. Hence, smaller but focused samples are more often needed than large samples. Quantitative research refers to the systematic empirical investigation of social phenomena via statistical, mathematical or computational techniques. The objective of quantitative research is to develop and employ mathematical models, theories and/or hypotheses pertaining to phenomena. The process of measurement is central to quantitative research because it provides the fundamental connection between empirical observation and mathematical expression of quantitative relationships. Quantitative data is any data that is in numerical form such as statistics, percentages, etc.     

Qualitative research is used to explore and understand people's beliefs, experiences, attitudes, behavior and interactions. Forms of the data collected can include interviews and group discussions, observation and reflection field notes, various texts, pictures, and other materials. Qualitative research often categorizes data into patterns as the primary basis for organizing and reporting results. In the academic social sciences the most frequently used qualitative research approaches include the following:

1.     Ethnographic Research, used for investigating cultures by collecting and describing data that is intended to help in the development of a theory. This method is also called “ethnomethodology” or "methodology of the people". An example of applied ethnographic research is the study of a particular culture and their understanding of the role of a particular disease in their cultural framework.

2.     Critical Social Research, used by a researcher to understand how people communicate and develop symbolic meanings.

3.     Ethical Inquiry, an intellectual analysis of ethical problems. It includes the study of ethics as related to obligation, rights, duty, right and wrong, choice etc.

4.     Foundational Research, examines the foundations for a science, analyzes the beliefs, and develops ways to specify how a knowledge base should change in light of new information.

5.     Historical Research allows one to discuss past and present events in the context of the present condition, and allows one to reflect and provide possible answers to current issues and problems. Historical research helps us in answering questions such as: Where have we come from, where are we, who are we now and where are we going?

6.     Grounded Theory is an inductive type of research, based or “grounded” in the observations or data from which it was developed; it uses a variety of data sources, including quantitative data, review of records, interviews, observation and surveys.

7.     Phenomenology describes the “subjective reality” of an event, as perceived by the study population; it is the study of a phenomenon.

8.     Philosophical Research is conducted by field experts within the boundaries of a specific field of study or profession, the best qualified individual in any field of study to use an intellectual analysis, in order to clarify definitions, identify ethics, or make a value judgment concerning an issue in their field of study.

Data analysis in qualitative research can be done by coding, recursive abstraction, and mechanical technique.

Quantitative methods are those which focus on numbers and frequencies rather than on meaning and experience.  Quantitative researcher asks a specific, narrow question and collects numerical data from participants to answer the question. The researcher analyzes the data with the help of statistics. The researcher is hoping the numbers will yield an unbiased result that can be generalized to some larger population. Qualitative research, on the other hand, asks broad questions and collects word data from participants. The researcher looks for themes and describes the information in themes and patterns exclusive to that set of participants. Quantitative research is used widely in social sciences such as psychology, economics, sociology, and political science, and less frequently in anthropology and history. Research in mathematical sciences such as physics is also 'quantitative' by definition, though this use of the term differs in context. In the social sciences, the term relates to empirical methods, originating in both philosophical positivism and the history of statistics, which contrast qualitative research methods. Qualitative methods produce information only on the particular cases studied, and any more general conclusions are only hypotheses. Quantitative methods can be used to verify, which of such hypotheses are true. Quantitative research is generally made using scientific methods, which can include:

1. The generation of models, theories and hypotheses
2. The development of instruments and methods for measurement
3. Experimental control and manipulation of variables
4. Collection of empirical data
5. Modeling and analysis of data

Data analysis in quantitative research can be done by using statistics and measurement.

  The comparison of quantitative and qualitative approach:

Quantitative approach	Qualitative approach
1.The method is standard.	1.The method is flexible.
2.Emphasize in formal method.	2.Emphasize in both formal and informal.
3.Using sample in the research	3.using population in the research
4.Using large scale in research	4.using small scale in research
5.use to examine theories.	5.use to find and to build theories.
6.there are inferential statistics.	6.there are describe statistics.
7.this is about experiment.	7.this is about hermeneutics (to translate and to be translated)
8.assessment using score in national examination.	8.assessment using class room based evaluation (portfolio).
9.shown in traditional learning	9.shown in innovative learning.
10.use quantity of data.	10.use quality of data.
11. there are manipulation data.	11. the data is contextual.
12.the research is impersonal.	12.the research is personal
etc	etc

Not only that, in the qualitative approach, there are originality, creativity, value/culture, dynamic, flowing, communication/expectation/assumption, relationship, process, life, update, experience, intuitive, improvisation, psychology, philosophy, humanism, ethnography, class room based research, explanation, language, honestly, and so on. In the quantitative approach, there are abstracted, symbol, rigid, perfectly prepare, model, formula, probability, representative, survey, and others.   

In the social sciences particularly, quantitative research is often contrasted with qualitative research which is the examination, analysis and interpretation of observations for the purpose of discovering underlying meanings and patterns of relationships, including classifications of types of phenomena and entities, in a manner that does not involve mathematical models. 

In most physical and biological sciences, the use of either quantitative or qualitative methods is uncontroversial, and each is used when appropriate. In the social sciences, particularly in sociology, social anthropology and psychology, the use of one or other type of method can be a matter of controversy and even ideology. Qualitative methods might be used to understand the meaning of the conclusions produced by quantitative methods. Using quantitative methods, it is possible to give precise and testable expression to qualitative ideas. This combination of quantitative and qualitative data gathering is often referred to as mixed-methods research.  Quantitative methods have come under considerable criticism.  In modern research, most psychologists tend to adopt a combination of qualitative and quantitative approaches, which allow statistically reliable information obtained from numerical measurement to be backed up by and enriched by information about the research participants' explanations.

C.      CONCLUSION

In a research there are two kinds of methods/approaches such as quantitative approach and qualitative approach. Besides that, there are also quantitative and qualitative data. Quantitative data usually deals with numbers and can be measured. Qualitative data usually deals with description and can be observed but not measured. Qualitative research is a method of inquiry employed in many different academic disciplines, traditionally in the social sciences, but also in market research and further contexts.  Quantitative research refers to the systematic empirical investigation of social phenomena via statistical, mathematical or computational techniques. The comparison of quantitative and qualitative approach: in the quantitative approach, the method is standard, emphasize in formal method, using sample in the research, using large scale, examine theories, there are inferential statistics, experiment, shown in traditional learning,  use quantity of data, there are manipulation data, the research is impersonal, etc. In the qualitative approach, The method is flexible, Emphasize in both formal and informal, using population in the research, using small scale in research, to find and to build theories, there are describe statistics, hermeneutics, shown in innovative learning, use quality of data, the data is contextual, the research is personal, etc. In modern research, most psychologists tend to adopt a combination of qualitative and quantitative approaches, which allow statistically reliable information obtained from numerical measurement to be backed up by and enriched by information about the research participants' explanations.

References:

http://en.wikipedia.org/Qualitative_research.htm.

http://en.wikipedia.org/Quantitative_research.htm.

http://www2.warwick.ac.uk/Quantitative%20and%20Qualitative%20Approaches.htm.