CONSTRUCT THE IDEAS IN MATHEMATICS LEARNING

Idea means a design that is arranged inside a human mind, can include opinion and ideals. Construct means build, helping to improve something. Construct the ideas means helping to improve a design that arranged in a human mind that makes a real idea.

Basically the idea is not easy to build. Moreover, developing ideas in mathematics learning. Not as easy as we imagine. In solving mathematical problems there is need for the development of ideas that allows us to solve the problem. Before we begin to work on problems in mathematics, we must understand a few important steps before we start.

First, we have to develop ideas related to the problem. This is useful to find all the possibilities that will appear when we solve the problem such as a double meaning or a hidden purpose in it. Developing an idea related a problem, never make us doing wrong the steps in our works.

The second is crafting ideas that have been developed into a draft of mind. So it becomes a whole design idea. Through constucting the ideas we can determine which steps we will take to resolve the problem.

Next is to solve the problem. In solving the problem, we can use the design idea that was conceived and developed before so that we'll find easier way to find the right answer.

By using the steps above, hopefully we can solve Mathematics problems better and  get a correct answer. In learning mathematics the most important is the process towards the right result, not on the final result regardless of the process. The processes that we do, can build our own ideas and can build up our knowledge. This process will add our experience in solving the math problems correctly. So that we build a world, ideas, and knowledge of our own without realizing it. And it would be easy to stick in our minds than solve out the problem without regard to the process.

In Indonesia, mathematics learning still focuses on teachers, so that students only be described as an empty barrel that every day continues to be filled and refilled. Students are difficult to grow, especially in developing their ideas, because every day they just accept, never do something.

Then, learning system in Indonesia is too open minded, still searching for the identity. So that the teaching methods sometimes adopted from other countries.

In mathematical problem solving in Indonesia, is still dominated by the teacher. Because they assumed that teachers teach as a professional obligation in accordance with government regulations and have to achieved all learning indicators of competency standards that have been applied.

However, many schools are implementing student center. Student center is a method of student-centered learning, students become a major role in the learning process, and the teacher only as a facilitator who facilitates the students when faced with learning difficulties. By applying this method of student center, students are expected to be able to develop ideas and construct their own ideas to solve mathematical problems according to their ability and their own way. Teachers are not allowed to impose a method to students. That way they can build their own knowledge, without being forced.

In Japan, the mathematical problem solving do not know "contstruct". They do not construct ideas before solving problems, but they are directly to solving the problem. It is can be said that students can not develop properly. Besides that, problem solving is also used as a character study. So that learning becomes semi-authoritarian and contstructivis not go well. Students are not given the opportunity to build their ideas and solve problems in their own way, so here the teacher took a leading role in the learning process. It is also influenced by the culture in Japan too.

While in Australia, they applied Mathematical Modelling method. Mathematicals modeling is a method of learning mathematics that uses models as a tool in the learning process. This method is used for teaching mathematics in primary schools. Due to the use of models or props students will be more easily aroused and easily capture the point that the teacher has been submitted. This model will facilitate students in learning mathematics. They can develop their skills by using models or learning aids.

According to Max Stephens, models used in the learning of mathematics should be varied and not monotonous, so students are more interested and not get bored Modelling problems need to be designed so that multiple solutions of varying mathematical sophistication are possible and such that students with a range of personal experiences and knowledge can participate In this way, the mathematical experiences of students become more challenging, authentic and meaningful.

. A good teacher in the learning does not focus on text books. Rather they should be able to develop a medium of learning for students. Learning mathematics in pattern-making is required. Because the pattern gives us the ability to find something we have not understood so that we can understand the meaning of the pattern. Involvement of students in the learning process of mathematics provide more knowledge and experience for them. So that they can easily capture the intent conveyed.

Mathematics can not be separated from the numbers. The number is mathematical knowledge. Function of the numbers is to measure the quantity, to decide, to the make something more specific, make a definition. Number is one of the pattern. The number is also often used to construct the problems in mathematics.

As a teacher we need to rationalize in teach mathematics to the students. It’s means that we have to be able to explain the phenomenom of mathematics education as a reference in teaching mathematics. Teacher personality is desperately needed in teach mathematics in class. Because teacher personality in teach mathematics is very influental to the students’ mental developing.

In addition, in constructing the ideas, teachers also need to know about the theories and international paradigm in learning mathematics. So that they can apply them appropriately in learning process at school. Teachers also must be able to see the difference between students' competence, so that they can deliver the right system to their students. Cultural differences are also very influential in the teaching of mathematics in schools.

Mathematics has links with the science of religion. Some concepts in mathematics are also used for religious purposes such as the construction of a Borobudur temple in Central Java, the level and shape are using the concept of a complex geometry and the construction is perfect. Religion has the highest position in life. It is effective if religion and mathematics knowledge can runs together and ballance.

Sources:

http://www.criced.tsukuba.ac.jp/math/apec/apec2012/0215/Max.ppt

General lecture on Monday, 12 March 2012 by Prof. Mohan Chinnappan, PhD.

PROBLEM IN DEVELOPING INTERNATIONAL LEVEL OF TEACHING LEARNING OF MATHEMATICS

In the era of globalization and the increasingly sophisticated technologies require an individual to have different abilities and skills more than other. Not only in economic or cultural course, but also education is very important. Creativity and problem solving skills are very necessary in the face of this globalization era. One means to improve the creative and problem-solving abilities in the field of education is through the learning of mathematics. However, the learning of mathematics were frequently encountered problems. Some problems in developing an international level of teaching learning of mathematics are as follows:

-          Low competitiveness at the international level, for example in the Olympics

-          The low average value of UNAS (National Final Exam)

-          The low interest in learning and achievement among students learn mathematics

-          Submission of material that is less attractive from teacher to student

-          Low quality studies supporting books

Some of the above are examples of common problems in developing an international level of teaching learning of mathematics.

Problems of low achievement and student interest in learning math because Mathematics learning in Indonesia is still dominated by the exercises are basic. It is not go into mathematics as a whole. Most teachers only explain a mathematical concept which only emphasized the importance of numeracy skills, algorithmic steps, and the memorization of formulas. This is why the students only get the basics mathematical skill.

If we linked to the internationalization of mathematics education, the achievement of basic skills is not enough. Mathematics learning, now and in the future, can not rest only on achieving basic skills, but instead should be designed to achieve a high level of mathematical competence (Parwati: 2008). If the teacher is still at the stage of explaining mathematical basics skills course, students most likely will just remain there.

In this globalization era as much as possible the teacher should be able to answer a variety of challenges at the international level. If the teacher remains at its foundation, traditional in nature, how they can develop and compete internationally. How could they make competent students in mathematics?

The next problem is the lack of teachers’ creativity in teaching. Most teachers are still fixated on the available curriculum. They only teach what is written in the syllabus. Most of them seldom add material that is actually very important but not listed in the syllabus.

Lack of teachers’ creativity in teaching demonstrated in the number of teachers who are still fixated on the material in the textbook for teaching materials. It also resulted in the ability of teachers to provide a question in the test. They are still hooked to the textbook. As a result, they give a question for test similar to the examples problem in the textbook. Furthermore, in providing a mathematical problem, teachers often assign only one correct answer. In addition students are also required to answer the questions given in accordance with what has been previously demonstrated by the teacher.

The above shows the teaching-learning process is still teacher centered. Teachers still dominate the teaching and learning activities. Also according to Marpaung (in repository.upi.edu) students are almost never given a chance to try out their own strategies to solve problems and the teacher did not dare to take decisions that are in the interest of the class curriculum. It can be concluded that the learning of mathematics in schools still require innovations that demands creativity of teachers in improving learning strategies.

Learning process in schools should be used students centered system. Here the teacher is not as central to the learning process, the teacher only as a facilitator and dynamic factor. All processes should be centered to student learning. Students are faced with a problem and they are required to complete it in accordance with their own creativity. Many ways and methods that can be develop creative skills in learning mathematics. One way is to apply the learning model based on creative problem solving. This learning model is expected to be able to nurture creative skills and problem solving skills through the habit of thinking and being creative in understanding and solving mathematical problems. In the end the habit of thinking and being creative will have a positive effect on student behavior in the face of their daily lives (repository.upi.edu : 2009). One way to improve the creativity of students in the study is to provide an open-ended problem. The presentation of open-ended problems will be able to train students to think of creativity as a problem presented requires students to think about the possibility of correct answers (Shimada, S., 1997).

The next problem is the lack of books for math investigations. Most of the books in circulation has a very low quality, because many books were written by people that do not involve a skilled person in mathematics education or mathematics teacher. As a result of the outstanding books do not meet the qualifications specified by the school. So,  a teacher  needs to develop students worksheets or LKS. LKS contains a summary of the material and practice questions that are routinely made ​​by the teachers themselves. By making the exercises by their own, teachers can determine the levels of difficulty of the questions that correspond to the ability of students. By using LKS,  a teacher could develop questions that are open-ended problem that not only to measured students' ability to calculate, but  also hone their creativity.  The books such as those described above can be used as a complement to learning (Parwati, 2008).

Developing competencies in creative thinking among students is very important in this era of global competition, because  the complexity of the problems in all aspects of modern life is increasing. Ability to think creatively considered a high order competencies  and can be viewed as a continuation of basic competence (commonly referred to as learning basic skills in mathematics) (Sudiarta: 2010).

REFERENCES:

Parwati, Ni Nyoman. 2008. Analisis Kebutuhan Pengembangan Model Pembelajaran Matematika Berpendekatan Tematik Berorientasi Pemecahan Masalah Terbuka pada Sekolah Dasar  di Provinsi Bali, Jurnal Pendidikan dan Pengajaran UNDIKSHA, Oktober 2008

repository.upi.edu

Shimada, S. & Becker, P., 1997. The Open-Ended Approach: A New Proposal for Teaching Mathematics. NY: NCTM.

Sudiarta, I G. P.  2010. Pengembangan Pembelajaran Berpendekatan Tematik Berorientasi  Pemecahan Masalah Matematika Terbuka untuk Mengembangkan  Kompetensi Berpikir Divergen, Kritis dan Kreatif, Jurnal Pendidikan dan Pengajaran UNDIKSHA.

The Review On The Educational Theories

Theory of Cognitive Development Jean Peaget

Jean Piaget's theory provide restrictions on intelligence, knowledge and students’ relationship with the environment. Intelligence is a continuous process of forming the necessary structures in continuous interaction with the environment. knowledge is highly subjective when an individual is still baby and in early childhood and will be objective when enters adulthood. Jean Piaget, designing models that describe how humans understand the world by collecting and organizing information.

According to Piaget (in Woolfolk: 2009) cognitive development is affected by maturation, activity and social transmission. Then Piaget divides cognitive development of children into four main periods are correlated with and more sophisticated with age:

1.      Sensory-motor period (years 0-2)

According to Piaget, baby born with some innate reflexes as well as and encouragement to explore their world. The scheme was originally formed through the differentiation of the innate  reflexes.

2.      Preoperational period (ages 2-7 years)

In this stage, children learn to use and represent objects with images and words. Their thinking still egocentric (the child difficult to see from others' viewpoints.)

3.      Concrete operational period (years 7-11)

In this stage a child has been able to sort certain objects by size, shape or other characteristics, then the child will begin to classify, to consider, understand, and observe an object.

4.      Formal operational period (11 years of age to adulthood).

Characteristics of this stage is the acquisition of the ability to think abstractly, reason logically, and draw conclusions from available information. In this stage, someone can understand such things as love, logical proof, and value.

According to Piaget (in Sugihartono, 2007: 109-110), the human mind has a structure called a schema or cognitive structure. By using the scheme, a person can process and identify the stimuli receives so that they can put it in a category/ corresponding concept.

By using the scheme, someone can adapt and coordinate their environment to form new schemata through assimilation process (taking an information from the outside world and then adjust the pre-existing structures) and accommodation (individual adjustment to a new thing), which are both regulated by equilibration (adjusting continuity between assimilation and accommodation).

Implications of Piaget's view of learning is the teacher must adapt the learning process is carried out with the cognitive stages of the students. Because if there is no adjustment between the two, teachers and students will have difficulty in teaching and learning and the learning objectives will not be achieved.

Piaget believes that good education is education that involves the child to experiment on their own, in a sense: 1.) Try everything to see what happens; 2) Manipulate signs and symbols; 3) Asking the question; 4) Find their own answers; 5) Match what he had found at some point with what he found at the other; 6) Comparing the findings with the findings of others.

            Realistic Mathematics Education (RME)

Realistic Mathematics Education (RME) is a mathematics learning approach developed by Freudenthal in Netherlands. Gravemeijer (1994: 82) said Realistic mathematics Education is rooted in  interpretation of Mathematics as an activity, such as problem-solving activities, find and organize the subject problem. Learning mathematics means doing mathematics, of which everyday life problem solving is an essential part.

Other concept in realistic mathematics education is state by Treffers (in Fauzan, 2002: 33-34). The key idea of ​​RME is Children should be given the opportunity to reinvent mathematics under the guidance of an adult (teacher). In Addition, the formal mathematical knowledge can be developed from children's informal knowledge.

The statements above describes a perspective of mathematics learning placed as a process for students to find their own mathematical knowledge based on informal knowledge they have. In this view of mathematics is presented not as “a finished good" that can be moved by the teacher into the minds of students.

Freudenthal mentions two types of Mathematization in mathematics learning:

1.      Horizontal mathematization, involves going from the world of life into the world of symbols, the examples are  identification, formulation and visualization problems with different ways by students.

2.      Vertical mathematization means moving within the world of symbol, the example of vertical mathematization are presentation of the relationships in the formula, refine and adjust the mathematical model, using different models, the formulation of mathematical models and generalization.

According Soedjadi (2001: 3), there are five characteristics in the learning of mathematics:

1.      Using the context, means that the environment around us or the knowledge that has been owned by students can be used as a part of contextual learning materials for students.

2.      Using the model, means that the mathematical problems or ideas can be expressed in the form of model from the real situations that lead to the abstract level.

3.      Using the contributions of students, means that the concept of problem solving or discovery is based on the idea of ​​students' contribution, in this case the students play a full part in the learning process.

4.      Interactive, means that learning activities are built by the students' interaction with students, students with teachers, students with the environment and so on, so that communication or interaction among class members to stay awake and facilitate the learning process because there is no boundary or divider that prevents them to interact.

5.      Intertwin, which means that teachers can use the different topics that can be integrated so as to bring an understanding of a concept at once.

In realistic mathematics learning, someone can build their own knowledge through the process of active learning. A concept does not instilled through the process of transferring, but someone who is learning should be given the freedom and encouragement to express their thoughts and construct knowledge for themselves. In addition, it is also necessary for interaction in the realistic mathematics learning, both among students, teachers and students, and so on.

The implications of the social aspect that is high enough in the learning activities of students, then teachers need to determine appropriate teaching methods and in accordance with those requirements. One method of teaching that can meet these objectives is the inclusion of a discussion in the learning activities of students.  Discussion activity is deemed able to encourage and to expedite interaction between students. In addition, the negotiation and evaluation between students and teachers are an important factor in this constructive learning.

            Constructivism Theory

Constructivist theory is defined as generative learning, which acts to create a  meaning of what is learned. Constructivism better understand learning as a human activity to build or create knowledge by giving meaning to the knowledge in accordance with their experience. Constructivism approach has some common concepts such as:

1.      Active learners build knowledge based on existing experience.

2.      In the context of learning, students should develop their own knowledge.

3.      The importance of fostering knowledge actively by students through the interplay between past learning with new learning.

4.      The most important element in this theory is student  actively build their knowledge by comparing new information with existing understanding.

5.      The imbalance is a major motivating factor learning.

6.      Teaching materials are provided to have a docking with the student experience to attract students.

There are two forms of constructivist, the construction of psychological (cognitive) and social constructivist. Cognitive constructivist emphasis on knowledge, confidence, self-concept, or identity who is focused on the people psychological live. In other words, cognitive constructivist focus on how individuals use the information, resources, and assistance of others to build and improve mental models and strategies in solving the problems they faced. In the learning, teacher intend to transfer concepts, ideas, and the knowledge of something to students, it will be interpreted and constructed by students themselves through the experience and knowledge. Social constructivist emphasis on the social interaction that occurs, the cultural, as well as activities that determine the development and individual learning to a problem. In  social constructivist, learning environment is influenced by a complex, real and suitable to be applied in the learning process. For example, the assignment to students to understand and gather information on various livelihood around him.

In an effort to implement constructivist learning theory, Tytler (in Sugihartono, 2007: 115) put forward several suggestions relating to the design of learning, as follows: (1) provide an opportunity for students to express his ideas with his own language, (2) provide an opportunity for students to thinking about his experiences to become more creative and imaginative, (3) provide an opportunity for students to try new ideas, (4) provide experiences that relate to the idea that has been owned by the student, (5) encourage students to think about changing their ideas, and (6 ) creating a conducive learning environment.

Sources:

Anita Woolfolk. Educational Psychology. Edisi Bahasa Indonesia. Yogyakarta: Pustaka Pelajar, 2009.

Fauzan, A. 2001. “Pengembangan dan Implementasi  Prototipe I  & II Perangkat Pembelajaran   Geometri   untuk   Siswa Kelas   IV   SD  Menggunakan Pendekatan RME”.  Paper presented at the National Seminar of Realistics Mathematic Education (RME) in UNESA Surabaya, 24 February 2001.

Gravemeijer, K. 1994. Developing Realistic Mathematics Education. Utrecht: Freudental Institute. Retrieved: http://ironerozanie.wordpress.com/2010/03/03/realistic-mathematic-education-rme-atau-pembelajaran-matematika-realistik-pmr/

Siswoyo, Dwi, dkk. 2008. Ilmu Pendidikan. Yogyakarta: UNY Press.

Soedjadi, R.. 1999. Kiat Pendidikan Matematika di Indonesia. Jakarta: Dirjen Dikti Depdikbud.

Sugihartono, dkk. 2007. Psikologi Pendidikan. Yogyakarta: UNY Press.