Education as an important role in preparing qualified human resources who competent and capable in the development of science and technology, so that education should be implemented as well as possible to get maximum results. Education should be managed both in quality and quantity. This can be achieved with the implementation of education that on time and appropriate to achieve the learning objectives, which take the form of teaching-learning process as the implementation of the school curriculum through teaching activities.
Many schools have implemented both the learning of mathematics by improving the quality of learners, learning mathematics easy and fun must be developed. Various concepts, methods, and strategies should be developed so that the creation of learning especially in mathematics that had been considered the student is not fun to be fun and there needs to be creative teachers. Teachers may use the methods developed in mathematical learning outside the classroom if they can help create a fun learning math.
Mathematics known as basic science, learning math will train critical skills, logical, analytical and systematic. But the role of mathematics not only to the extent that, like other fields, such as physics, economics, biology can not be separated from the role of mathematics. But the progress of physical science itself can not be achieved without the development of mathematics and mathematics itself.
The purpose of learning mathematics is the formation of the students' reasoning abilities, as reflected by the ability of critical thinking, logical, systematic, and has an objective character, honesty, discipline, in solving a problem both in the field of mathematics and other areas of everyday life.
Each state has its own way to give to their students learning mathematics, in this article will explain the idea of learning mathematics in the State of Indonesia, Japan, U.S., and Singapore.
1) Indonesia
According to the Law of the Republic of Indonesia Number 20 of 2003 on National Education System Chapter IX of the National Education Standards Article 35 paragraph 1 that national standards of education consists of content standards, processes, competence of graduates, educational personnel, facilities and infrastructure, management, financing, and educational assessment should be planned and periodically upgraded. It results in a requirement for the school as a reference for curriculum development, educational personnel, facilities and infrastructure, management, and financing.
In Indonesia, the learning of mathematics are applied almost all school text book oriented and tend to be less related to students' daily lives. Which tend to be abstract mathematical learning, while most teachers are still paying less attention to teaching thinking skills of students, or in other words a creative learning. As the methods used are less variable, do not make teaching meaningful, and as a result, students' motivation to be difficult to be grown and studied the pattern of tend to be rote and mechanical.
The new paradigm of education is now more emphasis on the learner as a human being who has the potential to learn and grow. Various approaches for learning mathematics is too influenced by the view that the mathematical tools at its disposal. This view tends to encourage teachers to be informed concepts / theorems and how to use it. Teachers tend to transfer knowledge to the minds of students and students passively receive it and not critical. Sometimes students answer questions with completely, but they can not reveal the reasons for their answers. Students can use the formula but do not know where it came from and why the formula is a formula that is used.
In the learning is never apart from the name of the curriculum, the curriculum in Indonesia is in use is the Education Unit Level Curriculum (KTSP/ Kurikulum Tingkat Satuan Pendidikan), curriculum and learning is very important and need each other. What is described in the curriculum must provide instruction in the classroom learning process. With the use of learning media such as computers, can bring objects to serve as an example in the form of an image or animation that is more interesting and memorable, so that students are learning more fun can be felt and not boring. In learning mathematics students usually have difficulty abstract nature of matter, the matter should have used a media or props, it's where the role teaching aid and computer as a learning tool.
There are several factors, they are factors that influence learning activities that include factors of teachers, students, facilities, equipment and media are available, as well as environmental factors. This is what sometimes impede the development of the learning process. Mastery of learning objectives is not the subject matter, but the process for changing student behavior in accordance with the will to achieve. For this reason the methods and strategies that teachers do not just lecture method, in Indonesia there are still many who use these methods, it should use methods, such as discussions, assignments, visits to specific objects or by media such as visual aids and computer.
Mathematics learning objectives are: (1) understand math concepts, explain the link between concept and apply the concepts or algorithms in flexible, accurate, efficient, and appropriately in problem solving, (2) using the pattern and nature of reasoning, mathematical manipulations in making generalizations , compile evidence, or explain mathematical ideas and statements, (3) solve problems that include the ability to understand the problem, design mathematical model, complete model and interpreted solution obtained, (4) communicate ideas with symbols, table, diagram, or other media to clarify circumstances or problems, (5) has the respect of the usefulness of mathematics in life, which has the curiosity, interest, and interest in studying mathematics and a tenacious attitude and confidence in problem solving (Depdiknas, 2006). Based on these objectives the government has made reforms and efforts to make improvements in the educational system, such as improving the curriculum, to enhance teachers' skills through courses. Nonetheless, student learning outcomes are still low, especially in mathematics', the reality of each examination (National Examination) the average student who did not pass the math. This is a problem for teachers to choose teaching methods that interest students to learn to create interest and motivation for students to excel which will also support the results of studying mathematics.
In the learning is never apart from the name of the curriculum, the curriculum in Indonesia is in use is the Education Unit Level Curriculum (KTSP/ Kurikulum Tingkat Satuan Pendidikan), curriculum and learning is very important and need each other. What is described in the curriculum must provide instruction in the classroom learning process. With the use of learning media such as computers, can bring objects to serve as an example in the form of an image or animation that is more interesting and memorable, so that students are learning more fun can be felt and not boring. In learning mathematics students usually have difficulty abstract nature of matter, the matter should have used a media or props, it's where the role teaching aid and computer as a learning tool.
There are several factors, they are factors that influence learning activities that include factors of teachers, students, facilities, equipment and media are available, as well as environmental factors. This is what sometimes impede the development of the learning process. Mastery of learning objectives is not the subject matter, but the process for changing student behavior in accordance with the will to achieve. For this reason the methods and strategies that teachers do not just lecture method, in Indonesia there are still many who use these methods, it should use methods, such as discussions, assignments, visits to specific objects or by media such as visual aids and computer.
Mathematics learning objectives are: (1) understand math concepts, explain the link between concept and apply the concepts or algorithms in flexible, accurate, efficient, and appropriately in problem solving, (2) using the pattern and nature of reasoning, mathematical manipulations in making generalizations , compile evidence, or explain mathematical ideas and statements, (3) solve problems that include the ability to understand the problem, design mathematical model, complete model and interpreted solution obtained, (4) communicate ideas with symbols, table, diagram, or other media to clarify circumstances or problems, (5) has the respect of the usefulness of mathematics in life, which has the curiosity, interest, and interest in studying mathematics and a tenacious attitude and confidence in problem solving (Depdiknas, 2006). Based on these objectives the government has made reforms and efforts to make improvements in the educational system, such as improving the curriculum, to enhance teachers' skills through courses. Nonetheless, student learning outcomes are still low, especially in mathematics', the reality of each examination (National Examination) the average student who did not pass the math. This is a problem for teachers to choose teaching methods that interest students to learn to create interest and motivation for students to excel which will also support the results of studying mathematics.
Can be seen, the low quality of education in terms of process, is the notion that during this process of education in Indonesia in the wake of the teachers tend to be limited to the mastery of subject matter or resting on a low level of cognitive development, which is able to develop creative thinking processes of education or the learning process is considered likely to place students as objects to be filled with a variety of information and materials rote. Communication occurs in one direction, namely the teacher to the students through expository approach is used as the main tool in the learning process.
The changes of this learning paradigm requires changes in the learning process and other matters including those relating to infrastructure. Facilities and infrastructure should be stimulated so that learning is centered on the learner can done optimally. In fact most of the facilities and infrastructure in various types and levels of education in Indonesia does not yet support the implementation of the desired learning. Current conditions in Indonesia showed a lot of schools do not have adequate infrastructure in terms of quantity and quality so that in particular in learning mathematics itself must have the facilities and infrastructure such as visual aids, computers and so on.
Facilities and infrastructure are vital to the advancement of learning in mathematics. Mathematics is a universal science that underlies the development of modern technology, have an important role in various disciplines and promote the human intellect.
As a professional person, a teacher has five main tasks of the learning plan, carry out the study, evaluating learning outcomes. Following up on the learning outcomes, the rest to guidance and counseling. ICTs can certainly play a role in the five main tasks are performed under the applicable curriculum.
The use of instructional media is very important, because the media can make the learning of procedural knowledge and declarative knowledge becomes more interesting and memorable, so that students perceived the learning experience more concrete. In addition the use of computers as instructional media can facilitate teachers in presenting the material and facilitate the students to absorb what the teacher explained. Learning through a computer is a form of learning which are individually designed in a way students interact directly with the subject matter in a special program through the computer system.
Each of the learning process must show active learners. In the most important mathematical learning is the process of thinking done. Students are trained to develop the ability of think logical, analytical, systematic and consistent. To assist in the thinking process or the drawing and animation can be used so that students will more easily and overcome difficulties in the natural. Thus, learning in Indonesia that are still monotonous with a variety of methods and strategies should be developed for the purpose of learning to run properly. Learning math is difficult indeed, but that difficulty can be easy when the student is interested in the methods applied by teachers can be fun, exciting attention and motivated to learn mathematics.
2) Japan and United States (U.S.)
Undergraduate and K-12 mathematics education in the United States have seen many reforms during the past decade. With the initial results of the Third International Mathematics and Science Study (TIMSS) appearing in October,1996, there is also considerable interest in mathematics education elsewhere, especially Japan. A major revision in the K-12 curriculum in Japan has recently occurred, and the Mathematical Society of Japan has formed the Working Group for Undergraduate Mathematics, a committee of educators and mathematicians, to examine undergraduate mathematics education in Japan. The success of the Japanese educational system in producing students who excel in mathematics is well-known and is pointed out in the results of TIMSS. This study was sponsored by the International Association for the Evaluation of Educational Achievement and involved approximately fifty nations world-wide. TIMSS focused on grades four, eight, and twelve, with Germany, Japan, and the United States receiving special attention. Unlike the National Council of Teachers of Mathematics Standards, which merely makes recommendations for K-12, Japan has a nationally set curriculum. Since a fifth grade mathematics class in Tokyo will be covering roughly the same material as a class In Nagasaki or Sapporo during any given time of the academic year, Japanese educators have an opportunity to collaborate and polish lessons on a nationwidescale. This is not the case in U.S. schools, where the curriculum is locally controlled. In Japan, the achievements of students reflect the benefits of coherent goals and focused teaching practices.
In Japan, Munbusho, the Japanese Ministry of Education, sets the number of class periods for the year, the length of the class periods, the subjects that must be taught, and the content of each subject for every grade in K-12. For this reason, changes in the Japanese educational system are usually introduced more cautiously than in the United States, and possible curriculum revisions are evaluated more carefully before being put into effect. Technology-based courses of the type that one often sees in U.S. classrooms are not as popular in Japan, and Japanese educators generally seem to prefer a more traditional, theoretical, and problem-solving based course. Even though the current curriculum standards encourage the use of calculators beyond the fifth grade, calculators are still not allowed in many Japanese classrooms. Computers seem to be more prevalent in the Japanese classroom than hand-held technology.
The elementary school curriculum is specified in Japan for grades 1-6. The goals and objectives of mathematics education at the elementary school level are to develop in children fundamental knowledge and skills with numbers and calculations, quantities and measurements, and basic geometric figures. In grades 1-3 children learn about the concept of numbers and how to represent them, the basic concepts of measurement, how to observe shapes of concrete objects and how to construct them, and how to arrange data and use mathematical expressions and graphs to express the sizes of quantities and investigate their mathematical relationships. They acquire an understanding of addition, subtraction, and multiplication, learn how to do basic calculations up to the multiplication and division of whole numbers, and learn how to apply these calculations. Children also become acquainted with decimal and common fractions during this time. The soroban or abacus is introduced in grade 3. Children learn basic concepts of measurement such as reading a clock, comparing quantities of length, area, and volume, and comparing sizes in terms of numbers. They are also taught the concepts of weight and time and shown how to measure fundamental quantities such as length.
By the end of grade 4, children are expected to have mastered the four basic operations with whole numbers and how to effectively apply them. They also should be able to do addition and subtraction of decimals and common fractions. In grades 5 and 6, children learn how to multiply and divide decimals and fractions. They are taught to understand the concept of area and how to measure the area of simple geometric figures and the size of an angle, as well as to understand plane and solid geometric figures, symmetry, congruence, and how to measure volumes. Children learn about the metric system during this time. Teachers show how to arrange data and use mathematical expressions and graphs to help children to become able to express the sizes of quantities. Letters such as x and a are introduced. Children also begin to learn about statistical data by using percentages and circle graphs (pie charts). It is recommended that calculators be introduced into the classroom in grade 5 to ease the computational burden.
Lower secondary school in Japan consists of grades 7-9. Preparation to get into the best high schools and universities begins at this time. There is tremendous pressure on students to perform well. Students are asked to learn a tremendous amount of material in grades 7-12, which is perhaps one of the major reasons why university and secondary school classrooms are often subdued. In contrast, elementary classrooms tend to be lively, with a great deal of interaction between students and teachers. In either case, classrooms are teacher-directed. The student-directed group learning that is found in some U.S. classrooms is virtually nonexistent in Japan.
In grade 7, students learn about positive and negative numbers, the meaning of equations, letters as symbols, and algebraic expressions. By the end of grade 8, they are able to compute and transform algebraic expressions using letter symbols and to solve linear equalities and simultaneous equations; they have also been introduced to linear functions, simple polynomials, linear inequalities, plane geometry, and scientific notation. In grade 9, students learn how to solve quadratic equations (those with real solutions) and are taught the properties of right triangles and circles, functions, and probability. In grade 7 and beyond it is recommended that calculators and computers should be efficiently used as the occasion demands.
In high school (grades 10-12), six mathematics courses are offered: Mathematics I, II, III and Mathematics A, B, and C. Although only Mathematics I is required of all students, those students intending to enter a university will usually take all six courses. In fact, Japanese high school students who take all of the courses offered will know more mathematics than many U.S. students do when they graduate from college. In Mathematics I, students are taught quadratic functions, trigonometric ratios, sequences, permutations and combinations, and probability. Mathematics II covers exponential functions, trigonometric functions, analytic geometry (equations of lines and circles), as well as the ideas of limits, derivatives, and the definite integral. Calculus is taught in Mathematics III, including functions and limits, sequences and geometric series, differential and integral calculus. More advanced topics such as Taylor series are usually not taught in Mathematics III. Mathematics A deals with numbers and algebraic expressions, equalities and inequalities, plane geometry, sequences, mathematical induction, and the binomial theorem. Computation and how to use the computer are also taught in this course. In Mathematics B, students learn about vectors in the plane and 3-space, complex numbers and the complex number plane, probability distributions, and algorithms. Mathematics C consists of a variety of topics, including matrix arithmetic (up to 3x3 matrices), systems of linear equations and their representation and solution using matrices, conic sections, parametric representation and polar coordinates, numerical computation including the approximate solution of equations and numerical integration, and some calculus-based statistics.
3) Singapore
Ever since Singapore scored first in both 4th and 8th grades in the Trends in International Math and Science Study (TIMSS) comparison assessments in 1995, 1999, and 2003, and in the top three in 2007, mathematics educators have been interested in the secret of Singapore’s success. While many factors have been catalogued – a coherent national curriculum, teacher training, a public belief in the importance of math to the national economy – without fail, all descriptions emphasize the importance of problem solving in the Singapore curriculum.
Interestingly, the first Singapore math curriculum, which was written in the 1980s, did not emphasize problem solving. It was not until 1991, and the writing of a new curriculum in 1992 that Singapore began emphasizing problem solving in its curriculum. The Singapore Ministry of Education uses a graphic to represent their vision for mathematics teaching: a pentagon, with problem solving in the center and these five interdependent, necessary elements surrounding it. Textbooks, written specifically to address this structure, provide constant support for understanding all five elements. Students are encouraged to consider how they think, how they communicate, and how they solve problems, so they can apply their skills to subsequent problems. In its latest efforts, the Ministry is working to increase student communication skills and metacognition during problem solving.
Despite the increased emphasis on problem solving in the United States, students in Singapore continue to perform better in math. There are five major reasons for this difference in performance : 1. Problem solving is embedded in Singapore texts, not as a separate activity but as central to every skill and concept discussion. 2. The problems that Singapore students work on are much more complex than those in typical American texts. Two- and three-step problems are the norm. 3. Non-routine, as well as routine problems are included in every grade level. 4. Students are taught specific problem-solving strategies in a carefully sequenced manner beginning in second grade. 5. Student attitudes are addressed and supported.
Each time a new concept is introduced in the Singapore textbooks, problem solving is central to the discussion. Consider the concepts of perimeter and area. Just as they do in the U.S., third graders in Singapore learn about these concepts. But in Singapore, students immediately consider them from several perspectives. So rather than just calculating the perimeter of a square with side length 3, students are asked:
A square of side 3 cm is formed using a piece of wire. The wire is straightened and then bent to form a triangle with equal sides. What is the length of each side of the triangle? (MIF 2nd grade)
When learning about area of rectangles, students don’t just calculate height times width, but are asked to find the side length of a square that has the same area as an 8 x 2 rectangle. In other words, whenever a skill or concept is taught, it is also applied in a problem setting. The problem may not even look like other problems that are modeled. Students learn that a single skill can be applied to a wide range of problems.
Complex word problems are the norm in the Singapore textbooks. Yet students begin with very simple problems.
There are 5 boxes of pencils. Each box has 12 pencils. How many pencils altogether?
By the end of the third grade chapter, however, they solve problems such as this one:
Shawn and Trish scored 36 goals in all. Shawn scored three times as many goals as Trish. How many goals did Trish score?
or
Flo saves 4 times as much money as Larry. Maria saves $12 less than Flo. Larry saves $32. How much does Maria save?
In other words, the Singapore textbooks take seriously the proposition that the purpose of math is to solve complex problems. Multiple-step word problems are introduced in the primary grades and become increasingly challenging in the higher grades. Thoughtfulness is displayed, though, in the simplicity of the initial problems. As students develop confidence, they tackle more complicated problems, including non-routine ones.
In addition to complex word problems, Singapore curriculum emphasizes non-routine problems those that go beyond the application of specific computation. Solutions to such problems often require a number of different strategies, or heuristics. The Singapore curriculum and textbooks build on the work of Polya by teaching specific strategies for problem solving, what Polya called heuristics. Many of these—such as “look for a pattern,” “draw a picture,” “simplify the problem”, and “work backwards”, are included in American texts. But too often, American texts teach these strategies by “type.” That is, students learn “look for a pattern” and then they are presented with problems that can be solved in that way. In Singapore, students are instead encouraged to consider which strategy will work best for a
particular problem. They are introduced to the strategies or heuristics and then given a variety of non-routine problems to solve. Let’s look now at a third grade example.
Mr. King has a total of 19 geese, chickens and ducks on his farm. He has 3 more chickens than geese. He has 2 fewer ducks than geese. How many ducks does he have?
Try to solve this. Then imagine a third grade student solving it. Singapore students are taught to draw models to represent the situation. Students might use the most basic strategy of guess and check or they may try it more systematically. For instance, one student took 19 counters, assigning 1 to the duck 3 to the geese and 6 to the chickens. The student then added
a counter to each row until all 19 had been used. Other students used a “work backwards” strategy. They know that if they take away 5 chickens and 2 geese, then they will be left with three equal piles. Since there are 12 counters or animals left, this means there must be 4 of each, hence 4 ducks. Notice that many of these strategies lead to the generalization students will use later in algebra to solve the problem. Encouraging multiple approaches and evaluating their effectiveness is the essence of good problem solving.
The most famous and developed of the heuristics (strategies) taught in Singapore is “model drawing,” sometimes called “bar modeling” in the U.S. Beginning in 2nd grade, students are taught to use rectangular shapes to model a word problem. These models: help students visualize abstract math relationships through pictorial representations, use rectangular blocks because they are easily divided, can be used before students know algebraic solutions, and can be used to model algebraic relationships. These models begin quite simply and typically model word problems. In the Grade 2 problem at right, a rectangle, divided into two parts, models a simple subtraction situation. There are 20 eggs, 7 are duck eggs. How many are chicken eggs? Other models include simple multiplication and division. By 5th grade, however these models increase significantly in complexity. In fact, model drawing proves to be a powerful tool for non-routine as well as routine problems. Consider the following problem:
25% of the fish in a bowl are guppies. The same number of guppies as were originally in the bowl are then added. What percentage of the bowl is now guppies?
Initially students are confused because there do not appear to be enough numbers. Yet a simple diagram can help. Students draw a rectangle, color one-fourth as guppies. Then they add a length to the rectangle that is the same size as the guppies portion. The picture shows clearly that 2 of the 5 parts of the bowl are guppies or 40%. Careful attention to the teaching of heuristics, to moving from simple to complex problems, and to sequencing the problems in such a way as to move from routine to non-routine problems is a hallmark of the textbooks. Still, problem-solving does not account for all the differences in performance between students in Singapore and the US. Attitudes toward mathematics and the development of metacognitive skills also play a role.
In Singapore, a country lacking natural resources and half the size of New York City, human capital is the most precious resource. The Singapore curriculum and textbooks recognize that developing problem-solving skills and creativity is a requirement for the 21st century. This belief, in addition to careful attention to teaching and cultivating problem-solving skills, are lessons worth considering as we try to make our students competitive in the global marketplace in which they live.
As has been explained above, if we want to built seriously the mathematics learning in schools, the method of active and constructive learning must be apply to our students to become quality of human resources. Schifter and Fosnot said that the process of using that method we need strong desire (kemauan), considering that the students and teachers habitually with old paradigm (teacher explain-student listening and follow the teacher’s instruction) and also the social-culture factors which different in other states.
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