Mathematics is one of the important science is taught in every country around the world. Since mathematics is a science that is widely used in everyday life. Mathematics taught in elementary school to university. In learning mathematics, in every country has an individual way according to the custom existing in the country. Because in thinking of mathematics have a diversity of knowledge or skill. This is because there are many purposes to learn mathematics. Therefore, the need for the development of mathematical thinking especially in learning mathematics. There are a variety of learning methods that have been formulated by various experts in the world, this method can be used in teaching mathematics in accordance with the material and conditions. The method can be used in between the methods of creative thinking, problem-based learning, problem solving, problem posing, etc. This method can be applied in various countries.
APEC-Ubon Ratchathani, International Symposium 2-5 November 2011.
Singapore Context: works of Peggy Foo
Singapore schools adopted the Teach Less, Learn More (TLLM) movement which is about teaching better, to engage our learners and prepare them for life, rather than teaching for tests and examinations. There is more quality in terms of classroom interaction, room for expression, and the learning of life-long skills such as thinking through innovative and effective approaches. At the same time, there is less focus on rote-learning, repetitive tests, and following prescribed answers and set formulae.
The emphasis on thinking permeates the teaching and learning of every subject including Mathematic. The revised syllabus continues to emphasize the conceptual understanding, proficiencies of skills and thinking skills in the teaching and learning of Mathematics. Teachers were asked to provide more opportunities for students to discover, reason and communicate Mathematics. Students were encouraged to engage in discussions and activities where they can explore possibilities and make connections.
The lesson study process consists of different parts – goal setting, research lesson planning, lesson teaching and evaluation, and consolidation of learning. For goal setting, a group of teachers identifies the research theme. It could be students’ weakness in an area of learning, or a topic which educators find challenging to teach. The lesson planned is then taught by one teacher while the other team members are present to observe the lesson and make notes as well as collect evidence of student learning and thinking. There is then a discussion involving everyone in the lesson study group, evaluating and reflecting on the lesson. The original lesson plan is then revised. Finally, there is a consolidation of learning for sharing purposes. This is seen as an important part of the lesson study process as the consolidating and sharing consist of the findings about teaching and learning, together with the teachers’ reflections. With the knowledge gained, teachers can use them in planning and conducting future lessons.
Japanese Context: the works of Isoda Masami
Learning through problem solving, a style of mathematics instruction that has fascinated U.S as well as researchers for decades. U.S researchers have called the Japanese approach “structured problem-solving”, because of the explicit lesson structure. In a structured problem solving lesson, students work on a carefully chosen problem that illuminates a new mathematical understanding to be developed. By grappling independently with the problem, and then sharing and building ideas as a class, students are able to be progress from their initial thinking to a new understanding of the mathematical concept. For example, students who know only how to calculate the area of a rectangle may initially be puzzled when asked to find the area of a parallelogram, and some may try to multiply its dimensions. However, if students investigate they can find ways to rearrange the parallelogram to make a rectangle that preserves the area of the parallelogram. Through comparing classmate various solution method and noticing that they all result in a rectangle of the same dimensions, students can understand how the area of parallelogram relate to the area of rectangle with the same height and base, the can eventually derive the formula for area of a parallelogram, rather than simply memorizing it as teacher given knowledge.
Not every mathematical topic seems to be well suited to structured problem solving, and Japanese a educators seem to have developed some shared knowledge about the topics that support structure problem solving . For example, the introduction of decimals would be challenging, since the major point of the lesson is for students to accept a particular approach and learn the conventional expression for it. In summary, textbook, or alternate teaching materials, may offer an important source for structure problem solving, by providing well designed tasks on topics suited to problem solving.
To plan a good problem solving lesson, teachers must anticipate the student solution approaches that will emerge, and plan which approaches to discuss with the whole class, in what order. In Japan, the teacher’s manual offers considerable information about student’s thinking. One of the features was information on student thinking; each sentence in the sections of the teacher’s manual on area of quadrilaterals was coded for whether it provided information on student thinking. The information on student thinking provided by the teacher’s manual was further subdivided into “single correct student answer” and “varied student thinking”. In teaching a structure problem solving, teacher do not simply state what students are to learn, but must orchestrate the presentation an discussion of ideas so that students grasp and hone the important ideas through their own work.
The Japanese educators taught public research lessons in the morning and, in the afternoon, taught small workshop in which teachers participated as students in problem solving lessons taught by the Japanese educators. For example, teachers noted that their own curiosity had been awakened by the experience of solving a problem, and they wanted to provide their students the same experience of being driven by mathematical curiosity.
Taiwaness Context: works of Fou Lai Lin
Fou Lai Lin has developed the innovations in mathematics education by adventuring through big problem in mathematics. The big problems in mathematics education can be generally described as (1) the challenge of integrating students’ perspectives into teaching; (2) the gap between theories/research and practices; and (3) the lack of learning theories for teachers and educators.
Integrating students’ perspectives into teaching to improve their learning is the central in mathematics education. Studies in problem solving have proposed teaching approaches to achieve this goal. Bell (1993) suggested diagnostic teaching approach to which teachers need to probe students’ misconceptions and then design the follow-up activities that can correct those misconceptions. Problem posing is a student-centered way to identify students’ prior knowledge and preferences in constructing problems through which teachers can further help them to learn new mathematics knowledge. Alternative to combining students’ perspectives in teaching can be collaborative problem solving , referring to an environment where a group of students express their personal opinions and ideas interactively to find solutions to a mathematics problem. Teachers usually face the pedagogical difficulties in that perceptions of students on the learning environments are often discrepant from the ones of teachers. For example, students may possess negative views on learning environments that can turn down their motivation in learning, but teachers are not aware of the situation and still think the arrangement is friendly for students and can motivate their learning.
Another central problem in mathematics education is the gap between theories/research and practices. Very often, the fragmentation of teaching activities leaves teachers on their own to face the difficulty in integrating theories and research into the context of their teaching. To solve the problem, researchers have attempted to shorten the distance by transforming the theories into practical publications which allow teachers to make sense of the theories and in turn be able to properly apply them into the context of teaching. In recent, teacher education also tries to design courses that integrate both theories/research and practices, enabling teachers to convert the learned theories and research effectively in teaching. The transfer of theories/research into teaching practices is not a linear and one-way process in which solutions to the problems encountered in teaching and learning can be directly obtained. The transferring process is complex, cyclic and sophisticated, involving the interplays with multiple tiers of participants thus causing the difficulties for educators, requiring them to bridge the gap in order to sufficiently implement professional developments and teacher education programs to facilitate the growth of teachers in profession.
The third problem is regarding the lack of learning theories for teachers and educators. Theories are fundamental and can be used to mediate the development of a variety of research designs and practical strategies to enhance the learning. In the last two centuries, salient theories on students’ learning from different perspectives have been established. In contrary, mathematics education is obviously short of learning theories for teachers and educators who stand in upper positions in influencing the learning of students. Establishing the learning theories for teachers and educators is necessary, but it is more difficult than establishing learning theories of students. This is because the complexity of the knowledge necessities educators and teachers to do their jobs well and a variety of factors can influence their learning.
Clearly, finding solutions to the three big problems is an emergent job in mathematics education. Here we offer some answers to the three problems by proposing five innovations derived from conducting a study which focuses on facilitating experienced mathematics teachers in designing conjecturing tasks in a Multi-tier learning environment (MLE). MLE refers to the learning environments involving students, teachers, and educators the core activity in problem solving. A conjecture is a guess, inference, statement, algorithm, theory or prediction that is made based on untested or unproven evidences or intuition. Successful mathematics learning should be proficient at five components: conceptual understanding, procedural fluency, strategic competence, adaptive reasoning, and productive disposition.
Indonesian Context: works of Marsigit
In Indonesia, based on the decision of the National Education No. 20 of 2003 confirmed that the education system in Indonesia should develop intelligence and skills of individuals, promotion attitude, patriotism, and social responsibility, should encourage a positive attitude of self-reliance and
of development. Improving the quality of teaching is one important task of raising the standard of education in Indonesia. It began in June 2006, based on Ministerial Decree No. 22, 23, 24 in 2006, the Government of Indonesia has implemented a new curriculum for primary and secondary education, called the SBC "School-Based"
SBC is the operational curriculum developed by and implemented in each educational unit. SBC consists of educational level of the education objectives, structure and level of the education curriculum, education calendar, and syllabus.
Secondary school-based mathematics curriculum outlines that the purpose of teaching and learning mathematics are as follows:
1. to understand mathematical concepts, to explain the relationship between them and to use them to solve the problem accurately and efficiently. to develop thinking skills in learning patterns and characteristic of mathematics,
2. to manipulate them in order to generalize, to prove and explain the ideas and mathematical proposition.
3. to develop problem solving skills that include understanding the problem, outlining mathematical models, solving them and predict outcome.
4. communicate mathematical ideas using symbols, tables, diagrams and other media.
5. to develop an appreciation of the use of mathematics in everyday lives, curiosity, consideration, and to encourage willingness and self-learning confidence in of mathematics.
This is why the goal of mathematics education from now is still to promote mathematical methods and to take into action. Above all, this cause it is necessary to conduct classroom-based research to investigate the driving factors necessary to the student's ability to develop mathematical methods
Today, the ability to have students not only can solve math problems quickly but students have the ability to think about what to do when faced with a math problem. Thus, students will think critically and creatively when faced with a math problem.
Mathematics should be applied to the natural situation, where any real problems that arise, and to solve them, it is necessary to use mathematical methods. Knowledge, skills, and mathematical methods to achieve the basic knowledge of science, information, and other fields of learning in which mathematical concepts are central, and to apply mathematics in real life situations. In mathematics learning, teachers can use the model of learning so that students can observe, identify and find the concept of matter being studied. To find the concept of matter being studied, students can have discussions with other students in small groups with teachers. In this teacher learning as having an important role to encourage students to develop mathematical methods. The students performed a mathematical method when they find it difficult or when they were asked by the teacher. Then the students conclude their discussion and presented in front of the class to get feedback from other students. For example, in learning about spherical surface area formula. First students make models of the ball, then identify its components. Students discuss in small groups that have been formed. Teachers encourage students to perform mathematical abstraction. By doing group work lead students to develop the mathematical concept of analogical thinking. Analogical thinking occurs when students can find the same ball with the ball finding the surface area that is covering the surface by turning the rope once. In the end, when the teacher asks students to write down the results, the students got the ball area equal to four times the size of the circle. Thus the students understand the concept of surface area of the ball, because students find the formula itself is not just memorize formulas that already exist.
