People will love something if they can enjoy it. Same condition with students, they will love Mathematics if they enjoy it. Many people said that Mathematics is subject that terrify, make stress, etc. They said like that because when they studied Mathematics, they were not enjoyed it. So, make them uncomfortable. By listening people answer about why they dislike Mathematics, we as the teacher in the next decade need method to minimize that problem and show that Mathematics is the universal science and it is the ground of other sciences. By research of observer we will get source about it.
First, from Mr. Marsigit and friends (Yogyakarta State University) in “Lesson Study : Promoting Student Thinking on the Concept of Least Common Multiple (LCM) Through Realistic Approach in the 4th Grade of Primary Mathematics Learning.” Since June 2006, Indonesia Government committed to implement the new curriculum for primary and secondary education called School-Based Curriculum. It is combines two paradigms, they are stressing on students competencies and concerning student’s learning process. The National Standard of Competencies as the minimum competencies that must be performed by the students, covering affective, cognitive, and psychomotor competencies. The Government has developed Contextual Teaching and Learning (CTL) and Realistic to support School-Based Curriculum implementations that means hope can develop students life skills by employing optimally the environment to support student’s activities. The scope of primary mathematics : numbers, geometry and measurements, and analyzing data.
Mathematics in primary school have 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 to access and employ information to preserve their live. Mathematical thinking is defined as students’ activities to communicate mathematical ideas in involves the using of symbols, tables, diagrams, and other sources hope the students are able to solve their problems. Contextual and realistic approaches are recommended to be developed by the teachers to encourage mathematical thinking in primary schools. Hope students can step-by step learn and master mathematics enthusiastically. To make their teaching learning of primary mathematics more effective, teachers also need to develop resources such as information technology, teaching aids, and other media.
Specifically,the primary mathematics curriculum outlines the aims of teaching learning of mathematics are as follows:
1. To understand the concepts of mathematics, to explain the relationships among them and to apply them to solve the problems accurately and efficiently.
2. To develop thinking skills to learn patterns and characteristics of mathematics, to manipulate them in order to generalize, to proof and to explain ideas and mathematics propositions.
3. To develop problems solving skills which covers understanding the problems, outlining mathematical models, solving them and estimating the outcomes.
4. To communicate mathematics ideas using symbols, tables, diagrams and other media.
5. 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.
As the Program for International Student Assessment (PISA) that concerned with the capacities of students to analyze, reason, and communicate ideas effectively as they pose, formulate and interpret mathematics in a variety of situations. Realistic Mathematics Education (RME) in the schema of Lesson Study let the teachers to improve instructional approach from traditional to progressive one. Isoda, M. (2006), outlined that mathematical thinking is open ideas; thus, it is very difficult to discuss its development without having a window to discuss. When we focus on each lesson, we easily focus on specific knowledge and skills (Understanding), and easily forget to develop Attitude, Mathematical Thinking and Representation. Freudenthal's view on mathematics (Freudenthal, 1991). Two of his important points of views are mathematics must be connected to reality (Mathematics must be close to children and be relevant to everyday life situations) and mathematics as human activity (Mathematics education organized as a process of guided reinvention, where students can experience a similar process compared to the process by which mathematics was invented, the meaning of invention is steps in learning processes while the meaning of guided is the instructional environment of the learning process). Two types of mathematization which were formulated explicitly in an educational context by Treffers, 1987, in Zulkardi, 2006, are 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. Vertical mathematization is the process of reorganization within the mathematical system itself. The process of reinvention in which both the horizontal and vertical mathematization take place in order to develop basic concepts of mathematics or formal mathematical language.
The learning process starts from contextual problems. Using activities in the horizontal mathematization, for instance, the student gains an informal or a formal mathematical model. By implementing activities such as solving, comparing and discussing, the student deals with vertical mathematization and ends up with the mathematical solution. Then, the student interprets the solution as well as the strategy which was used to another contextual problem.
The mathematics lessons that will be designed in Realistic Approach should represents the characteristics of how the students do matematization. The way of embed these characteristics into the lesson plan components can be seen in the following diagram (Zulkardi, 2006):
1. Introduction
2. Describing prepared contextual problems (problems situated in reality as follow):
Since the early of the year 2006, Shinta has two activitis i.e. swiming and gardening. She is periodically going to swim once a week and gardening every 8 days, as shown in the following callendar:
Question:
When Shinta is going for swimming and gardening on the same days?
3. Developing (group discussion)
4. Reason and explanation (presentation)
5. Conclusion (homework)
6. Clossing
The search in this lesson study is to uncover the idea of mathematics as a human activity that is stressed in realistic approach. Teacher organized the class as a process of guided reinvention (De Lange, 1996, in Zulkardi, 2006) that is to step in learning LCM by developing instructional environment example let the students to freely choose and develop their methods and aids to solve the problems. The teacher let the students to work individually and in group to perform horizontal mathematization and then anticipating the structure to more formal raise mathematization activities.
The characteristic of addition, subtraction, multiplication, division and so on as the nature of Mathematics to know the ability of students in Mathematics activity. The ceiling-beam (peran) teacher based on Peter Gould is to develop the teaching material. The teaching learning scheme are :
I. Lesson Plan
II. Apperception
III. Small group discussion
IV. Various method
V. Various interaction
VI. Various media
VII. Student’s reflection
VIII. Scheme of achieving competence
IX. Student’s conclusion
X. Student’s worksheet
The second is from Shigeo Katagiri (University of Tsukuba) in Mathematical Thinking and How to Teach It. The Aim of School Education described as follows in a report by the Curriculum Council: “To cultivate qualifications and competencies among each individual school child, including the ability to find issues by oneself, to learn by oneself, to think by oneself, to make judgments independently and to act, so that each child or student can solve problems more skillfully, regardless of how society might change in the future.” Of course, not every child will be able to act independently at the same level, but each school child must be able to act independently according to his or her own capabilities. To this end, teaching methods that focus on the individual are important. Mathematical thinking allows for: (1) An understanding of the necessity of using knowledge and skills, (2) Learning how to learn by oneself, and the attainment of the abilities required for independent learning. Although we have examined a specific example of the importance of teaching that cultivates mathematical thinking during each hour of instruction, for a teacher to be able to teach in this way, he must first have a solid grasp of “what kinds of mathematical thinking there are.” After all, there is no way a person could teach in such a way as to cultivate mathematical thinking without first understanding the kinds of mathematical thinking that exist. Let us consider the characteristics of mathematical thinking.
1. Focus on Sets
For instance, the states of “working to establish a perspective” and “attempting to analogize, and working to create an analogy” are ways of thinking. In other words, mathematical thinking means that when one encounters a problem, one decides which set, or psychological set, to use to solve that problem.
2. Thinking Depends on Three Variables
The type of thinking that use is determined by the problem (situation), the person, and the approach (strategy) used. In other words, the way of thinking depends on three variables: the problem (situation), the person involved, and the strategy.
3. Denotative Understanding
Concepts are made up of both connotative and denotative components. Even if the concept of mathematical thinking is expressed with words, as in “mathematical thinking is this kind of thing,” this will be almost useless when it comes to teaching. In other words, mathematical thinking should be captured denotatively.
4. Mathematical Thinking is the Driving Force Behind Knowledge and Skills
Mathematical thinking acts as a guiding force that elicits knowledge and skills, by helping one realize the necessary knowledge or skills for solving each problem. There is another type of mathematical thinking that acts as a driving force for eliciting other
types of even more necessary mathematical thinking. This is referred to as the “mathematical attitude.”
First of all, mathematical thinking can be divided into the following three categories:
I. Mathematical Attitudes
1) Attempting to grasp one’s own problems or objectives or substance clearly, by oneself
(1) Attempting to have questions
(2) Attempting to maintain a problem consciousness
(3) Attempting to discover mathematical problems in phenomena
2) Attempting to take logical actions
(1) Attempting to take actions that match the objectives
(2) Attempting to establish a perspective
(3) Attempting to think based on the data that can be used, previously learned
items, and assumptions
3) Attempting to express matters clearly and succinctly
(1) Attempting to record and communicate problems and results clearly and
Succinctly
(2) Attempting to sort and organize objects when expressing them
4) Attempting to seek better things
(1) Attempting to raise thinking from the concrete level to the abstract level
(2) Attempting to evaluate thinking both objectively and subjectively, and to
refine thinking
(3) Attempting to economize thought and effort
II. Mathematical Thinking Related to Mathematical Methods
1. Inductive thinking
2. Analogical thinking
3. Deductive thinking
4. Integrative thinking (including expansive thinking)
5. Developmental thinking
6. Abstract thinking (thinking that abstracts, concretizes, idealizes, and thinking that
clarifies conditions)
7. Thinking that simplifies
8. Thinking that generalizes
9. Thinking that specializes
10. Thinking that symbolize
11. Thinking that express with numbers, quantifies, and figures
III. Mathematical Thinking Related to Mathematical Contents
1. Clarifying sets of objects for consideration and objects excluded from sets, and
clarifying conditions for inclusion (Idea of sets)
2. Focusing on constituent elements (units) and their sizes and relationships (Idea of units)
3. Attempting to think based on the fundamental principles of expressions (Idea of
expression)
4. Clarifying and extending the meaning of things and operations, and attempting to think based on this (Idea of operation)
5. Attempting to formalize operation methods (Idea of algorithm)
6. Attempting to grasp the big picture of objects and operations, and using the result of this understanding (Idea of approximation)
7. Focusing on basic rules and properties (Idea of fundamental properties)
8. Attempting to focus on what is determined by one’s decisions, finding rules of relationships between variables, and to use the same (Functional Thinking)
9. Attempting to express propositions and relationships as formulas, and to read their
meaning (Idea of formulas).
The last, hope the teacher in the next decade can make students love Mathematics by using many method based on research of many observers.
0 komentar:
Posting Komentar