Mathematical Thinking Across Multicultural Context in APEC
By: Marsigit, M.A.
Reviewed by: Hafizh Praditya Mahardika/09301241001
(http://hafizhpradityamahardika.blogspot.com)
A. Introduction
In APEC that has been held in Thailand of the year 2011, i see there are many participant, that from Thailand, Indonesia, Vietnam, Brunei, Taiwan, etc. In that event discussed about developing mtahematics learning. How make mathematics lesson be undersatand by the students. And in the papers from various author in every participant, there are some defference mathematics learning in every countries. The papers contain the ideas from the authors, about problem mathematics learning in their countries, and how to solve that problem. They have good innovation to make mathematics learning developed. Like hand-on, textbooxt, how to read tables and chart, etc
B. Discussion
1. Indonesia Context : Teacher’s Simulation on Developing Problem Solving Based Mathematics Textbook in Vocational Senior High School Mathematics Teaching In Indonesia by Marsigit
Providing books or textbook is one of crucial policies to improve the quality of teaching learning process i.e. improving problem solving skills. The currently studies on mathematics education in Indonesia have indication that children’s achievement in the subject of mathematics and science is low. Children mastery on mathematics concepts and mathematics process skill is still low. This fact may be as the results of : (1) the shortage of laboratory activities; (2) lack of teachers having mastered science process skill approach; (3) contents on mathematics and science curriculum too crowded; (4) too many time consuming administration stipulation for teachers; (5) lack of laboratory equipment and laboratory human resource. The study also indicates that mismatch among the objectives education, curriculum, and evaluation system which can be identified by the following: (1) National Leaving Examination assess the children’s ability cognitively only; (2) streaming in Senoir Scondary School starting at grade 3; it is argued that the implementation of this system is late and consider individual defferences inappropriately; (3) University Entance Examination System is considered to trigger Elementary and Secondary School teachers apply goal oriented rather than process oriented in teaching mathematics and scince; (4) many teacher still have difficulty in elaborating the syllabus; (5) a number of mathematics topics are considered to be difficult for teachers to teach; (6) a significant number of children consider some mathematics topics as difficult to understand; (7) teachers consider that they still need guidelines for conducting teaching process by using science process skills approach. In Indonesia School-Based Curriculum should encourage the student to think logically, analitically, systematically, crtitically, creatively, and be abble to collaborate with others. And also it need to develop solving skills covering both closed and open problems. In solving problem, the students need to creatively develop many ways and alternatives, to develop mathematics models, and to estimate the result. The curriculum outlines the aims of teaching of mathematics are as follows: (1) to understand the concept of mathematics, to explain the relationships among them and to apply them to solve accurately and efficiently; (2) to develop thinking skills to learn pattern 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 which covers understanding the problems, outlining mathematical models, solving them, and estimating the outcomes; (4) to develop appreciations of the uses of mathematics in daily lifes, curiosity, consideration,and willingness to learn mathematics as well as tough and self-confidence.
For certain program of improving the quality of teaching learning mathematics in Senior High School, research evidences indicated that it was take time for teachers to shift from teacher-centered to student-centered. To promote problem solving activities the teachers need to en-culture their efforts in innovation teaching learning process which meet to academic students needs, encouraging students to be active learners, developing various strategic of teaching, developing various teaching materials, and developing teaching evaluation. The teachers perceived that in order to promote problem solving activities more effectively, they also need: (1) to make good atmosphere for teaching and learning; (2) to promote to implement various teaching methods and teching learning recources; (3) to give the chances for their students to permform their initiatives; (4) to promote cooperative learning; (5) to support the students to be active learner. Isoda Masami indicated that mathemtical thinking has been described in the context of problesolvinf, and he also stated that in the problem solving approach for mathematics teaching, the students meet an unknown problem which can be solved with previously learner mathematics. Polya (1957) outlined the stepsnof problem solving activities as follows: (1) understand the problem; (2) devising a plan; (3) carrying out the plan; and (4) looking back. He also stated that developed the stategies to problem solving activities as folloes: (1) trial and error; (2) making diagram; (3) trying the simple problem; (4) making table; (5) finding the pattern; (6) breaking down the goal; (7) considering the possibilities; (8) thinking logically; (9) reserving the order; and (10) identifying the impossibility.
Textbook for Vocationaly Senior High School mathematics should be understandable, meaningful, and containt of good examples. Most of them expressed that good text book should use simple, communicative, and standardized language. But in small number of them indicated that good textbook of mathematics should meet the following criteri: systematic, completed by good exercise, completed by good assessment system, interesting in display, performance and good illustration and good layout, relevant and applicable to daily life, contextual, curriculum-based, cotain good problem solving, and promote active learner.
In simulation in developing textbook were conducted as part of developing teaching learning resources in the Certification Program of the year 2011 at Faculty of Mathematics and Science, Yogyakarta State University, the mathematics teachers has given the procedure for simulate consist of the steps as follows:
· Whole class lecturing
· Reviewing in the developed textbook
· Small group discussion
· Teacher’s presentation
· Whole-class discussion
· Teacher’s filling in the quesionnare
· Teacher’s reflection
But in realistic that every developed single book should deal with student’s difficulties in learning mathematics, because there is no approach in daily life. The textbook was suggested that it provide such a space to get students feedback in order to communicate their ideas with their teacher. Therefore, the teacer though how to develop communicative textbook.
2. Brunei Context: Innovation in Problem Solving Based on Mathematics Textbook and E-Textbook by Madihah Khalid
In Brunei, almost teachers there use recommended textbooks or books supplied by the Ministry of Education in order to teach the students. Some of the techers only rely on those book, while others surf by internet. The techers use internet because there are many ideas on how to teach a mathematics topic effectively. There the teacher are encouraged to refer to the curriculum when planning the lesson following Madihah Khalid, where pupils are expected to:
· Communicate in order to learn and express their understanding
· Connect mathematical ideas to other concept in mathematics, to everyday experiences and ti other diciplines
· Demonstrate fluency with mental mathematics and estimation
· Develop and apply new mathematical knowledge through problem solving
· Develop mathematics reasoning and creativity
· Select and use technologies as tools for learning and solve problems
· Develop visualization skill to assist in the processing of information, making connections and solving problems
· Develope positive attitude and value toward mathematics
The teacher are supposed to emphasize the process and skills of communication, connection, visualization, and reasoning when teaching the topic. The students work cooperative one with other. To make learning process more interest, so in text book many insert picture, like pizza, monkey, human, cake, and other. That strategy was choosen because pupils will be easier to visualize and transfer their ideas to the manipulatives that will be used in the classroom. Since English is the second language of the students, creative ways were disccused to make sure pupils know the meaning of compare and size. This was to be lesson starter. In the end, it was decided that the use of ICT will be implemented in another lesson due to the amount of contet to be covered. For example, before lesson begining, the teacher plays musical chair game with the pupils. With game the teacher connected content on game between the materials. So the pupils will feel interest to following class, and they do not aware if they have study witihin. However the lesson can be considered a succes if the teacher consider pupil’s participation in class. The children were active, participative, and looked interested in the lesson. But they still need to improve in term in communication, reasoning, or mathematical thinking. On the other hand, Khalid thinks the teachers in the team also need prepare questions in advance besides the ones that were of high-order-thinking questions. Everything said, Khalid thinks language still plays a major part in the success of the lesson when it comes to communication. Khalid also thinks the class was a bit rushed because three problems were discussed and the students could not master all of the concepts taught well enough.
3. Vietnam Context : Practicing to Master The Tables and Charts to Develop Statistical Reasoning Ability of High School Student In Vietnam by Hoang Nam Hai
In the current era of information as we have encountered manydifferent kinds of data the information we get. Various forms ofinformation presentation is also the more we find, samplestatistics are presented in tables and graphs. Implementation ofthis discourse is that students should be able to translate the information presented earlier. Therefore, students' abilities inreading and processing of statistical data is needed. For that,create the students' ability to read and understand the statistics is the duty of every mathematics educators in particular, in order tocreate an educated man. Read and understand the statisticalinformation is defined as the ability to identify, describe, andmake judgments and conclusions of articles related to statisticalinformation. Statistical information is presented in a variety ofdiverse forms which can be viewed as a statistical, text, tables,and graphs. Ability to read and understand the data was based on three things, that : (1) being aware of and understanding statistical information; (2) explaning and reasoning from the statistical information including trends, causal relation; (3) applying and participating in the fields of socio-economics activities. There are assessed on several levels from level 1 to 6 as following:
· Level 1 : identifying and understanding the statistical information presented in tables and charts. Identifying in the natural features in the chart (axis, high-low performaning line, annotation, ...); identifying the contents of the data in different columns (turnover, profit...).
· Level 2 : understanding statistical data represented in tables and chart. Choosing the right kind of charts to represent the various and types. Knowing how to calculate the key figure as well as how to compare the figure of the data given in tables and charts.
· Level 3 : understanding the statistical figure represented in tables and charts. Selecting expertly what kind of chart to show which data, when to use tables, charts to shoe statistical figures. Practicing calculation, explaining data from tables and charts, and finding causal relationships between them.
· Level 4 : linking data in tables, charts, and explaining, using statistical reasoning skill to find out the causal relationships among the satistical information to make right and significant statistic judgment, conlusions.
· Level 5 : linking skillfully the datas in tables, charts, and explaining, using expertly statistical reasoning skills to find the causal relationship among the statistical information to make right and significant statistic judgment and conclusions.
· Level 6 : matering the statistical figures represented on tables and charts. Statistical reasoning ahieve at high levels, the conclusion are positively applied in all areas of socioeconomic activities.
Statistical reasonig is kind of based on the set of statistical data to identify, explain, analyse, and make statistical conclusion as well as discovering the laws of statistics with a same type of crowd. Developing the ability of statistical reasoning for high school students is to train citizens how to apply statistical, thus to practice ability of statistical reasoning for high school student, they need to rely on their own ability:
· Reading, explaining, and drawing conclusions from mathematical model represented statistical information such as formulas, tables, charts, and diagrams.
· Modeling information through statistical formulas, tables, and plots.
· Estimating and checking answer from problems of real life arise related statistically to determine validity and identify more cases, then choosing the most sensible and optimal way.
· Using the mathematical and statistical methods to solve all the problems related statistically and can see their limitations in real life.
· Appliying gained knowledge to real life.
Through the process, statistical reasoning ability of students is also gradually formed ang developed. The prerequsite for the development of statistical reasoning ability for high school is practicing ability of reading out statistical information, like identify information, explain, analyze, and also make conclusions.
4. Taiwan Context : Adventuring Through Big Problem as Mean of Innovation in Mathematics Education by Fou-Lai Lin
In advancing education in each field must have a menghambatinya aka encounter problems, not least in promoting mathematics education. Obstacles in question such as: integrating the perspective of the challenges in teaching students, the gap between theory and practice, the lack of theory of learning for teachers and educators. The first problem is about the challenges of integrating perspectives in advancing mathematics education, for the Bell (1993) suggested that pembalajaran diagnostic approach to investigate the cause of one student accepted concept and then design a follow-up activities that can improve their misconceptions. Bersentral on student learning can identify with the priorities of students and teachers can know students need more help. So it will know where the wrong concepts captured by these students. Because after each difference will inevitably happen to think the concept of teachers with students, due to the influence of the learning environment. So that teachers should motivate students more understanding of the learning environment and adjustment of teachers with students. The second problem is the gap between theory and practice, the teachers still have difficulty in applying the theory into actual practice. In mathematics there are many solutions of applied problems from the real world, but to bring these solutions into mathematical form is very difficult from the theory alone. In this study sought to shorten the distance between the transfer theory with practice by giving teachers the flexibility to determine the rhythm pembejalaran. Teachers should attempt such a way to combine theory with practice effectively. The third problem is the lack of theoretical learning on teachers and educators, such as property Pieget theory that essentially create a balance to explain the relationship between organisms with the environment and extend knowledge through various mechanisms.In supporting education, learning theory plays an important role, because the theory of learning both teachers and students will have more grip in the continuity and smoothness of teaching and learning activities. But building a theory on teachers and educators Study abroad is more difficult than on building on students.
Many of the innovations that can solve the problems facing teachers and educators, of course it was derived from studies of teachers who are experienced in designing tasks in a multi-level learning environment (MLE), which allows the active involvement of teachers and students work together. Innovations include:
1. The principle needs to guide teachers in designing a series of tasks.
According to Ruthven, Laborde, Leach, and Tiberghien (2009) there are three layers of the main theories, frameworks medium, and design tools and principles that are as penengahkerangka work which can serve to mediate the major contributions to the theory of the design process through coordination and contextualize theoretical insights about the epistemological and cognitive dimensions into the design of the sequence of teaching and research. Three principles tesebut will involve students actively in designing the framework of their knowledge.
2. Kebutuan will desian tool
Design tool that serves to identify and deal with certain aspects of the situation under design can support both the formulation of the initial design and subsequent improvements when implementing the design in the teachings.
3. The use of three entries as a starting material in the starting design
Three inputs such as students' misconceptions of the concept, theorem introduced in schools, and math facts. Three inputs are used by teachers to begin the task of identifying goals and initiated the design with ease.
4. N + strategy as a means for the teaching profession merancah pertumubuhan
N + Strategy refers to the approach to facilitate teachers in designing tasks using the type originally conjecturing they can not do. Therefore, the number of types Nadalah conjecturing that the teacher already knows how to design the structure. Plus (+) means the number of types of conjecturing they learn through participation in professional development. This strategy is very important, namely: problem solving can improve their math skills and broaden thinking in a flexible way, there are many different ways and finish the estimate of the same task, and some kind of conjecture is very important in mathematical formulations.
5. Combining the perspectives of students in the design
To combine the perspectives of students in Desai takes several stages, namely the input phase, attack phase, and phase review.Enter phase involves what is known about this problem and what it takes to solve the problem. Phase mangacu attack on students who already understand the intent of the problems encountered, and math activities that may occur in the attack phase of a complex and varied, usually associated with how people make and justify their allegations alleged to convince others. Since the phase of an attack involving a different formulation, the approach that students try to find a solution and depends on the individual students themselves, students' perspectives can be used as feedback to modify the design task. Tinajuan phase refers to the look back at what has happened to improve and expand students' thinking skills or trying to arrange a solution in a more general context.
The contribution of innovations in the above is to mejebantani dalampembelajaran math problems that arise, especially the three major problems earlier. With this innovation, both teachers and students can further enhance the quality of thinking in the perspective of both specific and general. Specialization and generalization allows students to make educated guesses, to justify the proposed allegations, and to convince others with empirical evidence or deductive. Similarly, educators and teachers are learning analogue to student learning. Educators and teachers need to implement specialization and
generalization to understand the environment for their teaching activities and then propose scaffolding strategies involving various types of allegations
C. Conclusion
In education there are many innovation that it can be developed. With hand-on, textbook, e-book, etc the teacher can make learning process more effectively. But it need seriously from both of the teacher and the student. In many countries like Thailand, Korea, Berunei, Vietnam also Indonesia there are many obstacle in learning process, espesially in matjematics learning process, so with APEC wished can exchange that sollution.
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