Mathematics Teaching Across Multicultural Context
by:
Felisitas Sayekti Purnama U
09301241007
Student of Mathematics Education 2009
Yogyakarta State University
I. Introduction
Mathematics is axiomatic, abstract, formal, and deductive. Mathematics is the science which is obtained by way of reasoning. Mathematics is formed as a result of human thought associated with the ideas, processes, and reasoning. For most students, mathematics is the science that is difficult to learn. However, without them knowing it, mathematics is a science that is very useful in human life. Consciously or unconsciously, every person uses math in their life. Mathematics is used in science, social studies, science and technology, and others. Every day, all mathematicians around the world are always trying to develop a mathematical since realized the importance of mathematics for improving the welfare of human life.
Each student in the whole world was required to be able to use mathematics and mathematical thinking in everyday life. Students can use math as a way of reasoning, logical thinking, critical, systematic, and objective. Indirect object of learning mathematics is that students have the ability to solve problems. To be able to solve a problem, they need to have the reasoning skills that can be obtained through the learning of mathematics. All students learn mathematics in the world according to culture in their respective countries. Learning mathematics in Japan will be different than in Russia. Learning mathematics in Thailand will be different with in Indonesia or in Singapore. Culture in each country makes a way of mathematics teaching in each country were different. However, mathematics teaching across multicultural context.
II. Mathematics Teaching Across Multicultural Context
Mathematics teaching is the basis for various types of teaching. When student learn mathematics, student can learn the logical and rational mode of thinking. Mathematics can apply in physics, science, technology, social, statistics, and economics. In Indonesia and another country, mathematics is taught from very young age, from kindergarten to university. That is because all countries recognize the importance of mathematics. Because each country has different culture, mathematics teaching in one country must be different with another country. Taiwan, Vietnam, Russian, Hong Kong, Brunei Darussalam, Thailand, Philippines, and Indonesia give a contribution in mathematics multicultural teaching. Following we will review some works of mathematics educationist from different context of culture in relation to the aspects of mathematical teaching.
1. Taiwan Context: the works of Fou Lai Lin
Fou Lai Lin elaborated with Hui-Yu Hsu, Kai-Lin Yang, Jian-Cheng Chen, and Kyeong-Hwa Lee explained about adventuring is an innovation in mathematics education. There are three big problems in mathematics education: challenge of integrating students’ perspectives into teaching practices, the gap between theories and practices, and the lack of learning theories for teachers and educators. Integrating students’ perspectives into teaching to improve their learning is the central in mathematics education. When students possess negative views on learning environments, they 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. The teachers must design better learning environments in alignment with students’ perspectives that can maximize the learning outcomes of students. Teachers have a trouble in integrating theories and research into the context of their teaching. To solve the problem, researchers 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. Establishing the learning theories for teachers and educators is necessary, but it is more difficult than establishing learning theories of students.
He gives a solution for three big problems. The solution is conjecturing tasks. Conjecturing tasks in a Multi-tier learning environment (MLE) can facilitate student`s study. MLE refers to the learning environments involving students, teachers, and educators. They work together as a community for their individual growth. MLE emphasizes the importance of interactions between the participants in learning environments and how interactions can contribute to individual growth. MLE can arrange professional development for mathematics teachers who were required to design tasks of conjecturing in problem solving.
Fou Lai Lin said that conjecturing can enhance conceptual understanding, enhance the procedural fluency, enhance strategic competence, enhance adaptive reasoning, and enhance productive disposition. Various types of conjecturing are:
a. conjecture with empirical induction from a finite number of discrete cases
b. conjecture related to empirical induction, but is made by observing continuous and dynamical cases for a general rule
c. conjecture analogy
d. conjecturing is abduction
e. conjecture made on the basis of a visual representation of a problem or a perceptual translation of its statement.
In his opinion, teacher must make an innovations derived from the study. The innovations are makes conjecturing sequence; design tool which serve to identify and address specific aspects of the situation under design can support both the initial formulation of a design; use students’ misconception, theorems introduced in school mathematics, and the mathematics facts to initiate the designs; N+ strategy to facilitate teachers in designing tasks using types of conjecturing they originally cannot do; and incorporate students’ perspectives into the designs.
2. Vietnam Context: the works of Hoang Nam Hai
Hoang Nam Hai explained about practicing to master the tables and charts to develop statistical reasoning ability of High School Student in Vietnam. He explained the concept of reading and understanding of statistical information and tables, statistic charts, the levels of evaluating students’ ability of mastering of tables, and statistics charts.
a. Reading and understanding of statistical information
In his opinion, reading and understanding of statistical information is defined as the ability to identify, explain and make his judgments and conclusions of the articles relating to statistical information. Reading and understanding of statistical tables and charts is the ability of each individual to identify, explain, and make their judgments and conclusions of the tables, charts presenting statistical data. The ability of reading and understanding of statistical information is presented in different forms based on three criteria: being aware of and understanding statistical information; explaining and reasoning from the statistical information including trends, causal relation; and applying and participating in the fields of socio-economic activities.
Level of reading and understanding of statistic information are:
a) Identifing and understanding the statistical information presented in tables, charts.
b) Understanding statistical data represented in tables and charts.
c) Understanding the statistical figures represented on tables and charts.
d) Linking data in tables, charts and explaning, using statistical reasoning skill to find out the causal relationship among the statistical information to make right and significant statistic judgments, conclusions.
e) Linking skillfully the given data in tables, charts and explaning, using expertly statistical reasoning skills to find the causal relationship among the statistical information to make right and significant statistic judgments and conclusions.
f) Mastering the statistical figures represented on tables and charts.
b. Reading and understanding of statistical chart
The data are presented by columns and rows in the tables. Student can be trained when teacher give two types of questions to practice. The first question is related to reading data and the second related to inferring from the data shown in statistical charts and charts.
c. Reading and understanding of statistical information is the prerequisite for development of statistical reasoning ability for high school students
Statistical reasoning is the kind of reasoning based on the set of statistical data to identify, explain, analyze and make statistical conclusions as well as discovering the laws of statistics with a same type of crowd.
3. Russian Context: the works of Ivan Vysotskiy
He explained about stochastic line in Russian Junior High School. In Russian, the probability and statistic theory learned in Junior High School. In school, the main of statistic concepts need to be learned on the intuitive elementary level because of students have no firm understanding about the random changeability around us.
Ivan said that in the beginning, student in Russian discuss how to represent data after collecting them into convenient form. The main methods are tables and charts. After that, they discuss about descriptive statistics. The main descriptive parameters (means, median, amplitude) are understandable for majority of sixth- and eighth graders. Teacher can illustrate the random changeability in the world. Then, students can see the data formalization and description on life like statistics material. The main descriptive parameters to be discussed in 7 – 8 grades: average, median, amplitude, deviation, variance.
In Russian, there is no necessity to discuss universes in school completely. This term might be used only for finite sets of objects they are talking about. The concept of a random experiment is fundamental for probability theory and teaching it. The main obstacle is the concept of the space of the elementary outcomes. In tertiary school they can accurately define the space including the measure and sigma-algebra on the appropriate set. This way is impossible in the school.
In Russian, combining elementary events they can obtain new events. The substantial educational element is to teach students to transform abstract description into verbal (synthetic) form and back.
Russian use three operations are union, intersection and transition to the complementary event. This is next way to get new events from others. Their experience shows that there are no serious difficulties to adopt these propositions for students of 8th grade (aged 13-14). The basic method to teach students to operate with events is Euler-Venn diagrams.
The conditional experiment or conditional probability is not necessary part of junior school course. Combinatory can be learned in school. Combinatory rules come from such problems in natural way. The same approach is fit for the teaching concept of a transposition (rearrangement). Students must learn how to write down all the transposition of three and four items. After that we can go to the formal description using factorial. Graphs are of great importance in primary learning combinatorial analysis. Making trees illustrates not only multiplying rule but gives natural algorithm for enumeration.
Scattering charts implemented in Russian school. When working with them it is the convenient case to talk about variety of relations in the world. Scattering chart studied mainly in 7 and 8 grades. Teacher in there could make the lesson more colorful and spectacular using computer. But, Russian always trying to make the material available for every teacher even if he has not have computer equipment when the lesson goes.
4. Hong Kong Context: the works of Cheng Chun Chor Litwin
Innovation on problem solving based mathematics textbooks and e-textbooks is Cheng Chun Chor Litwin`s opinion. He is using fractions learning to enhance mathematical thinking. Based on teachings of fraction additions and fractions with problem solving in two classes in a Hong Kong school, he tried to investigate the design of a lesson in promoting mathematical thinking.
He uses lesson study method. His model of this study is “plan and teach, review, and re-teach”. There are three phases in this study. First, the object of learning is identified and a lesson was designed and delivered. The aim of the lesson was to enable students to develop mathematical thinking, and the lesson study focus on the lesson design and pedagogical process. Second, the lesson design was review and revised, picking up effective teaching point and evaluate the lesson design and object of learning. Finally, the revised lesson was taught again, at the same grade or at a lower grade so as to develop the object of learning and its associated pedagogical process so that the teaching and learning is more effective.
He said that there are two types of questions used in this study for enhancing mathematical thinking with fractions. Types of questions are an investigation on fraction additions with positive integral value and their sums add to 1 and problem solving based on equivalence fractions or using algebraic equation. This type of question included finding fractions that has positive integral value and certain relation of the numerator and denominators.
The three phases of the lesson study in Hong Kong school is as follow:
a. Using equivalent fraction to design the question and use equivalent fractions to solve problem
b. Review and use question that require strategy more than using equivalent fractions
c. Question is introduced for using algebraic equation to solve the problem.
He said that there are two principles of conducting learning and through lesson study. Lesson study has one objectives is to study how students think and work so that it could hint the design of lesson which took place later on. Good lesson design in mathematics should be able to allow student to learn partly by themselves and partly through teachers‟ intervention, so that mathematical thinking happens. This principle is simple but it needs teacher professional expertise to connect student knowledge.
Another principle is the motivation for learning. When students can learn new ideas based on their own knowledge, it is the best motivation for learning. Design of lesson based on student previous knowledge not only could reduce student cognitive load, it also arouse student motivation as they can see that they could work out the answer.
5. Brunei Darussalam Context: the works of Madihah Khalid
Innovation in problem solving based on mathematics textbooks and e-textbooks is one of innovation in mathematics teaching. In Brunei Darussalam, teacher use recommended textbooks or books supplied by the Ministry of Education. Some of the teachers look at other books or internet in order to get ideas on how to teach a mathematics topic effectively. Madihah will examine the design of a lesson in the topic of “comparing fractions” at year 4 level. The research lesson was designed according to the recommendations from the recommended text-book with changes to make the lesson more interesting.
Teachers are supposed to emphasize the process and skills of communication, connection, visualization and reasoning when teaching this topic. Teacher can use computer to be used for ease of comparison for fractions with denominators bigger than 10 and then teacher refer to the textbook. The textbook shows examples of comparison of fractions with like denominators. Then student will compare fractions where the numerators and denominators are different. Teacher gives three problems to the student where placed in an envelope and each group of 4 – 5 students working collaboratively. The students took out problem 1 from the envelope and start working on the answer right away. This problem did not seem hard to them. They were able to answer using the multilink block and explain their answers well as to why one fraction is bigger than the other. For problem 2, the students were asked to use paper folding. They manage to solve the problem and showed it using the A4 paper as to which part is bigger. They could not explain why one is bigger than the other convincingly. In problem 3, where the problem is to determine which fraction of chocolate is more between 3/5 and 2/10, they took some time to decide on the suitable manipulative to show their solution. Many of student use the fraction block. Students could show using the fraction block, which is bigger. Some students even draw diagrams. They could also explain why one is bigger than the other although the explanation is not very satisfactory. Teacher is discuss the differences between the three problems and summarized the lesson.
Teacher in Brunei use chocolate bars was chosen as the item to be divided and compared in the problem instead of pizza and carrot cake as in the book. Student will be interest to study. The teacher decided to provide multilink blocks, A4 papers and fraction blocks as manipulative. Teacher hopes that the student can show which fraction is more than the other using manipulative. Students are expected to be able to give reasons as to why a fraction is bigger or smaller than the one it was compared to.
Madihah said that teacher can create something interesting and fun as the lesson starter with plays the musical chair game with the students. The purpose of the introduction is to define the word ‘compare’ and ‘size’. Before the game started, students were asked to guess who will win the game and why. After the game was played and the bigger girl won, the teacher connected the words bigger and smaller, stronger and weaker, quicker and slower to the word ‘compare’ and bigger and smaller or shorter and taller to ‘size’. She then proceeded clarify that in comparing fractions, the size of the fraction is important. She also asked the students why we need to compare fraction in real-life. The class enjoyed the introduction and observers thought the definition could be done in a simpler way.
The lesson can be success if student give a participation in class. The student were active, participative, and looked interested in the lesson. They still need to improve in terms of communication, reasoning, or mathematical thinking. Teacher need to prepare questions in advance besides the ones that were in the lesson plan, so that they can be more fluent and the questions asked were of higher-order-thinking questions.
6. Thailand Context: the works of Supot Seebut
Supot Seebut elaborated with Sasitorn Pusjuso, Sakda Noinang, and Utith Inprasit. They are explained the development of hand on and e-activities for learning mathematics models. Mathematical modeling could be defined as translating encountered problems into mathematical forms by seeing mathematics as a tool of solving problem. The process of mathematical modeling consists of four main stages:
a. Observing a phenomenon, delineating the problem situation inherent in the phenomenon, and discerning the important factors that affect the problem.
b. Conjecturing the relations among factors interpreting them mathematically to obtain a model for the phenomenon
c. Applying appropriate mathematical analysis to the model
d. Obtaining result and reinterpreting them in the context of the phenomenon under study and drawing conclusions.
Supot Seebut uses hand on activities for learning mathematics models. Hands-on activities help student to understand concept about mathematical models and they could use all activities in their classroom. These examples of modeling exercises have been to prepare teachers’ teaching about mathematical modeling in Master Teacher project and middle school level. He made an activity when he would like to know the number of fish in pond. This information would be valuable for stocking the pond and for studying the availability of fish in the pond. How would student approximate the size of the pond’s fish population? Student must use ratio and proportion.
In another activity, student must to know that how would put bus stop to get minimum total distance among all houses when a company opens new store or plant. One of the most important considerations is where to locate the facility so that the distance traveled by suppliers and costumer, or the distances its product must be shipped, are kept to a minimum. Like this situation, student can use simple geometry and graphing. Student can study mathematical model from hands-on activities using computers as an aid in a mathematical model lesson. E-activities about mathematical modeling were developed to support learning mathematical models online. E-Activities developed in nearly future would help more teachers and learners to perform mathematical modeling.
7. Philippines Context: the works of Soledad A. Ulep
Ulep explained about transforming a mathematics textbook practical work activity into a problem solving task though lesson study. He wants to introduce polynomial to students. Textbook can described the practical work activity because textbook with visual illustration. Student can imagine practical work activities of making a box with an open top. It pointed out that a cut made was actually the side of a square whose length was equal to the height of the box that would be formed. It gave the equation of the volume of the box in terms of this height. It referred to this equation as a polynomial function that was cubic. It showed that every polynomial defined a function. It then presented the definition of a polynomial function and cited that a linear function and a quadratic function are also polynomial functions.
After student can imagine practical work activities, they are formulating the problem solving task. They construct the problem solving task. They must:
a. Construct a rectangular open-top box of different sizes using grid paper, scissors, and tape.
b. Use one sheet of paper for each box.
c. There should be no overlap on the top edges and retain the original length and width of the grid paper.
d. Students do not waste the grid paper so they can make many boxes.
Textbook must use problems that would require student to think to provide a context for developing the concept of polynomial function. In lesson study, the problem was developed collaboratively, tried in actual classrooms, and improved based on actual implementation results. It could be argued that the students used more thinking skills when they solved the problem than when they would have been simply asked to follow a set of procedures to learn polynomial function.
8. Indonesia Context: the works of Marsigit
In Indonesia, student study mathematics in primary and secondary school to encourage think logically, analytically, systematically, critically, creatively and be able to collaborate with others. Marsigit said that contextual and realistic approaches are recommended to be developed by the teachers to encourage mathematical thinking in primary schools. Teachers also need to develop resources such as information technology, teaching aids and other media. Teacher can use textbook based problem solving to teaching learning in vocational senior high school. When teacher try to develop textbook, they find some difficulties. There are lack of skill and knowledge of writing mathematical textbook, managing and allocating the time, needs a budget, not easy to determine the theme of the book, not easy to collect supporting data, etc.
He said that a textbook should provide: trial and error schema, activities of making diagrams, activities of manipulating the tables, finding the pattern, breaking down the goal, and considering possibilities. The problem solving based mathematics textbook in the Vocational Senior High School can be developed based on the criteria that are: Trial and Error, Making diagram, Trying the simple problem, Making Table, Finding the pattern, Breaking down the goal, Considering the possibilities, Thinking Logically, Reversing the Order, and Identifying the impossibility.
III. Conclusion
Mathematics is a science that is very useful in human life. Consciously or unconsciously, every person uses math in their life. All students learn mathematics in the world according to culture in their respective countries. Learning mathematics in one country will be different in another country. Culture in each country makes a way of mathematics teaching in each country were different. However, mathematics teaching across multicultural context. Mathematics teaching in:
a. Taiwan: teacher make an innovations derived from the study. The innovations are makes conjecturing sequence; design tool which serve to identify and address specific aspects of the situation under design can support both the initial formulation of a design; use students’ misconception, theorems introduced in school mathematics, and the mathematics facts to initiate the designs; N+ strategy to facilitate teachers in designing tasks using types of conjecturing they originally cannot do; and incorporate students’ perspectives into the designs.
b. Vietnam: the concept of reading and understanding of statistical information and tables, statistic charts, the levels of evaluating students’ ability of mastering of tables, and statistics charts are reading and understanding of statistical information, reading and understanding of statistical chart, and reading and understanding of statistical information is the prerequisite for development of statistical reasoning ability for high school students
c. Russian: the main of statistic concepts need to be learned on the intuitive elementary level because of students have no firm understanding about the random changeability around us. Scattering charts implemented in Russian school. When working with them it is the convenient case to talk about variety of relations in the world. Scattering chart studied mainly in 7 and 8 grades. Teacher in there could make the lesson more colorful and spectacular using computer. But, Russian always trying to make the material available for every teacher even if he has not have computer equipment when the lesson goes.
d. Hong Kong: Innovation on problem solving based mathematics textbooks and e-textbooks, used lesson study method. The model of this study is “plan and teach, review, and re-teach”.
e. Brunei Darussalam: Innovation in problem solving based on mathematics textbooks and e-textbooks is one of innovation in mathematics teaching. In Brunei Darussalam, teacher use recommended textbooks or books supplied by the Ministry of Education. The lesson can be success if student give a participation in class. The students were active, participative, and looked interested in the lesson. They still need to improve in terms of communication, reasoning, or mathematical thinking. Teacher need to prepare questions in advance besides the ones that were in the lesson plan, so that they can be more fluent and the questions asked were of higher-order-thinking questions.
f. Thailand: Mathematical modeling could be defined as translating encountered problems into mathematical forms by seeing mathematics as a tool of solving problem. In Thailand, they uses hand on activities for learning mathematics models. Hands-on activities help student to understand concept about mathematical models and they could use all activities in their classroom. These examples of modeling exercises have been to prepare teachers’ teaching about mathematical modeling in Master Teacher project and middle school level. E-activities about mathematical modeling were developed to support learning mathematical models online. E-Activities developed in nearly future would help more teachers and learners to perform mathematical modeling.
g. Philippines: Philippines is transforming a mathematics textbook practical work activity into a problem solving task though lesson stud to introduce polynomial to students. Textbook can described the practical work activity because textbook with visual illustration. Textbook must use problems that would require student to think to provide a context for developing the concept of polynomial function. In lesson study, the problem was developed collaboratively, tried in actual classrooms, and improved based on actual implementation results. It could be argued that the students used more thinking skills when they solved the problem than when they would have been simply asked to follow a set of procedures to learn polynomial function.
h. Indonesia: Contextual and realistic approaches are recommended to be developed by the teachers to encourage mathematical thinking in primary schools. Teacher can use textbook based problem solving to teaching learning in vocational senior high school. The problem solving based mathematics textbook in the Vocational Senior High School can be developed based on the criteria that are: Trial and Error, Making diagram, Trying the simple problem, Making Table, Finding the pattern, Breaking down the goal, Considering the possibilities, Thinking Logically, Reversing the Order, and Identifying the impossibility.
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