ETHNO MATHEMATICS

Every country has its own culture. There is a different culture between Indonesia and Malaysia, also between Indonesia and Japan. There’s also different culture between one province and another province in Indonesia. It is because Indonesia is a country with thousands of islands spanning out from the West (Aceh, Sumatra) to the East (Merauke, Irian Jaya). It is located between two oceans, the Pacific and Indian, and two continents, Asia and Australia. The population would be about 350 million residents. In fact by 2011, the population had grown to 375 million. Those different cultures we usually called as ethnology.

Mathematics learning process is also different between one country and another country. It depends on the culture of the country. For example, in Europe we can use mathematics problem based on four seasons. But we can’t do this in Indonesia, since Indonesia only has two seasons. This difference usually called as ethno mathematics. Ethno mathematics consists of the culture of mathematics, mathematical culture, the uncultured of mathematics, and also the context of mathematics.

In mathematics education, ethno mathematics is the study of the relationship between mathematics and culture (D'Ambrosio, 1999, 146). Often associated with "cultures without written expression" (D'Ambrosio, 1997, may paraphrases Ascher 1986), it may also be defined as "'the mathematics which is practiced among identifiable cultural groups'" (Powell and Frankenstein, 1997 quoting D'Ambrosio). It refers to a broad cluster of ideas ranging from distinct numerical and mathematical systems to multicultural mathematics education. The goal of ethno mathematics is to contribute both to the understanding of culture and the understanding of mathematics, and mainly to lead to an appreciation of the connections between the two. ⁽¹⁾

Ethno mathematics comes from the philosophy of mathematics that affected to the paradigm of mathematics. This paradigm then developed into two kinds of theories. The first theory is about history, psychology, anthropology, and sociology. The second theorem is about mathematics. Then this two theories combined together become ethno mathematics that developed into mathematical context (formal mathematics, model of mathematics, model concrete, and the concrete of mathematics. Ethno mathematics also be in ice berg theorem.

The final project of studying ethno mathematics is a small research in mathematics learning process that’s affected by the culture of a country. This research based on a theory. Then from the theory we can uncover the culture phenomena, so we get the data and develop it into a construction of research and we can make a new theory. This research must contain of the unity, concept, method, scope, relationship, structure, subject, characteristics, value, and timeline.

The example of ethno mathematics around the world is on numerals. These are some explanation of the numerals in some countries: ⁽¹⁾

1.      English

For instance, in English, there are four different systems. The units words (one to nine) and ten are special. The next two are reduced forms of Anglo-Saxon "one left over" and "two left over" (i.e., after counting to ten). Multiples of ten from "twenty" to "ninety" are formed from the units words, one through nine, by a single pattern. Thirteen to nineteen, and in a slightly different way twenty-one through ninety-nine (excluding the tens words), are compounded from tens and units words. Larger numbers are also formed on a base of ten and its powers ("hundred" and "thousand"). One may suspect this is based on an ancient tradition of finger counting. Residues of ancient counting by 20s and 12s are the words "score", "dozen", and "gross". (Larger number words like "million" are not part of the original English system; they are scholarly creations based ultimately on Latin).

2.      German

The German language counts similarly to English, but the unit is placed first in numbers over 20. For example, "26" is "sechsundzwanzig", literally "six and twenty". This system was formerly common in English, as seen in an artifact from the English nursery rhyme "Sing a Song of Sixpence": Sing a song of sixpence, / a pocket full of rye. / Four and twenty blackbirds, / baked in a pie.

3.      French

In the French language as used in France, one sees some differences. Soixante-dix (literally, "sixty-ten") is used for "seventy". The words "quatre-vingt" (literally, "four-twenty", or 80) and "quatre-vingt-dix" (literally, "four-twenty ten" 90) are based on 20 ("vingt") instead of 10. Swiss French and Belgian French do not use these forms, preferring more standard Latinate forms: octante for 80 and nonante for 90.

4.      Mesopotamia

In ancient Mesopotamia the base for constructing numbers was 60, with 10 used as an intermediate base for numbers below 60.

5.      West Africa

Many West African languages base their number words on a combination of 5 and 20, derived from thinking of a complete hand or a complete set of digits comprising both fingers and toes. In fact, in some languages, the words for 5 and 20 refer to these body parts (e.g., a word for 20 that means "man complete"). The words for numbers below 20 are based on 5 and higher numbers combine the lower numbers with multiples and powers of 20. Of course, this description of hundreds of languages is badly oversimplified; better information and references can be found in Zaslavsky (1973).

Then, some examples of ethno mathematics in Indonesia are:

1.      The role of mathematics in Yogyakarta palace

2.      The role of mathematics in Javanese culture

3.      The role of mathematics in ‘wayang’ culture

4.      The role of mathematics in Borobudur and Prambanan building

5.      The role of mathematics on the counting of someone death

6.      The role of mathematics in fengshui

7.      The role of mathematics in Javanese calendar

8.      The relation of mathematics and forecasting

9.      Irrational mathematics

10.  The relation of mathematics and horoscop

11.  The role of mathematics on Javanese ‘primbon’.

⁽¹⁾http://en.wikipedia.org/wiki/Ethnomathematics

How to Develop Mathematics Instructional Design

Instructional design can be interpreted from different perspectives, such as discipline, as a science, as a system, and as a process. As a discipline, instructional design discusses about various research and theories learning process strategies. As a science, instructional design is a science that creates the specification development, implementation, assessment, and management of situations that provide learning facilities in the macro and micro scale for various subjects at various levels of complexity. As a system, instructional design is the development of learning system and the implementation including the properties and procedures to improve the quality of learning. Instructional design as a process is the development of the specification of learning by using learning theory and learning theory to guarantee the quality of learning. Instructional design is the overall process of learning needs and goals as well as its delivery system, includes development of materials and learning activities, materials testing and evaluation, and implementation of learning activity⁽¹⁾ .

Some definitions of instructional design are:

1.      Reigeluth (1983); instructional design is the abstract of the application of learning theory and facilitating someone learning process;

2.      Rothwell-Kazana (1992); formulate the design of learning by improving the quality of one's performance and impact to the organization;

3.      Gagne, et al (1992); developed the design concept of learning by stating that the design of learning helps the learning process one in which the learning process itself has immediate and long-term stage;

4.      Gentry (1994); is a process of formulating and determining learning objectives, strategies, techniques and media in order to achieve common goal;

5.      Raiser (2002); instructional design was a series of procedures to develop a system of education and training programs with a consistent and proven. ⁽³⁾

Thus we can conclude that mathematics instructional design is a practice of creating students’ mathematical experiences which make the acquisition of knowledge and skill more efficient, effective, and appealing.

An instructional design consists of instructional design in general and design of teaching learning mathematics. The elements of instructional design in general are various or type; background or the philosophy; and also the example. The elements of design of teaching learning mathematics are various or type; background; example; comparison or reach marking; and appropriate design.

Before a teacher starts to teach in a class, first he must determine the philosophy, nature; method; and ethic or esthetics, and also the psychology of mathematics. After that he can teach the students that can be in innovative learning or traditional learning. During teaching learning processes there are many aspects that influence it. They are class management, theory or paradigm, context or culture, assessment, learning resources (textbook), philosophy; pedagogy; psychology, media or teaching aid, students worksheet, curriculum or syllabi (lesson plan), method or approach, and also class management. Teacher also must be pay attention on the references and implementation of the learning process. The references consist of formal object and mathematics object. For example to reach an international level, there are some process start from publishing, objective mathematics, reflecting, subjective mathematics, criticizing, and back to publishing.

Good instructional design takes into account the role the on-screen presentation plays and the role the instructor plays in the experience for the learner. Good instructional design will help to avoid what our math student is experiencing as she tries to grapple with solving linear equations. Good instructional design therefore has to provide accessible skill attainment. ⁽²⁾

Hence, here I will explain developing lesson plan, student worksheet, and learning resources.

1.      Developing Lesson Plan

Lesson Plan is teacher’s detailed description of the course of instruction for one class. In Indonesia, we called lesson plan as RPP (Rencana Pelaksanaan Pembelajaran). A daily lesson plan is developed by a teacher to guide class instruction. Detail will vary on the preference of the teacher, subject being covered, and the need or curiosity of the children. There may be requirements mandated by the school system regarding the plan.

A well-developed lesson plan reflects interests and needs of students. It incorporates best practices for the educational field. The lesson plan correlates with the teacher’s philosophy of education, which is what the teacher feels in the purpose of educating the students.

Also, when teacher develop lesson plan, the teacher must know the contents of lesson plan:

-          Identity and Information of school, class, subject, time allocation, etc.

-          Aim of the material

-          Competency that will be developed during the learning process

-          Description of the competency

-          Learning resources that most used during learning process

-          Teaching contents or material that will be studied.

So, from explanations above, I think that on developing lesson plan, teacher must pay attention on the meaning of lesson plan, that lesson plan developed based on the teacher, students’ needs, and school mandate. So, it must be different between one teacher and other teachers. Also teacher must pay attention on the contents of lesson plan. So lesson plan can be work well to teachers and students.

2.      Developing Student’s Worksheet

One of the characteristics of innovative learning is developing student’s worksheet. It caused the paradigm of learning is learning mathematics is doing some exercises. So we need teaching aids, and on of teaching aids is student’s worksheet.  Student’s worksheet or LKS is a worksheet that contains the information and command/instruction from teachers to students to do a learning activity in the form of work, practice, or in form of the application of learning outcomes to achieve a goal.

A good student’s worksheet must complete the characteristics of worksheet, which is student’s worksheet can promote students to construct their own knowledge of mathematics. On developing student’s worksheet, teacher can use discovery methods, they are structure guide discovery and unstructured guide discovery. So, a student’s worksheet is not only collection of many exercises.

Then, there are four steps in developing good student’s worksheet:

a.       Determining the instructional aims by analyzing the students by identify who are our students and knowing the character of our students

b.      Collects the materials that we need based on instructional aims

c.       Arranging the students worksheet, at least there are materials, tasks, and exercises

d.      Checking the student’s worksheet is it consistent with the instructional aim or not.

3.      Developing Learning Resources

Learning resources are all good sources of data, people, and the particular form that can be used by learners in learning, either separately or be combined to facilitate learners in achieving learning goals or achieve a certain competence.

There are two types of learning resources:

-          Learning resources are designed (learning resources, by design), the learning resources that are specifically designed or developed as a component of an instructional system to provide facilities directed learning and formal.

-          Learning resources are used (learning resources, by utilization), the learning resources that are not designed specifically for the purpose of learning and its existence can be found, implemented and utilized for the purposes of learning. This learning resources may take from:

a.       Message: information, teaching materials, etc

b.      People: teachers, students, etc

c.       Materials: books, pictures, etc

d.      Tools or equipment : computer, white board, etc

e.       Approaches, methods, techniques of solving problems

f.       Environment: class, library

Thus, we can conclude that learning resources can be come from many ways, not only based on books and teacher’s note like traditional methods.

(1)  Pengertian Desain Pembelajaran. http://zuhairistain.blogspot.com/2009/04/pengertian-desain-pembelajaran_16.html.

(2)  Cognitive Load Theory and Instructional Design. http://www.usask.ca/education/coursework/802papers/chipperfield/

(3)  Desain Pembelajaran. http://tepenr06.wordpress.com/2011/09/06/desain-pembelajaran/ Page 5

How to Develop Mathematical Design

Instructional mathematical design is a practice of creating students’ mathematical experiences which make the acquisition of knowledge and skill more efficient, effective, and appealing. The purpose of a learning design is to achieve the best solution in solving the problem by utilizing a number of information available. Thus, a design emerged as the human need to solve a problem faced. There are four basic components in the planning of instructional design. The fourth term represents the following questions:

1.         Characteristics of students or teaching participants

2.         Destination

3.         Learning strategies

4.         Evaluation procedures

An instructional design consists of instructional design in general and design of teaching learning mathematics. The elements of instructional design in general are various or type; background or the philosophy; and also the example. The elements of design of teaching learning mathematics are various or type; background; example; comparison or reach marking; and appropriate design.

Before a teacher starts to teach in a class, first he must determine the philosophy, nature; method; and ethic or esthetics, and also the psychology of mathematics. After that he can teach the students that can be in innovative learning or traditional learning. During teaching learning processes there are many aspects that influence it. They are class management, theory or paradigm, context or culture, assessment, learning resources (textbook), philosophy; pedagogy; psychology, media or teaching aid, students worksheet, curriculum or syllabi (lesson plan), method or approach, and also class management. Teacher also must be pay attention on the references and implementation of the learning process. The references consist of formal object and mathematics object. For example to reach an international level, there are some process start from publishing, objective mathematics, reflecting, subjective mathematics, criticizing, and back to publishing.

Here, I will explain about the learning resources, media or teaching aid and students’ worksheet. Learning resources are all good sources of data, people, and the particular form that can be used by learners in learning, either separately or be combined to facilitate learners in achieving learning goals or achieve a certain competence.

There are two types of learning resources:

-    Learning resources are designed (learning resources, by design), the learning resources that are specifically designed or developed as a component of an instructional system to provide facilities directed learning and formal.

-    Learning resources are used (learning resources, by utilization), the learning resources that are not designed specifically for the purpose of learning and its existence can be found, implemented and utilized for the purposes of learning. This learning resources may take from:

a.  Message: information, teaching materials, etc

b. People: teachers, students, etc

c.  Materials: books, pictures, etc

d. Tools or equipment : computer, white board, etc

e.  Approaches, methods, techniques of solving problems

f.  Environment: class, library

Thus, we can conclude that learning resources can be come from many ways, not only based on books and teacher’s note like traditional methods.

Teaching aids is a tool used by teachers, facilitator, or tutors to help learners improve reading and other skills, illustrate or reinforce a skill, fact, or idea and relieve anxiety, fears, or boredom, since many teaching aids are like games.

The aim of teaching aid is to motivate the students to apply his full attention to materials. There are many kinds of teaching aids; they are mathematics card, interactive presentation, worksheet, etc.

One of the characteristics of innovative learning is developing student’s worksheet. It caused the paradigm of learning is learning mathematics is doing some exercises. So we need teaching aids, and on of teaching aids is student’s worksheet.  Student’s worksheet or LKS is a worksheet that contains the information and command/instruction from teachers to students to do a learning activity in the form of work, practice, or in form of the application of learning outcomes to achieve a goal.

A good student’s worksheet must complete the characteristics of worksheet, which is student’s worksheet can promote students to construct their own knowledge of mathematics. On developing student’s worksheet, teacher can use discovery methods, they are structure guide discovery and unstructured guide discovery. So, a student’s worksheet is not only collection of many exercises.

Then, there are four steps in developing good student’s worksheet:

a.     Determining the instructional aims by analyzing the students by identify who are our students and knowing the character of our students

b.     Collects the materials that we need based on instructional aims

c.     Arranging the students worksheet, at least there are materials, tasks, and exercises

d.    Checking the student’s worksheet is it consistent with the instructional aim or not.