My Exploration of Culturally Embedded Mathematics

My Exploration of Culturally Embedded Mathematics

Amrit Bahadur Poudel
M.ed Mathemtics
Kathmandu University

While I was in high school, the image of mathematics was a subject of discoveries of the brilliant people around the world. Learning mathematics was to learn to take the challenges. I had no idea about the purpose of studying mathematics in school. Another motivating factor was that it is a subject which is compulsory to most of the good streams of studies. My teacher never told us that mathematics is a social and culture product. How people use mathematics and how it help people in their public life was not of any concern for me. I never imagined that the people around me in my village are also the inventors and the ones using the mathematical knowledge.

After taking the classes in the master’s degree in Kathmandu University, I came to know that mathematics is not a culture free subject. I still remember the moments of discussions in the class about the contextualization in curriculum. I was shocked with the wrong belief about the concepts and understanding in math I had.
Contextualizing a curriculum is a big issue. There is no exact measurement to measure the level of contextualization. As I learnt from the M.Ed classes, contextualization is a process of adapting the concepts of math in relation to the social and cultural values of the places where the learners live. This helps the learners to understand the value of learning math and find an appropriate use of mathematical ideas in their immediate life.
In the present context, the curriculum of our country is centrally prepared and has an imposed notion of teaching. The interest of the Nepalese students and their level of understanding, culture and social norms are not duly considered. Students in Jumla are expected to understand and solve the problems of electricity and taxes and service charges without any additional support in an equal proportion to the students of urban area. Even the students of urban area have knowledge of high level calculation but are confused when given simple problems of daily shopping and banking transactions. Thus the present curriculum and the textbooks are not fulfilling the present need of our nation.

Thus our curriculum and the teaching strategies are in a transition phase to be changed. In such a situation I have got the opportunity to involve in the "Developing culturally contextualized Mathematics Resource Materials: Capturing Practices of Woman and Disadvantaged Comminutes”. I have been guided to explore the cultural, social and individual activities through ethnographical research methods. I have been encouraged to search for various cultural, social and individual practices and pedagogical implications of a culture sensitive curriculum. To develop a culture sensitive curriculum, we observed the every day activities of the stakeholders such as students, parents, teachers and school management committee members.
In Taukhel, I tried to observe the mathematical implications in the local community. It was not so easy like collecting the stones out of the grains of rice to collect the mathematical applications in the local community like this where I never had been before and never had such research experience besides studying the philosophy of math in text books and in the class. The evidences were not readymade so that I could ask the people and dig the place where the objects were hidden. It was like diving into an Ocean to discover a pearl. Even I knew that the people were not ready to try to explain their practices because it was difficult for them to define their day to day works into a formula. Though they used different mathematical concepts invented by themselves and their ancestors, they had no idea what exact pattern or system they have been using. Even they had difficulty to recall the day to day activities and other social norms where they use certain mathematical rules and patterns. Since they used their knowledge instantly whenever they required, they never kept any account of the knowledge they used in their life. They didn’t have any system of collecting important skills or formulas they frequently required, their mind worked as the dictionary of everything. So I needed to plan and study their way of life and social and cultural reality before I actually could start my research.
For this, I chose one girl-student, her parents, her teacher and one of the women from the school management committee. My respondent student (Sarita) found mathematics very tough subject among the subjects she studied in her school. Math has always been her headache in her studies. Though she wanted and tried to be good in math, it has been a most difficult job to accomplish in her life. However as I observed her activities at home I found her using many mathematical concepts carefully and successfully without actually knowing the formula or any formal definition. She could make appropriate proportion of salt while she cooked the food for her guests. This demonstrated that she had a good idea of proportion. If her teacher would have given the examples of making appropriate proportion of salt or such other examples which occur in student’s day to day life while teaching Proportions, it would have bee very useful for students to understand the concept of Proportion.
She used a circular stone when she played with her friends; they played a game called chatti with one leg. She made different rectangular partitions of a big rectangle where she can stand and jump from one room to another. While she played, she jumped over alternate rectangles with one leg and sometimes she jumped over two rectangles. Her friends understood the rules of the game and the shapes of the geometrical figures for their game but not for their math in the school. They never wrote the rules in a formal format but they conveyed to each other whenever they gathered to play. They sometimes changed the patterns and made new rules according to their comfort or according to the number of members. Thus they had good idea about the patterns and geometrical figures but they never connected the activities while they studied the same patterns in the school.
Sarita and her mother normally kept the glasses of tea in a fixed pattern whenever they cleaned the glasses. They could exactly manage to place all the glasses on a small row of a rack which at the beginning was difficult for me to understand. They were very much used to with the sequence; I must say that the sequence they used in their home was purely invented by them. She easily understood the sequence she used while she arranged the glasses at home but I guess she will possibly fail in the test of the “sequence and series” chapter if she studied Additional mathematics in grade ten. There were lots of patterns and mathematical ideas the people used in their day to day life. It was a moment of joy when I found that the simple people with simple life in poverty in the remote corner of the earth use such interesting mathematical ideas. I enjoyed my journey and will be my pleasure to meet the people again and discover the inventions of the mathematicians living in the economically poor part of Nepal with the simple instruments of iron and wood in their hands and beautiful ladies with the with eye catching bamboo nets on their back. I remember the days in the following ways:

Oh! Beautiful lady,
You didn’t use any paste,
But your smiles with silver line of teeth,
gave a memorable taste.

Oh! Beautiful lady,
You don’t care what the hell mathematics is,
But I remember the math in your steps,
when you danced in the cool breeze.

Oh! Beautiful lady,
I am so sorry,
I enclosed all the beautiful patterns you made without the protractor,
But, but,
I am so sorry,
I forgot to take a single snap of the inventor.

Oh! Beautiful lady,
You are making me crazy.

Mathematics for All

MATHEMATICS FOR ALL

Abstract

History is fundamentally open to change; liberation is an authentic goal; and the concept of “Mathematics for All” is to be brought into being all over the world. Swiss psychologist Jean Peaget proposed four cognitive developmental stages on the basis of age factor. A mother calculates the incomes and expenditures of her family for a day or a month. A farmer predicts whether it will rain or not to save his/ her crops from natural disasters. A carpenter makes different geometrical models. A businessperson calculates his/her profit or loss. A poor always makes plans for his two-time food. A rich makes plans to earn as much money as to become richer. A young has many aims and goals of his/her life to be fulfilled in the days to come. An old man or woman calculates his/ her time of living in this world. Economists used statistical mathematics (data) to change the economic status of the world. Two men put their giant footsteps on the moon. Galileo proved that the earth revolves around the sun, not the sun around the earth. The scientists such as Einstein, Newton, etc. used mathematics and changed the world where we are living happily or unhappily. Newton proved that all the objects in this universe attract each other with a certain force of attraction. So, Mathematics is everywhere…..Every human being is living upon with the mathematical calculations directly or indirectly. On the other hand, some people think that mathematics is only for those who learn mathematics. They think that only the academic mathematics based on prescribed curriculum is actual mathematics. The mathematics they are using in their everyday life is not the mathematics. In this paper, I will present how mathematics is connected with human beings, how people think about mathematics and also give a glimpse of ethnomathematics in the context of Nepal.

Public image of mathematics
What is the mental representation (not necessarily visual) of mathematics, originated from past experience as well as associated beliefs, attitudes and conceptions of people? Sane is an illiterate carpenter in my village. He is always busy in fulfilling the demand of his customers. In the Dashain festival last year, I went to my village and asked him to prepare my mathematical materials within three days. He accepted, but asked me why I needed those materials. My straight answer was “To teach my students mathematics.” He was surprised, and with his wide-open eyes he again asked me the reason that the school teachers in the village had never come to him to order the same. I convinced him how and why those materials could be used in the classrooms. I also made him clear that what he was making daily were all mathematical objects. This is how people are thinking mathematics is only for those who learn it in schools. Bal Chandra Luitel, a Nepali educator associated with Kathmandu University also insists that mathematics for him was a foreign subject in his early (school) education. My father was also convinced by some of his friends serving in British Army that Mathematics, Science and English are the main subjects that I had to learn. He always used to put pressure on me to emphasize these subjects saying “Learn Hisab, Shainish/Chainish and English” for I would do something in my life if I could go to study in the foreign country. Here he might be conscious about school mathematics or used to think that mathematics was discovered or invented by whites (Gore) and hoped that I could get opportunities in the field of mathematics in the foreign countries. Ubiratan D’Ambrosio, the most well-known mathematics educator in Brazil developed the idea of ethnomathematics into concept and a program which has emphasized the mathematics practiced by cultural groups, such as urban and rural communities, groups of workers, professional classes, children in a given age group, indigenous societies, and so many other groups that are identified by the objectives and traditions common to these groups (D’Ambrosio, p.1). It reflects that mathematics being practiced by different cultural groups is also a kind of mathematics. Then, Sane is also practicing ethnomathematics. Potters, black-smiths, farmers, carpenters, tailors, shopkeepers, etc. in the rural villages are practicing ethnomathematics on their own way.
Most of the school children think that mathematics is the toughest subject; and they learn it, because they have to learn it. If a mathematics teacher asks students to choose the optional subjects (in the context of Secondary Education in Nepal), he/she mostly finds less number of students to take mathematics. Students think that compulsory mathematics is already burden for them and they don’t want to increase more. If we ask them about the impression of mathematics teacher, they think that mathematics teachers are rigid, strict and rude to them. If we ask their parents about the impact of mathematics, most of them say that mathematics is difficult for their children and what they are learning is not useful in achieving their goals.
Henderson (1980) claims that the majority of people today are scared of mathematics (and mathematicians) and feel powerless in the presence of mathematical ideas (p. 12). Similarly, Paul Ernest claims that many people take mathematics negatively and think that mathematics is perceived to be “difficult, cold, abstract, and in many cultures, largely masculine” (Ernest, 1996, p.802). In the context of Nepali society, we rarely find women mathematics educators. In Nepal, people think that mathematics is a subject of male only, and females are underestimated and side-lined for the reason that they are culturally and traditionally weak in logical reasoning. Lim Chap Sam asserts three widely claimed mathematical myths in the literature: Mathematics is a difficult subject; Mathematics is only for the clever ones; and Mathematics as a male domain. If the public image of mathematics is negative, then according to Howson and Kahane (1990), the image of mathematicians is even worse. They are regarded as “arrogant, elitist, middle class, eccentric, male social misfits. They lack social antennae, common sense, and a sense of humour (p.3).” Here I want to decode two kinds of public images about mathematics: First, there are some people who think that mathematics is a pure and absolute subject being taught in schools and it has no connection with their cultures or they think that the mathematics they are using in their everyday life is not mathematics. This kind of image is mostly seen in illiterate people in the rural areas. Second, there are other type of people who think that mathematics is useful in our everyday life to some extent, but it has certain roots and limitations causing problems to them; as for example, investment in educating their children in mathematics for a long term does not return their money immediately. This kind of problems is mostly seen in the people who are literate or to some extent conscious about the practical application of mathematics.
In the context of Nepal, parents think that mathematics is being taught to their children to get higher percentage in the S.L.C. examination. They have realized that academic mathematics has no implications on their practical life; rather it is the gateway for their children to go to abroad for higher studies. The curriculum is designed scientifically and the books designed by the Government are also highly appreciable. But teachers enjoy the conventional methods of teaching on the content basis. Teachers are happy to ask the difficult questions, as if teachers themselves are competing each other. Students are being victimized in such a way that they are compelled to take extra tuition class to pass the exam. That’s why students feel that mathematics is a difficult subject in Nepal. A S.L.C. qualified student in mathematics in Nepal can not build up confidence to use mathematics in their everyday life. When I passed S.L.C., I had no idea how to find how much carpet is needed in the room. I have done many problems of carpeting a room, finding the cost, area, volume, etc. But I didn’t get its practical aspects in course of my schooling. If my children are unable to communicate in mathematics in their everyday life, what’s the use of mathematics for? Then, is mathematics for all or for those who are smarter ones?
Mathematics is for whom?
Certainly, I stand in favor of “Mathematics for all.” Every human being in this world is directly or indirectly practicing mathematics in their everyday life. The world has done many developmental works in the field of science and technology and others due to the contribution of mathematics. Even the great scientists Newton, Einstein, etc. contributed in the field of science with the help of mathematics. Computer is also rooted to mathematics. From poor villagers to rich people in the world are living their lives on the basis of mathematics. Whatever are the forms of mathematics, all human beings in this world are using mathematics knowingly or unknowingly. Our ancestors used to kill animals with arrows and sharp objects for their survivals. They also used mathematics because that objects were in different geometrical shapes. They might be using mathematical ideas and knowledge to find the location where they could find the animals. Thus if we talk about the ethnomathematics in our everyday practice, then we find mathematics everywhere.
The question here is that in what way people are taking mathematics. Is mathematics empowering people? What is mathematical empowerment, then? Do people really think that mathematics has contributed for individual as well as social empowerments? We can give many examples for mathematical empowerment in our society. Mathematics empowers people for social justice. Generally, it is seen that a person who has mathematical knowledge is powerful in society. A mathematically empowered person can see the pros and corns, profit and loss, and even predict or guess the possible consequences. He/ she can take a calculated risk in his/her future plan. On the other hand, people who have never gone to school to learn academic mathematics have empowered themselves by practicing culturally associated mathematics such as carpentry, tailoring, pottering, etc. People are empowered in any field due to mathematical knowledge. Is it necessary that people have to go to schools to learn the mathematical skills? According to critical educator Peter Mclaren, “Education system all over the world serves only the wealthy and privileged, and the dominant cultures mostly have hegemony over the oppressed, marginalized ethnic groups, and women.” Then, why should they go to schools? However, what they are learning by doing in their everyday practices such as buying, selling, counting, measuring, weighing, ordering, classifying, ciphering, sorting, inferring, modeling, etc. are all mathematical techniques and skills. The only key point here is that ethnomathematics is to be brought into being all over the world. People are in illusion that mathematics is not for all; and that it is for those who are males, clever ones, and learn mathematics at schools. If a priest wakes up on time early in the morning to worship God, and works according to schedule; if a robber makes a successful plan to loot a bank by taking a calculated risk; if every human being is aware of mathematics knowingly or unknowingly, then why don’t such truths of mathematics come into public?
Who are responsible for creating illusion of mathematics?
Why is there a lack of appreciation of mathematics? According to Kloz (1996), the director of the Public Understanding of Mathematics Forum claims that mathematics profession is the most misunderstood in all of academia. According to him, the public thinks that mathematics contemplate ancient proofs and work as lonely recluses. Moreover, the most common public image of a mathematician has been furnished by a physicist (example, Einstein) rather than a mathematician (Sam, p. 17). Here mathematicians should analyze how public has interpreted mathematics. They can analyze how the image of mathematics has been paralyzed by Physicists and others though they take the basic support of mathematics. Brown and Porter (1997 as cited in Sam, p.17) propose that the mathematicians themselves be blamed. This is because “mathematicians themselves are failing to define and explain their subject in a global sense to their students, to the public and to the government and industry” (p.11). In the context of Nepal, I want to include the Luitel’s experiences as a teacher-trainer in the government schools of Dhulikhel Municipality. He asserts that “…..we need to understand that most of the classroom activities that I observed were guided by externally mandated curricula and textbooks. In the text book, there was no bell and there were no examples of concentric (curved) parallel lines……However, a creative teacher could think beyond this slim document in order to address the complexity of teaching…….S/he could take the students to visit a Hindu temple to study the geometric properties of the bell…..In essence, a creative teacher could conduct many meaningful activities regardless of the perspective, monological nature of the curriculum process” (p.88). It is not my aim to blame any teachers, but teachers are also responsible for spoiling the public image of mathematics in Nepal. Parents insist that mathematics teachers teach their students in such a tricky way that the students are compelled to take extra tuitions, and tuition is exam-oriented. Students are taught how to pass the exam or score good marks. Some parents think that only sons are eligible for mathematics learning. Some think that mathematics has no practical outcomes until and unless their children learn science and technology, computer etc. Pure mathematics is of no use. Parents are there to choose their children’s future in education. This type of wrong practices of parents in Nepal has also created illusion on mathematical application. Who is responsible to make parents aware of such problems: teachers or government?
Now-a-days parents are taking education as consumption or investment. They expect either an immediate benefit from education as consumption or a future production from education as investment. But they think that mathematics education as a consumption is not yet able to provide an immediate benefit to them and mathematics as an investment is for the rich people only, not for the poor people. Poor people can not wait for a long time to get the returns from their investment on mathematics education. Because of these uncertainties in consumption or investment of mathematics education, public do not want their children to stick on learning mathematics. It is seen that students are not interested in recruiting into mathematics in Bachelor and Master Levels in Nepal. They do not see their future in learning mathematics. Then, who secures their future in the field of mathematics? These kinds of negative public images of mathematics need to be investigated and tested critically and empirically.
Conclusion
Here I assert that mathematics is not only the mathematics that our children learn at schools, it is also an ethnomathematics being practiced by different cultural groups in their everyday life. If we are using mathematics in any forms at least to survive and thrive, then we are mathematically empowered. If we are mathematically empowered, then we should be indebted to mathematics and mathematicians who have contributed a lot in bringing mathematics in this form so that others (e.g. physicists) are enjoying the contribution of mathematics to their fields. Whatever are the public images of mathematics and whoever be responsible for that, mathematics is serving all human beings all over the world. Thus, mathematics is for all. If it is for all, then it is our (i.e. mathematicians, educators, teachers, parents, and students’) duty and responsibility to respect and value mathematics.
To end the article, I want to include a poem about the image of Mathematics before and after joining Kathmandu University.

Mathematics was a part of my life,
Theorems were to memorize,
Calculations were stepwise,
And my purpose of learning was victimized.
Theorems were in the midst of fog,
So were the Calculus and Log,
Analysis and Algebra were capitalized,
And my purpose of learning was victimized.
I desired to be Newton in Math,
I tried to be Einstein,
But I found Mathematics as a Giant,
And my purpose of learning was victimized.
Now I find it in the patterns of stars in the sky,
And find in the cultures of our society,
Mathematics is not what I thought as anxiety,
It is really my life and property.
Mathematics is the backbone of other subjects,
Science and Technology, and Computers, and so on,
All have their origin at Mathematics,
I am proud to be a part of Mathematics.


References:
D’Ambrosio, U. (1985). Ethnomathematics: Link between traditions and modernity (Publisher Unknown).
Jaworski, B. (1994). Investigating Mathematics Teaching: A constructivist Enquiry. London, Washington, D.C.: The Falmer Press.
Luitel, B.C. (2003). Narrative Explorations of Nepali Mathematics Curriculum Landscapes: An Epic Journey (Unpublished). A research report for the degree of Master of Science (Mathematics Education). Australia: Curtin University of Technology.
Sam, L.C. (1999). Public Images of Mathematics. A thesis for the degree of Doctor of Philosophy in Education. U.K.: University of Exeter.
Mclaren, P. An introduction to critical pedagogy in the foundations of education. Los Angeles: University of California.
Sutton, J., & Krueger, A. (2002).EDThoughts: What we know about Mathematics Teaching and Learning. Aurora: McREL.

By: Indra Mani Shrestha June 30, 2007
M. Ed. (Math) (2007 Batch)
Kathmandu University

How to learn Mathematics?

How To Learn Mathematics?

Genreally, people run away when they approach mathematical problems. But mathematics is not as complicated as general people think. When people reach to depth and find beauty of mathematics, then it is doubtless they will find mathematics very beautiful, interesting and wonderful. Here are some important things to remember when learning mathematics.

Learning mathematics is like learning another language. One has to assume that I am leanring a lanugage rather solving tough problems. At first it will be hard but it will get progressively easier. A lot of concepts in mathematics are inter-related, so knowing one helps you understand many others. Being frustrated is not a problem, it is a natural part of the learning process, so don't give up. We should tackle all the basic stuff all at once and spend an hour everyday leanring one of basic things. It needs a regular practice which will ease your challenges.
Some steps for learning mathematics.

Create learning time. Make sure you have at least an hour a day to dedicate to learning mathematics.

Become acquainted with the vocabulary. Keep a mathematical dictionary by your side as you study. Many areas of mathematics require knowing a certain amount of mathematical vocabulary and it is less frustrating to be able to quickly look up the meanings.
Get at least two reference books. This way, you will have two different explanations and one of the explanations may make better sense to you than the other or a combination of both will help you to get it.

Tackle subjects along with their prerequisites. Many concepts are related and knowing one helps you understand the other. If you didn't grasp one concept as well as you should have earlier, set aside a little time to revisit it and learn some more and then combine it with the new concept. Often, the new concept will help the older one to gel in your mind.

Progress through the levels of mathematics. Work your way up to advanced mathematics through this progression: Basic algebra, basic geometry, basic calculus, intermediate algebra, regular calculus, number theory, linear algebra, advanced algebra, combinatorics, analysis, topology.
Practice with many problems. Do as many mathematics problems as you can lay your hands on - even those beyond the class. This will assist you in getting a good feel for the topics and will likely help much of mathematics become "second nature" to you.

By Laxman Nepali

M.ed Mathematics (Fourth Semister)