Chapter 1:Ideas for the use of ethnomathematics have been developed in many countries over many years. However, the actual theory of ethnomathematics was first introduced by Professor Ubiratan D'Ambrosio of São Paulo, Brazil at the International Congress of Mathematics in Australia in 1984. Prof. D'Ambrosio has consequently explained,

Ethnomathematics is the art or technique (techne), which is to explain, to understand, of playing in reality (mathema), inside of a proper cultural context (ethno) (D'Ambrosio, 1993).

Ethnomathematicans readily recognize that all cultures and all people have developed unique and sophisticated ways to explain, to know and to modify their own reality. They recognize that these ideas, as are the cultures that they are embedded in, are part of a natural, constant, and dynamic process of evolution and growth. It is not the premise of ethnomathematics to disdain nonwestern or non-European models or traditions, but to consider the validity of all explanations of reality as constructed by all peoples, cultures, and historical backgrounds. These forms of knowledge are not considered static or dead; they are part of a process of constant mutation, evolution, and growth, as part of their own unique cultural dynamism, and so therefore should be studied and catalogued.

Using an idea borrowed from cultural anthropology, no single form of problem solving is any better or worse than any other form (Hall, 1976). Each has evolved unique problems to resolve. Ethnomathematics is not solely mathematical; it is broader and includes other disciplines. By making use of the diverse methods that cultures and peoples use to find explanations to increase their own understanding of their world and time, ethnomathematics seeks to understand these unique realities.

Alternative forms of mathematics often come about as peoples work to explain and resolve practical problems in their daily lives. What is universal is that all cultures have found ways to search for knowledge. All cultures have the necessity, in fact have found, unique ways to quantify, compare, classify, measure and explain day to day phenomena (Borba, 1990).

Much of the formal mathematics as traditionally taught in many schools does not readily allow for prior knowledge and experience of learners entering the classroom. It does not allow for learners to gain easy access or give them the experiences necessary to arrive at real meaning, or something useful in their daily lives. Many students are denied an adequate environment in which to develop a successful attitude and mathematical abilities in school.

For example, one needs only observe children playing soccer. Or look at other children deeply engaged in the process of problem solving as found in the playing of a favorite video game. Or still, other children working with their parents in home or job related tasks, to see mathematics used enthusiastically and realistically. Yet, the connection between the academic i.e. in-school mathematics and the outside world is weak at best for many people. Ethnomathematicans see this as a lost opportunity. The ongoing worldwide research in ethnomathematics seeks to further study and document and explore these connections.

At the same time, it is evident that schools must find ways to speed up the process for children to construct their own knowledge. Humanity took thousands of years to accumulate the knowledge that we expect learners to become fluent with-in 12 to 18 years. Obviously we cannot offer it all at once for children. A constructivist view of education tells us that learners must be encouraged to build upon their own experiential context using a variety of modalities (concrete experiences, writing, verbalization and others). This is precisely how people do mathematics outside of the classroom. We must create learning environments that allow children time to reflect on these realities, which lead them enthusiastically to this knowledge in ways that connect to the child’s own unique experiences, by supporting them to build upon their own understandings and context.

By using an ethnomathematical approach in the acquisition of mathematical knowledge, the implementation of this idea in learning environments will provide children with experiences that make sense to them. Through examples, activities, explorations and methods that add relevancy to their learning experience learners better understand what it is they are learning and doing. The creation of a realistic learning environment awakens and sharpens the interest and curiosity of learners. Using mathematical models and the basic tools of mathematical modeling can make the connection of in-school mathematics to the mathematics found in learner’s daily lives.

As we learn how to assist children to gain access to positive mathematical experiences they need we make it possible for all children to successfully participate in society (NCTM, 1999, 2000). The present image and role that mathematics possesses in society requires a great deal of modification (Schmidt, 1999). This suggests that the role of mathematics as a function / selection machine, coldly determining which pupil goes on to university or to trade school, seriously needs adjustment. We need to move towards a pedagogy that includes an open system of mathematics experiences that insure that all learners have equal and full access to the opportunities needed to enable them to decide for themselves where and what they want to do with their own futures.

We must also argue for the importance of mathematics used for the construction of a sense of participation in society (i.e. citizenship). We call here for an emphasis mainly in the development of critical thinkers employing the independent participation by teachers and pupils alike. Ethnomathematics provides for the establishment of connections between the mathematics with other subjects and of daily or what some children call real life.

Teachers, while mediating the teaching-learning process learn to assist learners to clarify the importance of mathematics as an instrument for better understanding of their world. Ethnomathematics encourages us to stimulate the interest, curiosity, creativity, critical inquiry, and the ability to resolve problems in all students. Mathematical knowledge has been constructed by all humanity. Its use is not exclusive, nor is it to be used solely for mathematicians and scientists, but by all people.

The mathematics used in schools is the culmination of the many accumulated historical and cultural experiences. Western, what we refer to from now on as traditional 21st century mathematics developed over the last five centuries has its roots in Greek and Egyptian cultures. It developed and has used the necessary abilities to count, to locate, to measure, to represent and to explain, in accordance with the necessities and interests stemming from a Mediterranean and European historical paradigm. The mathematics as currently taught in the majority of the world’s schools was spread by European colonization and dominance over the past five hundred years, and is in accordance with the history and the unique necessities related to the historical moments from which it sprang. Therefore we use a form of mathematics which is every bit a culturally constructed form of knowledge. In an approach to this unique pattern of knowing, in which the pupil also takes part and effectively becomes the active agent of this process of teaching and learning.

Beginning with our own unique form of knowledge, we can learn to contribute towards the elimination of the myth that one and only one form of mathematics represents the truth. Thus, a form of knowledge has come to determine who becomes a member of the elite few, and it is used for entry into distinct social groups. As well, many higher level courses in mathematics function as a filter or sorter. Instead we seek here to demonstrate mathematics to be used as a tool for gaining full access to life’s diverse opportunities, is dangerous, and should be an anathema for the support of citizens living in an ideal democracy.

Much of the work developed for teachers is rooted in this traditional concept of mathematics. It also represents a mathematics that asks pupils to simply reproduce or memorize pre-prepared information: what Paulo Freire refers to as the banking model (Freire, 1970). Most mathematics curriculum is traditional in this sense. Many new proposals continue in this direction and may even include new applications and methodologies. However, these new applications applied to old pedagogy are doomed to fail. For they often make use of a paradigmatic system used for traditional and modern mathematics, and not for true empowerment. Students of today must to be able to use mathematics to better their lives; it is the overall quality of a successful mathematics education that depends on the following characteristics that enables them to become full participants in their society. Because we all live in a globalized society, students:

· Must be able to reason logically;

· Must be able to use their own creativity;

· Must strengthen their own unique on-board ability to quickly adapt and learn in new and innovative situations;

· Must be able to become active knowledge gatherers and disseminators;

· Must develop a sense of responsibility towards the environment, including the preservation of its rich and diverse cultures;

· Must be able to develop initiative for resolving problems;

· Must develop processes for daily professional life.

As a basic condition for critical performance in society, mathematics educators must be part of a globalized cultural context, where students have access to and develop knowledge of mathematics. For many people, a lack of mathematical knowledge makes it difficult to understand the complex challenges that face them. They are powerless to resolve basic problems in their own lives, and are shutout of more powerful careers and professions. Students need the opportunity to learn to take a position connected to the problems and methods needed for deciding within their own context, so that these learners can use a variety of mathematical tools, which enable a better understanding of the natural and social phenomena of the world. In this way, mathematics becomes a social instrument, and is integrated into the concept of citizenship where the mathematical content can be explored in order to develop logical reasoning (the capacity to think). All learners must go beyond the mathematics of memorization, and learn to develop a professional spirit of criticalness, showing mathematics as valuable knowledge in the life of all human beings.

The teacher’s work in this newly evolving context must be to learn to make critical analysis of mathematical content (NCTM, 1989; CDE, 1992). Teachers must learn to examine the true nature of the mathematics they offer their students. They must be able to show a real need and concern for other cultures and different historical contexts, and the mathematics as used by these diverse peoples. Teachers must be trained to establish comparisons between mathematical concepts of the past, and enable learners to develop their own connections.

Pedagogy

Ethnomathematics places tremendous value upon the empowerment of learners to resolve problems using situational or real life perspectives (Mendonça, 1999). Through the use of interdisciplinary activities, students learn to see mathematics as linked to other areas by using research tools that analyze problems and develop mathematical models. However, the selection of content alone is not enough to guarantee their meeting of mathematical objectives. The way in which subjects are organized by the classroom teacher (pedagogy) as equally important, since individual or personal responsibility, involvement and connection to the disciplines are important to the success of each new aspect of the mathematics learners find for themselves. This enables the formation of flexible, curious teachers and learners who readily engage in the basic research and critical exploration needed to collect the mathematical data and tools necessary for life in a globalized information-rich society.

However, the majority of programs training teachers to teach mathematics currently do not make this an easy objective. Most of the mathematics that was transmitted to them is hardly critical in nature, but passive, that is, with the emphasis on the attainment and memorization of basic facts. The universality of mathematical knowledge is often not revealed. Too often, mathematics is transmitted in a crystallized form, and learners rarely interact with the subject. Mathematics comes from using a dynamic process of communication, testing, and experimentation. It is really a series of interactive questions, and is alive and growing, not static or confined to mere arithmetic or algebra. This somewhat uncertain but dynamic science contrasts greatly with the traditional processes of becoming a student and lifelong learner.It does not mean that the mathematics of the past is limited, but it was applicable in a context much more limited than of that today (D'Ambrosio, 1993).

Today, such important ideas such as fractals and chaos theory can be shown as prime examples of mathematics that are all but disregarded by the standard school curriculum. These two areas alone offer truly engaging experiences and powerful ideas for young people. Yet they seem to be all but ignored by the politics of high stakes testing, accountability and back to basics. The great part of the traditional school-university curricula is based on the mathematics found in the 16th to 19th centuries, and when something referring to 20th century mathematics presents itself, it is more than often consolidated into what it became in the 19th century.

Many new teachers enter the profession with limited or an obsolete knowledge of mathematics. Yet the world they train children to live in is a dynamic information-rich, interactive, diverse and globalized world. Like their teachers, students condition their minds to this static, obsolete and mechanized knowledge while in school. Much of the mathematics curriculum and instruction found in schools and textbooks contributes to the pacification of their learners. The mastering of obsolete mathematics is often considered important for those who seek to go on to advanced classes in mathematics. Mindless memorization has been deemed sufficient for those who have succeeded in this form of mathematics. More often than not, it has been these same students who have been identified as brilliant and intelligent. This then forms the basis for numerous complaints related to current math reform. Learning to think and reason creatively is difficult work, difficult to assess, and of a higher order than memorization.

An ethnomathematics perspective can give a child the responsibility to learn basic information. Learners can learn how to make connections from the past to the present. In ethnomathematical contexts, as in real life, it is skill and creativity that are most important for success. If schools and communities allow the trust, freedom, space and encouragement for mathematical creativity in their teachers and students, they contribute to the formation of truly empowered and active citizens for the 21st century. The role of the teacher in an ethnomathematical learning process is not that of being the conductor of learning (sage on the stage), but of a facilitator or coach, for the student (a guide on the side).

So it is that educators must find ways to support and encourage teachers to incorporate the outside interests and culture of their students and communities. Teachers must be encouraged to organize engaging and useful projects that pay attention to the outside reality of the school. It is the practicality of an ethnomathematical perspective that must be organized so that each learner finds their own niche, giving and extending to them the intellectual resources necessary. Each learner must allow them the opportunity to construct knowledge from their own reality. Thus, mathematics becomes something good and useful to them. It then is something essential to the learner, community, and society. For example, a 15-year-old student who entered school as a 7-year-old has lived about 131,000 hours. Depending on where they live, they have passed approximately 9,000 hours pertaining to school tasks. They have watched about 16,000 hours of TV and slept approximately 44,000 hours. It is doubtful that in the 122,000 hours the individual spent outside of school, they did not stop learning. Learning most certainly happens as part of play, in living, and coexists even as we sleep. It is not confined to the six or so hours a day the child spends at school.

What we are attempting to describe here is the functioning of an ethnomathematical perspective. What is important to understand is that this perspective gives individuals a chance to learn mathematics in a natural and realistic context. By having students carry-out projects, work with people that they may not know or have chosen to work with, to form teams of diverse and talented individuals, with each one making a contribution to the project students learn to use mathematics with realistic relevance. Learners must develop the capacity for living and coexisting in a dynamic and diverse society, by respecting the strengths and weaknesses of others, by becoming critical and self-sufficient individuals, and with a willingness to be agents of social transformation.

Ethnomathematics is not only a different approach to mathematics; it is an area of knowledge. The focus of ethnomathematics is interdisciplinary. It presents a knowledge discipline that is complementary and inclusive of other academic disciplines. Ethnomathematics contributes to the giving of another image pertaining to in-school mathematics. Thus another primary objective of ethnomathematics is to sharpen the curiosity and creativity of the learner.

Using a pedagogical point of view that fully incorporates the history of mathematics, we can see that science evolves and is born from diverse cultural systems. We can construct relationships to what often are seen as distant, exotic or strange cultures that are not part of the learner’s universe. While taking care not to be artificial, it is important to become knowledgeable with the contributions made by the Mayans, Aztecs, Egyptians, Greeks, Babylonians and other peoples of antiquity. Western science is not the only form of thinking that developed answers to interesting problems, learning this is extremely important in connecting children to our human heritage. Also learning from the past can help us keep us from making similar mistakes in the present (Stuart, 2000).

This perspective tells us that we also need to introduce and study the unique mathematical perspectives and situations originating in our daily lives. Students must have a chance to practice, observe, reflect, and question things from a mathematical perspective. Children naturally construct their own knowledge base through the use of active questions, by making use of such things as patterns, quantification, geometric forms, space, and time. Adults call it play, Piaget referred to it as children’s work.

Constructivist theory tells us that children learn to draw conclusions from the interactions they have with objects and with peers, when using concrete experiences in their own personal environment. This paradigm shows us how we can learn to develop methods in which children experience the mathematics found in their environment. This natural necessity to discover and explore new ideas and situations touches upon the core emotions of the child. It often manifests itself in the form of games and play. During the early stages of learning, children often feel that they alone are making discoveries. The children become engaged, as they become personally involved in the search for explanations and the ways they invent in coming to understanding their world. It is this process of knowledge creation, which consequently allows the learner to create mathematical models, leading to a dynamic process of understanding and the ability to decode their own reality. Children in all cultures possess a natural drive and ability to understand mathematics. This is evident in preschool and primary children worldwide. Over the length of their schooling, however, many come to feel that mathematics is not useful, interesting or enjoyable. Ways must be found to overcome this aversion and give children the tools by which they can explore alternatives for solving problems in an academic setting.

Older children find games and social activities engaging (basketball, soccer, baseball, and computer games). They are willing to spend hours with these problem-situations. Yet most children are less inclined to spend time on the basics or traditional math tasks. Tasks often far less complicated than the actual games and activities they engage in outside of school. Finding ways to connect this out-of-school reality to formal in-school mathematics and the scientific paradigm is essential to empowered problem solving. Those who are called active problem-solvers are people willing and able to construct informal problem-solving knowledge outside of formal school environments. Children the world-over do this willingly. They can problem solve when they are alone or with friends, for hours at a time, with the absence of a teacher, or supervising adult.

It is well known that many computer game and sports players are successful at problem solving in the absence of a structured curriculum, and standards. Their assessment is often brutal, with life and death consequences, as in soccer or computer games, to name two universal examples. Often children learning to play computer games or soccer outside of the school environment are able to easily find ways to decode this reality with naturalness and spontaneousness. In these games they easily extend their vision of the world and develop criticalness about their experiences, often doing sophisticated problem-solving far more difficult and complex than that done in the formal curriculum. Ethnomathematics proposes to make mathematics creative and spontaneous as that found outside the school or academic environment of the child. What ethnomathematics hopes to accomplish is to unlock the very essence of pedagogical work within the classroom.

The life experiences that learners possess need to be taken into account. For example, consider a child who lives in a port city, close to a shipyard. This child might be better able to assimilate content much more easily when people use the examples of cargo containers, cranes, bridges, ships and boats. A Native American child growing up in the desert region of the American Southwest, might find examples and stories of ships and water difficult to comprehend, as would the child living in the coastal port city comprehend living in an eight sided log or multistoried adobe home. In the case of the urban child, it would be wise to include the child’s experience of leaving their home for the school, by designing experiences that take into account their daily trips to school - turning corners, crossing streets, mindful of traffic, public transportation and other urban realities.

Ethnomathematics is the acquisition of the daily necessities of life, the important people and places that are most certainly examples of other useful connections to mathematics for younger children. Asking children to observe, that is, to truly pay attention and to use the real experiences as they learn to observe and interact with the world around them is an important connection to mathematics and the scientific method. We can show comparisons, we can use the data gathered from their own reality and use this data to practice and learn algorithms, to perform and learn statistics. The explorations and comparisons related to the difference in the size of their steps can take-in the diversity of people and use this data to begin basic mathematical instruction. Thus, children arrive at the conclusion necessary to have a standard measure.

Children in one town in the state of São Paulo, Brazil, during their daily trips to and from school, needed to cross a bridge over a small river that ran through their city. Teachers there taught the children to take the time, to stop and linger a little, to observe what they saw. They asked their students questions such as, how many capivaras did you see? How many birds are there in the local ecosystem? How many and of what color flowers did you see? Who did you meet? Where were they going? The children learned to see mathematics in the very journey they make to school. They began to observe for instance the dredges that were working the river to prepare for the coming rains. They began to perceive what happened every time the dredge moved a shovel full of river bottom. The observations created new questions related to the amount of garbage and silt removed from the river. They quickly expanded to the mathematics dealing with the time of the job being done, the length and the width of the river to the fish that by chance were killed, and to the kinds of garbage found. The children developed a critical ability to ask questions related to their environment. They were able to make connections between the mathematics in their daily life with the mathematics learned in the school. In this way, the teacher did not lose the chance to observe where children focused their interests and excitement in learning to explore and work in the community. The community as well began to see the school as a resource for economic and community improvement.

Another danger of experiencing mathematics only by that found in academic or formal settings is that this ready and finished knowledge often leads us to believe that in younger children’s classrooms it should happen and be learned in just one way. Teachers must search for, and bring out the creativity in the children they work with; this creativity comes from within their own environmental context. This relates to the process of actively respecting, recognizing, and becoming involved with what is around us. In other words to truly learn to make a difference.

An ethnomathematics perspective suggests that it is the educator too, who must learn to see the world differently. We must learn to search, to observe, to pay attention to the true reality of our learners. This will enable children to observe and find the mathematics around them, and incorporate their own unique cultural and social reality. As stated earlier, we seek in this book, to offer a few practical solutions towards the resolution of this dilemma.

By only giving knowledge that is of questionable import, at the loss of much more engaging and empowering content is a travesty. Traditionally the child learns by constant repetition of everything that they have learned. They are judged to be successful by the amount he/she can parrot back to the teacher in a timely fashion. Ethnomathematics asks us to look deeply at the mathematics in collaboration with several disciplines. The majority of the content in mathematics has been learned for conditions that only exist pertaining to survival in formal academic and school environment. This is accomplished without little or any true connection to its usefulness in daily life. Adjusting this process may give certain vitality to the mathematics experience for both teachers and students.

The inclusion of practical activities in the use of both quantitative and qualitative aspects develops logical reasoning in learners. However, one of these aspects is not only enough to characterize the necessity of this inclusion. It is important to understand the total function played by mathematics, as found by the practical application of knowledge and in the development of reasoning. These aspects must be considered as non-separable elements as we use mathematics when we are deeply engaged in learning to beat a computer-based simulation or action game, or when we excel at basketball or soccer.

The inclusion of the real-world gives conceptual tools to learners. It serves to establish a true sense of continuity between school and real-life. It assists both educators and learners to find ways that enable them to construct a sense of intellectual autonomy and freedom. This autonomy is most certainly not the exclusive goal of traditional mathematics. These methodological concerns in mathematics education must become those of the teacher. They are in reply to the questions and concerns between student and teacher and form the basic fundamental approach to ethnomathematics as pedagogical action."