Chapter 6: Using an Ethnomathematics Program and the Methodology of Mathematical Modeling in Classrooms

We are living an age characterized by globalization, the rapid exchange of information, and technological advances, Like it or not, this new world requires new abilities for the next generation of participants in an extremely competitive society.

With the introduction of computers, new relationships between human development, culture, knowledge and the work environment were formed (Toffler, 1990). Like every other human institution, schools as we once knew them, are being forced to transform themselves. School environments now require a much larger and diversified range of vision than previously possessed. Finding ways to truly prepare and educate individuals for full participation for life in a diverse contemporary world of the future is the essence of empowerment.

With this in mind, a primary goal for educators should be to collaborate with, and the development of the talents of diverse peoples. All children should be encouraged to learn about living and working in a society that will continue to be increasingly diverse over time. Given this environment, the school must return to some basic core values, yet assist learners as well to grow beyond just “the basics”. These values must assist in the formation of citizens with the ability to truly think critically. These same values insist that all people must be free to construct their own area of performance or expertise.

This is a transitional moment in human history. We live in a time where change is increasing at an exponential rate. We are all being lead by mass (electronic) media sources that do not necessarily have our best interests at heart.. We have to prepare children to coexist together in powerful domains of capitalism, competition, science and technology. Continued modifications to the social and cultural structure we live in, as well as the relationships between diverse peoples that search for an egalitarian education without any form of social exclusion are paramount. It is crucial that a continued conscious effort be made to encourage all students, especially young women and minorities, in their pursuit mathematically empowering tools and techniques (NTCM, 1989). The development of higher order thinking and learning abilities in our students should become a primary goal of any educational plan, especially in mathematics.

A new type of teacher is necessary as well. This new living and working environment indicates that the very nature of the teaching profession must needs change. Teachers must be duly prepared to handle new technologies, to truly respect and understand other cultures, to learn more than one language, and to develop the ability to respond to a variety of new problems, processes and rhythms that will be demanded by the society on the threshold of a new age.

For these reasons access to and the making good use of calculators and computers has become essential in the preparation of young people. Using these tools to interpret statistical information present in the students’ environment and to develop an understanding using mathematical games is important in developing sense of empowerment. At the same time, recreational mathematics develops a capacity in learners for the use of free time that challenge students to apply intellectual reasoning and participate in challenging exercises that reinforce cultural bases for thinking and learning.

These features are of the utmost importance for the development of a realistic and empowering mathematical teaching-learning environment. However, there exists a need to integrate and contextualize lessons and activities that take learners from simple and or traditional exercises to the ability to search, sort, create and report new forms of information.. Students must learn to analyze and reflect, and deeply understand the groundswell of information coming to them from a variety of contexts. (National Curricular Parameters for the Area of Mathematics, Brazil, 1998). For living together in a society based on social justice, individuals must:

a)    Be introduced to the world of work and a consumer society;
b)    Learn about their body, personal health and hygiene, and information about sexuality (including orientation).
c)    Have knowledge about the preservation of the environment;
d)    Have access to information that allows them to reflect and debate on questions of social urgency.

Currently, these subjects are gaining a certain "status” or importance. They are described as “transversal themes” or “questions” that might be used to assist learners in the acquisition of a “global” dimension to their thinking. Using this context, one goal for schooling should be to create the necessary environment whereby students practice both the “making” and “using” questions for discussions and debates. Statistics, issues and events related to health, the environment and violence are subjects that escape traditional learning environments should be used to transform the discipline, into an environment of multidisciplinary and/or interdisciplinary contexts where people truly make use of mathematics (D'Ambrosio, 1999).

According to the Brazilian National Curricular Parameters for Mathematics developed by the Ministry of the Education and Sport in Brazil, a mathematics education emphasizes an agenda which: “The learning of mathematics is linked to understanding, that is, it signifies and internalizes meaning; by building a real understanding. It seeks a means to identify connections with other subjects and events (NCPAM, 1998).”

Thus, a traditional mathematics curriculum handles content by what Paulo Freire refers to as "compartmentalization", using a system of rigid repetition and mindless memorization. This system gives learners few opportunities to develop strong connections to real-life uses of the subject. For students, a true understanding of mathematics results when real connections are established between it and what NCPAM refers to as "transversal themes" that is, interdisciplinary connections made between mathematics and other curriculum subjects. Mathematics experiences can be connected to daily activities.

There is no real path that exists that could be identified as the best for education in a global society. Despite years of study, we still do not know enough about how the human brain functions, nor do we truly understand the intricacies of culture and history on the way we learn and think for any discipline or area of study. There is no "best curriculum” for all children. However, current knowledge offers range of intriguing possibilities or best practices that will assist the work of the teacher and student in the classroom to construct pedagogical practicalities useful for life in the 21st century. In this way, ethnomathematics as pedagogical action can enable us to use themes found in the surrounding culture, in the history of mathematics, to make use of concrete materials, to integrate games, artistic and culturally-based designs, and mathematical modeling, as features that can supply real context and develop a strong foundation for learners. Ethnomathematics as pedagogical action serves as the instrument for the formulation of further hypotheses and the construction of problem-solving strategies as learners develop with out a universal learning theory or educational “best practice”.

When we consider ethnomathematics as an alternative for program development, the use of mathematical modeling methodologies in schools makes pedagogical sense for schools.  Learners need time for exploration of questions related to social urgency. It is through practical and engaging perspectives for learners the use of such transversal themes such as ethics, citizenship, sexuality, environment, health, cultural plurality, work, consumption and the information society come to engage learners in make true use of mathematics.

Mathematics should be presented in a meaningful context, whether relating to the students' own world and interests or to problems that confronted people in the past and/or in to other parts of our world (Zaslavsky, 1996). Student cultural backgrounds can be integrated into the mathematical learning experience (NTCM, 1989; Zaslavsky, 1996). The ideas considered here provide examples that give meaning to and assist in organizing content form. The designs as developed here explore what we refer to as "transversal subjects". They are part of the reality whose approach estimates the interaction of the mathematics in the analysis of reality-based situations. This is done in the measure where it offers subsidies for the understanding of subject matter that will be approached in solving real-world problems. In this context, learners gain practice in solving the kinds of problems that people deal with in their daily lives.

These designs promise that students will truly come to understand social questions. It promises to assist learners in becoming aware of the world around them. It does this by including the learning of new concepts, procedures and developing of new personal attitudes. However, the idea of “transversality” requires a commitment to share and communicate with teachers in the other curriculum areas. It is essential, therefore that teachers plan social or community-based questions that can be approached from different outlooks.

The use of an ethnomathematics program using as a foundation a methodology of mathematical modeling, enables a connection between the in-school realities and the out of school uses to knowledge and reasoning. Students actually become ethnomathematics researchers by using mathematical modeling techniques, where they learn to seek and use mathematical models that translate to a deep understanding of real or cultural situations – in other words, mathematics becomes truly useful to them.

An ethnomathematics program plays a fundamental part in its recognition of the cultural complexity and reality of students and their individual community settings and reality. It forms the foundation as a practical pedagogy, by allowing students themselves to assist the teacher in assuming some responsibility for forming their own understanding and comprehension. Students are encouraged by the teachers to reflect and discuss the daily aspects of their practical pedagogical concerns. In this way, by using the diversity and differences between cultures as a positive tool, the methodology of mathematical modeling is used to develop comprehension, understanding and usefulness.

The formalization of a problem in mathematical terms serves as the starting point for more the more complex and difficult mathematical modeling to take place. During the period of idea elaboration students learn the relevant mathematical procedures, algorithms, and practical aspects necessary to carry out their research project. Having excellent mathematical knowledge at the outset does not necessarily mean that the student will be a good modeler, or does the lack of it mean they will not. One of the basic conditions for the learning of the mathematical modeling is to study, learn about, and use a number of historical models and essential ideas.

Euclidean geometry, Newtonian mechanics, and the geometric point of view are historical examples of modeling in mathematics and practically all mathematical theories (D'Ambrosio, 1993).

When the students choose the subject they then become responsible as well for their learning, they become personally invested in the outcome. They are motivated and spontaneous in the search, which adds a certain unique richness and flexibility to their designs. The questions they themselves raise become rich sources for problems that in turn motivate the learning of additional mathematical concepts for the respective solutions.

Thus, the real challenge for the teacher is to encourage students in a direction that insures that the students explore systems relevant to their research question, so that they are understood and have meaning. This is important so that there is a certain guided structure or direction created, so students can construct mathematical models that will form the basis of the content to be learned. This informal structure creates conditions that encourage the systematization of concepts, and which become part of natural learning experience of the student. We see this in the informal or out of school aspect of many learners’ lives – as they learn to play a sport, a job, or master a video game.


The Methodology of Modeling

What it is important here is not the teaching of modeling per se, but the teaching of mathematics using modeling as a methodology. This is an important distinction. The mathematics as pertaining to different school levels can be taught where various degrees of difficulty gradually can be analyzed. This then creates an environment supporting different approaches to problem-solving activities connecting directly to the learners’ reality or culture is sound pedagogical practice.

The objective here is to teach mathematics through researching, exploring and developing activities that support the organization of mathematical concepts used to develop mathematical models. This naturally becomes part of the content or an established organized program. This methodology uses mathematical modeling and is a process whereby concepts are modified and customized to fit the student’s unique research questions, interests, situation and context. It is uses methods similar to coaching versus direct one-size fits all instruction. Through a continual process of modification, the classic mathematical modeling is used as a methodology pertaining to regular school courses this is what we call “mathematical modeling”.


Mathematical Modeling as a Methodology

The methodology using mathematical modeling follows a distinct sequence of defined stages. An applied example is chosen initially to be worked with students with the objective of making them familiar with the dynamics of the process, supplying them an outline or introduction to the program that they themselves will develop. It is necessary that the professor has some experience with mathematical modeling and already has constructed some models for the students to observe and analyze.

What follows, is a description of an ethnomathematics program using a methodology for mathematical modeling used for classroom settings. It can be used as a starting point for exploration of different mathematical themes, especially geometry. There are numerous interdisciplinary, historical, and cultural aspects that can be discussed while students work in groups and accomplish explorations and research related to the accomplishment of other activities that involving the geometric construction of mathematical models, constructions, experiments, or other situations that the teacher considers adequate.

As mentioned earlier, a unique aspect of this paradigm is that the students themselves are able to choose the subject to be researched. This almost guarantees engagement and “buy-in”. Suggestions are listed during whole-class discussions. As a beginning point, a teacher “influenced” list of subject choices can be written on the blackboard. But students should be allowed to add to it, and discuss the pros and cons to each. The teacher may also place subjects on the list of particular or topical interest During 1999, students may want to look at Y2K, the millennium, or global warming. Subjects are often obtained by engaging students in bibliographical research. Later, the teacher weaves commentaries related to suggested subjects, indicating the many advantages and disadvantages of subjects included in the list. Many times, subjects chosen by the groups are those which the teacher communicates an emphasis or can show importance.

At a first, it can be convenient to work with just one subject for all the class. Giving to each different group sub-problems that were considered in the initial development or “brain-storming” stage of the problem. Each group can approach diverse aspects of the larger project, while in relation to the same initial subjects. Some experiences carried through with other subjects in the same class or school show that the greatest difficulty is in the attention the teacher must give to each group. After some experience working with modeling, teachers often feel able to take care of certain subjects more than others do; this of course is a matter of style and interest of the teacher and students. This is most certainly appropriate if his or her own enthusiasm or personal research goals motivate the teacher. However, the choice of situations that are unknown, therefore become challenging and rewarding adventures in the teaching-learning process.

This method encourages students to survey the problems as they are directly related through the study and use of integrated bibliographical resources (using books, periodicals, magazines, the Internet and other sources). As well, students make and conduct interviews with specialists or experts of the field. Often teams bring an interesting aspect to the class by inviting the specialists to speak on the subject of study at school.

The growing data collection is enhanced by ongoing or direct experimentation. It is also enhanced with the introduction of statistical research methods. Students soon find that a combination of these procedures will yield more interesting results.

It is interesting to point out that many times data found using this research method is of an ethnomathematical nature in itself. Students often see how some forms of mathematics proceed from cultural necessity and have become part of the customs of a community without any concern with the "scientific nature" of its origin. Students then take this knowledge by relating or connecting it to the universal form of mathematical knowledge. Thus, the school experience makes real use of the curiosity of the pupil. Students see and experience mathematical power (Parker, 1990) directly in the environmental context where it originates and is used. This activity and method also fulfills a task that is increasingly forgotten in the universal rush to “improve schools”. In an era of accountability and curriculum standards, solely emphasizing test scores as an indicator for academic achievement. Students in this model see mathematics as a tool for freeing and developing their creativity.

Thus, the establishment of student research groups that give students the opportunity to become familiar with problems based on their research is essential and integral to this process. As part of the process the teacher presents other problems related or not with the subject in question. Therefore the analogy made between diversified problems shows the importance and power of mathematics as a common language that is identified though the elements of the group, and is based on the common interests and experience of the students.

When forming their mathematical models, each group must be sure to establish:
a)     The involved essential "0" variable in the problem;
b)     The hypotheses that simplifies the problem;
c)     The systematization of concepts used in the resolution of mathematical models and that will become part of the considered content;
d)     The conclusion obtained after the interpretation of the solution and the validation of the mathematical models with the searched system.

In a constructivist manner, the teacher assists the students by offering questions begin with more simplified versions of the models than those already known by students, the students can modify it themselves. Developing models become even more complex or general, aim at the particular mathematical content or curriculum standards that form required educational program. Solutions students as researchers discover and develop themselves are interpreted using their own data. When possible, they are shown how to verify and validate models and analyze the consequences. It is of basic importance that learners are shown how to find graphical expressions for their solutions and verifications for their hypothesis.  The capacity of the students and to make forecasts allows them to come to a better agreement of the systems they learn to analyze. If the model is satisfactory they can then use it to make forecasts, analyses or other forms of action on the reality.

It is intended that by using this form of project development students learn a as a methodology for teaching-learning in mathematics, that:

a)     Learners gain experience using both modeling and mathematical modelação, to investigate and discuss accomplishments and various innovative proposals used in mathematics;
b)    Learners come to a certain "sense" with mathematical content through an approach that allows them to experience mathematics used in the daily lives of people in various contexts;
c)    As well, teachers give mathematical meaning to the experiences in the life of students;
d)    Teachers extend social concepts from the study of a given situation-problem, using mathematics as a language to understand, simplify, decide, forecast, modify and transform the reality that students experience;
e)    To create a interdisciplinary predisposition for the learning of related mathematics to other subjects, making true integration possible with other fields of human knowledge.

In relation to the above, Claudia Zaslavsky has said, “Students need to learn about the mathematical involvement on their own and within their societies and groups. This knowledge should increase students' self esteem and self-confidence (Zaslavsky, 1996)”.

It is important that we rethink education by integrating culturally related meanings, by making use of mathematical tasks created outside of school. The form of activities that require reflection related to mathematical concepts, from problem situations, while of a practical / cultural nature generates meaning appropriate in this context. Like watching young people learn to excel at computer games or soccer, this process uses a true sense of engagement in the development of the understanding of concepts and mathematical models becomes readily apparent to all participants.

The in-school activity as outlined here constitutes a certain ethnomathematical context that encourages students to find the contents and mechanisms for the construction of significant mathematical experience related to their own research. The projects students complete in this context include the elaboration of activities where they themselves construct and communicate solid mathematical arguments to each other and an audience. They do this in defense of familiar mathematical ideas or explorations learned as part of the research and exploration. This same process in the classroom communicates explicit practical ideas pertaining to school, as a real and daily activity, in the measure where its language and procedure become familiar to students.

As part of this ethnomathematical context, the pedagogical work considered here is part of the ongoing process involving the interpretation of contexts in which mathematical knowledge is present. What this form of work encourages is a more flexible interrelationship between culture, the student's learning and social environment and the mathematics that needs to be learned in the school setting. It is more realistic, and if done correctly is certainly more rigorous for both teacher and student. The personal interests and unique needs of the student for the determined subject are seen as the choice mechanism for creating directions and activities supporting their personal growth. Because this system is part of a larger reality, through which models the students build and research, mathematics then becomes a powerful tool students use to explain, understand and act upon their own world. As we have shown above, mathematics easily articulates with transversal themes. We shall consider this relationship to the development of citizenship and information below.


Citizenship
Mathematics objectively collaborates in the formation of citizens (cite). Therefore an infusion of a sense of citizenship and participation in the learning process itself is very much a central aspect of ethnomathematics. Some responsibility lies upon the individual student there for e to make sure they know basic facts and algorithms once they have mastered them in the academic setting. However without the development of any formal connections between the many academic fields and diverse areas of knowledge in mathematics education, little can be done to contribute for the general formation of pupils with a view towards the development of citizens who can fully participate in their own communities. Too often low achieving students are not allowed to see the power and beauty of mathematics. Ethomathematics as pedagogical action tells us that we must find ways for all students to participate, not just the advanced or gifted ones in mathematics.

It is well known that mathematics is a discipline provoking a certain sense of failure and dread in many students. As well, it is often an undeniable fact that mathematics is used more often than not as a sorting mechanism for the exclusion of a great part of the population in relation to careers in science and politics. It is often used as a selective academic process or filter

All educational paradigms look to give mathematics its place in the development of citizens. In fact, for me, of all the movements that might answer my concept of citizenship that of making a more coherent relationship between citizenship and education, is an ethnomathematics program (Sebastiani Ferreira, 1993).

Ethnomathematics makes possible the release of universal mathematical truths and respects the learning of the non-academic citizen. Therefore, we must search for a new form of citizenship constructing itself through social action and politics for all people. A citizenship not determined solely by society’s elite who consider themselves the true “holders of knowledge", but with the input of all members of the given society. This can only be done if we extend the use of tools for reasoning and thinking to members of society, not just the easy to teach or talented members of a community.

Information nowadays, involves enormous amounts of information propagated by all forms of media, books and electronic sources. These data sources present us with varied forms of information, profoundly influencing our professional as well as daily life. Information most certainly influences our happiness and our culture. Yet most traditional models of literacy do not take these realities into account when teaching students to “read”, “write”, or to “figure”. Empowerment means that students will be able to analyze statistics and graphs they read in the media. It means they can make sense of non-print media messages and propaganda. It means that they can apply mathematical knowledge to the problems of society, and take appropriate action as citizens of their communities.

The process of data collection as introduced here allows for the development of a particular form of thought and reasoning useful in assisting students to participate in their world.. These processes include the ability to collect, organize, present, and interpret information (data). By making use of mathematical modeling methodologies, students are better equipped to formulate models that will accumulate an ample variety of mathematical concepts. It also allows for the development of certain critical attitudes useful in making daily forecasts and decisions based on the information presented by the world-wide media.

This information can be used to further awaken interests in students for developing a sense of social conscience in matters that offer new questions and ideas related to their reality. It can be used in developing a significant context for reinforcing and learning new concepts and procedures and to make connections to other knowledge areas. The experience allows for mathematical contents to be developed by the students themselves that supply necessary statistical instruments and data that can be explored by the accomplishment of research that is interesting to the students themselves.


Information

Today we live in a world characterized by an enormous glut of information. It is often difficult for us to make true sense of reality - either because there are too many choices, or because we are often manipulated by visual images that convey information in an emotional and manipulative format. Many people simply “tune out” to the overwhelming noise and confusion. We do not possess the tools to make sense of this information. A true sense of empowerment in such a world, means that we are able to analyze statistics and graphs we encounter in the media. It means that we are better able to apply mathematical knowledge to the problems of our lives, society, and to take appropriate action as citizens of our own communities.

This information wave (Toffler, 1990) requires us to develop particular forms of thinking and reasoning, in order to be true participants in our communities. In order to do this it is necessary for us to learn to collect, organize, and interpret data. We must be able to communicate rationally and to make healthy decisions using statistical language or tools. By making use of modeling methodologies, we learn to formulate models that embrace a variety of mathematical content that supply amplitude and applications of basic concepts. We develop certain attitudes that position us critically to make previsions that make further decisions based on the information presented daily to us by the various mediums of communication we are in contact with on a day to day basis.

The information we are accustomed to use in our professional or out of school life is more often of a different quality and quantity than that found in the typical textbook driven school environment. Yet teachers are charged with the responsibility of preparing children to live in a world that is vastly different than that of the average classroom / school. The majority of modern literacy theories designate reading as the primary information retrieval source. Yet, the reality for most of us is that the vast majority of inhabitants on this planet currently receive their information from non-print media sources (i.e. film, television, radio, billboards, signage, gossip etc). Schooling has very little connection to other areas of understanding, thinking and learning. An ethnomathematical perspective seeks to make this connection, by truly using the true reality to assist learners to live and work, to contribute to life in this modern 21st century reality.