Capitulo 2 ? Modelagem Matemática como Metodologia de Ensino

Chapter 2: Defining Mathematical Modeling

Mathematics is a dynamic, changing, and active system. Current research in ethnomathematics only goes to further this supposition. To learn mathematics, therefore, does not mean that we passively receive or memorize ready-made concepts in prepackaged form, what might be called “fast-food”, or what Paulo Freire referred to as the banking model of education.

Any real learning of mathematics is intimately related to the gender, age, work, reality, culture, unique thoughts and emotions of each individual. The true learning of mathematics is like that of learning to be an artist, or athlete, as in learning to play soccer or in the mastering of a video game.

The construction of mathematical concepts should incorporate the true reality of each individual. This should begin by placing new situations and problems in front of the child for them to master with in their own context of their unique cultural and experiential reality. It is only on this basis that new math concepts can be constructed for the constructure or destructure of an individual’s link to the larger mathematical universe. It is with a sight towards the understanding and resolution of problems that we do mathematics. Beginning the journey from the child or learner’s experience is as logical as it is pedagogically sound. For these reasons, it is the intellectual activity of the child that must come as close as it can to that done by the very people who use mathematics in the real world. Like all empowered users and learners of mathematics, learners must be given experiences to enable them to learn how to:

1. Break a problem situation into manageable parts;
2. Create a hypothesis;
3. Test the hypothesis;
4. Correct the hypothesis;
5. Make transference and generalizations to their own reality.

In mathematics, as an outcome of the banking model, students often face prior experiences as somewhat passive learners and come to math class with a level of anxiety and fear. It is a well-documented fact that the majority of the population avoids mathematics at all costs. The need to feel empowered, the need to participate in making mathematical decisions and the need to direct where one’s own learning should go is a basic fundamental pedagogical need. This is why computer gaming enamors so many people, and why we can see it as worldwide phenomenon. Whether teachers like it or not, the reality is that many young people feel comfortable using this technology. Through an intense interest in computer games and toys their anxiety level is greatly diminished and the sense of accomplishment is strengthened. The use of technology allows students to be creative and utilize alternative forms of intelligences. It allows them to build on their self-esteem. Thus, mathematics activities should vary in the use of a variety of tools and technologies (CDE, 1992, NCTM, 1989) to enable opportunities for:

Appropriate project work;
Group and individual assignments;
Discussion between teacher and students and among students;
Practice on mathematical methods;
Exposition by the teacher.

Because of this it is not acceptable that intellectual activity of a child is based exclusively on memorization and testing, or for that matter with the application of prepackaged knowledge which serves to increase math avoidance in any given population (Schmidt, 1999). The ethnomathematical conception of the teacher in a given teaching-learning environment is that of mediator of the child’s learning. Teaching then becomes a much more complex task than that of a simple "giver", banker or dispenser of knowledge.

The developing ethnomathematical conception of the teacher and learner is that of an active collaboration in the learning process. The task then is much more stimulating than that of mere consumers of knowledge. Of course, children cannot possibly construct every bit of knowledge, or know were they must go without some form of guidance or coaching. In ethnomathematics it is the purpose of the teacher to introduce them to new tools and techniques, to help them practice becoming proficient in using them, to ultimately guide them towards more sophisticated uses and applications. That is, to coach, advise, collaborate and mentor them, in the discovery of solutions to questions and explorations.

The work of the educator then partially becomes that of a learning environment organizer. The teacher creates new or evolving situations allowing learners to form new mathematical concepts. A learning environment that truly involves students in the mathematics empowers people to truly understand and interpret their own ideas and findings. The growing experience and knowledge student possess takes-on new forms and evolves into new products, which often stands as a creation incorporating the historic nature of mathematical concepts, by assisting students to reflect on the processes that they have learned with expectations require them to demonstrate their thoughts and ideas to others. As they gain more confidence in developing new ideas for themselves, they can show the presence of mathematics in their daily lives.

One creates further explanations and ways to work within a mathematically based reality, what Paulo Freire called a transforming action. This action looks to reduce its degree of complexity, through the choice of a system that it represents. In this isolated system, representations of this reality are often derived enabling the elaboration of ongoing strategies that explore, explain, and increase comprehension. The study of this system (action) allows us to reflect on the possibilities inherent within it and for it to become the object of critical analysis by the learners themselves. This process in which we consider, analyze and make ongoing reflections on a system is called modeling. The applications are inserted as part of the modeling and are always made through the system’s reality in which they instituted the model. This of course affects all future analyses and reflections or, "the ongoing system is an integral part of the considered reality; that is, it is a set of items including the inter-relatedness between these essential components (D'Ambrosio, 1990)."

In acquiring knowledge using modeling, we work with approaches by which we construct new representations. It is necessary that the established relations constructed between the system and its representations make the research valid. This can be done through analyses and reflections on various models which often end-up guiding this practical pedagogy.

Through the significant and motivating pedagogical activities, the child’s learning becomes a mathematical reality. The measure by which this is incorporated into the knowledge and concepts that are essential for their future performance in society are true marks of success. However, this incorporation is not given through simple or blind adhesion to the teacher or materials. The program has as a main objective the development of quality reflections and participation by the learner, so that they may be better able to trace parallels between their experiences and the out of school reality. Thus modeling fits into an evolving ethnomathematically-based program, and because it includes thinking in a critical and historical nature, uses representations that must underlie the modeling process (D'Ambrosio, 1993).

A child’s involvement in ethnomathematics requires an inner transformation that manifests itself as action when questions are both formulated and answered. This engaged hunger for an answer or next step is precisely what we observe when people are engaged in such activities as basketball, soccer, or computer gaming. Consequently teachers learn to instigate an inquiry process by demonstrating possible connections for the learner to the required mathematical content. This relationship is not subject, content or assessment driven, per se. Children are deeply involved in the questions being posed, and actively contribute to the direction of their learning when they perceive what is making true meaning for them. Then students are better able to gain true mathematical problem-solving experience and thereby develop autonomous attitudes and mathematical abilities. They are enabled and can recognize, indeed learn to pose new questions for themselves. Again, this is exactly what goes on when adults observe computer or soccer players engaged-in intense problem-solving situations.

Educators must develop the skills to work as true facilitators that create conditions for increased involvement to make mathematics accessible for all students (NCTM, 1999). Ethnomathematics suggests that teachers must create a climate where by errors that naturally occur are part-of the teaching-learning process. In this way, modeling becomes significant and good models can be elaborated and demonstrated. Students learn as they apply models and see pedagogically based ethnomathematics as something truly practical. This form of modeling produces, indeed develops, further models equally interesting and useful to students. This process enables learners to learn how to engage a process of continual modification of their decisions, to alter their reflections, to develop mature discussions, to create ongoing analysis and to develop conclusions for use with their models.

Educators facilitate real-life conditions, making them accessible and engaging. Through ongoing dialogue and discussion, the teachers reduce undue pressure on their students. This enables teachers to really observe and listen to their students' ideas and thoughts, encouraging a more authentic form of assessment and evaluation. All of this is done in a climate where by errors are part of the natural learning process that can be considered positive learning opportunities.

After these questioning activities are initiated, models are further elaborated by the students and then demonstrated. In this way, students learn to apply modeling as an educational methodology and experience ethnomathematics as a practical pedagogy. This process produces and develops useful models that enable them to further develop modifications through collaborative decision making. After they have made further reflections, discussions, analyses and conclusions about the presented models they then present their findings to their peers and colleagues.

When we consider that children actively construct meaning, especially in situations found outside of their own true reality, the waste of their enthusiasm and innate energy, does not occur. This natural desire and enthusiasism to discover new ideas, and the mastery of difficult concepts allows learners to construct and master their growing mathematical knowledge. Mathematical modeling is the study of problems or situations like a language for understanding, simplification and resolution with sights for a possible forecast or modification of the studied object (Bassanezi, 1994). Mathematical modeling as a learning strategy values the child’s own knowledge and stimulates true academic performance in pupils. Ethnomathematics as pedagogical action therefore discloses the potentiality of mathematics by using modeling as an efficient methodology.

In this type of work, mathematics then becomes a realistic tool for students allowing them to develop and use questions and techniques that are part of their own real-world environment. The main purpose of this process is to develop a capacity to analyze and interpret data, to test formulated hypotheses, to create models, and to verify their efficiency. The process of creating or redefining conditions so students can really understand a phenomenon and have opportunities to act for its transformation is realistic.

Mathematics is a dynamic discipline, the study of which has fundamental importance for all people. The way that we introduce students to mathematical modeling is to expose them to a diversity of engaging problems and models that include mathematical interpretations of problems which in turn are representations of the systems under study. Therefore, when we analyze a given situation for its mathematical perspective, the teaching-learning process itself stimulates thought, not just the mere memorization of basic facts and algorithms. Students come to deeply understand the mathematics being used and to become involved in the creation of new mathematical instruments and the formulating of new theories.

The great majority of mathematical questions are used to explain and to make forecasts on phenomena in our real world. What is interesting is that often many of these explanations can be unique from one culture to another. What is interesting is that many of these different perspectives can be used in the representation of these situations and for the formulation of mathematical models.

Mathematical models are almost always a system of, for example, algebraic and differential equations, obtained through the establishment of relations between variables considered essential in an analyzed phenomena (Bassanezi, 1994).

However, a model is not just a set of quantitative variables that make qualitative representations on the system being analyzed. Thus, models are not always accurate. When this becomes evident, we must explore all the details of the model and examine new or emerging hypotheses, check the preciseness of calculations, and then make necessary adjustments to the model. When this is done, forecasts can be developed that validate the hypotheses.

Considering these relationships we might conclude that learning to model consists of applying the techniques necessary to solve the connected mathematical problems in the systems under study. Part of the process of learning to model allows us to learn to verify the parameters for the resolution of these same problems and models. The process also looks to see if they occur as the implications in the interrelationship of the effected selections in the systems through a holistic study of the given reality. It is not possible to explain, to know, to understand, or to carry-out the reality outside of a truly holistic context. They are no more than partial or incomplete visions of reality (D'Ambrosio, 1993).

At the same time, it is important not to confuse mathematical modeling with a mere group of formulas, theories or techniques that resolve mathematical models. These can be easily memorized, learned and later forgotten as disconnected irrelevant pieces of data, preventing this unique form of conceptual challenge, logical and critical reasoning that serve as the essence of the process of modeling.
Often when a question is initially placed in a model, some variables might not seem to make sense. Educators then need to prepare students, to search for further understanding, to encourage positive behavior, and influence them in developing the system they wish to represent. Models will begin to take on a degree of familiarity and to come into focus. During this ongoing resolution process participants develop a continual relationship between the model and the modeler.

Normally, the modeler has a general idea related to the type of model that they seek to elaborate. At this stage, it is normal that students do not possess a clear or detailed notion about what they hope to find. Generally, the modeler constructively defines the characteristics of the model as they go along, so that they can have a general idea as to the objectives that they seek to attain. It is a process of investigation and discovery, more akin to anthropology, archeology or biology. It is a process that gradually uncovers mathematics like a mystery, where over time the answer is opened-up to researchers or the private investigators.

The mathematical model starts to clarify itself and then to develop itself and acquire the form that it eventually will take on when students organize and share their findings as a group of researchers. During this stage, the model might be confused with the larger system whose basic form is determined and developed over time. During the finishing process, adjustments are made to the final details of the model, so that they more closely reflect the true reality they come to represent.

The finished model contains certain mathematical errors that may not be in agreement with the system being modeled. If this is the case, the project must be rechecked, using once again, the same developmental steps of building the model, and carefully making necessary alterations to the model to achieve accuracy. This improved model then can become part of the overall analyzed system.

The function of the teacher in this program is important. Instructors serve as a behavioral model by mediating action and demonstrating various related mathematical models that must be learned by students. Students are generally moved to study in the same direction that the teacher coaches them for constructing their models. Therefore, both must work and communicate clearly together. Through the ongoing analysis of the models they develop together, the teacher becomes the mediator between students and the mathematics needed to build and use the models. The work of the teacher relates to finding ways that assist students to determine how they themselves can bring their models to a final conclusion.

By encouraging students with their ideas, the teacher familiarizes students with the larger mathematical system. The teacher collaborates with them in the elaboration of their models. Thus, the teacher assists students in the development of their growing understanding, By asking them questions, especially when it seems that the possibility of obtaining mathematical ideas seems small, the teacher coaches them in the direction of their research. A detailed appreciation of the hidden systems can then be made after their research is fully completed. This can be done while assisting students with their ideas, and in the collaboration and elaboration of their own models.

To formulate a representation for a system, it is necessary to develop and maintain a harmonious and open relationship between the teacher and pupil. A model, to be truly successful, requires patience and sensitivity in its elaboration so that the teacher can supply the responses necessary to questions and assist them in the understanding of the reality that allow them to enhance their vision of the real world they are "discovering". It is easy for the teacher, for the sake of saving time, to give direct answers, instead of allowing the students to struggle a little with the work.

However, to have a successful model, the teacher, like any coach, requires patience and sensitivity in determining how to elaborate and supply answers to the questions raised. In the same way that the mathematical model is prepared, we can substitute the mathematical language for natural language which functions as a bridge between the language of the students and mathematics. Between mathematics and one’s mother language exists a relationship of mutual inception (Machado, 1990)

In the end, modeling receives detailed treatment, through the ongoing process of revision and correction. The modelers learn to look for the best way in which to explain their models. This process includes a critical interpretation of their solutions and the validation of their own considered reality. For the use of this methodology, we then define certain stages between the teacher and individual students in the project. Each step or phase requires meetings between the teacher-as-mediator and the student research group. During these meetings, students present their objectives and goals, and choose the subjects for their respective research. The whole class meetings between teachers and students allow the students themselves to form working groups around similar research topics. During this time, they all assist the teacher who guides their work. The students are expected to elaborate and assemble a final presentation of their findings.

The activities to be developed with students during the process of modeling can be grouped in three distinct phases:

1.Initial Phase: Preparation of the Modeling

Interdisciplinary meeting between teachers involved in the project;

Meeting with the students involved in the project;

Explanation and elaboration of the dynamics of the process;

Choice of the subjects for elaboration of the modeling;

Presentation of the relation between subjects;

Justification for the group of the subject choice.

2.Intermediate Phase: Emphasizing the Development and Elaboration of Modeling

To find hidden themes;

To find-out where to go as the direction arrives in relation to the chosen subject;

Indicate the bibliographical databases for the research plan;

Collect basic information and statistical data;

Lectures or discussions by professionals representing diverse sectors for the prompting and motivation of pupils;

Visits to colleges and universities, businesses, museums, public organizations and offices, cooperatives, etc;

Interview specialists, and professionals in diverse areas of knowledge;

Interviews with representatives from the community;

Notation of principal ideas realized in reading pertinent literature that is pertinent to the chosen theme;

To know the themes under various aspects and forms;

Development and detailed organization of basic research objectives;

Formulation, sorting out and analysis of the mathematical models;

Submit text and chapters periodically for analysis by the teacher;

Check on tables and graphics;

Verify citations and bibliographical references;

Revise the mathematical models and accompanying final text;

Final elaboration of report; Teacher checks to see that students communicate their findings correctly, that they are absolutely sure, so they can defend their chosen themes in their final presentation.

3.Final Phase: Presentation of Modeling and Delivery of the Final Report

Presentation and defense of the theme;

Delivery of the report;

Presentation of the final self-evaluation of the group for the consideration of the project;

Production of the final report for the teacher specifying the development of the methodology and the objectives that were reached;

The difficulties presented when developing the objectives that had been reached.

We can develop a methodology for mathematical modeling by utilizing the following procedures:

Steps in Mathematical Modeling

1) Choice of the Subject
At this stage, participants survey possible themes or ideas for development. Such themes might include: a study of various sectors of production, alternative economic situations, politics, society, agriculture, education, arts, health, etc. The subjects must have breadth so that they can encourage questions in various directions. Once a theme is selected, students are divided into groups and organized around their chosen research interests. The choice of the subject must be the guide, it is important that students are involved in the process and that they feel motivated by the subjects and problems that they raise themselves. Having chosen the subject, they do not have the accurate notion of the type of mathematics that appears. Thus, they must count or measure, make tables, and do basic statistics to appear in a preliminary table of data so they see a beginning to the modeling process.

2) Research on the Subject
The group participants are encouraged to visit museums, industries, cooperatives, laboratories, farms, universities, libraries, periodicals and magazines, and other agencies (resources) in accordance with the necessities of the chosen research subject. Their object is to search for an understanding of the topic that they will study. The gathering of new information must be carried through by using bibliographical references in books, magazines, the Internet, and by making interviews with people whose life experiences vary across diverse cultures and points of view. This stage of the research has as an objective of collecting the quantitative as well as the qualitative data that will assist in the formulation of the hypothesis. This knowledge must then be analyzed and interpreted by the students as the foundation for their mathematical models. Needless to say, the construction of a database is pivotal for the modeling process.

3) Question Elaboration
The questions that should be considered by students stem from the actual questions taken from their own research. It is related to the experiences and content that they know and/or are comfortable with. The first questions are often crude and simple in nature. They can be solved through the introduction of basic mathematics and need not emphasize the elaboration of questions, i.e. what is considered, or should be rejected. Further considerations related to the relationship of the question with the problem being studied are also considered at this time. To begin with these initial questions often starts a magnification process of the ideas that will draw the students into the search for generalizations and analogies with further correlated situations. The gathering of questions for the system will be used to analyze and reflect upon in later stages.

4) Elaboration of the Mathematical Model
This next stage is important in preparing students in forming the answers found between those found in the real world and those of a seemingly abstract conceptual nature. Students at this stage need a great deal of support by the teacher. Teachers proceed first by assisting students in their interpretation of the data developed in their field research. Researchers organize and analyze the data collection process. Questionnaires are organized to use as specific methods of sampling, so that an analysis of the relations between variables can be considered. The study of these relations are essential for the agreement of the studied event. This activity leads toward formulating the hypothesis, establishing this form for the mathematical models that usually are elaborated through the formulation of mathematical content. A major goal for the model at this point is to learn new mathematical content, which is necessary for students to further study and test their hypothesis. This gives them the tools that they need for ongoing analysis of their developing model. They also make use of and gain the tools necessary for writing, and deciding which features are useful in discarding unimportant data and unnecessary elements they may encounter. This procedure represents one conceptual aspect important to the process of modeling that seeks to develop the formation of a mental picture of the situation being shaped, with the objective of experiencing it mentally, and internalizing the necessary learning concepts.

5) Formulating the Mathematical Problems
Formulation of the mathematical problem appears as a direct consequence of analyzing a series of examples by teacher and student groups. The teacher does not have to clarify the questions related to the research subject to be decided at this time. However, the teacher must mediate and clarify the process for learners so that the groups easily organize doubts and concerns. The teacher makes suggestions for further study of the subject. The teacher must be dynamic, and if questions do not appear as part of the group work, the teacher encourages the researchers to continue on with their search. The teacher must point the group towards a path that induces the students to search the proper problems. The transference of verbal relations (natural language) to mathematical symbology is a task that demands a great deal of effort on the part of students and teachers. The teacher gives careful attention to the symbology that students know, as well as their actual relation to standardized symbols, and for the parameters or the supplied data that directs the students in the formulation of the mathematical problems in their developing model. The formulation of a problem in mathematical terms represents a critical phase in the training period for more difficult aspects of modeling. It must be met with care by the teacher and be used to encourage increased curiosity and creativity by the students.

6) Resolution of the Mathematical Problems
This phase leads the group towards the finalization of their decision. At this stage, the problem does not need to be solved accurately because a variety of assumptions or approaches are necessary in the resolution of the problem. Teachers must be careful in not over anticipating the mathematical difficulties that students can have at this point. As well, they should not over emphasize the resolution of mathematical models around one particular technique or of a theory. At this phase, the mathematical concepts that had been identified in the solution of the mathematical models can be systemized.

7) Interpretation of the Solution
Discussions must encourage individual group members to reach an understanding through the further refinement, interpretation and solution of mathematical models. While the teacher monitors each group, the groups must work independently on their designs. When the teacher observes further evidence of common problems and interests by all groups, they consider using this information to develop a collective lesson covering the necessary content. The interpretation of the mathematical solution often involves the review of mathematical concepts and is carried through in an analytical and graphical way.

8) Comparing the Model with Reality
In this phase, the researchers compare their findings of the mathematical model with the analyzed system. Now, the validation of the models must be as coherent as possible. If by chance the model is not good, the system must be revised through the elaboration of more significant systems or, if necessary, new research must be undertaken. If the model is found to be satisfactory, researchers can use it to make forecasts, analyses or take any other form of action. A model is considered good if its capacity to forecast is accessible to the verification of the system under study.

9) Report and Defense of the Subject
At the end of each stage, groups display the results of the research for the classroom, so they can further collaborate and develop suggestions for the continuation or modification of the works. At the final or completion end of the process, the work will be shared with the larger group using a thesis defense style presentation which includes a detailed report. The models created for questioning are shared and the hypotheses and the conclusions they devised are shown to the other groups or class.

10) Evaluation
In the presentation and defense of the subject, the group participants must act as a kind of examination board. Each group is evaluated by its performance and each individual student is evaluated by the elements of its group. This is done in a manner that transcends self-evaluation. The teacher, as well, evaluates the presentations and reports presented for the groups.

The Methodology

As a methodology for teaching, mathematical modeling contains other important objectives as well. The principal objective is the development and interest in the doing of the research. This means the further development of the abilities related to searching, sorting, creating and reporting new forms of data and information. The communication and reflection on these findings by looking at the discovery of mathematical models that are found to translate into real situations, with the proposition of a possible decision in relation to the analyzed system.

Specific objectives utilize a specific method for the realization of the written work and include:

Organizational development using the scientific method;

The inclusion of content knowledge in planning;

Bringing the student to look for adequate strategies for presentation;

Orientation of the students towards what is correctly communicated,

Collaborative group work;

Integrating participating students in the project;

The development of proposed themes, including parting points and the incentives used to develop a capacity for self-realization in a globalized society;

To prepare students for a process of life-long learning;

To offer them the capacity for learning, which includes an ample understanding in more diverse spheres of life - personal, social and professional;

Development of a critical reflection with actual themes that afflict society;

Proportional solutions for problems brought in relation to the studied themes;

To develop the ability to be educated with incentives to read, a critical capacity, and specific abilities of behavior during insecure or difficult situations which are constantly in our daily lives.

Resolution of Mathematical Models: Further Expectations

At the end of the project, again it is important that student groups apply the scientific method for the elaboration of a report. The strategies and methodologies used in the final presentation should be based on reality and facilitate the use of communication and dialogue through the use of a variety of technology or media. The report includes the defense of the arguments which respect divergent opinions, an acquired capacity for discussion, reflection, analysis, and detection of problems and appropriate solutions. It must be sure to be relevant to the developed original themes.

It is hoped that during this process, students and educators alike acquire an understanding and development of a critical sense of citizenship based on equity. This process is designed to give learners experience in becoming tolerant, accepting, yet critical citizens and professionals. This is important so they learn to contribute in a way that accelerates the process of social transformation as outlined by Freire.

With the conclusion of the project, students have had the opportunity to develop the capacities and abilities such as - autonomy, self-confidence, argumentation, criticalness, and flexibility towards often-imposed changes in life in a globalized society. As well, theses experiences assist learners in acquiring interactions with others as represented by diverse members of the group. This kind of scientific exploration allows learners the opportunity to practice working collectively in projects set towards a single goal, as found in the real world working and learning context.

Critical Consciousness

Considering the empowering and ethnomathematical focus of this work, the hope is that after the novice learns modeling, they will enter the first developmental level of critical consciousness. A lack of critical consciousness can create a distinct disassociation for the learner between what they are learning and what they see in the real world. Passive learners are used to problems that are not part of their personal life, reality or society. In other words, the activities are in danger of having minimal, if any, real use outside of the classroom context. In contrast, what modeling seeks to accomplish is to give the learner experience in seeing value and usefulness to what they learned. It seeks to develop a critical position in support of problems and the proposed ways to resolve them. It is necessary that we assure that all students have access to a knowledge-base and tools, so they can become truly empowered, to participate in the cultural, social, and technological transformations going on around them.

As well, this design contributes towards an equally critical exercise of citizenship. Through educational experiences that sharpen critical abilities of the child, the acquisition of a new sense of empowered knowledge acquisition allows one to develop more consistent strategies for full participation in a globalized society. It is one that faces social, economic and technological challenges of unimaginable proportions. With the utilization of diverse pedagogical features that the modern world offers and through frequent interdisciplinary activities, modelers improve their ability to undertake organized research and analyze problems.

In view of this design we have developed a method by which learners challenge, stimulate, research, communicate and develop an extending world vision, different forms of thinking, and different abilities, leading learners towards reflection and action. The design we have presented here, that requires educators to assist learners to develop objectives, do research, organize collected data, present their written work, all combine towards giving learners real-life contact with mathematics and learning. The process promotes in the student the cognitive capacity that develops their growing ability to organize and elaborate information and knowledge.

The design also promotes the interaction between diverse disciplines and education themes, integrating educators, while also making the addition of mathematical content. Therefore, it occurs that during all the above processes, an example of cooperation between students and teachers as a model of integration of school-community experiences. The application of ethnomathematics as pedagogical action allows learners to make new discoveries. The program respects, indeed values the learner and looks for ways that develop the talent of educators and students alike.

updated 9 october 2002