Chapter 3

In this chapter we wish to discuss ideas that allow the educator and student to implement the history of mathematics by integrating subjects not necessarily found in the traditional curriculum.  Such integration suggests that we use a variety of materials and learning theories.

Processes related to the necessities of living in society, groups and cultures are mathematical in nature. In relation to unique cultural process, we might think about the kinds of questions we refer to as "motivating problems", problems common to mathematics and many socio-cultural applications. Every society generates unique problems related to its individual surroundings. All people are engaged in a search to satisfy both their material and intellectual needs. If we want to understand the direction in which mathematical knowledge is both used and produced by a society, then these two aspects - the material and the intellectual - cannot be separated.

Our modern technological information-based society has generated certain basic essentials. To take care of these necessities, diverse new sciences have come into existence and are used to find the answers to numerous challenges stemming from the implementation, use and integration of technology. Common cultural icons such as airplanes, freeways, and skyscrapers lend themselves easily to mathematical modeling by students living in this context. Many students find examples of modern architecture engaging. Factors such as size, scale and the use of diverse materials can most certainly lend themselves to mathematical explorations. In the construction of large buildings, it is the mathematics that supplies the theoretical instruments and intellectual power used to determine the necessary structure and design . Many architects frequently refer to their work as "solving problems" of space and aesthetics. This can be one example of the application of mathematics.

We should not forget that skyscrapers are a necessity that our modern society suddenly built for no reason what so ever. They are representations of structures that came about because thousands, then millions of people came to live together in increasingly smaller places as are found in our modern megalopolises such as New York, Chicago, São Paulo, Tokyo, and Kuala Lumpur. The modern notion that human beings can live in extremely large groups, was totally strange, indeed impossible to imagine for peoples in divers cultures two or three hundred years ago.

There are now about 6 billion people living on Earth. Strangely, there still seems to be sufficient space for us to live in small groups. However, many people find they cannot live in small towns because basic options like the freedom to be who you are, recreation and jobs are not found as they might be as they are in many of the big cities of the world. Questions related to "lack of space" are really not that at all, but are related to cultural values or human rights as shared by many citizens.

All cultures have developed mathematical tools to solve problems important to them. We continue to argue here, as do many in the field of ethnomathematics, that teachers and students need a set of ethnomathematical values and processes that enable them to make sense of information and will allow them to engage in the dynamic changing world why live in.  It has become increasingly difficult to convince many students of the utility of mathematics for resolving certain types of problems.

If the teacher does not consider alternative points of view important, or if they are not concerned primarily with their teaching; or if the students themselves do not think mathematics is important, neither group is going to be successful.  Possibly, the teacher teaches only because of, or what is required by the curriculum.  Possibly, the student "learns" only by showing up in class, and doing the minimal requirements, taking tests, and then forgetting about it altogether. In this way, the basic assumptions and understandings are affirmed, but are condensed into predetermined expectations, disconnected from the outside reality or needs of society.  It is from within this context that mathematics assumes an important dimension. For the chosen methods become a process of interaction between cultures; that is an interaction between unique and diverse ways of thinking and of organizing the world.

For a student to learn to operate in an empowered fashion from within their reality, new forms of learning and valuing of mathematics need be found to enable students and teachers to become immersed in their culture, where these factors are valued as well. For this to occur, it is necessary that schools come to respect and understand the context of the world that students will come to own.

Ethnomathematics can take us further; it asks us to find out,  for instance why skyscrapers have become a necessity in cities worldwide.  This notion, of living in enormous interconnected, indeed interdependent groups, presents a difference for members of diverse cultures. How does this fit into the classroom?

Thus it is that students can come to understand how mathematics exists in all cultural contexts and how it can become an important way by which we are able to act on our own individual reality. This action is accomplished with the intention to both transform and to preserve the traditions. For example, the mathematics used by the carpenter, the doctor, the mason, or the engineer can offer insights for the classroom. As well as the mathematics used by a child who lives on the street, or plays sports or video games are contexts valuable in making connections.

In our globalized society, many diverse cultures are integrated and are becoming part of an interactive world. Others have become marginalized, often the mathematical traditions are lost to history without having been saved or understood. In this cultural dynamic, new forms of knowledge are produced as well as learned and need to be saved. Mathematical knowledge is part of this dynamic process of action on this intellectual and material reality.

Machado (1994) approached the relationship between mathematics and maternal language. He has pointed-out with respect to the fact that human beings are born immersed in cultures all having similar aspects of interaction (language-alphabet and number). Therefore, the constituent elements of two main systems of representation of reality: the alphabet and numbers are learned at the same time by all people. This occurs before arriving at school, and more often than not, without rigid distinctions to the borders between systems or forms of communication.

When Kamii (1985) talks of the importance of a child learning to help arrange the supper table, she is not only commenting on the chance that the child has "to develop a sense of reason". The child also is analyzing the fact that they are learning the process to make a "one for one" correspondence with place settings and family members. This is one example of an early activity that has a place in the daily culture of every child. As well, it is an activity that gives legitimacy to the child’s early effort to think and to reflect on the action that they are carrying through.

D'Ambrosio (1988) spoke of an interaction cycle passing for thought and reality. He expressed ideas related to the dialectic between the child’s process of thinking and action. His cycle suggests a process that repeats and transforms the inner life of the child and has the power to change the culture around them. We see this as young children learn to play games and become literate in a family that may have limited access to media, technology or games. A culture of inquiry requires some form of mathematical knowledge. This mathematical knowledge transforms in an intimate manner, the way in which the learner percieves their envrionment. At the same time, it may allow them increased access to the greater context in which they are embedded.

Here is an example related to the use of calculators. Hand-held calculators not only calculate, but also, if used correctly, allow the user to expand on their abilities.  Most people soon realize that basic addition and subtraction facts are faster if memorized. However, some algorithms that have been previously taught in great detail in many primary schools can now be relegated to the calculator. This is especially true for such activities as double or triple digit long-division. Graphing calculators allow students to quickly see the dynamic possibilities of algebra as they compare and change various aspects of equations and quickly graph them. This is possible because the technology allows them to really see mathematical ideas quickly and easily.

At the same time, technology changes the way that we now see mathematics. Therefore it becomes less important if a child knows how to make a certain calculation error-free, and is related more to how the concept itself is understood and produced by this activity. An ethnomathematics-based math course most certainly expects children to know the math. It goes further, however, and it holds up higher expectations that expect them to become critical analyzers of their answers, their work, and their growing ideas. This is something a curriculum focussing on "back-to-basics" or standardized high stakes testing, fails to accomplish for most learners.


Mathematics History and The Implications for Teaching and Learning Mathematics

An unbiased look at the history of mathematics is more than merely helpful in giving learners the opportunity to see the connection and reality of human legacy of mathematics.  When we study the history of mathematics that people have used in various contexts, we see the mathematical accomplishments left by all peoples and cultures.  We come to see that mathematics is something that all people have done (indeed can do) successfully.  This is done with enthusiasm no matter what the cultural context, or technological tools available.  The value in learning about past or alternative scientific achievements keeps us from having to "reinvent the wheel".  Learning to appreciate unique accomplishments not just as curiosities, but as valued attempts at problem solving, transforms these accomplishments from cultural artifacts made by lessor or "underdeveloped" cultures into useful examples of thinking, learning, communicating and doing of mathematics.  When we are able to assist students to see and value culturally based patterns of thought, because of the historical context in which these cultures were living and contributing to our overall human heritage, we can give much more than one tool for problem solving.  

For example, the ancient Greeks produced many great thinkers. However, Aristotle and Plato did not think about negative numbers or fractions.  Indeed, evidence suggests that for them negative numbers did not exist at all.  The ancient Mayans and other original inhabitants of the Americas developed a sophisticated culture including massive architectural wonders.  All of which was undertaken without the use of the wheel.

These situations seem strange to us now, but we can only really understand the original ideas and accomplishments that led to the way we think today if we try and understand the way the ancients once looked at numbers and the ratios.  The how and why they made and used their mathematics from within their own particular paradigm represents valuable lessons for all of humanity.  Then we can see that our own thinking is only limited by the paradigm that we are operating from within as well. Understanding the mathematical culture of our students that allows us the insight into how other people think. This allows us to construct positive experiences that enable us to connect to the overall human mathematical legacy.

For example, the theorem of similar triangles eventually solved the ancient problem of measuring the distance from the Earth to the moon - a problem faced by ancient astronomers in many places in the world.  This was proved long before rockets or radio were invented.

The mathematics that each culture has developed has been used to act on each unique reality or context. Thus, mathematics is dialectic in nature, and really serves as a cultural tool as well as a cultural bi-product. Integrating the history of mathematics is satisfying to both teachers and students. Often there is more agreement between teachers and students through the use of sociological and historical aspects of mathematics. Besides encouraging the study of mathematics as a cultural product with a dynamic body of knowledge, students and teachers are reminded that mathematics is not a static collection of rules and disconnected methods.


Integration of Subjects

The history of mathematics, through historical examples, can supply elements allowing us to integrate various academic subjects through the mathematics being studied. It may be exciting to compare the various algebraic resolutions for solving geometric problems made by the Babylonians, for example, the examination of solutions using geometric methods used to decide appropriate approaches for algebraic equations, as invented by Arab mathematicians in the Middle Ages.  May also be of interest. Using analytical geometry, in which two mathematical contents: algebra and geometry are used together, also allows for the integration of two mathematical modes of thought: algebraic and geometric. This integration of subjects as modes of thought has at least two essential aspects:

a.    Technical Character: for the integration of subjects we offer students a greater chance to understand the mathematical concepts. Example: When working with trigonometry, it is possible to approach geometry through the use of functions. Two mathematical goals can be accomplished simultaneously in this way because students have an integrated understanding of the subject. When learners encounter academic difficulties, they use their experience in the other related subject to recall and to obtain a better understanding.

b.    General Character: mathematics is a collection of rules and methods that were constructed through cultural dynamics and the study of patterns.  Some concepts have been simultaneously molded, reformulated and constructed.  It is not possible now for us to really understand how Greek geometry was in opposition to Greek arithmetic. The concept of function sometimes was understood algebraically and other times in terms of commanded pairs; that is, these two aspects were explained through different points of view.

We would like to present a way to promote the integration of subjects where students are first divided into small groups and work with a given problem. Instead of leaving groups to work alone, the teacher moves from group to group and coaches students towards finding solutions to the given problem. Students are expected to apply the theory that was given in the class lesson. Students are expected to not abuse the trust and freedom given to them to decide solutions to the problem in the manner they wish. Final work is judged rigorously by both peers and teacher using a rubric (see appendix for samples).

In this way, different solutions appear. For example, one group develops a solution based on a geometric method, another group begins using an algebraic-based method, while yet another group is encouraged to follow their intuition by using graphical and numerical methods. Ownership of all material and work remains with the students who are allowed to develop increased enthusiasm and to grow confident in their mathematical problem-solving abilities. Working with students in this way encourages the sharing of diverse solutions between problem-solving strategies that each group employed. Students freely, respectfully and safely discuss the limitations and virtues of each method with each other. They learn to work with a surety that it is possible for more than one method to supply alternative aspects of the concept by which they work.
To work in this way presents certain advantages:

a.    Students using real situations in their personal experiences and day to day world can formulate the problem themselves. Or the teacher may want to take advantage of an event, and bring a question found on television, radio, Internet or other news sources.

b.    Instead of teaching separately and later trying to integrate the content, the integration directly happens within the learning process at the same time as the solutions are developed and debated.

c.    There often develops a very enthusiastic and lively dynamic in classrooms. Students debate the fine points of the subject at hand. With the integration of more than one content area, students are able to practice extending a growing vision of mathematics. They begin to perceive mathematics as a dynamic and useful body of knowledge. They are led towards learning how mathematics is a didactic and useful feature, which offers them a chance to understand, to produce and to apply the knowledge they need to solve real problems in their lives.

d.    In this way, if the history of mathematics is understood as part of the process of learning mathematics, students have access to many ideas and come to regard them as a natural part of their lives.

The Use of Concrete or Manipulative Materials

The use of concrete manipulatives in the classroom can be explained in a couple of ways. If we are working, for example, with small children, the constructivist Piagetian-based theory of intellectual development says that it is best to work using "concrete" or hands-on materials. Therefore, according to this pedagogical point of view, the child until about 13 or 14 years is not truly capable of thinking abstractly on things that are not "mentally concrete". Using object representations by which the child has had direct contact enables them to think and learn. For this reason, it is import that we offer all learners a variety of concrete or manipulative materials as they develop new ideas. This is actually true as well for all learners no matter what age.

As we work with adolescents, again using ideas based-on Piagetian- constructivist theory it may at first seem that " concrete" objects are not quite as important. However, we know that in practical and professional terms, this is not true. Scientists and mathematicians often use writing, models and drawings to express and save their ideas. They prefer to let "the paper do the thinking" for them, or to "use objects to think with" (Papert, 1993). Many students, who are having trouble with mathematics, operate under the false assumption that smart-kids do it all in their head.  It is of primary importance that all learners have access to tools and manipulatives to assist them in both learning and problem solving.

The open and free uses of these types of materials are of great importance in demonstrating, teaching, learning, and problem-solving environment of an ethnomathematics-based classroom. Our students must be able to freely make use of these materials, at any time. Mathematicians, scientists and researchers also make use of cardboard objects, buttons, trays and even work with computerized drawings and software tools to obtain a better understanding of the object under study.

According to Bruner (1974), when trying to understand an object or idea, children often attempt to see how an object functions in its environment; how it tastes, smells, looks, moves, etc.  It is what some kids might describe as finding out "how it works" or using free play (Fuys & Tischler, 1979, p. 12).  When we use concrete features and manipulative materials, we let students carry through or experiment in their own concrete way.

We often create clear pictures for a variety of ideas and objects we work with through interacting with concrete materials and experiences. With concrete objects and ideas, we can start to elaborate representations written on the object. We can start to think mainly through these representations. This would be called the "iconic phase" of the cognitive process.

Concrete  --> Iconic  --> Symbolic

The final stage begins when learners have gained the ability to think using abstract representations, and the accompanying rules. This phase is called symbolic. Again it is important to note that this interpretation does not apply only to children. We all naturally proceed through these levels when learning new ideas, talents, or even when moving to a new city or country.

Think about the traveler who visits a new city. While reading and studying before hand he encounters the city, may encounter many stories, read maps, and make plans for getting to know the new location before hand. Upon arriving, the map actually looks different; where once it was merely two dimensional, it now takes on a quality all its own, indeed the street names and locations now have meaning, and he can clearly see what they mean having experienced it first hand. Learning mathematics is much like this. Sadly, many people never get past the first impressions, indeed never are allowed to experience or navigate what Seymour Papert once called "mathland". The same of course can be said of a visit to "mathland".  Using concrete examples, manipulatives, and a variety of materials which allow learners to increase their understanding of mathematics.

The conditions that allow this type of work are of major social importance. The school setting, conditions and values all act together to influence the form of knowledge that a culture comes to value. If we work to solve a problem, we stop in the phase that in allows us to resolve the problem, no matter at what phase we are in: concrete, iconic or symbolic.


Three Major Theories of Education

There are three major philosophical theories of education are used as a basis for delivering instruction and content: Behaviorism, Cognitivism, and Constructivism. Each philosophy has contributed to mathematics education in its own right. Each is used in the delivery of content and influences instruction even through the philosophies may in fact may be in direct opposition to each other (Barrick, 1980). The present "math wars" in mathematics education stem from the conflict of these fundamental educational philosophies.

Behaviorism
B.F. Skinner based his philosophy and psychology on the idea that the learner is like a programmable machine. One does not really have to care what happens inside the machine. One merely feeds the machine the necessary information, and can expect a logical output. The focus is simply on shaping the learner’s response to stimuli.  The teacher’s job is to "condition" the learner to respond in a certain way to certain stimuli. The learner interacts with the learning environment, but does so passively.  

Under behaviorist psychology, the instructor has tight control on the instructional environment and information that the student must interact and master. The instructor assesses student progress based on their behavioral outcomes. A Skinnerian-based pedagogy is usually used without any complex integration of muscle memory and cognitive processing in interaction with the environment (Barrick, 1980).

In education, the behavior to be shaped and maintained is usually verbal, and is to be brought under control of both verbal and non-verbal stimuli.  If our current knowledge of the acquisition and maintenance of verbal behavior is to be applied to education, some sort of teaching machine is needed. The machine itself, of course, does not teach. It simply brings the student into contact with the person who composed the material it presents. It is a laborsaving device because it can bring one programmer into contact with an indefinite number of students (Skinner, B.F. in Barrick, 1999).

Three ways that behavioral shaping has been applied to mathematics learning environments has been through the use of timed tests, rewards and punishments, and various textbook learning programs.

Cognitivism
In cognitivism, learning occurs when a learner processes information. Learning is viewed as input, processing, storage, and retrieval of information. At the center of a cognitivist pedagogy is the structuring of experiences that involve the learner as an active participant in the process of learning. Effective learning takes place when instructional activities rewrite a deep level of thinking, and manipulating content. Major theorists are Jerome S. Bruner and Lev. S. Vygotsky.

The first object of any act of learning, over and beyond the pleasure it may give, is that is should serve us in the future. Learning should not take us somewhere, it should allow us later to go further more easily (Bruner, S. in Barrick, 2000).

Cognitivist pedagogy provides for a range of activities and content for any group of learners to customize their own learning within a framework of larger goals and objectives with in a course of study. Emphasis is on delivering the content in various forms: lecture, lab, homework, study groups, etc.

Constructivism
Constructivism is very similar to cognitivism in that the learner plays an integral part in the learning environment itself. The major difference between the two theories is that in constructivist pedagogy, it is the learner who is viewed as the creator and processor of the educational experience. Learning is learner driven, with the student to student dialogue as the key component. The instructor is a facilitator or coach of the learner-defend and run environment, where the teaching-learning process is a transaction of  the mutual responsibility of learner and teachers (Knowles, in Barrick, 2000). The primary theorists of constructivism are John Dewey, Malcolm Knowles, Jean Piaget, and Maria Montessori. The aim of a constructivist education is the development of reflective, creative, responsible thought (Dewey, in Barrick, 2000).


Using Concrete Actions

How do these three theories related to ethnomathematics and modeling?  Numerous factors related to thinking have natural biological connections. Human development, including cognitive functioning, is directly related to our biological or physical maturity. The possibility of passing one determined phase children are not fully capable of functioning in an iconic (abstract) phase when working with a situation that is new or out of context. In other words, context is everything. Portions of each theory seem appropriate for numerous socio-cultural developmental reasons.

Thus, it is not an easy task to decide which theory to follow. Indeed it is unwise to use only one type of theory all the time with all learners. Therefore, it is only by the study and reflection on our experiences in classrooms that these decisions can be made to the benefit of our students. Learning theory offers important points in the use of concrete or manipulative materials, that do not pass the psycho-cognitive curriculum test for the teaching and learning of mathematics.

When we speak of the "concrete", we also seek to relate real situations and problems to mathematical activity. Problems whose solutions allow us either to use or learn an action as related to our own reality become the true test of engagement for learners. The importance of the "concrete" is much more a sociological aspect than it is a psychological one.

The necessity of understanding mathematics as a cultural product has been argued here as the essence of ethnomathematics as pedagogical action. It is from this perspective, that learners can try to understand and use knowledge gained from interacting with real and concrete problems. Some mathematical educators say that the concrete only applies in relation to manipulative materials. However, this does not supply an accurate dimension on the amplitude or importance of the concrete examples. The idea of manipulatives corresponds to concrete ideas. When we mention pure and applied mathematics, we are engaged in a form of reality, making reference to the mathematics applied to concrete problems, and real uses of mathematics.

Many believe that concrete connections are in opposition to learning abstract mathematical concepts. We might better understand the use of the "concrete" from the developmental perspective by referring to the contextualization of the mathematics used. When we work with concrete manipulative materials, we work in a given or created context. A context then exists that aids further in the understanding of the process by manipulating or using concrete materials that will most certainly assist in the learning process. Whereas in the past, much of the emphasis has been on the cognitive-developmental aspects of mathematics learning, we are interested here in the sociological implications as well.

By using concrete socio-cultural connects, we can further the mathematical abilities of our learners. It is a sociological perspective that we seek to emphasize here.  Much of the recent research in mathematics education is related to the cognitive processes, and its related importance, with little emphasis on the sociological effects of the interaction between the learner and concrete objects or ideas used in thinking.

It is from this point of view that a problem such as the construction of a bridge is concrete in nature. An exploration of different realities necessary for constructing a bridge in San Francisco, São Paulo, or Sydney may offer many opportunities for learners to explore the mathematics as needed to solve important geographic, social and cultural problems necessary to build a structure in these diverse cities. The social as well as the physical forces that act on a community to enable it to build a bridge influence the mathematics and materials used. A similar type of bridge built in different places possesses characteristics and materials that enable mathematical study. The data are not mere numbers written in a diagram, but are of real importance especially to the unique socio-cultural setting of the structure. This most certainly includes the opportunity to mathematically examine the significance of materials used in the construction of the bridge and other factors related to the climatic and seismic realities that offer ample opportunities to develop and learn mathematics. The construction of a bridge can be seen as an action that modifies a reality. To use mathematics to explore a problem like a given bridge is only one among many possibilities.

 
The Use of Games

The world of all children is a reality of games. Since the first years of life, children are engaged in playful activities. Play is children’s work, and is part of the basic human endowment. Games also form part of the adult, as well as the infantile world. Play is one of the basic motivating elements used to awaken interest in children, adolescents and adults for the teaching and learning of mathematics.

It is no secret that teachers the world-over collect and share a variety of games used to explore and customize daily learning activities. So that this happens in alignment with established curricular goals, it is of the utmost importance that teachers guide the ideas, concepts and processes inherent in any given mathematically powerful game.

Many teachers have learned that students help to construct mathematical knowledge within the context of a game. Much more than this, teachers have knowledge of the games and activities needed to make use in reaching curricular intentions and objectives. It is through the use of mathematics-based games that educational objectives and standards can be enhanced and understood through practicing different points of view. Play is important in learning mathematics:

1.    Play allows children to learn about mathematical characteristics in a setting that is both engaging and stress free;

2.    Play provides a situation (context) in which children can indirectly learn and use the vocabulary of size, shape, and color with other children or while talking about their activity w/classmates or the teacher;

3.    Play creates a situation whereby children can practice the learning of basic facts and other important forms of knowledge in a fast, yet meaningful way;

4.    While engaged in play activity, children may experience and investigate aspects of mathematics other than those directly involved in the planned activity, thus enriching their mathematical background;

5.    Play can be fun, and allows many children to experience mathematics in a setting free from anxiety (Fuys and Tischler, 1979).

Games often develop the desire and the interest of the player to learn the proper skill or action to keep their engagement with mathematics through involvement in competition or challenge. This often motivates the player to know their limits and to discover possibilities for overcoming these limits, acquiring confidence and courage to risk.

Any observation of a group of adolescents playing computer games demonstrates that the competitive element is often channeled towards themselves. The drive to "beat the game" develops advanced levels of problem solving and the creation of a situation that encourages perseverance. Youngsters often play in a group, connecting via internet with friends across town or country to beat the game.

Games engage students' interests by challenging them competitively, by either winning against an opponent or completing an interesting or engaging task (Zaslavsky, 1996). This aspect can be understood with the following example: the involvement of a pupil who has to answer 20 traditional multiplication exercises using positive and negative whole numbers. Compare the level of engagement and interest with the involvement of a child who, playing a similar game on the computer, has to correctly decide 50 questions of the same type in order to beat the computer. Certainly, the child using the computer to practice is somewhat more motivated to carry through the task.  The engagement is clearly seen, and the game does not need to be on the computer; children are equally engaged with board or card games of most any kind (i.e. Monopoly, Risk, chess, etc).  The primary idea is that the pupil accomplishes his work in the context of an activity that is engaging, interesting, and amusing allowing the pupil to be motivated rather than bored.

Another form of using games allows pupils to enter and engage themselves into a world where things function in ways similar to mathematical content that we teach. For example, if we want to teach students to add and to subtract using positive and negative whole numbers, we might want to use a game called "Real Estate Banker" (See appendix). In this game, students accumulate money, they learn about debt and interest.  Students soon develop certain strategies that serve as the basis for a future formal usage of mathematics within a business context.

However, this formalization cannot always happen in a simple way. Therefore, games generally are constructed and played without taking into consideration the existing link between the functioning of these two very different contexts. When we use the usual notation for positive and negative whole numbers, it is possible that this abstract context cannot be interpreted the same in the learner’s mind, as is in the Real Estate Bank game itself that creates a useful hook or context for remembering content.

It is the knowledge that the student themselves construct in their own mindswhile they interact and play that allows them to learn and actively apply new ideas. Urban street children in many countries sell candy or other things in the street – often quickly as autos wait for traffic lights to change. These children learn to develop strategies that enable them to keep accounts, and make change quickly. Often calculating rapidly in their heads. Yet, when they come to school, they possess enormous difficulties in accomplishing simple algorithmic exercises because the context has most assuredly changed for them.

Games exist in many ways for use as a strategy in teaching and learning mathematics in the classroom:

a.    Structured games (Dienes, 1974) bring the idea that students must only work and assimilate mathematical concepts that we desire they learn. In this way, the conception of structured games is next to the ideal for much of modern mathematics, which emphasizes formal aspects of mathematical knowledge;
b.    Games are important for the understanding of differences between rules and laws, two important aspects of our daily lives;
c.    Games that come from outside of the daily experience of children form a good basis for ethnomathematical notions and development of ideas that mathematics is produced by more than one culture.

The use of games in an ethnomathematics-based program most certainly contributes to the educational process of learning mathematics. Therefore, games present forms and propitious features that give understanding for many existing mathematical structures. Games offer practice in implementing difficult concepts that need to be assimilated.  Games serve as facilitators of learning, and allow learners to:

a. Develop the capacity to elaborate and develop new questions
b. To search for different solutions;
c. To rethink situations;
d. To evaluate attitudes;
e. To elaborate strategies;
f. To find and reorganize new relations;
g. To risk solutions; and
h. To validate or answer problems.

Most games develop problem solving skills by requiring students to work out strategies, to think ahead, and to evaluate various moves in a quick and efficient manner. Children exercise their skills in geometry and measurement, number sense, and in combination with computation and probability (Zaslavsky, 1996).

In the 1960’s and 1970’s in the western world, the idea of structured games greatly influenced the movement of modern mathematics and software development (computer gaming). From 1980, structured forms of curriculum games gradually gained importance. The recognition of the cultural identity and mathematical contribution made by diverse peoples has contributed towards a change of attitude in mathematical curriculum. An ethnomathematical point of view is essential in coming to understand the processes of learning mathematics.


Recreational Mathematics

To understand, to learn and to use mathematics demands much more than learning mere algorithms and their application in the resolution of a problem situation. A proposal for teaching and learning mathematics must make the construction of concepts through significant situations possible. Thus, we must find ways to further elaborate activities that engage ideas and mathematical representations from other sources, beyond overly didactic books and informative texts full of disconnected tables and graphs.

Thus, a great number of mathematical educators are not worried about mathematics being taught using a fundamental approach so students develop a sense of intuition or enjoyment. The use of practical, interesting and simple problems to reach these objectives is of a great benefit to students (Kasimatis, et al, 1999). Teachers must have the knowledge of mathematics so students can have a productive period of training, and learn to understand, think and reason, to better reflect and act on their own reality. This is the essence of personal empowerment.

The primary objective of education is to connect learners to the objective of seeing mathematics as one of the main accomplishments made by human beings. This knowledge allows the integration and connection of learners to the greater universe that encircles us (Zaro; Hillebrand, 1990). Unfortunately for many people, mathematics has lost its philosophical beauty. It seems that schooling makes it difficult to give students opportunities to observe and master the real world that surrounds them. Ethnomathematics as a pedagogical action allows teachers to show learners how to relate mathematical knowledge to real events.