Chapter 5: Applying Mathematical
Modeling to Transversal Themes
In mathematics, so that we can offer to our students, an education that
permits them to open ways that allow them to form their own personal realization
and for a better social well being. In answer to the "back to basics" movement
that places for a too much emphasis on learning to calculate and not enough
of learning to learn, NCSM (The National Council of Supervisors of Mathematics),
produced as far back as 1977, a document that to widely defined the basic
competencies in mathematics. This document had many repercussions in the time
of its publishing. In collaboration with an elaboration of an Agenda for
Action, NCSM considered the social and technological changes those alternative
pedagogical orientations that progressively were incorporating into the teaching
of mathematics.
In 1988, in its annual meeting in Chicago NCSM developed what is considered
the basic minimal contents for mathematics that students need in their future
activities in the documentation of Basic Mathematical Skills for the 21st
Century. NCSM considered essential the competencies necessary for considering
individuals in work is for the participation in society. To attain these objectives,
the students need to:
• To acquire a profound understanding
the understanding of mathematical principles,
• To develop the capacity to reason clearly,
• To develop the capacity to communicate in an effective
manner,
• To recognize the application of mathematics in the surrounding
world,
• To develop the capacity to face the mathematical problems
with confidence.
NCSM identified important areas of competence in mathematics that students
need to present determine abilities, for them to perform as responsible citizens
for the 21st century. Because they are directly related with the presuppositions
of an ethnomathematics program, highlight the following areas:
Problem Solving: Problem solving is a process of understanding
when new or non-familiar situations are encountered. Resolving written problems
is a form of problem solving, yet, plus it is important that students are
presented with problems that are not contextualized. Students need to have
conditions where they come to alternative solutions and resolve problems
that are presented to them with more than just a solution being asked of
them.
Communicating Mathematical Ideas: Students need the ability
to communicate mathematically. Included is the capacity to comprehend mathematical
ideas that are transmitted verbally, by writing or by some form of depiction.
Students need to learn to express mathematical ideas after they talk, or
they write about them. They need to do this with the help of drawings, graphics,
diagrams, and concrete materials. To develop the capacity to express mathematics
during class time students can be given support to learn to discuss, to reason.
Mathematical Reasoning: The importance of logical reasoning
in mathematics is salient to what students need in becoming capable of coming
to conclusions and to share data with conditions. Another emphasis is the
validation, by the student themselves who need to justify their thinking
and their process of solving problems and models or, therefore using already
known facts, a sense of suitableness to allow them to generalize their new
knowledge. The student needs also to gain the capacity to identify patterns
e to make guesses, and to use counter-examples for invalidating a guess.
Applying Mathematics to Daily Life: Students need to be encouraged
to represent mathematically situations in real life using graphics, diagrams,
tables and expressions, that process the represented data, by using mathematical
models to obtain results that need to be interpreted in connection with a
real or given context.
Probability and Statistics: Students need to possess basic
abilities in probability and statistics. In one general way students need
to have be able to plan and to utilize a collection of data to respond to
questions in the daily life and to comprehend how mathematics is utilized
to help make previsions in diverse situations as: elections, business, lotteries,
sporting events and population growth. Because we are living in a world where
information is processed and presented statistically, probability and statistics
can become part of the culture of mathematics for all students.
By utilizing an ethnomathematics program, there exists the possibilities
to convey a way to teach and develop automatically in students for them to
que acquire abilities to produce mathematics. This program enables people
to participate in the production of mathematics, therefore, include the formulation
and experimentation of their own hypothesis, the dialogue between students,
so that they can come to exchange ideas, answer and to validate their hypotheses.
Making use of mathematical modeling in the classroom contributes to facilitating
learning mathematics, by getting students to discover shared by the students,
they acquire the confidence necessary to explore their ideas and to reach
a common consciences to see the necessary solutions for each type of problem.
Introducing new themes such as statistics; downplays the emphasis in acquiring
mechanical processes like algorithms and calculations, in general the presence
of realistic problems and games brings mathematics closer to the universe
of the student and permits what they come to see as an important social discipline
( Lellis, Imenes; 1994).
Ethnomathematics and Information
Today, there is an enormous quantity of information that driven by the media,
books and other sources, that are generating even more data, that they can
influence in our daily professional activity, in our lives and in our culture.
Empowerment means that students will be able to analyze the statistics and
graphs they read in the media, to apply their mathematical knowledge to the
problems of society, and to take appropriate action as citizens of their communities
(Zaslavsky, 1996).
The data itself is developing particular forms of thinking and reasoning
that can help us to resolve problem situations in what is necessary to collect,
organize and present data, to interpret, communicate results and to make decisions
using the language of statistics. By utilizing a mathematical modeling methodology
we formulate models that include varieties of mathematical contexts to enlarge
and apply concepts. In so doing we further develop a perspective that places
the learner in position to critically make decisions that are based on the
information presented daily in the media.
A information can be used to awaken the interest of the student by using
social issues and can be utilized through a significant context for learning
the concepts and mathematical methods that connect with other areas of understanding.
Mathematical contents establish provide necessary instruments for a statistical
approach, which can be explored by doing research that is of interest to students.
By looking more to the future, we recognize that the use of tools such as
calculators and computers as applications of a statistical method that is
continuously expanding. The creative resolution of problems, the rigorous
reasoning and the efficient communication, adds to this importance. To perform
these functions with efficiency, in the next century, students need to see
a greater connection between mathematics, with the development of basic capacities
for problem solving that the gets them ready to understand the concepts that
are in these regulations that are based upon becoming more participative
citizens in their own social context.
Ethnomathematics and Citizenship
The construction of a real participatory citizenship in a child demands
a practical form of education that needs to develop a deep sense of the social
reality. This must be done in relation to the personal, collective and environmental
life of the learner. All children deserve the right to be able to actively
participate and exercise his / her own act of citizenship. This means that
all children have to learn the mathematical tools necessary to achieve success
in their own cultural context (NCTM Statement, 1998, in appendix). To be effective,
all learners have to be literate (including numerate in mathematical context).
Literacy does not involve only reading and writing, it must involve the coming
to understanding, making sense of, and the otherwise active use of mathematical
data, as well. In this context, mathematics plays an essential part in formulating
one’s own citizenship.
Mathematics is certainly a discipline that provokes repercussions, with
a great responsibility for shut out a great part of the population to actively
participate in society. In the whole selective process that in society utilizes
when possibly more competitors that need or when a number of competitors that
demand that mathematics is need to solve a problem.
All of the educational paradigms look for mathematics to take a place in
the education of citizens. All the movements that in fact respond to the concept
of citizenship that makes a relation more consistent between citizenship and
education, is an Ethnomathematics Program (Sebastiani Ferreira, 1993).
Ethnomathematics makes possible the liberation of universal mathematical
ideas with respect to the non-academic learning forms of citizenship. Therefore,
we need to look for a citizenship that can be constructed in social action
and the politics and that is not determined by the elite considerations as
“the holders of knowledge".
The Ethnomathematics Program and Marginalized Social Groups
The pertaining members of marginalized social groups like the Movimento
dos Trabalhadores Rurais Sem-Terra (MST), in Brazil, they need to know
how to read and write, as well they need to have an understanding about basic
mathematical concepts. Mathematical literacy is necessary as well. Therefore
to establish direct connections between mathematics education and the culture
of marginalized social groups, making possible what they have dignity and
guarantee the direct exercise of citizenship.
Brief History
In Brazil in the 1980’s, problems of land distribution worsened. The greatest
concentration of land was with 4.5 million small rural parcels (of about 100
hectares) with 20% of the total population working 78% outside of the rural
workforce. And 50,000 large rural properties (more than 1000 hectares) occupying
45% of the area and absorbing only 4% of the rural workforce. With the end
of the economic immigration of the 1970’s, and with the recession in the
1980’s, there occurred even greater unemployment in the rural areas. From
this grew a number of violent conflicts in the countryside: there are 4.2
thousand between 1987 and 1994, which left hundreds of victims.
The Brazilian government has utilized the politics of “assentamentos” in
relation to public lands in areas considered unproductive and appropriating
them as part of agrarian reform. In the last dozen years there have been secured
in Brazil a little less than 300,000 families, with less than 7% of those
that were necessary in accordance with the MST, the most important organization
that fought for agrarian reform, and was the leader in social mobilization
in the countryside. According to the MST, there are at least 4.5 million
families in Brazil to place. The properties of the landowners that were also
organized in political entities responded against the repression and invasion
of the land by the MST. In April 1997, the MST made in Brasilia, the largest
popular demonstration against the government of Fernando Henrique Cardoso,
called: the “National March for Agrarian Reform”, with approximately 100,000
participants.
An Approach Ethnomathematics
Literacy in mathematics suggests also that the culture and traditions of
these groups are highly valued. The popular methods that these groups that
are socially marginalized combine to realize mathematical calculations that
need to be: discussed, thought through, analyzed and compared to mathematical
methods considered both “official and traditional”.
The members of these groups, can establish comparisons between the different
understandings, and principally, can make conditions that find the method
that they find to more adequate for solving problems, when they are faced
with real situations.
The approach to ethnomathematics for the teaching and learning of mathematics
the needs of the “homeless” in Brazil by connecting to mathematics education
and the culture of the community and their productive activity. The mathematics
brings a consideration, of “day to day” activities of children and the adults.
In this way, the lands, the vegetable gardens, the plantings, the harvests
and the calendars, are the real problems that can be transformed into mathematical
models to be solved.
The approximation of a mathematics education with the reality and the culture
of the homeless or rural poor: the schools are connected to cultivation and
rural necessities and the walls of the school are used to develop and teach
mathematics related to this cultural necessity.
It is important that the mathematics is to be found outside of the classroom.
Therefore, the learning process of the students occurs primarily as the real
practice of learning mathematics. Finally, when they turn to mathematical
books, for understanding they are formally linked to the theory and fundamentals.
The mathematics teachers always use the same theme for all the students,
varying the activities by observing their individual: research, data collection
and the exercises in the classroom.
Mathematical Model: The Vegetable Garden
The students go to their vegetable garden to determine the area and perimeter.
They begin by measuring it with a garden hose or a rope that can be used like
a trena. They find a way to make correspondences to units of non-conventional
measurement with conventional units of measurement. Determining the area of
each canteiro, as well as the number of necessary canteiros to make a vegetable
garden allows them to reflect on the dimensions and the forms that represent
ideas for the canteiros. For them that can maximize the area of each form
that has a better use of the planed space for the vegetable garden.
Young children register the quantity and the variety of vegetables and collected
in each canteiro.
When students return to the classroom, the have acquired a combined depiction
since they raise tomatoes, lettuce, carrots, cabbage, and egg plant. Afterwards
they brought the data and information together of all that was harvested in
order to obtain the total of vegetables that were planted. With the majority
of students, the teacher work with the notion of major or minor, therefore,
not only can they be compared to the work with vegetables, but also in comparison
of the number of onions or the number of egg plant, etc. At this time it is
expedient to introduce the symbology of mathematics necessary for effectuating
these representations. The four basic operations of mathematics (addition,
subtraction, multiplication and division) also are works from within this
context. There exists the preoccupation with the formulation and problem solving
that represents each situation. So, the presented accounts of the students
are contextualized não É adequate application to this methodology.
The calculations are realized with the utilization of reasoning and the mathematical
understanding that they have. New mathematical concepts are going to be incorporated
into the prior concepts of the students who are going to construct their
own mathematical understanding, based on situations taken from their own
reality. The students have an understanding stemming from a real necessity
using calculations developed from an effectual mathematical determination.
The objective of the activities e os problems, that are part of the teaching
of each classroom, are created in conjunction with teachers and their students.
A Mathematical Model: The Production of Milk
The students speak with the parents who work with the production of milk,
to raise referential data production, the price of each liter of milk and
the costs of what are necessary to determine production. The calculations
related to daily, weekly, monthly and annual production levels. To permit
them to determine the value levied about the selling of their products. After
the value is determined for the sales, the students subtract their expenditures,
such as growth and answers for livestock; subtracting the taxes, for them
to calculate the value of profits obtained by assentamento. Therefore, after
these calculations, the students determined the percentage of profits that
each family of assentamento can receive.
Conclusion
Today, in Brazil, the discussion and efficiency of agrarian
reform of the economic solution: the development of production, as a social
solution: the boost in businesses and also that solution ecological: the
equilibrium between the city and the countryside. For some the production
is done on small properties that are not competitive, above all, in this
time of economic globalization, cannot be stimulated. For others, that divergent,
the small properties continually to be responsible for a major number of
businesses in the countryside and for the major production of foodstuffs
of internal consumption. It behooves mathematics educators to study these
questions, because after they can associate mathematical concepts with your
use of the real world, prepare the students to think over the agirem of reality.
The assentamentos, of countries and the teacher is they are united in the
defense of the study of children. The study encounters a lot of importance
with the study (work?) therefore provides with the necessary tools for acquiring
understanding. The work of undertaking the development of what a mathematics
activity is, therefore the children calculate the hours: daily, weekly and
monthly, that they permit to determine the daily, weekly and monthly importance
of their salaries. Learning to determine the hours in relation to this numeration
system in conjunction with the historically developed schedule, the researches
transform themselves into users of other practical forms of mathematics.
Therefore the children themselves learn to take responsibility and organize
their own time to study, to have fun, eat, to rest, to work for them and
the many other realities of daily life. Other mathematical activities can
be developed, using data of birthdays, calendar do plantio, school calendar,
hours that the teacher teaches, and all of what they produce by assentamento.
Therefore the students find a way to identify the relation of these situations
with mathematical patterns. The evaluation of the children is made
by using a written evaluation and with participation in the classroom and
the activities outside of class. A letter of participation is given to all
participating students that also realized a self-evaluation. The parents
also are also part of the evaluation process, evaluating the evolution of
the students and the atuation of the teacher. Constata-se the evaluation
of the indexes de assiduidade e uma reduction of the indexes of repeating.
The Ethnomathematics and the Transversal Themes
This form of mathematics education is strongly connected to citizenship,
and requires that questions of aspects social they are presented for learning
the reflection for part of the students. The inclusion of real social questions
in the school curriculum is not a recent preoccupation by educators, yet it
offers an important connection for learners. Learners come to see that their
work has value, and that it can contribute to the well being of their community.
These questions are related to a social need for students to see the possibility
of participation as the force for collaboration in the prevention and resolution
of problems that are economic, environmental, social, political and psycho-emotional
in nature.
In this context, the objectives of the contents of the transversal themes
need to have incorporated into mathematics education are other existing areas
in the school curriculum. Themes such as violence, prejudice, cultural plurality,
the environment, health and questions related to sexual orientation are themes
developed by school children in Brazil. They are not given a place in the
school curriculum with new areas of study or disciplines that need to have
work with projects in context to mathematics education and other areas of
understanding. The transversal themes are amply and can generate motivating
problems that translate the preoccupations of actual society, using important
questions; urgent that presents various forms of world citizenship. These
projects have the objective the discussion, reflection and analyses of social
problems, that can be presented proposals for solutions for social questions.
These themes need to have presentations by the students as a methodology and
strategy for the teaching and learning of mathematics. For these students
that can work with themes taken from their own reality, and is necessary that
they have an adequate conceptualization of mathematics, that actually becomes
instrumental for the reflection and the analysis of the proposed problems.
Effectively, your natural environment can be analyzed
in all of their complexity with way of models that to a major or minor degree,
approximates the real fact of the representation about what we really think
about and what we really strategize about in life. These models that compare
the distinct manners em todas as etapas da evoluTion da humanidade e em todas
as culturas, constituem a mathematics e as distintas ethnomathematics s (D’Ambrosio,
1999, p. 116).
Applying Mathematical Modeling in the School Using Paper Recycling
The work for this project was accomplished in October 1996, at Escola Estadual
de Primeiro e Segundo Graus Dr. Coriolano Burgos, in Amparo, in the state
of São Paulo, Brazil. This school is similar to high school in North
America. The students project and documents were shown in the central room
of school, on October 23, 1996, which is the national day of the Preservation
of the Patrimony. There was also exhibition of this research in the ìDelegacia
de Ensino de Amparoî for 40 mathematics teachers from Amparo and region.
This Delegacia is similar to Central School District Offices in North America.
As well this work was demonstrated at a workshop for 40 mathematics teachers,
on July, 1998, in São Leopoldo, in the state of Rio Grande do Sul,
Brazil, in the VI ENEM ( National Meeting of Mathematics Education).
Choosing the theme: Paper Recycling
Justification of study
The project began by generating a discussion about the great amount of notebook
pages that the students wasted in the classrooms and the subsequent affects
on the environment.
Main Objectives
• To identify uses for recycled paper;
• To consider the importance of recycled paper; and
• To create interdisciplinary links between mathematics
and other areas of the curriculum.
• Specific Objectives
• To estimate the amount of trees saved with recycled
paper;
• To develop understanding in statistics including the
development, construction and analysis of data and graphs;
• To instill sense of conscientiousness in the students
about the importance of avoiding waste of paper;
• To develop other subjects related to recycled paper
and other applications of mathematics.
The Project
Counting the total notebook pages thrown away made the development of data.
For purposes of this research the students counted only whole pages and it
was done once each morning, in the afternoon and at night. In Brazil, schools
work in the morning, from 7 a.m. to noon; in the afternoon, from 1 p.m. to
6 p.m.; and at night, from 7 p.m. to 11 p.m. After one week of data collection,
the students began the construction, comparison and analysis of the data and
graphs. These graphs demonstrated the amount of trees that would be cut down
by the continued waste of paper. As well, the graphs show the recycling of
paper would save the number of trees. The students also prepared explanatory
posters, the statistical graphs and other documents that they were requested
for the accomplishment of this project, under direction of Mr. Rosa. The
great majority of the students at school visited the exhibition and was able
to verify the amount of paper that each class wasted. Through subsequent
data comparison students initiated an understanding related to recycling
of paper.
Introduction
Many supermarket sacks, notebooks, cardboard boxes and packaging are manufactured
from reused paper. This process of obtaining new products starting from used
materials is called recycling. The recycling process allows the making of
newsprint. Practically, all the newsprint used in newspapers is made with
recycled paper. Wood fiber is the raw material used for the production of
paper, and recycling of that material contributes significantly to saving
trees in the world. Since recycling is done throughout the whole world, a
great number of trees will be saved on our planet. Improvement in paper
recycling techniques will provide to obtain paper of good quality and still
save a number of trees. Other possible benefits of obtaining of recycled
paper is one of the alcohol production, which is of great importance in the
chemical industry in Brazil, and the manufacture of proteins and oils needed
for the production of cellulose which is used in paper manufacturing.
Mathematical Modeling
- How to calculate the weight of notebook pages?
- Inside notebook pages (white paper only): 56 g/m2
- The calculation of the area of notebook pages:
Area of notebook page = 0.203 x 0.279
Area of notebook page = 0.056637 m2
Calculation of the weight of the notebook page:
1 m2 _____________________ 56 g
0.0056637 ________________ X
1/ = 56/
0.056637 X
X = 3.171672 g
Each notebook page weighs approximately 3.17 g
Comparative Data
The students calculate that each notebook page weighs approximately 3.17
grams.
The students found that is necessary to cut down 1 tree to produce 58.81
kg of paper.
There is an average waste of 17 notebook pages each day, for each class
in the school.
There are 62 classes that waste a total of 1,074 notebook pages a day, that
is to say, 3,40637 grams or 3.40637 kg.
In just one month (22 school days), there was a waste of 74.94026 kg of
notebook pages, making necessary the cutting down of 1.27 trees to produce
this amount of paper.
Other Facts
- In the south of Brazil, more than 80% of the trees of the
Atlantic forest have been cut down (1991).
- The city of São Paulo, Brazil, the third largest city
in the world, with 10,000,000 people, produces 500,000 newspapers daily.
Therefore for these newspapers 1,700 trees are cut down every day (1980).
- The city of São Paulo collects more than 13,000 tons
of garbage daily. A significant total of this is paper that could be recycled
(1988).
- Brazil prints 6,000,000 newspaper daily and for that production
10,200 tree are cut down every day (1980).
- In the United States, the goal for the end of decade of the
nineties is to reuse 40% of the whole used paper (1990).
- The students also found the following
Of each hundred old trees
They remain five witnesses accusing
The incredible secular torturer
Remain now five, not more. The rest are ghosts
Of the proud primitive forest .
(Carlos Drummond
de Andrade, famous Brazilian poet, 1902-1987)
However, seven years after the publication of Drummond’s poem in 1984, the
data became inaccurate. Today there are no longer five, but just three survivors
in each hundred primitive trees. People need to notice that not reusing paper
is at least a luxury that is an irresponsible act.
Conclusion
After the analysis of the results, the students were alerted to the need
of avoiding the waste of paper. The data demonstrated to them the problem
of deforestation and environmental consequences. The most significant fact
that the students discovered was that in just a month (approximately 22 school
days), there were approximately 75 kg of notebook pages thrown away. The students
found that is translated into the cutting down of 1.27 trees to make this
75-kg of paper. The data and experience gained from this project allowed
the students to improve the cleanliness of school. They also verified the
effects of wasting paper on the family budget.
Graph 1 - Wasted Pages per Class
Graph 3 - Paper Production in Relation to the Amount of Paper in Kg and
Number of Trees Consumed
One way in which we transform an industrial society, into an information
society; the mathematical understanding becomes even more important for individuals
that can have access to a mathematics education that prepares them for work
and for an active social life. To obtain an equitable society in the next
century, we have to abolish the many difficulties in front of us in mathematics
education. This way we can realize a work in context with higher expectations
in relation to work-learning
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