Chapter 4:      The Solution That Looks For A Problem: Mathematical Modeling And Its Applications For Teaching And Learning In Mathematics

The determination of the time or location where the first expression of mathematical modeling took place is probably not possible.  Mathematics has expressed itself through modeling since the earliest of recorded history.  The invention of the wheel for the Sumerians in 3000 B.C. is one of the earliest mathematical models considered.  When a tree trunk was first observed rolling down a decline, the idea to use objects to roll heavy loads was born.  Loads were placed on logs (rolling objects) instead of carrying this same cargo on their backs, and eventually allowed the Egyptians to transport objects and to build their remarkable architectural wonders.  This invention led to the construction and development of architectural wonders and the development of new mathematical problems inherent in the solving and construction of impressive buildings and monuments.

Another example of an early mathematical model was activity originating with the work of Eratosthenes (276-196 B.C.).  Eratosthenes created a mathematical model that allowed him to accurately measure the circumference of the Earth.  Eratosthenes was a Cyrenian citizen (present day Libya), and spent a great part of his youth in Athens, where as a youth, he was a distinguished and popular athlete.  While most noted as an author of books about astronomy and geometry, he also wrote poetry and theater pieces.  As a man of many accomplishments, he is especially noted for having first determined the circumference of the earth, the size of the sun and moon, as well as their distances from the Earth.

Eratosthenes’ Measurement of the Circumference of the Earth

One of the questions that defied ancient mathematicians and astronomers were problems related to the determination of the size of the sun and the moon.  To arrive at these measures, Eratosthenes found it necessary to first know the circumference of the Earth; many of the most prominent mathematicians of the time were employed in the problem of measuring the Earth.  It was Eratosthenes; however, who discovered how to answer to this problem.

Brief History

Eratosthenes found that when he planted a rod in the ground at a 90˚ angle, and observed its shadow, a variation in the length of the shadow occurred which was dependent upon where in the world one observed this phenomena.  If we try this ourselves we can easily observe that in the morning, the size of the shade cast by the rod is longest and diminishes over time until the sun is directly overhead at noon.  After noon, the shadow begins to lengthen in the opposite direction until sunset.  We call noon the time when the shade is at the least length.

If we measure the shadow of this rod at noon on a series of successive days, we also will see a variation in the length of the shadow.  Early peoples possessed this knowledge.  It is not difficult for anyone to ascertain that the hottest part of the day occurs when the sun is at its highest point in the sky.  These same early peoples also, out of some necessity, saw that the noontime offered the least amount of shade, with the evenings and mornings offering longer shadows.  We also know that this has served as the basis for time keeping for many cultures. 

In numerous locations in the Northern Hemisphere, for example, many people once observed that the summer solstice is the day when shade is at its minimum.  This phenomenon came to define the beginning of the summer, and of course formed the basis for celebrations and other religious events in many cultures throughout Asia, North America, and Europe.  In the same way, the beginning of the winter has been defined by the winter solstice, the day when shade at noon is maximized in the northern hemisphere.  The term solstice comes of the Latin and means “sun static”, that is when the sun seems to stand still. 

Eratosthenes knew that the summer solstice occurred on the 22 of June in the city of Syene (present day Aswan) situated on the Nile River in Egypt.  As is true everywhere at noon on the solstice, the sun was at its zenith, at which time he placed a rod firmly in the ground.  Because the sun was at its highest point, the rod did not cast a shadow.  Eratosthenes then traveled north to Alexandria, approximately at the same time that the sun was at its zenith position in Syene, and placed a rod in one of the main squares of the city.  He measured the angle formed by the rod and the path formed by its shadow.  Using this procedure, he determined the measurement of the angle as 1/50 of the measure of the circumference of the Earth.

 

Mathematical Model

We know that two parallel straight lines intercepting a third transversal straight-line form internal alternating angles and that these angles constitute congruent angles.  It is easy to imagine that the projected shadows for both the rods placed in the two cities for Eratosthenes can be drawn out, to any length.  Thus, the traced line of the rod in Syene was considered parallel to an imaginary line traced by the rod in Alexandria, and the line of the projected shadow in Alexandria can be considered a transversal straight line between these two straight lines.  Knowing the values of the angles as well as the distance between Alexandria and Syene, we then determine the measure of the circumference of the Earth just as Eratosthenes did.

Eratosthenes made use of the following data:

·        C is the center of the Earth;

·        The rod not forming a shadow was in Aswan;

·        The shadow angle formed by the rod and its shade was in Alexandria;

·        B is the angle with vertex in the center of the Earth with an extension of the rods between Alexandria and Syene (Aswan). 

·        The angle as measured is equal to 1/50 of the measure of the circumference of the Earth.

As the sun’s rays are parallel as they strike the Earth, straight lines r and s form parallel lines and the internal alternating angles.  Therefore, the angles “a” and “b” are congruent.  Therefore:

Angle a = Angle b

 

As the angle measured is 1/50 of the circumference of the Earth, and the angles “a” and “b” are equal, therefore the distance between Syene and Alexandria is 1/50 of the circumference of the Earth.  Eratosthenes possessed the knowledge that the distance between Syene and Alexandria was approximately 5,000 stadia (1 km = 6.3 stadia, 1 stadium = approximately 607 feet / 185 meters).  Therefore, Eratosthenes concluded that the circumference of the Earth was approximately equal to:

50 x 5,000 = 250,000 stadia

Using this ratio, we can easily determine this angle in kilometers:

Conclusion

The extraordinary work of Eratosthenes determined that the circumference of the Earth was approximately 40,000 km (24,854.85 miles).  Modern measurements made by specialized equipment have determined that this measure is 40,075 km (24,901.45 miles).  We know that the measure calculated by Eratosthenes was made more than 2000 years ago, yet it possesses a margin of error that is of relatively minor importance.  An astonishing accomplishment considering the time and place Eratosthenes in which made these calculations.

Recent Developments

It was only during the last two decades that mathematical modeling formed an identity with ongoing research in the field, defined objectives, and an established sense of its nature and potentiality as a method of inquiry.  Mathematical modeling uses real situations to explore the mathematics as a language for understanding, simplifying, and resolving problems associated to a given context.  The models allow users to make forecasts, as well as ongoing modifications to the model.  Mathematics is only one of the instruments used to reach this objective.  Often in the mathematical modeling process, prerequisite questions do not exist.  By first coming to understand the context itself, modelers work to resolve problems from within this perspective.  As a more organic process of learning and pedagogy, previously those participating in the process will learn unknown and unclear mathematical content.

Mathematical modeling can be best understood as a new paradigm in mathematics education.  So students may better face real-world challenges, the paradigm has as its primary objective the development of a critical perspective and creativity.  Therefore, students come to possess the essential tools they need to interact within their own reality and to transform it in order to contribute towards the greater good.  Ethnomathematics, using mathematical modeling as described here, is similar to the contrast between the Newtonian and Quantum paradigms in physics and scientific research.  Wheatley (1992) describes it as:

Scientists in many different disciplines are questioning whether we can adequately explain how the world works by using the machine imagery created in the 17^th century, most notably by Sir Isaac Newton.  In the machine model, one must understand parts.  Things can be taken apart, dissected literally or representationally, and then put back together without any significant loss.  The Newtonian model of the world is characterized by materialism and reductionism – that is a focus on things rather than relationships…  The quantum view of reality strikes against most of our notions of reality.  Even to a scientist, this is admittedly bizarre.  But it is a world where relationship is the key determiner of what is observed and how particles manifest themselves…  Many scientists now work with the concept of fields – invisible forces that structure space or behavior.  In the new science of quantum physics, physical reality is not just tangible, it is also intangible.  Fields are invisible, yet they are the substance of the universe… in organizations, which is the more important influence on behavior- question: It depends… what is critical is the relationship between the person and the setting.  That relationship will always be different and will always evoke different potentialities.  It all depends on the players and the moment (Wheatley in Payne, 1998 pp. 142-143).

It then becomes an individual’s reality, or a composite of facts that collaborate to form a perception of real problems within unique cultural contexts that are important here.  Ethnomathematics and mathematical modeling can be used to study this relationship - the relationship between an individual and group of people and the perception of the problem.  It seeks to study the “players and the moment” as it were.  How an individual interacts with their environment and the kinds of mathematics used or developed within this interaction can be best documented by using mathematical modeling.

This same individual can then  intervene to form the resolution of problems with the objective of improving the quality of life of the community.  In this way, the developing work is in direct relation to the social and conceptual aspect and not to the traditional mechanics of “doing” mathematics.  One problem can be solved in a variety of ways.  Many times this “solution that looks for a problem” may require other forms of human knowledge that begin an efficient way for interdisciplinary study.

The real world shapes the abstract mathematical world; imagination is formed by our interaction with the real.  Often the validation of a mathematical solution does not satisfy the experimental data that has been received from a real-world context, causing certain modifications to be made in a mathematically or abstract fashion.  The given mathematical model satisfactorily represents the reality being described by this model.  Thus, the modleing cycle is formed:

The Modeling Cycle

The basic problem in the application of mathematics in mathematical modeling is not a question of mathematical knowledge per se, but the questions related to new discoveries of how people learn, and the way humans think.  The perspective that ethnomathematics lends to this endeavor is that this learning is culturally influenced and produces infinite variety and beauty.  Mathematical modeling, when linked to ethnomathematics, allows us to study and catalogue these findings.

We have developed an educational model that leads to a strategy of action that results in the creativity of each person when processing information (D’Ambrosio, 1990).

 

The Basic Cycle of Human Behavior

Outline for Applying Mathematical Modeling

As in the use of the scientific method, it is especially important that students understand the necessity to follow a defined and organized set of rules, or protocol, when performing mathematical modeling.  Each individual stage is important for the successful implementation and conclusion of the design.  Therefore, the following stages are suggested:

1.      Choice of theme;

2.      Justification of research question;

3.      Justification for the specific choice of the subject;

4.      Development of brief history related to curious, important, and interesting aspects in relation to the chosen theme;

5.      Development of the central questions relating to the theme;

6.      Problem formulation, including the formulation of a hypothesis and related variables;

7.      The development of the central or orienting question that approaches the subject in a general way;

8.      The development of the essential mathematical aspects that enable modelers to answer the central question;

9.      Further formulation of the problem focusing on the experimental hypotheses and variables;

10.  Examination of possible mathematical models related to the subject;

11.  Solution of mathematical models;

12.  Validation or testing of the mathematical models;

13.  Further discovery of related information (using the Internet, museums, libraries);

14.  Documentation of collected information (using video, recording, and other data entry tools);

15.  Exploration of additional methodologies and strategies to develop the research hypothesis (direct interviews, public visits to the industries and establishments);

16.   Chronology of the work, including an indication of the time needed to accomplish each stage;

17.  Final conclusion of the process (initial, intermediate and final stages).

18.  Completion of bibliographical references

The Mathematical Modeling Web

 

Thus, mathematical modeling is based on a process that encourages the elaboration of students’ abilities to connect their growing understanding of life (reality or context) with mathematics.  This process promises that by using ethnomathematics to grow a vision of the world can encourage the development of truly autonomous thought.  This autonomous thought is the kind that Paulo Freire spoke of as being that which allows learners to become fully contributing and empowered citizens.

Ethnomathematics as pedagogical action is therefore the use of mathematical modeling as educational methodology that allows us to construct knowledge in a serious and systematic manner.  The program as described here seeks to develop curricular experiences that enable modelers to learn to reflect upon experiences using realistic events.  Students learn to look for methods of truly understanding the given situation, and do not settle for rote learning.  Students learn to interpret data and information from various sources.  They weave connections with realities that give legitimacy to pedagogical action.  From this philosophical base, this program significantly attributes to the socio-cultural context of mathematics the historical situation that permits an action reflecting a deep transformation of the student’s intellectual autonomy. 

Another motive for this program is to reanimate the mathematics education community.  D’Ambrosio’s challenge to us is that it is our own unique understanding of mathematics that needs to be reconstructed.  This is a deeper understanding that is in conjunction with our participation as a community in the actual construction of mathematical knowledge by our students.

 

Practical Examples of Applications Using Modeling

Mathematical Uncertainty as Mathematical Power

As the title to this chapter suggests, the work we produce here works in reverse – that is it is the solution that looks for a problem that interests us.  We began to be interested in what kind of mathematics that is going to appear when it is used on a daily basis, as opposed to the traditional formal mathematical scientific research use of mathematics.  We notice this kind of problems solving done when children play soccer or computer  and/or video games.  We saw evidence of this in the literature (Guerdes, 2000; Ascher, 1998: Zaslavsky, 2001) in relation to cultures in Africa.  The program at the Instituto de Ciências Exatas at the Pontifícia Universidade Católica de Campinas (PUCC) in Brazil gave us the tools to develop and solve a mathematical model, which we will outline here[1].  In the coffee experiment (see Appendix I), PUCC teacher researchers explored the interdisciplinary links between language & literature, mathematics, geography, biology, and history in relation to coffee production and consumption. 

No subject should be taught in isolation.  There is broad international agreement (TIMSS, NCTM, PCN[i]) that students need opportunities to learn more than basic mathematical algorithms and need to extend their understanding to include how mathematics connects to other disciplines, to problems in society and the environment, and to diverse people around the world.  Mathematical modeling and ethnomathematics has been developed to help teachers to assist students to discover the relationship between mathematics, real world, and daily life.  All students need to be encouraged to develop and apply higher–level critical thinking skills.  In so doing, they must learn mathematical concepts, they need to learn to question, to take risks, to verbalize ideas, to listen to their own as well as other’s ideas, and to critique their own and other’s ideas.  This form of critical thinking must be practiced regularly.  One of the best ways to practice critical thinking in our students is to apply ethnomathematics as pedagogical action and mathematical modeling as methodology and strategy to teaching mathematics.

Mathematical modeling encourages the development of a learning environment that allows for the continuation of mathematical learning in the context of its relationship with other disciplines and areas of human understanding.  The ongoing development of this form of understanding and problem solving connects mathematics to a larger area to explore for learners, researchers, and teachers.  Mathematical modeling creates an environment that senses the applicability and practicality in concrete situations, and that will resolve day-to-day problems that are diagnosed and elaborated during the process of doing research.  Many educators understand that mathematical modeling as a teaching-learning strategy contains certain difficulties.  To add new content and goals into an existing program is difficult to accomplish in any school environment.  The lack of resources, space and time available that the teacher environment dictates is an important concern, as well, the current push to teach towards standards and assessments creates another obstacle for curricular reform.

Modeling has only one main objective; it is the amplification of the knowledge related to the mathematics of the subject being studied.  The development of this critical capacity and subsequent abilities in the participants necessary to do the research, is also time consuming, and can take away from the formal curriculum and goals of the school.

Practical Examples of Applications Using Modeling Mathematical

What we share here in the Appendices (I, II, and III) serve as examples of projects we have developed and used.  They represent at least three attempts at integrating ethnomathematics and modeling that were developed by students in secondary and university levels.  The amount of mathematics taught and learned was extraordinarily different than that of the required curriculum of the time.  In 1998, a team of public secondary mathematics teachers in Brazil developed The Coffee Experiment (Rosa, 2000).  At its completion, it was directly connected to the Brazilian National Standards (cite: Parametros).  At this stage, we seek to give three examples of ethnomathematics based on history & culture.

See in Appendices:

• The Coffee Experiment
• The Size of the Earth
• Population