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Learning Environment Through Modelling and Computing
7.9 LEARNING ENVIRONMENT THROUGH MODELLING AND COMPUTING Regina Lino Franchi Methodist University of Piracicaba, Brazil Abstract-Mathematical modelling and computing are important in mathematics education and mathematical modelling because, in addition to enabling the learning of mathematics in a contextual way, it gives students the opportuniiy to develop their potential. Computing facilitates visualizations and offers precise and quick answers and, because it enables mathematical work through experiment form, it allows the student to exploit the concepts in an autonomous way, trying out changes and variations and drawing conclusions. The joint work of computing with modelling has brought new possibilities to modelling. Many of the dificulties in the modelling process were overcome by the ease of data collection and handling, and by the representations management with software and using the Internet. This paper presents a characterization and utilization of mathematical learning environment through mathematical modelling and computing.
1. INTRODUCTION The use of mathematical modelling in classes makes it possible to work with mathematics in a contextual way, so that it can contribute significantly to the process of acquisition of mathematical knowledge by the students (Franchi, 1993). In doing so it gives students the opportunity to develop their potential. There is also a trend in education to include computing. Talking specifically about mathematics, the possibility of calculations, simulation, using visual, auditory and animating resources, the precision and speed of the answers has led to reflections about the processes of construction of mathematics concepts by the students, about the validity of working some techniques and about other possibilities of work in differentiated environments for mathematics learning. Modelling and computing have been tried and discussed by teachers and researchers in mathematics education. There have also been attempts to evaluate the possibility to work in contexts that involve these two trends simultaneously. The present time is marked by the search of theory from the developed practice with Modelling and computing in learning situations. This paper intends to contribute to this debate by the characterization of learning environments with mathematical modelling and computing. The paper also shows examples of activities in this kind of environment.
474
Learning Environment Through Modelling and Computing
2. CHARACTERIZATION OF LEARNING ENVIRONMENTS THROUGH MATHEMATICAL MODELLING AND COMPUTING In spite of the great number of experiences of modelling use in learning contexts, there is no consensus amongst researchers about the characteristics of modelling in mathematical education and about the possibilities of use of this kind of strategy. From the point of view of applied mathematics it is said that the characterization of the modelling is clear. The main objective for applied mathematics is the model. Thus, the activity is successful when it gets solutions that are validated and that can give answers to questions with accuracy. For mathematical education, the process of construction of the model acquires special importance. Modelling can be thought as a strategy of mathematical learning. In this case, mathematical concepts are worked by its relation with the phenomenon in study. The concepts are introduced through the modelling process, inserted in the context of the studied problem and then the study of techniques and deepening of theories may be carried out. It is also possible to evaluate the modelling process from its relation with the development of the student's potentialities, such as: capacity to identify, to formulate and to solve problems, capacity to search for information, to use resources varied in the search for solutions, creativity, cooperation, team work, initiative, capacity to analyze and to compare possible solutions, capacity do decide and to evaluate the consequences of the carried out actions and so forth. These abilities are stimulated, demanded and developed during the different stages of the model construction process. The use of modelling can still contribute for the social-critical students' formation if the possibility to reflect about the context of the phenomenon that has been studied is considered. Modelling can contribute for the understanding of the reality and at the same time make it possible to act on it. Some researchers have tried to identify what, in fact, characterizes the modelling presence in learning situations. They search for indications of modelling presence by the development of thematic projects external to mathematics (that do not always result in the construction of a model) or by the use of hypotheses that may simplify the representation and the use of approaches culminating with the effective model creation (see Bean, 2001 p56). Barbosa (200 1) gives an important contribution for this debate characterizing what he called learning environments with modelling. Based on different types of activities described in the research literature on modelling and education and based on Skovsmose's definition (2000) of learning environments, he defines: "Modelling is a learning environment in which the students are invited to inquire and/or to investigate situations with reference in the reality, by means of the mathematics" (Barbosa, 2001 p3 1). This characterization includes many different possibilities of modelling use that have already been included in such literature. Relating to computing, there is a certain agreement that the development of activities using the computer in classroom characterizes a computing environment. However, the use of this resource does not necessarily imply in changes of pedagogical position. We may essentially have a lecture, in spite of the use of the computer. Borba and Penteado (2001), discoursing about educational environments
Franchi
475
with the presence of computing, point to the importance of educative practices that emphasize the construction acquiring of meaning by those involved. Based on the above statements, we characterize, in a form analogous to Barbosa: an environment of mathematical learning through computing is the one in which the students are invited to inquire and/or to investigate mathematics objects, by means of computing. The situations that stimulate the investigation, experimentation and formation of hypotheses can come from an internal or external context of mathematics (Franchi, 2002). It is also important to consider a combined work of computing with modelling. Computing has brought new possibilities for modelling. Many of the difficulties of the modelling process were overcome by the facility of data collection and treatment and by the ease of representations possible through particular software. The model can be constructed with more fieedom without being concerned about complex mathematics, or that that mathematics might be difficult for certain school levels. So, we can characterize a learning environment through mathematical modelling and computing as the one in which the students are invited to inquire and/or to investigate situations with reference on the reality, by means of mathematics, using computing in the stages of modelling process or yet the one in which the students are invited to inquire and/or investigate mathematics objects, by means of computing, motivated by situations with reference on the reality (Franchi, 2002). On the first case we can consider, for instance, a study of a real situation, by means of mathematics, using computing to organize and to represent collected data or to solve formulated equations. On the second case we can consider, for instance, a study of a mathematical subject that can characterize a phenomenon of the reality, using software to work with characteristics and relative content proprieties. 3. WORK POSSIBILITIES IN LEARNING ENVIRONMENT THROUGH MATHEMATICAL MODELLING AND COMPUTING
The following description is about a study carried out about the work possibilities with the theme “Dengue” in a learning environment through mathematical modelling and computing. More than the knowledge of the subject, the main objective of the description is to give an example of how the study of a reality theme (of interest to some community) makes possible the creation of a mathematical learning environment through mathematical modelling and computing. Dengue is a disease transmitted by the bite of the mosquito Aedes Aegypti. The mosquito reproduces in any container used to store water in shady or sunny areas. Cans, tyres, pots of aquatic plants or water boxes are common places for these mosquitoes to reproduce themselves. Its life cycle is divided into two phases: the aquatics phase (egg, eclosion, larva and pupa) and the aerial phase (birth and adult). The main factors that can influence the mosquito propagation are the temperature, humidity, regional topography, and the human activities related to the places where the mosquito may reproduce. The usual ways used to control the mosquito propagation are: mechanical control (elimination of the places where they may reproduce), chemical control (like spray) and biological control (cohabit of animal species that feeds on the mosquito larva).
476
Learning Environment Through Modelling and Computing
There are many work possibilities related to the theme. For instance, the growth of the mosquito population can be studied (from the aquatic phase to the adult phase), the biological larva control or the control of the places where they may reproduce. Related to the control of the places where they may reproduce it is possible to study the conservation of water pools, pots or tanks. In the following example, a set of pools with water movement (by cascades) was considered, and the periodic chlorine addition in the tanks that make up the set. The formulated question was: how to control the amount of chlorine in each tank of the set? Time (hours)
I
I chlorine (grams)
0
1
1
1
1000.0 900.0 I 810.0
5
1
590.5
6 7
1 1
531.4 478.3
348.67 Table 1. Amount of Chlorine.
1200 1000
Gi Y
800
'
y = -0,l I 7 2 2 + 5,11212- 104,53~+999,77 R2= 1 y = 3,35392 - 97,827~ + 99555 R2= 0,9998
~
Q)
-r-E 0 0
600 400
y = -64,288~+945,25 R2= 0,979
~
~
0
10
5 time(h)
Figure 1. Graph Amount of chlorine (8) x time (h).
15
477
Franchi
Hypotheses considered were: the volume of water in the tank remains the same (the rate of water entering is equal to the exit rate); the chlorine is all added to the tank just once at the beginning of the process and then mixed to the water; the chlorine is eliminated only by the dilution due to the water renewal in the tank. From experimental data, found by research, the following problem was formulated: a tank is completely full and contains one hundred cubic metres of water with 1 Kg of chlorine. Pure water is placed in the tank in a constant outflow while an escape valve eliminates the water excess. How is the amount of chlorine in the tank at time t determined? At first, the experimental data (Table 1) were considered using mathematical software. The graphs were plotted and fitting produced some possibilities of mathematical expressions (Figure 1). Any of the expressions was considered applicable between the initial and final data. However, it is not possible to extrapolate about the phenomena beyond the data. So, a theoretical study was necessary. We considered A(t) the amount of chlorine in the tank in instant t and K the rate of pure water entering and chlorinated water exiting by the escape valve. (NIOO).K kg chlorine goes out of the tank in each instant, having therefore a dilution along the process. After one hour, the amount of chlorine in the tank is:
Transforming this discrete analysis into a continuous analysis ( At
_ dA - --K . A dt
100
+ 0 ): (3)
The expression for A is then: A
= 1000.~-0.1053.'
(4)
This expression is equivalent of the exponential function found by the software. Testing this solution with the data of the table we get very similar values. The retrieve function is an approach of what happens in the reality. Mathematically the solution indicates the t-axis as a horizontal asymptote. This means that A + 0 when t + 00 . Actually, within a determined tolerance range, it is possible to find a value o f t for which the amount of chlorine is practically zero. It is possible to make predictions with this model for beyond the data set shown in the table. For example, an analysis of the conditions of conservation of the tank: after how many hours will the amount of chlorine in the water be less than 2g/m3? Or yet: with what regularity must chlorine be added in order to keep the water of the tank in a level considered acceptable for the treatment in question? Another approach for the same problem (discrete analysis) can be considered and the expression found (also equivalent to the previous ones) is: A(n) = A(O).(I-K/100)n (5) A simpler analysis about the adequacy of the curves found graphically can be made by taking in account the characteristics of the problem. The second degree
478
Learning Environment Through Modelling and Computing
polynomial curve is not adjusted because, according to this function, after some time the amount of chlorine will increase and it is not in accordance with the problem. The third degree curve also is not adjusted because this curve tends to assume negative values and it is not in accordance with the problem either. The activity can be characterized as a learning environment through modelling and computing. The students were invited to inquire and to investigate that the situation studied accorded with reality, in doing so both mathematics and computing were used as tools. Initially the computer was used to look for information about the theme. After the problem was formulated, the computer was used to organize and to represent collected data. Computing resources enabled us to fit data and obtain different expressions for possible solutions. In order to interpret the answers found by software, a theoretical study was carried out and the results were compared with reality. In this specific problem it was not necessary to use mathematical software to solve the equation but, if the problem had not been so simple, this recourse would have been used. In these kinds of activities, the different approaches, analyses and comparisons necessitate a frequent movement from modelling to computing and from computing to modelling. It can enrich the processes of mathematical knowledge construction, contributing to the learning development of the students. It is possible to perceive in the description, different levels of complexity in treating the same problem and the different mathematical concepts that can be systemized by the process of model construction. The choice of this, or that, approach depends on the context of the activity development (type of course or school level). For an average level course, for example, the polynomial and exponential functions could be studied using the software recourses. For the mathematical model, a discrete approach could be made which results in an exponential function whose expression can be compared with the one found using software. In case of an undergraduate course, a continuous treatment makes it possible the study of derivative concepts, differential equation and integral. The derivative concept is used to equate the variation rate of the amount of chlorine, the types of differential equations and the processes of resolution explored. The concept of integration as inverse of the differentiation is also applied. The work with the theme can be completed by the development of activities related to other questions already mentioned. All the studied problems may contribute for the knowledge and understanding of the reality related to the chosen theme allowing to critical analysis and action on this reality. 4. FINAL CONSIDERATIONS
The previous description is an example of the countless possibilities of work in mathematical learning environment through mathematical modelling and computing. In the described situation the students cover the modelling process in all of its stages, since the problem formulation about the theme, the construction of the model and its validation. This kind of activity contributes much to the development of the student’s potential, stimulating and facilitating the activity within a reality. So such activities are interesting and useful within mathematical education.
Franchi
479
However, the characterization of learning environments through modelling and computing admits the possibility to work with activities that do not involve the whole modelling process or that do not result in complete model construction. Certain activities in these environments can be equally interesting in some situations, contributing for the student's intellectual development. It is admits the possibility that students could be invited to investigate a real situation in which the questions have already been formulated. In this case they have to be encouraged to search for different resources (like mathematics and computing) for the resolution. It is also possible to inquire of the proper mathematical content, with reference in something external to mathematics, using computing resources. In this case the model is not a result of the activity, but it is previously given and used as motivation for the study. In learning environments through modelling and computing, teachers and students are co-participants of the process. The performance of each of them in the organization of the activities can be more or less intense depending on the type of the developed activity. The choice of the way to implement these environments in curricular activities depends on the context, varying in accordance with different possibilities and limitations of it.
REFERENCES Barbosa, J. C.(200 1) Modelagem matematica: concepc6e.s e experithias de futuros professores. 2001. 253 f.Tese (Doctoral Thesis in Mathematics Education), Instituto de GeociCncias e CiCncias Exatas, Universidade Estadual Paulista, Rio Claro. Bassanezi, R. C.( 1994) Modelagem Matematica. Dynamis, Blumenau, 2, 7, 55-83. Bean, D.(2001) 0 que C modelagem matematica? Educaca'o matemutica em revista. Sa'o Paulo: SBEM, 8,9110. Borba, M. C., Penteado, M. G.(2001) Informatica e educaca'o matematica. Belo Horizonte: AutCntica. Franchi, R. H. 0. L. (1993) Modelagem Matematica como estratdgia de aprendizagem do Calculo Diferencial e Integral nos cursos de Engenharia. Masters disseration in Mathematics Education, Universidade Estadual Paulista, Rio Claro. Frachi, R. H. 0. L. (2002) Uma proposta curricular de matematica para cursos de engenharia utilizando modelagem matematica e informatica. 189 f. Tese (Doctoral Thesis in Mathematics Education), Universidade Estadual Paulista, Rio Claro. Skovsmose, 0. (2000) Cenirios de investigaqgo. Boletim de educaca'o matematica, Rio Claro, 14,66-9 1.
1. INTRODUCTION The use of mathematical modelling in classes makes it possible to work with mathematics in a contextual way, so that it can contribute significantly to the process of acquisition of mathematical knowledge by the students (Franchi, 1993). In doing so it gives students the opportunity to develop their potential. There is also a trend in education to include computing. Talking specifically about mathematics, the possibility of calculations, simulation, using visual, auditory and animating resources, the precision and speed of the answers has led to reflections about the processes of construction of mathematics concepts by the students, about the validity of working some techniques and about other possibilities of work in differentiated environments for mathematics learning. Modelling and computing have been tried and discussed by teachers and researchers in mathematics education. There have also been attempts to evaluate the possibility to work in contexts that involve these two trends simultaneously. The present time is marked by the search of theory from the developed practice with Modelling and computing in learning situations. This paper intends to contribute to this debate by the characterization of learning environments with mathematical modelling and computing. The paper also shows examples of activities in this kind of environment.
474
Learning Environment Through Modelling and Computing
2. CHARACTERIZATION OF LEARNING ENVIRONMENTS THROUGH MATHEMATICAL MODELLING AND COMPUTING In spite of the great number of experiences of modelling use in learning contexts, there is no consensus amongst researchers about the characteristics of modelling in mathematical education and about the possibilities of use of this kind of strategy. From the point of view of applied mathematics it is said that the characterization of the modelling is clear. The main objective for applied mathematics is the model. Thus, the activity is successful when it gets solutions that are validated and that can give answers to questions with accuracy. For mathematical education, the process of construction of the model acquires special importance. Modelling can be thought as a strategy of mathematical learning. In this case, mathematical concepts are worked by its relation with the phenomenon in study. The concepts are introduced through the modelling process, inserted in the context of the studied problem and then the study of techniques and deepening of theories may be carried out. It is also possible to evaluate the modelling process from its relation with the development of the student's potentialities, such as: capacity to identify, to formulate and to solve problems, capacity to search for information, to use resources varied in the search for solutions, creativity, cooperation, team work, initiative, capacity to analyze and to compare possible solutions, capacity do decide and to evaluate the consequences of the carried out actions and so forth. These abilities are stimulated, demanded and developed during the different stages of the model construction process. The use of modelling can still contribute for the social-critical students' formation if the possibility to reflect about the context of the phenomenon that has been studied is considered. Modelling can contribute for the understanding of the reality and at the same time make it possible to act on it. Some researchers have tried to identify what, in fact, characterizes the modelling presence in learning situations. They search for indications of modelling presence by the development of thematic projects external to mathematics (that do not always result in the construction of a model) or by the use of hypotheses that may simplify the representation and the use of approaches culminating with the effective model creation (see Bean, 2001 p56). Barbosa (200 1) gives an important contribution for this debate characterizing what he called learning environments with modelling. Based on different types of activities described in the research literature on modelling and education and based on Skovsmose's definition (2000) of learning environments, he defines: "Modelling is a learning environment in which the students are invited to inquire and/or to investigate situations with reference in the reality, by means of the mathematics" (Barbosa, 2001 p3 1). This characterization includes many different possibilities of modelling use that have already been included in such literature. Relating to computing, there is a certain agreement that the development of activities using the computer in classroom characterizes a computing environment. However, the use of this resource does not necessarily imply in changes of pedagogical position. We may essentially have a lecture, in spite of the use of the computer. Borba and Penteado (2001), discoursing about educational environments
Franchi
475
with the presence of computing, point to the importance of educative practices that emphasize the construction acquiring of meaning by those involved. Based on the above statements, we characterize, in a form analogous to Barbosa: an environment of mathematical learning through computing is the one in which the students are invited to inquire and/or to investigate mathematics objects, by means of computing. The situations that stimulate the investigation, experimentation and formation of hypotheses can come from an internal or external context of mathematics (Franchi, 2002). It is also important to consider a combined work of computing with modelling. Computing has brought new possibilities for modelling. Many of the difficulties of the modelling process were overcome by the facility of data collection and treatment and by the ease of representations possible through particular software. The model can be constructed with more fieedom without being concerned about complex mathematics, or that that mathematics might be difficult for certain school levels. So, we can characterize a learning environment through mathematical modelling and computing as the one in which the students are invited to inquire and/or to investigate situations with reference on the reality, by means of mathematics, using computing in the stages of modelling process or yet the one in which the students are invited to inquire and/or investigate mathematics objects, by means of computing, motivated by situations with reference on the reality (Franchi, 2002). On the first case we can consider, for instance, a study of a real situation, by means of mathematics, using computing to organize and to represent collected data or to solve formulated equations. On the second case we can consider, for instance, a study of a mathematical subject that can characterize a phenomenon of the reality, using software to work with characteristics and relative content proprieties. 3. WORK POSSIBILITIES IN LEARNING ENVIRONMENT THROUGH MATHEMATICAL MODELLING AND COMPUTING
The following description is about a study carried out about the work possibilities with the theme “Dengue” in a learning environment through mathematical modelling and computing. More than the knowledge of the subject, the main objective of the description is to give an example of how the study of a reality theme (of interest to some community) makes possible the creation of a mathematical learning environment through mathematical modelling and computing. Dengue is a disease transmitted by the bite of the mosquito Aedes Aegypti. The mosquito reproduces in any container used to store water in shady or sunny areas. Cans, tyres, pots of aquatic plants or water boxes are common places for these mosquitoes to reproduce themselves. Its life cycle is divided into two phases: the aquatics phase (egg, eclosion, larva and pupa) and the aerial phase (birth and adult). The main factors that can influence the mosquito propagation are the temperature, humidity, regional topography, and the human activities related to the places where the mosquito may reproduce. The usual ways used to control the mosquito propagation are: mechanical control (elimination of the places where they may reproduce), chemical control (like spray) and biological control (cohabit of animal species that feeds on the mosquito larva).
476
Learning Environment Through Modelling and Computing
There are many work possibilities related to the theme. For instance, the growth of the mosquito population can be studied (from the aquatic phase to the adult phase), the biological larva control or the control of the places where they may reproduce. Related to the control of the places where they may reproduce it is possible to study the conservation of water pools, pots or tanks. In the following example, a set of pools with water movement (by cascades) was considered, and the periodic chlorine addition in the tanks that make up the set. The formulated question was: how to control the amount of chlorine in each tank of the set? Time (hours)
I
I chlorine (grams)
0
1
1
1
1000.0 900.0 I 810.0
5
1
590.5
6 7
1 1
531.4 478.3
348.67 Table 1. Amount of Chlorine.
1200 1000
Gi Y
800
'
y = -0,l I 7 2 2 + 5,11212- 104,53~+999,77 R2= 1 y = 3,35392 - 97,827~ + 99555 R2= 0,9998
~
Q)
-r-E 0 0
600 400
y = -64,288~+945,25 R2= 0,979
~
~
0
10
5 time(h)
Figure 1. Graph Amount of chlorine (8) x time (h).
15
477
Franchi
Hypotheses considered were: the volume of water in the tank remains the same (the rate of water entering is equal to the exit rate); the chlorine is all added to the tank just once at the beginning of the process and then mixed to the water; the chlorine is eliminated only by the dilution due to the water renewal in the tank. From experimental data, found by research, the following problem was formulated: a tank is completely full and contains one hundred cubic metres of water with 1 Kg of chlorine. Pure water is placed in the tank in a constant outflow while an escape valve eliminates the water excess. How is the amount of chlorine in the tank at time t determined? At first, the experimental data (Table 1) were considered using mathematical software. The graphs were plotted and fitting produced some possibilities of mathematical expressions (Figure 1). Any of the expressions was considered applicable between the initial and final data. However, it is not possible to extrapolate about the phenomena beyond the data. So, a theoretical study was necessary. We considered A(t) the amount of chlorine in the tank in instant t and K the rate of pure water entering and chlorinated water exiting by the escape valve. (NIOO).K kg chlorine goes out of the tank in each instant, having therefore a dilution along the process. After one hour, the amount of chlorine in the tank is:
Transforming this discrete analysis into a continuous analysis ( At
_ dA - --K . A dt
100
+ 0 ): (3)
The expression for A is then: A
= 1000.~-0.1053.'
(4)
This expression is equivalent of the exponential function found by the software. Testing this solution with the data of the table we get very similar values. The retrieve function is an approach of what happens in the reality. Mathematically the solution indicates the t-axis as a horizontal asymptote. This means that A + 0 when t + 00 . Actually, within a determined tolerance range, it is possible to find a value o f t for which the amount of chlorine is practically zero. It is possible to make predictions with this model for beyond the data set shown in the table. For example, an analysis of the conditions of conservation of the tank: after how many hours will the amount of chlorine in the water be less than 2g/m3? Or yet: with what regularity must chlorine be added in order to keep the water of the tank in a level considered acceptable for the treatment in question? Another approach for the same problem (discrete analysis) can be considered and the expression found (also equivalent to the previous ones) is: A(n) = A(O).(I-K/100)n (5) A simpler analysis about the adequacy of the curves found graphically can be made by taking in account the characteristics of the problem. The second degree
478
Learning Environment Through Modelling and Computing
polynomial curve is not adjusted because, according to this function, after some time the amount of chlorine will increase and it is not in accordance with the problem. The third degree curve also is not adjusted because this curve tends to assume negative values and it is not in accordance with the problem either. The activity can be characterized as a learning environment through modelling and computing. The students were invited to inquire and to investigate that the situation studied accorded with reality, in doing so both mathematics and computing were used as tools. Initially the computer was used to look for information about the theme. After the problem was formulated, the computer was used to organize and to represent collected data. Computing resources enabled us to fit data and obtain different expressions for possible solutions. In order to interpret the answers found by software, a theoretical study was carried out and the results were compared with reality. In this specific problem it was not necessary to use mathematical software to solve the equation but, if the problem had not been so simple, this recourse would have been used. In these kinds of activities, the different approaches, analyses and comparisons necessitate a frequent movement from modelling to computing and from computing to modelling. It can enrich the processes of mathematical knowledge construction, contributing to the learning development of the students. It is possible to perceive in the description, different levels of complexity in treating the same problem and the different mathematical concepts that can be systemized by the process of model construction. The choice of this, or that, approach depends on the context of the activity development (type of course or school level). For an average level course, for example, the polynomial and exponential functions could be studied using the software recourses. For the mathematical model, a discrete approach could be made which results in an exponential function whose expression can be compared with the one found using software. In case of an undergraduate course, a continuous treatment makes it possible the study of derivative concepts, differential equation and integral. The derivative concept is used to equate the variation rate of the amount of chlorine, the types of differential equations and the processes of resolution explored. The concept of integration as inverse of the differentiation is also applied. The work with the theme can be completed by the development of activities related to other questions already mentioned. All the studied problems may contribute for the knowledge and understanding of the reality related to the chosen theme allowing to critical analysis and action on this reality. 4. FINAL CONSIDERATIONS
The previous description is an example of the countless possibilities of work in mathematical learning environment through mathematical modelling and computing. In the described situation the students cover the modelling process in all of its stages, since the problem formulation about the theme, the construction of the model and its validation. This kind of activity contributes much to the development of the student’s potential, stimulating and facilitating the activity within a reality. So such activities are interesting and useful within mathematical education.
Franchi
479
However, the characterization of learning environments through modelling and computing admits the possibility to work with activities that do not involve the whole modelling process or that do not result in complete model construction. Certain activities in these environments can be equally interesting in some situations, contributing for the student's intellectual development. It is admits the possibility that students could be invited to investigate a real situation in which the questions have already been formulated. In this case they have to be encouraged to search for different resources (like mathematics and computing) for the resolution. It is also possible to inquire of the proper mathematical content, with reference in something external to mathematics, using computing resources. In this case the model is not a result of the activity, but it is previously given and used as motivation for the study. In learning environments through modelling and computing, teachers and students are co-participants of the process. The performance of each of them in the organization of the activities can be more or less intense depending on the type of the developed activity. The choice of the way to implement these environments in curricular activities depends on the context, varying in accordance with different possibilities and limitations of it.
REFERENCES Barbosa, J. C.(200 1) Modelagem matematica: concepc6e.s e experithias de futuros professores. 2001. 253 f.Tese (Doctoral Thesis in Mathematics Education), Instituto de GeociCncias e CiCncias Exatas, Universidade Estadual Paulista, Rio Claro. Bassanezi, R. C.( 1994) Modelagem Matematica. Dynamis, Blumenau, 2, 7, 55-83. Bean, D.(2001) 0 que C modelagem matematica? Educaca'o matemutica em revista. Sa'o Paulo: SBEM, 8,9110. Borba, M. C., Penteado, M. G.(2001) Informatica e educaca'o matematica. Belo Horizonte: AutCntica. Franchi, R. H. 0. L. (1993) Modelagem Matematica como estratdgia de aprendizagem do Calculo Diferencial e Integral nos cursos de Engenharia. Masters disseration in Mathematics Education, Universidade Estadual Paulista, Rio Claro. Frachi, R. H. 0. L. (2002) Uma proposta curricular de matematica para cursos de engenharia utilizando modelagem matematica e informatica. 189 f. Tese (Doctoral Thesis in Mathematics Education), Universidade Estadual Paulista, Rio Claro. Skovsmose, 0. (2000) Cenirios de investigaqgo. Boletim de educaca'o matematica, Rio Claro, 14,66-9 1.