dynamic software for geometry and mathematics

Dynamic software for Geometry and mathematics - Expo-Sciences International 15-17 September 2013

DYNAMIC SOFTWARE FOR GEOMETRY AND MATHEMATICS

Fernando Moreno, 2013

Universidad de Medellín, Colombia

[email protected]

From the academic Civil engineering undergraduate experience it has been

developed in the research process, a didactic strategy using Software Cabri II plus

that allows the student to create appropriate geometric models to solve practical

and particular problem, which can extend from the generality of the software to

other environments where geometry is a fundamental part of mathematical

modeling of the problem, addressing Euclidean geometry and its applications, from

the conception of dynamic geometry.

These tools are used in a differential way to the traditional in the University of

Medellin, to favor the development of abstract thinking and understanding of

applied mathematical concepts of engineering based on physical challenges,

generating significant learning experiences. Showing how the experience of a

geometric construction allows the approach of a concept in a deductive and

intuitive manner with the geometrical view, as fostering mathematical proof for

the interpretation of theorems and its condense to improve the understanding of

the mathematical applications for student attrition, getting the student to falls in

love with his career.

We have approached the proposal involving a change of chip in the relationship of

the teacher with the student and the appropriation of the technological tools that

make the trainee manager of their own knowledge, where the teacher is assumed

as a facilitator.

It is concluded that study of geometry and calculation through the dynamic

software facilitates theoretical understanding by the students and is a potent

instigator to creativity when it comes to proposing solutions; furthermore the

development of geometrical thinking involves at the same time the development of

modeling skills. This proven experimentally by prototype scale models designed

with this methodology, products winners of national contest, competitions on

university creativity and innovation and nodal encounters merits national seed

research in Colombia.

Key words: Geometric construction, mathematical modeling, dynamic geometry, didactic mathematics.

[email protected]


“Open a private space, in particular geometry and math in general, to find good problems

with new resources” (Saidon, 1998)

Applications made with Cabri II Plus dynamic software, include two modules in this work, the first is the design of a functional and structural bridge based on geometric concepts which are posteriors to the engineering approach, this was taken to a physical scale model and its behavior was analyzed in the laboratory.

The second module consists on showing the behavior of dissolved chemical concentrations in solvent fluids which are constantly flowing in tanks, this way is possible to understand graphically the behavior of the exchange rate of such solutions respect to time, using the concept of "change reason" applied to integral calculus and expressing it with geometric constructions through dynamic geometry software.

Using Cabri II Plus software as a simulation pattern, additionally for the first module a scale prototype was constructed. This, based on classroom experience and developed in the Dynamic Software seedbed, and now argued theoretically giving continuity to the research training process.

PROBLEM STATEMENT AND JUSTIFICATION

The problem begins with the following questions:

What geometrical concepts are useful in structural design?

What visual input can be set to facilitate the understanding of chemical solutions problem situations?

Analyzing the problem of many people in the student community in physical issues, in particular with the geometric visualization of these, an opportunity to interpret its geometric nature was found, basing on software Cabri II Plus modeling tools, which allowed us to focus on provide geometric solutions for a bridge construction according the given specifications and geometrical model for the chemical concentrations variation in a tank, using integral calculus applications.

The main objective is to show how dynamic geometry can be used creatively to promote understanding of concepts applied to specific problems. To do so, determinate through the dynamic geometry resource and using the software Cabri II Plus, the construction requirements of civil structures considering that there is not a deep understanding of concepts related to civil engineering, minimizing impediments for design and physical modeling these structures.

Moreover, the intention is to provide a contribution to the university community which helps improve the perception and understanding of the application of "rate of change" concept in the integral calculus approach, using the resource of Dynamic Geometry Software.


MATHEMATICS TEACHING

● New technological tools.

From the teachers perception, New technologies define discipline development and individual study primacy, under a self-taught methodology which allows to know and discerning problems of both man and contemporary society, these tools make teachers raise new concerns about their mathematics teaching strategies and the communication with the students effectiveness.

Youth receive constantly opinions of society issues, science and culture in general, due to the advances in the way we access information makes them to keep their senses willing for individuals diversity knowledge, social groups and the interaction in which the same being is immersed. This is important, while what is seeing is being used and is imagined as a formative guide, while recognizing that the main subject current and future development is the same.

The new technologies make institutions to assume the responsibility to respond for this new order, because of the social requirement about students having a basic culture, being able to extend their learning and understand the environment in which they are.

By introducing technology in these knowledge institutions is necessary to look like a social laboratory as a technical mission. (Castro, 1991)

Then, it is necessary for all educators to consider the possibilities offered by new

technologies - particularly the computational tools, such as Cabri II Plus, displaying

all creativity and imagination forms to find better ways to take them to the

classroom and use them to promote the integral development of students´s logical

and abstract thinking.

● Cabri Software.

Cabri was taken as a mediation tool allowing also observe students' difficulties

when they analyze the different representations of the function.

"Some of the detected mistakes happen because of the confusion between different representations, due to the characteristics transfer of one representation to another. The most relevant confusions are: graphic-design, graphic verbal, interval-point visual confusion and confusion caused by personal experience, that certainly can act simultaneously making it difficult to interpret or graph construction ". (Fabra Lasalvia & Deulofeu Piquet, 2000)

The possibility to move figures (dragging) maintaining the same structural relationships, is a manipulation form, the execution of computer representations, which contributes to the realism of those geometric objects.… (Armella, 2000)


According to (Torregrosa & Quesada, 2007) in his analysis of the theoretical model

proposed by (Duval, 1998) where refers to the basic problems of the geometry

teaching, and which defines the geometric activity involves three cognitive

processes: visualization, reasoning and construction. Keep in mind that these

processes should be developed separately, also proposed to recognize the different

processes of both reasoning and visualization as evidence that the coordination

between visualization and reasoning can occur only after this work of

differentiation.

Visualization:

Visual identification and 2D and 3D configurations.

Apprehension mode.

Reasoning:

Natural language to name, describe or argue.

Definitions, theorems for deductive organization

Construcción:

Ruler and Compass

Geometric Software

Figure 1. Software Environment geometry.

Figure 2. Cognitive Processes, Duval.


The demonstration on dynamic geometric according Duval:

It has a traditional approach to geometry

• Creating doubts about the validity of the empirical observations.

• Attempting to create a need for deductive demonstration.

Dynamic tools.

• Produce sereval settings easily and quickly.

• No need for greater conviction / verification.

One of the most important aspects that he rescues (Duval, 1998) es que estas herramientas permiten enriquecer y potenciar las tareas de argumentación entre los alumnos, que habitualmente ocupan un lugar menor en la clase, debido a que básicamente la intervención en clase del docente se puede limitar a aceptar verdadero dichos argumentos y validarlos.

In research carried out on "Training and Research on the use of Information Technology and Communication in Mathematics for ESO and Highschool". A General Direction of Academic Organization of the Ministry of Education of the Madrid Comunity Project Directorate and the University Institute of Education Sciences of the Autonomous University of Madrid and later expanded in Andalucia, Castilla y León and Castilla La Mancha.

Researchers José María Arias Saenz Cabezas and Ildefonso Maza have concluded that improves student performance in math by 25% on average between Cabrí, Derive, Excel and Internet. (Being Cabrí percentage superior to other programs, with an estimated 30%) (Laborde, 2006)

According (Villiers, 1999) such computer programs were designed with the specific intent to offer to students with a microworld environment for the experimental exploration and to have a dynamic geometry software which encourage: Verification of true conjecture and the false conjectures counterexamples.

GEOMETRIC APPLICATIONS LABORATORY.

The first module for use:

If support is placed at the light midpoint which form the end supports of a beam,

the acting loads on this beam are distributed more evenly. For this was applied

Construction and the "bisector of a segment of" concept. The arcs are a structural

shape that fits greater lights, and allows the application of the moment’s law for

the gravitational loads distribution. For this was applied a circumference

construction by three points, and the arc subtended at the circumference support

points.


When radial voltages are applied which bond the arch truss together with the

rolling surface, the charge located at the light center point is distributed in the

truss and is concentrated at the nodes of such an item. For this construction was

applied slope of the line to the bow, in the nip a perpendicular was drawn and is

illustrated precisely the points at which they should have a tension cables.

(Álvarez, 2003)

For the second module:

Generate a relationship with the use of mathematical functions allowed in Cabri II

Plus and dynamic display in this software, all this in combination with elements

and processes of descriptive geometry and graphic expression. (Wellman, 2003)

Figure 3. Geometric Model Bridge in Cabri II

Figure 4. Modeling Chemical Concentrations application, Cabri II


RESULTS.

Data collected description, its presentation should be in narrative form, without adding tables or graphs. In the case of research proposal indicates expected results; if corresponds to Ongoing research indicates to partial results, if corresponds to completed Research indicates outcomes.

For the first module, the following results were generated

1. Three constructs are postulated whose geometric shape features contain structural concepts applied and proven by physics. 2. Was defined in the Cabri software designing an arch bridge, which has benefits in their model dynamics and exclusive application of geometric concepts. 3. We confirmed the functionality of the methodology testing the model in the X CONCURSO DE ESTRUCTURAS EIA to take first place in its category.

For module two:

1. Postulated geometric constructions whose characteristics simulate the behavior of the variation of the chemical concentration in a tank. 2. It is posible to display the fundamental theorem of calculus. 3. Intuitively its built difference concept centered numerical differentiation.

CONCLUSIONS OF EXPERIENCES.

For the first module:

The geometry study of a structural element optimizes dynamic tools based on the ability of its analysis and the approach of possible structural hypotheses.

A fundamental parameter of any structure is its shape, and optimize its functionality without having to do with the type of material used.

It is possible to check experimentally that Euclidean geometry concepts applied to physical behavior topic can define a structure.

For the second module:

The geometry and calculation study through dynamic software facilitates theoretical understanding by students, and is a potent instigator to creativity when it comes to proposing solutions. The geometrical thinking development involved while developing skills modeling and interpretation of specific realities.


RECOMMENDATIONS.

In this investigative process, concerns remain unresolved, and are the daily

teaching and learning process, thats why it is necessary to give continuity to this

ideas laboratory in the academic exercise, in the University of Medellin is from the

monitoring practice given by students for students, in order to find answers to the

following question.

¿Does the study of geometry prepares for life?

Picture 1. Prototype Arch Bridge in greater efficiency. By Fernando Moreno

Photo 2. Training students, College Camilo Mora. By Fernando Moreno


BIBLIOGRAPHY

Torregrosa, G., & Quesada, H. (2007). Coordination of cognitive processes in geometry.

Revista latinoamericana de investigación en matemática educativa.

Álvarez, E. (2003). Elementos de Geometría. Medellin.: Universidad de Medellin.

Armella, l. M. (2000). Instrumentos Matemáticos Computacionales. Recuperado el 11 de

agosto de 2013, de REDuteka: http://www.eduteka.org/Tema3.php

Castro, M. (1991). La educación en la era de la informática: qué da resultado y qué no /

coord. ISBN 1-886938-48-2, 30-42.

Duval, R. (1998). Geometry from a cognitive point of wiew. Registres sémiotiques et

apprentissages itellectuels.

Fabra Lasalvia, M., & Deulofeu Piquet, J. (2 de julio de 2000). Construcción de gráficos de

funciones: “Continuidad y prototipos”. Revista Latinoamericana de Investigacion en

Matemática Educativa., 207-230.

Laborde, J. M. (2006). Explorar Nuevas Dimensiones. Nuevas Dimensiones. Bogotá.

Saidon, L. (s.f.). Centro Babbage.

Saidon, L. (1998). En “Haciendo Geometría I”. Centro Babbage.

Villiers, M. d. (1999). Algunos desarrollos en enseñanza de la geometria (3).

Wellman, L. (2003). Aplicaciones de Vectores en el espacio. En Geometría Descriptiva (pág.

Pag336.).

dynamic software for geometry and mathematics

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