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Yuriko Yamamoto Baldin
yuriko@dm.ufscar.br
Department of Mathematics
Universidade Federal de So Carlos
BRAZIL
Abstract: The main objective of this paper is to present considerations about the necessary preparation of teachers who aim to use dynamic geometry software as didactical tool in spatial geometry lessons, after getting familiar with diverse teaching approaches with technology in plane geometry classes. The natural impulse to extend the teaching methodologies of two dimensional setting needs to be reformulated since the instrumental characteristic of three dimensional setting requires from the teacher the knowledge beyond the subject of lessons, at elementary school level. The paper will discuss the issues based on the concept of pedagogical content knowledge of Schulman.
Keywords: Teaching geometry; teaching with technology; pedagogical content knowledge; teacher preparation.
INTRODUCTION
The use of dynamic geometry software (DGS) as didactical tool has permitted teachers and researchers to study different teaching/learning methodologies, including those that improve the communication of mathematical concepts and ideas in traditional classes as well as that develop studentcentered activities. According to (Unesco, 2002, p.34), technology can be used to support traditional forms of learning as well as to transform learning.the brightest promise of technology in education is as a support for new innovative, and creative forms of teaching and learning.
Especially important are the technologybased activities that allow the students to learn mathematics by experimentation (BorweinBailey, 2003). In particular, the geometry classes take significant advantages with the use of DGS to develop enriched teaching/learning environments, because the instrumental characteristics of most of such software are appropriate to accomplish the internalization process of incorporating mathematical knowledge. The research done in (Mariotti, 2002) about the notion of geometric construction by using the external tools, drag mode and history command of Cabri, confirmed that The availability of an external tool referring to the procedure of the construction in its temporal sequence very often contributed to the production of a description and a correct justification of a construction(Laborde, 2002, p.29). The role of computer programs as mediators of students knowledge in learning mathematics is discussed in (Laborde, ibid) and the references therein. It is well stressed that the simple contact with technology is not sufficient, and the design of tasks proposed to students is critical in order to foster a conceptualization process. Therefore it is very important to introduce prospective and inservice teachers to the instrumental dimension of the use of technology, making them aware of the critical role of interface features (Laborde, ibid, p.36)
It is a challenge for teachers, either prospective or inservice, to choose and to design appropriate technologybased activities for their classes, implying necessary reflection about the teacher preparation courses. The knowledge required to teach with technology goes beyond the specific content of the subject of the lessons at basic school level. It demands an integration of the subject and the specific mathematical knowledge of the instrumental dimension of the used technology (Baldin, 2002). Moreover, either with or without the technology, the right approach (teaching methodology) to mathematical knowledge in learning environment requires from the teacher the understanding of the context. The concept of pedagogical content knowledge, introduced in (Schulman, 1986), turns out to be a framework to understand the intertwining of the key ingredients of the formation of teachers prepared to use the technology as asset in planning the lessons. In (VealMaKinster) the authors say The pedagogical content knowledge is formed by the synthesis of three knowledge bases: subject matter knowledge, pedagogical knowledge and knowledge of context. According to (Schulman, ibid, p.9), the pedagogical content knowledge include an understanding of how particular topics, problems, or issues are organized, presented, and adapted to the diverse interests and abilities of learners, and presented for instruction. Therefore, when using technology, the teacher must have the ability to distinguish the instrumental characteristics among the available software, in order to assure the adequate choice of teaching methodology and to provide an efficient learning environment. This consideration demands a careful preparation of teachers that can perceive the context of the use of technology and the learning stage of their classes.
In the context of teaching geometry with the use of DGS, we raise the following issues:
What are the main characteristics of CabriGeometry that facilitates the learning environment for plane geometry?
What are the characteristics of Cabri 3D that prevent the extension to spatial geometry of methodologies used in plane geometry? How Cabri 3D can be used in most effective way?
The objective of this paper is to discuss some differences between CabriGeometry and Cabri 3D that clarify the different methodologies of teaching geometry in basic school classes, and give example of design of spatial geometry lessons that take advantage of three dimensional DGS.
THE TRANSITION FROM CABRIGEOMETRY TO CABRI 3D.
As mentioned before, the instrumental power of Cabrigeometry at geometric constructions becomes evident through its functions as construction tools, the drag mode that stimulates the discovery by experimentation and the history of commands that permits to keep track of the paths of thinking and actions. Such characteristics suggest that Cabrigeometry can contribute to create a meaningful learning environment. According to (Unesco, 2002, pp 2324), the view of learning process based on the studentcentered learning environment has emerged from cognitive learning research, with confluence of several learning theories, which give support to the student as active user of technology in the process of construction of knowledge.
In the case of Cabrigeometry, a DGS suited for plane geometry, the instrumental difficulty is lessened through careful design of activities that could introduce simultaneously the geometric objects and the function tools of the software. In other words, the interface between the student as user and the computer can be part of the process of geometric abstraction, because the correspondence of figures of plane geometry and the drawn ones in the screen is exact. The connection from concrete models (paper pieces, wood pieces, geoplane, etc) to their representation in a computer screen is straightforward, that is, what you see in the object you touch is the figure you see on the screen. Other feature is the dragging mode that can modify the attitude of teachers and students towards school geometry. It introduces the element of test and validation in the construction, of modification of constructed figures to explore invariants and properties, of being auxiliary in conjecturing loci and paths of moving objects, of visualization of simulations in geometrically modelled problems, among others. This facility can be introduced at early stages of educational schemes, working with elementary objects such as points, lines, circles, triangles, polygons, etc. Therefore, teacher and student can experiment the instrumental dimension of 2dimensional DGS at the same time of the construction of first concepts of the plane geometry and of the exercise of abstraction. The most productive design of lessons is based on problemsolving in which the students are challenged to experiment and conjecture strategies and results.
The planning of spatial geometry lessons would have, at first, the expectation of adopting similar approaches of plane geometry. For example, such approaches could be the use of dialogue between the function tools and geometric objects to build mathematical concepts, or the use of drag mode to discover geometric properties. These approaches point to the construction of studentcentered learning environment and to the learning by experimentation.
However, the first difficulty of teacher is to interpret the didactical role of Cabri 3D. The natural connection between concrete models and the geometric objects on the computer screen does not exist in threedimensional setting. The use of three dimensional DGS requires the ability to interpret the projected images on a two dimensional screen. This demands an extra abstraction of mind to reconstruct the real object from the plane images produced by different projections. Therefore, the software can not be used to introduce the study of spatial geometry in a studentcentered environment. Neither can be used by a teacher without background knowledge of the theory of projections, a topic out of curriculum of basic schools, yet necessary to a user of the software.
The instrumental difficulty of Cabri 3D is an actual handicap to extend the successful teaching approaches with DGS of plane geometry to spatial geometry. Taking advantages of three dimensional DGS challenges the teacher to plan other classroom strategies.
In the next section, we describe a teaching experience that has overcome somehow the problem raised above, showing that the pedagogical content knowledge can make differences in successful teaching.
2 TEACHING EXPERIENCES.
A teaching experience, aimed at the pedagogical integration of context and technology, has been carried out during the school year of 2007 at a public school in Brazil, with 80 7th graders. The use of drawing tools (straightedge and compass) in classroom activities together with laboratory type construction tasks has shown to be very motivating for students that could compare both results, taking the advantage of history command of Cabri retrieved in paperpencil constructions. The lessons on geometric constructions, without measuring, have had impact on students understanding about logical reasoning and mathematics. The student could follow the correct order of construction paths through the history command of Cabri, and the teacher has succeeded in introducing abstract topics as theorems and their proofs, when accessible, in expository type classes. The integration of specific content knowledge with computerized learning environment resulted in a better teacherstudent communication and assessment tasks. In general, students become more interested in content knowledge when they can construct it and understand it as active user of technology. The use of drawing tools is a handson activity that enhances the connection between the abstract concepts of mathematics and the visualized geometric objects. The measuringfree mathematics has surprised the students about the possibility of doing mathematics without calculations (algebra). The methodology has influenced also on disciplinary attitudes of some students.
The geometry lessons for 40 6th graders were about concepts of convex and nonconvex polyhedrons. The possibility of using Cabri 3D has motivated the teacher to design lessons that could take the advantage of the software, considering the pedagogical adequacy of activities.
The first lesson was conducted with traditional expository approach, in which the teacher drew a cube and some other solids on the board. This has evidenced the above mentioned difficulty of students about the interpretation of projected images of three dimensional objects on a plane (board, notebook).
On the other hand, the Cabri 3D has interesting planification feature of solids and their plane sections. That makes possible the confection of different and nonconventional models.
The design of geometry lessons had three distinct aspects:
Traditional expository lessons with the presentation and the development of basic elements of spatial geometry;
Manipulation of concrete models of geometric solids, with the objective of getting the students familiar with actual objects and capable of interpret different views in space and their representation on a plane (board or notebook);
Use of Cabri3D by teacher to empower the possibilities of different solids and to help students to understand the concepts developed in traditional expositions.
In the making of concrete models, materials like cardboard, straws and wires have been used, besides the models produced by Cabri3D upon solids studied in expository lessons. The teacher has used a big TV connected to a PC to project the figures in the classroom. Although the technology was not actually used by the student its use in appropriate instances has produced impact on students understanding about the projected representation of solids, seen in textbooks and other illustrations. The development of the nontrivial concept of convexity of solids has been facilitated by the combined use of software and concrete models. The subsequent activities have included the study of diverse plane sections on standard and nonstandard solids and the exercise of abstracting the visualization of some of these sections. The use of software has facilitated the introduction to other topics of spatial geometry such as pyramids, prisms, cylinders, cones and spheres. The teaching approach that provided the contact with concrete models before the visualization of projected images was appropriate to overcome the instrumental difficulty of software.
CONCLUDING REMARKS
The outcome of teaching experience allowed a better understanding of the meaning of the concept of pedagogical content knowledge and its importance in the preparation of teachers. The careful reflection on the design of different teaching strategies adopted for plane and spatial geometry lessons could not be done without a previous research by the teacher about the foundations of geometry and the theory of projections. The importance of bringing the students interest in the first place of the planning made it possible to understand the most effective use of technology in the learning environment. The analysis of assessment of studentsachievement will be considered in future work.
REFERENCES
Baldin, Y.Y. (2002). On some important aspects in preparing teachers to teach with technology. Proceedings of ICTMT2, Crete, Greece.
Borwein, J., Bailey, D. (2003) Mathematics by Experiment: Plausible Reasoning in the 21st Century, Natick, MA, A K Peters.
Laborde, C. (2003). Technology used as a Tool for Mediating Knowledge in the Teaching of Mathematics: the case of Cabrigeometry. Proceedings of the 8th ATCM, Chung Hua University. Hsinchu, Taiwan, R.O.C
Mariotti, A. (2002). Technological advances in mathematics learning. Handbook of International Research in Mathematics Education. Lynn English (ed) Mahwah NJ: Lawrence Erlbaum.
Schulman, L.S. (1986). Those Who Understand Knowledge Growth in Teaching. Educational Researcher, 15 (2), 414.
Veal, W.R., MaKinster,J.G.. Pedagogical Content Knowledge Taxonomies, //wolfweb.unr.edu/homepage/crowther/ejse/vealmak.html
UNESCO (2002). Information and communication technologies in teacher education, a planning guide. Division of Higher Education, Unesco, Paris, France.
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