Effect of Using Cabri II Environment by Prospective Teachers on Fractal Geometry Problem Posing

Transcription

1 Effect of Using Cabri II Environment by Prospective Teachers on Fractal Geometry Problem Posing Reda Abu-Elwan, PhD Mathematics Education Sultan Qaboos University Abstract The use of Cabri II in mathematics education has considerably spread in the last few years. Nevertheless, many teachers haven t yet completely overcome their fears and suspicions about using it in geometry teaching. Furthermore, there is a generational gap which raises further difficulties: while most of the in-service teachers in Oman don t have confidence in the new technologies or with their application in educational activities. This study investigates how Cabri II might be effective in developing prospective teacher's skills in developing new fractal problems for school geometry level. The research was designed to introduce prospective mathematics teachers into a learning experience with a Dynamic Geometry Environment (Cabri II); making them work in small groups on developing fractals problems based on Cabri II dynamic geometry. Twenty of mathematics prospective teachers participated in six activity sessions in topics of Circles, triangles fractals problem posing. The experience showed that the participants, after getting used to Cabri II, were able to apply their competence in the construction of interactive educational materials for a classroom situation. All of the creative fractals shapes in proposed problems and images were developed by the participants, for that purpose, the Cabri II was used. Introduction Technology in mathematics has become an important factor in mathematics teachers' preparation programs. Research indicates that teachers who are effective at integrating technology possess specialized knowledge about technology, pedagogy and content, and the intersections of those three domains (Koehler, Mishra, & Yahwa, 2007; Suharwoto & Lee, 2005). Educational technology researchers have advanced the construct of technological, pedagogical and content knowledge (Mishra & Koehler, 2006) as a theoretical and empirically-based framework about teachers knowledge needed to effectively integrate technology in their teaching. In Sultan Qaboos University; Technology in mathematics and teaching considered to be a main dimension of Mathematics teacher's preparation program. Part of Methods of teaching secondary mathematics course is to use Cabri II environment for the learning of teaching geometry and Fractal geometry is a new topic in the same course. Fractal geometry is a new language for the complex forms and patterns found in nature. It represents a change in the way that scientists "do science". It provides new tools to describe, model, analyze, and measure the natural world, and wonderful new connections within the world of mathematics. Fractal geometry is exciting, visual, relevant to many disciplines, and lends itself naturally to technology supported activities. Both students and teachers with relatively little mathematical background can approach a large number of current research problems in this area and appreciate the integration across traditional disciplines. Students enrolled of this course should create new fractal problems using Cabri II environment as a main project. This study showed that students have the abilities to create new fractals problems based on Euclidean geometry and Cabri II software. Background Geometry has always been a rich area in which students can discover patterns and formulate conjectures. The use of Cabri II environment enables students to examine many cases, thus extending their ability to formulate and explore conjectures (NCTM, 2000; 309) as well as developing new questions for a given geometry problem. In Principles and Standards for School Mathematics (NCTM, 2000), NCTM presents its technology principle (pp ). The technology principle has three components: 1) technology enhance mathematics learning, 2) technology supports effective mathematics learning, 3) technology influences 1

2 what mathematics is taught. The range of accessible problems is extended by technological tools as students are able to execute routine procedures quickly and accurately, allowing additional time for conceptualizing and modeling. Learning is assisted by feedback, which is supplied immediately by Cabri II. Problem posing in mathematics "Problem posing and problem solving is obviously closely related. On the one hand, problem posing draws heavily on the processes of problem solving, such as identifying the key elements of a problem and how they relate to one another and to the goal of the problem. On the other hand, problem posing takes children beyond the parameters of the solution process" (English, 1997, 173). Problem posing instruction involves instruction with student generation or formulation of problems to solve (Silver, 1994). It involves the creation of original problems which may be associated with particular conditions. This study dealt only with the generation of new problems. The following questions may be asked to guide additional inquiries into fractal construction: What other geometric process lead to interesting fractals? Can inquiry into the patterns of any generated fractal relate to your fractal? Will a fractal be formed if the dimensions of the construction matrix are changed? Could you have used these ideas in a different way to solve the problem? How might you change some of these ideas to make a different problem? What if you not given all these ideas? What might the problem become then? What if we were adding some new ideas? What ideas might we add? What new questions might we ask then? Fractal geometry problems Fractal geometry is a relatively new and important area of mathematics which differs significantly from traditional geometry. Fractals have many applications in a variety of fields of study aside from mathematics, such as art, engineering, physics and computer science. Unlike classical geometric shapes, which are linear or continuous curves (straight lines, triangles, smooth curves and circles) alone do not represent the world in which we live; the shapes of fractal geometry (clouds, trees, mountains, coastlines and snowflakes) are nonlinear. Fractals are more representative of the natural world we live because they appear to exist within it. However fractal geometry does not appear in many traditional curricula even in Omani mathematics curricula. Teaching Fractal Geometry is consistent with implementation of the guidelines and the recommendations of The National Council of Teachers of Mathematics, NCTM. In the process of creating fractals students will be able to examine the following geometric concepts and skills recommended by NCTM (1989): For grades K-4, topics in geometry and spatial sense such as describing, modeling, drawing, classifying, combining, dividing, and changing shapes. For grades 5-8, identifying and comparing geometric figures in one, two, and three dimensions and the applications of geometric properties and relationships in problem solving and real life situations. For grades 9-12, such topics in geometry as representing problem situations with geometric models, classifying figures in terms of congruence and similarity. Fractals can be divided into two categories: Natural fractals and mathematically structured fractals. Natural Fractals One does not have to look very far to find examples of natural fractals in the everyday environment of students. Examples of natural fractals include coastlines and curvy rivers (as illustrated on wall maps), flowers such as a rose or carnation, tree branches, rock formations, mountain ranges, seaweed and other aquatic plants, coral, and parts of the human anatomy such as curly hair, veins, and intestines. Mathematically Structured Fractals Included in this category is simulations of fractal patterns that are computer generated; which are a class of fractals created consistent with some mathematical rules and principles such as Cantor Dust, Mandelbrot Set, Sierpinski Triangle, and Koch Curve. This study dealt with Mathematically Structured Fractals as an environment of problem posing. 2

3 These fractals are generated by iteration of an event or shape repeatedly. Popular examples of these simulated fractals are the Mandelbrot set, Koch snowflake, and Julia sets. These and other fractals are available through different sites on the World Wide Web (Ex. & ). Mandelbort was the first to uncover the beauty of the computer-generated images. This can only be seen because of computer technology, since these images are the result of millions of iterations. Within a study of fractal geometry, students can make conjectures about relationships between figures or number patterns and they can then form their own generalizations and problems. This study is focused on that structured fractals created through Cabri II environment. Cabri II Environment National Council of Teachers of Mathematics (NCTM) suggests, in Principals and Standards for School Mathematics, that interactive Geometry and Geometer's Sketchpad. This dynamic geometry software offer opportunities for the users to manipulate and, precisely speaking, to act directly on, geometrical diagrams particularly by grapping and dragging certain geometrical objects (e.g. points) with the mouse. These new opportunities of direct access to and interactions with geometric diagrams open a new opportunities of experimentation (Balacheff& Kaput, 1996). one aspect of the well powerful of dynamic geometry is that in the new experimental field it granted, the geometric drawings, as opposed to figures by virtue of the distinction made by, Laborde (1993), preserve the invariant properties salient to the geometric configurations. Whilst grabbing and dragging the geometric objects ( e.g. a point that in turn changes the shape of a triangle). The "dragging" facilitates the reasoning process in helping student to move backward and forward between particular instances of geometric relations and general theories about invariant relationships. Holzl said that "The drage mode alerts the relational character of geometric objects" (Holzl, 1996, p. 171), for example; if one constructs an equilateral triangle ABC in which points A and B are given, then C cannot be dragged whereas A and B can. From a relational point of view there is no need to distinguish the points A, B and C, as each pair of them determines the original equilateral triangle. From a functional viewpoint (essentially Cabri) the situation looks different: A and B determine the position of C but in return C dose not determine the position of A and B. As Whiteley (2000) spoke about his reflection on his experiences in both studying and teaching geometry, the whole point about using dynamic geometry programs is learning to see differently and therefore think differently. Cabri presents the learner with two worlds (Sutherland& Balasheff, 1999): a theoretical world, which is that of geometry, and a mechanical, manipulative world, which is the phenomenological domain of Cabri. Cabri objects are part of a computer environment which can be considered "half world" (Noss & Hoyles, 1996; 6) in between theory and practice. With this respect Cabri figures are a midway between empirical and genetic objects. On one hand, as empirical objects they can be manipulated and the effect of this manipulation can be seen on the screen as it happens. On the other hand, dragging figures in Cabri allows one "to see the one as a multitude, other than one among others" (Pimm, 1995; 59). The researcher has observed in his own courses the power of dynamic geometry (Cabri II) to explore many examples and help student-teachers makes generalizations in geometry. The power of dragging is an important factor to generate new conditions of a figure on a screen, in addition to other features of Cabri II. Objectives and Research Questions Research question of this study is: "Does experience with Cabri II environment enhance student-teachers' abilities to formulate new fractal geometry problems?" In this study I define "formulate" as "to generate a new questions based on a fractal geometry figures developed by student-teachers". Instruments: To investigate student-teachers abilities of using Cabri II environment in problem posing, I use "Task Analysis" as a tool for that purpose; it showed how dose student could construct fractal geometry figure and what kind of formulated problem he developed. 3

4 To investigate student-teachers skills of problem posing in fractals, I use "Interview" with each group of them, asking them about their work (procedures, the role of dragging to construct new situation for a fractal figure, and what they can do to modify ill formulated problem?. Method Several educational tasks were designed and final generated problems have been analyzed regarding the quality of well problem posing skills criteria. Objectives of the educational tasks were: To construct a fractal figure, based on fractals properties of fractal dimensions and self similarity. To develop new questions for that constructed fractal figure, To explain how Cabri II would help to solve the posed problem. Subjects and contexts The subjects were 20 student-teachers enrolled of methods of teaching mathematics course, each two student-teachers working together to create new fractal figure based on their knowledge of: Fractal geometry principles, their skills of using Cabri II software utilities, and problem posing strategies. Working in cooperative groups allows students to check their results with other group members. After the students finished their individual fractals, they can collect data about the patterns they observe. They can use these data to make conjectures and test them in their group before presenting their fractals to the whole class. Initially, the teacher s role is to explain how to construct fractals. Training session of using Cabri II in 2 hours a week of 4 times, while Problem Posing Strategies and examples was 4 hours, Fractal geometry topics taught in 6 hours. All teaching process and training has been done by researcher. The phase of constructing new fractal geometry problems using Cabri II has been done in the final stage of the course. All problems about fractal geometry necessarily imply designing and creating macros. They stimulate pupils algorithmically thinking and they can be a basis for beginning programming activities as well. All these problems involve pupils actively in the phase of the construction of the fractal, allowing them to invent new ones by themselves. Drawing fractals on a computer has several advantages over drawing fractals by hand: (1) students can create more fractals in the allotted time, and (2) correcting mistakes is easier and less frustrating. Actually, students rarely make mistakes when they construct fractals with a computer. Technology allows them to copy stages without much backtracking, whereas when they manually build a fractal with transformations, copying requires going through all the steps from stage 1 on. Utilizing the Euclidean rules of construction such as angle bisectors, parallel and perpendicular lines, midpoints, and perpendicular bisectors the students examined the properties and construction of some of the elementary figures of fractal geometry. Accepted fractal problem criteria were: Using Cabri II environment, context of the formulated problem should be related to Fractal geometry topics, and new generated problems are solvable. Results and Discussion The following tasks showed some of the problems of the basic construction that studentteachers have drawn using Cabri II: Task #1: Students were asked to use circles properties to construct "fractal Carpet". They started drawing a single circle with specific radius, then using Cabri II environment to connect three of that circle tangent, using circle center of the three tangent circles to construct the main unit as in figure (3), coloring is option as in fig. (5), using millions of iterations to construct fig. (7). Generated questions like: Is it possible to complete Carpet in fig. 7? Led to formulate a problem done by "Ali and Murshed":" based of the given initial circle, could you expect 4

5 numbers of circles needed to complete the 10 th stage of the figure, General formula for n circles?" This task showed that both Ali and Murshed tried to generate a figure of fractal carpet using the properties of circles, and iteration geometry for a basic generator of one single circle, as I asked them; about fig. 6? They said that constructing new question to complete the square carpet need a new conditions, that Cabri allowed them to use it. (5) (4) (3) (2) (1) (6) (7) Task#2: (8) This task done in fig. 8 by Muna and Badria, they tried to construct an iterated figure using equal sides square, dividing each side in 1:2 ratio, then iterate that some stages, formulating a question related to generalization. But they formulate another question "what if we divide sides in a ratio of 2:3, does the generalization will differ?" In questioning them about fractal properties in that figure, they failed to prove the existing of similarity on the screen, it considered a good idea, but not fractal geometry problem posing. Task#3: This task in fig. 9 done by Hamad, showed his tray to construct a fractal triangle, using the idea of iteration of a two triangle with one diamond, this unit was the basic of the whole triangle in fig. 10. (9) 5

6 Hammad generate a problem based of that figure: "what if the initial triangle would be right triangle how could be the 5 th stage of constructed triangle?" another question based of the original figure (10) was" use dragging properties to formulate other figures by same fractal pattern. (10) The context of fractal geometry provided an environment which encouraged problem posing. The role of Cabri II in the problem posing skills is highlighted. It is a tool for extending investigations. Exploring figures with Cabri is effective in encouraging creativity. Students were very engaged with geometric construction of the fractals utilizing the principles and rules of Euclidian construction, as well as using Cabri dynamic geometry software. Also, several students went beyond what was required and created their own fractals. References Balacheff, N. & Kaputt, J. J. (1996). Computer-based learning environments in mathematics In A. J. Bishop, K. Clements, C. Keitel, J. Kilpatrick & C. Laborde (Eds.), International handbook of mathematics education: part one (pp ). Dordresht: Kluwer Academic Publshers. English, Lyne D. (1997). Promoting a Problem Posing Classroom, Teaching Children Mathematics, November, p.173 Holzl, R. (1996). How dose "dragging" affect the learning of geometry. International Journal of Computers for Mathematical Learning, 1, pp Koehler, M.J., Mishra, P., & Yahya, K. (2007). Tracing the development of teacher knowledge in a design seminar: Integrating content, pedagogy and technology. Computers & Education, 49, Laborde, C. (1993). The computer as a part of the learning environment: The case of geometry. In C. Keitel & K. Ruthven (Eds.), Learning through computers: Mathematics and educational technology, Berline, pp Mishra, P. & Koehler, M.J. (2006) Technological Pedagogical Content Knowledge: A framework for teacher knowledge. Teachers College Record, 108(6), National Council of Teachers of Mathematics (1989). Curriculum and Evaluation Standards for School Mathematics. Reston, Virginia: The National Council of Teachers of Mathematics Inc. National Council of Teachers of Mathematics (2000). Principles and Standards for School Mathematics. Reston, Virginia: The National Council of Teachers of Mathematics Inc. Pimm, D. (1995). Sympols and meanings in school mathematics. Prenceton: NJ university press. Polly, D. & Brantley-Dias, L. (2009). TPACK: Where do we go now? Tech Trends, 53(5), Silver,E. A. (1994). On mathematical problem posing. For the learning of mathematics. 14 (1), pp Suharwoto, G. & Lee, K. (2005). Assembling the pieces together: What are the most influential components in mathematics preservice teachers development of technology pedagogical content knowledge (TPCK)? In C. Crawford, et al (Eds.), Proceedings of Society for Information Technology and Teacher Education International Conference Chesapeake, VA: AACE Sutherland, R. & Balasheff, N. (1999). Didactical complexity of computational environments for the learning of mathematics. International Journal of Computers for Mathematical Learning. V. 4, pp Whiteley, s W. (2000). Dynamic geometry program and the practice of geometry. Paper distributed st the Ninth International Congress on Mathematical Education (ICME9), 31 July- 7 August, Tokyo. 6