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Nov 16, 2015
EduMath 19 (12/2004)
Reconstructive Modelling inside Dynamic Geometry Systems In honour of Jean-Marie LABORDE, the creator of Cabri Gomtre, on the occasion of his 60th birthday.
Prof. Dr. Heinz Schumann Faculty III, Mathematics/Informatics, University of Education Weingarten
1. Introduction It is undisputed that non-mathematical applications, including
mathematical modelling, in the mathematics classroom of general schools are an important argument in legitimating the teaching of mathematics.
For example, Corbitt and Edwards, in the NCTM Year-book 1979 Applications in school mathematics, describe the mathematical modelling process as follows: The mathematical-modelling process can be summarized schematically, as shown. It involves (1) the formulation of a real-world problem in mathematical terms, that is, the construction of a mathematical model; (2) the analysis or solution of the resulting mathematical problem; and (3) the interpretation of the mathematical results in the context of the real-world problem. (Diagram 1)
Diagram 1 (Mathematical modelling process)
The list of mathematical competencies (OECD 1999, p. 43) of the PISA
Framework includes modelling, which is defined in more detail in Modelling skill :
This includes structuring the field or situation to be modelled; mathematising (translating reality into mathematical structures); de-mathematising (interpreting mathematical models in terms of reality); working with a mathematical model; validating the model; reflecting, analyzing and offering a critique of a model and its results; communicating about the model and its results (including the limitations of such results); and monitoring and controlling the modelling process.
Geometrical modelling is a wide and interesting application of modelling, and elementary geometry is a powerful and even interdisciplinary modelling tool (see, e.g. Cummins, J. et al. Geometry: Concepts and Applications, 2001).
The traditional media used for geometrical modelling are supplemented today by computer graphics tools (Diagram 2). In our opinion, only directly manipulative computer graphics tools like Dynamic Geometry Systems (DGS) should be recommended for secondary grade pupils.
Physical, materials like paper, cardboard, wood,
metal, string etc. for creating 2- and 3-dimensional geometrical models
Traditional tools for drawing, measuring and calculating for geometrical modelling using
Computer graphics tools particularly DGS for creating
2- and 3-dimensional geometrical models
Diagram 2 (media-specific modelling)
Interface problems (see arrows in the diagram) between media-specific modelling options, in particular between computer-assisted and traditional
EduMath 19 (12/2004)
modelling in the geometry classroom, will not be considered here. (It is conceivable, as a contribution to media training in the geometry classroom, to introduce the currently available modelling media and to analyze their advantages and shortcomings.)
When using media-specific modelling aids and modelling tools, the following skill is required (OECD 1999, p. 49):
Aids and tools skill. This includes knowing about, and being able to make use of, various aids and tools (including information technology tools) that may assist mathematical activity as well as knowing about the limitations of such aids and tools.
Traditional media have the advantages of offering an integrative manner of perception and of training the coordination of visual and tactile activities, which is becoming increasingly important in view of the many hours children spend watching TV or playing with the computer mouse and keyboard. On the other hand, traditional modelling is limited in terms of the complexity, authenticity, up-dating, versatility, portability, reproduction, distribution, ease of communication and administration of the models created: the options for self-correcting; and the power of the modelling tools. Of course, sufficient skill in using the computer technology and computer tools is required if the advantages of the computer in the development and use of virtual geometrical models are to be fully utilized.
We are going to develop a method for geometrical modelling on the basis of DGS which intends to strengthen the part of geometrical construction in modelling (Schumann 2003). In our opinion, this application of the computer is an example of its effective integration in geometry teaching. DGS are powerful tools for modelling what can be interpreted as two-dimensional elements of our everyday environment. Technically, tools must be capable of importing image files of such elements into DGS. The imported images can then be reconstructed by modelling making use of the adequate characteristics of DGS, which offer far wider options than conventional modelling tools. Solutions to authentic modelling problems consist of the reconstruction of the geometry of elements found in nature or created by man, either intentionally or
unintentionally. As a result of the reconstruction process, we obtain static or dynamic virtual models.
Diagram 3 presents an outline of methods and ways of working with DGS in the context of geometry teaching in lower and middle secondary schools (Schumann 2001); modelling in DGS is supported by all other methods and options listed in the diagram.
Dynamic Geometry Systems
(Cabri II+, Cinderella 2, Geometers Sketchpad etc.)
Exploring Modular working
Diagram 3 (methods and ways of working with DGS)
When using DGS in geometrical modelling, we strive for the following general teaching goals in geometry:
Training of geometrical perception, training the geometrical eye (teaching goal concerning the phenomenology of perception)
Appreciating the usefulness of geometry (affective teaching goal)
Applying and enhancing knowledge in geometry: concepts, statements and methods (Cognitive teaching goal)
Experimental exploration and analysis of phenomena, which can be geometrically modelled (meta-cognitive teaching goal)
Training the use of DGS (technical teaching goal)
The process of modelling is illustrated in Diagram 4.
EduMath 19 (12/2004)
Element of the real-world environment
Image to be modeled inside DGS
Static/Dynamic Model Figural/Kinematic Structure imagined
Image file of real-world element input
RECONSTRUCTING using tools inside DGS
Exploring by drawing and measuring/calculating; figural analysis/ functional analysis
Diagram 4 (modelling process in DGS)
The modelling process, i.e. the creation of static and/or dynamic models, is described below in the form of a school project guideline.
Guideline for creating a static model:
1. Look for an object in your environment, in printed media or on the web which you think can be analyzed and reconstructed using the tools of 2-dimensional geometry.
2. Make a picture of that object with a digital camera, by scanning or by making a digital copy and load the image file into your DGS.
3. Analyze the picture of the object in DGS by drawing, measuring and calculating. Look for geometrical figures and rules. Attempt to find arguments for the rules.
4. Reconstruct the identified figure or configuration using the tools of the DGS.
5. Validate your reconstruction by comparing it with the original image.
6. Publish your verified model, together with a description and the picture of the original structure, e.g. on the web or in a poster.
Guideline for creating a dynamic model:
1. Look for a moving object in your environment, in printed media or on the
web which you think can be analyzed, reconstructed and simulated using the tools of geometry in the plane.
2. Make a picture of that object using a digital camera, a camcorder, by scanning or by making a digital copy, and load the image files or the digitized video into your DGS.
3. Analyze the picture/s of the moving object by drawing and measuring. Look for geometrical figures and rules. Observe the function of the moving object, or get information on its function. Attempt to find arguments for the rules and functions.
4. Reconstruct a functioning model using the tools of DGS.
5. Verify your reconstruction by means of simulations. Check whether the functionality of the reconstruction is sufficiently similar to the functionality of the original.
6. Publish your verified dynamic model, together with a description and the picture of the original object, e.g. on the web.
The description of mathematical modelling in the PISA-Skills applies, in principle, also to reconstructive modelling in DGS.
There is one major limitation to modelling in DGS: Only those objects can be modeled which can be described by the tools of 2-dimensional elementary geometry and which can be reconstructed using the tools and methods of DGS. (Real 3D modelling as a reconstruction method necessitates digital recording of a three-dimensional object or structure. This type of digitalization technology is not available for school geometry, and there are no 3D tools for school geometry