1. Virtual Reality serving Mathematics Education: Dynamical Visualization of Multivariable Functions Yehuda Badihi 1, David Zeitoun 2 Th. Dana Picard 1 1.Jerusalem College of Technology 2. Orot College of Education 1 Supported by the Israel Science Foundation, research grant number 1340/05 2 Supported by a Meital R&D grant

2. 2 Representations of a mathematical object Multiple representations of the same objects The given representation may be static, despite the dynamical nature of the rule being taught Transfers between different representations Complete mastering requires a permanent transfer from one kind of representation to another kind. MEITAL 2010

3. 3 Strange 3D plots: treatment of discontinuities MEITAL 2010

4. 4 Strange 3D plots: Mathematical Differences MEITAL 2010

5. 5 Visualisation of the plots: Polar Coordinates MEITAL 2010

6. 6 Incomplete is better than false

7. MEITAL 2010 7 Right coordinates are not a panacea

8. MEITAL 2010 8 Why to look for another technology? Conventional representations: Use a global viewing and zooming. It does not change the nature of the representation We wish: A dynamic viewing based on flying through given path: – dynamical change of the values of the parameters. – Permanent recalculation of “positions”.

9. MEITAL 2010 9 Virtual Reality – what’s that? Synthetic physical computer-based environment. 3D and Stereoscopic Imaging of the environment as close as possible to real world perception The viewing and interaction with the environment is represented as in the reality (by head and eye movements etc.). Audio surrounded representation of the objects. Basic notions VR requires advanced technology (hardware and software).

10. MEITAL 2010 10 The Main principle In order for the Student to experience a “world picture” of the function, it is necessary for him/her to be placed and exposed to the best representation of the function, including its problematic characteristics (discontinuities, singular points etc.). “Virtual Reality” Technology and Mathematical Imaging

11. MEITAL 2010 11 The VR technology provides a feeling of being inside the picture by effective presence, orientation and temporary detachment from the surroundings outside the scenario. Virtual reality and Immersion

12. MEITAL 2010 12 Path Lines for the visualization of the Discontinuities

13. MEITAL 2010 13 Presence Being located as an integral part of the virtual environment In other words: sensual interaction, both active and passive, with the environment Properties/skills and derivatives

14. MEITAL 2010 14 Orientation Possibility to “feel” a world image –Personal place inside the world –Absolute and relative positioning of objects –Depth perception (3D) –Within the range of eye catching –High resolution => separation of relevant details Properties/skills and derivatives

15. MEITAL 2010 15 Immersion: live experience Reactions to stimuli in virtual world similar to their real world counterpart Disconnection from the stimuli in the real environment The more deeply lively experienced the learning process the more internalized its results (Ausburn and Ausburn, 2004; Barnett et al, 2005). Have "a physical model that you can feel in your hands“ (JCT student, 2008). Properties/skills and derivatives

16. MEITAL 2010 16 Among the principles for virtual imaging of mathematical functions 1.VR perception will enhance the physical meaning of the function 2.The VR imaging provides to the learner tools for a precise, sensual distinction without in-between, of changes in the function behavior 3.VR imaging will faithfully show all the points of the graph of the function’ including those which are hard to visualize with more classical technical means (CAS, plotters, etc.)

17. MEITAL 2010 17 Among the principles for virtual imaging of mathematical functions 4. Audio Modality: complements visual effects 5. The color resolution has a low level of influence on the confidence Possible exception: if color perception and understanding is an integral part of the requested skills

18. MEITAL 2010 18 Among the principles for virtual imaging of mathematical functions 6. Presence of an external observer Part of the virtual environment “hidden” communication 7. The added value of a stereophonic experience is greater than the stereoscopic visual experience Sound added to the dynamical presentation of movement 8. The usage of VR should slow down the extinction of the learnt material

19. MEITAL 2010 19 Basic configuration: presentation, presence and orientation

20. MEITAL 2010 20

21. MEITAL 2010 21 Where are we currently? Double development –Interactive views from outside: not yet VR, GUI driven –VR system Future tasks: –Sound addition –Permanent maintenance –Write the support material

22. MEITAL 201022 A few references A. Arcavi (2003). The role of visualization in the learning of Mathematics, Educ. St. in Math. 52. C. Sik-Lanyi, Z. Lanyi,A. Tilinger (2003), Using Virtual reality to Improve Space and Depth Perception, J. of Information Technology Education 2. L. Ausburn, F. Ausburn (2004). Desktop Virtual Reality: A Powerful New Technology for Teaching and Research in Industrial Teacher. J. of Industrial Teacher Educ. 41. Dana-Picard Th. (2007). Motivating constraints of a pedagogy-embedded Computer Algebra System, Int. J. of Sc. and Math. Educ. 5 (2)