Yehuda Badihi 1, David Zeitoun 2 Th. Dana Picard 1 1.Jerusalem College of Technology

Virtual Reality serving

Mathematics Education:

Dynamical Visualization

of Multivariable Functions

Yehuda Badihi 1, David Zeitoun 2 Th. Dana Picard 1

1.Jerusalem College of Technology2. Orot College of Education1 Supported by

the Israel Science

Foundation, research

grant number 1340/05

2 Supported by a Meital

R&D grant

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.

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Strange 3D plots: treatment of discontinuities

2 2

2 2( , )

x yf x y

x y

3 3

2 2( , )

x yf x y

x y

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Strange 3D plots: Mathematical Differences

2 2

2 2( , ) ;

(0,0);

cos( ); sin( );

0 2 ;0 ;

( , ) cos(2 );

.

x yf x y

x y

Discontinuity in

x y

f

Depends onthe path

3 3

2 2

3 3

3 30

( , ) ;

(0,0);

cos( ); sin( );

0 2 ;0 ;

( , ) (cos ( ) sin ( ));

lim (cos ( ) sin ( )) 0

x yf x y

x y

Continuity in

x y

f

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Visualisation of the plots: Polar Coordinates

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Incomplete is better than false

22

22

),(yx

yxyxf

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Right coordinates are not a panacea

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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”.

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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).

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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

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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

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Path Lines for the visualization of the Discontinuities

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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

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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

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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

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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.)

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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

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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

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Basic configuration: presentation, presence and orientation

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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

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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)