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
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
3
D plots: treatment of discontinuities
2 2
2 2
(,)
x y
f x y
x y
3 3
2 2
(,)
x y
f x y
x y
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Strange
3
D plots: Mathematical Differences
2 2
2 2
(,);
(0,0);
cos( );sin( );
0 2;0;
(,) cos(2 );
.
x y
f x y
x y
Discontinuityin
x y
f
Depends onthe path
3 3
2 2
3 3
3 3
0
(,);
(0,0);
cos( );sin( );
0 2;0;
(,) (cos ( ) sin ( ));
lim (cos ( ) sin ( )) 0
x y
f x y
x y
Continuityin
x y
f
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Visualisation of the plots: Polar Coordinates
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Incomplete is better than false
2
2
2
2
)
,
(
y
x
y
x
y
x
f
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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.
•
3
D 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 (
3
D)
–
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
)
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