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Nov 14, 2013 (3 years and 8 months ago)

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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.

MEITAL 2010

3

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



MEITAL
2010

4

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

   
   
   
  




 
   
 
 
MEITAL 2010

5

Visualisation of the plots: Polar Coordinates

MEITAL
2010

MEITAL
2010

6

Incomplete is better than false

2
2
2
2
)
,
(
y
x
y
x
y
x
f



MEITAL
2010

7

Right coordinates are not a panacea

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”.

MEITAL
2010

9

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).

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

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

MEITAL
2010

12

Path Lines for the visualization of the
Discontinuities

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

MEITAL
2010

14

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

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

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.)

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

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



MEITAL
2010

19

Basic configuration: presentation, presence
and orientation

MEITAL
2010

20

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

MEITAL
2010

22

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
)