Transcript matserv.pmmf.hu

Anna Rybak, University of Bialystok,
Poland
István Lénárt, ELTE University, Hungary
Play computer game ...
and learn spherical geometry
The main question
Children like computer games.
Is it possible to teach topics from beyond the
curriculum using computer game?
Five-in-a-line
Experiment
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Main idea: From game on the plane (circle and cross)
through game on the sphere (five-in-a-line) to models
Main aim: to investigate and learn some properties of figures
on the sphere
Groups that took part in the experiment:
- students of Math Institute (math teachers-to-be)
- kids 11 and 12 years old (members of computer science
circle from middle school in Bialystok)
- practising math teachers (participants of seminar for
teachers in the Institute of Computer Science, University of
Bialystok)
Time: 2 lessons (1,5 hour) for each group
First step
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Playing circle and cross on the plane 3x3,
then 4x4.
First mathematical question:
What should we know about geometry in
order to play circle and cross?
Answer: We should know what shape is
called “straight line”.
Next step
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Presentation of the game five-in-a-line on the sphere.
Second mathematical question:
What should we know about geometry in order to play
this game?
Answer: We should know what shape is called “straight
line” on the sphere.
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So: What is a straight line on the sphere?
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Answer: Equator!!!
Remark:
Students had big troubles with answering this question. Kids
immediately answered.Teachers were in the middle between
these two groups.
Very important question
Why is the equator a spherical straight
line?
One boy from 5th grade of middle school (11 years old)
answered immediately:
Because it divides a sphere into two
identical parts, just like planar straight
line divides a plane into two identical
parts.
Students did not find this property of spherical straight
line, teachers needed a hint.
Next steps
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Playing the game in pairs
Question from a leader:
Which properties of spherical straight line
are different from properties of planar
straight line?
Answer:
Straight line on a plane is infinite, on a
sphere is finite.
Next steps
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Remark from a leader:
So maybe properties of other figures on a
sphere are different from properties of figures
on a plane.
Discussion about “spherical” figures:
segments, triangles, other polygons.
Looking for different figures on the sphere
from game.
Investigation of the sum of internal angles in
the spherical triangle
It is possible to rotate the sphere, so participants can easily estimate
measures of angles in the marked triangle. They discover that the sum
is not equal 180°!
Then it is possible to mark other triangles and to discover that sum
of angles is not constant.
Very important moment:
from game to models
Remark from a leader:
You could only estimate angles in some
special triangles using a model of the
sphere on the screen of computer. It is
possible to construct any triangle on 3Dmodel and measure angles of
constructed triangle.
Presentation of models and tools
Problems for investigations with
use of models and tools
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What is the value of the sum of interior
angles in a spherical triangle?
Does “spherical π” exist i.e. Is the ratio of
circle's circumference to its diameter is
constant on a sphere?
Can we use a spherical square as a unit of
area on a sphere?
Justify all your answers.
Survey (for kids)
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What do you think about the game that you learnt
about today?
Would you prefer the game four-in-a-line?
What do you think about geometry that you learnt
about today?
Did the game help you to learn geometry on
a sphere?
What activity you prefer: playing game or working
with models? Why?
Would you like to learn more about geometry on a
sphere?
How do you think: is this geometry useful for
anything?
The most interesting answers
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This geometry is a little bit strange because of
figures.
This geometry is much more difficult and interesting
in comparison with geometry on a plane.
Game is interesting, requires a lot of thinking, it is
much more difficult to put points on a straight line
on a sphere than on a plane.
This game is not easy. It is necessary to think a lot
if one wants to win.
This game is strange, but interesting.
It requires a lot of logic.
The most interesting answers
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I prefer game.
Game is always more interesting, but we can
get more knowledge from models, so I prefer
to work with models.
I prefer to work with models because it is
possible to find much more properties.
I prefer playing football.
Survey (for teachers)
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What do you think about the game – from didactical
point of view?
What do you think about such methodology of
introducing spherical geometry?
Can spherical geometry be taught at school? Why?
Does the game help in learning spherical
geometry? Why?
What do you prefer: to play game or to work with
models? Why?
The most interesting answers
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Game helps to develop imagination, concentration,
looking for strategy.
Method is good: from playing to learning.
These ideas could be taught at mathematical
circles, for interested students.
Work with models lets you touch the tools, models
are 3-D, picture on the screen is flat.
Conclusions
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It is possible to use computer game (not
educational computer game) for introducing quite
new topic (from beyond curriculum) to middle
school, make this topic interesting and understable
for kids.
It is possible to use two kinds of media: electronic
and manipulative in the same teaching process and
to raise kids' interest in using manipulative media
as more useful in investigations.
Conclusions
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The results of joining different kinds of media and
learning new topics are better when learners are
younger.
Computer game may raise kids' interest for 3-D
geometry and non-Euclidean geometries.
At the same time, it helps concept formation in
Euclidean plane geometry too.
Thank you for your attention