2. Geogebra Global Gathering
Lagoudakos-Stavrou-Kokkonas MatHistory using Geogebra 1
Abstract
MATHistory using GEOGEBRA
The reference to historical facts is a strong student’s mobilization tool in
learning and studying math procedure, because it stimulates and
maintains young’s people interest, making math less scary even to
students who aren’t Math lovers.
The verification of some students, that even great mathematicians have
faced difficulties in solving problems, finally turns into confidence and
comfort on their side. It becomes obvious that in an exploratory process
- speculation - doubts - mistakes - incomplete formalities – deadlocks,
even different approaches, are not only legitimate but also an essential
component of Mathematics .
The reference to the History of Mathematics helps the whole class to
develop metacognitive skills. Students collaborate, investigate, ask,
speculate and finally present their conclusions.
Practicing in solving problems, inspired by the Mathematics of History,
encourages more students than those who may solve problems which
seem indifferent from human’s daily needs.
Within this framework GEOBEBRA as teaching tool helps us a lot. Well-
designed Geogebra applications make students and teachers enable to
deal with problems such as:
1. Measuring the Height of pyramid by Thales.
2. The proof of the Pythagorean Theorem.
3. The measurement of the radius of the earth by Eratosthenes.
4. The verification of 2 ’s irrationality.
5. The calculation of a better approximation of π from Archimedes,
using the method of inscribing and circumscribing regular
polygons.
3. Geogebra Global Gathering
Lagoudakos-Stavrou-Kokkonas MatHistory using Geogebra 2
The problems are incorporated into teaching methods and become
matters of investigation, since Geogebra software is open to everyone
and enables: Measurement - Construction -Movement - Parameters
change- Dynamic use of all elements of a shape or construction.
The final rejection or verification of a guess is now possible in a short
time. Therefore we have more space and time for experimentation -
training and demonstration in class.
4. Geogebra Global Gathering
Lagoudakos-Stavrou-Kokkonas MatHistory using Geogebra 3
Story No 1
The myth
We are located in Egypt. Thales with his followers
stand in front of the Great Pyramid of Cheops. It's
really huge. But what is it’s height?
This is a problem raised by the priests. They knew the
correct answer and the question was a challenge for his
abilities as a mathematician. It is easy to calculate the
height of a tree or a building but the calculation is not
so easy for a pyramid. How can this be done
accurately?
Then the solution is presented in front of his eyes, over
the shadows of his own and his partner’s body.
Thales was a short guy compared to his huge travel companion, who
had almost twice Thales’s height, but so did their shadows.
A simple observation was not only the impetus for the solution to the
problem but also a sparkle for a new generalized topic. The concept of
similar shapes had just started to take form.
The associated application
It is a simple simulation of
the problem. The answer to
this problem is given in
steps, so students’ work can
be customized to their
personal abilities - the
needs of each one. At the
end there is a suggestion to
move the sun, which can
cause further discussions.
Another application using the 3D version of Geogebra is the next
challenge for the instructor.
Thales of Mellitus
(624-547 b.C)
5. Geogebra Global Gathering
Lagoudakos-Stavrou-Kokkonas MatHistory using Geogebra 4
Story No 2
The myth
A philosophical school was founded in southern Italy.
Essentially it was a brotherhood whose structure was
highly hierarchical. Just like numbers are ordered so was
the construction of that company. Number one in the
hierarchy was Pythagoras, then were two of his students
who followed him, and so on. Every new student should
monitor classes for a long time in silence until he earned
the right to speak and participate as equal to others.
The most important of Pythagoras’s discoveries, unlike what many
believe, was not the theorem itself that eventually took his name. The
Pythagorean Theorem was known before him. What he actually did was
to justify why this odd requirement happened to be true: "In a right
triangle the square of the hypotenuse is equal to the sum of the squares
of the vertical sides, and vice versa". For the first time in history , he
presented the general proof of this proposal, not just for a particular
triangle, but for all the existing right triangles!!! .
The associated application
Three rigorous mathematical proofs of the
Pythagorean Theorem are presented here.
The original proof as presented in Euclid's
“Elements”, is included in this presentation.
There is also a mechanical "proof" which only
demonstrates the use of the theorem and is
based on the acceptance of the proposal
through the confirmation of the application’s
use.
It is well known that there are more than 400 proofs of this theorem. It’s
a challenge for all of us to gather these proofs in a GEOGEBRA Book.
Pythagoras of Samos
( 572-497 b.C)
6. Geogebra Global Gathering
Lagoudakos-Stavrou-Kokkonas MatHistory using Geogebra 5
. Story No 3
The myth
In Plato's "Menon" the writer refers to a dialogue between
Socrates and a disciple. In this dialogue is introduced for
the first time a new teaching technique called “maieutic”.
The teacher is trying through a series of consecutive
questions to elicit from the student something new, based
on already conquered knowledge. The mistake causes the
birth of a new idea, an idea that can, through proper
guidance, lead to new results.
At the same time this dialogue refers to the first irrational
number ever presented in the History of Mathematics: the wonderful
number 2 . Thus, starting from the problem of doubling the square,
we move to the need of calculating the side of the desired square.
The associated application
Students are following the steps to resolve the
problem as presented in Plato's dialogue.
Mistakes are the springboards for the final
solution. Here are questions that the student is
required to answer, exploiting the
functionalities of the program. The growth tool
gives learners the opportunity to perceive
what the irrational number means. The
concept of the approach is presented as a
necessity.
It is advisable to engage students with whatever follows the myth.
Mythical figures such as Hippasus and also the end of the Pythagorean
brotherhood is a good opportunity for a cross-disciplinary approach to
knowledge.
Socrates of Athens
(470 – 399 b.C)
7. Geogebra Global Gathering
Lagoudakos-Stavrou-Kokkonas MatHistory using Geogebra 6
Story No 4
The myth
One of the first stories of applied mathematics:
It is said that Eratosthenes knew two important facts.
Firstly, that in the city of Syinis (today’s Aswan) there
was a well, on whose base the sun sets vertically, on
the summer solstice’s (June 21st
) midday, with no
shadows at all. Secondly, that the orientation
between Syini - Alexandria was exactly on the axis
North - South with a distance of about 800 km
between these two cities.
So from these two facts , he could calculate something which is
considered very difficult, even in our days. That is the measurement of
the earth’s radius with incredible accuracy!!
Therefore, the significant role of mathematics in history becomes
obvious, as well as practical and useful.
The associated application
This is a rather static application
without much use of the dynamic
functionalities of the software.
The progressive revelation of the
solution will give rise to a class
debate.
This simple application combined
with the visual material of June
21st
, gives teachers the
opportunity to do their lesson out of the classroom and students to
calculate - as other "Eratosthenes" - the radius of the earth.
Eratosthenes of Cyrene
(275-193 b.C)
8. Geogebra Global Gathering
Lagoudakos-Stavrou-Kokkonas MatHistory using Geogebra 7
Story No 5
The myth
Perhaps the most famous mathematician of
ancient times is Archimedes of Syracuse, Sicily. He
is probably the first global scientist, who worked on
physics, technology, mathematics, constructions,
with almost everything humans knowledge at that
time!
He was the first to implement a new method in
order to calculate the length of the circle. The
famous “method of exhaustion”, owes its name to Archimedes. Not only
because it exhausts the mathematician who applies it, but also because
using a successive application of a computational algorithm he is able to
measure lengths, areas and volumes.
It is said that he applied this method by inscribing and circumscribing
regular polygons into a circle, so he managed to calculate its length and
area . The accuracy of his calculations is due to the construction of a
regular polygon with 96 sides, starting from the triangle and
continuously doubling the number of sides. In this way it is considered to
be the first one who calculated the number π, ending his work with the
relationship: 10 10
3 π 3
71 70
The associated application
This application subscribes and
circumscribes regular polygons
in a circle. It is rather obvious
that the length of the circle is
greater than the perimeter of
the inscribed polygon and less
than the perimeter of the
circumscribed polygon.
Archimedes of Syracuse
( 287-212 b.C)
9. Geogebra Global Gathering
Lagoudakos-Stavrou-Kokkonas MatHistory using Geogebra 8
Also it is clear that, with an increasing number of sides of the polygons,
the true value of the circle circumference is approached effectively by
the perimeters of these two polygons. The number of sides is increased
simply using the slider. The program gives us the opportunity to increase
the number of sides of the polygons until we get Archimedes’s number
of sides (96 sides). Furthermore, we can ask calculations to be made
with whatever decimal approximation we wish.
10. Geogebra Global Gathering
Lagoudakos-Stavrou-Kokkonas MatHistory using Geogebra 9
From the History of Mathematics
to Mathematics of History
All the above mentioned topics can be really presented during formal
teaching hours. They can also be part of students' group work, in the
context of a wider research project.
The knowledge acquired by our students during the school years is more
likely to be lost as time goes by. It was a shocking experience for us,
when we met some classmates and we discovered that Civil Engineers
or Architects and the majority of well-educated friends of ours, have
forgotten most of their high school mathematical knowledge.
Eventually have they actually used all of this knowledge?
We guess that they have probably used some school topics only as a tool
for their pre- university exams and in some prerequisite courses for
their degree…
But these have remained as abilities and skills: to collaborate, to search,
to evaluate information to be presented, to persuade, to develop, to
realize sometimes you have to start from scratch, to believe in your
abilities, after you have deeply understand your pros but also your cons
or weaknesses.
So, some problems such as those mentioned above, are suitable to
enable students to form groups dealing with them, answering the
mathematical part and on the other hand, to put the solution in a right
time frame, right place frame, right media frame, right economic frame
and right social environment frame. Finally, we are facing any problem,
with different methods than that was first used.
11. Geogebra Global Gathering
Lagoudakos-Stavrou-Kokkonas MatHistory using Geogebra 10
So ,we are using tools offered by mathematics and technology – any kind
of tools - regardless of the topic or student’s age. All of these tools are
serving the proof of an open problem.
Under these facts, Geogebra is an easily handling tool for anybody . It is
also an open tool that is given for free. The verification or rejection of a
guess can be done quickly by creating the appropriate Geogebra
application.
In our educational community, Geogebra offers generously whatever
thoughts are born in our minds and at the same time gives the
opportunity to exchange thoughts and ideas. There is definitely a front
line educational-technological community that will support you in your
educational concerns and experiments.
12. Geogebra Global Gathering
Lagoudakos-Stavrou-Kokkonas MatHistory using Geogebra 11
REFERENCES
1. Mathematicians: by E.T. Bell (University of Crete
Publications).
2. History of Mathematics: by R. Mankiewicz (Alexandria
Publications)
3. Geometry and its workers in Ancient Greece: by D.
Tsimpourakis
4. History of mathematics: by Victor katz (University of Crete
Publications).
5. The parrot theorem: by Denis Guedj (Polis Publications)
6. Maieutic of Socrates and its implementation in Greek High-
School: by E. Kothali - M. Chounti - N. Korobili - A. Bovi- Z.
Kostiani - D. Papadopoulos (Euclid C Magazine Jan 28 1991-
Hellenic Mathematical Society)
7. Geogebra Help Manual versions 3.0 - 4.2 - 5.0
