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  2. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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