| Abstract |
When one teaches Maths, he frequently comes across the question asked by his pupils: “Where am I going to use the things I keep learning in my daily life?”. The same question was also put to Euclid by one of his disciples, as it is quoted by Ioannis Stovaios in his Anthology (5th century A.D.), when Euclid started teaching his disciple the theorems of Geometry with the latter expressing his agony by asking “…and now what will my benefit be, after I have learnt it?”. Those people who have a substantial relation with Maths can realize that the questions are caused to the pupils due to its special nature. In essence Maths is a theoretical science which has many applications on every human activity, though. So long as these applications were inexistent, very few people would be able to acknowledge the value of the Mathematical Science. Iamvlichos considers that “Maths excels among visible and lacks among intelligible in beauty, order and precision. It lies within an intermediate symmetry and concordance and possesses the power to forward us by sending us to the indivisible ideas, since it is akin to them”. The pupils, however, who must work with their thinking lying on the conceivable, seek after the value of Maths within the visible and the tangible. Therefore, when it is taught, the pursuit of the actual intermediate symmetry would be the means for the avoidance of the problems encountered in the school classes. Such a symmetrical logic would form the syllabus in the curricula so as for the miscellaneous mathematical concepts to be included in the latter according to their historical significance. From the discovery of asymmetry to the foundation of the total of real numbers there has been a recorded history of 2400 years of speculation and creativity. But what is the importance these concepts have in our school reality? The irrational numbers and the total of the real ones are taught in the school book of the 2nd grade of Junior High occupying only two pages from a total of 254. Therefore, it can be easily realised that the quantity of the syllabus and the effort made for its completion in a very short time does not allow for further consolidation of that particularly important piece of theory. This choice disrupts the aforementioned intermediate symmetry.
The exploitation of history in Maths teaching has been emphasized by many researchers in Greece and internationally. Its usefulness for the mental deepening during the teaching of mathematical concepts has been evident exactly where teaching has not been confined only to the creativity of the last two centuries.
In this dissertation I am not going to argue over the issue of the need for history use in the daily practice of Maths classes, which has already been done by a large number of very successful researchers. Instead, notes and techniques closely related to school Maths, collected from historical sources, are going to be presented in a chronological sequence, as far as this is possible.
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More specifically this sequence will start from ancient Egypt. There, Maths, or Mathematical Arts according to Aristotle, is terrestrial. It is where the term “Geometry” – the measurement of Earth - starts off and the first recorded needs for the measurement of croplands came up. It is also there where the first formulae for the measurement of the area and the first approximative procedures for the area of the quadrilateral and the cycle came up. Next in sequence comes Greece, where the Mathematical technicals was engrafted with the philosophical thought of the ancient Greeks and was led to the sphere of concepts (axiomatic foundation). It was actually there where the Mathematical Science was created. Within this setting the discovery of asymmetry, Eudoxu’s theory of ratios and the method of exhaustion of Eudoxus and Archimedes were presented. Right from that era we are going to draw the techniques of the quadrature of the polygonal passages and the cycle, as well as Trigonometry, which was developed as a tool for the study of the movement of planets and stars. Next we are going to deal with the trigonometry which was developed by Indian, Arabian even European mathematicians covering a span until the years of Regiomontanus in the 15th century. We are also going to see the transition from the calculation of the arcs to that of the strings of the cycle along with the transition to the modern forms of trigonometric functions. With regards to the latter some other ways of their definition without the use of angles and arcs are going to be added, after the presentation of the modern form of infinitesimal calculus will have been completed. From the 17th century Europe, just before the introduction of the infinitesimal calculus, Cavalieri’s method for the calculation of the area of the cycloid curve will be presented along with Riemann’s integral for the calculation of the area dating back from the 19th century. By using modern argumentation, including that of the limit, it will eventually become evident that this particular integral emerged from the ancient method of exhaustion. Right from that period the foundation of real numbers as a commutative body with a principle of completeness came up. In addition, the foundation of real numbers through Dedekind’s cats will remind us of the theory of proportions of Eudoxus. Finally, the dissertation is completed with the exploration of the concept of the area in school Maths, where a full foundation of this particular concept with the use of triangulation is provided.
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