Papers by Georgios Κ . Ntontos
European Journal of Education Studies, 2024
In this paper an attempt is made to design a mathematical machine based on the Pythagorean theory... more In this paper an attempt is made to design a mathematical machine based on the Pythagorean theory of fitting an equivalent parallelogram (given angle and side) with another parallelogram. An equivalent square or rhombus is constructed given a rectangle or a parallelogram respectively. It is then proved that this mechanism can simultaneously draw a parabolic arc. The parameters of this mathematical machine are checked. In each case, an attempt is made to find the focus and the directrix of the corresponding parabola. The important help of the GeoGebra software for the initial mechanical design of the machine is highlighted, but also the important differences of this simulation from the physical construction.
Παρουσιάζουμε την αθλητική βιογραφία (περιληπτικά) τριών συμπατριωτών μας (Αναστάσιος, Κων/νος & ... more Παρουσιάζουμε την αθλητική βιογραφία (περιληπτικά) τριών συμπατριωτών μας (Αναστάσιος, Κων/νος & Σπύρος Ντόντος), οι οποίοι πρώτευσαν την 10ετία του 50 και του 60 στον Κλασσικό Αθλητισμό με τις φανέλες της Εθνικής Ελλάδος, της Εθνικής Ενόπλων και της Α.Ε.Κ., για να θυμηθούν οι παλαιότεροι και να μάθουν οι νεότεροι. Αποτελεί απόσπασμα από το βιβλίο του Σωτηρίου Ι. Νοτάρη «Α.Ε.Κ. Κλασσικός Αθλητισμός», με τη συμπλήρωση στοιχείων, για τον Σπύρο Ντόντο. Λόγω του πρόωρου θανάτου του Σπύρου, αρκετά στοιχεία δεν ήταν γνωστά στον εκδότη, τα οποία μάζεψε, από την οικογένειά του, ο Γεώργιος Κ. Ντόντος και μας τα προσέφερε μαζί με όλο το άλλο υλικό.
European Journal of Education Studies, 2024
This paper aims to highlight two historical parabolographs and a third one that is at the design ... more This paper aims to highlight two historical parabolographs and a third one that is at the design level and to prove the equivalence of the respective definitions of parabola they contain. Specifically, using the Geogebra math software, each mechanism is attempted to articulate another, so that by moving the cursor of the original mechanism to the common output of the two articulated mechanisms, the same parabola is drawn. This proves the equivalence of definitions of parabolic curve as well as of suitably articulated mathematical machines.
This paper presents the analysis and construction of two mathematical machines that can precisely... more This paper presents the analysis and construction of two mathematical machines that can precisely design a specific conic section, the Parabola. The subject of geometric curves has mathematicians been concerned, from antiquity to the present day, both with their usefulness and with the legitimacy of their use to solve geometrical problems.
There is also the special concept of parabola, a conic section that has been studied for many centuries through various contexts. There was also some planning for the use of mathematics to teach the concept of parabola to secondary school students.
The mechanism of Bonaventura Cavalieri (1598-1647) and Frans van Schooten (1615-1660) was analyzed and constructed after simulation using Geogebra Dynamic Geometry software and AutoCAD. During the analysis and construction, mathematical concepts were found which are hidden within the machines and constitute their basic building block.
We aspire that the attempt to analyze, simulate, construct, and use mathematical machines and mechanisms in teaching improves students’ attitudes towards mathematics and the evidence process. Besides, experience with curve design mechanisms has always played an important role in the development of analytical geometry, algebraic symbolism, calculus and concept of functions.
Thesis Chapters by Georgios Κ . Ntontos
Μηχανές δημιουργίας γεωμετρικών καμπυλών και εφαρμογή τους στην διδακτική πράξη, 2024
In this research, a brief historical review of mathematical machines and mechanisms is first pres... more In this research, a brief historical review of mathematical machines and mechanisms is first presented, finally focusing on the two historical 17th century parabolographs designed and constructed by Bonaventura Franciscus Cavalieri and Frans van Schooten. The two mechanisms were analyzed, simulated and reconstructed. Through dynamic geometry software, extensions and possibilities of the mechanisms that are not apparent were sought. With apparent elements of the van Schooten parabolograph, the local curvature of the drawn parabolic arc was calculated. The “inverse tangent problem” was formulated, which is solved geometrically by the van Schooten parabolograph, essentially solving the corresponding differential equation. The equivalence of the parabola’s definitions, embodied by these parabolographs, was confirmed mechanically making them mechanically equivalent.
By the mediation of these mechanisms, the concept of parabola was taught to a part of 10thgrade class which was divided into two groups. Two planned four-hour experimental lessons were carried out at the school area. Individual results were analyzed qualitatively and quantitatively, focusing on two cognitive indicators and three cognitive structure indicators through activities that were initially converted into ordinal variables without special weighting. A nonparametric test of the median of each of the five qualitative indicators was also conducted for each group. A non-parametric test of the correlation between the cognitive indicators and the activity indicators of each group was also performed.
The two cognitive and the three structure indicators through activities, provided a more complete picture of each student's level of understanding, and a more complete process for assessing how they understand. Mathematical machines, consisting of rigid modular rods, as well as dynamic geometry software, emerge as cognitive bridges between empirical justification and deductive reasoning. Mechanical linkages and their dynamic simulations emerge as suitable contexts to connect empirical and deductive reasoning, as balanced linking tools in the teaching of Algebra and Geometry in secondary education.
Conference Presentations by Georgios Κ . Ntontos
1st Mathematical Student Conference of Western Greece, 2024
This article describes a concept now unknown in classrooms, that of mathematical machines. It beg... more This article describes a concept now unknown in classrooms, that of mathematical machines. It begins with a historical trail highlighting the almost parallel development of Mathematics and Mechanical Engineering, with indicative reference to some of the people who contributed to the development of these two scientific fields. Four mathematical machines are then presented, analysing their structure and the result of their use as simulations have been built using the Geogebra software. The construction of one of the four machines with natural materials is described. Finally, the impressions of the history, analysis and synthesis of these instruments through simulation and through the attempt to construct them with natural materials are discussed.
34th Hellenic Conference of Mathematical Education, 2017
The way in which the values of two variables are linked, within a measurement space, it is likely... more The way in which the values of two variables are linked, within a measurement space, it is likely to be known, to be or not to be determined even if we know that there is a type of connection. In these cases, we can sometimes state that there is correlation of the two variables or there is co-variation, alternatively we search for the ratio of the two variables. Moreover, theorems, which link the correlation of the variables to the correlation of their rates, are mentioned and proved by studying the characteristics of the ratio and rate factors.
The work conducted aimed to analyze the thought process of 10th grade
students, in the framework ... more The work conducted aimed to analyze the thought process of 10th grade
students, in the framework of two quantities’ covariation: change of water
level in a container being filled with a constant amount of water. The
students worked in a laboratory environment, performing the experiment
and afterwards plotting the results in a table of values and graphical plot.
The students' abilities to correlate both the table of values, and the graphical
representation with the rate of change of height in relation to volume, and
postulate predictions, was the main aim of this work.
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Papers by Georgios Κ . Ntontos
There is also the special concept of parabola, a conic section that has been studied for many centuries through various contexts. There was also some planning for the use of mathematics to teach the concept of parabola to secondary school students.
The mechanism of Bonaventura Cavalieri (1598-1647) and Frans van Schooten (1615-1660) was analyzed and constructed after simulation using Geogebra Dynamic Geometry software and AutoCAD. During the analysis and construction, mathematical concepts were found which are hidden within the machines and constitute their basic building block.
We aspire that the attempt to analyze, simulate, construct, and use mathematical machines and mechanisms in teaching improves students’ attitudes towards mathematics and the evidence process. Besides, experience with curve design mechanisms has always played an important role in the development of analytical geometry, algebraic symbolism, calculus and concept of functions.
Thesis Chapters by Georgios Κ . Ntontos
By the mediation of these mechanisms, the concept of parabola was taught to a part of 10thgrade class which was divided into two groups. Two planned four-hour experimental lessons were carried out at the school area. Individual results were analyzed qualitatively and quantitatively, focusing on two cognitive indicators and three cognitive structure indicators through activities that were initially converted into ordinal variables without special weighting. A nonparametric test of the median of each of the five qualitative indicators was also conducted for each group. A non-parametric test of the correlation between the cognitive indicators and the activity indicators of each group was also performed.
The two cognitive and the three structure indicators through activities, provided a more complete picture of each student's level of understanding, and a more complete process for assessing how they understand. Mathematical machines, consisting of rigid modular rods, as well as dynamic geometry software, emerge as cognitive bridges between empirical justification and deductive reasoning. Mechanical linkages and their dynamic simulations emerge as suitable contexts to connect empirical and deductive reasoning, as balanced linking tools in the teaching of Algebra and Geometry in secondary education.
Conference Presentations by Georgios Κ . Ntontos
students, in the framework of two quantities’ covariation: change of water
level in a container being filled with a constant amount of water. The
students worked in a laboratory environment, performing the experiment
and afterwards plotting the results in a table of values and graphical plot.
The students' abilities to correlate both the table of values, and the graphical
representation with the rate of change of height in relation to volume, and
postulate predictions, was the main aim of this work.
There is also the special concept of parabola, a conic section that has been studied for many centuries through various contexts. There was also some planning for the use of mathematics to teach the concept of parabola to secondary school students.
The mechanism of Bonaventura Cavalieri (1598-1647) and Frans van Schooten (1615-1660) was analyzed and constructed after simulation using Geogebra Dynamic Geometry software and AutoCAD. During the analysis and construction, mathematical concepts were found which are hidden within the machines and constitute their basic building block.
We aspire that the attempt to analyze, simulate, construct, and use mathematical machines and mechanisms in teaching improves students’ attitudes towards mathematics and the evidence process. Besides, experience with curve design mechanisms has always played an important role in the development of analytical geometry, algebraic symbolism, calculus and concept of functions.
By the mediation of these mechanisms, the concept of parabola was taught to a part of 10thgrade class which was divided into two groups. Two planned four-hour experimental lessons were carried out at the school area. Individual results were analyzed qualitatively and quantitatively, focusing on two cognitive indicators and three cognitive structure indicators through activities that were initially converted into ordinal variables without special weighting. A nonparametric test of the median of each of the five qualitative indicators was also conducted for each group. A non-parametric test of the correlation between the cognitive indicators and the activity indicators of each group was also performed.
The two cognitive and the three structure indicators through activities, provided a more complete picture of each student's level of understanding, and a more complete process for assessing how they understand. Mathematical machines, consisting of rigid modular rods, as well as dynamic geometry software, emerge as cognitive bridges between empirical justification and deductive reasoning. Mechanical linkages and their dynamic simulations emerge as suitable contexts to connect empirical and deductive reasoning, as balanced linking tools in the teaching of Algebra and Geometry in secondary education.
students, in the framework of two quantities’ covariation: change of water
level in a container being filled with a constant amount of water. The
students worked in a laboratory environment, performing the experiment
and afterwards plotting the results in a table of values and graphical plot.
The students' abilities to correlate both the table of values, and the graphical
representation with the rate of change of height in relation to volume, and
postulate predictions, was the main aim of this work.