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Session Overview |
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| 9:30am - 10:30am | Keynote III: Keynote (Tomás Recio) | |||
| Jacob-Volhard-Hörsaal | ||||
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Dynamic Geometry and Mathematics: few trains on a two-way track Universidad de Cantabria, Spain
Dynamic geometry is designed as a helpful tool for mathematics comprehension. This can be thought as a single direction: from dynamic geometry to mathematics. But, conversely, some non elementary mathematics seem to be required to understand (and to improve) dynamic geometry performance.
Noticing this mutual interaction is neither very popular, nor strictly new (e.g. consider the paradigmatic case of Cinderella). On the other hand, we would argue in the talk how to trace back the origins of such interaction in order to include some fancy names such as, say, Babbage or Watt, ending up with Nash.... Yet, we think it is perhaps convenient to insist now and again on the importance of traveling back and forth along this two way track, for the benefit of mathematics education. This would be the main idea in my lecture, exemplified by some situations I have been recently dealing with, in order to improve dynamic geometry features for locus computation and automatic theorem proving.
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| 10:30am - 11:00am | Coffee Sa AM: Coffee break and Poster presentation | |||
| Georg-Cantor-Haus | ||||
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Several Aspects of Using Computer Supports for Mathematics Learning of Foreign Students from the CIS (Countries of Independent States) G. S. Skovoroda Kharkiv National Pedagogical University, Kharkiv, Ukraine
The article is devoted to peculiarities of foreign students training in a modern computer environment. Special attention is given to the problems which are relating in studying of mathematical disciplines. The problems of graphical solution of mathematical and computer tasks with using modern computer technology and visual facilities are discussed.
From real world to derivative – How to effectively include mathematical modeling and GeoGebra in mathematics education 1University of Novi Sad, Serbia; 2Technical Colegge of Applied Sciences, Serbia
Direct application of innovative teaching method based on principles of mathematical modeling is illustrated by example of the teaching unit related to introducing the concept of the first derivative processed by using mathematical modeling as teaching method and GeoGebra as software tool. Step by step procedure of making mathematical model is explained – from starting preparation, to implementation of the mathematical model and drawing conclusions from it.
The positive effects of mathematical modeling and GeoGebra as software tool to better understanding, the creation of advanced mathematical thinking, and the application of mathematical theory on solving real world problems are presented and illustrated.
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| 11:00am - 1:00pm | AKMUI II: Vorträge | |||
| VSP 1.03 | ||||
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app@school – App-Entwicklung als Lehr-Lern-Szenario in der Schule
Bei app@school geht es darum, dass die Schülerinnen und Schüler mithilfe analoger und digitaler Werkzeuge eine eigene mobile app konzipieren, entwickeln und publizieren. Indem der gesamte Produktionsprozess einer Applikation vollzogen wird, werden sowohl fachlich-mediale als auch soziale Kompetenzen angebahnt.
Um dieses Ziel zu verwirklichen haben wir das theoretisch fundiertes Lehr-Lern-Szenario app@school entworfen, das einerseits den pädagogischen Projektgedanken aufgreift, also auf prozessuales Lernen in (Experten)Gruppen abzielt, und sich andererseits an den Abläufen agiler Softwareentwicklung orientiert. Als Entwicklungswerkzeug verwenden wir Stencyl (http://stencyl.com), da es eine visuelle Gestaltung aller Programmbestandteile (Levels, Actors, Logik) ermöglicht. Der Einsatz dieses Tools erlaubt darüber hinaus die ideale Verzahnung der einzelnen Expertengruppen (Grafiker, Leveldesigner, Programmierer, PR-Manager, Projektmanager), die jeweils an einem Module der mobile app arbeiten. Im Rahmen eines Vortrags auf der Tagung des AK »Mathematik und Informatik« würde sowohl das didaktische Design von app@school als auch erste Ergebnisse dessen schulpraktischer Erprobung Anfang diesen Jahres zur Diskussion gestellt. Interaktives Konstruieren im länderübergreifendem bilingualen Mathematikunterricht University of Education Ludwigsburg, Germany
Für das Fach Mathematik wird ein Unterrichtsszenario aus dem INTACT-
Kontext dargestellt, in dem Schülergruppen aus zwei verschiedenen Ländern Dreieckskonstruktionen mit DGS durchführen und dazu Beschreibungen in der Fremd- bzw. der Muttersprache anfertigen. Diese Beschreibungen werden zwischen den beiden Schülergruppen der verschiedenen Länder ausgetauscht, um die Dreiecke zu rekonstruieren. Dabei sind sowohl geometrische wie auch sprachliche, interkulturelle und soziale Fertigkeiten und Kompetenzen gefordert und gefördert. Nutzbar sind die Materialien, die über eine speziell entwickelte Lernplattform zur Verfügung gestellt werden, nicht nur mit interaktiven Whiteboards sondern auch mit anderen Geräten, wie z.B. Tablet-PCs, Smartphones usw. Im multilingualen Projekt „INTACT - Interactive teaching materials across culture and technology“ der Pädagogischen Hochschule Ludwigsburg mit den Partnerländern Spanien, Irland, Ungarn, Rumänien und Portugal werden interaktive Materialien für den bilingualen Unterricht verschiedener Fächer zur Verfügung gestellt und so eine virtuelle Kooperation zwischen Schulen der ganzen Welt ermöglicht. INTACTwird für drei Jahre aus Fördermitteln der EU im Rahmen des Förderprogramms Lebenslanges Lernen - "COMENIUS Multilateral Projects" gefördert. Pen&Paper-Programmierung - Neue Chancen für digitale Medien
Der Begriff des Algorithmus gehört seit langem zu den fundamentalen
mathematischen Ideen, die im Mathematikunterricht einen spiraligen Aufbau haben. Für die Sekundarstufen bedeutet die Thematisierung für die Schüler die Chance, selbstständig Algorithmen zu entwickeln, um so einerseits Einsichten in mathematische Strukturen zu gewinnen und zu vertiefen und andererseits einen reflektierten Umgang mit den heutigen digitalen Medien zu fördern. In meinem Vortrag stelle ich die von mir entwickelte Pen&Paper-Programmiersprache Adi vor. Ursprünglich als Ausgangspunkt zum Erlernen einer Programmiersprache gedacht, nutze ich diese Sprache als Vehikel, um einer möglichen Interpretation des Begriffs "Algorithmisches Denken" nachzugehen. | |||
| 11:00am - 1:00pm | Automated Deduction: Automated Deduction in Dynamic Geometry Tools (working group) Session Chair: Predrag Janičić | |||
| VSP 1.04 | ||||
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Automatic theorem proving in Dynamic Geometry contexts: what is it good for? Two diverse points of view 1Universidad de Cantabria Spain; 2Architectural technical high school, Serbia
The talk will be dual, in the sense of addressing the same issue from two different perspectives. One, that of a secondary mathematics teacher, with experience on the introduction of ATP tools (GCLC and GeoGebra) at the classroom. Then, the point of view of a university ATP researcher and developer, with sustained involvement in math education. In both cases the main point will be to provide some answers to the key question:
What is ATP good for? The most efficient ATP systems in geometry are usually algebraic. Although these provers do not generate classical readable proofs, they give an yes/no answer and can generate non-degeneracy conditions (NDGs) that need to hold for the statement to be satisfied. Although ATPs can be applied to check the main statement of the theorem, we advocate that it is much more beneficial for pupils to apply automation on intermediate steps, and so to verify if their conclusions are correct and if they are on the right track to prove the main theorem. In such setting, pupils still need to produce global proof steps, while the ATP system takes care of details that are usually straightforward, but tedious to justify. Pupil cooperates with the machine and relies on its help, but his understanding of the geometry problem is still crucial for success. Trough this activity student develops his thinking and analytical skills, but, unlike with classical pen-and-paper proofs, in a very rigorous setting, since every conclusion must be precisely formulated and pass the automated test. Also, a careful analysis of NDGs may reveal many subtle issues about the theorem that is being proved. In this talk we will present some of our experience and ideas about using these techniques (implemented both in GCLC and GeoGebra) in high-school geometry classes.
Automated algebraic calculations of geometric figures in dynamic geometry systems University of Education Weingarten, Germany
Using Methods based on Automated Deduction in Geometry (ADG), it is possible to perform algebraic calculations on interactively constructed figures. Thus, a parameter of such a figure can be calculated algebraically as a function of other parameters of this figure. This opens up a new computer-assisted connection of synthetic elementary geometry to algebra. On the other hand, the question: "WHAT HOW depends on WHOM?" is cleared up when a figure is dynamically varied. - In this lecture there are given some selected examples in a constraint-based DGS developed for educational purposes. Their didactic relevance is explained and some resulting mathematics education problems are discussed.
Teaching loci and envelopes in GeoGebra 1University of Vigo, Spain; 2Johannes Kepler University, Austria
GeoGebra is open source mathematics education software being used in thousands of schools worldwide. Since version 4.2 (December 2012) it supports symbolic computation of locus equations as a result of joint effort of mathematicians and programmers helping the GeoGebra developer team. The joint work, based on former researches, started in 2010 and continued until present days, now enables fast locus and envelope computations even in a web browser in full HTML5 mode. In conclusion, classroom demonstrations and deeper investigations of dynamic analytical geometry is ready to use on tablets or smartphones as well.
In our talk we consider some typical grammar school topics when investigating loci is a natural way of defining mathematical objects. Such topics include definition of a parabola and other conics in different situations like synthetic definitions or points and curves associated with a triangle. In most grammar schools, however, no other than quadratic curves are discussed, but generalization of some exercises and also every day problems will introduce higher order algebraic curves. Thus our talk will mention the cubic curve ``strophoid'' as locus of heights of a triangle when one of the vertices moves on a circle. Also quartic ``cardioid'' and sextic ``nephroid'' can be of every day interest when investigating mathematics in a coffee cup. We will also focus on GeoGebra specific tips and tricks when constructing a geometric figure to be available for getting the locus equation. Among others, simplification and synthetization (via the intercept theorem) will be mentioned.
Extending the range of computable objects in Dynamic Geometry by using Quantifier Elimination University of Vigo, Spain
Traditionally, loci in Dynamic Geometry Systems (DGS) have been displayed either tracing the locus point, either through a special command that, roughly, enhances such tracing procedure. Some DGS incorporate more sophisticated approaches being able to return algebraic knowledge about loci. Nevertheless, these enhancements deal with a restricted type of loci, those where the position of the locus points is completely determined by another point, the mover, which must lie on a linear path. If the mover point is not unique, or it is not bound to a line, standard DGS can only offer a tracing strategy, just returning a graphical answer. Consider, for instance, two circles and a point on each circle. The locus of their midpoint is a 2-dimensional region, that cannot be easily described with current DGS.
In this talk I illustrate these limitations when computing loci. Furthermore, a discussion on the application of Cylindrical Algebraic Decomposition software will be given. | |||
| 11:00am - 1:00pm | Computer-Aided II/II: Computer-Aided Experiments and Explorations in the Math Classroom (working group) Session Chair: János Karsai | |||
| VSP 1.02 | ||||
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Programming in High School as a Learning Resource of Zeros of 2º degree Polynomial Function 1Pontifícia Universidade Católica de São Paulo, Brazil; 2Faculdade de Tecnologia de São José dos Campos - ETEP Cetec Educacional S.A
This paper presents a research work where has as objective to check if the proposal of an algorithm converted into a computer program can help high school students in the learning of the zeros of the 2nd degree polynomial function. The research was conducted in two stages. The first stage was with a 1st year high school student in order to verify if the activities were appropriate and, in the second part, we have selected four participants for the second stage. After the analysis of the first stage development, the activities were improved to the second one, composed of three activities, among which the software Visualg 2.0. The APOS theory by Ed Dubinsky, theoretical support of the research, presents the action levels, process, object and scheme, that allow the verification of the individual`s capacity to develop actions over an object and think about its properties. The research participants had improvements in their learning, because besides developing a computer program to determine the zeros of 2nd degree polynomial function, they have started to elaborate other functions previewing their possible solutions, presenting all the levels of the APOS theory. As research methodology we have adopted the Design Experiments. We have justified its use, for adjustments could be done during the work development. Analyzing the activities which were done we have concluded that the students have achieved a satisfactory learning level over the object of study.
Key words: 2nd degree polynomial function, APOS theory, Algorithm.
Computer-Aided Exploring the Mathematics behind Technical Problems – Examples of Classroom Practices Beuth Hochschule Berlin, Germany
The mathematics in technical problems can be discovered by computer-aided experiments. Examples are presented from four different courses in the areas of statics, elasticity, finite elements and partial differential equations. It will be reported on the implementation within the curriculum at Beuth University of Applied Sciences Berlin, the classroom experiments and the teacher’s role.
The Role of Technology in Supporting Students' Conceptual Understanding in Linear Algebra Humboldt-Universität zu Berlin, Germany
Current trends in research on the impact of technologies in mathematics education emphasize their increased role in supporting students' conceptual understanding in comparison with numerous previous studies about technology contribution in procedural understanding. This talk exemplifies the role of Dynamic Geometry Systems utilizing students' conceptual understanding of dot product of vectors in the transition between upper high school and university education. Students' conceptual understanding is identified as constituting a structured network of: concept definitions and concept images (Tall & Vinner, 1981) of dot product of vectors developed by students; three modes of description and thinking (Hillel, 2000; Sierpinska, 2000) of dot product of vectors: arithmetic, geometric and axiomatic-structural; and concept's applications in problem solving situations. Authentic video recordings and students' written works serve as two collected data sets for qualitative analysis of students' interactions in the designed Dynamic Geometry Environment, within the framework of instrumental genesis (Drijvers et al., 2010). The study is part of a larger design-based research (The Design-Based Research Collective, 2003) undergoing seven phases in a cyclic manner, ending with evaluation and dissemination of created teaching and learning materials as visual dynamic applets and worksheets.
Levels of reasoning with coherences between dynamically linked representations of functions Heidelberg University of Education, Germany
What students describe when they explore computerbased multiple representations of functions does not necessarily reflect how, or whether at all, they understand. In fact, the case of one student explaining his observations while exploring properties of a multiple representation environment suggests that such observations - even if correctly stated - could be based on a superficial perception of the properties only. A qualitative content analysis of further interviews results in a three level model of reasoning with multiple representation learning environment. The theoretical base of this analysis is formed by cognitive theories that describe the learning process in mathematics as a process of abstracting from superficial aspects of representations to structural coherencies between them.
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| 1:00pm - 2:15pm | Lunch II: Lunch (incl. Coffee break V) | |||
| Georg-Cantor-Haus | ||||
| 2:15pm - 4:15pm | AKMUI III: Vorträge & Arbeitsgruppen | |||
| VSP 1.02 | ||||
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Terme besser verstehen mit neuen Medien
Der Umgang und das Verstehen und Interpretieren von Termen sind ein wichtiger Bestandteil des Mathematikunterrichts. Der Umgang mit Termen findet im Wesentlichen auf einer formalen Ebene statt. Durch die Benutzung neuer Medien eröffnen sich für die Schülerinnen und Schüler neue Zugänge zu Termen. Dies wird im Vortrag sowohl mit Beispielen aus der Sek. I als auch aus der Sek. II gezeigt.
Vom Funktionenmikroskop zur digitalen Funktionenlupe
Das Funktionenmikroskop von A. Kirsch war ein Klassiker für die Erarbeitung eines Grundverständnisses von Steigung und Differenzierbarkeit im Sinne lokaler Glättung. Damals ein aufwändiger Foliensatz in Lehrerhand, konnte die Grundidee des ‚Hineinzoomens‘ später mit gängigen Funktionenplottern digital umgesetzt werden.
In dem Vortrag wird nun eine interaktive ‚Funktionenlupe‘ mit GeoGebra vorgestellt, die mit zwei Graphikfenstern und Ortslinien einen Zugang lokal zur Steigung des Funktionsgraphen und global zur Ableitung der Funktion bietet. Sie ermöglicht einen entdeckenden, anschaulichen und (zunächst) kalkülfreien Einstieg in die Analysis und einen Aufbau von Grundvorstellungen von Steigung und Ableitungsfunktion bis hin zur Krümmung. | |||
| 2:15pm - 4:15pm | Workshop 2 & 3 | |||
| VSP 1.04 | ||||
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Mapping mathematics learning resources University of Education Weingarten, Germany
In this workshop, we shall work collectively to write down a map of the sources of learning resources, be them open or not, adaptable or not, nicely made or not.
The aim is to assemble a broad panorama of the places where mathematics learning resources can be found. We shall do so in a web-based map which can be followed live during the workshop. The wish is to leverage the diversity of attendees at the conference so as to obtain some clarity in the local practices in teachers of their surroundings. The result of this workshop would support such harvesting projects as Open-Discovery-Space, i2geo.net, probably Elixier, and other initiatives. Moreover, it would help users of such repositories to get a broader overview and thus decide better what to visit at their next resourcing excursions. Controlling Lego Mindstorms robots by Cinderella's scripting interface PH Ludwigsburg, Germany
By installing the Mindstorms plugin, the interactive geometry software Cinderella can be used as a remote control for Lego Mindstorms robots.
In this workshop we will explain how to install and use the plugin. We will give a short introduction to the CindyScript programming language. Simple examples will show how to read sensor values from a Lego NXT and use them to manipulate geometric constructions. Vice versa, interactive geometric constructions are used as remote controls for motors. There will be a few Mindstorms controller on location for own explorations. | |||
| 2:15pm - 4:15pm | Workshop 4 & 5 | |||
| VSP 1.03 | ||||
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Development of Dynamic Demonstrations with Mathematica University of Szeged, Hungary
Some years ago, one of the main improvement in Mathematica was the dynamic functionality. Real-time graphics and functions for dynamic variables (Manipulate…) meant a breakthrough in the classroom usage of CAS systems. Currently, this technique has been greatly improved. In addition, the so called CDF ("Computable Document Format") has been introduced. In the workshop, the participants can get acquainted with the elements of dynamic features of Mathematica and they learn how to prepare simple CDF documents.
Colleagues are kindly encouraged to send problems, topics for discussion in the frame of the workshop. The participants should bring their laptops with Mathematica installed. A 30 days trial version can be downloaded from the website of Wolfram Research. Planned length: 3-4 hours.
Sketchometry - DGS for tablets, smartphones and interactive whiteboards University of Bayreuth, Germany
Sketchometry is a Dynamic Geometry System (DGS) for the student in classroom. In contrast to many other DGS which are primarily used by teachers as a presentation tool, sketchometry tries to be a simple tool that invites students to experiment and discover.
Sketchometry can be used on tablets, smartphones, interactive whiteboards as well as on desktop computers. The software has a touch-optimized user interface for easy sketching of geometric constructions. In the workshop we will discover sketchometry by creating various classroom examples. Please, bring your own device: tablet, smartphone or notebook.
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| 4:15pm - 4:45pm | Coffee Sa PM: Coffee break and Poster presentation | |||
| Georg-Cantor-Haus | ||||
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Visual introduction to bifurcations University of Szeged, Hungary
Investigating the dependence on parameters is essential in studying dynamical systems. In particular, the bifurcation theory is getting more and more important in most fields of engineering and sciences. Nevertheless, these theories are hardly included in standard university curricula.
We will give an intuitive introduction with the help of dynamic demonstrations developed in Mathematica. We consider elementary examples of both difference and differential equations presenting different types of bifurcation. During the whole treatment, we keep in mind the real didactic “contradiction” that the students do not or only partly have the required knowledge. The interactive demonstrations will be available on our website www.model.u-szeged.hu.
MEASURES OF GEOMETRIC OBJECTS AS THE LIMIT VALUES 1University of Novi Sad, Serbia; 2Elementary school of Nis; 3Faculty of Mechanical Engineering Nis
The idea for this work is based on the visualization of the limit process in elementary school. The circumference of a circle is obtained visually starting from the perimeter of regular polygon. The volume of cylinder is obtained visually starting from the volume of prism.
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| 4:45pm - 5:45pm | AKMUI IV: Arbeitsgruppen | |||
| 4:45pm - 5:45pm | Collaborative: Collaborative use of DGS and KETpic (working group) Session Chair: Setsuo Takato | |||
| VSP 1.04 | ||||
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KETCindy - Fine Combination of KETpic and Cinderella 1Toho University, Japan; 2Kogakuin University, Japan; 3Nagano National College of Technology, Japan; 4Kisarazu National College of Technology, Japan; 5Graduate School of Information Science, Nagoya University, Japan
According to the result of our questionnaire survey, one major opinion of collegiate mathematics teachers in Japan is that there is no necessity to use high-quality graphics in education. However, from our experience, graphics use seems to play a crucial role in some classroom situations. Though we have been attempting some statistical efficiency assessments for teaching materials containing graphics, it is not so easy to separate the effect of using graphics from other factors like context of classroom or communications between students. The aim of this research is to verify the effect of using high-quality graphics in collegiate mathematics education through some cognitive scientific experiments. The experiments were designed in the following two ways:
(1) Detecting the change of students’ brain activity after their seeing effective figures through EEG (ElectroEncephaloGram) measurement (2) Observing the difference in time needed for students to answer a question between before and after their seeing effective figures In fact, we picked up the case of the comparison of growth degree between exponential function y=2^x and power function y=x^4. We prepared some graphs of these functions by gradually changing the scale in y direction, so that students can recognize that the growth of y=2^x is greater than that of y=x^4 when x becomes sufficiently large. We showed these graphics step by step to three students and detected their brain activities through EEG (ElectroEncepharoGram) measurement. As a result, the judgment of these students changed when they saw a triggering figure, and some change in the trend of EEG signal and solution time was observed at that time. These results indicate that using effective figures should have great influence on students’ reasoning processes.
A Collaborative Laboratory for Geometry: A Case Study at Portugal and Serbia 1CISUC/University of Coimbra, Portugal; 2Faculty of Mathematics/University of Belgrade, Belgrade, Serbia; 3CISUC/Department of Mathematics, University of Coimbra, Coimbra, Portugal; 4School of Science and Tecnology/University of Trás-os-Montes e Alto Douro, Vila Real, Portugal
The Web Geometry Laboratory} (WGL) platform is a collaborative blended-learning Web-environment for geometry, it integrates a dynamic geometry systems (DGS) and it provides a collaborative environment for students and teachers. Its use is possible in the context of a classroom or remotely. Apart from its development its evaluation is being done through a series of case studies, sustained through a qualitative approach (interpretive research), being conducted in Portugal and Serbia (Prototype in hilbert.mat.uc.pt/WebGeometryLab).
An initial case study in Portugal, with groups of secondary students (17 years old) was done, using various gathering information techniques: quizzes; tests; direct observation; record interactions on the platform; challenges. We analysed the use of the WGL collaborative environment by the students. Another case study, in Serbia, was conducted in the context of remote access to the platform (homework). The study included 50 secondary students (15 years old). All students attended the traditional classes in school. Half of the students used WGL platform for homework and the other half did their homework the traditional way. We investigated the impact of collaborative work to the motivation level and level of achievement. Using an action research approach, the platform is being developed. These studies revealed some aspects that could be enhanced, e.g. a chat feature. More and wider case studies are being prepared allowing the validation and further development of the WGL platform. These studies also indicates that there is a significant improvement in the motivation of students and a slight improvement in their achievement when using the WGL platform. The WGL platform will include in future stages of development the implementation of an adaptive environment allowing the construction of students' profiles and learning paths. A final stage will be the integration of a geometric automated theorem prover and its use in the learning process.
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| 4:45pm - 5:45pm | Creativity: Creative Mathematical Thinking and Digital Tools (working group) Session Chair: Péter Körtesi | |||
| VSP 1.03 | ||||
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Convex hull of the maximum volume of a space curve in the special case Deoma, Russian Federation
Let СN be the closed three-dimensional polygon with 2N edges (N > 3), the perimeter L(СN) and the convex hull of СN volume V(CN). We want to find maximum V(CN) for given L(СN) in the special case when the convex hull may be divided into tetrahedra having one common edge. Let C be the rectifiable closed three-dimensional curve with the length L(C) and the volume of the convex hull V(C). C may be obtained using the СN limit at infinity. We want to find maximum V(C) for given L(C).
We assume that the convex hull maximum volume is achieved if two conditions are satisfied: at first, the slope angle θ between the curve and the Z-axis is constant, the segment which intersects the Z-axis is perpendicular to it. The second condition is: the projection of the convex hull to the XY plane has the maximum area. For V(L) there is an exact evaluation. The sign of the equality holds if and only if the curve is congruent to the curve obtained in the paper. There are two solids of equal volume. One solid is axisymmetric, the second solid is centrally symmetric. The area of the convex hull projection on the XY plane has an exact evaluation. The sign of the equality holds if and only if the curve is congruent to the curve obtained in the paper. Finding the maximum area of the convex hull of CN projection in case N = 4 + 2n is reduced to finding the function extremum under the obtained conditions. These equations may be solved analytically for n < 12 and numerically for an arbitrary n. The solution has been found and checked using DGS GInMA. There are some examples of the maximum volume convex hulls V(CN) and maximum V(C) for given L(C). | |||
| 6:00pm | Dinner: Conference dinner | |||
| Bergschenke (conference dinner location) | ||||
