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Motion with LEGOs and Dynamic Geometry
Abstract and Figures
A basic task of engineering is the problem of assembling mechanisms that make an element move along a prescribed path. Modern electrical engineering allows devices to move with incredible precision by means of microcontrollers, but many applications still rely on classic methods that use belts, chains, shafts or ball bearings.
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... The course was highly inspired by the book (Bryant & Sangwin, 2008) and used several concepts from its chapter "How to draw a straight line?". Preliminary ideas from the papers (Kovács & Kovács, 2017;Kovács, 2019bKovács, , 2019dKovács, , 2018Kovács, Recio, & Vélez, 2019;Kovács, 2019e;Kovács & Kovács, 2018) were extensively used during the course. ...
... The course was highly inspired by the book (Bryant & Sangwin, 2008) and used several concepts from its chapter "How to draw a straight line?". Preliminary ideas from the papers (Kovács & Kovács, 2017;Kovács, 2019bKovács, , 2019dKovács, , 2018Kovács, Recio, & Vélez, 2019;Kovács, 2019e;Kovács & Kovács, 2018) were extensively used during the course. ...
A summary of an experimental course on algebraic curves is given that was held for young learners at age 11. The course was a part of Epsilon camp, a program designed for very gifted students who have already demonstrated high interest in studying mathematics. Prerequisites for the course were mastery of Algebra I and at least one preliminary year in a prior Epsilon camp. The summary gives an overview of the flow of teaching, the achieved results and some evaluation of the given feedback.
... Cancelling the leading term of (8) by (12) and continuing reduction: . . . Equation added: ...
A summary of an experimental course on algebraic curves is given that was held for young learners at age 11. The course was a part of Epsilon camp that is dedicated for very gifted students who already showed high interest in studying mathematics. Preliminaries of the course were accomplishing Algebra I and at least one preliminary year in a former Epsilon camp. The summary gives an overview of the flow of teaching, the achieved results and some evaluation of the given feedback.
... The student then seeks answers adapted to the context, to solve the problem or to accomplish the task, without having to bear all the weight of the logical artillery of the «orthodox presentation of mathematical texts» (see Section 2.6). With a dynamic geometry software, for example, we can refer to the many ways that promote investigating geometrical properties of a figure or generalizing some observed/conjectured geometric properties (cf. the nine tools of Kovács, Recio & Vélez, 2018a), and to the combine use of LEGOs and the software to link proving, computation and experimental views in modelling tasks (Kovács, 2018). ...
Our article aims to define the notion of instrumental proof based on didactic, epistemological and cognitive considerations. We raise issues and challenges related to the use of this type of proof in mathematical work and mathematical thinking. The theory of mathematical working spaces serves as a construct on which we address questions about proof, reasoning and epistemic necessity, taking advantage of the possibilities offered by the development geneses and fibrations in an instrumented perspective. The coordination of the semiotic, discursive and instrumental geneses of the working space founds discursive-graphic proofs, mechanical proofs and algorithmic proofs that are activated at school in the subject-milieu interactions. We end with a discussion on some consequences of the computer-assisted modelling of the learning conditions of mathematics, and we conclude on a necessary reconciliation of heuristics and validation. Notre article vise à définir la notion de preuve instrumentale en partant de considérations didactiques, épistémologiques et cognitives. Nous soulevons des enjeux et des défis liés à ce type de preuve au regard du travail mathématique et de la pensée mathématique. La théorie des espaces de travail mathématique sert de charpente sur laquelle nous abordons des questions sur la preuve, le raisonnement et la nécessité épistémique, profitant des possibilités qu'offrent le développement des genèses et des fibrations dans une perspective instrumentée. La coordination des genèses sémiotique, discursive et instrumentale de l'espace de travail fondent des preuves discursivo-graphiques, des preuves mécaniques et des preuves algorithmiques qui s'activent à l'école dans l'interaction sujet-milieu. Nous terminons par une discussion de quelques conséquences de la modélisation des conditions d'apprentissage des mathématiques assisté par des dispositifs informatiques, et nous concluons sur un rapprochement nécessaire entre heuristique et validation.
A basic task of engineering is the problem of assembling mechanisms that make an element move along a prescribed path. In the modern era many applications still rely on classic methods that use belts, chains, ball bearings or shafts-all of these applications can be well modeled by mathematical means. In the talk we focus on mathematical modeling of mechanical engineering by using computer algebra for calculation and a LEGO compilation for supplementary experiments. We discuss some novel methods in the dynamic geometry software GeoGebra Discovery to study the motion of linkages from both the computation and the experimental views at the same time.
Our article aims to define the notion of instrumental proof based on didactic, epistemological and cognitive considerations. We raise issues and challenges related to the use of this type of proof in mathematical work and mathematical thinking. The theory of mathematical working spaces serves as a construct on which we address questions about proof, reasoning and epistemic necessity, taking advantage of the possibilities offered by the development geneses and fibrations in an instrumented perspective. The coordination of the semiotic, discursive and instrumental geneses of the working space founds discursive-graphic proofs, mechanical proofs and algorithmic proofs that are activated at school in the subject-milieu interactions. We end with a discussion on some consequences of the computer-assisted modelling of the learning conditions of mathematics, and we conclude on a necessary reconciliation of heuristics and validation.
A summary of an experimental course on algebraic curves is given that was held for young learners at age 11. The course was a part of Epsilon camp that is dedicated for very gifted students who already showed high interest in studying mathematics. Preliminaries of the course were accomplishing Algebra I and at least one preliminary year in a former Epsilon camp. The summary gives an overview of the flow of teaching, the achieved results and some evaluation of the given feedback.
STEM education connects engineering and mathematics in an impressive way. A basic task of engineering is the problem of assembling mechanisms that make an element move along a prescribed path. In the modern era many applications still rely on classic methods that use belts, chains, ball bearings or shafts-all of these applications can be well modeled by mathematical means. In the workshop focused on mathematical modeling of mechanical engineering, a new approach is presented that extends the classic way of learning and teaching geometry from the earliest years of secondary education by using computer algebra for calculation and a LEGO Technic compilation for further experiments. We discuss some novel methods in the dynamic geometry software GeoGebra to study the motion of linkages from both the computation and the experimental views at the same time. As a result, STEM education can take profit from the new features for automated reasoning in GeoGebra that is recently available either offline or online free of charge.
STEM education connects engineering and mathematics in an impressive way. A basic task of engineering is the problem of assembling mechanisms that make an element move along a prescribed path. In the modern era many applications still rely on classic methods that use belts, chains, ball bearings or shafts-all of these applications can be well modeled by mathematical means. In the talk focused on mathematical modeling of mechanical engineering, a new approach is presented that extends the classic way of learning and teaching geometry from the earliest years of secondary education by using computer algebra for calculation and a LEGO Technic compilation for further experiments. We discuss some novel methods in the dynamic geometry software GeoGebra to study the motion of linkages from both the computation and the experimental views at the same time. As a result, STEM education can take profit from the new features for automated reasoning in GeoGebra that is recently available either offline or online free of charge.
La educación STEM conecta la ingeniería y las matemáticas de una manera impresionante. Una tarea básica de la ingeniería es ensamblar mecanismos que hacen que un elemento describa una trayectoria prescrita. En la era moderna, muchas aplicaciones todavía se basan en métodos clásicos que usan correas, cadenas, rodamientos de bolas o ejes; todas estas aplicaciones pueden modelarse con medios matemáticos. En la charla se presenta un nuevo enfoque que amplía la forma clásica de aprender y enseñar geometría, desde los primeros años de la educación secundaria, mediante el uso de GeoGebra para el cálculo, y una compilación de LEGO Technic para experimentos adicionales. Discutimos algunos métodos novedosos en el software de geometría dinámica GeoGebra para estudiar el movimiento de los mecanismos, usando la vista algebraica y la geométrica simultáneamente. Como resultado, la educación STEM puede sacar provecho de estas nuevas funcionalidades, disponibles en GeoGebra, para el razonamiento automatizado.
We present a compilation of LEGO Technic parts to provide easy-to-build constructions of basic planar linkages. Some technical issues and their possible solutions are discussed. Fine details -- like deciding whether the motion is an exactly straight line or not -- are forwarded to the dynamic mathematics software tool GeoGebra.
GeoGebra Automated Reasoning Tools (GGB-ART) are a collection of GeoGebra tools and
commands ready to automatically derive, discover and/or prove geometric statements in a dynamic
geometric construction. The aim of this workshop is to present, through examples, the use of GGBART
and to argue about its potential impact in the classroom.
Keywords: Automated theorem proving and discovery, GeoGebra, Dynamic geometry software,
Elementary geometry in education.
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A utomated generation of Kempe linkages for algebraic curves in a dynamic geometry system. (Unpublished Bachelor's thesis)
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Kobel, A. (2008). A utomated generation of Kempe
linkages for algebraic curves in a dynamic geometry system. (Unpublished Bachelor's thesis). University of Saarbrücken, Saarbrücken,
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Reasoning on linkages. Paper presented at the GeoGebra Global Gathering
- T Recio
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- M P Vélez
Recio, T., Kovács, Z. & Vélez, M. P. (2017, July).
Reasoning on linkages. Paper presented at the
GeoGebra Global Gathering, Linz, Austria.
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Retrieved December 4, 2017 from https://
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