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Deciding geometric properties symbolically in GeoGebra

Authors:
Markus Hohenwarter at Johannes Kepler University of Linz
Markus Hohenwarter
Zoltán Kovács at Private University College of Education of the Diocese of Linz
Zoltán Kovács
Tomas Recio at Universidad Antonio de Nebrija
Tomas Recio
  • Universidad Antonio de Nebrija
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Abstract

Deciding geometric properties symbolically in GeoGebra
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All content in this area was uploaded by Zoltán Kovács on Aug 06, 2016
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Descriptive geometry
is no more threatened
by dynamic geometry
software than stairs
by elevators.
Vera Viana
director of the
Portuguese Descriptive Geometry and
Drawing Teachers Association
This is the point.
One technology
doesn't replace another,
it complements.
Books are no more
threatened by Kindle
than stairs by elevators.
Stephen Fry
English comedian, actor, writer,
presenter and activist
the Elbphilharmonie in Hamburg
(under construction)
Chou's Example 230
Midpoint parallelograms?
Chess-playing
automaton
“The Turk”
Wolfgang von Kempelen
1770
Chess-playing
automaton
“The Turk”
(hoax!)
Wolfgang von Kempelen
1770
Chess-playing
automaton
“The Turk”
(hoax!)
Wolfgang von Kempelen
1770
References I
Ancsin, G., Hohenwarter, M., Kovács, Z. (2013). GeoGebra goes web. The
electronic journal of mathematics and technology (Volume 7, Number 6, pp. 412-
418).
Botana, F., Hohenwarter, M., Janičić, P., Kovács Z., Petrović, I., Recio, T.,
Weitzhofer, S. (2015). Automated theorem proving in GeoGebra: current
achievements. Journal of Automated Reasoning (Vol. 5, Number 1, pp.. 39-59).
Chou, S. C. (1987). Mechanical geometry theorem proving. Norwell, MA, USA:
Kluwer Academic Publishers.
Decker, W., Greuel, G.M., Pfister, G. & Schönemann, H. (2012). Singular 3-1-6 –
A computer algebra system for polynomial computations.
http://www.singular.uni-kl.de.
Hohenwarter, M. (2002). Ein Softwaresystem für dynamische Geometrie und
Algebra der Ebene. Master’s thesis. Salzburg: Paris Lodron University.
References II
Houghton, T. (2014). The impact of GeoGebra – Evidence. Prezi presentation at
https://prezi.com/bsi-qdd6jr6r/the-impact-of-geogebra-evidence
Howson, G. & Wilson, B. (Eds.) (1986). School mathematics in the 1990s. ICMI Study
Series Volume 2. Cambridge University Press.
Kovács, Z. (2015a). Computer based conjectures and proofs in teaching Euclidean
geometry. Dissertation. Linz: Johannes Kepler University.
Kovács, Z. (2015b). The Relation Tool in GeoGebra 5. In Botana, F., Quaresma, P. (Eds.),
Post-conference Proceedings of the 10th International Workshop on Automated Deduction in
Geometry (ADG 2014), 9-11 July 2014, Lecture Notes in Artificial Intelligence 9201 (pp. 53-
71). Springer.
Lin, F. L., Yang, K. L., Lee, K. H., Tabach, M. & Stylianides, G. (2012). Principles of task
design for conjecturing and proving. In Hanna, G. & de Villiers, M. (Eds.), Proof and
Proving in Mathematics Education. The 19th ICMI Study (pp. 305-326). Springer.
Parisse, B. (2013). Giac/Xcas, a free computer algebra system. Available at http://www-
fourier.ujf-grenoble.fr/~parisse/giac.html
The role of automated reasoning of geometry statements in mathematics instruction
Conference Paper
Full-text available
  • Feb 2017
The poster briefly describes the precise meaning of automated reasoning tools (ART) in elementary geometry, and announces the recent implementation of such ART in GeoGebra, a popular dynamic geometry software, available in a variety of platforms, with millions of users worldwide. Moreover, it claims there is a clear and urgent need to elaborate a research project that should address and evaluate the potential impact in mathematics instruction of such tools.
View
Computer Based Conjectures and Proofs in Teaching Euclidean Geometry
Thesis
Full-text available
  • Jul 2015
The aim of this dissertation is to identify effective methods on extending teaching of conjectures and proofs of Euclidean geometry theorems in secondary schools by utilizing computers, and to develop corresponding technology and software tools by enhancing existing dynamic geometry systems. Based upon analysis of the possible aims of mathematics teaching and an overview of the existing software tools and the theory, • a general report highlights areas where computers can indeed help in the teaching process, and areas of typical dangers of abuse of computers are also presented, • an enhancement of the dynamic mathematics software GeoGebra is introduced to give more support for teachers and students concerning proofs, • an analysis of an effective mathematical method in theorem proving for the Euclidean geometry is introduced in an effort to help non-expert readers to learn the basics of the applied mathematical algorithms, • some typical classroom situations are shown by utilizing GeoGebra to support teaching of certain topics in Euclidean geometry. The results in this dissertation can be considered as a prototype, but they are elaborated enough to be integrated in classroom use already and also for developing it further to provide a wider range of use for teaching proofs.
View
The Relation Tool in GeoGebra 5
Chapter
Full-text available
  • Jun 2015
  • Lect Notes Comput Sci
GeoGebra is open source mathematics education software being used in thousands of schools worldwide. Its new version 5 supports automatic geometry theorem proving by using various methods which are already well known, but not widely used in education software. GeoGe-bra's new embedded prover system chooses one of the available methods and translates the problem specified by the end user as the input for the selected method, similarly to portfolio solvers. The available methods include Wu's method, the Buchberger-Kapur method, the Area method and Recio's exact check method, some of them as embedded algorithms, others as outsourced computations. These methods can also be hidden from end users who are provided with an intuitive graphical user interface , the Relation Tool. Since GeoGebra maintains the development in an open-sourced way by collaborating with the OpenGeoProver, Singular and Giac projects, further enhancements can be expected by a larger community, including implementing other methods, too.
View
Automated Theorem Proving in GeoGebra: Current Achievements
Article
Full-text available
  • Jun 2015
GeoGebra is an open-source educational mathematics software tool, with millions of users worldwide. It has a number of features (integration of computer algebra, dynamic geometry, spreadsheet, etc.), primarily focused on facilitating student experiments, and not on formal reasoning. Since including automated deduction tools in GeoGebra could bring a whole new range of teaching and learning scenarios, and since automated theorem proving and discovery in geometry has reached a rather mature stage, we embarked on a project of incorporating and testing a number of different automated provers for geometry in GeoGebra. In this paper, we present the current achievements and status of this project, and discuss various relevant challenges that this project raises in the educational, mathematical and software contexts. We will describe, first, the recent and forthcoming changes demanded by our project, regarding the implementation and the user interface of GeoGebra. Then we present our vision of the educational scenarios that could be supported by automated reasoning features, and how teachers and students could benefit from the present work. In fact, current performance of GeoGebra, extended with automated deduction tools, is already very promising—many complex theorems can be proved in less than 1 second. Thus, we believe that many new and exciting ways of using GeoGebra in the classroom are on their way.
View
GeoGebra goes Web
Article
Full-text available
  • Oct 2013
The introduction of smartphones with broadband Internet access allows students to access educational materials from almost everywhere. While the open source software GeoGebra is widely used on desktop and laptop computers, it is currently not available for the majority of mobile touchscreen devices like Apple’s iPhone/iPad or Google’s Android devices. In a former paper we described the project GeoGebraMobile which aimed to overcome this limitation by making GeoGebra applets accessible to students on a wide range of mobile devices. In our present paper we outline several enhancements of this former work leading to the new GeoGebraWeb project, report on its current status, and describe future plans.
View
Principles of Task Design for Conjecturing and Proving
Article
Full-text available
  • Jan 2011
Principles of task design should have both the fundamental function of a clear relation to the learner’s rules, learning powers or hypothetical learning trajectories and the practical function of easy evaluation of many similar tasks. Drawing on some theories and practical tasks in the literature, we developed a total of 11 principles of task design for learning mathematical conjecturing, transiting between conjecturing and proving, and proving. To further validate the functioning of those principles, more empirical research is encouraged.
View
View
View
School mathematics in the 1990s. ICMI Study Series
  • Jan 1986
  • G Howson
  • B Wilson
Howson, G. & Wilson, B. (Eds.) (1986). School mathematics in the 1990s. ICMI Study Series Volume 2. Cambridge University Press.
The impact of GeoGebra – Evidence. Prezi presentation at https://prezi.com/bsi-qdd6jr6r/the-impact-of-geogebra-evidence
  • Jan 2014
  • T Houghton
Houghton, T. (2014). The impact of GeoGebra – Evidence. Prezi presentation at https://prezi.com/bsi-qdd6jr6r/the-impact-of-geogebra-evidence
Giac/Xcas, a free computer algebra system
  • Jan 2013
  • B Parisse
Parisse, B. (2013). Giac/Xcas, a free computer algebra system. Available at http://wwwfourier.ujf-grenoble.fr/~parisse/giac.html