Geogebra Reasoning Tools for Humans and for Automatons

See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/347256855

GeoGebra Reasoning Tools for Humans and for Automatons

Presentation · December 2020 DOI: 10.13140/RG.2.2.26851.58407

CITATION READS 1 142

2 authors:

Zoltán Kovács Tomas Recio Private Pädagogische Hochschule der Diözese Linz Universidad Antonio de Nebrija

160 PUBLICATIONS 520 CITATIONS 197 PUBLICATIONS 1,310 CITATIONS

SEE PROFILE SEE PROFILE

Some of the authors of this publication are also working on these related projects:

Automatic reasoning in Geometry View project

GeoGebra documentation View project

All content following this page was uploaded by Zoltán Kovács on 15 December 2020.

The user has requested enhancement of the downloaded file. Motivation Geometer Automaton Abstract Didactical implications

GeoGebra Reasoning Tools for Humans and for Automatons

Zoltán Kovács1 Tomás Recio2

1The Private University College of Education of the Diocese of Linz, Austria

2University Antonio de Nebrija, Madrid, Spain

Asian Technology Conference in Mathematics December 14-16, 2020

Zoltán Kovács, Tomás Recio GeoGebra Reasoning Tools for Humans and for Automatons Motivation Geometer Automaton Abstract Didactical implications Outline

1 Motivation Antecedents

2 Geometer Automaton Introductory examples Further examples The future. . .

3 Didactical implications

Zoltán Kovács, Tomás Recio GeoGebra Reasoning Tools for Humans and for Automatons Motivation Geometer Automaton Abstract Didactical implications Summary

We present two recent tools: 1 for humans: the GeoGebra Discovery toolset (Relation , LocusEquation , Prove and ProveDetails) 2 for automatons: 1 the Discover(X) command and tool and the 2 WebDiscovery (Automated Geometer). We conclude the presentation with reﬂections on the pros and cons and on the potential impact of these reasoning tools in the educational world.

Zoltán Kovács, Tomás Recio GeoGebra Reasoning Tools for Humans and for Automatons Motivation Geometer Automaton Antecedents Didactical implications Our work is much related to ATCM! ATCM 1995

“. . . geometry problem solving instead of geometry theorem proving should be emphasized. . . ” “algebra and geometry should be kept in pace in the teaching”

Wen-Tsün Wu (1919-2017) invited speaker Wu, W. (1995): Geometry problem-solving and its contemporary signiﬁcance. In: Electronic Proceedings of the First ATCM. 18–21 Dec. 1995. The Assoc. of Math. Educators, 67–73. Singapore

Zoltán Kovács, Tomás Recio GeoGebra Reasoning Tools for Humans and for Automatons Motivation Geometer Automaton Antecedents Didactical implications Our work is much related to ATCM! ATCM 1995

“. . . geometry problem solving instead of geometry theorem proving should be emphasized. . . ” “algebra and geometry should be kept in pace in the teaching”

Zoltán Kovács, Tomás Recio GeoGebra Reasoning Tools for Humans and for Automatons Motivation Geometer Automaton Antecedents Didactical implications Our work is much related to ATCM! ATCM 2019

Fu, H., Zhang, J., Zhong, X., Zha, M. and Liu, L. (2019): Robot for Mathematics College Entrance Examination. In: Electronic Proceedings of the 24th ATCM, Leshan, China

“. . . pioneering mathematicians [Tarski, Wu, Zhang] (who have) made profound progress. . . ” Concluding that: “Stimulated by them, many new methods appeared such as the resultant elimination method, Gröbner basis and so on. These methods are called algebraic elimination methods. . . ” And declaring that their aim is: “. . . to develop a mathematics robot for automated solving of elementary mathematics problems just like human being, which can automatically answer both standard and non-standard questions. . . ”

Zoltán Kovács, Tomás Recio GeoGebra Reasoning Tools for Humans and for Automatons Motivation Geometer Automaton Antecedents Didactical implications Our work is much related to ATCM! ATCM 2019

Fu, H., Zhang, J., Zhong, X., Zha, M. and Liu, L. (2019): Robot for Mathematics College Entrance Examination. In: Electronic Proceedings of the 24th ATCM, Leshan, China

Zoltán Kovács, Tomás Recio GeoGebra Reasoning Tools for Humans and for Automatons Motivation Geometer Automaton Antecedents Didactical implications Our work is much related to ATCM! ATCM 1995/2019

In both ATCM invited speeches, with almost 25 years of difference, there are specific mentions to 1 the direct ancestors and origins of our tools, based on algebraic elimination methods, 2 the goals (automated solving of elementary problems in geometry) 3 the impact in education: Wu already mentions the need to change the main goal from “proving” to “solving”, a subtle different focus, related to the fact that proving is a closed task while solving addresses an open question—that might require proving, but it is the student who has to decide what to prove and for what purpose. . .

Zoltán Kovács, Tomás Recio GeoGebra Reasoning Tools for Humans and for Automatons Motivation Geometer Automaton Antecedents Didactical implications Our work is much related to ATCM! ATCM 1995/2019

Automaton (αυτοματον): “A mechanism that is relatively self-operating. . . Plural: automatons or automata” (Merriam-Webster dictionary) Geometer automaton: a mechanism that successfully performs, in a self-operating mode, the tasks that human geometers are expected to accomplish: to conjecture, prove, refute, discover...... geometric properties!

Zoltán Kovács, Tomás Recio GeoGebra Reasoning Tools for Humans and for Automatons Motivation Introductory examples Geometer Automaton Further examples Didactical implications The future. . . GeoGebra Reasoning Tools for humans and for automatons

Howson, G., Wilson, B. (1986): ICMI Study series: School mathematics in the 1990’s. Kuwait: Cambridge University Press

GitHub page:

https://github.com/kovzol/geogebra-discovery

Online release: https://autgeo.online

Chou, S. C. (1987): Mechanical geometry theorem proving. D. Reidel Publishing Company, Dordrecht, Netherlands Zoltán Kovács, Tomás Recio GeoGebra Reasoning Tools for Humans and for Automatons Motivation Introductory examples Geometer Automaton Further examples Didactical implications The future. . . GeoGebra Reasoning Tools for automatons Discover command and tool

Zoltán Kovács, Tomás Recio GeoGebra Reasoning Tools for Humans and for Automatons Motivation Introductory examples Geometer Automaton Further examples Didactical implications The future. . . Further examples A problem from Spanish Civil Service Math Teacher’s examination

Kovács, Z., Recio, T., Vélez, M.P. (2019): GeoGebra Automated Reasoning Tools. . . ACA 2019, 25th Conference on Applications of Computer Algebra

Zoltán Kovács, Tomás Recio GeoGebra Reasoning Tools for Humans and for Automatons Motivation Introductory examples Geometer Automaton Further examples Didactical implications The future. . . Further examples A generated problem that has been accepted to the home round of the Czech-Slovak mathematical olympiad 2020/2021, category A (highest)

Bak, P. (2020): Automated Generation of Planar Geometry Olympiad Problems. Master Thesis in Informatics. Institute of Computer Science, Pavol Jozef Šafárik University in Košice, Slovakia

Zoltán Kovács, Tomás Recio GeoGebra Reasoning Tools for Humans and for Automatons Motivation Introductory examples Geometer Automaton Further examples Didactical implications The future. . . Further examples International Mathematical Olympiad 2017, shortlisted problems, G3

Zoltán Kovács, Tomás Recio GeoGebra Reasoning Tools for Humans and for Automatons Motivation Introductory examples Geometer Automaton Further examples Didactical implications The future. . . Further examples A diverse generalization of the Steiner-Lehmus theorem

Let ABC be a triangle with the ﬁxed side AB. Where to put point C to ensure that the proportion of the angle bisectors through A and B, respectively, is 2 : 1?

Zoltán Kovács, Tomás Recio GeoGebra Reasoning Tools for Humans and for Automatons Motivation Introductory examples Geometer Automaton Further examples Didactical implications The future. . . Further examples A collection of geometric statements and algebraic curves (benchmark)

Prover test database (288 test cases) including several well-known non-trivial theorems like Ceva, Desargues, Menelaos, Napoleon, Pappus, Ptolemy and Wallace-Simson LocusEquation test database (180 test cases) including several examples from kinematics and envelope equations (up to degree 20)

Making conjectures on geometric inequalities (with proofs) Prototype available on Raspberry Pi (with free Mathematica version) Intelligent geometry book “i-geometrybook” Readable proofs? Not really.

Zoltán Kovács, Tomás Recio GeoGebra Reasoning Tools for Humans and for Automatons Motivation Geometer Automaton Didactical implications Rethink the curriculum!

“. . . even if the students will not have to deal with computers till they leave school, it will be necessary to rethink the curriculum, because of the changes in interests that computer have brought.” Howson, G., Wilson, B. (1986): ICMI Study series: School mathematics in the 1990’s

Zoltán Kovács, Tomás Recio GeoGebra Reasoning Tools for Humans and for Automatons Motivation Geometer Automaton Didactical implications The role of reasoning and proving As an auxiliary tool, what opportunities/diﬀerences involve using GeoGebra Reasoning Tools?

It can guide student exploration, provide hints, answer (partially) questions. . . It can help building up diagrams, locus. . . also for teachers! Math proofs: an identity feature. . . but. . . What if the output of reasoning can be automatically obtained? Many parts of Euclidean geometry can be successfuly proven with GeoGebra now. . . like with a pocket calculator! “The message is clear: classical proof must move over. . . ” Davis, P.J. (1995): The Rise, Fall, and Possible Transﬁguration of Triangle Geometry: A Mini-History. Am. Math. Monthly 102(3) “. . . geometry problem solving instead of geometry theorem proving should be emphasized. . . ” Wu, W. (1995): Geometry problem-solving. . .

Zoltán Kovács, Tomás Recio GeoGebra Reasoning Tools for Humans and for Automatons Motivation Geometer Automaton Didactical implications Mills grind slowly The impact in education is very slow

Gelernter, H. (1959): Realization of a geometry theorem proving machine. Proc. Int. Conf. on Information Processing. Paris, UNESCO house. 273-282. QED Manifesto (1994): “QED is . . . a project to build a computer system that eﬀectively represents all important mathematical knowledge and techniques.” Chen, X., Wang, D. (2007): Towards an electronic geometry textbook. In: Botana, F., Recio, T. (eds.): Automated Deduction in Geometry, LNAI 4869, 1-23. Springer. Richard, P.R., Fortuny, J.M., Hohenwarter, M., Gagnon, M. (2007): geogebraTUTOR : une nouvelle approche pour la recherche sur l’apprentissage compétentiel et instrumenté de la géométrie à l’école secondaire. In: Bastiaens, T., Carliner, S. (eds.): Proceedings of E-Learn 2007 – World Conference on E-Learning in Corporate, Government, Healthcare, and Higher Education, 428-435. AACE, Canada. Davenport, J., Fleuriot, J., Quaresma, P., Recio, T., Wang, D. (2019): Intelligent Geometry Tools, Proc. Second International Workshop on Automated Reasoning: Challenges, Applications, Directions, Exemplary Achievements, Natal, Brazil, EPTCS 311. Shalosh B. Ekhad, XIV (ca. 2050): Plane Geometry: An elementary textbook. Downloaded from the future by Doron Zeilberger

Zoltán Kovács, Tomás Recio GeoGebra Reasoning Tools for Humans and for Automatons Motivation Geometer Automaton Didactical implications

Zoltán Kovács, Tomás Recio GeoGebra Reasoning Tools for Humans and for Automatons Appendix For Further Reading For Further ReadingI

Hohenwarter, M., Kovács, Z., Recio, T. Using GeoGebra Automated Reasoning Tools to Explore Geometric Statements and Conjectures. In: Proof technology in mathematics research and teaching, Series: Mathematics Education in the Digital Era, Vol. 14. Springer, Cham, 2019. Kovács, Z., Recio, T., Richard, P. R., Van Vaerenbergh S., Vélez, M. P. Towards an Ecosystem for Computer-Supported Geometric Reasoning. International Journal of Mathematical Education in Science and Technology. 2 November 2020.

View publication stats Zoltán Kovács, Tomás Recio GeoGebra Reasoning Tools for Humans and for Automatons