Virginia Mathematics Teacher Spring 2017

When clicking “More...” GeoGebra starts to translate the geometric figure into a set of algebraic equations with integer coefficients in the background, but assuming the input points to be completely general and arbitrary. GeoGebra then manipulates on the equations by usually performing billions of atomic computations which are hardly of classroom interest, hence those details will not be shown to the user. Finally GeoGebra concludes that the equations can be interpreted as an evidence about the geometric statement on the midpoint parallelogram, that is, it is true in general, independently of the initial points. Modern DGSdynamic geometry systems including this ATP feature, can be considered as a kind of “geometry calculators”. Their ATP algorithms may be hidden for the user with, but just the final result is visible. In this sense the usual way of proving is usually substituted by a yes or no answer computed by the machine. However, the traditional way of proving should not be discouraged at all. When the truth about a conjecture is already known by mechanical computation, the real intellectual challenge will start: to find an elegant chain of reasons to show why that conjecture is true. A well known quote from Paul Halmos highlights that “the only way to learn mathematics is to do mathematics”. What does it mean to do mathematics? According to Bruno Buchberger, it is “knowledge derivation and problem solving by reasoning.” (Buchberger et al., 1998) To support this idea, Buchberger introduced the creativity spiral “algorithms → computational results → conjectures → theorems → algorithms → …”, and so forth, to describe mathematics as an infinite chain of recurring activities. In his model proving is the activity which connects conjectures and theorems. In conclusion we can say that without knowing why a conjecture is true learning mathematics is not really possible, either. Acknowledgments. A preliminary version of this paper was presented at ICME-13 in Hamburg, Germany (Hohenwarter et al., 2016). References Buchberger, B. and the Theorema Working Group (1998). Theorema: Theorem proving for the masses using Mathematica. Invited Talk at

the Worldwide Mathematica Conference, Chicago, June 18-21, 1998. 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). Springer. Hohenwarter, M., Kovács, Z. & Recio, T. (2016). Deciding geometric properties symbolically in GeoGebra. https://www.researchgate.net/ publication/305916853_Deciding_geometri c_properties_symbolically_in_GeoGebra Hickey, W. (2013): The 12 Most Controversial Facts In Mathematics. http:// www.businessinsider.de/the-most- controversial-math-problems-2013-3? op=0#-63 Kovács, Z. (2015): Midpoint parallelograms. GeoGebra Materials. https:// www.geogebra.org/m/pBuQA4Eo Kovács, Z. (2016): Midpoint parallelograms. GeoGebra Materials. https:// www.geogebra.org/m/BrnY77nE 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. and de Villiers, M., editors, Proof and Proving in Mathematics Education . The 19th ICMI Study, pp. 305- 326. Springer.

Zoltan Kovacs Johannes Kepler University Zoltan@geogebra.org

Virginia Mathematics Teacher vol. 43, no. 2

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