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Drawing topology using Ariadne
Moritz Sümmermann¹
To cite this version:
Moritz Sümmermann¹. Drawing topology using Ariadne. Eleventh Congress of the European Society for Research in Mathematics Education, Utrecht University, Feb 2019, Utrecht, Netherlands. �hal- 02428844�
Drawing topology using Ariadne
Moritz L. Sümmermann¹
1University of Cologne, Germany; [email protected] Keywords: Topology, Learning environment, Mathematics education, Computer software.
Research overview
The focus of this work is to build a learning environment, making it possible to learn about paths and homotopies without the use of formalism. Here, learning environment means a microworld given through a software in the sense of Papert (Papert, 1987). To help achieve this goal I have developed Ariadne, a software tool for the visualization of and interaction with paths. These paths can be constructed on a wide variety of surfaces, from the plane to manifolds of arbitrary genus, and punctured versions thereof. A more detailed account of Ariadne‘s capabilities is given in
(Sümmermann, 2019).
Figure 1: The Pochhammer Contour, a non-nullhomotopic green path starting and ending at the magenta dot with winding number zero around both black punctures, constructed in Ariadne.
Topology in general is not present in the school curriculum, which limits the extent of research in the field of topology education. It is, however, a very important part of modern mathematics, so there have been some attempts to visualize topology, either without (Strohecker, 1996; Sugarman, 2014) or with software (Culler, Dunfield, Goerner, & Weeks, n.d.; Scharein, 1998). There has been no attempt to implement interactive continuous deformations as represented by homotopies, which is the focus of Ariadne. It also follows a different approach didactically, as its purpose is not only to visualize concepts already known to the user, but to teach the user these concepts by letting him interact with the visualization. The theoretical framework behind the design of Ariadne is based on the design principles of Devlin, 2013 and the Artefact Centric Activity Theory from (Ladel &
Kortenkamp, 2013).
Ariadne is split into a two- and a three-dimensional mode. Both are usable on any touchscreen device, such as tablet-PCs or smartphones. In 2D, the user can construct points, paths and homotopies of paths on the plane with an arbitrary number of punctures, as well as compute the
winding number around these punctures. This allows the user to tackle questions on the existence and equivalence of paths, and thus the treatment of the fundamental group. The same can be done for closed orientable surfaces of genus g in three-dimensional mode.
For the 3D-mode, a mixed reality environment is implemented. This mode facilitates the interaction with two-dimensional surfaces in three-dimensional space, such as the sphere or the torus, and thus alleviates handling issues inherent to the two-dimensional touchscreen. The three-dimensional mode also allows the construction of paths on the universal cover of the chosen surface, which is for most surfaces the hyperbolic plane. Ariadne is being evaluated through individual interviews with students from all age groups, in which they are being posed questions to assess their understanding of the used concepts. Further research directions are a didactical analysis of the topological notions involved in Ariadne to ensure that the answer quality is representative for the understanding of the content, planned to be implemented as a qualitative empirical study with mathematicians. The questions can then be refined based on this analysis.
Poster contents
The poster contains a short summary of the mathematical objects involved using some formulas and pictures, so it is clear what mathematics are conveyed with Ariadne. This is by no means exhaustive, but intends to sensitize the audience to the subtleties of the concepts involved. In the center of the poster is a tablet-PC, which the conference participants can use to test Ariadne for themselves. Another part of the poster is a list of sample questions which can be answered with the help of Ariadne, as a demonstration of Ariadne’s capabilities. The last part is a short overview on the technicalities of the program for those interested in the mechanisms of action behind Ariadne.
References
Culler, M., Dunfield, N. M., Goerner, M. & Weeks, J. R. (n.d.). SnapPy, a computer program for studying the geometry and topology of 3-manifolds. Available at http://snappy.computop.org (15.8.2018) .
Devlin, K. (2013). The Music of Math Games. American Scientist 101 (2), 87.
Ladel, S. & Kortenkamp, U. (2013). An activity-theoretic approach to multi-touch tools in early maths learning. The International Journal for Technology in Mathematics Education 20 (1), 3 – 8.
Papert, S. (1987). Learning Environments and Tutoring Systems. In R. W. Lawler & M. Yazdani (ed.),Artificial Intelligence and Education, Vol. 1 (pp. 79 – 94). Ablex Publishing Corp.
Scharein, R. G. (1998). Interactive Topological Drawing. Unpublished doctoral dissertation, Department of Computer Science, The University of British Columbia.
Strohecker, C. (1996). Design of an Environment for Learning about Topology and Learning about Learning. In Proceedings of the Second International Conference on the Learning Sciences. . Sugarman, C. (2014). Using Topology to Explore Mathematics Education Reform. Unpublished
master's thesis, Harvey Mudd College.
Sümmermann, M. L. (2019). Ariadne – A Digital Topology Learning Environment. The International Journal for Technology in Mathematics Education 26 (1), 21 – 26.
