Advertisement

Digital Experiences in Mathematics Education

, Volume 5, Issue 2, pp 124–144 | Cite as

Dragging as a Geometric Construction Tool: Continuity Considerations Inspired by Students’ Attempts

  • Marita BarabashEmail author
Article

Abstract

Straightedge-and-compass construction problems in Euclidean geometry are known, at times, to be unsolvable. In such cases, additional mathematical tools, such as the Archimedean spiral, may yield a solution. Dynamic geometry (DG) environments suggest adding another mathematical tool as an alternative to the classical approach to geometric constructions: Continuous variation of one of the parameters of the problem. Adoption of this approach raises both mathematical and didactic issues for discussion. This paper presents several solved problems of triangle construction given data that includes angle bisectors. Some of these problems have been shown to be otherwise unsolvable; others may be solvable, but, to the author’s best knowledge, have not been solved using only classical tools. Presented for solution to a special group of prospective teachers, these problems provided them with the experience of facing the need to justify a DG-based solution mathematically, distinguish a ‘dynamic illustration’ from a self-consistent algorithm and refer to mathematical constraints or ‘rules of the game’ – altogether a challenging and fruitful learning opportunity.

Keywords

Construction problems in geometry Dragging Dynamic tools in geometry Angle bisectors in triangle 

Notes

References

  1. Barabash, M. (2005). A non-visual counterexample in elementary geometry. The College Mathematics Journal, 36(5), 397–400.CrossRefGoogle Scholar
  2. Barabash, M., Gurevich, I., & Yanovsky, L. (2009). Usage of computerised environment in the undergraduate course ‘Plane transformations and constructions in geometry. International Journal for Technology in Mathematics Education, 16(2), 49–62.Google Scholar
  3. Courant, R., & Robbins, H. (1978). What is mathematics? An elementary approach to ideas and methods. Oxford: Oxford University Press.Google Scholar
  4. Davis, P. (1995). The rise, fall, and possible transfiguration of triangle geometry: a mini-history. The American Mathematical Monthly, 102(3), 204–214.CrossRefGoogle Scholar
  5. Delone, B., & Zhitomirsky, O. (1959). Zadachnik po geometrii (A problem book in geometry). Moscow, Russia: Physmatgis (in Russian).Google Scholar
  6. Epstein, D., & Levy, S. (1995). Experimentation and proof in mathematics. Notices of the American Mathematical Society, 42(6), 670–974.Google Scholar
  7. Katz, V. (1998). A history of mathematics: An introduction (2nd ed.). Reading: Addison-Wesley.Google Scholar
  8. Leung, A., Baccaglini-Frank, A., & Mariotti, M. (2013). Discernment of invariants in dynamic geometry environments. Educational Studies in Mathematics, 84(3), 439–460.CrossRefGoogle Scholar
  9. Zajic, V. (2003). Triangle from angle bisectors. http://www.cut-the-knot.org/triangle/TriangleFromBisectors.shtml. Accessed 29/1/19

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Achva Academic CollegeBeer TouviaIsrael

Personalised recommendations