Advertisement

Geometrical Conceptualization

  • Harry SilfverbergEmail author
Chapter

Abstract

This chapter reviews the challenges of learning and teaching geometric conceptualization. We describe how geometry appears differently during the various stages of learning in institutionalized education. The earlier conceptual stages of geometry always form the foundation for the next interpretation of geometry that the learner will encounter. This stratified structure of geometrical thinking is visible in geometrical conceptualization and teaching geometry as a prolific dialogue between the concrete and the conceptual approaches. This chapter places particular focus on the central nature of prototypical conceptualization and figurativeness in the early stages of learning geometry, the gradual development of the ability to categorize and form definitions, and the roles these have in the process of shaping a pupil’s structured conceptual geometric knowledge. Theoretical observations are illustrated with examples.

Keywords

School geometry Conceptual knowledge Learning Primary Secondary 

Bibliography

  1. Atebe, H.U. (2008). Students’ van Hiele levels of geometric thought and conception in plane geometry: A collective case study of Nigeria and South Africa. PhD thesis. Rhodes University, South Africa. Retrieved from http://eprints.ru.ac.za/1505/1/atebe-phd-vol1.pdf.
  2. Barnbrook, G. (2002). Defining language. A local grammar of definition sentences. Philadelphia, PA: John Benjamin’s Publishing Company.CrossRefGoogle Scholar
  3. Bell, C. (2011). Proofs without words: A visual application of reasoning and proof. The Mathematics Teacher, 104(9), 690–695.Google Scholar
  4. Blair, S. (2004). Describing undergraduates’ reasoning within and across Euclidean, taxicab, and spherical geometries. Ph.D. diss: Portland State University.Google Scholar
  5. Bobis, J., Sweller, J., & Cooper, M. (1993). Cognitive load effects in a primary-school geometry task. Learning and Instruction, 3(1), 1–21. Retrieved from.  https://doi.org/10.1016/S0959-4752(09)80002-9. CrossRefGoogle Scholar
  6. Chan, K. K., & Leung, S. W. (2014). Dynamic geometry software improves mathematical achievement: Systematic review and meta-analysis. Journal of Educational Computing Research, 51(3), 311–325.CrossRefGoogle Scholar
  7. Clements, D. H., Swaminathan, S., Hannibal, M. A. Z., & Sarama, J. (1999). Young children's concepts of shape. Journal for Research in Mathematics Education, 30(2), 192–212.CrossRefGoogle Scholar
  8. de Villiers, M. (1995). The handling of geometry definitions in school textbooks. Editorial, appeared in PYTHAGORAS 38, pp. 3–4, Dec 1995. Retrieved from http://mzone.mweb.co.za/residents/profmd/geomdef.htm.
  9. Dieudonné, J. (1981). The universal domination of geometry. Two-Year College Mathematics Journal, 12(4), 227–231.CrossRefGoogle Scholar
  10. Erbas, A. K., & Yenmez, A. A. (2011). The effect of inquiry-based explorations in a dynamic geometry environment on sixth grade students’ achievements in polygons. Computers & Education, 57(4), 2462–2475.CrossRefGoogle Scholar
  11. Fischbein, E. (1993). The theory of figural concepts. Educational Studies in Mathematics, 24(2), 139–162.CrossRefGoogle Scholar
  12. Fischbein, E., & Nachlieli, T. (1998). Concepts and figures in geometrical reasoning. International Journal of Science Education, 20(10), 1193–1211.CrossRefGoogle Scholar
  13. Gal, H., & Linchevski, L. (2010). To see or not to see: Analyzing difficulties in geometry from the perspective of visual perception. Educational Studies in Mathematics, 74, 163–183.CrossRefGoogle Scholar
  14. Gawlick, T. (2005). Connecting arguments to actions –dynamic geometry as means for the attainment of higher van Hiele levels. Zentralblatt für Didaktik der Mathematik, 37(5), 361–370.CrossRefGoogle Scholar
  15. Guven, B., & Baki, A. (2010). Characterizing student mathematics teachers’ levels of understanding in spherical geometry. International Journal of Mathematical Education in Science and Technology, 41(8), 991–1013.CrossRefGoogle Scholar
  16. Hannafin, R. D., & Scott, B. N. (2001). Teaching and learning with dynamic geometry programs in student-centered learning environments. A mixed method inquiry. Computers in the Schools, 17(1–2), 121–141.CrossRefGoogle Scholar
  17. Harel, G., & Sowder, L. (1998). Students’ proof schemes: Results from exploratory studies. In A. H. Schoenfeld, J. Kaput, & E. Dubinsky (Eds.), CBMS issues in mathematics education Vol. 7, Research in collegiate mathematics education III (pp. 234–283). Rhode Island: American Mathematical Society.Google Scholar
  18. Hershkowitz, R. (1990). Psychological aspects of learning geometry. In P. Nesher & J. Kilpatrick (Eds.), Mathematics and cognition. A research synthesis by the International Group for the Psychology of Mathematics Education (pp. 70–95). Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  19. Houdement, C., & Kuzniak, A. (2003). Elementary geometry split into different geometrical paradigms. In M. A. Mariotti (Ed.), Proceedings of CERME 3. Belaria, Italy.: http://www.dm.unipi.it/~didattica/CERME3/proceedings/
  20. Kuzniak, A., & Rauscher, J. C. (2011). How do teachers’ approaches to geometric work relate to geometry students’ learning difficulties. Educational Studies in Mathematics, 77(1), 129–147.  https://doi.org/10.1007/s10649-011-9304-7. CrossRefGoogle Scholar
  21. Leikin, R., & Winicki-Landman, G. (2000a). On equivalent and non-equivalent definitions: Part 1. For the Learning of Mathematics, 20(1), 17–21.Google Scholar
  22. Leikin, R., & Winicki-Landman, G. (2000b). On equivalent and non-equivalent definitions: Part 2. For the Learning of Mathematics, 20(2), 24–29.Google Scholar
  23. Mammarella, I. C., Giofrè, D., & Caviola, S. (2017). Learning geometry: The development of geometrical concepts and the role of cognitive processes. In D. C. Geary, D. B. Ochsendorf, R. Mann, & K. Koepke (Eds.), Acquisition of complex arithmetic skills and higher-order mathematics concepts. Mathematical cognition and learning (Vol. 3, pp. 221–246). London: Academic Press.Google Scholar
  24. Mammarella, I. C., Giofrè, D., Ferrara, R., & Cornoldi, C. (2013). Intuitive geometry and visuospatial working memory in children showing symptoms of nonverbal learning disabilities. Child Neuropsychology, 19(3), 235–249.CrossRefGoogle Scholar
  25. Markovits, Z., & Hershkowitz, R. (1997). Relative and absolute thinking in visual estimation processes. Educational Studies in Mathematics, 32, 29–47.CrossRefGoogle Scholar
  26. Mason, J., Burton, L., & Stacey, K. (1982). Thinking mathematically. London, UK: Addison-Wesley.Google Scholar
  27. Matsuo, N., & Silfverberg, H. (2011). Some aspects of the Japanese and Finnish 8th graders’ geometrical knowledge: Focusing on two concepts and one relation. In The proceedings of 44th annual research conference of Japan Society of Mathematical Education. Tokyo: Japan Society of Mathematical Education.Google Scholar
  28. NCLD. (1999). National Center for Learning Disabilities (NCLD) (1999). Retrieved from http://www.ldonline.org/article/6390#anchor520397.
  29. Nelsen, R. B. (1993). Proofs without words: Exercises in visual thinking. Washington: MAA.Google Scholar
  30. Nelsen, R. B. (2000). Proofs without words II: More exercises in visual thinking. Washington: MAA.Google Scholar
  31. Nelsen, R. B. (2015). Proofs without words III: Further exercises in visual thinking (classroom resource materials). Washington: MAA.CrossRefGoogle Scholar
  32. Okazaki, M. & Fujita, T. (2007). Prototype phenomena and common cognitive paths in the understanding of the inclusion relations between quadrilaterals in Japan and Scotland. In J.H. Woo, H.C. Lew, K.S. Park, & D.Y. Seo (Eds.). Proceedings of the 31st conference of the international group for the psychology of mathematics education, Vol. 4 (ss. 41–48). Seoul: PME.Google Scholar
  33. Patsiomitou, S., & Emvalotis, A. (2010). Students movement through van Hiele levels in a dynamic geometry guided reinvention process. Journal of Mathematics and Technology, 2010(1–3), 18–48 http://www.ijar.lit.az/pdf/jmt/1/JMT2010(1-3).pdf.
  34. Parzysz, B. (2003). Pre-service elementary teachers and the fundamental ambiguity of diagrams in geometry problem-solving. In M. A. Mariotti (Ed.), Proceedings of CERME 3. Belaria, Italy.: http://www.dm.unipi.it/~didattica/CERME3/proceedings/
  35. Rosch, E., & Mervis, C. B. (1975). Family resemblanches: Studies in the internal structure of categories. Cognitive Psychology, 7, 573–605.CrossRefGoogle Scholar
  36. Sigler, A., Segal, R., & Stupel, M. (2016). The standard proof, the elegant proof, and the proof without words of tasks in geometry, and their dynamic investigation. International Journal of Mathematical Education in Science and Technology, 47(8), 1226–1243.CrossRefGoogle Scholar
  37. Silfverberg, H. (1999). Peruskoulun oppilaan geometrinen käsitetieto. Acta Universitatis Tamperensis; 710. Tampereen yliopisto.Google Scholar
  38. Silfverberg, H., & Joutsenlahti, J. (2007). Miten opettajaopiskelijat ymmärtävät käsitteen kulma? In K. Merenluoto, A. Virta, & P. Carpelan (Eds.), Opettajankoulutuksen muuttuvat rakenteet. Ainedidaktinen symposium 9.2.2007. Kasvatustieteiden tiedekunnan julkaisuja B:77 (pp. 239–247). Turku: Turun yliopisto.Google Scholar
  39. Silfverberg, H. & Matsuo, N. (2008a). Comparing Japanese and Finnish 6th and 8th graders’ ways to apply and construct definitions. Research Paper. In O. Figueras, J.L. Cortina, S. Alatorre, T. Rojano, & A. Sepúlveda (Eds.) Proceedings of the Joint Meeting of PME32 (International group for the Psychology of Mathematics Education) and PME-NA XXX, Morelia Mexico, July 17-12, 2008. Vol.4 (ss. 257-264). Morelia: PME.Google Scholar
  40. Silfverberg, H. & Matsuo, N. (2008b). Comparative study of Finnish and Japanese students’ understanding of class inclusion and defining in a geometrical context. In A. Kallioniemi (Eds.) Uudistuva ja kehittyvä ainedidaktiikka. Ainedidaktinen symposiumi 8.2.2008 Helsingissä. Osa 2. Tutkimuksia 299 (ss. 608–620). Helsinki: University of Helsinki.Google Scholar
  41. Tall, D., & Vinner, S. (1981). Concept image and concept definition in mathematics with particular reference to limits and continuity. Educational Studies in Mathematics, 12(2), 151–169.CrossRefGoogle Scholar
  42. Trzcieniecka-Schneider, I. (1993). Some remarks on creating mathematical concepts. Educational Studies in Mathematics, 24(3), 257–264.CrossRefGoogle Scholar
  43. van Hiele, P. M. (1957). De problematiek van het inzicht. Unpublished thesis. Utrecht: University of Utrecht.Google Scholar
  44. van Hiele-Geldof, D. (1957). De didaktiek van de meetkunde in de eerste klas van het V.H.M.O. Unpublished thesis. Utrecht: University of Utrecht.Google Scholar
  45. Vinner, S. (1991). The role of definitions in the teaching and learning mathematics. In D. Tall (Ed.), Advanced mathematical thinking (pp. 65–81). Mathematics Education Library. Dordrecht, Netherlands: Kluwer Academic Publishers.Google Scholar
  46. Vinner, S., & Dreyfus, T. (1989). Images and definitions for the concept of function. Journal for Research in Mathematics Education, 20(4), 356–366.CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Teacher EducationUniversity of TurkuTurkuFinland

Personalised recommendations