Educational Studies in Mathematics

, Volume 74, Issue 2, pp 163–183 | Cite as

To see or not to see: analyzing difficulties in geometry from the perspective of visual perception

  • Hagar Gal
  • Liora Linchevski


In this paper, we consider theories about processes of visual perception and perception-based knowledge representation (VPR) in order to explain difficulties encountered in figural processing in junior high school geometry tasks. In order to analyze such difficulties, we take advantage of the following perspectives of VPR: (1) Perceptual organization: Gestalt principles, (2) recognition: bottom-up and top-down processing; and (3) representation of perception-based knowledge: verbal vs. pictorial representation, mental images and hierarchical structure of images. Examples given in the paper were mostly taken from Gal's study (2005) which aimed at identifying and analyzing Problematic Learning Situations (after Gal & Linchevski, 2000) in junior high school geometry classes. Gal's study (2005) suggests that while this theoretical perspective became part of teachers' pedagogic content knowledge, the teachers were aware of their students' thinking processes and their ability to analyze and cope with their students’ difficulties in geometry was improved.


Visualization Visual perception Visual perception processing Geometry Problematic learning situation Teacher preparation 



We are grateful to Prof. Shimon Ullman, from Weizmann Institute, Israel, for reading an earlier version of this paper and providing us with his comments and suggestions. We are also grateful to the anonymous reviewers who read the paper and gave us important comments, which were used to improve the paper.


  1. Anderson, J. R. (1995). Cognitive psychology and its implications (4th ed.). New York: Freeman.Google Scholar
  2. Bishop, A. (1983). Space and geometry. In R. Lesh & M. Landau (Eds.), Acquisition of mathematics concepts and processes (pp. 175–203). New York: Academic Press.Google Scholar
  3. Boden, M. (2006). Mind as machine. A history of cognitive science. Oxford: Clarendon Press.Google Scholar
  4. Bryant, P. (1974). Perception and understanding in young children. New York: Basic Books.Google Scholar
  5. Burger, W., & Shaughnessy, J. (1986). Characterizing the van Hiele levels of development in geometry. Journal for Research in Mathematics Education, 17, 31–48.CrossRefGoogle Scholar
  6. Carpenter, T., & Fennema, E. (1992). Cognitively guided instruction: Building on the knowledge of students and teachers. International Journal of Educational Research, 17(5), 457–469.CrossRefGoogle Scholar
  7. Cheng, Y. H., & Lin, F. L. (2005). One more step toward acceptable proof in geometry. Cyprus: Symposium Proposal EARLI.Google Scholar
  8. Clements, D. H., & Battista, M. T. (1992). Geometry and spatial reasoning. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 420–464). New York: Macmillan.Google Scholar
  9. Del Grande, J. (1990). Spatial sense. The Arithmetic Teacher, 37(6), 14–20.Google Scholar
  10. Duval, R. (1998). Geometry from a cognitive point of view. In C. Mammana & V. Villani (Eds.), Perspectives on the teaching of geometry for the 21st century (pp. 37–52). Dordrecht: Kluwer.Google Scholar
  11. Duval, R. (1999). Presentations, vision and visualization: Cognitive functioning in mathematical thinking. Proceedings of the North America chapter of the International Group for the Psychology of Mathematics Education, 21st, Cuernavaca, Morelos, Mexico.Google Scholar
  12. Fielker, D. S. (1979). Strategies for teaching geometry to younger children. Educational Studies in Mathematics, 10, 85–133.CrossRefGoogle Scholar
  13. Fischbein, E. (1993). The theory of figural concepts. Educational Studies in Mathematics, 24(2), 139–162.CrossRefGoogle Scholar
  14. Gal, H. (1998). What do they really think? What students think about the median and bisector of an angle in the triangle, what they say and what their teachers know about it. In: A. Olivier & K. Newstead (Eds.), Proceedings of the 22nd International Conference for the Psychology of Mathematics Education, 2, (pp 321–328). South Africa: Stellenbosch.Google Scholar
  15. Gal, H. (2005). Identifying problematic learning situations in geometry instruction, and handling them within the framework of teacher training. Thesis for a Doctor of Philosophy degree. Jerusalem: Hebrew University of Jerusalem (Hebrew).Google Scholar
  16. Gal, H., & Linchevski, L. (2000). When a learning situation becomes a problematic learning situation: The case of diagonals in the quadrangle. In: T. Nakahara & M. Koyama (Eds.), Proceedings of the 24th International Conference for the Psychology of Mathematics Education, 2, (pp 297–304). Hiroshima, Japan.Google Scholar
  17. Gal, H., & Linchevski, L. (2002). Analyzing geometry problematic learning situations by theory of perception. In: A. Cockburn & E. Nardi (Eds.), Proceedings of the 26th International Conference for the Psychology of Mathematics Education, 2, (pp 400–407). United Kingdom: Norwich.Google Scholar
  18. Gal, H., & Linchevski, L. (2009). Changes in teachers' ways of coping with problematic learning situations in geometry instruction. In R. Even & D. L. Ball (Eds.), The professional education and development of teachers of mathematics: The 15th ICMI Study. New ICMI Study Series, 11 (pp. 192–193). New York: Springer.Google Scholar
  19. Gal, H., & Vinner, S. (1997). Perpendicular lines: What is the problem? Pre-service teachers’ lack of knowledge on how to cope with students’ difficulties. In: E. Pehkonen (Ed.), Proceedings of the 21st International Conference for the Psychology of Mathematics Education, 2, (pp 281–288). Finland: Lahti.Google Scholar
  20. Gibson, J. J. (1979). The ecological approach to visual perception. Boston: Houghton Mifflin.Google Scholar
  21. Gutierrez, A. (1996). Visualization in 3-dimentional geometry: In search of a framework. In: L. Puig & A. Gutierrez (Eds.), Proceedings of the20th International Conference for the Psychology of Mathematics Education, 1, (pp 3–19). Valencia, SpainGoogle Scholar
  22. von Helmholtz, H. (1909/1962). Treatise on physiological optics. New York: Dover.Google Scholar
  23. Hershkowitz, R. (1987). The acquisition of concepts and misconcepts in basic geometry - Or when “a little learning is a dangerous thing.” In: J. Novak (Ed.), Proceedings of the Second International Seminar on Misconceptions and Educational Strategies in Science and Mathematics, 3, (pp 238–251).Google Scholar
  24. Hershkowitz, R. (1989a). Geometric concept images of students and teachers. Thesis submitted for a Doctor of Philosophy degree. Jerusalem: Hebrew University of Jerusalem (Hebrew).Google Scholar
  25. Hershkowitz, R. (1989b). Visualization in geometry: Two sides of the coin. Focus on Learning Problems in Mathematics, 11(1), 61–76.Google Scholar
  26. Hershkowitz, R., Ben Haim, D., Holes, C., Lappan, G., Mitchelmore, M., & Vinner, S. (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.Google Scholar
  27. Hoffer, A. (1977). Mathematics resource project: Geometry and visualization. Palo Alto: Creative Publication.Google Scholar
  28. Hoffer, A. (1983). Van Hiele-based research. In R. Lesh & M. Landau (Eds.), Acquisition of mathematics concepts and processes (7th ed., pp. 205–227). New York: Academic Press.Google Scholar
  29. Klahr, D., Chase, W., & Lovelace, E. (1983). Structure and process in alphabetic retrieval. Journal of Experimental Psychology. Learning, Memory, and Cognition, 9, 462–477.CrossRefGoogle Scholar
  30. Kouba, V., Brown, C., Carpenter, T., Lindquist, M., Silver, E., & Swafford, J. (1988). Results of the fourth NAEP assessment of mathematics: Measurement, geometry, data interpretation, attitudes and other topics. Arithmetic Teacher, 35(9), 10–16.Google Scholar
  31. Kosslyn, S. M. (1973). Scanning visual images: Some structural implications. Perception & Psychophysics, 14, 90–94.Google Scholar
  32. Lehrer, R., Jenkins, M., & Osana, H. (1998). Longitudinal study of children’s reasoning about space and geometry. In R. Lehrer & D. Chazan (Eds.), Designing learning environments for developing understanding of geometry and space (pp. 137–167). Mahwah: Erlbaum.Google Scholar
  33. Lin, F. L. (2005). Modeling students’ learning on mathematical proof and refutation. In: H. Chick & J. Vincent (Eds.), Proceedings of the 29th International Conference for the Psychology of Mathematics Education, 1, (pp 3–18) Melbourne, Australia.Google Scholar
  34. Marr, D. (1982). Vision. A computational investigation into the human representation and processing of visual information. San Francisco: W.H. Freeman and Company.Google Scholar
  35. Metzler, J., & Shepard, R. N. (1974). Transformational studies of the internal representations of three dimensional objects. In R. L. Solso (Ed.), Theories of cognitive psychology: The Loyola symposium. Hillsdale: Erlbaum.Google Scholar
  36. Neisser, U. (1967). Cognitive psychology. New York: Appleton.Google Scholar
  37. Paivio, A. (1986). Mental representations: A dual coding approach. Oxford: Oxford University Press.Google Scholar
  38. Paivio, A. (2007). Mind and its evolution. A dual coding theoretical approach. Mahwah: Lawrence Erlbaum Associates, Publishers.Google Scholar
  39. Palmer, S. E. & Rock, I. (1994). Rethinking perceptual organization: the role of uniform connectedness. Psychonomic Bulletin and Review 1(1).Google Scholar
  40. Peterson, M., & Enns, J. (2005). The edge complex: Implicit memory for figure assignment in shape perception. Perception and Psychophysics, 67(4), 727–740.Google Scholar
  41. Peterson, M. & Grant, E. (2003). Memory and learning in figure-ground perception. In: B. Ross & D. Irwin (Eds.) Cognitive vision: Psychology of learning and motivation, 42, (pp 1-34).Google Scholar
  42. Presmeg, N. (1986). Visualization and mathematical giftedness. Educational studies in mathematics, 17, 297–311.CrossRefGoogle Scholar
  43. Presmeg, N. (1992). Prototypes, metaphors, metonymies and imaginative rationality in high school methematics. Educational Studies in Mathematics, 23, 595–610.CrossRefGoogle Scholar
  44. Presmeg, N. (1997). Generalization using imagery in mathematics. In L. English (Ed.), Mathematical reasoning—analogies, metaphors and images (pp. 299–313). Mahwah: Lawrence Erlbaum Associates, Publishers.Google Scholar
  45. Pylyshyn, Z. W. (2003). Seeing and visualizing—It's not what you think. Cambridge: The MIT Press.Google Scholar
  46. Reed, S. (1974). Structural descriptions and the limitations of visual images. Memory and Cognition, 2, 329–336.Google Scholar
  47. Roland, P. E., & Friberg, L. (1985). Localization of cortical areas activated by thinking. Journal of Neurophysiology, 53, 1219–1243.Google Scholar
  48. Rubin, E. (1915/1958). Figure and ground. In D. Beardslee & M. Wertheimer (Eds.), Readings in perception (pp. 194–203). Princeton: D. van Nostrand Company, Inc.Google Scholar
  49. Santa, J. L. (1977). Spatial transformations of words and pictures. Journal of Experimental Psychology: Human Learning and Memory, 3, 418–427.CrossRefGoogle Scholar
  50. Sinnett, S., Spence, C., & Soto-Faraco, S. (2007). Visual dominance and attention: The Colavita effect revisited. Perception & Psychophysics, 69(5), 673–686.Google Scholar
  51. Shepard, R. N., & Metzler, J. (1971). Mental rotation of three-dimensional objects. Science, 171, 701–703.CrossRefGoogle Scholar
  52. 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
  53. TIMSS (1999). The trends in international mathematics and science study (TIMSS) and the progress in international reading literacy study (PIRLS) international study center.
  54. Wallrabenstein, H. (1973). Development and signification of a geometry test. Educational Studies in Mathematics, 5, 81–89.CrossRefGoogle Scholar
  55. Wertheimer, M. (1958). Principles of perceptual organization. In D. Beardslee & M. Wertheimer (Eds.), Readings in perception (pp. 115–135). Princeton: D. van Nostrand Company, Inc.Google Scholar
  56. Wolfe, J., Butcher, S., Lee, C., & Hyle, M. (2003). Changing your mind: On the contributions of top-down and bottom-up guidance in visual search for feature singletons. Journal of Experimental Psychology: Human Perception and Performance, 29(2), 483–502.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  1. 1.David Yellin College of Education, JerusalemJerusalemIsrael
  2. 2.The Hebrew University of JerusalemJerusalemIsrael

Personalised recommendations