ZDM

, Volume 47, Issue 3, pp 465–481

The act and artifact of drawing(s): observing geometric thinking with, in, and through children’s drawings

Original Article

Abstract

In mathematics education, as in other domains, drawing serves as means to access, assess, and attend to children’s understanding. While theoretical accounts of drawings are often based on developmental stage theories, we examine insights gained by considering children’s geometric thinking and reasoning from embodied cognitive perspectives. We ask, what if the act of drawing serves as a means by which children become aware of geometric concepts and relationships, rather than being viewed as a product of that awareness? In this paper, we examine three vignettes and inquire into the ways that children come to draw in geometric contexts. We suggest that the children’s choice to draw as a mode of thinking, the different ways they draw, the manners in which they attend to the mathematics as they draw, and the conceptions that arise with their drawings, contribute in significant ways to their geometric understanding.

References

  1. Arcavi, A. (2003). The role of visual representations in the learning of mathematics. Educational Studies in Mathematics, 52(3), 215–241.CrossRefGoogle Scholar
  2. Bartolini Bussi, M. G. (2007). Semiotic mediation: fragments from a classroom experiment on the coordination of spatial perspectives. ZDM—The International Journal on Mathematics Education, 39(1–2), 63–71.CrossRefGoogle Scholar
  3. Burger, W. F., & Shaughnessy, J. M. (1986). Characterizing the van Hiele levels of development in geometry. Journal for Research in Mathematics Education, 17(1), 31–48.CrossRefGoogle Scholar
  4. Cain, P. (2010). Drawing: the enactive evolution of the practitioner. Chicago: University of Chicago Press.Google Scholar
  5. Carlsen, M. (2009). Reasoning with paper and pencil: the role of inscriptions in student learning of geometric series. Mathematics Education Research Journal, 21(1), 54–84.CrossRefGoogle Scholar
  6. Carruthers, E., & Worthington, M. (2006). Children’s mathematics: making marks, making meaning (2nd ed.). London: Sage Publications.Google Scholar
  7. Châtelet, G. (2000/1993). Les enjeux du mobile. Paris: Seuil. [English translation by R. Shore and M. Zagha: Figuring space: philosophy, mathematics and physics. Dordrecht: Kluwer, 2000].Google Scholar
  8. Clements, D. H., & Sarama, J. (2009). Learning and teaching early math: the learning trajectories approach. New York: Routledge.Google Scholar
  9. Crespo, S. M., & Kyriakides, A. O. (2007). To draw or not to draw: exploring children’s drawings for solving mathematics problems. Teaching Children Mathematics, 14(2), 118–125.Google Scholar
  10. Darling Kelly, D. (2004). Uncovering the history of children’s drawing and art. Westport: Praeger Publishers.Google Scholar
  11. David, M. M., & Tomaz, V. S. (2012). The role of visual representations for structuring classroom mathematical activity. Educational Studies in Mathematics, 80(3), 413–431.CrossRefGoogle Scholar
  12. Davis, G., & Hyun, E. (2005). A study of kindergarten children's spatial representation in a mapping project. Mathematics Education Research Journal, 17(1), 73–100.CrossRefGoogle Scholar
  13. de Freitas, E., & Sinclair, N. (2012). Diagram, gesture, agency: theorizing embodiment in the mathematics classroom. Educational Studies in Mathematics, 80(1–2), 133–152.Google Scholar
  14. Department for Children, Schools and Families [DCSF]. (2008). Mark making matters: Young children making meaning in all areas of learning and development. Nottingham: DCSF.Google Scholar
  15. Depraz, N., Varela, F. J., & Vermersch, P. (Eds.). (2003). On becoming aware: a pragmatics of experiencing. Philadelphia: John Benjamins Publishing.Google Scholar
  16. Diezmann, C. M., & English, L. D. (2001). Promoting the use of diagrams as tools for thinking. In A. A. Cuoco & F. R. Curcio (Eds.), The roles of representation in school mathematics. 2001 NCTM yearbook. (pp. 77–89). Reston: NCTM.Google Scholar
  17. Diezmann, C. M., & McCosker, N. T. (2011). Reading students’ representations. Teaching Children Mathematics, 18(3), 162–169.Google Scholar
  18. Duval, R. (1999). Representation, vision and visualization: cognitive functions in mathematical thinking. Basic issues for learning. In F. Hitt and M. Santos (Eds.). Proceedings of the 21st North American PME conference (pp. 3–26). Cuernavaca, Morelos, Mexico.Google Scholar
  19. Duval, R. (2014). Commentary: linking epistemology and semio-cognitive modeling in visualization. ZDM—The International Journal on Mathematics Education, 46(1), 159–170.CrossRefGoogle Scholar
  20. Edens, K., & Potter, E. (2008). How students “unpack” the structure of a word problem: graphic representations and problem solving. School Science and Mathematics, 108(5), 184–196.CrossRefGoogle Scholar
  21. Edwards, L. D. (2009). Gesture and conceptual integration in mathematical talk. Educational Studies in Mathematics, 70(2), 127–141.CrossRefGoogle Scholar
  22. Goldin, G. A. (2002). Representation in mathematical learning and problem solving. In L. D. English (Ed.), Handbook of international research in mathematics education (pp. 197–218). Mahwah: Lawrence Erlbaum Associates, Publishers.Google Scholar
  23. Grafton, S. T., & Hamilton, A. F. (2007). Evidence for a distributed hierarchy of action representation in the brain. Human Movement Science, 26(4), 590–616.CrossRefGoogle Scholar
  24. Holt, L. E., & Beilock, S. L. (2006). Expertise and its embodiment: examining the impact of sensorimotor skill expertise on the representation of action-related text. Psychonomic Bulletin and Review, 13(4), 694–701.CrossRefGoogle Scholar
  25. Hostetter, A. B., & Alibali, M. W. (2008). Visible embodiment: gestures as simulated action. Psychonomic Bulletin and Review, 15(3), 495–514.CrossRefGoogle Scholar
  26. Inan, H. Z., & Dogan-Temur, O. (2010). Understanding kindergarten teachers’ perspectives of teaching basic geometric shapes: a phenomenographic research. ZDM—The International Journal on Mathematics Education, 42, 457–468.CrossRefGoogle Scholar
  27. Kaput, J. J. (1998). Representations, inscriptions, descriptions and learning: a kaleidoscope of windows. Journal of Mathematical Behavior, 17(2), 265–281.CrossRefGoogle Scholar
  28. Lakoff, G., & Núñez, R. E. (2000). Where mathematics comes from: how the embodied mind brings mathematics into being. New York: Basic Books.Google Scholar
  29. MacDonald, A. (2013). Using children’s representations to investigate meaning-making in mathematics. Australasian Journal of Early Childhood, 38(2), 65–73.Google Scholar
  30. Malchiodi, C. A. (1998). Understanding children’s drawings. New York: Guilford Press.Google Scholar
  31. Maturana, H., & Varela, F. (1992). The tree of knowledge: the biological roots of human understanding (Revised edition). Boston: Shambhala.Google Scholar
  32. Mulligan, J. T., & Mitchelmore, M. C. (2009). Awareness of pattern and structure in early mathematical development. Mathematics Education Research Journal, 21(2), 33–49.CrossRefGoogle Scholar
  33. National Council of Teachers of Mathematics. (2006). Curriculum focal points for prekindergarten through Grade 8 mathematics. Reston: NCTM.Google Scholar
  34. National Film Board of Canada (NFB). (1969). Notes on a triangle [Film]. Montreal: The Board.Google Scholar
  35. National Research Council. (2009). Mathematics learning in early childhood: paths toward excellence and equity. In Committee on early childhood mathematics, Christopher, T., Cross, Taniesha, A., Woods, and Heidi Schweingruber. (Eds.). Center for Education, Division of Behavioral and Social Sciences and Education. Washington, DC: The National Academies Press.Google Scholar
  36. Nemirovsky, R., & Ferrara, F. (2009). Mathematical imagination and embodied cognition. Educational Studies in Mathematics, 70(2), 159–174.CrossRefGoogle Scholar
  37. Nunokawa, K. (2006). Using drawings and generating information in mathematical problem solving processes. Eurasia Journal of Mathematics, Science and Technology Education, 2(3), 33–54.Google Scholar
  38. Piaget, J., & Inhelder, B. (1967). The child’s conception of space. London: Routledge and Kegan Paul.Google Scholar
  39. Ping, R., & Goldin-Meadow, S. (2010). Gesturing saves cognitive resources when talking about nonpresent objects. Cognitive Science, 34(4), 602–619.CrossRefGoogle Scholar
  40. Presmeg, N. (2006). Research on visualization in learning and teaching mathematics: Emergence from psychology. In A. Gutierrez & P. Boero (Eds.), Handbook of research on the psychology of mathematics education: past, present, and future (pp. 205–236). Rotterdam: Sense Publishers.Google Scholar
  41. Rivera, F. D. (2014). From math drawings to algorithms: emergence of whole number operations in children. ZDM—The International Journal on Mathematics Education, 46(1), 59–77.CrossRefGoogle Scholar
  42. Rivera, F., Steinbring, H., & Arcavi, A. (Eds.) (2014). Visualization as an epistemological tool. ZDM—The International Journal on Mathematics Education, 46(1), 1–2.Google Scholar
  43. Sarama, J., & Clements, D. H. (2009). Early childhood mathematics education research: Learning trajectories for young children. New York: Routledge.Google Scholar
  44. Schneckloth, S. (2008). Marking time, figuring space: gesture and the embodied moment. Journal of Visual Culture, 7(3), 277–292.CrossRefGoogle Scholar
  45. Sinclair, N., & Gol Tabaghi, S. (2010). Drawing space: mathematicians’ kinetic conceptions of eigenvectors. Educational Studies in Mathematics, 74(3), 223–240.CrossRefGoogle Scholar
  46. Thom, J. S. (2011). Nurturing mathematical reasoning. Teaching Children Mathematics, 18(4), 234–243.Google Scholar
  47. Thom, J. S., & Roth, W. M. (2011). Radical embodiment and semiotics: toward a theory of mathematics in the flesh. Educational Studies in Mathematics, 77(2–3), 267–284.CrossRefGoogle Scholar
  48. Thom, J., Roth, W. M., & Bautista, A. (2010). In the flesh: living, growing conceptual domains in a geometry lesson. Complicity: An International Journal of Complexity and Education, 7, 77–87.Google Scholar
  49. Thompson, E. (2010). Mind in life: Biology, phenomenology, and the sciences of mind. Cambridge: Harvard University Press.Google Scholar
  50. Varela, F. J., Thompson, E., & Rosch, E. (1991). The embodied mind: cognitive science and human experience. Cambridge: The MIT Press.Google Scholar
  51. Wheatley, G. H. (2007). Quick draw (2nd ed.). Bethany Beach: Mathematics Learning.Google Scholar
  52. Woleck, K.R. (2001). Listen to their pictures. An investigation of children’s mathematical drawings. In A.A. Cuoco and F.R. Curcio (Eds), The roles of representation in school mathematics. 2001 NCTM yearbook. (pp. 215–227). Reston: NCTM.Google Scholar
  53. Woodward, M. (2012). A monstrous rhinoceros (as from life): toward (and beyond) the epistemological nature of the enacted pictorial image. Plymouth: Transtechnology Research.Google Scholar
  54. Zahner, D., & Corter, J. E. (2010). The process of probability problem solving: use of external visual representations. Mathematical Thinking and Learning, 12(2), 177–204.CrossRefGoogle Scholar

Copyright information

© FIZ Karlsruhe 2015

Authors and Affiliations

  1. 1.University of VictoriaVictoriaCanada
  2. 2.University of AlbertaEdmontonCanada

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