Skip to main content
SpringerLink
Log in

Designing spatial visual tasks for research: the case of the filling task

  • Published:
Educational Studies in Mathematics Aims and scope Submit manuscript

Abstract

This article focuses on the development and problematization of a task designed to foster spatial visual sense in prospective and practicing elementary and middle school teachers. We describe and analyse the cyclical stages of developing, testing, and modifying several “task drafts” related to ideas around dilation and proportion. Challenged by participant non-actions and non-responses, we as task designers identified and anticipated sources of difficulties, which motivated repeated modification of the task to further the intended learning goals. The task in its present form incorporates numerous considerations including choices around materials, wording of questions and prompts, and sequencing of experiences. It also reflects our enriched understanding of exploration strategies and the roles of manipulatives and technology in spatial visual tasks designed for adult learners.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

References

  • Arcavi, A. (2003). The role of visual representations in the learning of mathematics. Educational Studies in Mathematics, 52, 215–241.

    Article  Google Scholar 

  • Arzarello, F., Micheletti, C., Olivero, F., Robutti, O., & Paola, D. (1998). A model for analysing the transition to formal proofs in geometry. In A. Olivier & K. Newstead (Eds.), Proceedings of the 22nd Conference of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 24–31). Bellville, South Africa: Kwik Kopy Printing.

    Google Scholar 

  • Ball, D. L., Hill, H. C., & Bass, H. (2005). Knowing mathematics for teaching: Who knows mathematics well enough to teach third grade and how can we decide? American Educator Fall, 2005(14–22), 43–46.

    Google Scholar 

  • Battista, M. T., Wheatley G. H., & Talsma, G. (1982). The importance of spatial visualization and cognitive development for geometry learning in preservice elementary teachers. Journal for Research in Mathematics Education, 13(5), 332–340.

    Article  Google Scholar 

  • Bryant, P. (2009). Key understandings in mathematics learning paper 5: Understanding space and its representation in mathematics. University of Oxford. Available at: www.nuffieldfoundation.org.

  • Clements, D. H. (1999). Geometric and spatial thinking in young children. In J. Copley (Ed.), Mathematics in the early years (pp. 66–79). Reston, VA: National Council of Teachers of Mathematics.

    Google Scholar 

  • Clements, D. H. (2002). Computers in early childhood mathematics. Contemporary Issues in Early Childhood, 3(2), 161–181.

    Article  Google Scholar 

  • Conference Board of the Mathematical Sciences. (2001). The mathematical education of teachers. Providence RI and Washington DC: American Mathematical Society and Mathematical Association of America.

    Google Scholar 

  • Cooney, T. J. (1999). Conceptualizing teachers’ ways of knowing. Educational Studies in Mathematics, 38, 163–187.

    Article  Google Scholar 

  • Del Grande, J. (1990). Spatial sense. Arithmetic Teacher, 37(6), 14–20.

    Google Scholar 

  • Eisenberg, T., & Dreyfus, T. (1991). On the reluctance to visualize in mathematics. In W. Zimmermann & S. Cunningham (Eds.), Visualization in teaching and learning mathematics (pp. 25–38). USA: Mathematical Association of America.

    Google Scholar 

  • Gadanidis, G., Sedig, K., & Liang, H.-N. (2004). Designing online mathematical investigation. Journal of Computers in Mathematics and Science Teaching, 23(3), 273–296.

    Google Scholar 

  • 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.

    Article  Google Scholar 

  • Glaser, B. G. (2002). Conceptualization: On theory and theorization using grounded theory. International Journal of Qualitative Methods, 1(2), 23–38.

    Google Scholar 

  • Glaser, B. G., & Strauss, A. L. (1967). The discovery of grounded theory: Strategies for qualitative research. Chicago: Aldine.

    Google Scholar 

  • Goldenberg, E. P., & Cuoco, A. A. (1998). What is dynamic geometry? In R. Lehrer & D. Chazan (Eds.), Designing learning environments for developing understanding of geometry and space (pp. 351–368). Mahwah, NJ: Lawrence Erlbaum Associates.

    Google Scholar 

  • Goldin, G. (1998). Representational systems, learning, and problem solving in mathematics. Journal of Mathematical Behavior, 17(2), 137–165.

    Article  Google Scholar 

  • Guven, B., & Kosa, T. (2008). The effect of dynamic geometry software on student mathematics teachers’ spatial visualization skills. The Turkish Online Journal of Educational Technology, 7(4), 11–18.

    Google Scholar 

  • Hershkowitz, R., Ben-Chaim, D., Hoyles, C., Lappan, G., Michelmore, M., & Vinner, S. (1989). Psychological aspects of learning geometry. In P. Nesher & J. Kilpatrick (Eds.), Mathematics and cognition (ICMI Study Series) (pp. 70–95). Cambridge: University Press.

    Google Scholar 

  • Koedinger, K. R. (1992). Emergent properties and structural constraints: Advantages of diagrammatic representations for reasoning and learning. Paper presented at the AAAI Spring Symposium Series: Reasoning with Diagrammatic Reasoning.

  • Lampert, M. (1992). Problems in teaching authentic mathematics. In F. Oser, D. Andreas, & J. Patry (Eds.), Effective and responsible teaching: A new synthesis (pp. 295–314). San Francisco: Jossey-Bass.

    Google Scholar 

  • Lanius, C. S., & Williams, S. E. (2003). Proportionality: A unifying theme for the middle grades. Mathematics Teaching in the Middle School, 8(8), 392–396.

    Google Scholar 

  • 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. 137–168). Mahwah, NJ: Lawrence Erlbaum Associates.

    Google Scholar 

  • Ma, L. (1999). Knowing and teaching elementary mathematics: Teachers’ understanding of fundamental mathematics in China and the United States. Hillsdale, NJ: Lawrence Erlbaum Associates.

    Google Scholar 

  • NCTM (National Council of Teachers of Mathematics) (2000). Principles and standards for school mathematics. Reston, VA.

  • Presmeg, N. (1997). Generalization using imagery in mathematics. In L. English (Ed.), Mathematical reasoning. Analogies, metaphors and images (pp. 299–312). Mahwah, NJ: Erlbaum.

    Google Scholar 

  • Reiber, L. P., & Matsko, M. J. (2001). Serious design for serious play in physics. Educational Technology, 1, 14–24.

    Google Scholar 

  • Robichaux, R. R. (2005). The improvement of spatial visualization: A case study. Journal of Integrative Psychology, 4(2).

  • Sierpinska, A. (2004). Research in mathematics education through a keyhole: Task problematization. For the Learning of Mathematics, 24(2), 7–15.

    Google Scholar 

  • Simon, M. A. (1995). Reconstructing mathematics pedagogy from a constructivist perspective. Journal for Research in Mathematics Education, 26(2), 114–145.

    Article  Google Scholar 

  • Simon, M. A. (2006). Key developmental understandings in mathematics: A direction for investigating and establishing learning goals. Mathematical Thinking and Learning, 6, 91–104.

    Article  Google Scholar 

  • Simon, M. A., & Tzur, R. (2004). Explicating the role of mathematical tasks in conceptual learning: An elaboration of the hypothetical learning trajectory. Mathematical Thinking and Learning, 6(2), 91–104.

    Article  Google Scholar 

  • Sinclair, M. P. (2003). Some implications of the results of a case study for the design of pre-constructed, dynamic geometry sketches and accompanying materials. Educational Studies in Mathematics, 52(3), 289–317.

    Article  Google Scholar 

  • Skemp, R. R. (1976). Relational understanding and instrumental understanding. Mathematics Teaching, 77, 20–26.

    Google Scholar 

  • Smith, G. G., Gerretson, H., Olkun, S., Yuan, Y., Dogbey, J., & Erdem, A. (2009). Stills, not full motion, for interactive spatial training: American, Turkish and Taiwanese female pre-service teachers learn spatial visualization. Computers & Education, 52, 201–209.

    Article  Google Scholar 

  • Thanheiser, E. (2009). Preservice elementary school teachers’ conceptions of multidigit whole numbers. Journal for Research in Mathematics Education, 40(3), 251–281.

    Google Scholar 

  • Wheatley, G. H. (1990). Spatial sense and mathematics learning. Arithmetic teacher, 37(6), 10–11.

    Google Scholar 

  • Whiteley, W. (2004). Teaching to see like a mathematician. In G. Malcolm (Ed.), Multidisciplinary approaches to visual representations and interpretations. Studies in multidisciplinarity (vol 2) (pp. 279–292). Amsterdam: Elsevier.

    Google Scholar 

  • Yakimanskaya, I. S., Wilson, P. S., & Davis, E. J. (1991). The development of spatial thinking in schoolchildren. Reston, VA: National Council of Teachers of Mathematics.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. York University, 4700 Keele St., Toronto, ON, M3J 1P3, Canada

    Margaret Sinclair, Ami Mamolo & Walter J. Whiteley

Authors
  1. Margaret Sinclair
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ami Mamolo
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Walter J. Whiteley
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Margaret Sinclair.

Additional information

Brief description of article: An analysis of research funded by SSHRC that used designed tasks to investigate teachers’ development of visual and spatial skills related to geometry.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Sinclair, M., Mamolo, A. & Whiteley, W.J. Designing spatial visual tasks for research: the case of the filling task. Educ Stud Math 78, 135–163 (2011). https://doi.org/10.1007/s10649-011-9315-4

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10649-011-9315-4

Keywords

Search

Navigation