Visualising Cubic Reasoning with Semiotic Resources and Modeling Cycles

  • Caroline Yoon
  • Tessa Miskell
Part of the Semiotic Perspectives in the Teaching and Learning of Mathematics Series book series (SEMIPTL)


Diagrams and physical manipulatives are often recommended as useful semiotic resources for visualising area and volume problems in which nonlinear reasoning is appropriate. However, the mere presence of diagrams and physical manipulatives does not guarantee students will recognise the appropriateness of nonlinear reasoning.


Small Fish Modelling Cycle Large Fish Real World Context Fishing Trip 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Arzarello, F., Paola, D., Robutti, O., & Sabena, C. (2009). Gestures as semiotic resources in the mathematics classroom. Educational Studies in Mathematics, 70(2), 97–109.CrossRefGoogle Scholar
  2. Blum, W., & Niss, M. (1991). Applied mathematical problem solving, modelling, applications, and links to other subjects – State, trends and issues in mathematics instruction. Educational Studies in Mathematics, 22(1), 37–68.CrossRefGoogle Scholar
  3. De Bock, D., van Dooren, W., Janssens, D., & Verschaffel, L. (2002). Improper use of linear reasoning: An in-depth study of the nature and the irresistibility of secondary school students’ errors. Educational Studies in Mathematics, 50, 311–334.CrossRefGoogle Scholar
  4. De Bock, D., Verschaffel, L., Janssens, D., van Dooren, W., & Claes K. (2003). Do realistic contexts and graphical representations always have a beneficial impact on students’ performance? Negative evidence from a study on modeling non-linear geometry problems. Learning and Instruction, 13, 441–463.CrossRefGoogle Scholar
  5. Diezmann, C. (2005). Primary students’ knowledge of the properties of spatially oriented diagrams. In H. Chick & J. Vincent (Eds.), Proceedings of the 30th Annual Conference of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 281–288). Melbourne, Australia: PME.Google Scholar
  6. Duval, R. (2006).A cognitive analysis of problems of comprehension in a learning of mathematics. Educational Studies in Mathematics, 61, 103–131.Google Scholar
  7. Esteley, C., Villarreal, M., & Alagia, H. (2004). Extending linear models to non-linear contexts: An in depth study about two university students’ mathematical productions. In M. J. Høines & A. B. Fuglestad (Eds.), Proceedings of the 28th Conference of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 343–350). Bergen, Norway: International Group for the Psychology of Mathematics Education.Google Scholar
  8. Gibson, D, (1998). Students’ use of diagrams to develop proofs in an introductory analysis course. In A. Schoenfeld, J. Kaput, & E. Dubinsky (Eds.), Research in collegiate mathematics education (Vol. 7, pp. 284–307). Providence, RI: American Mathematical Society.CrossRefGoogle Scholar
  9. Kaput, J. J. (1989). Linking representations in the symbol systems of algebra. In S. Wagner & C. Kieran (Eds.), Research issues in the learning and teaching of algebra (pp. 167–194). Reston, VA: National Council of Teachers of Mathematics.Google Scholar
  10. Koedinger, K. (1994). Emergent properties and structural constraints: Advantages of diagrammatic representations for reasoning and learning. In B. Chandrasekaran & H. Simon (Eds.), Reasoning with diagrammatic representations (pp. 151–156). Menlo Park, CA: AAAI Press.Google Scholar
  11. Lamon, S. (2007). Rational numbers and proportional reasoning: Toward a theoretical framework for research. In F. K. J. Lester (Ed.), Second handbook of mathematics teaching and learning (pp. 629–667). Greenwich, CT: Information Age Publishing.Google Scholar
  12. Lesh, R., & Doerr, H. (Eds.). (2003). Beyond constructivism: Models and modeling perspectives on mathematics problem solving, learning, and teaching. Hillsdale, NJ: Lawrence Erlbaum Associates.Google Scholar
  13. Lesh, R., Hoover, M., Hole, B., Kelly, A., & Post, T. (2000). Principles for developing thought-revealing activities for students and teachers. In A. E. Kelly & R. A. Lesh (Eds.), Handbook of research design in mathematics and science education (pp. 591–646). Mahwah, NJ: Lawrence Erlbaum Associates.Google Scholar
  14. Liljedahl, P. (2005). Mathematical discovery and affect: The effect of AHA! Experiences on undergraduate mathematics students. International Journal of Mathematical Education in Science and Technology, 36(2–3), 219–236.CrossRefGoogle Scholar
  15. Modestou, M., Elia, I., & Gagatsis, A. (2008). Behind the scenes of pseudo proportionality. International Journal of Mathematical Education in Science and Technology, 39(3), 313–332.CrossRefGoogle Scholar
  16. Niss, M., Blum, W., & Galbraith, P. (2007). Introduction. In W. Blum, P. Galbraith, M. Niss, & H.-W. Henn (Eds.), Modelling and applications in mathematics education: The 14th ICMI study (New ICMI Studies Series) (Vol. 10, pp. 3–33). New York, NY: Springer.CrossRefGoogle Scholar
  17. Nunokawa, K. (2006). Using drawings and generating information in mathematical problem solving. Eurasia Journal of Mathematics, Science and Technology Education, 2(3), 33–35.Google Scholar
  18. Pantziara, M., Gagatsis, A., & Elia, I. (2009). Using diagram as tools for the solution of non-routine mathematical problems. Educational Studies in Mathematics, 72, 39–60.CrossRefGoogle Scholar
  19. Peirce, C. S. (1931/1958). In C. Hartshorne, P. Weiss, & A. Burks (Eds.), Collected papers (Vols. I–VIII). Cambridge, MA: Harvard University Press.Google Scholar
  20. Polya, G. (1957). How to solve it. Princeton, NJ: University Press.Google Scholar
  21. Radford, L. (2009). Why do gestures matter? Sensuous cognition and the palpability of mathematical meanings. Educational Studies in Mathematics, 70(2), 111–126.CrossRefGoogle Scholar
  22. Schoenfeld, A. (1994). Mathematical thinking and problem solving. Hillsdale, NJ: Lawrence Erlbaum Associates, Inc.Google Scholar
  23. Shaughnessy, J. M. (1992). Research in probability and statistics: Reflections and directions. In A. D. Grouws (Ed.), Handbook of research in mathematics teaching and learning (pp. 465–494). New York, NY: Macmillan.Google Scholar
  24. Stillman, G., Galbraith, P., Brown, J., & Edwards, I. (2007). A framework for success in implementing mathematical modelling in the secondary classroom. In J. Watson & K. Beswick (Eds.), Mathematics: Essential research, essential practice: Proceedings of the 30th Annual Conference of the Mathematics Research Group of Australiasia, Hobart (pp. 688–707). Adelaide, Australia: MERGA.Google Scholar
  25. Thomas, M. O. J. (2008). Developing versatility in mathematical thinking. Mediterranean Journal for Research in Mathematics Education, 7(2), 67–87.Google Scholar
  26. Thompson, D. W. (1992). On growth and form: The complete revised edition. New York, NY: Dover Publications.CrossRefGoogle Scholar
  27. Treffers, A. (1987). A model of goal and theory description in mathematics instruction: The Wiskobas project. Dordrecht, The Netherlands: Reidel.Google Scholar
  28. van Dooren, W., De Bock, D., Hessels, A., Janssens, D., & Verschaffel, L. (2005). Not everything is proportional: Effects of age and problem type on propensities for over generalization. Cognition and Instruction, 23, 57–86.CrossRefGoogle Scholar
  29. van Leeuwen, T. (2005). Introducing social semiotics. New York, NY: Routledge.Google Scholar
  30. Yoon, C., Radonich P., & Sullivan, N. (in press). Volume and space: The snapper problem and variations. New Zealand: New Zealand Council for Educational Research.Google Scholar

Copyright information

© Sense Publishers 2016

Authors and Affiliations

  • Caroline Yoon
    • 1
  • Tessa Miskell
    • 2
  1. 1.Department of MathematicsUniversity of AucklandNew Zealand
  2. 2.Department of MathematicsUniversity of AucklandNew Zealand

Personalised recommendations