Skip to main content
SpringerLink
Log in

Geometrical representations in the learning of two-variable functions

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

Abstract

This study is part of a project concerned with the analysis of how students work with two-variable functions. This is of fundamental importance given the role of multivariable functions in mathematics and its applications. The portion of the project we report here concentrates on investigating the relationship between students’ notion of subsets of Cartesian three-dimensional space and the understanding of graphs of two-variable functions. APOS theory and Duval’s theory of semiotic representations are used as theoretical framework. Nine students, who had taken a multivariable calculus course, were interviewed. Results show that students’ understanding can be related to the structure of their schema for R3 and to their flexibility in the use of different representations.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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
Fig. 6
Fig. 7

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(3), 215–241.

    Article  Google Scholar 

  • Asiala, M., Brown, A., DeVries, D. J., Dubinsky, E., Mathews, D., & Thomas, K. (1996). A framework for research and development in undergraduate mathematics education. In J. Kaput, E. Dubinsky & A. H. Schoenfeld (Eds.), Research in collegiate mathematics education II (pp. 1–32). Providence: American Mathematical Society.

    Google Scholar 

  • Baker, B., Cooley, L., & Trigueros, M. (2000). The schema triad—a calculus example. Journal for Research in Mathematics Education, 31, 557–578.

    Article  Google Scholar 

  • Bishop, A. J. (1980). Spatial abilities and mathematics education: A review. Educational Studies in Mathematics, 11(3), 257–269.

    Article  Google Scholar 

  • Bishop, A. J. (1983). Space and geometry. In R. Lesh & M. Landau (Eds.), Acquisition of Mathematics Concepts and Processes (pp. 175–203). New York: Academic.

    Google Scholar 

  • Breidenbach, D., Hawks, J., Nichols, D., & Dubinsky, E. (1992). Development of the process conception of function. Educational Studies in Mathematics, 23, 247–285.

    Article  Google Scholar 

  • Brown, A., DeVries, D., Dubinsky, E., & Thomas, K. (1998). Learning binary operations, groups, and subgroups. Journal of Mathematical Behavior, 16(3), 187–239.

    Article  Google Scholar 

  • Cooley, L., Trigueros, M., & Baker, B. (2007). Schema thematization: a framework and an example. Journal for Research in Mathematics Education, 38, 370–392.

    Article  Google Scholar 

  • Czarnocha, B., Dubinsky, E., Prabhu, V., & Vidakovic, D. (1999). One theoretical perspective in undergraduate mathematics education research. In O. Zaslavsky (Ed.), Proceedings of the 23rd Conference of PME (Vol. 1, pp. 95–110). Haifa: PME.

    Google Scholar 

  • Czarnocha, B., Dubinsky, E., Loch, S., Prabhu, V., & Vidakovic, D. (2001). Conceptions of area: in students and in history. The College Mathematics Journal, 32(2), 99–109.

    Article  Google Scholar 

  • Dreyfus, T. (1991). On the status of visual reasoning in mathematics and mathematics education. In F. Furinghetti (Ed.), Proceedings of the 15th Conference of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 33–48). Genova: Universita de Genova.

    Google Scholar 

  • Dubinsky, E. (1991). Reflective abstraction in advanced mathematical thinking. In D. Tall (Ed.), Advanced mathematical thinking (pp. 95–123). Dordrecht: Kluwer.

    Google Scholar 

  • Dubinsky, E. (1994). A theory and practice of learning college mathematics. In A. Schoenfeld (Ed.), Mathematical thinking and problem solving (pp. 221–243). Hillsdale: Erlbaum.

    Google Scholar 

  • Dubinsky, E., Weller, K., McDonald, M. A., & Brown, A. (2005). Some historical issues and paradoxes regarding the concept of infinity: An APOS analysis, part 1. Educational Studies in Mathematics, 58(3), 335–359.

    Article  Google Scholar 

  • Dubinsky, E., & Yiparaki, O. (2000). On student understanding of AE and EA quantification. In E. Dubinsky, A. H. Schoenfeld & J. Kaput (Eds.), Research in collegiate mathematics education IV (pp. 239–289). Providence: American Mathematical Society.

    Google Scholar 

  • Duval, R. (1999). Representation, vision and visualization: Cognitive functions in mathematical thinking. In F. Hitt & M. Santos (Eds.), Proceedings of the XXI Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 3–26). Columbus: ERIC.

    Google Scholar 

  • Duval, R. (2006). A cognitive analysis of problems of comprehension in a learning of mathematics. Educational Studies in Mathematics, 61(1), 103–131.

    Article  Google Scholar 

  • Gagatsis, A., Christou, C., & Elia, I. (2004). The nature of multiple representations in developing mathematical relationships. Quaderni di Ricerca in Didattica, 14, 150–159.

    Google Scholar 

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

    Article  Google Scholar 

  • 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). London: Erlbaum.

    Google Scholar 

  • Gutiérrez, A. (1996). Visualization in 3-dimensional geometry: in search of a framework. In L. Puig & A. Gutiérrez (Eds.), Proceedings of the 20th Conference of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 3–19). Valencia: Universidad de Valencia.

    Google Scholar 

  • Hitt, F. (2002). Representations and mathematics visualization. North American Chapter of the International Group for the Psychology of Mathematics Education. Mexico City: Cinvestav-IPN.

    Google Scholar 

  • McDonald, M. A., Mathews, D. M., & Strobel, K. H. (2000). Understanding sequences: A tale of two objects. In E. Dubinsky, A. Schoenfeld & J. Kaput (Eds.), Research in collegiate mathematics education IV (pp. 77–102). Providence: AMS.

    Google Scholar 

  • Piaget, J., & García, R. (1983). Psicogénesis e historia de la ciencia. Mexico: Siglo XXI Editores.

    Google Scholar 

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

    Google Scholar 

  • Presmeg, N. C. (2006). Research on visualization in learning and teaching mathematics. In A. Gutiérrez & P. Boero (Eds.), Handbook of research on the psychology of mathematics education: Past present and future (pp. 205–235). Rotterdam: Sense Publishers.

    Google Scholar 

  • Sfard, A. (1992). Operational origins of mathematical objects and the quandary of reification—the case of function. In E. Dubinsky & G. Harel (Eds.), The concept of function: Aspects of epistemology and pedagogy (pp. 59–84). Washington DC: MAA.

    Google Scholar 

  • Trigueros, M. (2000). Students’ conception of solution curves and equilibrium in systems of differential equations. In M. L. Fernandez (Ed.), Proceedings of the XXII Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 93–97). Columbus: ERIC.

    Google Scholar 

  • Van Nes, F., & De Lange, J. (2007). Mathematics education and neurosciences: Relating spatial structures to the development of spatial sense and number sense. The Montana Mathematics Enthusiast, 4(2), 210–229.

    Google Scholar 

  • Yerushalmy, M. (1997). Designing representations: reasoning about functions of two variables. Journal for Research in Mathematics Education, 28, 431–466.

    Article  Google Scholar 

  • Zazkis, R., Dubinsky, E., & Dauterman, E. (1996). Using visual and analytic strategies: A study of students' understanding of permutation and symmetry groups. Journal for Research in Mathematics Education, 27, 435–457.

    Article  Google Scholar 

Download references

Acknowledgements

This project was partially supported by Asociación Mexicana de Cultura AC.

We are thankful to Ed Dubinsky for his comments and suggestions.

Author information

Authors and Affiliations

  1. Departamento de Matemática, Universidad de Puerto Rico en Mayagüez, Mayagüez, 00680, Puerto Rico

    Rafael Martínez-Planell

  2. Depto. de Matemáticas, Instituto Tecnológico Autónomo de México, Río Hondo #1, Tizapán, San Angel, México DF, 01000, México

    Maria Trigueros

Authors
  1. Maria Trigueros
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Rafael Martínez-Planell
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Maria Trigueros.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Trigueros, M., Martínez-Planell, R. Geometrical representations in the learning of two-variable functions. Educ Stud Math 73, 3–19 (2010). https://doi.org/10.1007/s10649-009-9201-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10649-009-9201-5

Keywords

Search

Navigation