## 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 R^{3} and to their flexibility in the use of different representations.

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## References

Arcavi, A. (2003). The role of visual representations in the learning of mathematics.

*Educational Studies in Mathematics, 52*(3), 215–241.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.Baker, B., Cooley, L., & Trigueros, M. (2000). The schema triad—a calculus example.

*Journal for Research in Mathematics Education, 31*, 557–578.Bishop, A. J. (1980). Spatial abilities and mathematics education: A review.

*Educational Studies in Mathematics, 11*(3), 257–269.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.Breidenbach, D., Hawks, J., Nichols, D., & Dubinsky, E. (1992). Development of the process conception of function.

*Educational Studies in Mathematics, 23*, 247–285.Brown, A., DeVries, D., Dubinsky, E., & Thomas, K. (1998). Learning binary operations, groups, and subgroups.

*Journal of Mathematical Behavior, 16*(3), 187–239.Cooley, L., Trigueros, M., & Baker, B. (2007). Schema thematization: a framework and an example.

*Journal for Research in Mathematics Education, 38*, 370–392.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.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.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.Dubinsky, E. (1991). Reflective abstraction in advanced mathematical thinking. In D. Tall (Ed.),

*Advanced mathematical thinking*(pp. 95–123). Dordrecht: Kluwer.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.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.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.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.Duval, R. (2006). A cognitive analysis of problems of comprehension in a learning of mathematics.

*Educational Studies in Mathematics, 61*(1), 103–131.Gagatsis, A., Christou, C., & Elia, I. (2004). The nature of multiple representations in developing mathematical relationships.

*Quaderni di Ricerca in Didattica, 14*, 150–159.Goldin, G. A. (1998). Representational systems, learning, and problem solving in mathematics.

*Journal of Mathematical Behavior, 17*, 137–165.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.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.Hitt, F. (2002).

*Representations and mathematics visualization. North American Chapter of the International Group for the Psychology of Mathematics Education*. Mexico City: Cinvestav-IPN.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.Piaget, J., & García, R. (1983).

*Psicogénesis e historia de la ciencia*. Mexico: Siglo XXI Editores.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.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.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.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.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.Yerushalmy, M. (1997). Designing representations: reasoning about functions of two variables.

*Journal for Research in Mathematics Education, 28*, 431–466.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.

## Acknowledgements

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

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

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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

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