### Flexible Conceptions of Perspectives and Representations: An Examination of Pre-Service Mathematics Teachers’ Knowledge

#### Abstract

The concept of multiple representations of functions and the ability to make translations among representations are important topics in secondary school mathematics curricula (Moschkovich, Schoenfeld, & Arcavi, 1993; NCTM, 2000). Research related to students in this domain is fruitful, while research related to teachers is underdeveloped. This research looks in fine-grained ways at the nature of understanding exhibited by 59 pre-service mathematics teachers as they approached four problems that called for translations between representations of functions. Teachers’ written and verbal responses were examined according to the extent to which they utilized essential constructs of process and object perspectives. Findings suggest that teachers exhibited three conceptions – flexible, disconnected or constrained. Specifically, teachers demonstrated: (1) constructs of both perspectives and operated within algebraic and graphical representations, (2) constructs of both perspectives, but not transitional enough to link them across problems, or (3) constructs of one perspective and did not operate within algebraic and graphical representations.

#### Keywords

#### Full Text:

PDFDOI: http://dx.doi.org/10.18404/ijemst.23592

#### References

Blume, G.W & Heckman, D.S (1997). What do students know about algebra and function? In P. Kenney & E. Silver (Eds.), Results from the sixth mathematics assessment of the National Assessment of Educational Progress (pp. 225-277). Reston, VA: NCTM.

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

Chiu, M. M., Kessel, C., Moschkovich, J., & Muñoz-Nuñez, A. (2001). Learning to graph linear functions: A case study of conceptual change. Cognition and Instruction, 19(2), 215-252.

Cooney, T.J., Beckman, S. & Lloyd, G.M. (2010). Developing essential understanding of functions for teaching mathematics in grades 9-12. Essential Understanding Series. Reston, VA: NCTM.

Cuoco, A.A. (Ed.) (2001). The roles of representation in school mathematics, 63rd yearbook of National Council of Teachers of Mathematics (NCTM). Reston, VA: NCTM.

Dreyfus, T., & Vinner, S. (1989). Images and definitions for the concept of function. Journal for Research in Mathematics Education, 20(4), 356-366.

Dufour-Janvier, B., Bednarz, N., & Belanger, M. (1987). Pedagogical considerations concerning the problem of representation. In C. Janvier (Ed.), Problems of Representation in the teaching and learning of mathematics (pp. 125-148). Hillsdale, NJ: Lawrence Erlbaum Associates.

Even, R. (1990). Subject-matter knowledge for teaching and the case of functions. Educational Studies in Mathematics, 21, 521-544.

Even, R. (1993). Subject-matter knowledge and pedagogical content knowledge: Prospective secondary teachers and the function concept. Journal for Research in Mathematics Education, 24, (2), 94-116

Goldenberg, E. Paul (1988). Mathematics, Metaphors, and Human Factors” Mathematical, Technical, and Pedagogical Challenges in the Educational Use of Graphical Representation of Functions. Journal of Mathematical Behavior, 7, 135-173.

Greenes, C. E. (2008). Algebra and algebraic thinking in school mathematics, 70th yearbook of the National Council of Teachers of Mathematics (NCTM).

Kleiner, I. (1989). Evolution of the function concept: A brief survey. College Mathematics Journal, 20, 282-300. Knuth, E. (2000). Student understanding of the cartesian connection: An exploratory study. Journal of Research in Mathematics Education, 31, 500-508.

Lacampagne, C., Blair, W., Kaput, J. (1993). The Algebra Initiative Colloquium, vols. 1 and 2. U.S. Department of Education.

Leinhardt, G., Zaslavsky, O., & Stein, M.K. (1990). Functions, graphs and graphing: Tasks, learning, and teaching. Review of Educational Research, 60 (1), 1-64.

Lesh, R., Post, T., & Behr, M. 1987. Representations and translation among repre- sentations in mathematics learning and problem solving.” In Problems of representation in the teaching and learning of mathematics, edited by C. Janvier, pp. 33–40. Hillsdale, NJ: Lawrence Erlbaum Associates.

Lloyd, G. M., & Wilson, M. R. (2002). Using a card sort to determine one’s understanding of function. In J. Sowder & B. Schappelle (Eds.), Lessons learned from research (pp. 209-212). Reston, VA: National Council of Teachers of Mathematics.

Lloyd, G. , Herbel-Eisenmann, B. & Star, J.R. (2011). Developing essential understanding of expressions, equation and functions, grades 6-8. Reston, VA: National Council of Teachers of Mathematics.

Moschkovich, J. (1999). Students’ use of the x-intercept as an instance of a transitional conception. Educational Studies in Mathematics, 37, 169-197.

Moschkovich, J., Schoenfeld, A., & Arcavi, A. (1993). Aspects of understanding: On multiple perspectives and representations of linear relations and connections among them. In T. Romberg, E. Fennema, & T. Carpenter (Eds.), Integrating research on the graphical representation of function (pp. 69-100). Hillsdale, NJ: Lawrence Erlbaum Associates.

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

National Council of Teachers of Mathematics (2006). Curriculum focal points for prekindergarten through grade 8 mathematics: A quest for coherence. Reston, VA: NCTM

National Council of Teachers of Mathematics (2009). Focus in High School Mathematics: Reasoning and Sense Making. Reston, VA: NCTM.

Norman, A. (1992). Teachers’ mathematical knowledge of the concept of function. In E. Dubinsky & G. Harel (Eds.), The concept of function: Aspects of epistemology and pedagogy (pp. 215-232). Washington, D.C.: Mathematical Association of America.

Romberg, T., Fennema, E. Carpenter, T. (1993). Toward a common research perspective. In T. Romberg, E. Fennema & T. Carpenter (Eds.), Integrating research on the graphical representation of function. Hillsdale, NJ: Lawrence Erlbaum Associates.

Schoenfeld, A., Smith III, J., & Arcavi, A. (1993). Learning: The microgenetic analysis of one student’s evolving understanding of a complex subject matter domain. In R. Glaser (Ed.), Advances in Instructional Psychology, vol. 4, (pp. 55-61). Hillsdale, NJ: Lawrence Erlbaum Associates.

Schwartz, B. & Dreyfus, T. (1995). New actions upon old objects: A new ontological perspective of functions. Educational Studies in Mathematics, 29, 259-291.

Sfard, A. (1991). On the dual nature of mathematical conceptions: Reflections on processes and objects on different sides of the same coin. Educational Studies in Mathematics, 22(1), 1-36.

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, D.C.: Mathematical Association of America.

Star, J.R. & Rittle-Johnson, B. (2009). Making algebra work: Instructional strategies that deepen student understanding, within and between representations. ERS Spectrum, 27, 11-18).

Stein, M.K., Baxter, J., & Leinhardt, G. (1990). Subject-matter knowledge and elementary instruction: A case from functions and graphing. American Educational Research Journal, 27, (4), 639-663.

Tirosh, D., Even, R. & Robinson, N. (1998). Simplifying algebraic expressions: Teacher awareness and teaching approaches. Educational Studies in Mathematics, 35, 51-64.

Vinner, S. (1989). The avoidance of visual considerations in calculus students. Focus on Learning Problems in Mathematcs, 11(2), 149-156.

Yerushalmy, M., & Schwartz, J. (1993). Seizing the opportunity to make algebra mathematically and pedagogically interesting. In T. Romberg, E. Fennema, & T. Carpenter (Eds.), Integrating research on the graphical representation of function (pp. 69-100). Hillsdale, NJ: Lawrence Erlbaum Associates.

This work is licensed under a Creative Commons Attribution 4.0 License.