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Teachers’ conceptions of representation in middle school mathematics

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Abstract

Current reform efforts call for an emphasis on the use of representation in the mathematics classroom across levels and topics. The aim of the study was to examine teachers’ conceptions of representation as a process in doing mathematics, and their perspectives on the role of representations in the teaching and learning of mathematics at the middle-school level. Interviews with middle school mathematics teachers suggest that teachers use representations in varied ways in their own mathematical work and have developed working definitions of the term primarily as a product in problem solving. However, teachers’ conception of representation as a process and a mathematical practice appears to be less developed, and, as a result, representations may have a peripheral role in their instruction as well. Further, the data suggested that representation is viewed as a topic of study rather than as a general process, and as a goal for the learning of only a minority of the students—the high-performing ones. Implications for mathematics teacher education, prospective and practicing, are discussed.

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References

  • Ball, D. L. (1993). Halves, pieces and twoths: Constructing and using representational contexts in teaching fractions. In T. Carpenter, E. Fennema, & T. Romberg (Eds.), Rational numbers: An integration of research (pp. 328–375). Hillsdale, NJ: Lawrence Erlbaum Associates, Inc.

    Google Scholar 

  • Ball, D. L. (1997). What do students know? Facing challenges of distance, context, and desire in trying to hear children. In B. Biddle, T. Good, & I. Goodson (Eds.), International handbook on teachers and teaching (pp. 769–817). Dordrecht, The Netherlands: Kluwer.

    Google Scholar 

  • Ball, D. L. (2001). Teaching with respect to mathematics and students. In T. Wood, B. S. Nelson, & J. Warfield (Eds.), Beyond classical pedagogy: Teaching elementary school mathematics (pp. 11–22). Mahwah, NJ: Lawrence Erlbaum Associates, Inc.

    Google Scholar 

  • Ball, D. L., & Cohen, D. (1999). Developing practice, developing practitioners: Toward a practice-based theory of professional education. In G. Sykes & L. Darling-Hammond (Eds.), Teaching as the learning profession: Handbook of policy and practice (pp. 3–32). San Francisco, CA: Jossey-Bass.

    Google Scholar 

  • Bergqvist, T. (2005). How students verify conjectures: Teachers’ expectations. Journal of Mathematics Teacher Education, 8, 171–191.

    Article  Google Scholar 

  • Cai, J. (2005). US and Chinese teachers’ constructing, knowing and evaluating representations to teach mathematics. Mathematical Thinking and Learning, 7, 135–169.

    Article  Google Scholar 

  • Cifarelli, V. (1998). The development of mental representations as a problem solving activity. Journal of Mathematical Behavior, 17, 239–264.

    Article  Google Scholar 

  • Cobb, P. (2003). Modeling, symbolizing, and tool use in statistical data analysis. In K. Gravemeijer, R. Lehrer, B. van Oers, & L. Verschaffel (Eds.), Symbolizing, modeling and tool use in mathematics education (pp. 171–198). Dordrecht, The Netherlands: Kluwer.

    Google Scholar 

  • Cobb, P., Stephen, M., McClain, K., & Gravemeijer, K. (2002). Participating in classroom mathematical practices. The Journal of the Learning Sciences, 10(1 & 2), 113–163.

    Google Scholar 

  • Cobb, P., Yackel, E., & McClain, K. (2000). Symbolizing and communicating in mathematics classrooms: Perspectives on discourse, tools, and instructional design. Mahwah, NJ: Lawrence Erlbaum Associates.

    Google Scholar 

  • Cobb, P., Yackel, E., & Wood, T. (1992). A constructivist alternative to the representational view of mind in mathematics education. Journal for Research in Mathematics Education, 29, 306–333.

    Google Scholar 

  • Creswell, J. (1998). Qualitative inquiry and research design: Choosing among five traditions. London: Sage.

    Google Scholar 

  • Cuoco, A. (2001). The roles of representation in school mathematics (2001 Yearbook). Reston, VA: NCTM.

    Google Scholar 

  • Dreyfus, T., & Eisenberg, T. (1996). On different facets of mathematical thinking. In R. J. Sternberg & T. Ben-Zeev (Eds.), The nature of mathematical thinking (pp. 253–284). Hillsdale, NJ: Lawrence Erlbaum Associates.

    Google Scholar 

  • 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 mathematical problem solving (pp. 109–122). Hillsdale, NJ: Lawrence Erlbaum Associates.

    Google Scholar 

  • Eisenberg, T., & Dreyfus, T. (1994). On understanding how students learn to visualize function transformations. In A. Schoenfeld, E. Dubinsky, J. Kaput, & C. Kessel (Eds.), Research in collegiate mathematics education, IV (pp. 45–68). Providence, RI: American Mathematical Society.

    Google Scholar 

  • English, L. D. (1997). Mathematical reasoning: Analogies, metaphors and images. Mahwah, NJ: Lawrence Erlbaum Associates.

    Google Scholar 

  • Fosnot, C., & Dolk, M. (2002). Young mathematicians at work. Portsmouth, NH: Heinemann.

    Google Scholar 

  • Gibson, D. (1998). Students’ use of diagrams to develop proofs in an introductory analysis course. CBMS Issues in Mathematics Education, 7, 284–307. Providence, RI: American Mathematical Society.

  • 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. English (Ed.), Handbook of international research in mathematics education (pp. 197–218). Mahwah, NJ: Lawrence Erlbaum Associates, Inc.

    Google Scholar 

  • Gravemeijer, K., Lehrer, R., van Oers, B., & Verschaffel, L. (2003). Symbolizing, modeling and tool use in mathematics education. Dordrecht, The Netherlands: Kluwer.

    Google Scholar 

  • Greeno, J. (1987). Instructional representations based on research about understanding. In A. Schoenfeld (Ed.), Cognitive science and mathematics education (pp. 61–88). New York: Academic Press.

    Google Scholar 

  • Greeno, J., & Hall, R. (1997). Practicing representation. Phi Delta Kappan, 78(5), 361–368.

    Google Scholar 

  • Grosslight, L., Unger, E. J., & Smith, C. (1991). Understanding models and their use in science: Conceptions of middle and high school students and experts. Journal of Research in Science Teaching, 28(9), 799–822.

    Article  Google Scholar 

  • Hall, R. (1989). Exploring the episodic structure of algebra story problem solving. Cognition and Instruction, 6, 223–283.

    Article  Google Scholar 

  • Hall, R., & Rubin, A. (1998). There’s five little notches in here: Dilemmas in teaching and learning the conventional structure of rate. In J. Greeno & S. Goldman (Eds.), Thinking practices in mathematics and science learning (pp. 189–235). Mahwah, NJ: Lawrence Erlbaum Associates, Inc.

    Google Scholar 

  • Hall, R., & Stevens, R. (1995). Making space: A comparison of mathematical work in school and professional design practices. In S. Star (Ed.), The cultures of computing (pp. 118–145). London: Basil Blackwell.

    Google Scholar 

  • Hill, H. C., Sleep, L., Lewis, J., & Ball, D. L. (2007). Assessing teachers’ mathematical knowledge. In F. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 111–155). Reston, VA: NCTM.

    Google Scholar 

  • Izsák, A. (2003). “We want a statement that is always true”: Criteria for good algebraic representations and the development of modeling knowledge. Journal for Research in Mathematics Education, 34(3), 191–227.

    Article  Google Scholar 

  • Izsák, A., & Sherin, M. G. (2003). Exploring the use of new representations as a resource for teacher learning. School Science and Mathematics, 103, 18–27.

    Article  Google Scholar 

  • Janvier, C. (1987). Translation processes in mathematics education. In C. Janvier (Ed.), Problems of representation in the teaching and learning of mathematics (pp. 27–32). Hillsdale, NJ: Lawrence Erlbaum Associates.

    Google Scholar 

  • Jones, K. (1997). Student–teachers’ conceptions of mathematical proof. Mathematics Education Review, 9, 21–32.

    Google Scholar 

  • Kaput, J. J. (1991). Notations and representations as mediators of constructive process. In E. von Glasersfeld (Ed.), Radical constructivism in mathematics education (pp. 53–74). Dordrecht, The Netherlands: Kluwer.

    Google Scholar 

  • Kaput, J. J. (1992). Technology and mathematics education. In D. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 515–556). New York: Macmillan.

    Google Scholar 

  • Kaput, J. J. (1998). Representations, inscriptions, descriptions and learning: A kaleidoscope of windows. Journal of Mathematical Behavior, 17(2), 265–281.

    Article  Google Scholar 

  • Kaput, J. J., Noss, R., & Hoyles, C. (2008). Developing new notations for a learnable mathematics in the computational era. In L. English (Ed.), Handbook of international research in mathematics education (2nd ed., pp. 693–715). New York: Routledge/Taylor & Francis.

    Google Scholar 

  • Knuth, E. (2002). Secondary school mathematics teachers’ conceptions of proof. Journal for Research in Mathematics Education, 33(5), 379–405.

    Article  Google Scholar 

  • Lampert, M. (2001). Teaching problems and the problems of teaching. New Haven: Yale University Press.

    Google Scholar 

  • Lampert, M., & Ball, D. (1998). Teaching, multimedia and mathematics: Investigations of real practice. New York: Teachers College Press.

    Google Scholar 

  • Leinhardt, G. (1989). Math lessons: A contrast of novice and expert competence. Journal for Research in Mathematics Education, 20(1), 52–75.

    Article  Google Scholar 

  • Leinhardt, G. (2001). Instructional explanations: A commonplace for teaching and location for contrast. In V. Richardson (Ed.), Handbook for research on teaching (pp. 333–357). Washington, DC: American Education Research Association.

    Google Scholar 

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

    Google Scholar 

  • Lesh, R., Behr, M., & Post, T. (1987). Rational number relations and proportions. In C. Janvier (Ed.), Problems of representation in the teaching and learning of mathematics (pp. 41–58). Hillsdale, NJ: Lawrence Erlbaum Associates.

    Google Scholar 

  • Meira, L. (2003). Mathematical representations as systems of notations-in-use. In K. Gravemeijer, R. Lehrer, B. van Oers, & L. Verschaffel (Eds.), Symbolizing, modeling and tool use in mathematics education (pp. 87–104). Dordrecht, The Netherlands: Kluwer.

    Google Scholar 

  • Monk, S. (2003). Representation in school mathematics: Learning to graph and graphing to learn. In J. Kilpatrick, W. G. Martin, & D. Schifter (Eds.), A research companion to principles and standards for school mathematics (pp. 250–262). Reston, VA: NCTM.

    Google Scholar 

  • Morris, A. (2008). Assessing pre-service teachers’ skills for analyzing teaching. Journal of Mathematics Teacher Education, 9, 471–505.

    Article  Google Scholar 

  • National Council of Teachers of Mathematics. (1989). Curriculum and evaluation standards. Reston, VA: Author.

    Google Scholar 

  • National Council of Teachers of Mathematics. (1991). Professional standards for the teaching of mathematics. Reston, VA: Author.

    Google Scholar 

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

    Google Scholar 

  • Newell, A., & Simon, H. (1972). Human problem solving. Englewood Cliffs, NJ: Prentice Hall.

    Google Scholar 

  • Nunokawa, K. (1994). Solver’s structures of a problem situation and their global restructuring. Journal of Mathematical Behavior, 13, 275–297.

    Article  Google Scholar 

  • Ochs, E., Jacoby, S., & Gonzales, P. (1994). Interpretive journeys: How physicists talk and travel through graphic space. Configurations, 2(1), 151–171.

    Article  Google Scholar 

  • Olson, D. (1994). The world on paper. Cambridge: Cambridge University Press.

    Google Scholar 

  • Pimm, D. (1987). Speaking mathematically: Communication in mathematics classrooms. London: Routledge.

    Google Scholar 

  • Presmeg, N. (1997). Metaphoric and metonymic signification in mathematics. Journal of Mathematical Behavior, 17(1), 25–32.

    Article  Google Scholar 

  • Putnam, R., & Borko, H. (1997). Teacher learning: Implications of new views of cognition. In B. Biddle, T. Good, & I. Goodson (Eds.), International handbook of teachers and teaching (pp. 1223–1296). Dordrecht, The Netherlands: Kluwer.

    Google Scholar 

  • Roth, W. M., & McGinn, M. (1998). Inscriptions: Toward a theory of representing as social practice. Review of Educational Research, 68(1), 35–59.

    Google Scholar 

  • Schoenfeld, A. H. (1985). Mathematical problem solving. New York: Academic Press.

    Google Scholar 

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

    Article  Google Scholar 

  • Sfard, A. (2000). Steering (dis)course between metaphors and rigor: Using focal analysis to investigate an emergence of mathematical objects. Journal for Research in Mathematics Education, 31(3), 296–327.

    Article  Google Scholar 

  • Sherin, M. (2002). When teaching becomes learning. Cognition and Instruction, 20(2), 119–150.

    Article  Google Scholar 

  • Silver, E., Ghousseini, H., Gosen, D., Charalambous, C., & Font Strawhun, B. (2005). Moving from rhetoric to praxis: Issues faced by teachers in having students consider multiple solutions for problems in the mathematics classroom. Journal of Mathematical Behavior, 24, 287–301.

    Article  Google Scholar 

  • Smith, M., Hughes, E., Engle, R., & Stein, M. (2009). Orchestrating mathematical discussions. Mathematics Teaching in the Middle School, 14(9), 549–556.

    Google Scholar 

  • Stein, M. K., Engle, R., Smith, M., & Hughes, E. (2008). Orchestrating productive mathematical discussions: Five practices for helping teachers move beyond show and tell. Mathematical Thinking and Learning, 10, 313–340.

    Article  Google Scholar 

  • Stylianou, D. A. (2002). On the interaction of visualization and analysis—the negotiation of a visual representation in problem solving. Journal of Mathematical Behavior, 21(3), 303–317.

    Article  Google Scholar 

  • Stylianou, D. A. (2008). Representation as a cognitive and social practice. In: O. Figueras (Ed.), Proceedings of the joint meeting of the 32nd Annual Meeting for the Psychology of Mathematics Education and Psychology of Mathematics Education - North America (vol. 4, pp. 289–296). Mexico, Morelia: Centro de Investigación y de Estudios Avanzados del IPN and Universidad Michoacana de San Nicolas de Hidalgo.

  • Stylianou, D. A., Kenney, P. A., Silver, E. A., & Alacaci, C. (2000). Gaining insight into students’ thinking through assessment tasks. Mathematics Teaching in the Middle Grades, 6, 136–144.

    Google Scholar 

  • Stylianou, D. A., & Silver, E. A. (2004). The role of visual representations in advanced mathematical problem solving: An examination of expert-novice similarities and differences. Journal of Mathematical Thinking and Learning, 6(4), 353–387.

    Article  Google Scholar 

  • Swan, M. (1993). Assessing a wider range of students’ abilities. In N. Webb & A. Coxford (Eds.), Assessment in the mathematics classroom. 1993 Yearbook (pp. 26–39). Reston, VA: NCTM.

    Google Scholar 

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Acknowledgments

The research reported here was supported in part by the National Science Foundation under Grant # REC-0447542. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. I would like to acknowledge the contributions of Kara Imm and Nabin Chae during the data collection and the coding of the interview data.

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  1. City College School of Education, The City University of New York, Convent Avenue at 138th street, NAC 6/207, New York, NY, 10031, USA

    Despina A. Stylianou

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Stylianou, D.A. Teachers’ conceptions of representation in middle school mathematics. J Math Teacher Educ 13, 325–343 (2010). https://doi.org/10.1007/s10857-010-9143-y

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