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Coordinating Multiple Representations in a Reform Calculus Textbook

  • Briana L. Chang
  • Jennifer G. Cromley
  • Nhi Tran
Article

Abstract

Coordination of multiple representations (CMR) is widely recognized as a critical skill in mathematics and is frequently demanded in reform calculus textbooks. However, little is known about the prevalence of coordination tasks in such textbooks. We coded 707 instances of CMR in a widely used reform calculus textbook and analyzed the distributions of coordination tasks by chapter and for the type of task demanded (perception vs. construction). Results suggest that different coordination tasks are used earlier and later in learning and for different topics, as well as for specific pedagogical and scaffolding purposes. For example, the algebra-to-text coordination task was more prevalent in the first chapter, suggesting that students are being eased into calculus content. By contrast, requests to construct graphs from algebraic expressions were emphasized in later chapters, suggesting that students are being pushed to think more conceptually about functions. Our nuanced look at coordination tasks in a reform textbook has implications for research in teaching and learning calculus.

Keywords

Calculus Conceptual understanding Functions Multiple representations Textbooks 

Notes

Acknowledgements

The research reported herein was supported by grant number R305A120471 from the U.S.Department of Education. The opinions are those of the authors and do not represent the policies ofthe U.S. Department of Education. Portions of this study were presented at the 2013 ResearchPresession of the annual meeting of the National Council of Teachers of Mathematics on April 15,2013 in Denver, Colorado. We are grateful to Shaaron Ainsworth, Tim Fukawa-Connelly, WilliamZahner, and Theodore Wills for providing helpful comments in discussions of and on earlier versions of the paper. We also thank the reviewers whose comments helped improve and clarify this manuscript.

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References

  1. Acevedo Nistal, A., Van Dooren, W., Clarebout, G., Elen, J. & Verschaffel, L. (2009). Conceptualising, investigating, and stimulating representational flexibility in mathematical problem solving and learning: A critical review. ZDM Mathematics Education, 41(5), 627–636.CrossRefGoogle Scholar
  2. Acevedo Nistal, A., Van Dooren, W. & Verschaffel, L. (2012). What counts as a flexible representational choice? An evaluation of students’ representational choices to solve linear function problems.  Instructional Science, 40(6), 999–1019.Google Scholar
  3. Adu-Gyamfi, K. & Bossé, M. J. (2014). Processes and reasoning in representations of linear functions. International Journal of Science and Mathematics Education, 12(1), 167–192.CrossRefGoogle Scholar
  4. Ainsworth, S. (2006). DeFT: A conceptual framework for considering learning with MR. Learning and Instruction, 16(3), 183–198.CrossRefGoogle Scholar
  5. Ainsworth, S., Bibby, P. & Wood, D. (2002). Examining the effects of different multiple representational systems in learning primary mathematics. Journal of the Learning Sciences, 11(1), 25–61.CrossRefGoogle Scholar
  6. Andrà, C., Lindström, P., Arazarello, F., Holmqvist, K., Robutti, O. & Sabena, C. (2015). Reading mathematics representations: An eye-tracking study. International Journal of Science and Mathematics Education, 13(suppl. 2), 237-259. doi: 10.1007/s10763-013-9484-y
  7. Bell, A. & Janvier, C. (1981). The interpretation of graphs representing situations. For the Learning of Mathematics, 2(1), 34–42.Google Scholar
  8. Bossé, M. J., Adu-Gyamfi, K. & Chandler, K. (2014). Students’ differentiated translation processes. International Journal for Mathematics Teaching and Learning. Retrieved from http://www.cimt.plymouth.ac.uk/journal/.
  9. Bossé, M. J., Adu-Gyamfi, K. & Cheetham, M. R. (2011). Assessing the difficulty of mathematical translations: Synthesizing the literature and novel findings. International Electronic Journal for Mathematics Education, 6(3), 113–133.Google Scholar
  10. Davis, J. D. (2009). Understanding the influence of two mathematics textbooks on prospective secondary teachers’ knowledge. Journal of Mathematics Teacher Education, 12(5), 365–389.CrossRefGoogle Scholar
  11. De Bock, D., van Dooren, W. & Verschaffel, L. (2015). Students’ understanding of proportional, inverse proportional, and affine functions: Two studies on the role of external representations. International Journal of Science and Mathematics Education, 13(Suppl. 1), 47–69.CrossRefGoogle Scholar
  12. Dick, T. & Edwards, B. (2008). MR and local linearity: research influences on the use of technology in calculus curriculum reform. In G. Blume & M. K. Heid (Eds.), Research on technology and the teaching and learning of mathematics (Cases and perspectives, Vol. 2, pp. 255–278). Charlotte, NC: Information Age.Google Scholar
  13. Dreyfus, T. (1990). Advanced mathematical thinking. In P. Nesher & J. Kilpatrick (Eds.), Mathematics and cognition: A research synthesis by the international group for the psychology of mathematics education (pp. 113–134). Cambridge, United Kingdom: Cambridge University Press.CrossRefGoogle Scholar
  14. Dubinsky, E. & Harel, G. (1992). The nature of the process conception of function. In G. Harel & E. Dubinsky (Eds.), The concept of function: Aspects of epistemology and pedagogy (pp. 85–106). Washington, DC: Mathematical Association of America.Google Scholar
  15. Dugdale, S. (1993). Functions and graphs: Perspectives on student thinking. In T. A. Romberg, E. Fennema & T. P. Carpenter (Eds.), Integrating research on the graphical representation of functions (pp. 101–130). Hillsdale, NJ: Erlbaum.Google Scholar
  16. Duval, R. (2000). Basic issues for research in mathematics education. In T. Nakahara & M. Koyama (Eds.), Proceedings of the 24th Conference of the International Group for the Psychology of Mathematics Education (pp. 55–69). Hiroshima, Japan: Nishiki.Google Scholar
  17. Duval, R. (2006). A cognitive analysis of problems of comprehension in a learning of mathematics. Educational Studies in Mathematics, 61(1/2), 103–131.CrossRefGoogle Scholar
  18. Eisenberg, T. (1991). Functions and associated learning difficulties. In D. O. Tall (Ed.), Advanced mathematical thinking (pp. 140–152). Dordrecht, The Netherlands: Kluwer.Google Scholar
  19. Elia, I., Gagatsis, A., Panaoura, A., Zachariades, T. & Zoulinaki, F. (2009). Geometric and algebraic approaches in the concept of “limit” and the impact of the “didactic contract”. International Journal of Science and Mathematics Education, 7(4), 765–790.CrossRefGoogle Scholar
  20. Elia, I., Panaoura, A., Eracleous, A. & Gagatsis, A. (2007). Relations between secondary pupils’ conceptions about functions and problem solving in different representations. International Journal of Science and Mathematics Education, 5(3), 533–556.CrossRefGoogle Scholar
  21. Elia, I., Panaoura, A., Gagatsis, A., Gravvani, K. & Spyrou, P. (2008). Exploring different aspects of the understanding of function: Toward a four-facet model. Canadian Journal of Science, Mathematics & Technology Education, 8(1), 49–69.CrossRefGoogle Scholar
  22. Even, R. (1998). Factors involved in linking representations of functions. The Journal of Mathematical Behavior, 17(1), 105–121.CrossRefGoogle Scholar
  23. Fan, L. & Zhu, Y. (2007). Representation of problem-solving procedures: A comparative look at China, Singapore, and U.S. mathematics textbooks. Educational Studies in Mathematics, 66(1), 61–75.CrossRefGoogle Scholar
  24. Fleiss, J. L. (1981). Statistical methods for rates and proportions (2nd ed.). New York, NY: Wiley.Google Scholar
  25. Gagatsis, A. & Shiakalli, M. (2004). Ability to translate from one representation of the concept of function to another and mathematical problem solving. Educational Psychology: An International Journal of Experimental Educational Psychology, 24(5), 645–657.CrossRefGoogle Scholar
  26. Ganter, S. L. (2001). Changing calculus: A report on evaluation efforts and national impact from 1988–1998. Washington, DC: Mathematical Association of America.Google Scholar
  27. Geiger, M., Stradtmann, U., Vogel, M. & Seufert, T. (2011, August-September). Transformations between different forms of representations in mathematics. Paper Presented at the Biennial Meeting of the European Association for Research on Learning and Instruction, Exeter, England.Google Scholar
  28. Healey, J. F. (2009). The essentials of statistics: A tool for social research (2nd ed.). Belmont, CA: Wadsworth.Google Scholar
  29. Hughes-Hallett, D., Gleason, A.M., Lock, P.F., Flath, D.E., Lomen, D.O., Lovelock, D., …Tucker, T.W. (2010). Applied calculus (4th ed.). Hoboken, NJ: Wiley.Google Scholar
  30. Janvier, C. (Ed.). (1987). Problems of representation in the teaching and learning of mathematics. Hillsdale, NJ: Erlbaum.Google Scholar
  31. Johansson, M. (2005). Mathematics textbooks—the link between the intended and implemented curriculum? Paper presented at the 8th International Conference: Reform, Revolution, and Paradigm Shifts in Mathematics Education, Johor Bharu, Malaysia.Google Scholar
  32. Jones, D. L. & Tarr, J. E. (2007). An examination of the levels of cognitive demand required by probability tasks in middle grades mathematics textbooks. Statistics Education Research Journal, 6(2), 4–27.Google Scholar
  33. Kendal, M. & Stacey, K. (2003). Tracing learning of three representations with the differentiation competency framework. Mathematics Education Research Journal, 15(1), 22–41.CrossRefGoogle Scholar
  34. Kilpatrick, J., Swafford, J. & Findell, B. (Eds.). (2001). Adding it up: Helping children learn mathematics. Washington, DC: National Academy Press.Google Scholar
  35. Kölloffel, B., de Jong, T. & Eysink, T. H. S. (2005). The effects of representational format in simulation-based inquiry learning. Paper presented at the 11th Conference of the European Association for Research on Learning and Instruction, Nicosia, Cyprus.Google Scholar
  36. Laughbaum, E. D. (1999). On teaching intermediate algebra from a function approach. Virginia Mathematics Teacher, 25(2), 36–39.Google Scholar
  37. Leikin, R., Leikin, M., Waisman, I. & Shaul, S. (2013). Effect of the presence of external representations on accuracy and reaction time in solving mathematical double-choice problems by students of different levels of instruction. International Journal of Science and Mathematics Education, 11(5), 1049–1066.Google Scholar
  38. Leinhardt, G., Zaslavsky, O. & Stein, M. K. (1990). Functions, graphs, and graphing: Tasks, learning, and teaching. Review of Educational Research, 60(1), 1–64.CrossRefGoogle Scholar
  39. Lithner, J. (2000). Mathematical reasoning in task solving. Educational Studies in Mathematics, 41(2), 165–190.CrossRefGoogle Scholar
  40. Lithner, J. (2003). Students’ mathematical reasoning in university textbook exercises. Educational Studies in Mathematics, 52(1), 29–55.CrossRefGoogle Scholar
  41. Lithner, J. (2004). Mathematical reasoning in calculus textbook exercises. Journal of Mathematical Behavior, 23(4), 405–427.CrossRefGoogle Scholar
  42. Mahir, N. (2010). Students’ interpretation of a function associated with a real-life problem from its graph. Problems, Resources, and Issues in Mathematics Undergraduate Studies, 20(5), 392–404.Google Scholar
  43. McGee, D. L. & Martinez-Planell, R. (2014). A study of semiotic registers in the development of the definite integral of functions of two and three variables. International Journal of Science and Mathematics Education, 12(4), 883–916.CrossRefGoogle Scholar
  44. McGee, D.L. & Moore-Russo, D. (2014). Impact of explicit presentation of slopes in three dimensions on students’ understanding of derivatives in multivariable calculus. International Journal of Science and Mathematics Education, 1–28. doi:  10.1007/s10763-014-9542-0
  45. Mesa, V. (2010). Strategies for controlling the work in mathematics textbooks for introductory calculus. Research in Collegiate Mathematics Education, 16, 235–265.Google Scholar
  46. Monoyiou, A. & Gagatsis, A. (2008). A coordination of different representations in function problem solving. Proceedings of the 11th International Congress of Mathematics Education. Retrieved from http://tsg.icme11.org/document/get/200
  47. Moschkovich, J., Schoenfeld, A. H. & Arcavi, A. (1993). Aspects of understanding: On multiple perspectives and representations of linear relations and connections among them. In T. A. Romberg, E. Fennema & T. P. Carpenter (Eds.), Integrating research on the graphical representations of functions (pp. 69–100). Hillsdale, NJ: Erlbaum.Google Scholar
  48. Nicol, C. C. & Crespo, S. M. (2006). Learning to teach with mathematics textbooks: How preservice teachers interpret and use curriculum materials. Educational Studies in Mathematics, 62(3), 331–355.CrossRefGoogle Scholar
  49. Nitsch, R., Fredebohm, A., Bruder, R., Kelava, A., Naccarella, D., Leuders, T. & Wirtz, M. (2015). Students’ competencies in working with functions in secondary mathematics education—empirical examination of a competence structure model. International Journal of Science and Mathematics Education, 13(3), 657-682. doi:  10.1007/s10763-013-9496-7
  50. Nyikahadzoyi, M.R. (2015). Teachers’ knowledge of the concept of a function: A theoretical framework. International Journal of Science and Mathematics Education, 13(Suppl. 2), 261-283. doi:  10.1007/s10763-013-9486-9
  51. Ramsey, F. L. & Schafer, D. W. (2002). The statistical sleuth: A course in methods of data analysis (2nd ed.). Belmont, CA: Duxbury Press.Google Scholar
  52. Remillard, J. (2005). Examining key concepts in research on teachers’ use of mathematics curricula. Review of Educational Research, 75(2), 211–246.CrossRefGoogle Scholar
  53. Reys, R., Reys, B., Lapan, R., Holliday, G. & Wasman, D. (2003). Assessing the impact of standards-based middle grades mathematics curriculum materials on student achievement. Journal for Research in Mathematics Education, 34(1), 74–95.CrossRefGoogle Scholar
  54. Rezat, S. (2009). The utilization of mathematics textbooks as instruments for learning. In V. Durand-Guerrier, S. Soury-Lavergne & F. Arzarello (Eds.), Proceedings of the 6th Congress of the European Society for Research in Mathematics Education (pp. 1260–1269). Lyon, France: INRP.Google Scholar
  55. Roth, W. & Bowen, G. M. (2003). When are graphs worth ten thousand words? An expert-expert study. Cognition and Instruction, 21(4), 429–473.CrossRefGoogle Scholar
  56. Schmidt, W. H., McKnight, C. C., Houang, R. T., Wang, H., Wiley, D. E., Cogan, L. S. & Wolfe, R. G. (2001). Why schools matter: A cross-national comparison of curriculum and learning. San Francisco, CA: Jossey-Bass.Google Scholar
  57. Shepherd, M. D., Selden, A. & Selden, J. (2012). University students’ reading of their first-year mathematics textbooks. Mathematical Thinking and Learning, 14(3), 226–256.CrossRefGoogle Scholar
  58. Sood, S. & Jitendra, A. K. (2007). A comparative analysis of number sense instruction in reform-based and traditional mathematics textbooks. The Journal of Special Education, 41(3), 145–157.CrossRefGoogle Scholar
  59. Thompson, P.W. (1994). Students, functions, and the undergraduate curriculum. In E. Dubinsky, A. Schoenfeld, & J. Kaput (Eds.), Research in collegiate mathematics education 1, CBMS issues in mathematics education (pp. 21–44). Providence, RI: American Mathematical Society.Google Scholar
  60. Uitenbroek, D. G. (1997). SISA binomial. Southampton, England: D.G. Uitenbroek. Retrieved January 01, 2013 from http://www.quantitativeskills.com/sisa/distributions/binomial.htm.
  61. Van Dooren, W., De Bock, D. & Verschaffel, L. (2012). How students understand aspects of linearity. In T. Tso (Ed.), Proceedings of the 36th conference of the international group for the psychology of mathematics education (pp. 179–186). Taipei, Taiwan: PME.Google Scholar
  62. Vincent, J. & Stacey, K. (2008). Do mathematics textbooks cultivate shallow teaching? Applying the TIMSS Video Study criteria to Australian eighth-grade mathematics textbooks. Mathematics Education Research Journal, 20(1), 82–107.CrossRefGoogle Scholar
  63. Waisman, I., Leikin, M., Shaul, S. & Leikin, R. (2014). Brian activity associated with translation between graphical and symbolic representations of functions in generally gifted and excelling in mathematics adolescents. International Journal of Science and Mathematics Education, 12(3), 669–696.CrossRefGoogle Scholar
  64. Weinberg, A. & Wiesner, E. (2011). Understanding mathematics textbooks through reader-oriented theory. Educational Studies in Mathematics, 79, 49–63.CrossRefGoogle Scholar
  65. Yerushalmy, M. & Shwartz, J. L. (1993). Seizing the opportunity to make algebra mathematically and pedagogically interesting. In T. A. Romberg, E. Fennema & T. P. Carpenter (Eds.), Integrating research in the graphical representation of function (pp. 41–48). Hillsdale, NJ: Erlbaum.Google Scholar
  66. Zhu, Y. & Fan, L. (2006). Focus on the representation of problem types in intended curriculum: a comparison of selected mathematics textbooks from Mainland China and the United States. International Journal of Science and Mathematics Education, 4(4), 609–626.CrossRefGoogle Scholar

Copyright information

© Ministry of Science and Technology, Taiwan 2015

Authors and Affiliations

  • Briana L. Chang
    • 1
  • Jennifer G. Cromley
    • 2
  • Nhi Tran
    • 1
  1. 1.Psychological, Organizational and Leadership StudiesTemple UniversityPhiladelphiaUSA
  2. 2.Cognitive Science of Teaching and LearningUniversity of Illinois at Urbana-ChampaignChampaignUSA

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