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An analytical framework for categorizing the use of CAS symbolic manipulation in textbooks

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Abstract

The symbolic manipulation capabilities of computer algebra systems, which we refer to as CAS-S, are now becoming instantiated within secondary mathematics textbooks in the United States for the first time. While a number of research studies have examined how teachers use this technology in their classrooms, one of the most important factors in how this technology is used in the classroom is how it is embedded within curricular resources such as textbooks. This study introduces readers to an analytical framework for examining CAS-S within textbooks and presents the results of its application to three secondary U.S. mathematics textbook units involving polynomial functions. The framework consists of two components: application of CAS-S and reflection on CAS-S uses. The analyses identified differences among the three textbook units in pedagogical intent and task connectedness involving CAS-S. The majority of CAS-S tasks were coded as involving low procedural complexity and there were few instances in which the technology was used in the construction of proofs. Textbook developers asked students to reflect on the visible CAS-S result as opposed to the invisible process leading to those results. The implications of these results as well as the potential transformative role of CAS-S are discussed.

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Notes

  1. We use the terminology textbook to refer to the set of materials produced for students to use in or outside of the classroom that consists of but is not limited to the presentation of mathematical objects such as definitions as well as exercises or problems for students to complete.

  2. The first author was not involved in the development of these three curricula. The second author worked as a graduate research assistant on the Core-Plus Mathematics Project while revisions to the second edition materials were being made. She did not directly contribute to any of the student problems or teacher solutions that were analyzed for this study.

  3. We use the terminology support materials for the textbook or teacher support materials to consist of teacher’s guides or teacher’s editions of textbooks as well as other ancillary materials such as assessment materials that textbook developers produce to support teachers’ implementation of textbooks. Teacher’s guides or teacher’s edition of textbooks often contain answers to exercises appearing in the student textbook as well as pedagogical suggestions about how to implement textbook activities.

  4. Heid and Edwards use the terminology curriculum in their paper, but leave it undefined. However, their use of this terminology in their paper is consistent with our use of textbooks here so we use this terminology in this section.

  5. Consistent with the design intended by curriculum authors, it is assumed that students in a particular course have completed all prior courses in the textbook sequence up to the lesson or investigation in question.

  6. Students ages 14–18 in the U.S. typically study algebra for two years. The first year is often described as beginning algebra and includes the study of linear, quadratic, and exponential functions. The second year of algebra is often referred to as advanced algebra and includes the study of matrices, polynomials, conic sections, rational functions, and trigonometry. CPM is an integrated curriculum yet the algebra content in Course 3 is generally aligned with content in second year algebra texts.

  7. For example, the CPM unit analyzed here appeared in the third textbook in the series, so occasionally analyses moved to the second or first course to determine if students learned these skills with paper-and-pencil.

  8. Five of the six coders used one Coding Sheet per Task, instead of one coding sheet per CAS episode; this caused an issue in interpreting coding decisions for one particular task that involved multiple parts that spanned both sections of the framework. Consequently Part e of Task 13 (see Fig. 6) was excluded from this analysis; Parts a-d were included to keep as much of the data as possible in the sample.

  9. In consultation with a statistician, with the fixed number of raters and categorical ratings, it was most appropriate to interpret this result according to Fleiss’ kappa (Wikipedia, 2011).

  10. We acknowledge an anonymous reviewer’s comment for calling attention to the need for such design research, which we believe could build upon results of this study.

References

  • Artigue, M. (2002). Learning mathematics in a CAS environment: The genesis of a reflection about instrumentation and the dialectics between technical and conceptual work. International Journal of Computers for Mathematical Learning, 7, 245–274.

    Article  Google Scholar 

  • Boers, M. A., & Jones, P. L. (1994). Students’ use of graphics calculators under examination conditions. International Journal of Mathematical Education in Science and Technology, 25, 491–516.

    Article  Google Scholar 

  • Braswell, J. S., Lutkus, A. D., Grigg, W. S., Santapau, S., Tay-Lim, B., & Johnson, M. S. (2001). The nation’s report card: Mathematics 2000. Washington, D.C.: National Center for Education Statistics.

    Google Scholar 

  • Buchberger, B. (1990). Should students learn integration rules? ACM SIGSAM Bulletin, 24(1), 10–17.

    Article  Google Scholar 

  • Cedillo, T., & Kieran, C. (2003). Initiating students into algebra with symbol-manipulating calculators. In J. T. Fey, A. Cuoco, C. Kieran, L. McMullin, & R. M. Zbiek (Eds.), Computing algebra systems in secondary school mathematics (pp. 219–239). Reston, VA: National Council of Teachers of Mathematics.

    Google Scholar 

  • Chval, K. B., & Hicks, S. J. (2009). Calculators in K-5 textbooks. Teaching Children Mathematics, 15(7), 430–437.

    Google Scholar 

  • Cuoco, A., & Goldenberg, P. (2003, June). CAS and curriculum: Real improvement or déjà vu all over again? Paper presented at the Third Computer Algebra in Mathematics Education (CAME) Symposium at Reims, France. Retrieved January 12, 2012, from http://www.lkl.ac.uk/research/came/events/reims/3-Presentation-CuocoGoldenberg.pdf

  • Davis, J. D. (2011). Categorizing CAS use within one reform-oriented United States mathematics textbook. International Journal for Technology in Mathematics Education, 18(3), 121–126.

    Google Scholar 

  • Davis, J. D. (2012). An examination of reasoning and proof opportunities in three differently organized secondary mathematics textbook units. Mathematics Education Research Journal, 24, 467–491.

  • Doerr, H. M., & Pratt, D. (2008). The learning of mathematics and mathematical modeling. Research on technology and the teaching and learning of mathematics: Research syntheses (pp. 259–285). Charlotte, NC: Information Age Publishing.

  • Drijvers, P. (2003). Algebra on screen, on paper and in the mind. In J. Fey, A. Cuoco, C. Kieran, L. McMullin, & R. M. Zbiek (Eds.), Computer algebra systems in secondary school mathematics education (pp. 241–267). Reston, VA: National Council of Teachers of Mathematics.

    Google Scholar 

  • Drijvers, P., Doorman, M., Boon, P., Reed, H., & Gravemeijer, K. (2010). The teacher and the tool: Instrumental orchestrations in the technology-rich mathematics classroom. Educational Studies in Mathematics, 75, 213–234.

    Article  Google Scholar 

  • Education Development Center. (2009). Algebra 2. Boston: Pearson Education.

    Google Scholar 

  • Edwards, T. (2000). Computer symbolic algebra: Catalyst for change in secondary school classrooms. Unpublished manuscript, The Ohio State University.

  • Fey, J. T. (Ed.). (1984). Computing and mathematics: The impact on secondary school curricula. College Park, MD: National Council of Teachers of Mathematics.

    Google Scholar 

  • Fey, J. T., Hirsch, C. R., Hart, E. W., Schoen, H. L., Watkins, A. E., Ritsema, B. E., et al. (2009). Core-plus mathematics: Contemporary mathematics in context. Course 3, teacher edition (2nd ed.). New York: Glencoe.

  • Flanders, J., Lassak, M., Sech, J., Eggerding, M., Karafiol, P. J., McMullin, L., et al. (2010). The University of Chicago school mathematics project: Advanced algebra. Teacher's edition (3rd ed., Vol. 2). Chicago: Wright Group.

  • Fleiss’ kappa. (2011). In Wikipedia. Retrieved April 6, 2011, from http://en.wikipedia.org/wiki/Fleiss%27_kappa.

  • Garry, T. (2003). Computing, conjecturing, and confirming with a CAS tool. In J. T. Fey, A. Cuoco, C. Kieran, L. McMullin, & R. M. Zbiek (Eds.), Computer algebra systems in secondary school mathematics education (pp. 137–150). Reston, VA: National Council of Teachers of Mathematics.

    Google Scholar 

  • Grouws, D. A., & Smith, M. S. (2000). NAEP findings on the preparation and practices of mathematics teachers. In E. A. Silver & P. A. Kennedy (Eds.), Results from the seventh mathematics assessment of the National Assessment of Educational Progress (pp. 107–139). Reston, VA: National Council of Teachers of Mathematics.

    Google Scholar 

  • Guin, D., & Trouche, L. (1999). The complex process of converting tools into mathematical instruments. The case of calculators. International Journal of Computers for Mathematical Learning, 3, 195–227.

    Article  Google Scholar 

  • Heid, M. K. (1988). Resequencing skills and concepts in applied calculus using the computer as a tool. Journal for Research in Mathematics Education, 19(1), 3–25.

    Article  Google Scholar 

  • Heid, M. K., & Blume, G. W. (2008). Algebra and function development. In M. K. Heid & G. W. Blume (Eds.), Research on technology and the teaching and learning of mathematics of mathematics: Syntheses, cases, and perspectives. Vol. 1: Research syntheses (pp. 55–108). Charlotte, NC: Information Age.

  • Heid, M. K., & Edwards, M. T. (2001). Computer algebra systems: Revolution or retrofit for today’s mathematics classrooms? Theory Into Practice, 40(2), 128–136.

    Article  Google Scholar 

  • Henningsen, M., & Stein, M. K. (1997). Mathematical tasks and student cognition: Classroom-based factors that support and inhibit high-level mathematical thinking and reasoning. Journal for Research in Mathematics Education, 28(5), 524–549.

    Article  Google Scholar 

  • Hillel, J. (1991). Computer algebra systems as cognitive technologies: Implications for the practice of mathematics education. In C. Keitel & K. Ruthven (Eds.), Learning from computers: Mathematics education and technology (pp. 18–47). Berlin: Springer.

    Google Scholar 

  • Kieran, C., & Drijvers, P. (2006). The co-emergence of machine techniques, paper-and-pencil techniques, and theoretical reflection: A study of CAS use in secondary school algebra. International Journal of Computers for Mathematical Learning, 11(2), 205–263.

    Article  Google Scholar 

  • Kieran, C., & Saldanha, L. (2008). Designing tasks for the codevelopment of conceptual and technical knowledge in CAS activity. In G. W. Blume & M. K. Heid (Eds.), Research on technology and the teaching and learning of mathematics. Vol. 2: Cases and perspectives (pp. 393–414). Charlotte, NC: Information Age Publishing.

  • Kilpatrick, J. (1997). Five lessons from the new maths era. Paper presented at Reflecting on Sputnik: Linking the Past, Present and Future of Educational Reform, Symposium Hosted by the Center for Science, Mathematics, and Engineering Education, National Academy of Sciences, Washington, D.C. Retrieved October 19, 2009, from http://www.nationalacademies.org/sputnik/kilpat4.htm

  • Lagrange, J. B. (2003). Learning techniques and concepts using CAS: A practical and theoretical reflection. In J. T. Fey, A. Cuoco, C. Kieran, L. McMullin, & R. M. Zbiek (Eds.), Computer algebra systems in secondary school mathematics education (pp. 259–283). Reston, VA: National Council of Teachers of Mathematics.

    Google Scholar 

  • Lagrange, J. B. (2005). Using symbolic calculators to study mathematics. In D. Guin, K. Ruthven, & L. Trouche (Eds.), The didactical challenge of symbolic calculators: Turning a computational device into a mathematical instrument (pp. 113–135). New York: Springer.

    Chapter  Google Scholar 

  • Mayes, R. L. (1995). The application of a computer algebra system as a tool in college algebra. School Science and Mathematics, 95(2), 61–68.

    Article  Google Scholar 

  • Morgan, C., Mariotti, M. A., & Maffei, L. (2009). Representation in computational environments: Epistemological and social distance. International Journal of Computers for Mathematical Learning, 14(3), 241–263.

    Article  Google Scholar 

  • Otte, M. (1986). What is a text? In B. Christiansen, A. G. Howsen, & M. Otte (Eds.), Perspectives on math education (pp. 173–202). Dordrecht: Kluwer.

    Chapter  Google Scholar 

  • Peddiwell, J. A. (1939). The saber-tooth curriculum. New York: McGraw-Hill Book.

    Google Scholar 

  • Peschek, W. (2005). The impact of CAS on our understanding of mathematics education. Paper presented at the Fourth Computer Algebra in Mathematics Education Symposium in Roanoke, VA. Retrieved January 12, 2012, from http://www.lkl.ac.uk/research/came/events/CAME4/CAME4-topic2-Peschek-paper.pdf

  • Randolph, J. J. (2008). Online Kappa Calculator. Retrieved April 6, 2011, from http://justus.randolph.name/kappa

  • Remillard, J. T., & Bryans, M. B. (2004). Teachers' orientations toward mathematics curriculum materials: Implications for teacher learning. Journal for Research in Mathematics Education, 35(5), 352–388.

    Article  Google Scholar 

  • ResearchWare, Inc. (2009). HyperResearch: Qualitative Analysis Tool (Version 2.8.3) [computer software]. Randolph, MA.

  • Ruthven, K. (1990). The influence of graphic calculator use on translation from graphic to symbolic forms. Educational Studies in Mathematics, 21, 431–450.

    Article  Google Scholar 

  • Senk, S. L., Thompson, D. R., Viktora, S. S., Usiskin, Z., Ahbel, N. P., Levin, S., et al. (1996). Advanced algebra (2nd ed., Teacher’s Edition, Part 2). Glenview, IL: Scott Foresman.

  • Stein, M. K., Remillard, J., & Smith, M. S. (2007). How curriculum influences student learning. In F. K. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 319–369). Charlotte, NC: Information Age Publishing.

    Google Scholar 

  • Tall, D., Smith, D., & Piez, C. (2008). Technology and calculus. In M. K. Heid & G. W. Blume (Eds.), Research on technology and the teaching and learning of mathematics of mathematics: Syntheses, cases, and perspectives. Vol. 1: Research syntheses (pp. 207–258). Charlotte, NC: Information Age.

    Google Scholar 

  • Thomas, M. O. J., Monaghan, J., & Pierce, R. (2004). Computer algebra systems and algebra: Curriculum, assessment, teaching, and learning. In K. Stacey, H. Chick, & M. Kendal (Eds.), The future of the teaching and learning of algebra: The 12th ICMI study (pp. 155–186). New York: Kluwer.

    Google Scholar 

  • Usiskin, Z., Anderson, K., & Zotto, N. (Eds.). (2010). Future curricular trends in school algebra and geometry: Proceedings of a conference. Chicago: Information Age.

  • Valverde, G. A., Bianchi, L. J., Wolfe, R. G., Schmidt, W. H., & Houang, R. T. (2002). Using TIMSS to investigate the translation of policy into practice through the world of textbooks. Dordrecht, The Netherlands: Kluwer.

    Google Scholar 

  • Zbiek, R. M., & Hollebrands, K. F. (2008). A research-informed view of the process of incorporating mathematics technology into classroom practice by in-service and prospective teachers. In M. K. Heid & G. W. Blume (Eds.), Research on technology and the teaching and learning of mathematics. Vol. 1: Research syntheses (pp. 287–344). Charlotte, NC: Information Age.

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  1. Western Michigan University, Kalamazoo, MI, USA

    Jon D. Davis

  2. University of Wisconsin, Madison, WI, USA

    Nicole L. Fonger

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Davis, J.D., Fonger, N.L. An analytical framework for categorizing the use of CAS symbolic manipulation in textbooks. Educ Stud Math 88, 239–258 (2015). https://doi.org/10.1007/s10649-014-9581-z

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