# Prospective teachers’ views on the use of calculators with Computer Algebra System in algebra instruction

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## Abstract

Although growing numbers of secondary school mathematics teachers and students use calculators to study graphs, they mainly rely on paper-and-pencil when manipulating algebraic symbols. However, the Computer Algebra Systems (CAS) on computers or handheld calculators create new possibilities for teaching and learning algebraic manipulation. This study investigated the views of Turkish prospective secondary mathematics teachers on the use of advanced calculators with CAS in algebra instruction. An open-ended questionnaire and group interviews revealed prospective teachers’ views and beliefs about when and why they prefer three possible uses of CAS—black box, white box, or Symbolic Math Guide (SMG). The results showed that participants mainly preferred the white box methods and especially SMG to the black box method. They suggested that while the black box method could be used after students mastered the skills, the general white box method and SMG could be used to teach symbolic manipulation.

## Keywords

Prospective teachers Graphing calculators Computer Algebra Systems Algebra instruction Secondary mathematics education## 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.CrossRefGoogle Scholar - Ball, D. L., Lubienski, S. T., & Mewborn, D. S. (2001). Research on teaching mathematics: The unsolved problem of teachers’ mathematical knowledge. In V. Richardson (Ed.),
*Handbook of research on teaching*(4th ed., pp. 433–456). New York: Macmillan.Google Scholar - Barzel, B. (2005).
*New technology? New ways of teaching—No time left for that!*Retrieved June 9, 2009, from http://www.lkl.ac.uk/research/came/events/CAME4/CAME4-topic3-Barzel-paper.pdf. - Borko, H., & Putnam, R. T. (1996). Learning to teach. In D. Berliner & R. Calfee (Eds.),
*Handbook of educational psychology*(pp. 673–708). New York: Macmillan.Google Scholar - Buchberger, B. (1990). Should students learn integration rules?
*Sigsam Bulletin,**24*(1), 10–17.CrossRefGoogle Scholar - CAME 2005 theme group 3. Teachers and CAS: Summary of the theme group discussion. (2005). Retrieved June 9, 2009, from http://www.lkl.ac.uk/research/came/events/CAME4/CAME4-topic3-discussion-group-summary.pdf.
- Cuoco, A. & Goldenberg, P. (2003).
*CAS and curriculum: Real improvement or dĕja vu all over again?*Retrieved June 9, 2009, from http://www.lkl.ac.uk/research/came/events/reims/3-Presentation-CuocoGoldenberg.pdf. - Doerr, H. M., & Zangor, R. (2000). Creating meaning for and with the graphing calculator.
*Educational Studies in Mathematics,**41*, 143–163.CrossRefGoogle Scholar - Drijvers, P. (2000). Students encountering obstacles using a CAS.
*International Journal of Computers for Mathematical Learning,**5*, 189–209.CrossRefGoogle Scholar - Drijvers, P. (2003). Algebra on screen, on paper, and in the mind. In J. T. 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. (2004). Learning algebra in a computer algebra environment.
*The International Journal of Computer Algebra in Mathematics Education,**11*(3), 77–89.Google Scholar - Drijvers, P. & Trouche, L. (2008). From artifacts to instruments: A theoretical framework behind the orchestra metaphor. In G. W. Blume & M. K. Heid (Eds.), Research on technology and the teaching and learning of mathematics: Vol. 2. Cases and perspectives (pp. 363–391). Charlotte, NC: Information Age.Google Scholar
- Edwards, M. T. (2001).
*The electronic*“*other*”:*A study of calculator-based symbolic manipulation utilities with secondary school mathematics students*. Unpublished Doctoral Dissertation, Ohio State University, Columbus, OH.Google Scholar - Edwards, M. T. (2002). Symbolic manipulation in a technological age.
*Mathematics Teacher,**95*(8), 614–620.Google Scholar - Edwards, M. T. (2003). Calculator-based computer algebra systems: Tools for meaningful algebraic understanding. In J. T. Fey, A. Cuoco, C. Kieran, L. McMullin, & R. M. Zbiek (Eds.),
*Computer algebra systems in secondary school mathematics education*(pp. 117–134). Reston, VA: National Council of Teachers of Mathematics.Google Scholar - Forgasz, H. J., & Griffith, S. (2006). Computer algebra system calculators: Gender issues and teachers’ expectations.
*Australian Senior Mathematics Journal,**20*(2), 18–29.Google Scholar - Heid, M. K. (2003). Theories for thinking about the use of CAS in teaching and learning mathematics. In J. T. Fey, A. Cuoco, C. Kieran, L. McMullin, & R. M. Zbiek (Eds.),
*Computer algebra systems in secondary school mathematics education*(pp. 33–52). Reston, VA: National Council of Teachers of Mathematics.Google Scholar - Heid, M. K. (2005). Technology in mathematics education: Tapping into visions of the future. In W. J. Masalski (Ed.),
*Technology-supported mathematics learning environments: Sixty-seventh yearbook*(pp. 345–366). Reston, VA: National Council of Teachers of Mathematics.Google Scholar - 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.CrossRefGoogle Scholar - Hoch, M. & Dreyfus, T. (2005).
*Structure sense in high school algebra: The effect of brackets.*ERIC Documentation Reproduction Service No. ED 489 561.Google Scholar - Kendal, M., & Stacey, K. (2001). The impact of teacher privileging on learning differentiation with technology.
*International Journal of Computers for Mathematical Learning,**6*, 143–165.CrossRefGoogle Scholar - Kendal, M., Stacey, K., & Pierce, R. (2005). The influence of a computer algebra environment on teachers’ practice. In D. Guin, K. Ruthven, & L. Trouche (Eds.),
*The didactical challenge of symbolic calculators: Turning a computational device into a mathematical instrument*(pp. 83–112). New York: Springer.CrossRefGoogle Scholar - Kieran, C. (2007). Interpreting and assessing the answers given by the CAS expert: A reaction paper.
*The International Journal for Technology in Mathematics Education,**14*(2), 103–107.Google Scholar - Kutzler, B. (2000).
*The algebraic calculator as a pedagogical tool for teaching mathematics*. Retrieved August 30, 2007, from http://www.kutzler.com/downloads/the_algebraic_calculator_as_a_pedagogical_tool.pdf. - Kutzler, B. (n.d.).
*Two-tier exams as a way to let technology in*. Retrieved August 30, 2007, from http://www.kutzler.com/downloads/art_exam.pdf. - Lagrange, J.-B. (2005a). Transposing computer tools from the mathematical sciences into teaching: Some possible obstacles. In D. Guin, K. Ruthven, & L. Trouche (Eds.),
*The didactical challenge of symbolic calculators: Turning a computational device into a mathematical instrument*(pp. 67–82). New York: Springer.CrossRefGoogle Scholar - Lagrange, J.-B. (2005b). Using symbolic calculators to study mathematics: The case of tasks and techniques. 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.CrossRefGoogle Scholar - Lagrange, J.-B. (2007). Didactic time, epistemic gain and consistent tool: Taking care of teachers’ needs for classroom use of CAS.
*The International Journal for Technology in Mathematics Education,**14*(2), 87–94.Google Scholar - Lumb, S., Monaghan, J., & Mulligan, S. (2000). Issues arising when teachers make extensive use of computer algebra.
*The International Journal of Computer Algebra in Mathematics Education,**7*(4), 223–240.Google Scholar - Mahoney, J. F. (2002). Computer algebra systems in our schools: Some axioms and some examples.
*Mathematics Teacher,**95*(8), 598–605.Google Scholar - Miles, M. B., & Huberman, A. M. (1994).
*Qualitative data analysis: An expanded sourcebook*(2nd ed.). Thousand Oaks, CA: SAGE Publications, Inc.Google Scholar - National Council of Teachers of Mathematics. (1991).
*Professional standards for teaching mathematics*. Reston, VA: National Council of Teachers of Mathematics.Google Scholar - National Council of Teachers of Mathematics. (2000).
*Principles and standards for school mathematics*. Reston, VA: National Council of Teachers of Mathematics.Google Scholar - Pajares, M. F. (1992). Teachers’ beliefs and educational research: Cleaning up a messy construct.
*Review of Educational Research,**62*, 307–332.Google Scholar - Peschek, W. (2007). The impact of CAS on our understanding of mathematics education.
*The International Journal for Technology in Mathematics Education,**14*(2), 95–101.Google Scholar - Pierce, R., & Stacey, K. (2004). Monitoring progress in algebra in a CAS active context: Symbol sense, algebraic insight and algebraic expectation.
*The International Journal of Computer Algebra in Mathematics Education,**11*(1), 3–11.Google Scholar - Prawat, R. S. (1992). Teachers’ beliefs about teaching and learning: A constructivist perspective.
*American Journal of Education,**100*(3), 354–395.CrossRefGoogle Scholar - Richardson, V. (1996). The role of attitudes and beliefs in learning to teach. In J. Sikula, T. J. Buttery, & E. Guyton (Eds.),
*Handbook of research on teacher education*(2nd ed., pp. 102–119). New York: Macmillan.Google Scholar - Schneider, E. (2000). Teacher experiences with the use of a CAS in a mathematics classroom.
*The International Journal of Computer Algebra in Mathematics Education,**7*(2), 119–141.Google Scholar - Schneider, E. (n.d.).
*Changes of teaching mathematics by computer algebra systems*(*CAS*). Retrieved June 9, 2009, from http://www.fmd.uni-osnabrueck.de/ebooks/gdm/PapersPdf1997/Schneider.pdf. - Stacey, K., Kendal, M., & Pierce, R. (2002). Teaching with CAS in a time of transition.
*The International Journal of Computer Algebra in Mathematics Education,**9*(2), 113–127.Google Scholar - Stipek, D. J., Givvin, K. B., Salmon, J. M., & MacGyvers, V. L. (2001). Teachers’ beliefs and practices related to mathematics instruction.
*Teaching and Teacher Education,**17*, 213–226.CrossRefGoogle Scholar - Tabach, M., Hershkowitz, R., & Arcavi, A. (2008). Learning beginning algebra with spreadsheets in a computer intensive environment.
*Journal of Mathematical Behavior,**27*, 48–63.CrossRefGoogle Scholar - Thompson, A. G. (1984). The relationship of teachers’ conceptions of mathematics and mathematics teaching to instructional practice.
*Educational Studies in Mathematics,**15*(2), 105–127.CrossRefGoogle Scholar - Thompson, A. G. (1992). Teachers’ beliefs and conceptions: A synthesis of the research. In D. A. Grouws (Ed.),
*Handbook of research on mathematics teaching and learning*(pp. 127–146). New York: Macmillan.Google Scholar - Wilf, H. S. (1982). The disk with the college education.
*The American Mathematical Monthly,**89*(1), 4–8.CrossRefGoogle Scholar - Yoder, A. J. (2000).
*The relationship between graphing calculator use and teachers*’*beliefs about learning algebra*. ERIC Documentation Reproduction Service No. ED 446 987.Google Scholar - Zbiek, R. M. (2002). Influences on mathematics teachers’ transitional journeys in teaching with CAS.
*The International Journal of Computer Algebra in Mathematics Education,**9*(2), 129–137.Google Scholar