Skip to main content

Advertisement

Log in

What happens when CAS procedures are objectified?—the case of “solve” and “desolve”

  • Published:
Educational Studies in Mathematics Aims and scope Submit manuscript

Abstract

Inspired by the entering of computer algebra systems (CAS) in the Danish upper secondary school mathematics program, this article addresses, from a theoretical stance, what may happen when traditional procedures are outsourced to CAS. Looking at the commands “solve” and “desolve,” it is asked what happens when such CAS procedures are objectified in the students’ minds, and what the nature might be of the resulting “objects.” The theoretical analyses draw on a selection of classical mathematics education frameworks on conceptualization and are related to the research literature on technology in mathematics education. The article suggests the following characteristics as elements of negative effects: loss of distinctive features of concept formation, a consequential reclassification of mathematical objects, instability of CAS solutions as objects, and prevailing a posteriori reasoning on students’ behalf when relying solely on CAS in their mathematical work.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

Notes

  1. Personal communication with “maths counsellors” in Danish upper secondary school (Jankvist & Niss, 2015).

  2. With reference to Schoenfeld, “The word problem is used here in this relative sense, as a task that is difficult for the individual who is trying to solve it.” (1985, p. 74). We do not distinguish sharply between problem, exercise, and task. Rather we focus on the relation between student’s CAS-based work and their mathematical conceptualization.

  3. Personal communications with “maths counsellors.”

  4. Inspired by the phenomenon of the experimenter’s regress (Collins & Pinch, 1993, p. 97).

  5. Inspired by R.E.M.’s song “It’s the end of the world as we know it (and I feel fine)” (1987).

  6. For example, Buchberger’s (2002) so-called “purists.”

References

  • Arnon, I. (1998). In the mind’s eye: How children develop mathematical concepts—extending Piaget’s theory. Unpublished doctoral dissertation. Haifa, Israel: School of Education, Haifa University.

  • Arnon, I., Nesher, P., & Nirenburg, R. (2001). Where do fractions encounter their equivalents? Can this encounter take place in elementary school? International Journal of Computers for Mathematical Learning, 6(2), 167–214.

    Article  Google Scholar 

  • 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(3), 245–274.

    Article  Google Scholar 

  • Artigue, M. (2010). The future of teaching and learning mathematics with digital technologies. In C. Hoyles & J.-B. Lagrange (Eds.), Mathematics education and technology—rethinking the terrain. The 17th ICMI study (pp. 463–475). Boston, MA: Springer.

  • Asiala, M., Brown, A., DeVries, D., Dubinsky, E., Mathews, D., & Thomas, K. (1996). A framework for research and curriculum development in undergraduate mathematics education. In Research in collegiate mathematics education II. CBMS issues in mathematics education (Vol. 6, pp. 1–32). Providence, RI: American Mathematical Society.

  • Bodin, A. (1993). What does to assess mean? The case of assessing mathematical knowledge. In M. Niss (Ed.), Investigations into assessment in mathematics education. An ICMI study (pp. 113–141). Dordrecht, the Netherlands: Springer.

    Chapter  Google Scholar 

  • Bosch, M., & Gascón, J. (2014). Introduction to the anthropological theory of the didactic (ATD). In A. Bikner-Ahsbahs & S. Prediger (Eds.), Networking of theories as a research practice in mathematics education (pp. 67–83). Cham, Switzerland: Springer.

    Google Scholar 

  • Buchberger, B. (2002). Computer algebra: The end of mathematics? SIGSAM Bulletin, 36(1), 3–9.

    Article  Google Scholar 

  • Collins, H., & Pinch, T. (1993). The golem. What everyone should know about science. Cambridge: Cambridge University Press.

    Google Scholar 

  • Dreyfus, T. (1994). The role of cognitive tools in mathematics education. In R. Biehler, R. W. Scholz, R. Strässer, & B. Winkelmann (Eds.), Didactics of mathematics as a scientific discipline (pp. 201–211). Dordrecht, the Netherlands: Kluwer.

  • Dreyøe, J., Schewitsch, P. S., & Hansen, P. S. (2017). Design of written assessment items in mathematics with the integration of digital technologies—focusing on the Danish folkeskole, upper secondary school (stx), and teacher education. Unpublished master’s thesis. Copenhagen, Denmark: Danish School of Education, Aarhus University.

  • Drijvers, P., Godino, J. D., Font, V., & Trouche, L. (2013). One episode, two lenses: A reflective analysis of student learning with computer algebra from instrumental and onto-semiotic perspectives. Educational Studies in Mathematics, 82(1), 23–49.

    Article  Google Scholar 

  • Dubinsky, E. (1991). Reflective abstraction in advanced mathematical thinking. In D. Tall (Ed.), Advanced mathematical thinking (pp. 95–126). Dordrecht, the Netherlands: Springer.

  • Dubinsky, E., Dautermann, J., Leron, U., & Zazkis, R. (1994). On learning fundamental concepts of group theory. Educational Studies in Mathematics, 27(3), 267–305.

    Article  Google Scholar 

  • Grønbæk, N. (2017). Outsourcing and insourcing of CAS in the teaching of mathematics. Plenary presentation at Matematikdidaktikkens Dag 2017, Copenhagen, Denmark.

    Google Scholar 

  • Guin, D., Ruthven, K., & Trouche, L. (Eds.). (2005). The didactical challenge of symbolic calculators. Turning a computational device into a mathematical instrument. Boston, MA: Springer.

  • Guin, D., & Trouche, L. (1998). The complex process of converting tools into mathematical instruments: The case of calculators. International Journal of Computers for Mathematical Learning, 3(3), 195–227.

    Article  Google Scholar 

  • Hoyles, C., & Lagrange, J.-B. (2010). Introduction. In C. Hoyles & J.-B. Lagrange (Eds.), Mathematics education and technology—rethinking the terrain. The 17th ICMI study (pp. 1–11). Boston, MA: Springer.

  • Iversen, S. M., Misfeldt, M., & Jankvist, U. T. (2018). Instrumental mediations and students’ identities. Recherches en Didactique des Mathématiques, 38(2), 133–155.

    Google Scholar 

  • Jankvist, U. T., & Misfeldt, M. (2015). CAS-induced difficulties in learning mathematics? For the Learning of Mathematics, 35(1), 15–20.

    Google Scholar 

  • Jankvist, U. T., Misfeldt, M., & Marcussen, A. (2016). The didactical contract surrounding CAS when changing teachers in the classroom. REDIMAT – Journal of Research in Mathematics Education, 5(3), 263–286.

    Article  Google Scholar 

  • Jankvist, U. T., & Niss, M. (2015). A framework for designing a research-based “maths counsellor” teacher program. Educational Studies in Mathematics, 90(3), 259–284.

    Article  Google Scholar 

  • Kieran, C. (2007). Learning and teaching algebra at the middle school through college levels. In F. K. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 707–762). Charlotte, NC: Information Age.

  • Lagrange, J.-B. (2005). 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). Boston, MA: Springer.

  • Misfeldt, M., & Jankvist, U. T. (2018). Instrumental genesis and proof: Understanding the use of computer algebra systems in proofs in textbooks. In M. Tabach & H.-S. Siller (Eds.), Uses of technology in K-12 mathematics education: Tools, topics and trends (pp. 375–385). Cham, Switzerland: Springer.

  • Niss, M. (2010). Modeling a crucial aspect of students’ mathematical modeling. In R. Lesh, P. L. Galbraith, C. R. Haines, & A. Hurford (Eds.), Modeling students’ mathematical modeling competencies: ICTMA 13 (pp. 43–59). Boston, U.S.A.: Springer.

    Chapter  Google Scholar 

  • Rasmussen, C. L. (2001). New directions in differential equations. A framework for interpreting students’ understandings and difficulties. The Journal of Mathematical Behavior, 20(1), 55–87.

    Article  Google Scholar 

  • Rhine, S., Harrington, R., & Starr, C. (2018). How students think when doing algebra. Charlotte, NC: Information Age Publishing.

    Google Scholar 

  • Schoenfeld, A. H. (1985). Mathematical problem solving. Orlando, FL: Academic Press.

  • 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), 1–36.

    Article  Google Scholar 

  • Sfard, A., & Linchevski, L. (1994). The gains and the pitfalls of reification—the case of algebra. Educational Studies in Mathematics, 26(2–3), 191–228.

    Article  Google Scholar 

  • Shaffer, D. W., & Kaput, J. J. (1998). Mathematics and virtual culture: An evolutionary perspective on technology and mathematics education. Educational Studies in Mathematics, 37(2), 97–119.

    Article  Google Scholar 

  • Tall, D. (1994). Computer environments for the learning of mathematics. In R. Biehler, R. W. Scholz, R. Strässer, & B. Winkelmann (Eds.), Didactics of mathematics as a scientific discipline (pp. 189–199). Dordrecht, the Netherlands: Kluwer.

  • Tall, D., & Vinner, S. (1981). Concept image and concept definition in mathematics with particular reference to limits and continuity. Educational Studies in Mathematics, 12(2), 151–169.

    Article  Google Scholar 

  • Trouche, L. (2005a). Calculators in mathematics education: a rapid evolution of tools, with differential effects. In D. Guin, K. Ruthven, & L. Trouche (Eds.), The didactical challenge of symbolic calculators. Turning a computational device into a mathematical instrument (pp. 9–39). Boston, MA: Springer.

  • Trouche, L. (2005b). An instrumental approach to mathematics learning in symbolic calculator environments. In D. Guin, K. Ruthven, & L. Trouche (Eds.), The didactical challenge of symbolic calculators. Turning a computational device into a mathematical instrument (pp. 137–162). Boston, MA: Springer.

  • Verillon, P., & Rabardel, P. (1995). Cognition and artifacts: A contribution to the study of thought in relation to instrumented activity. European Journal of Psychology of Education, 10(1), 77–101.

    Article  Google Scholar 

  • Weigand, H.–G. (2014). Looking back and ahead-didactical implications for the use of digital technologies in the next decade. Teaching Mathematics and its Applications, 33(1), 3–15.

    Article  Google Scholar 

  • Zandieh, M. J. (2000). A theoretical framework for analyzing student understanding of the concept of the derivative. In E. Dubisnky, A. H. Schoenfeld, & J. Kaput (Eds.), CBMS issues in mathematics education (Vol. 8, pp. 103–122). Providence, RI: American Mathematical Society.

Download references

Acknowledgements

We thank reviewers and editor for their valuable comments. This article was partly written in the frame of project 8018-00062B under Independent Research Fund Denmark.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Uffe Thomas Jankvist.

Additional information

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Jankvist, U.T., Misfeldt, M. & Aguilar, M.S. What happens when CAS procedures are objectified?—the case of “solve” and “desolve”. Educ Stud Math 101, 67–81 (2019). https://doi.org/10.1007/s10649-019-09888-5

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10649-019-09888-5

Keywords

Navigation