Instrumenting Mathematical Activity: Reflections on Key Studies of the Educational Use of Computer Algebra Systems

  • Kenneth Ruthven
Article

Abstract

This paper examines the process through which students learn to make functional use of computer algebra systems (CAS), and the interaction between that process and the wider mathematical development of students. The result of ‘instrumentalising‘ a device to become a mathematical tool and correspondingly ‘instrumenting’ mathematical activity through use of that tool is not only to extend students' mathematical technique but to shape their sense of the mathematical entities involved. These ideas have been developed within a French programme of research – as reported by Artigue in this issue of the journal – which has explored the integration of CAS – typically in the form of symbolic calculators – into the everyday practice of mathematics classrooms. The French research –influenced by socio-psychological theorisation of the development of conceptual systems- seeks to take account of the cultural and cognitive facets of these issues, noting how mathematical norms – or their absence – shape the mental schemes which students form as they appropriate CAS as tools. Instrumenting graphic and symbolic reasoning through using CAS influences the range and form of the tasks and techniques experienced by students, and so the resources available for more explicit codification and theorisation of such reasoning. This illuminates an influential North American study– conducted by Heid – which French researchers have seen as taking a contrasting view of the part played by technical activity in developing conceptual understanding. Reconsidered from this perspective, it appears that while teaching approaches which ‘resequence skills and concepts’ indeed defer – and diminish –attention to routinised skills, the tasks introduced in their place depend on another –albeit less strongly codified – system of techniques, supporting more extensive and active theorisation. The French research high lights important challenges which arise in instrumenting classroom mathematical activity and correspondingly instrumentalising CAS. In particular, it reveals fundamental constraints on human-machine interaction which may limit the capacity of the present generation of CAS to scaffold the mathematical thinking and learning of students.

computer algebra systems computer and calculator uses in education mathematical thinking mathematics teaching and learning secondary and post-secondary education 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

REFERENCES

  1. Artigue, M. (1997). Le logiciel DERIVE comme révélateur de phénomènes didactiques liésà l'utilisation d'environnements informatiques pour l'apprentissage. Educational Studiesin Mathematics 33(2): 133–169.CrossRefGoogle Scholar
  2. Artigue, M. (1999). The teaching and learning of mathematics at the university level:Crucial questions for contemporary research in education. Notices of the AmericanMathematical Society 46(11): 1377–1385.Google Scholar
  3. Artigue, M. (2001). Learning mathematics in a CAS environment: The genesis of a reflectionabout instrumentation and the dialectics between technical and conceptual work. In P. Kent (Ed.), Proceedings of the 2nd Biennial Symposium of the Computer Algebra in Mathematics Education Group: <http://ltsn.mathstore.ac.uk/came/events/freudenthal/1-Presentation-Artigue.pdf>.Google Scholar
  4. Artigue, M. (2002). Learning mathematics in a CAS environment: The genesis of a reflectionabout instrumentation and the dialectics between technical and conceptual work. International Journal of Computers for Mathematical Learning 7(3).Google Scholar
  5. Bosch, M. and Chevallard, Y. (1999). La sensibilité de l'activité mathématique auxostensifs. Objet d'étude et problématique. Recherches en Didactique des Mathématiques 19 (1): 77–124.Google Scholar
  6. Guin, D. and Trouche, L. (1999). The complex process of converting tools into mathematicalinstruments: The case of calculators. International Journal of Computers forMathematical Learning 3(3): 195–227.CrossRefGoogle Scholar
  7. Heid, M.K. (1988). Resequencing skills and concepts in applied calculus using thecomputer as a tool. Journal for Research in Mathematics Education 19(1): 3–25.CrossRefGoogle Scholar
  8. Heid, M.K. (2001). Theories that inform the use of CAS in the Teaching and Learningof Mathematics. In P. Kent (Ed.), Proceedings of the 2nd Biennial Symposium ofthe Computer Algebra in Mathematics Education Group: <http://ltsn.mathstore.ac.uk/ came/events/freudenthal/3-Presentation-Heid.pdf>.Google Scholar
  9. Judson, P. (1990). Elementary business calculus with computer algebra. Journal ofMathematical Behavior 9(2): 153–157.Google Scholar
  10. Lagrange, J.B. (1999). Complex calculators in the classroom: Theoretical and practicalreflections on teaching pre-calculus. International Journal of Computers for MathematicalLearning 4(1): 51–81.CrossRefGoogle Scholar
  11. Lagrange, J.B. (2000/01). L'intégration des instruments informatiques dans l'enseignement: Une approche par les techniques. Educational Studies in Mathematics 43(1): 1–30.CrossRefGoogle Scholar
  12. Newman, D., Griffin, P. and Cole, M. (1989). Social mediation goes into cognitivechange. In The Construction Zone:Working for Cognitive Change in School. Cambridge: Cambridge University Press.Google Scholar
  13. Palmiter, J. (1991). Effects of computer algebra systems on concept and skill acquisitionin calculus. Journal for Research in Mathematics Education 22(2): 151–156.CrossRefGoogle Scholar
  14. Ruthven, K. (1995). Extending MaxBox with a computational algebra system. Mathematicsin School 24(4): 16–19.Google Scholar
  15. Ruthven, K. (2001). Theorising the dialectic between technique and concept in mathematicalactivity: Reflections on recent French studies of the educational use of computeralgebra systems. In P. Kent (Ed.), Proceedings of the 2nd Biennial Symposium ofthe Computer Algebra in Mathematics Education Group: <http://ltsn.mathstore.ac.uk/ came/events/freudenthal/1-Reaction-Ruthven.pdf>.Google Scholar
  16. Smith, J.P., diSessa A.A. and Roschelle, J. (1993/94). Misconceptions reconceived: Aconstructivist analysis of knowledge in transition. Journal of the Learning Sciences 3(2):115–163.CrossRefGoogle Scholar
  17. Suchmann, L.A. (1987). Plans and Situated Actions: The Problem of Human-Machine Communication. Cambridge: Cambridge University Press.Google Scholar
  18. Tall, D. (1996). Functions and calculus. In A.J. Bishop et al. (Eds.), InternationalHandbook of Mathematics Education. Dordrecht: Kluwer.Google Scholar
  19. Thurston, W.P. (1994). On proof and progress in mathematics. Bulletin of the AmericanMathematical Society 30(2): 161–177.Google Scholar
  20. Trouche, L. (2000). La parabole du gaucher et de la casserole à bec verseur: Étudedes processus d'apprentissage dans un environnement de calculatrices symboliques. Educational Studies in Mathematics 41(3): 239–264.CrossRefGoogle Scholar
  21. Vergnaud, G. (1997). The nature of mathematical concepts. In T. Nunes and P. Bryant (Eds.), Learning and Teaching Mathematics: An International Perspective. Hove: Psychology Press.Google Scholar
  22. Verillon, P. and Rabardel, P. (1995). Cognition and artifacts: A contribution to the studyof thought in relation to instrumented activity. European Journal of Psychology ofEducation 10(1): 77–101.CrossRefGoogle Scholar

Copyright information

© Kluwer Academic Publishers 2002

Authors and Affiliations

  • Kenneth Ruthven
    • 1
  1. 1.Faculty of EducationUniversity of CambridgeCambridgeUnited Kingdom

Personalised recommendations