The Mathematical Education (한국수학교육학회지시리즈A:수학교육)

Korean Society of Mathematical Education (한국수학교육학회)
권호 표지
DOI QR코드

DOI QR Code

A Qualitative analysis of students' factorization of xn-1 using a CAS application

CAS 어플리케이션을 이용한 xn-1의 인수분해 일반화 과정에 대한 질적 분석

  • Received : 2013.06.17
  • Accepted : 2013.08.19
  • Published : 2013.08.31
Download PDF
⟨ Previous Next ⟩

Abstract

The purpose of the study was to investigate how students generalize and prove the factoring of $x^n-1$ using a Computer Algebra System application and the role of CAS in this process. The theoretical framework consists of the anthropological and the instrumental approach. In particular, the basis of the Task-Technique-Theorization(T-T-T) frame adapted form Chevallard's anthropological approach of Didactics is utilized. We found that Technique-Theorization emerges in mutual interaction between paper-and-pencil techniques and computer algebra techniques. And this interaction led to the students' theoretical reflection and conceptual understanding. In this process, we could identify three epistemic role of CAS : the role of checking the result, the role of cognitive stimulation and the role of extending thinking. Therefore CAS plays on a epistemic role of checking the result of a task, stimulating the student' cognition and extending their thinking as well as pragmatic role of producing the result of a input.

Keywords

References

  1. 강인애 (2003). PBL의 이론과 실제. 서울: 문음사.(Kang, I. (2003). The theory and practice of PBL. Seoul: Moonumsa.)
  2. 양창길, 김철수 (1996). 극한에 대한 실험 수학적 접근. 과학교육 13, 47-66. (Yang, C., & Kim, C. (1996). Experimental mathematics approaches on limits. Journal of education 13, 47-66.)
  3. 조영주, 김경미 (2010). 컴퓨터 대수 환경에서 매개변수 개념에 대한 고등학생의 이해에 관한 사례 연구. 수학교육 논문집 24(4), 949-974.(Cho, Y., & Kim, K. (2010). The case study of high school students' understanding of the concept of parameter in a computer algebra environment. Communications of Mathematical Education 24(4), 949-974.)
  4. 한세호 (2009). 고등학교 수학학습에서 컴퓨터대수체계(CAS)의 도구발생. 박사학위 논문, 건국대학교. (Han, S. (2009). Instrumental genesis of computer algebra system in high school students' mathematical learning. Doctoral dissertation, KU.)
  5. Artigue, M. (1998). Teacher training as a key issue for the integration of computer technologies. In Tinsley & Johson (Eds.), Information and Communications Technologies in School Mathematics (121-129). Chapman and Hall.
  6. 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. https://doi.org/10.1023/A:1022103903080
  7. Bell, F. H. (1980). Teaching Elementary School Mathematics. Iwoa: Wm. C. Brown Company Publisher.
  8. Boers-van Oosterum, M. A. M. (1990). Understanding of variables and their uses acquired by students in traditional and computer-intensive algebra. Dissertation. MA: University of Maryland.
  9. Borwein, J., Bailey, D., & Girgensohn, R. (2003). Experimentation in Mathematics: Computational Paths to Discovery. Boston, MA: AK Peters.
  10. Brown, R. (1998). Using computer algebra systems to introduce algebra. The International Journal of Computer Algebra in Mathematics Education 5, 147-160.
  11. Chevallard, Y. (1999). L'analyse des pratiques enseignantes en theorie anthropologique du didactique, Techerches en Didactique des Mathe matiques 19(2), 221-266.
  12. Drijvers, P. (2003). Learning Algebra in a Computer Algebra Environment: design research on the understanding of the concept of parameter. Utrecht: CD-${\beta}$ Press.
  13. Graham, A., & Thomas, M. (2000). Building a versatile understanding of algebraic variables with a graphic calculator. Educational Studies in Mathematics 41, 265-282. https://doi.org/10.1023/A:1004094013054
  14. Guin, D., Ruthven, K., & Trouche, L. (2004). The Didactical Challenge of Symbolic Calculator: Turning a Computational Device into a Mathematical Instrument. Dordrecht: Kluwer Academic Publisher.
  15. Hiebert J., & Lefevre P. (1986). Conceptual and procedural knowledge in mathematics: an introductory analysis. In J. Hiebert (Ed.), Conceptual and procedural knowledge: the case of Mathematics (1-27). Lawrence Erlbaum Associates.
  16. 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, 205-263. https://doi.org/10.1007/s10758-006-0006-7
  17. Kieran, C., & Guzman, J. (2010). Role of Task and Technology in Provoking Teacher Change: A Case of Proofs and Proving in High School Algebra. In R. Leikin, R. Zazkis (Eds.), Learning Through Teaching Mathematics (127-152), New York: Springer.
  18. Kuhn, T. S. (1999). 과학혁명의 구조 (김명자 역), 서울: 까치. (원저1962년출판)
  19. Lagrange, J. B. (2000). L'integration d'instruments informatiques dans l'enseignement: une approche par les techniques (The integration of technological instruments in education: an approach by means of techniques) Educational Studies in Mathematics 43, 1-30. https://doi.org/10.1023/A:1012086721534
  20. Lagrange, J. B. (2003). Learning techniques and concepts using CAS: A practical and theoretical reflection. In J. T. Fey (Ed.), Computer Algebra Systems in Secondary School Mathematics Education (269-283). Reston, VA: NCTM.
  21. Lagrange, J. B. (2005a). Curriculum, classroom practices and tool design in the learning of functions through technology-aided experimental approaches, International Journal of Computers for Mathematical Learning 10, 143-189. https://doi.org/10.1007/s10758-005-4850-7
  22. Lagrange, J. B. (2005b). 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 (113-135). New York: Springer.
  23. Lakatos, I. (1976). Proofs and Refutations, the Logic of Mathematical Discovery. Cambridge: University Press.
  24. Mounier, G., & Aldon, G. (1996). A problem story: Factorisations of $x^n$-1. International DERIVE Journal 3, 51-61.
  25. Noss, R., & Hoyles, C. (1996). Windows on Mathematical Meanings: Learning cultures and computers. Dordrecht: Kluwer Academic Publisher.
  26. O'Callaghan, B. R. (1998). Computer-intensive algebra and students' conceptual knowledge of functions. Journal for Research in Mathematics Education 29, 21-40. https://doi.org/10.2307/749716
  27. Piaget, J. (1980). Les formes elementaires de la dialectique. Paris: Gallimard.
  28. Pozzi, S. (1994). Algebraic reasoning and CAS: Freeing students form syntax? In H. Heugl, & B. Kutzler(Eds.), Derive in education: Opportunities and strategies (171-188). Bromley, UK: Chartwell-Bratt.
  29. Prensky, M. (2005). Listen to the natives. The Association for Supervision and Curriculum Development: Educational Leadership, Dec 2005-Jan 2006, 9-13.
  30. Rabardel, P., & Samurcay, R. (2001). From Artifact to Instrument-Mediated Learning. New Challenges to Research on Learning, International symposium organized by the Center for Activity Theory and Developmental Work Research. University of Helsinki, March, 21-23.
  31. Repo, S. (1994). Understanding and reflective abstraction: Learning the concept of the derivative in the computer environment. The International Derive Journal 1(1), 97-113.
  32. Sierpinska, A., & Lerman, S. (1996). Epistemologies of mathematics and mathematics education, In Bishop et. al. (Eds.), International Handbook of Mathematics Education (827-876). Dordrecht: Kluwer Academic Publishers.
  33. Trouche, L. (2004). Managing complexity of human/machine interactions in computerized learning environments: Guiding students' command process through instrumental orchestrations. International Journal of Computers for Mathematical Learning 9, 281-307. https://doi.org/10.1007/s10758-004-3468-5
  34. Trouche, L. (2005). Construction et conduite des instruments dans les apprentissages mathematiques: Necessite des orchestrations. Recherches en didactique des mathematiques 25, 91-138.
  35. Verillon, P., & Rabardel, P. (1995). Cognition and artifacts: A contribution to the study of thought in relation to instrument activity. European Journal of Psychology of Education 9(3), 77-101.
  36. Verillon, P. (2000). Revisiting Piaget and Vigotsky: In search of a learning model for technology education. Journal of Technology Sutdies 26(1), 3-10.