Advertisement

ZDM

, Volume 44, Issue 5, pp 641–651 | Cite as

From the didactical triangle to the socio-didactical tetrahedron: artifacts as fundamental constituents of the didactical situation

  • Sebastian Rezat
  • Rudolf Sträßer
Original Article

Abstract

Research on the use of artifacts such as textbooks and digital technologies has shown that their use is not a straight forward process but an activity characterized by mutual participation between artifact and user. Taking a socio-cultural perspective, we analyze the role of artifacts in the teaching and learning of mathematics and argue that artifacts influence the didactical situation in a fundamental way. Therefore, we believe that understanding the role of artifacts within the didactical situation is crucial in order to become aware of and work on the relationships between the teacher, their students and the mathematics and, therefore, are worthwhile to be considered as an additional fundamental aspect in the didactical situation. Thus, by expanding the didactical triangle, first to a didactical tetrahedron, and finally to a “socio-didactical tetrahedron”, a more comprehensive model is provided in order to understand the teaching and learning of mathematics.

Keywords

Artifacts Cultural–historical activity theory Didactical tetrahedron Didactical triangle Digital technologies Instrumental approach Mathematics textbooks Sociocultural perspective Teachers’ practice Tools 

Notes

Acknowledgments

We sincerely thank the three unknown reviewers, who reviewed the first version of this paper, for their constructive and pertinent comments and suggestions. We particularly appreciate the work of Vincent Geiger who turned our text into readable Australian English.

References

  1. Adler, J. (2000). Conceptualizing resources as a theme for teacher education. Jorunal of Mathematics Teacher Education, 3(3), 205–224.CrossRefGoogle Scholar
  2. Ainley, J., Eveleigh, F., Freeman, C., & O’Malley, K. (2010). ICT in the teaching of science and mathematics in year 8 in Australia: report from the IEA Second International Technology in Education Study (SITES) survey. ACER Research Monographs. http://research.acer.edu.au/acer_monographs/6. Accessed 17 July 2012.
  3. 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
  4. Bartolini Bussi, M. G., & Mariotti, M. A. (2008). Semiotic mediation in the mathematics classroom: Artifacts and signs after a Vygotskian perspective. In L. D. English (Ed.), Handbook of international research in mathematics education (pp. 750–787). Mahwah: L. Erlbaum Associates.Google Scholar
  5. Bauersfeld, H. (1980). Hidden dimensions in the so-called reality of a mathematics classroom. Educational Studies in Mathematics, 11(1), 23–41. doi: 10.1007/bf00369158.CrossRefGoogle Scholar
  6. Brousseau, G. (1997). Theory of didactical situations in mathematics. Dordrecht: Kluwer Academics Publishers.Google Scholar
  7. Bruner, J. S. (1960). The process of education. Harvard: Harvard University Press.Google Scholar
  8. Chevallard, Y. (1985). La Transposition Didactique. Du savoir savant au savoir enseigné. Grenoble: Pensées sauvages.Google Scholar
  9. Churchhouse, R. F., Cornu, B., Ershov, A. P., Howson, A. G., Kahane, J. P., van Lint, J. H., et al. (1984). The influence of computers and informatics on mathematics and its teaching. An ICMI discussion document. L’Enseignement Mathématique, 30, 161–172.Google Scholar
  10. Damlamian, A., & Sträßer, R. (2009). ICMI Study 20: educational interfaces between mathematics and industry. ZDM—The International Journal on Mathematics Education, 41(4), 525–533.CrossRefGoogle Scholar
  11. Dörfler, W. (2007). Matrices as Peircean diagrams: A hypothetical learning trajectory. In D. Pitta-Pantazi & G. Philippou. Larnaca (Eds.), European research in mathematics education. Proceedings of the fifth congress of the European Society for Research in Mathematics Education (pp. 852–861). Cyprus: European Society for Research in Mathematics Education (ERME), Department of Education, University of Cyprus.Google Scholar
  12. 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: Volume 2. Cases and perspectives (pp. 363–391). Charlotte: Information Age.Google Scholar
  13. Engeström, Y. (1987). Learning by expanding. An activity-theoretical approach to developmental research. Helsinki: Orienta-Konsultit Oy.Google Scholar
  14. Engeström, Y. (1998). Reorganizing the motivational sphere of classroom culture: an activity-theroretical analysis of planning in teacher team. In F. Seeger, J. Voigt, & U. Waschescio (Eds.), The culture of the mathematics classroom (pp. 76–103). Cambridge: Cambridge University Press.Google Scholar
  15. Freudenthal, H. (1991). Revisiting mathematics education. China lectures. Dordrecht: Kluwer Academic Publishers.Google Scholar
  16. Geiger, V. (in print). The role of social aspects of teaching and learning in transforming mathematical activity: Tools, tasks, individuals and learning communities. In S. Rezat, M. Hattermann, & A. Peter-Koop (Eds.), TransformationA Big Idea in Mathematics Education. New York: Springer. Google Scholar
  17. Griesel, H., & Postel, H. (1983). Zur Theorie des Lehrbuchs – Aspekte der Lehrbuchkonzeption. Zentralblatt für Didaktik der Mathematik, 83(6), 287–293.Google Scholar
  18. Gueudet, G., & Trouche, L. (2009). Towards new documentation systems for mathematics teachers? Educational Studies in Mathematics, 71(3), 199–218.CrossRefGoogle Scholar
  19. Gueudet, G., & Trouche, L. (2012). Teachers’ work with resources: Documentational geneses and professional geneses. In G. Gueudet, B. Pepin, & L. Trouche (Eds.), From text to ‘lived’ resources (Vol. 7, pp. 23–41, Mathematics Teacher Education). Dordrecht: Springer.Google Scholar
  20. Love, E., & Pimm, D. (1996). ‘This is so’: a text on texts. In A. J. Bishop, K. Clements, C. Keitel, J. Kilpatrick, & C. Laborde (Eds.), International handbook of mathematics education (Vol. 1, pp. 371–409). Dordrecht: Kluwer.Google Scholar
  21. Olive, J., Makar, K., Hoyos, V., Kor, L., Kosheleva, O., & Sträßer, R. (2010). Mathematical knowledge and practices resulting from access to digital technologies. In C. Hoyles & J. B. Lagrange (Eds.), Mathematics education and technology—rethinking the terrain (pp. 133–177). New York: Springer.Google Scholar
  22. Pepin, B., & Haggarty, L. (2001). Mathematics textbooks and their use in English, French and German classrooms: a way to understand teaching and learning cultures. ZDM—The International Journal on Mathematics Education, 33(5), 158–175.CrossRefGoogle Scholar
  23. Rabardel, P. (2002). People and Technology: a cognitive approach to contemporary instruments. http://ergoserv.psy.univ-paris8.fr. Accessed 17 July 2012.
  24. Remillard, J. T. (2005). Examining key concepts in research on teachers’ use of mathematics curricula. Review of Educational Research, 75(2), 211–246.CrossRefGoogle Scholar
  25. Rezat, S. (2006). A model of textbook use. In J. Novotná, H. Moraová, M. Krátká, & N. A. Stehlíková (Eds.), Proceedings of the 30th Conference of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 409–416). Prague: Charles University, Faculty of Education.Google Scholar
  26. Rezat, S. (2009). Das Mathematikbuch als Instrument des Schülers. Eine Studie zur Schulbuchnutzung in den Sekundarstufen. Wiesbaden: Vieweg+Teubner.CrossRefGoogle Scholar
  27. Rezat, S. (2011). Interactions of teachers’ and students’ use of mathematics textbooks. In G. Gueudet, B. Pepin, & L. Trouche (Eds.), From text to ‘lived’ resources. Mathematics curriculum materials and teacher development (pp. 231–246). New York: Springer.Google Scholar
  28. Schmidt, W. H., McKnight, C. C., Houang, R. T., Wang, H., Wiley, D. E., Cogan, L. S., et al. (2001). Why schools matter: A cross-national comparison of curriculum and learning. San Francisco: Jossey-Bass.Google Scholar
  29. Sträßer, R. (2009). Instruments for learning and teaching mathematics. An attempt to theorise about the role of textbooks, computers and other artefacts to teach and learn mathematics. In M. Tzekaki, M. Kaldrimidou, & C. Sakonidis (Eds.), Proceedings of the 33rd conference of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 67–81). Thessaloniki: PME.Google Scholar
  30. Tall, D. (1986). Using the computer as an environment for building and testing mathematical concepts. A tribute to Richard Skemp. Warwick. http://www.warwick.ac.uk/staff/David.Tall/themes/computers.html. Accessed 17 July 2012.
  31. Vygotsky, L. (1997). The instrumental method in psychology. In R. W. Rieber & J. Wollock (Eds.), The collected works of L. S. Vygotsky. Volume 3. Problems of the theory and history of psychology (pp. 85–89). New York: Plenum Press.Google Scholar
  32. Wartofsky, M. W. (1979). Modelsrepresentation and the scientific understandig (Vol. 48, Boston Studies in the Philosophy of Science). Dordrecht: Reidel.Google Scholar
  33. Wertsch, J. V. (1998). Mind as action. New York: Oxford University Press.Google Scholar
  34. Yackel, E., & Cobb, P. (1996). Sociomathematical norms, argumentation, and autonomy in mathematics. Journal for Research in Mathematics Education, 27(4), 458–477.CrossRefGoogle Scholar

Copyright information

© FIZ Karlsruhe 2012

Authors and Affiliations

  1. 1.Justus-Liebig-University of GiessenGiessenGermany

Personalised recommendations