From the didactical triangle to the socio-didactical tetrahedron: artifacts as fundamental constituents of the didactical situation
- 978 Downloads
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.
KeywordsArtifacts Cultural–historical activity theory Didactical tetrahedron Didactical triangle Digital technologies Instrumental approach Mathematics textbooks Sociocultural perspective Teachers’ practice Tools
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.
- 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.
- 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
- Brousseau, G. (1997). Theory of didactical situations in mathematics. Dordrecht: Kluwer Academics Publishers.Google Scholar
- Bruner, J. S. (1960). The process of education. Harvard: Harvard University Press.Google Scholar
- Chevallard, Y. (1985). La Transposition Didactique. Du savoir savant au savoir enseigné. Grenoble: Pensées sauvages.Google Scholar
- 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
- 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
- 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
- Engeström, Y. (1987). Learning by expanding. An activity-theoretical approach to developmental research. Helsinki: Orienta-Konsultit Oy.Google Scholar
- 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
- Freudenthal, H. (1991). Revisiting mathematics education. China lectures. Dordrecht: Kluwer Academic Publishers.Google Scholar
- 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.), Transformation—A Big Idea in Mathematics Education. New York: Springer. Google Scholar
- Griesel, H., & Postel, H. (1983). Zur Theorie des Lehrbuchs – Aspekte der Lehrbuchkonzeption. Zentralblatt für Didaktik der Mathematik, 83(6), 287–293.Google Scholar
- 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
- 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
- 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
- Rabardel, P. (2002). People and Technology: a cognitive approach to contemporary instruments. http://ergoserv.psy.univ-paris8.fr. Accessed 17 July 2012.
- 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
- 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
- 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
- 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
- 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.
- 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
- Wartofsky, M. W. (1979). Models—representation and the scientific understandig (Vol. 48, Boston Studies in the Philosophy of Science). Dordrecht: Reidel.Google Scholar
- Wertsch, J. V. (1998). Mind as action. New York: Oxford University Press.Google Scholar