Abstract
This chapter seeks to provide an integrating theoretical framework for understanding the somewhat disparate and disconnected literatures on “modelling” and “technology” in mathematics education research. From a cultural–historical activity theory, neo-Vygtoskian perspective, mathematical modelling must be seen as embedded within an indivisible, molar “whole” unit of “activity.” This notion situates “technology”—and mathematics, also—as an essential part or “moment” of the whole activity, alongside other mediational means; thus it can only be fully understood in relation to all the other moments. For instance, we need to understand mathematics and technology in relation to the developmental needs and hence the subjectivity and “personalities” of the learners. But, then, also seeing learning as joint teaching–learning activity implies the necessity of understanding the relation of these also to the teachers, and to the wider institutional and professional and political contexts, invoking curriculum and assessment, pedagogy and teacher development, and so on. Historically, activity has repeatedly fused mathematics and technology, whether in academe or in industry: this provides opportunities, but also problems for mathematics education. We illustrate this perspective through two case studies where the mathematical-technologies are salient (spreadsheets, the number line, and CAS), which implicate some of these wider factors, and which broaden the traditional view of technology in social context.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Bartolini Bussi, M. G. (1998). Joint activity in mathematics classrooms: A Vygotskian analysis. In F. Seeger, J. Voigt, & U. Waschescio (Eds.), The culture of the mathematics classroom (pp. 13–49). Cambridge, UK: Cambridge University Press.
Bernstein, B. (2000). Pedagogy, symbolic control, and identity. Oxford, UK: Rowman and Littlefield.
Black, M. (1962) (Ed.). Models and metaphors: Studies in language and philosophy. Ithaca, NY: Cornell University Press.
Black, L., Williams, J., Hernandez-Martinez, P., Davis, P., & Wake, G. (2010). Developing a “leading identity”: The relationship between students’ mathematical identities and their career and higher education aspirations. Educational Studies in Mathematics, 73(1), 55–72.
Blum, W., Galbraith, P. L., Henn, H.-W., & Niss, M. (Eds.). (2007). Modelling and applications in mathematics education. New York, NY: Springer.
Blunden, A. (2008). Forward to Hegel’s logic (being Part 1 of the Encyclopaedia of the Philosophical Science of 1830). In W. Wallace (translation, original 1873) accessed from Marxist Internet Archive, online, 2009.
Brown, A. M. (2011). Truth and the renewal of knowledge: The case of mathematics education. Educational Studies in Mathematics, 75(3), 329–343.
Bruner, J. S. (1960/1977). The process of education. Cambridge, MA: Harvard University Press.
Burkhardt, H. (1981). The real world and mathematics. Glasgow, Scotland: Blackie.
Cobb, P. (2007). Putting philosophy to work: Coping with multiple theoretical perspectives. In F. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 1–38). Reston, VA: National Council of Teachers of Mathematics.
Cobb, P., Yackel, E., & McClain, K. (Eds.). (2000). Communicating and symbolizing in mathematics classrooms: Perspectives on discourse, tools, and instructional design. Mahwah, NJ: Lawrence Erlbaum.
Cole, M. (1996). Cultural psychology: A once and future discipline. Cambridge, UK: Cambridge University Press.
Confrey, J., & Maloney, A. (2007). A theory of mathematical modelling in technological settings. In W. Blum, P. L. Galbraith, H.-W. Henn, & M. Niss (Eds.), Modelling and applications in mathematics education (pp. 57–68). New York, NY: Springer.
Davydov, V. V. (1990). Types of generalization in instruction. Reston, VA: National Council for Teachers of Mathematics.
Drijvers, P. (2003). Learning algebra in a computer algebra environment. Design research on the understanding of the concept of parameter (Doctoral dissertation). Utrecht University, The Netherlands. Retrieved from http://igitur-archive.library.uu.nl/dissertations/2003-0925-101838/inhoud.htm.
Engeström, Y. (1991). Non scolae sed vitae discimus: Toward overcoming the encapsulation of school learning. Learning and Instruction, 1(3), 243–259.
Freudenthal, H. (1983). Didactical phenomenology of mathematical structures. Dordrecht, The Netherlands: Kluwer.
Geiger, V., Faragher, R., & Goos, M. (2010). CAS-enabled technologies as “agents provocateurs” in teaching and learning mathematical modelling in secondary school classrooms. Mathematics Education Research Journal, 22(2), 48–68.
Gravemeier, K. (2007). Emergent modelling as a precursor to mathematical modelling. In W. Blum, P. L. Galbraith, H.-W. Henn, & M. Niss (Eds.), Modelling and applications in mathematics education (pp. 137–144). New York, NY: Springer.
Gravemeier, K., Lehrer, R., van Oers, B., & Verschaffel, L. (2002). Symbolizing, modelling and tool use in mathematics education. Dordrecht, The Netherlands: Kluwer.
Hanna, G., & Jahnke, H. N. (2007). Proving and modelling. In W. Blum, P. L. Galbraith, H-W. Henn, & M. Niss (Eds.), Modelling and applications in mathematics education (pp. 142–152). New York, NY: Springer.
Holland, D., & Quinn, N. (Eds.). (1987). Cultural models in language and thought. Cambridge, UK: Cambridge University Press.
Hoyles, C., Noss, R., & Pozzi, S. (2001). Proportional reasoning in nursing practice. Journal for Research in Mathematics Education, 32, 4–27.
Joseph, G. G. (2010). Crest of the peacock: Non-European roots of mathematics (3rd ed.). Princeton, NJ: Princeton University Press.
Kaiser, G., Blum, W., Ferri, R., & Stillman, G. (Eds.). (2011). Trends in teaching and learning of mathematical modelling. New York, NY: Springer.
Kaiser, G., & Sriraman, B. (2006). A global survey of international perspectives on modelling in mathematics education. ZDM—Zentralblatt fur Didaktik der Mathematik, 38(3), 302–310.
Kent, P., Guile, R., Hoyles, C., & Bakker, A. (2007). Characterising the use of mathematical knowledge in boundary-crossing situations at work. Mind, Culture, and Activity, 14(1–2), 64–82.
Krutetskii, V. (1976). The psychology of mathematical abilities in schoolchildren. Chicago, IL: University of Chicago Press.
Lakoff, G., & Johnson, M. (1980). Metaphors we live by. Chicago, IL: Chicago University Press.
Lakoff, G., & Núñez, R. (2000). Where mathematics comes from. New York, NY: Basic Books.
Lampert, M. (1990). When the problem is not the question and the solution is not the answer. American Educational Research Journal, 27, 29–63.
Latour, B. (1987). Science in action. Milton Keynes, UK: Open University Press.
Lave, J. (1988). Cognition in practice. Cambridge, UK: Cambridge University Press.
Lave, J., & Wenger, E. (1991). Situated learning: Legitimate peripheral participation. Cambridge, UK: Cambridge University Press.
Leontiev, A. N. (1978). Activity, consciousness, and personality. Englewood Cliffs, N.J.: Prentice-Hall.
Leontiev, A. N. (1981). Problems of the development of mind. Moscow, Russia: Progress Publishers.
Lesh, R., & Doerr, H. (2003). Beyond constructivism: Models, and modeling perspectives on mathematics problem solving, learning, and teaching. Mahwah, NJ: Lawrence Erlbaum.
Lesh, R., Galbraith, P. L., Haines, C. R., & Hurford, A. (Eds.). (2010). Modeling students’ mathematical modelling competences. New York, NY: Springer.
Lesh, R., & Zawojewski, J. (2007). Problem solving and modelling. In F. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 763–804). Reston, VA: National Council of Teachers of Mathematics.
Noss, R., Bakker, A., Hoyles, C., & Kent, P. (2007). Situating graphs as workplace knowledge. Educational Studies in Mathematics, 65(3), 367–384.
Noss, R., & Hoyles, C. (2011). Modeling to address techno-mathematical literacies in work. In G. Kaiser, W. Blum, R. B. Ferri, & G. Stillman (Eds.), Trends in teaching and learning of mathematical modelling (pp. 75–78). New York, NY: Springer.
Pollak, H. (1969). How can we teach applications of mathematics? Educational Studies in Mathematics, 2(2–3), 393–404.
Polya, G. (1957). How to solve it. Princeton, NJ: Princeton University Press.
Roth, W. -M., & Lee, Y. (2007). “Vygotsky’s neglected legacy”: Cultural-historical activity theory. Review of Educational Research, 77, 186–232.
Roth, W.-M., & Radford, L. (2011). A cultural-historical perspective on mathematics teaching and learning. Rotterdam, The Netherlands: Sense.
Ryan, J., & Williams, J. S. (2007). Children’s mathematics 4–15. Milton Keynes, UK: Open University Press.
Schoenfeld, A. (1992). Learning to think mathematically: Problem solving, metacognition, and sense making in mathematics. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 334–370). Reston, VA: National Council of Teachers of Mathematics.
Sfard, A. (1998). Two metaphors for learning and the dangers of choosing just one. Educational Researcher, 27(2), 4–13.
Sfard, A. (2008). Thinking as communicating. Cambridge, UK: Cambridge University Press.
Strässer, R. (2000). Mathematical means and models from vocational contexts: A German perspective. In A. Bessot & J. Ridgway (Eds.), Education for mathematics in the workplace (pp. 65–80). Dordrecht, The Netherlands: Kluwer.
Strässer, R. (2007). Everyday instruments: On the use of mathematics. In W. Blum, P. L. Galbraith, H.-W. Henn, & M. Niss (Eds.), Modelling and applications in mathematics education (pp. 171–178). New York, NY: Springer.
Streefland, L. (1991). Fractions in realistic mathematics education. Dordrecht, The Netherlands: Kluwer.
Treffers, A. (1987). Three dimensions: A model of goal and theory description, the Wiskobas project. Dordrecht, The Netherlands: Kluwer.
Van Oers, B. (2002). The mathematization of young children’s language. In K. Gravemeier, R. Lehrer, B. van Oers, & L. Verschaffel (Eds.), Symbolizing, modelling and tool use in mathematics education (pp. 29–58). Dordrecht, The Netherlands: Kluwer.
Vygotsky, L. S. (1978). Mind in society: The development of higher psychological processes. Cambridge, MA: Harvard University Press.
Vygotsky, L. S. (1986). Language and thought. Cambridge, MA: MIT Press.
Wake, G. (2007). Considering workplace activity from a mathematical modelling perspective. In W. Blum, P. L. Galbraith, H.-W. Henn, & M. Niss (Eds.), Modelling and applications in mathematics education (pp. 395–402). New York, NY: Springer.
Wartofsky, M. (1979). Models, representations and the scientific understanding. Dordrecht, The Netherlands: Reidel.
Watson, A., & Winbourne, P. (Eds.). (2007). New directions for situated cognition in mathematics. New York, MA: Springer.
Wenger, E. (1998). Communities of practice. Cambridge, UK: Cambridge University Press.
Williams, J. S. (2011). Towards a political economy of education. Mind, Culture, and Activity, 18, 276–292.
Williams, J. S., & Wake, G. D. (2007a). Black boxes in workplace mathematics. Educational Studies in Mathematics, 64, 317–343.
Williams, J. S., & Wake, G. D. (2007b). Metaphors and models in translation between college and workplace mathematics. Educational Studies in Mathematics, 64, 345–371.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer Science+Business Media New York
About this chapter
Cite this chapter
Williams, J., Goos, M. (2012). Modelling with Mathematics and Technologies. In: Clements, M., Bishop, A., Keitel, C., Kilpatrick, J., Leung, F. (eds) Third International Handbook of Mathematics Education. Springer International Handbooks of Education, vol 27. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-4684-2_18
Download citation
DOI: https://doi.org/10.1007/978-1-4614-4684-2_18
Published:
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4614-4683-5
Online ISBN: 978-1-4614-4684-2
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)