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ZDM

, Volume 50, Issue 1–2, pp 45–59 | Cite as

Modelling and the representational imagination

  • Corey Brady
Original Article
First Online:

Abstract

This article examines the work of 30 in-service teachers engaged with modelling activities during a course within an Ecuadorian master’s degree program in mathematics teaching. These teachers experienced a sequence of activities designed to explore imaginative aspects of mathematical modelling and problem solving, inviting perspectives from life outside of school. They built rich connections between real-world phenomena and a range of ideas about functions and representations of them. An analysis of the teachers’ work identifies a modelling resource—the representational imagination—describing its nature and the implications for models of classroom modelling aiming to support this resource.

Keywords

Modelling cycle Perception Linear function Representational imagination 
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References

  1. Arleback, J. B., Doerr, H. M., & O’Neil, A. H. (2013). A modeling perspective on interpreting rates of change in context. Mathematical Thinking and Learning, 15(4), 314–336.CrossRefGoogle Scholar
  2. Blum, W. (2015). Quality teaching of mathematical modelling: what do we know, what can we do?. In The Proceedings of the 12th international congress on mathematical education (pp. 73–96). Springer.Google Scholar
  3. Brady, C. (2013) Perspectives in motion (unpublished doctoral dissertation). University of Massachusetts, Dartmouth.Google Scholar
  4. Brady, C., Dominguez, A., Glancy, A., Jung, H., McLean, J., & Middleton, J. (2016). Models and modeling working group. In Proceedings of the 38th annual conference of the North American chapter of the International Group for the Psychology of Mathematics Education. PME-NA.Google Scholar
  5. Brady, C., Eames, C., & Lesh, R. (2015). Connecting real-world and in-school problem-solving experiences. Quadrante: Revista de Investigaçião em Educaçião Matemática [Special Issue, Problem Solving], 26(2), 5–38.Google Scholar
  6. Brady, C., McLean, J., Dominguez, A., Glancy, A., & Jung, H. (2017). A comparative analysis of learners’ models from multiple perspectives via a cross-institutional collaborator network. In 18th International conference on the teaching of mathematical modelling and applications (ICTMA-18), July 24–30, Cape Town, South Africa.Google Scholar
  7. Cramer, K. (2003). Using a translation model for curriculum development and classroom instruction. In R. Lesh & H. Doerr (Eds.), Beyond constructivism. Models and modeling perspectives on mathematics problem solving, learning, and teaching. Mahwah, NJ: Lawrence Erlbaum.Google Scholar
  8. Cobb, P., Confrey, J., DiSessa, A., Lehrer, R., & Schauble, L. (2003). Design experiments in educational research. Educational researcher, 32(1), 9–13.CrossRefGoogle Scholar
  9. diSessa, A. A., Hammer, D., Sherin, B., & Kolpakowski, T. (1991). Inventing graphing: Meta-representational expertise in children. Journal of Mathematical Behavior, 10(2), 117–160.Google Scholar
  10. Doerr, H. M., & English, L. D. (2003). A modeling perspective on students’ mathematical reasoning about data. Journal for Research in Mathematics Education, 34(2), 110–136.CrossRefGoogle Scholar
  11. Eames, C., Brady, C., Jung, H., & Glancy, A. (2018). Designing powerful environments for examining and supporting teachers’ models and modeling. In W. Blum, R. Borromeo Ferri (Eds.), Teacher competencies for mathematical modeling. Springer (In press)Google Scholar
  12. English, L. D., Arleback, J. B., & Mousoulides, N. (2016). Reflections on progress in mathematical modelling research. In A. Gutierrez, G. Leder & P. Boero (Eds.), The second handbook of research on the psychology of mathematics education (pp. 383–413). Rotterdam: Sense Publishers.CrossRefGoogle Scholar
  13. English, L. D., & Gainsburg, J. (2016). Problem solving in a 21st-century mathematics curriculum. In L. D. English & D. Kirshner (Eds.), Handbook of international research in mathematics education (3rd edn., pp. 313–335). New York, NY: Taylor & Francis.Google Scholar
  14. English, L. D., Jones, G. A., Bartolini Bussi, M. G., Lesh, R., Tirosh, D., & Sriraman, B. (2008). Moving forward in international mathematics education research. In L. D. English & D. Kirshner (Eds.), Handbook of international research in mathematics education: Directions for the 21st century (pp. 872–905). New York, NY: Routledge.Google Scholar
  15. Glaser, B. G. (1965). The constant comparative method of qualitative analysis. Social Problems, 12(4), 436–445.CrossRefGoogle Scholar
  16. Gooding, D. (1990). Experiment and the making of meaning: Human agency in scientific observation and experiment. Boston, MA: Kluwer.Google Scholar
  17. Goodwin, C. (1994). Professional vision. American Anthropologist, 96(3), 606–633.CrossRefGoogle Scholar
  18. Hatch, J. A. (2002). Doing qualitative research in education settings. Albany, NY: SUNY Press.Google Scholar
  19. Hegedus, S. J., & Moreno-Armella, L. (2009). Intersecting representation and communication infrastructures. ZDM Mathematics Education, 41(4), 399–412.CrossRefGoogle Scholar
  20. Hjalmarson, M. A., Diefes-Dux, H. A., & Moore, T. J. (2008). Designing model development sequences for engineering. In J. S. Zawojewski, H. A. Diefes-Dux & K. J. Bowman (Eds.), Models and modeling in engineering education: Designing experiences for all students (pp. 37–54). Rotterdam: Sense.Google Scholar
  21. Kaiser, G., Blomhøj, M., & Sriraman, B. (2006). Mathematical modelling and applications: empirical and theoretical perspectives. ZDM-Zentralblatt Für Didaktik Der Mathematik, 38(2), 82–85.CrossRefGoogle Scholar
  22. Kaput, J. (1994). The representational roles of technology in connecting mathematics with authentic experience. In R. Biehler, R. W. Scholz, R. Strasser & B. Winkelmann (Eds.), Didactics of mathematics as a scientific discipline (pp. 379–397). New York, NY: Springer.Google Scholar
  23. Kelly, A. E., Lesh, R. A., & Baec, J. Y. (2008). Handbook of design research methods in education: Innovations in science, technology, engineering, and mathematics learning and teaching. New York, NY: Routledge.Google Scholar
  24. Latour, B. (1999). Pandora’s hope: essays on the reality of science studies. Cambridge, MA: Harvard University Press.Google Scholar
  25. Lehrer, R., & Schauble, L. (2015). The development of scientific thinking. Handbook of Child Psychology and Developmental Science, 2, 671–714].Google Scholar
  26. Lesh, R., Cramer, K., Doerr, H., Post, T., & Zawojewski, J. (2003). Model development sequences. In R. Lesh & H. Doerr (Eds.), Beyond constructivism. Models and modeling perspectives on mathematics problem solving, learning, and teaching. Mahwah, NJ: Lawrence Erlbaum.Google Scholar
  27. Lesh, R., & Doerr, H. (2003). Beyond constructivism: A models and modeling perspectives on mathematics teaching, learning, and problem solving. Mahwah, NJ: Lawrence Erlbaum.Google Scholar
  28. Lesh, R., & Fennewald, T. (2013). Introduction to part I, modeling: what is it? Why do it? In R. Lesh, P. L. Galbraith, C. R. Haines & A. Hurford (Eds.), Modeling students’ mathematical modeling competencies (pp. 5–10). New York: Springer.CrossRefGoogle Scholar
  29. Lesh, R., Hamilton, E., & Kaput, J. (Eds.). (2007). Foundations for the future in mathematics education. Mahwah, NJ: Lawrence Erlbaum Associates.Google Scholar
  30. Lesh, R., & Harel, G. (2003). Problem solving, modeling, and local conceptual development. Mathematical Thinking & Learning, 5(2/3), 157–189.CrossRefGoogle Scholar
  31. Lesh, R., Hoover, M., Hole, B., Kelly, A., & Post, T. (2000). Principles for developing thought revealing activities for students and teachers. In: A. Kelly & R. Lesh (Eds.), Handbook of Research Design in Mathematics and Science Education. Abingdon: Routledge.Google Scholar
  32. Lesh, R., Middleton, J. A., Caylor, E., & Gupta, S. (2008). A science need: Designing tasks to engage students in modeling complex data. Educational studies in Mathematics, 68(2), 113–130.CrossRefGoogle Scholar
  33. MacLeod, M., & Nersessian, N. J. (2018). Modeling complexity: cognitive constraints and computational model-building in integrative systems biology. History and philosophy of the life sciences, 40(1), 17.Google Scholar
  34. Middleton, J. A., Lesh, R., & Heger, M. (2003). Interest, identity, and social functioning: Central features of modeling activity. In R. Lesh & H. Doerr (Eds.), Beyond constructivism. Models and modeling perspectives on mathematics problem solving, learning, and teaching. Mahwah, NJ: Lawrence Erlbaum.Google Scholar
  35. Moreno-Armella, L., & Brady, C. (2017). Technological supports for mathematical thinking and learning: Co-action and designing to democratize access to powerful ideas. ICME Monograph. Springer.Google Scholar
  36. Nemirovsky, R., Tierney, C., & Wright, T. (1998). Body motion and graphing. Cognition and Instruction, 16(2), 119–172.CrossRefGoogle Scholar
  37. Nersessian, N. J. (2010). Creating scientific concepts. Cambridge, MA: MIT.Google Scholar
  38. Ochs, E., Gonzales, P., & Jacoby, S. (1996). “When I come down I’m in the domain state”: Grammar and graphic representation in the interpretive activity of physicists. Studies in Interactional Sociolinguistics, 13, 328–369.Google Scholar
  39. Stevens, R., & Hall, R. (1998). Disciplined perception: Learning to see in technoscience. In: M. Lampert & M. Blunk (Eds.), Talking Mathematics in School: Studies of Teaching and Learning. New York, NY: Cambridge University Press.Google Scholar
  40. Stroup, W. (2002). Understanding qualitative calculus: A structural synthesis of learning research. International Journal of Computers for Mathematical Learning, 7(2), 167–215.CrossRefGoogle Scholar

Copyright information

© FIZ Karlsruhe 2018

Authors and Affiliations

  1. 1.Vanderbilt UniversityNashvilleUSA

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