Skip to main content
Log in

Modelling and the representational imagination

  • Original Article
  • Published:
ZDM Aims and scope Submit manuscript


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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12

Similar content being viewed by others


  • 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.

    Article  Google Scholar 

  • 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.

  • Brady, C. (2013) Perspectives in motion (unpublished doctoral dissertation). University of Massachusetts, Dartmouth.

  • 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.

  • 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 

  • 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.

  • 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 

  • Cobb, P., Confrey, J., DiSessa, A., Lehrer, R., & Schauble, L. (2003). Design experiments in educational research. Educational researcher, 32(1), 9–13.

    Article  Google Scholar 

  • 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 

  • 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.

    Article  Google Scholar 

  • 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)

  • 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.

    Chapter  Google Scholar 

  • 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 

  • 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 

  • Glaser, B. G. (1965). The constant comparative method of qualitative analysis. Social Problems, 12(4), 436–445.

    Article  Google Scholar 

  • Gooding, D. (1990). Experiment and the making of meaning: Human agency in scientific observation and experiment. Boston, MA: Kluwer.

  • Goodwin, C. (1994). Professional vision. American Anthropologist, 96(3), 606–633.

    Article  Google Scholar 

  • Hatch, J. A. (2002). Doing qualitative research in education settings. Albany, NY: SUNY Press.

    Google Scholar 

  • Hegedus, S. J., & Moreno-Armella, L. (2009). Intersecting representation and communication infrastructures. ZDM Mathematics Education, 41(4), 399–412.

    Article  Google Scholar 

  • 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 

  • 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.

    Article  Google Scholar 

  • 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 

  • 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 

  • Latour, B. (1999). Pandora’s hope: essays on the reality of science studies. Cambridge, MA: Harvard University Press.

    Google Scholar 

  • Lehrer, R., & Schauble, L. (2015). The development of scientific thinking. Handbook of Child Psychology and Developmental Science, 2, 671–714].

    Google Scholar 

  • 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 

  • 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 

  • 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.

    Chapter  Google Scholar 

  • Lesh, R., Hamilton, E., & Kaput, J. (Eds.). (2007). Foundations for the future in mathematics education. Mahwah, NJ: Lawrence Erlbaum Associates.

    Google Scholar 

  • Lesh, R., & Harel, G. (2003). Problem solving, modeling, and local conceptual development. Mathematical Thinking & Learning, 5(2/3), 157–189.

    Article  Google Scholar 

  • 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.

  • 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.

    Article  Google Scholar 

  • 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.

  • 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 

  • 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.

  • Nemirovsky, R., Tierney, C., & Wright, T. (1998). Body motion and graphing. Cognition and Instruction, 16(2), 119–172.

    Article  Google Scholar 

  • Nersessian, N. J. (2010). Creating scientific concepts. Cambridge, MA: MIT.

    Google Scholar 

  • 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 

  • 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.

  • Stroup, W. (2002). Understanding qualitative calculus: A structural synthesis of learning research. International Journal of Computers for Mathematical Learning, 7(2), 167–215.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to Corey Brady.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Brady, C. Modelling and the representational imagination. ZDM Mathematics Education 50, 45–59 (2018).

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: