More than meets the eye: patterns and shifts in middle school mathematics teachers’ descriptions of models

Abstract

Modeling is a major topic of interest in mathematics education. However, the field’s definition of models is diverse. Less is known about what teachers identify as mathematical models, even though it is teachers who ultimately enact modeling activities in the classroom. In this study, we asked nine middle school teachers with a variety of academic backgrounds and teaching experience to collect data related to one familiar physical phenomenon, cooling liquid. We then asked each participant to construct a model of that phenomenon, describe why it was a model, and identify whether a variety of artifacts representing the phenomenon also counted as models during a semi-structured interview. We sought to identify: what do mathematics teachers attend to when describing what constitutes a model? And, how do their attentions shift as they engage in different activities related to models? Using content analysis, we documented what features and purposes teachers attended to when describing a mathematical model. When constructing their own model, they focused on the visual form of the model and what quantitative information it should include. When deciding whether particular representational artifacts constituted models, they focused on how the representations reflected the system under study, and what purposes those representations could serve in further understanding that system. These findings suggest teachers may have multiple understandings of models, which are active at different times and reflect different perspectives. This has implications for research, teacher education, and professional development.

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

Fig. 1
Fig. 2
Fig. 3

References

  1. Ärlebäck, J. B. (2009). Towards understanding teachers’ beliefs and affects about mathematical modelling. In V. Durand-Guerrier, S. Soury-Lavergna, & F. Arzarello (Eds.), Proceedings of CERME 6 (pp. 2096–2105). Lyon, France: INRP.

  2. Barbosa, J. C. (2006). Mathematical modelling in classroom: A socio-critical and discursive perspective. ZDM-International Journal on Mathematics Education, 38(3), 293–301. doi:10.1007/BF02652812.

    Article  Google Scholar 

  3. Bautista, A., Wilkerson-Jerde, M. H., Tobin, R., & Brizuela, M. B. (2014). Mathematics teachers’ ideas about mathematical models: A diverse landscape. PNA, 9(1), 1–27.

    Google Scholar 

  4. Biembengut, M. S., & Hein, N. (2010). Mathematical modeling: implications for teaching. In R. Lesh, P. L. Galbraith, C. R. Haines, & A. Hurford (Eds.), Modeling students’ mathematical modeling competencies (pp. 481–490). New York: Springer, US. doi:10.1007/978-1-4419-0561-1.

    Google Scholar 

  5. Blomhøj, M., & Kjeldsen, T. (2006). Experiences from an in-service course for upper secondary teachers. ZDM, 38(2), 163–177.

    Article  Google Scholar 

  6. Blum, W. (2002). ICMI study 14: Applications and modelling in mathematics education: Discussion document. Educational Studies in Mathematics, 51(1/2), 149–171.

    Article  Google Scholar 

  7. Blum, W. (2015). Quality teaching of mathematical modelling: What do we know, what can we do? In S. J. Cho (Ed.), The proceedings of the 12th international congress on mathematical education: Intellectual and attitudinal changes (pp. 73–96). Springer International Publishing.

  8. Blum, W., & Leiß’s, D. (2007). How do students and teachers deal with modelling problems? In C. Haines, P. Galbraith, W. Blum, & S. Khan (Eds.), Mathematical modelling: Education, engineering and economics (pp. 222–231). Chichester: Horwood.

    Google Scholar 

  9. Blum, W., & Niss, M. (1991). Applied mathematical problem solving, modelling, applications, and links to other subjects—State trends and issues in mathematics instruction. Educational Studies in Mathematics, 22(1), 37–68.

    Article  Google Scholar 

  10. Borromeo-Ferri, R., & Blum, W. (2010). Insights into teachers’ unconscious behaviour in modeling contexts. In R. Lesh, P. L. Galbraith, C. R. Haines, & A. Hurford (Eds.), Modeling students’ mathematical modeling competencies (pp. 531–538). New York, NY: Springer. doi:10.1007/978-1-4419-0561-1.

    Google Scholar 

  11. Borromeo-Ferri, R., & Lesh, R. (2013). Should interpretation systems be considered to be models if they only function implicitly? In G. A. Stillman, G. Kaiser, W. Blum, & J. P. Brown (Eds.), Teaching mathematical modelling: Connecting to research and practice (pp. 57–66). New York: Springer.

    Google Scholar 

  12. Chapman, O. (2007). Mathematical modelling in high school mathematics: Teachers’ thinking and practice. In P. L. Galbraith, H-W. Henn, & M. Niss (Eds.), Modelling and applications in mathematics education: The 14th ICMI study (Vol. 10, pp. 325–332). New York: Springer.

    Google Scholar 

  13. Chi, M. T. H. (1997). Quantifying qualitative analyses of verbal data: A practical guide. Journal of the Learning Sciences, 6(3), 271–315. doi:10.1207/s15327809jls0603_1.

    Article  Google Scholar 

  14. de Oliveira, A. M. P., & Barbosa, J. C. (2013). Mathematical modeling and the teachers’ tensions. In R. Lesh, P. L. Galbraith, C. R. Haines, & A. Hurford (Eds.), Modeling students’ mathematical modeling competencies (pp. 511–517). Netherlands: Springer.

    Google Scholar 

  15. diSessa, A. A. (1993). Toward an epistemology of physics. Cognition and Instruction, 10, 105–225. doi:10.1080/07370008.1985.9649008.

    Article  Google Scholar 

  16. diSessa, A. A. (2002). Why “conceptual ecology” is a good idea. In M. Limón & L. Mason (Eds.), Reconsidering conceptual change: Issues in theory and practice (pp. 28–60). The Netherlands: Kluwer Academic Publishers.

    Google Scholar 

  17. Doerr, H. M. (2007). What knowledge do teachers need for teaching mathematics through applications and modeling? In W. Blum, P. Galbraith, H. W. Henn, & M. Niss (Eds.), Modelling and applications in math education (New ICMI study series (Vol. 10, pp. 69–78). New York, NY: Springer.

    Google Scholar 

  18. Doerr, H. M., & English, L. D. (2006). Middle grade teachers’ learning through students’ engagement with modeling tasks. Journal of Mathematics Teacher Education, 9(1), 5–32. doi:10.1007/s10857-006-9004-x.

    Article  Google Scholar 

  19. Erdogan, A. (2010). Primary teacher education students’ ability to use functions as modeling tools. Procedia Social and Behavioral Sciences, 2(2), 4518–4522.

    Article  Google Scholar 

  20. Ginsburg, H. P. (1997). Entering the child’s mind: The clinical interview in psychological research and practice. New York: Cambridge University Press.

    Google Scholar 

  21. Gravemeijer, K. (2002). Preamble: From models to modeling. In K. Gravemeijer, R. Lehrer, B. van Oers, & L. Verschaffel (Eds.), Symbolizing, modeling and tool use in mathematics education (pp. 7–22). Dordrecht: Kluwer Academic Publishers.

    Google Scholar 

  22. Hammer, D., & Berland, L. K. (2014). Confusing claims for data: A critique of common practices for presenting qualitative research on learning. Journal of the Learning Sciences, 23(1), 37–46.

    Article  Google Scholar 

  23. Hammer, D. M., Elby, A., Scherr, R. E., & Redish, E. F. (2005). Resources, framing, and transfer. In J. Mestre (Ed.), Transfer of learning from a modern multidisciplinary perspective (pp. 89–120). Greenwich, CT: Information Age Publishing.

    Google Scholar 

  24. Holmquist, M., & Lingefjard, T. (2003). Mathematical modeling in teacher education. In Q.-X. Ya, W. Blum, S. K. Housten, & Q.-Y. Jiang (Eds.), Mathematical modeling in education and culture (ICTMA 10) (pp. 197–208). Chichester: Horwood Publishing.

    Google Scholar 

  25. Hoyles, C., Noss, R., Kent, P., & Bakker, A. (2010). Improving mathematics at work: The need for techno-mathematical literacies. New York: Routledge.

    Google Scholar 

  26. Hoyles, C., Noss, R., & Pozzi, S. (2001). Proportional reasoning in nursing practice. Journal for Research in Mathematics Education, 32(1), 4–27.

    Article  Google Scholar 

  27. Imbrie, P., Zawojewski, J., Hjalmarson, M., Diefes-Dux, H., Follman, D., & Capobianco, B. (2004, June). Model eliciting activities: An in class approach to improving interest and persistence of women in engineering. Paper presented at 2004 annual conference of the American Society for Engineering Education, Salt Lake City, Utah. https://peer.asee.org/12973.

  28. Izsak, A. (2004). Students’ coordination of knowledge when learning to model physical situations. Cognition and Instruction, 22(1), 81–128. doi:10.1207/s1532690Xci2201_4.

    Article  Google Scholar 

  29. Kaiser, G., Blomhøj, M., & Sriraman, B. (2006). Towards a didactical theory for mathematical modelling. ZDM, 38(2), 82–85.

    Article  Google Scholar 

  30. Kaiser, G., & Maaß, K. (2007). Modelling in lower secondary mathematics classroom—Problems and opportunities. In W. Blum, P. Galbraith, H. W. Henn, & M. Niss (Eds.), Modelling and applications in math education (New ICMI study series (Vol. 10, pp. 275–284). New York, NY: Springer.

    Google Scholar 

  31. Kaiser, G., & Sriraman, B. (2006). A global survey of international perspectives on modelling in mathematics education. ZDM, 38(3), 302–310.

    Article  Google Scholar 

  32. Kuntze, S. (2011). In-service and prospective teachers’ views about modelling tasks in the mathematics classroom—Results of a quantitative empirical study. In G. Kaiser, W. Blum, R. Borromeo-Ferri, & G. Stillman (Eds.), Trends in teaching and learning of mathematical modelling (pp. 279–288). Netherlands: Springer.

    Google Scholar 

  33. Lehrer, R., & Schauble, L. (2000). Modeling in mathematics and science. In R. Glaser (Ed.), Advances in instructional psychology: Educational design and cognitive science (pp. 101–159). Hillsdale, NJ: Lawrence Erlbaum.

    Google Scholar 

  34. Lesh, R., & Caylor, B. (2007). Introduction to the special issue: Modeling as application versus modeling as a way to create mathematics. International Journal of Computers for Mathematical Learning, 12(3), 173–194.

    Article  Google Scholar 

  35. Lesh, R., & Doerr, H. M. (2003a). Beyond constructivism: Models and modeling perspectives on mathematics problem solving, learning, and teaching. Mahwah: Lawrence Erlbaum Associates.

    Google Scholar 

  36. Lesh, R., & Doerr, H. M. (2003b). A modeling perspective on teacher development. In R. Lesh & H. M. Doerr (Eds.), Beyond constructivism: Models and modeling perspectives on mathematics problem solving, learning, and teaching (pp. 125–139). Mahwah: Lawrence Erlbaum Associates.

    Google Scholar 

  37. Lingefjärd, T. (2002). Mathematical modeling for preservice teachers: A problem from anesthesiology. International Journal of Computers for Mathematical Learning, 7(2), 117–143.

    Article  Google Scholar 

  38. Maaß, K. (2009). What are teachers’ beliefs about effective mathematics teaching? In J. Cai, G. Kaiser, B. Perry, & N.-Y. Wong (Eds.), Effective mathematics teaching from teachers’ perspectives: National and cross-national studies (pp. 141–162). Rotterdam: Sense Publishers.

    Google Scholar 

  39. Maaß, K. (2010). Classification scheme for modelling tasks. Journal für Mathematik-Didaktik, 31(2), 285–311.

    Article  Google Scholar 

  40. Maaß, K. (2011). How can teachers’ beliefs affect their professional development? ZDM, 43(4), 573–586.

    Article  Google Scholar 

  41. Maaß, K., Artigue, M., Doorman, L. M., Krainer, K. & Ruthven, K. (2013). Implementation of inquiry-based learning in day-to-day teaching [Special Issue]. ZDM, 45(6), 779–923.

    Article  Google Scholar 

  42. Maaß, K., & Gurlitt, J. (2011). LEMA—Professional development of teachers in relation to mathematical modelling. In G. Kaiser, W. Blum, R. Borromeo-Ferri, & G. Stillman (Eds.), Trends in teaching and learning of mathematical modelling (pp. 629–639). Netherlands: Springer.

    Google Scholar 

  43. National Research Council (NRC). (2012). A framework for K-12 science education: Practices, crosscutting concepts, and core ideas. Washington, DC: The National Academies Press.

    Google Scholar 

  44. Ng, K. E. D. (2013). Pre-service secondary school teachers’ knowledge in mathematical modelling—A case study. In G. A. Stillman, G. Kaiser, W. Blum, & J. P. Brown (Eds.), Teaching mathematical modelling: Connecting to research and practice (pp. 427–436). Dordrecht: Springer.

    Google Scholar 

  45. Organisation for Economic Co-operation and Development (OECD). (2013). Draft PISA 2015 mathematics framework. OECD Publishing. doi:10.1787/9789264190511-en.

  46. Posner, G. J., Strike, K. A., Hewson, P. W., & Gertzog, W. A. (1982). Accommodation of a scientific conception: Toward a theory of conceptual change. Science Education, 66(2), 211–227.

    Article  Google Scholar 

  47. Pratt, D., & Noss, R. (2002). The microevolution of mathematical knowledge: The case of randomness. Journal of the Learning Sciences, 11(4), 453–488.

    Article  Google Scholar 

  48. Rees, W. G., & Viney, C. (1988). On cooling tea and coffee. American Journal of Physics, 56(5), 434–437.

    Article  Google Scholar 

  49. Smith, J. P., Disessa, A. A., & Roschelle, J. (1994). Misconceptions reconceived: A constructivist analysis of knowledge in transition. Journal of the Learning Sciences, 3(2), 115–163.

    Article  Google Scholar 

  50. Soon, T. L., & Cheng, A. K. (2013). Pre-service secondary school teachers’ knowledge in mathematical modelling—A case study. In G. A. Stillman, G. Kaiser, W. Blum, & J. P. Brown (Eds.), Teaching mathematical modelling: Connecting to research and practice (pp. 373–384). Dordrecht: Springer.

    Google Scholar 

  51. Sriraman, B., & Kaiser, G. (2006). A global survey of international perspectives on modelling in mathematics education. ZDM, 38(3), 302–310. doi:10.1007/s11858-007-0056-x.

    Article  Google Scholar 

  52. Teixidor-i-Bigas, M., Schliemann, A. D., & Carraher, D. (2013). Integrating disciplinary perspectives: The Poincaré Institute for Mathematics Education. The Mathematics Enthusiast, 10(3), 519–561.

    Google Scholar 

  53. Verschaffel, L., De Corte, E., & Borghart, I. (1997). Pre-service teachers’ conceptions and beliefs about the role of real-world knowledge in mathematical modeling of school word problems. Learning and Instruction, 7(4), 339–359.

    Article  Google Scholar 

  54. Wagner, J. F. (2010). A transfer-in-pieces consideration of the perception of structure in the transfer of learning. Journal of the Learning Sciences, 19(4), 443–479. doi:10.1080/10508406.2010.505138.

    Article  Google Scholar 

Download references

Acknowledgments

This research was supported in part by the National Science Foundation, Grant #DRL-0962863. We would like to thank Ken Wright, the anonymous reviewers, and the Editor of JMTE for their feedback on prior versions of this manuscript. Findings presented in this paper represent the work of the authors and not necessarily the funding agency, colleagues, or reviewers.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Michelle H. Wilkerson.

Appendices

Appendix 1: Interview protocol

Part 0: Email sent to participants

We are going to investigate the temperature of a hot cup of coffee that is left on the table. Please answer the following question and email back your response: What is your prediction of how the temperature of the coffee will change as time goes by? Please explain why you think that. For now, please rely only on your own understandings and ideas. Refrain from consulting resources such as books, the Internet, or other people.

Once we received their response, we mailed each participant a thermometer and asked him or her to conduct an experiment:

Now, actually try it out. Investigate the temperature of a hot cup of coffee that is left on the table and send me the data you collect along with any details you think I should know about the conditions in which the experiment took place. Please keep any notes you make and bring to our interview, or take pictures of them and email them.

  • Part I: Interview description task

  • Here is the data you collected.

  • What do you notice?

  • How would you communicate what you notice to somebody else?

  • Part II: Interview construction task

  • Can you create one model for the things you noticed?

  • What makes you call that a model?

  • How would you decide whether that’s a good model?

  • If the goal of the model were to predict coffee temperature, how would you change your model?

  • If there were more information available, would that affect your model? (And: what other kind of information would you use?)

  • If someone else made a similar investigation, could your model be used to describe the other data? [Possible follow up: could you use your same process to describe the other data?]

  • Part III: Interview sorting task

  • Present teachers with different representations of the coffee cooling situation (See Table 2).

  • Put them all on the table randomly.

  • Please create two groups: those that represent a model and those that do not.

  • Is there any other way that you would group the models?

  • Explain why you grouped them in this way. [Related or as follow up: What criteria did you use to make your grouping?]

  • For each group of representations they created, ask:

  • Is this a model of the data?

  • Why do you think this represents a model?

  • Is there any aspect of the representation that you think does not show the information provided in the best possible way?

  • If yes, what would you change about the representation?

  • If it’s not a representation of the model, is there something you would add?

  • Would you use this in your classroom? Why?

Appendix 2: Nature of coding disagreements

In this paper, some of the findings we report are based on aggregate patterns revealed by thematic coding analysis. We agree with Hammer and Berland (2014) that it is important to not only describe the degree of coding agreement in such analyses, but also describe the nature of coding disagreement. In this analysis, there were two types of disagreement of note.

In the first type of disagreement, the second coder assigned more codes to transcript excerpts than the first coder did, in general. Specifically, the second coder identified more utterances she deemed worthy of being assigned a code than the first coder. This type of disagreement was rare (4 out of 66 possible), but constituted all but one of the disagreements after discussion. Two of the disagreements were feature codes, and two were purpose codes. Since the first coder was responsible for analyzing the rest of the data, this suggests that our findings may slightly underestimate the total number of features and purposes attended to by participants during the interview. However, this would not affect our findings about shifts in attention from features to purposes.

The second type of disagreement emerged around whether one particular utterance referred to Including a Description of the Situation or Including Trend. The participant wanted to “model what is happening to the temperature over time”; one coder argued the participant was referring to the decrease in temperature as a trend, while the other argued the participant was referring to the coffee cooling situation more generally. This disagreement happened only once during interrater procedures.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Wilkerson, M.H., Bautista, A., Tobin, R.G. et al. More than meets the eye: patterns and shifts in middle school mathematics teachers’ descriptions of models. J Math Teacher Educ 21, 35–61 (2018). https://doi.org/10.1007/s10857-016-9348-9

Download citation

Keywords

  • Mathematical modeling
  • Mathematical models
  • Middle school
  • Teacher knowledge