More than meets the eye: patterns and shifts in middle school mathematics teachers’ descriptions of models
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.
KeywordsMathematical modeling Mathematical models Middle school Teacher knowledge
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.
- 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.Google Scholar
- 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
- 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.Google Scholar
- 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
- 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.CrossRefGoogle Scholar
- 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.CrossRefGoogle Scholar
- 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
- Hoyles, C., Noss, R., Kent, P., & Bakker, A. (2010). Improving mathematics at work: The need for techno-mathematical literacies. New York: Routledge.Google Scholar
- 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.
- 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.CrossRefGoogle Scholar
- 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.CrossRefGoogle Scholar
- 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
- 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
- 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
- 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
- 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
- Organisation for Economic Co-operation and Development (OECD). (2013). Draft PISA 2015 mathematics framework. OECD Publishing. doi: 10.1787/9789264190511-en.
- 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
- 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