# 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.

## Keywords

Mathematical modeling Mathematical models Middle school Teacher knowledge## Notes

### 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.

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