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 semistructured 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.
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Acknowledgments
This research was supported in part by the National Science Foundation, Grant #DRL0962863. 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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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.
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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/s1085701693489
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Keywords
 Mathematical modeling
 Mathematical models
 Middle school
 Teacher knowledge