## Abstract

Though there is an extensive research literature on understanding and assessing individual modeling competencies, less attention has been given to characterizing the social context of the classroom in which modeling occurs. Yet a classroom’s culture of modeling—its negotiated system of beliefs and values about the nature of modeling and what constitutes a complete, satisfactory solution to a modeling problem—has a shaping influence on its members’ participation in modeling activities. Devising a means of assessing classroom modeling cultures is thus a crucial task for both research and praxis. In this article, we present an approach to operationally defining and assessing classroom modeling cultures, which is (a) based in quantitative analyses of discourse during whole-class presentations of modeling solutions, and (b) grounded in research in individual modeling competencies. We show how our assessment distinguishes the classroom modeling cultures of different classrooms and how it captures shifts over time in the modeling culture of a given classroom. Finally, we accompany our quantitative results with interpretive analyses of presentation discourse, to triangulate and to attribute meaning to the patterns our assessment detects. Our primary data sources include video recorded presentation and Q&A sessions for three modeling activities in each of two US middle school (Grades 5–8) classrooms. Our assessment approach is a significant contribution because it operationalizes the construct of a classroom’s modeling culture, in terms that highlight potential connections with the research literature of individual modeling competencies. It therefore invites modeling research that coordinates learning and development across individual and social levels.

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## References

Anhalt, C., & Cortez, R. (2016). Developing understanding of mathematical modeling in secondary teacher preparation.

*Journal of Mathematics Teacher Education*,*19*(6), 523–545.Art Wolfe Inc. (2021). Aerial of porcupine caribou herd [Photograph]. Retrieved from https://artwolfe.com/showcase/porcupine-caribou-herd-arctic-national-wildlife-refuge-alaska-usa/

Brady, C., Eames, C. L., & Lesh, D. (2015). Connecting real-world and in-school problem-solving experiences.

*Quadrante,**24*(2), 5–38.Brady, C., & Jung, H. (2019a). Group presentations as a site for collective modeling activity.

*Proceedings of the 41st Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education*, St Louis, MO, USA.Brady, C., & Jung, H. (2019b).

*Class presentations of modelling solutions: A setting for individual and group modelling competencies*. Paper presented at the The 19th International Conference on the Teaching of Mathematical Modelling and Applications (ICTMA 19), Hong Kong, China.Bleiler-Baxter, S. K., Barlow, A. T., & Stephens, D. C. (2016). Moving beyond context: Challenges in modeling instruction. In C. Hirsch (Ed.),

*Annual perspectives in mathematics education: Mathematical modeling and modeling mathematics*(pp. 53–64). National Council of Teachers of Mathematics.Blomhøj, M., & Højgaard Jensen, T. (2003). Developing mathematical modelling competence: Conceptual clarification and educational planning.

*Teaching Mathematics and its Applications*,*22*(3), 123–139.Blomhøj, M., & Højgaard Jensen, T. (2007). What’s all the fuss about competencies? In W. Blum, P. L. Galbraith, H.-W. Henn, & M. Niss (Eds.),

*Modelling and applications in mathematics education: The 14th ICMI study*(pp. 45–56). Springer.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.Blumer, H. (1969).

*Symbolic interactionism: Perspective and method*. Univ of California Press.Boaler, J. (2000). Exploring situated insights into research and learning.

*ZDM‐Mathematics Education*,*31*(1), 113–119.Borromeo Ferri, R. (2006). Theoretical and empirical differentiations of phases in the modelling process.

*ZDM‐Mathematics Education*,*38*(2), 86–95. https://doi.org/10.1007/BF02655883Borromeo Ferri, R. (2007). Modelling problems from a cognitive perspective. In C. Haines, P. Galbraith, W. Blum, & S. Khan (Eds.),

*Mathematical modeling: Education, engineering, and economics*(pp. 260–270). Woodhead Publishing Limited.Borromeo Ferri, R. (2018).

*Learning how to teach Mathematical modeling in school and teacher education*. Springer International Publishing.Cai, J., Cirillo, M., Pelesko, J.A., Borromeo Ferri, R., Borba, M., Geiger, V., Stillman, G., English, L. D., Wake, G., Kaiser, G., & Kwon, O. N. (2014). Mathematical modeling in school education: Mathematical, cognitive, curricular, instructional and teacher education perspectives. In P. Liljedahl, C. Nicol, S. Oesterle & A. Darien (Eds.),

*Proceedings of the Joint Meeting of PME 38 and PME-NA 36*(pp. 145–172). PME-NA, Vancouver, Canada.Carmona, G. (2004).

*Designing an assessment tool to describe students’ mathematical knowledge.*Unpublished doctoral dissertation, Purdue University, West Lafayette, IN.Cartwright, N., & Bradburn, N. (2011).

*A theory of measurement*. National Academies Press.Chamberlin, M.T. (2000). Fun on the Field. Retrieved February 13, 2019, from https://engineering.purdue.edu/ENE/Research/SGMM/CASESTUDIESKIDSWEB/track.htm.

Cobb, P., & Bauersfeld, H. (1995).

*The emergence of mathematical meaning: Interaction in classroom cultures*. Lawrence Erlbaum Associates.Cobb, P., Stephan, M., McClain, K., & Gravemeijer, K. (2001). Participating in classroom mathematical practices.

*Journal of the Learning Sciences*,*10*(1-2), 113–163.Cobb, P., Confrey, J., DiSessa, A., Lehrer, R., & Schauble, L. (2003). Design experiments in educational research.

*Educational Researcher*,*32*(1), 9–13.Cole, M. (1995). Cultural–historical psychology: A meso-genetic approach. In L. M. Martin & E. Tobach (Eds.),

*Social psychology: Theory and practice of doing and knowing*(pp. 168–204). Cambridge University Press.Consortium for Mathematics and its Applications & Society for Industrial and Applied Mathematics [CMA and SIAM]. (2016). Guidelines for assessment and instruction in mathematical modeling education. Retrieved from http://www.siam.org/reports/gaimme.php. Accessed 30 April 2021.

Cronbach, L. J., & Meehl, P. E. (1955). Construct validity in psychological tests.

*Psychological Bulletin*,*52*(4), 281–302.Czocher, J. (2016). Introducing modeling transition diagrams as a tool to connect mathematical modeling to mathematical thinking.

*Mathematical Thinking and Learning*,*18*(2), 77–106.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.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.Engeström, Y. (1993). Developmental studies of work as a testbench of activity theory: The case of primary care medical practice. In S. Chaiklin & J. Lave (Eds.),

*Understanding practice: Perspectives on activity and context*(pp. 64–103). Cambridge University Press.English, L. D. (2003). Reconciling theory, research, and practice: A models and modelling perspective.

*Educational Studies in Mathematics*,*54*(2-3), 225–224.English, L. D. (2006). Mathematical modeling in the primary school: Children’s construction of a consumer guide.

*Educational Studies in Mathematics*,*63*(3), 303–323.English, L. D. (2009). Promoting interdisciplinarity through mathematical modelling.

*ZDM‐Mathematics Education*,*41*(1-2), 161–181.English, L. D. (2016). STEM education K-12: Perspectives on integration.

*International Journal of STEM Education*,*3*(1), 3.English, L. D., Jones, G., Bussi, M., Tirosh, D., Lesh, R., & Sriraman, B. (2008). Moving forward in international mathematics education research. In L. D. English (Ed.),

*Handbook of international research in mathematics education: Directions for the 21st century*(2nd ed.pp. 872–905). Routledge.Ford, M. J. (2015). Implications of choosing “practice” to describe science in the Next Generation Science Standards.

*Science Education*,*99*(6), 1041–1048.Frejd, P. (2013). Modes of modelling assessment. A literature review.

*Educational Studies in Mathematics*,*84*(3), 413–438.Geertz, C. (1973).

*The interpretation of cultures: Selected essays*. Basic Books.Glaser, B. G. (1965). The constant comparative method of qualitative analysis.

*Social Problems*,*12*(4), 436–445.Greeno, J. G. (1994). Gibson’s affordances.

*Psychological Review*,*101*(*2*), 336–342.Greeno, J. G., & van de Sande, C. (2007). Perspectival understanding of conceptions and conceptual growth in interaction.

*Educational Psychologist*,*42*(1), 9–23.Gresalfi, M., & Hand, V. M. (2019). Coordinating situated identities in mathematics classrooms with sociohistorical narratives: A consideration for design.

*ZDM‐Mathematics Education*,*51*(3), 493–504.Gresalfi, M., Martin, T., Hand, V., & Greeno, J. (2009). Constructing competence: An analysis of student participation in the activity systems of mathematics classrooms.

*Educational Studies in Mathematics*,*70*(1), 49–70.Hall, R., & Jurow, A. S. (2015). Changing concepts in activity: Descriptive and design studies of consequential learning in conceptual practices.

*Educational Psychologist*,*50*(3), 173–189.Holland, D. C., Lachicotte Jr., W., Skinner, D., & Cain, C. (1998).

*Identity and agency in cultural worlds*. Harvard University Press.Hutchins, E. (1995).

*Cognition in the Wild*. MIT press.Hutchins, E. (2012). Concepts in practice as sources of order.

*Mind, Culture, and Activity*,*19*(3), 314–323.Jung, H., & Brady, C. (2016). Roles of a teacher and researcher during in situ professional development around the implementation of mathematical modeling tasks.

*Journal of Mathematics Teacher Education,**19*(2–3), 277–295.Kaiser, G. (2007). Modelling and modelling competencies in school.

*Mathematical modelling (ICTMA 12): Education, engineering and economics*, 110–119.Kaiser, G., & Sriraman, B. (2006). A global survey of international perspectives on modelling in mathematics education.

*ZDM‐Mathematics Education,**38*(3), 302–310.Kaiser, G., & Brand, S. (2015). Modelling competencies: Past development and further perspectives. In

*Mathematical modelling in education research and practice*(pp. 129–149). Springer.Keller, E. F. (2002).

*Making sense of life: Explaining biological development with models, metaphors, and machines*. Harvard University Press.Kelly, A. E., Lesh, R., & Baek, J. Y. (Eds.). (2008).

*Handbook of innovative design research in science, technology, engineering, mathematics (STEM) education*. Taylor & Francis.Knorr Cetina, K. (1999).

*Epistemic cultures: How the sciences make knowledge*. Harvard University Press.Kobiela, M., & Lehrer, R. (2015). The codevelopment of mathematical concepts and the practice of defining.

*Journal for Research in Mathematics Education*,*46*(4), 423–454.Lamon, S. J., Parker, W. A., & Houston, S. K. (2003).

*Mathematical modelling: A way of life-ICTMA 11*. Woodhead Publishing.Lave, J., & Wenger, E. (1991).

*Situated learning: Legitimate peripheral participation*. Cambridge University Press.Lehrer, R. (2021). Promoting transdisciplinary epistemic dialogue. In M.-C. Shanahan, B. Kim, K. Koh, P. Preciado-Babb, & M. A. Takeuchi (Eds.),

*The learning sciences in conversation: Theories, methodologies, and boundary spaces*. Routledge.Lehrer, R., & Schauble, L. (2000). Developing model-based reasoning in mathematics and science.

*Journal of Applied Developmental Psychology*,*21*(1), 39–48.Lesh, R. (2003). How mathematizing reality is different from realizing mathematics. In

*Mathematical Modelling: A Way of Life-ICTMA 11*(pp. 37–52). Woodhead Publishing.Lesh, R. (2010). Tools, researchable issues & conjectures for investigating what it means to understand statistics (or other topics) meaningfully.

*Journal of Mathematical Modelling and Application*,*1*(2), 16–48.Lesh, R. & English, L. (n.d.). Caribou Count. Retrieved February 13, 2019, from https://engineering.purdue.edu/ENE/Research/SGMM/Problems/FactoryLayout/MEA/CASESTUDIESKIDSWEB/caribou.htm.

Lesh, R., & Harel, G. (2003). Problem solving, modeling, and local conceptual development.

*Mathematical Thinking and Learning*,*5*(2-3), 157–189.Lesh, R. E., & Doerr, H. M. (2003). Beyond constructivism: Models and modeling perspectives on mathematics problem solving, learning, and teaching. Lawrence Erlbaum Associates Publishers.

Lesh, R., & Lehrer, R. (2003). Models and modeling perspectives on the development of students and teachers.

*Mathematical Thinking and Learning*,*5*(2-3), 109–129.Lesh, R., & Sriraman, B. (2005). Mathematics education as a design science.

*ZDM‐Mathematics Education*,*37*(6), 490–505.Lesh, R., Lamon, S., Lester, F., & Behr, M. (1992). Future directions for mathematics assessment. In R. Lesh & S. Lamon (Eds.),

*Assessment of authentic performance in school mathematics*(pp. 389–436). Routledge.Lesh, R., Hoover, M., Hole, B., Kelly, A., & Post, T. R. (2000). Principles for developing thought-revealing activities for students and teachers. In

*Research design in mathematics and science education*(pp. 591–646). Lawrence Erlbaum Associates, Inc.Lesh, R. A., Hamilton, E., & Kaput, J. J. (Eds.). (2007).

*Foundations for the future in mathematics education*. Lawrence Erlbaum Associates.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.Lesh, R., Galbraith, P. L., Haines, C. R., & Hurford, A. (2010).

*Modeling students’ mathematical modeling competencies (ICTMA 13)*. Springer Science + Business Media.Maaß, K. (2005). Barriers and opportunities for the integration of modelling in mathematics classes: Results of an empirical study.

*Teaching Mathematics and Its Applications: International Journal of the IMA,**24*(2–3), 61–74.Maaß, K. (2006). What are modelling competencies?

*ZDM‐Mathematics Education,**38*(2), 113–142.Magiera, M. T., & Zawojewski, J. S. (2011). Characterizations of social-based and self-based contexts associated with students’ awareness, evaluation, and regulation of their thinking during small-group mathematical modeling.

*Journal for Research in Mathematics Education,**42*(5), 486–520.Miles, M. B., & Huberman, A. M. (1994).

*Qualitative data analysis: An expanded sourcebook*. Sage.Mislevy, R. J. (2007). Validity by design.

*Educational Researcher*,*36*(8), 463–469.Mosier, C. I. (1947). A critical examination of the concepts of face validity.

*Educational and Psychological Measurement*,*7*(2), 191–205.National Governors’ Association Center for Best Practices & Council of Chief State School Officers [NGA and CCSSO]. (2010).

*Common core state standards for mathematics*. Authors.National Research Council. (2001).

*Knowing what students know: The science and design of educational assessment*. National Academies Press.Ng, K. (2013). Initial perspectives of teacher professional development on Mathematical modelling in Singapore: A framework for facilitation. In G. A. Stillman, G. Kaiser, W. Blum, & J. P. Brown (Eds.),

*Teaching mathematical modelling: Connecting to research and practice*(pp. 415–425). Mathematics and Mathematics Academic Group, National Institute of Education, Nanyang Technological University.Niss, M. (2003). Mathematical competencies and the learning of mathematics: The Danish KOM project. In

*Proceedings of the 3rd Mediterranean conference on mathematical education*(pp. 115–124).Niss, M., Blum, W., & Galbraith, P. L. (2007). Introduction. In W. Blum, P. L. Galbraith, H. Henn, & M. Niss (Eds.),

*Modelling and applications in mathematics education. The 14th ICMI study*(pp. 3–32). Springer.Noble, T., Nemirovsky, R., Dimattia, C., & Wright, T. (2004). Learning to see: Making sense of the mathematics of change in middle school.

*International Journal of Computers for Mathematical Learning*,*9*(2), 109–167.Pickering, A. (1995).

*The mangle of practice: Time, agency, and science*. University of Chicago Press.Rubano, C. (2020). Western arctic caribou population and migration patterns continue to change [Photograph]. Retrieved from https://www.knom.org/wp/blog/2020/12/30/western-arctic-caribou-population-and-migration-patterns-continueto-change/

Schukajlow, S., Kaiser, G., & Stillman, G. (2018). Empirical research on teaching and learning of mathematical modelling: A survey on the current state-of-the-art.

*ZDM‐Mathematics Education*,*50*(1-2), 5–18.Sophian, C. (1997). Beyond competence: The significance of performance for conceptual development.

*Cognitive Development*,*12*(3), 281–303.Stake, R. E. (1995).

*The art of case study research*. Sage Publications.Stake, R. E. (2013).

*Multiple case study analysis*. Guilford press.Star, S. L. (2010). This is not a boundary object: Reflections on the origin of a concept.

*Science, Technology & Human Values*,*35*(5), 601–617.Star, S. L., & Griesemer, J. R. (1989). Institutional ecology, translations’ and boundary objects: Amateurs and professionals in Berkeley's Museum of Vertebrate Zoology, 1907-39.

*Social Studies of Science*,*19*(3), 387–420.Voigt, J. (1994). Negotiation of mathematical meaning and learning mathematics.

*Educational Studies in Mathematics*,*26*(2), 275–298.Vorhölter, K. (2018). Conceptualization and measuring of metacognitive modelling competencies: Empirical verification of theoretical assumptions.

*ZDM‐Mathematics Education,**50*(1), 343–354.Yackel, E., & Cobb, P. (1996). Sociomathematical norms, argumentation, and autonomy in mathematics.

*Journal for Research in Mathematics Education*,*27*(4), 458–477.Zawojewski, J. S. (2013). Problem solving versus modeling. In R. Lesh, P. L. Galbraith, C. R. Haines, & A. Hurford (Eds.),

*Modeling students’ mathematical modeling competencies*(pp. 237–243). Springer.

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Full electronic versions of the model-eliciting activity problems used in the study to be included as electronic supplementary materials.

## Funding

This material is based upon work supported by the US National Science Foundation under Grant #1652372 and #1615207, by Marquette University under an Explorer Challenge grant, and by the departmental support at the University of Florida.

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Brady, C., Jung, H. Modeling presentations: toward an assessment of emerging classroom cultures of modeling.
*Educ Stud Math* (2021). https://doi.org/10.1007/s10649-021-10056-x

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### Keywords

- Classroom modeling culture
- Assessment
- Practices
- Sociocultural perspectives
- Modeling actions
- Presentations