**Priscila Dias Corrêa**

**Abstract**: This study aims to investigate the mathematical proficiency promoted by mathematical modelling tasks that require students to get involved in the processes of developing mathematical models, instead of just using known or given models. The research methodology is grounded on design-based research, and the classroom design framework is supported by complexity science underpinnings. The research intervention consists of high-school students, from a grade 11 mathematics course, aiming to solve four different modelling tasks in four distinct moments. Data was collected during the intervention from students’ written mathematical work and audio and video recordings, and from recall interviews after the intervention. Data analysis was conducted based on a model of mathematical proficiency and assisted by interpretive diagrams created for this research purpose. This research study offers insight into mathematics teaching by portraying how mathematical modelling tasks can be integrated into mathematics classes to promote students’ mathematical proficiency. The study discusses observed expressions and behaviours in students’ development of mathematical proficiency and suggests a relationship between mathematical modelling processes and the promotion of mathematical proficiency. The study also reveals that students develop mathematical proficiency, even when they do not come to full resolutions of modelling tasks, which emphasizes the relevance of learning processes, and not only of the products of these processes.

**Keywords**: C*lassroom-based research; Complexity science; Design-based research; High-school level; Mathematical modelling; Mathematical proficiency.*

**Please Cite:** Corrêa, P. D. (2021). The mathematical proficiency promoted by mathematical modelling. *Journal of Research in Science, Mathematics and Technology Education*, 4(2), 107-131.

DOI: https://doi.org/10.31756/jrsmte.424

ReferencesAlmeida, L., Silva, K. d., & Vertuan, R. (2013).Modelagem matemática na educação básica.Contexto. Anderson, D., Nashon, S. M., & Thomas, G. P. (2009). Evolution of research methods for probing and understanding metacognition.Research in Science Education,39(2), 181-195. https://doi.org/10.1007/s11165-007-9078-1 Bahmaei, F. (2011). Mathematical modelling in primary school, advantages and challenges.Journal of Mathematical Modelling and Application,1(9), 3-13. Bassanezi, R. C. (1994). Modelagem como estratégia metodológica no ensino da matemática.Boletim de Educação da SBMAC. Biembengut, M. S. (2009). 30 anos de modelagem matemática na educação brasileira: Das propostas primeiras às propostas atuais.Alexandria Revista de Educação em Ciência e Tecnologia,2(2), 7-32. Bleiler-Baxter, S. K., Barlow, A. T., & Stephens, D. C. (2016). Moving beyond context: Challenges in modelling instruction. In C. R. Hirsch, & A. R. McDuffie (Eds.),Mathematical modelling and modelling mathematics(pp. 53-64). The National Council of Teachers of Mathematics. Blum, W., & Borromeo Ferri, R. (2009). Mathematical modelling: Can it be taught and learnt?Journal of Mathematical Modelling and Application,1(1), 45-58. Blum, W., & Borromeo Ferri, R. (2016). Advancing the teaching of mathematical modelling: Research-based concepts and examples. In C. R. Hirsch, & A. R. McDuffie (Eds.),Mathematical modelling and modelling mathematics(pp. 65-76). The National Council of Teachers of Mathematics. Bonotto, C. (2010). Engaging students in mathematical modelling and problem posing activities.Journal of Mathematical Modelling and Application,1(3), 18-32. Bracke, M., & Geiger, A. (2011). Real-world modelling in regular lessons: A long-term experiment. In G. Kaiser, W. Blum, R. Borromeo Ferri, & G. Stillman (Eds.),Trends in teaching and learning of mathematical modelling(pp. 529-549). Springer Netherlands. https://doi.org/10.1007/978-94-007-0910-2_52 Britzman, D. (2003).Practice makes practice: A critical study of learning to teach.State University of New York Press. Brown, J. P., & Edwards, I. (2011). Modelling tasks: Insight into mathematical understanding. In G. Kaiser, W. Blum, R. B. Ferri, & G. Stillman (Eds.),Trends in teaching and learning of mathematical modelling(pp. 187-197). Springer Netherlands. https://doi.org/10.1007/978-94-007-0910-2_20 Cavey, L. O., & Champion, J. (2016). Learning secondary school mathematics through authentic mathematical modelling tasks. In C. R. Hirsch, & A. R. McDuffie (Eds.),Mathematical modelling and modelling mathematics(pp. 131-141). The National Council of Teachers of Mathematics. Cirillo, M., Pelesko, J. A., Felton-Koestler, M. D., & Rubel, L. (2016). Perspectives on modelling in school mathematics. In C. R. Hirsch, & A. R. McDuffie (Eds.),Mathematical modelling and modelling mathematics(pp. 3-16). The National Council of Teachers of Mathematics. Corrêa, P. D. (2019). Observing high-school students' mathematical understanding and mathematical proficiency in the context of mathematical modelling.Proceedings of the 2018 Annual Meeting of the Canadian Mathematics Education Study Group,1, 143-151. Davis, B., & Simmt, E. (2003). Understanding learning systems: Mathematics education and complexity science.Journal for Research in Mathematics Education,34(2), 137-167. https://doi.org/10.2307/30034903 Dias Corrêa, P. M. (2017).Observing high-school students’ mathematical understanding and mathematical proficiency in the context of mathematical modeling[Doctoral Dissertation, University of Alberta]. ERA. https://doi.org/10.7939/R3RV0DC5M Doerr, H. M. (2006). Teachers' ways of listening and responding to students' emerging mathematical models.ZDM,38(3), 255-268. https://doi.org/10.1007/BF02652809 Dym, C. L. (2004).Principles of mathematical modelling.Academic Press.Elasticity of demand. (n.d.). Retrieved from Wrights Land of Economics: https://wrightslandofeconomics.wikispaces.com/Elasticity+of+Demand+(Tax+Incidence) English, L., & Doerr, H. (2004). Learning through interacting with students' ways of thinking.Mathematics Education For the Third Millennium: Towards 2010. Proceedings of the 27th Annual Conference of the Mathematics Education Research Group of Australasia,1, 215-222. Frejd, P. (2012). Teachers' conceptions of mathematical modelling at Swedish upper secondary school.Journal of Mathematical Modelling and Application,1(5), 17-40. Galbraith, P. (2011). Models of modelling: Genres, purposes or perspectives.Journal of Mathematical Modelling and Application,1, 3-16. Gann, C., Avineri, T., Graves, J., Hernandez, M., & Teague, D. (2016). Moving students from remembering to thinking: The power of mathematical modelling. In C. R. Hirsch, & A. R. McDuffie (Eds.),Mathematical modelling and modelling mathematics(pp. 97-106). The National Council of Teachers of Mathematics. Henningsen, M., & Stein, M. K. (1997). Mathematical tasks and students cognition: Classroom-based factors that support and inhibit high-level mathematical thinking and reasoning.Journal for Research in Mathematics Education,28(5), 524-549. https://doi.org/10.2307/749690 Ikeda, T., & Stephens, M. (2010). Three teaching principles for fostering students' thinking about modelling: An experimental teaching program for 9th grade students in Japan.Journal of Mathematics Modelling and Application,1(2), 49-59. Kaiser, G., & Stender, P. (2013). Complex modelling problems in co-operative, self-directed learning environments. In G. A. Stillman, G. Kaiser, W. Blum, & J. P. Brown (Eds.),Teaching mathematical modelling: Connecting to research and practice(pp. 277-293). Springer Netherlands. https://doi.org/10.1007/978-94-007-6540-5_23 Kawasaki, T., Moriya, S., Okabe, Y., & Maesako, T. (2012). The problems of mathematical modelling introduction on mathematics education in Japanese school.Journal of Mathematical Modelling Application,1(5), 50-58. Kilpatrick, J., Swafford, J., & Findell, B. (2001). The strands of mathematical proficiency. In J. Kilpatrick, J. Swafford, & B. Findell,Adding it up: Helping children learn mathematics(pp. 115-155). National Academy Press. Lamon, S. J. (2003). Beyond constructivism: An improved fitness metaphor for the acquisition of mathematical knowledge. In R. Lesh, & H. M. Doerr (Eds.),Beyond constructivism - Models and modelling perspectives on mathematics problem solving, learning, and teaching(pp. 435-447). Erlbaum. Lesh, R., Young, R., & Fennewald, T. (2013). Modelling in K-16 mathematics classrooms – and beyond. In R. Lesh, P. L. Galbraith, C. R. Haines, & A. Hurford (Eds.),Modelling students’ mathematical modelling competencies(pp. 275-283). Springer US. https://doi.org/10.1007/978-1-4419-0561-1_24 Meyer, J. F., Caldeira, A., & Malheiros, A. P. (2011).Modelagem em educação matemática.Autêntica. Palharini, B., & Almeida, L. M. (2015). Mathematical modelling tasks and the mathematical thinking of students. In G. A. Stillman, W. Blum, & M. S. Biembengut (Eds.),Mathematical modelling in education research and practice: Cultural, social and cognitive influences(pp. 219-228). Springer International Publishing. https://doi.org/10.1007/978-3-319-18272-8_17 Pirie, S., & Kieren, T. (1994). Growth in mathematical understanding: How can we characterize it and how can we represent it?Educational Studies in Mathematics,26(2/3), 165-190. https://doi.org/10.1007/BF01273662 Ricks, T. E. (2009). Unifying current trends in mathematics education research using complexity theory.Southeastern Teacher Education Journal,2(2), 63-74. Siller, H.-S., & Kuntze, S. (2011). Modelling as a big idea in mathematics - Knowledge and views of pre-service and in-service teachers.Journal of Mathematical Modelling and Application,1(6), 33-39. Silver, E., Mesa, V. M., Morris, K. A., Star, J. R., & Benken, B. M. (2009). Teaching mathematics for understanding: An analysis of lessons submitted by teachers seeking NBPTS certification.American Educational Research Journal,46(2), 501-531. https://doi.org/10.3102/0002831208326559 Stender, P., & Kaiser, G. (2016). Fostering modelling competencies for complex situations. In C. R. Hirsch & A. R. McDuffie (Eds.),Mathematical modelling and modelling mathematics(pp. 107-115). The National Council of Teachers of Mathematics. Stillman, G. (2001). The impact of school-based assessment on the implementation of a modelling/applications-based curriculum: An Australian example.Teaching Mathematics and its Applications,20(3), 101-108. https://doi.org/10.1093/teamat/20.3.101 Tall, D. (2013).How humans learn to think mathematically: Exploring the three worlds of mathematics. Cambridge University Press. . The Design-Based Research Collective. (2003). Design-based research: An emerging paradigm for educational inquiry.Educational Researcher,32(1), 5-8. https://doi.org/10.3102/0013189X032001005 Tuck, J. (n.d.).The relationship between price elasticity and total revenue. Retrieved from Small Business Chron: http://smallbusiness.chron.com/relationship-between-price-elasticity-total-revenue-24544.html Viecili, C. R. (2006).Modelagem matemática: Uma proposta para o ensino de matemática[Master Thesis, Pontifícia Universidade Católica do Rio Grande do Sul]. PUCRS Repository. http://hdl.handle.net/10923/3029 Vorhölter, K., Kaiser, G., & Borromeo Ferri, R. (2014). Modelling in mathematics classroom instruction: An innovative approach for transforming mathematics education. In Y. Li, E. A. Silver, & S. Li (Eds.),Transforming mathematics instruction: Multiple approaches and practices(pp. 21-36). Springer International Publishing. https://doi.org/10.1007/978-3-319-04993-9_3 Vygotsky, L. (1978).Mind in society: The development of higher psychological processes.Harvard University Press. Zawojewski, J. (2013). Problem solving versus modelling. In R. Lesh, P. Galbraith, C. Haines, & A. Hurford,Modelling students' mathematical modelling competencies: ICTMA 13(pp. 237-243). Springer. Zbiek, R. M., & Conner, A. (2006). Beyond motivation: Exploring mathematical modelling as a context for deepening students' understandings of curricular mathematics.Educational Studies in Mathematics,63(1), 89-112. https://doi.org/10.1007/s10649-005-9002-4 Zorzan, A. S. (2007). Teaching-learning: Some trends in mathematical education.Revista de Ciências Humanas,8(10), 77-93.