The pedagogy of thinking mathematically using math modeling in the classroom

Abstract

Mathematical modeling is the process of addressing a real-life task using mathematical concepts. Used in applied mathematics, engineering, the natural and social sciences, educators are recognizing that the use of mathematical modeling as a teaching tool gives students the opportunity to work conceptually on topics that are inherently motivating and engaging and are connected to real-life situations. This can support learning in the mathematics classroom in multiple ways, by using learning for transfer, problem-solving, and conceptual understanding. Students are able to incorporate cross-curricular skills and use real-life examples, which can make mathematics significantly more engaging. Modeling also provides students of varying ability levels to access content knowledge and construct new learning with mathematics that will extend beyond the classroom which is important especially in heterogenous classrooms and increasingly an increasingly diverse student population. Students who understand modeling have the potential to become stronger mathematicians and also use mathematics effectively later in life-whether working in a mathematics-related field or not. Providing professional development for teachers is crucial to making sure that modeling will be enacted with fidelity in the classroom. For pre-service teachers to effectively implement modeling in the classroom, they should both experience modeling as students themselves, learn how to create modeling scenarios in methods coursework, and then have multiple opportunities to enact modeling lessons with students. In-service teachers also need to have professional learning experiences with modeling, including creation of modeling activities, ways to implement modeling scenarios effectively, options for grouping students during modeling (including ways to work with language learners and students with special education needs), ways to grade modeling work, and how to connect modeling to curricular demands. Mathematical modeling is a highly-effective, engaging method to connect students to real-life mathematics, but teachers need professional learning supports if modeling is to be implemented effectively.

Keywords

• mathematical modeling
• modeling cycle
• professional content knowledge
• problem solving
• mathematics education

References

1. Association of Mathematics Teacher Educators (AMTE). (2017). Standards for Preparing Teachers of Mathematics. https://amte.net/standards
2. Blomhøj, M., & Kjeldsen, T. H. (2006). Teaching mathematical modelling through project work. ZDM, 38(2), 163-177. https://doi.org/10.1007/BF02655887
3. Blum, W. (2015). Quality teaching of mathematical modelling: What do we know, what can we do? The proceedings of the 12th international congress on mathematical education,
4. Blum, W., & Borromeo Ferri, R. (2009). Mathematical modelling: Can it be taught and learnt? Journal of mathematical modelling and application, 1(1), 45-58.
5. Blum, W., & Borromeo Ferri, R. (2016). Advancing the teaching of mathematical modeling: Research-based concepts and examples. In C. R. Hirsch, A. R. McDuffie, & National Council of Teachers of Mathematics (Eds.), Annual perspectives in mathematics education, Mathematical modeling and modeling mathematics. National Council of Teachers of Mathematics.
6. Bonotto, C. (2010). Realistic mathematical modeling and problem posing. In Modeling Students' Mathematical Modeling Competencies (pp. 399-408). Springer.
7. Borromeo Ferri, R., & Blum, W. (2009). Mathematical modelling in teacher education–experiences from a modelling seminar. CERME 6–WORKING GROUP, 11, 2046.
8. Bostic, J. D. (2015). A blizzard of a value. Mathematics Teaching in the Middle School, 20(6), 350-357. Carlson, M. A., Wiskstrom, M. H., Burroughs, E. A., & Fulton, E. W. (2016). A case for mathematical modeling in the elementary school classroom. National Council of Teachers of Mathematics.

9. Cetinkaya, B., Kertil, M., Erbas, A. K., Korkmaz, H., Alacaci, C., & Cakiroglu, E. (2016). Pre-service teachers’ developing conceptions about the nature and pedagogy of mathematical modeling in the context of a mathematical modeling course. Mathematical Thinking and Learning, 18(4), 287-314. https://doi.org/10.1080/10986065.2016.1219932
10. Chamberlin, M. T., & Chamberlin, S. A. (2010). Enhancing preservice teacher development: Field experiences with gifted students. Journal for the Education of the Gifted, 33(3), 381-416. https://doi.org/10.1177/016235321003300305
11. Chamberlin, S., Payne, A. M., & Kettler, T. (2020). Mathematical modeling: a positive learning approach to facilitate student sense making in mathematics. International Journal of Mathematical Education in Science and Technology, 1-14. https://doi.org/10.1080/0020739X.2020.1788185
12. Chamberlin, S. A., & Parks, K. (2020). A comparison of student affect after engaging in a mathematical modeling activity. International Journal of Education in Mathematics, Science and Technology, 8(3), 177-189. https://doi.org/10.46328/ijemst.v8i3.721
13. Cirillo, M., Pelesko, J. A., Felton-Koestler, M. D., & Rubel, L. H. (2016). Perspectives on modeling in school mathematics. In C. R. Hirsch, A. R. McDuffie, & National Council of Teachers of Mathematics (Eds.), Annual perspectives in mathematics education, Mathematical modeling and modeling mathematics. National Council of Teachers of Mathematics.
14. Clark, K. K., & Lesh, R. (2003). A modeling approach to describe teacher knowledge. In R. Lesh & H. D’oerr (Eds.), Beyond constructivism: Models and modelling perspectives on mathematics teaching, learning, and problem solving (pp. 159-174). Lawrence Erlbaum Associates, Inc.
15. Cobb, P., & Jackson, K. (2011). Assessing the quality of the Common Core State Standards for Mathematics. Educational Researcher, 40(4), 183-185. https://doi.org/10.3102/0013189X11409928
16. Doerr, H. (2016). Designing sequences of model development tasks. National Council of Teachers of Mathematics.
17. Doerr, H., & Lesh, R. (2003). Foundations of a models and modeling perspective on mathematics teaching, learning, and problem solving. In H. Doerr & R. Lesh (Eds.), Beyond constructivism: Models and modelling perspectives on mathematics teaching, learning, and problem solving (pp. 3-33). Lawrence Erlbaum Associates, Publishers.
18. Eames, C. L., Brady, C., & Lesh, R. (2016). The OECD PISA: An assessment of mathematical literacy and modeling processes. In C. R. Hirsch, A. R. McDuffie, & National Council of Teachers of Mathematics (Eds.), Annual perspectives in mathematics education, Mathematical Modeling and Modeling Mathematics. National Council of Teachers of Mathematics.
19. Einstein, A., & Infeld, L. (1938). The evolution of physics: The growth of ideas from early concepts to relativity and quanta. Simon and Schuster.
20. English, L. D., Fox, J. L., & Watters, J. J. (2005). Problem posing and solving with mathematical modeling. Teaching Children Mathematics, 12(3), 156.
21. English, L. D., & Watters, J. J. (2005). Mathematical modelling in the early school years. Mathematics Education Research Journal, 16(3), 58-79. https://doi.org/10.1007/bf03217401
22. Geiger, V., Ärlebäck, J. B., & Frejd, P. (2016). Interpreting curriculua to find opportunities for modeling: Case studies from Australia and Sweden. In C. R. Hirsch, A. R. McDuffie, & National Council of Teachers of Mathematics (Eds.), Annual perspectives in mathematics education, Mathematical modeling and modeling mathematics. National Council of Teachers of Mathematics.
23. Gould, H. (2016). What a modeling task looks like. In C. R. Hirsch, A. R. McDuffie, & National Council of Teachers of Mathematics (Eds.), Annual perspectives in mathematics education, Mathematical modeling and modeling mathematics. National Council of Teachers of Mathematics.
24. Hansen, R., & Hana, G. M. (2015). Problem posing from a modelling perspective. In M. Singer, N. F. Ellerton, & J. Cai (Eds.), Mathematical Problem Posing (pp. 35-46). Springer
25. Hattie, J. (2012). Visible learning for teachers: Maximizing impact on learning. Routledge.
26. Hiebert, J., Carpenter, T. P., Fennema, E., Fuson, K., Wearne, D., Murray, H., Olivier, A., & Human, P. (1997). Making sense: Teaching and learning mathematics with understanding. Heinemann.
27. Jacobs, G. J., & Durandt, R. (2016). Attitudes of pre-service mathematics teachers towards modelling: a South African inquiry. EURASIA Journal of Mathematics, Science and Technology Education, 13(1), 61-84. https://doi.org/10.12973/eurasia.2017.00604a
28. Karali, D., & Durmus, S. (2015). Primary school pre-service mathematics teachers’ views on mathematical modeling. EURASIA Journal of Mathematics, Science and Technology Education, 11(4), 803-815. https://doi.org/10.12973/eurasia.2015.1440a
29. Lesh, R., Galbraith, P. L., Haines, C. R., & Hurford, A. (2010). Modeling students’ mathematical modeling competencies (Vol. 10). Springer.
30. Lesh, R., Hoover, M., Hole, B., Kelly, A., & Post, T. (2000). Principles for developing thought-revealing activities for students and teachers. In A. Kelly & R. Lesh (Eds.), The handbook of research design in mathematics and science education (pp. 591-646). Lawrence Erlbaum and Associates.
31. Lesh, R., & Yoon, C. (2007). What is distinctive in (our views about) models and modelling perspectives on mathematics problem solving, learning and teaching? In W. Blum, P. L. Galbraith, H.-W. Henn, & M. Niss (Eds.), Modelling and applications in mathematics education : the 14th ICMI study (pp. 161-170). Springer.
32. Lesh, R., & Zawojewski, J. (2007). Problem solving and modeling. In F. K. Lester (Ed.), Second handbook of research on mathematics teaching and learning: A project of the National Council of Teachers of Mathematics. IAP.
33. Meyer, D. (2005). Missing the promise of mathematical modeling. Mathematics Teacher, 108(8), 6. https://doi.org/10.5951/mathteacher.108.8.0578
34. Meyer, D. (2015). Missing the promise of mathematical modeling. Mathematics Teacher, 108(8), 578-583. National Research Council. (1989). Everybody counts: A report to the nation on the future of mathematics education. National Academies Press.
35. Niss, M., Blum, W., & Galbraith, P. L. (2007). Introduction. In P. L. Galbraith, H.-W. Henn, & M. Niss (Eds.), Modelling and applications in mathematics education: The 14th ICMI study (Vol. 10, pp. 3-32). Springer Science & Business Media.
36. Pollak, H. (2003). A history of the teaching of modeling. In G. A. Stanic (Ed.), A history of school mathematics (Vol. 1, pp. 647-671). National Council of Teachers of English.
37. Polya, G. (1957). How to solve it: A new aspects of mathematical methods. Prentice University Press. https://nook.barnesandnoble.com/products/9781400828678/
38. Remillard, J. T. (2005). Examining key concepts in research on teachers’ use of mathematics curricula. Review of Educational Research, 75(2), 211-246. https://doi.org/10.3102/00346543075002211
39. Schmidt, W. H., & Houang, R. T. (2012). Curricular coherence and the Common Core State Standards for Mathematics. Educational Researcher, 41(8), 294-308. https://doi.org/10.3102/0013189X12464517
40. Schoenfeld, A. (1992). Learning to think mathematically: Problem solving, metacognition, and sense making in mathematics. In D. Grouws (Ed.), Handbook of Research on Mathematics Teaching and Learning (pp. 15). Macmillian.
41. Schukajlow, S., Kolter, J., & Blum, W. (2015). Scaffolding mathematical modelling with a solution plan. ZDM, 47(7), 1241-1254. https://doi.org/10.1007/s11858-015-0707-2
42. Sevis, S. (2016). Unpacking teacher knowledge for bridging in- and out-of-school mathematics using mathematically-rich and contextually-realistic problems (Publication Number 10143631) [Ph.D., Indiana University]. ProQuest Dissertations & Theses Global. Ann Arbor.
43. Skemp, R. R. (1976). Relational understanding and instrumental understanding. Mathematics Teaching, 77(1), 20-26.
44. Tran, D., & Dougherty, B. J. (2014). Authenticity of mathematical modeling. Mathematics Teacher, 107(9), 672-678. https://doi.org/10.5951/mathteacher.107.9.0672
45. Zawojewski, J. (2010). Problem solving versus modeling. In R. Lesh, P. L. Galbraith, C. R. Haines, & A. Hurford (Eds.), Modeling students’ mathematical modeling competencies (Vol. 10, pp. 237-244). Springer.