Mathematical modeling refers to an iterative process of applying mathematical concepts to genuine real-world situations that do not have a single anticipated mathematical path or solution (Cirillo et al., 2016; Zawojewski, 2016). The process for finding a solution is developed by the person considering the problem, while using different mathematical processes and reflecting on their decisions in the context of the problem along the way.
In order to help visualize the theorized phases of this process, several researchers have developed mathematical modeling cycles. Below are two of my favorite examples of these cycles, which show the complex and rich nature of these tasks:
(Bliss et al., 2014)
(Anhalt, Cortez, & Bennett, 2018)
Mathematical modeling tasks can take many forms, with differing levels of freedom for student choices. The tasks can be completely created by students, with the teacher ensuring that an appropriate question is developed, or they can be highly developed by the teacher, providing limited room for student choice. Teachers can also choose to provide opportunities for students to experience the entire modeling process, or just a portion of it (for more info see atomistic and holistic tasks below).
Students see the importance of mathematical understanding (Blum & Ferri, 2016)
Raise student interest and build awareness of real-world applications (Cirillo et al., 2016)
Deepen understanding of mathematical concepts (Anhalt et al., 2018)
Bolster problem solving skills, encourage creative and critical thinking (Gann et al., 2016)
Develop ability to critically analyze (Anhalt et al., 2018)
Necessitate flexibility, creating and modifying solution paths (Gann et al. , 2016)
Opportunities to express, modify, and refine ways of thinking (Lesh et al., 2003)
Potential to incorporate additional STEM content (Maiorca & Stohlmann, 2016)
Using a holistic approach, which provides an opportunity for students go through the entire modeling process, may not always be possible due to time constraints. Instead an atomistic approach may be easier to fit within a lesson, as it focuses on enacting a portion of the modeling process (Geiger et al., 2016). An atomistic approach can also allow for easier entry into mathematical modeling for both teachers and students, providing opportunities to gain experience before moving to larger challenges.
For example, using an atomistic approach, students could be provided with a task that asks them to consider possible variables that are present in a real-world situation. This would focus on the research and brainstorming portion of modeling, allowing them to make sense of a situation without continuing the process by finding a solution to a question.
Alternatively, another atomistic approach could be to present students with potential solutions to a question. This enables students to consider the validity of the solutions, discuss their strengths and weaknesses, and explore possible modifications. This type of activity would focus on the skills needed in later phases of the modeling process, encouraging analysis and model assessment.
Background information and the question to be considered, or the freedom to determine a question to pursue
Support by posing questions that:
facilitate communication
prompt discussion
assist student reflection
balance the mathematical demand of the situation (Carlson et al., 2016)
Informal opportunities to check student understanding through discussions or periodic reporting from the groups
Go through a cycle of posing questions, building solutions, and validating their conclusions (Carlson et al., 2016)
Direct their activities to come to a valid solution
Make decisions that support their goal and appropriately model the situation
Reflect upon their decisions to better understand the implications and refine their models (Maiorca & Stohlmann, 2016)
Anhalt, O., C., Cortez, R., & Bennett, A. B. (2018). The emergence of mathematical modeling competencies: An investigation of prospective secondary mathematics teachers. Mathematical Thinking and Learning, 20(3), 202-221.
Bliss, K. M., Fowler, K. R., & Galluzzo, B. J. (2014). Math modeling: Getting started and getting solutions. Society for Industrial and Applied Mathematics (SIAM).
Blum, W. (2011). Can modelling be taught and learnt? Some answers from empirical research. In G. Kaiser, W. Blum, R. B. Ferri, & G. Stillman (Eds.), Trends in teaching and learning of mathematical modelling (Vol. 1, pp. 15-30). Springer.
Blum, W., Ferri, R. (2016). Advancing the teaching of mathematical modeling: Research-based concepts and examples. In C. R. Hirsch & A. R. McDuffie (Eds.), Mathematical modeling and modeling mathematics (pp. 53-64). National Council of Teachers of Mathematics.
Carlson, M., Wickstrom M., Burroughs E., & Fulton, E. (2016). A case for mathematical modeling in the elementary school classroom. In C. R. Hirsch & A. R. McDuffie (Eds.), Mathematical modeling and modeling mathematics (pp. 121-130). National Council of Teachers of Mathematics.
Cirillo, M., Bartell, T., & Wager, A. (2016). Teaching mathematics for social justice through mathematical modeling. In C. R. Hirsch & A. R. McDuffie (Eds.), Mathematical modeling and modeling mathematics (pp. 53-64). National Council of Teachers of Mathematics.
Gann, C., Avineri, T., Graves, J., Hernandez, M., & Teague, D. (2016). Moving students from remembering to thinking: The power of mathematical modeling. In C. R. Hirsch & A. R. McDuffie (Eds.), Mathematical modeling and modeling mathematics (pp. 97-106). National Council of Teachers of Mathematics.
Geiger, V., Arleback, J. B., & Frejd, P. (2016). Interpreting curricula to find opportunities for modeling: Case studies from Australia and Sweden. In C. R. Hirsch & A. R. McDuffie (Eds.), Mathematical modeling and modeling mathematics (pp. 207-215). National Council of Teachers of Mathematics.
Hirsch, C. R., & McDuffie, A. R. (2016). Annual perspectives in mathematics education 2016: Mathematical modeling and modeling mathematics. National Council of Teachers of Mathematics.
Lesh, R., Cramer, K., Doerr, H. M., Post, T., & Zawojewski, J. S. (2003). Model development sequences. In R. Lesh & H. M. Doerr (Eds.), Beyond constructivism: Models and modeling perspectives on mathematics problem solving, learning, and teaching (pp. 35-58). Lawrence Erlbaum Associates, Inc.
Maiorca, C., & Stohlmann, M. (2016). Inspiring students in integrated STEM education through modeling activities. In C. R. Hirsch & A. R. McDuffie (Eds.), Mathematical modeling and modeling mathematics (pp. 153-162). National Council of Teachers of Mathematics.
Zawojewski, J. (2016). Teaching and learning about mathematical modeling. In C. R. Hirsch & A. R. McDuffie (Eds.), Mathematical modeling and modeling mathematics (pp. 51-52). National Council of Teachers of Mathematics.
