1. Understanding Mathematical Modeling: A Framework for Faculty Practice Dr. Todd Abel Dr. Tracie McLemore Salinas Department of Mathematical Sciences

2. What is Mathematical Modeling? How do you conceptualize mathematical modeling in your work as a teacher educator? 3/7/2016

3. Meaningful Modeling Discussions - IMSPC 3/7/2016 Modeling Working Group activities Discussions across stakeholders – K-12 teachers, Mathematicians, Mathematics Educators Informed by Mathematical modeling in the CCSSM & COMAP Alignment of modeling practices with classroom culture ideals Creating & Reviewing Common conceptualizations of modeling Progressions of student practices in modeling Characterizations of modeling tasks Framework for modeling

4. What is Mathematical Modeling? NCTM Process Standards Communication Connections Reasoning & Proof Problem Solving Representation 3/7/2016

5. What is Mathematical Modeling? NCTM Process Standards CCSSM Mathematical Practices 3/7/2016

6. What is Mathematical Modeling? In our work, it became important to consider the use of the word model in mathematics teaching and learning. Our distinction between models and modeling can be seen in the diagram below. 3/7/2016 Modeling begins in the real world and develops mathematical representations that have meaning for the real world context. Models begin in mathematics and develops real world representations that have meaning for the mathematical context. Real World Mathematics

7. Conceptualizing Mathematical Modeling “A mathematical model is a representation of a system or scenario that is used to gain qualitative and/or quantitative understanding of some real- world problems and to predict future behavior.” (Bliss, Fowler, & Galluzzo, 2014) 3/7/2016

8. Conceptualizing Mathematical Modeling 3/7/2016 Common Core State Standards, 2010 Math World Real World mathematize contextualize

9. Why Mathematical Modeling? Modeling is a vertically aligned mathematical practice. – In Common Core State Standards for Mathematics: Standards for Mathematical Practice: “Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace.” (CCSSM p. 7) High School Conceptual Category: “Modeling links classroom mathematics and statistics to everyday life, work, and decision- making. Modeling is the processing of choosing and using appropriate mathematics and statistics to analyze situations, understand them better, and to improve decisions.” (CCSSM p. 72)

10. Why Mathematical Modeling? Modeling is a vertically aligned mathematical practice. – In 2-year and 4-year post-secondary institutions NC Community College Developmental Math Redesign: “A module should begin, whenever possible, with a rich application with which students can connect and from which skills will emerge.” (NCCCS Developmental Math Modules, 2011) Modeling integrated into precalculus and calculus courses at 4-year colleges, and modeling or applied mathematics courses required of most NC mathematics undergraduates.

11. Why Mathematical Modeling? Modeling is a vertically aligned mathematical practice. – Teacher preparation Conference Board of Mathematical Sciences The Mathematical Education of Teachers II (2013): “[Preservice and inservice] teachers should have the opportunity to model with mathematics and to mathematize situations by focusing on the mathematical aspects of a situation and formulating them in mathematical terms” (p. 33)

12. Teaching and Learning Mathematical Modeling Either from experience or conjecture – what barriers exist to teaching and learning mathematical modeling? How do you find that future or in-service teachers conceptualize mathematical modeling? 3/7/2016

13. Teaching and Learning Mathematical Modeling – Some Discoveries Mathematical modeling doesn’t appear in most major mathematics methods course texts. 3/7/2016

14. Teaching and Learning Mathematical Modeling – Some Discoveries Mathematicians distinguished more clearly between the Interpretation and Validation phases of modeling than did teachers. Discussions focused on the important transition between these steps and the defining feature of context. 3/7/2016 Did I compute the answer correctly? Did I answer the question that was asked? Did I use the right parameters? Interpret Did I ask the correct question in the first place? Do I need to refine my question? Can I “break the model”? Validate Mathematical Context Real World Context

15. Teaching and Learning Mathematical Modeling - Some Discoveries The Modeling Cycle depiction may inadvertently proceduralize rather than organize modeling activities. Further, the iterative nature of the modeling process may not be clearly drawn from the depiction. 3/7/2016

16. Teaching and Learning Mathematical Modeling – Some Discoveries Students do not develop a connected understanding of the concepts of formulas, algorithms, and models and related ideas. Consequently they: often seek a single correct model (vs. the best model), or overlook the nuance related to variability, or fail to consider when a modeling process is complete and robust. Teachers often consider modeling to be an activity that is only done to apply previously-learned material instead of considering how it is integrated into teaching and learning. 3/7/2016

17. Toward a Framework for Modeling Foundational Documents – Proficiency Matrix for Student Practices in Mathematics (Hull, Balka, & Harbin Miles, 2014) 3/7/2016 (I) – Initial(IN) – Intermediate(A) - Advanced Model with Mathematics Use models to represent and solve a problem, and translate the solution to mathematical symbols. Use models and symbols to represent and solve a problem, and accurately explain the solution representation. Use a variety of models, symbolic representations, and technology tools to demonstrate a solution to a problem

18. Toward a Framework for Modeling Foundational Documents Guidelines for Assessment and Instruction in Statistics Education (GAISE) 3/7/2016

20. Teaching Mathematical Modeling How might this framework be useful for conceptualizing the modeling processes and helping teachers understand modeling? How might we avoid proceduralization of the modeling process? 3/7/2016

21. How Do We Teach Modeling? – Final Points Students need experience with mathematical modeling, therefore pre- service teachers need experiences with mathematical modeling. Mathematical modeling provides a rich area in which mathematicians can engage with mathematics education and mathematics educators. Modeling is a tool for teaching not just an application for mathematics already learned. Multiple passes through the process permits initial application of existing knowledge and motivation/introduction of new content as model is refined and improved. 3/7/2016

22. Final Thoughts Thank You! Todd Abel abelta@appstate.edu Tracie McLemore Salinas salinastm@appstate.edu 3/7/2016

23. Works Cited/Resources Bliss, K.M., Fowler, K.R., & Galluzzo, B.J. (2014). Math modeling: Getting started and getting solutions. Philadelphia, PA: SIAM. http://m3challenge.siam.org/resources/modeling-handbook Hull, T., Harbin Miles, R., & Balka, D. (2014). Realizing Rigor in the Mathematics Classroom. Corwin Press. Franklin, C., Kader, G., Mewborn, D., Moreno, J., Peck, R., Perry, M. and Scheaffer, R. (2007). Guidelines for assessment and instruction in statistics education (GAISE) report. Alexandria: American Statistical Association. National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Common Core State Standards. Washington, DC: Authors. NC Community College Developmental Mathematics Modules. (2011). Available at: http://ncmatyc.matyc.org/wp- content/uploads/file/BetaVersionDevelopmental%20Math%20Modules%20-%20NCCCS%5B1%5D.pdf 3/7/2016