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Development in Mathematical Modeling

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Exploring Mathematical Modeling with Young Learners

Part of the book series: Early Mathematics Learning and Development ((EMLD))

Abstract

The chapters in this section represent noteworthy steps in a research agenda with implications far beyond traditional conceptions of models and modeling, addressing key questions such as: How does the K-12 mathematics curriculum need to adapt to prepare students for the rapidly changing nature of “mathematical thinking” outside of school? What does it mean to “understand” the most important “big ideas” in elementary (K-16) mathematics? How do these ideas and understandings develop? How can these developments be documented and assessed, in their earliest manifestations? How can assessments of students’ most important conceptual achievements be based on operational definitions that do not simply reduce them to checklists of factual and procedural knowledge? Our goal here is to briefly describe how these chapters are situated within this larger context.

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References

  • Ärlebäck, J. B. (2009). On the use of realistic Fermi problems for introducing mathematical modelling in school. The Montana Mathematics Enthusiast, 6(3), 331–364.

    Article  Google Scholar 

  • Ärlebäck, J. B., Doerr, H. M., & O’Neil, A. H. (2013). A modeling perspective on interpreting rates of change in context. Mathematical Thinking and Learning, 15(4), 314–336.

    Article  Google Scholar 

  • 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). Reston, VA: National Council of Teachers of Mathematics.

    Google Scholar 

  • Blomhøj, M., & Højgaard Jensen, T. (2003). Developing mathematical modelling competence: Conceptual clarification and educational planning. Teaching Mathematics and Its Applications, 22(3), 123–139.

    Article  Google Scholar 

  • 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). New York, NY: Springer.

    Chapter  Google Scholar 

  • 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). Cham, Switzerland: Springer.

    Chapter  Google Scholar 

  • Blum, W., & Leiß, D. (2007). How do students and teachers deal with modelling problems? In C. Haines, P. Galbraith, W. Blum, & S. Khan (Eds.), Mathematical modeling: Education, engineering, and economics (pp. 222–231). Chichester, UK: Horwood.

    Google Scholar 

  • Bonotto, C. (2009). Artifacts: Influencing practice and supporting problems posing in the mathematics classroom. Proceedings of PME, 33(2), 193–200.

    Google Scholar 

  • Borromeo Ferri, R. (2006). Theoretical and empirical differentiations of phases in the modelling process. ZDM— International Journal on Mathematics Education, 38(2), 86–95. https://doi.org/10.1007/BF02655883

    Article  Google Scholar 

  • Borromeo 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). Cambridge, UK: Woodhead Publishing Limited.

    Google Scholar 

  • Brady, C., Eames, C., & Lesh, R. (2015). Connecting real-world and in-school problem-solving experiences. Quadrante: Revista de Investigaçião em Educaçião Matemática [Special Issue, Problem Solving], 26(2), 5–38.

    Google Scholar 

  • Collins, A. (1992). Toward a design science of education. In New directions in educational technology (pp. 15–22). Berlin, Heidelberg: Springer.

    Chapter  Google Scholar 

  • 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.

    Article  Google Scholar 

  • diSessa, A., Hammer, D., Sherin, B., & Kolpakowski, T. (1991). Inventing graphing: Meta-representational expertise in children. Journal of Mathematical Behavior, 10(2), 117–160.

    Google Scholar 

  • Djepaxhija, B., Vos, P., & Fuglestad, A. B. (2017). Assessing mathematizing competences through multiple-choice tasks: Using students’ response processes to investigate task validity. In G. Stillman, W. Blum, & G. Kaiser (Eds.), Mathematical modelling and applications: Crossing and researching boundaries in mathematics education (pp. 601–611). Cham, Switzerland: Springer.

    Chapter  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Eames, C., Brady, C., Jung, H., Glancy, A., & Lesh, R. (2018). Designing powerful environments to examine and support teacher competencies for models and modelling. In Lehrerkompetenzen zum Unterrichten mathematischer Modellierung (pp. 237–266). Wiesbaden, Germany: Springer Spektrum.

    Chapter  Google Scholar 

  • English, L. D. (2003). Reconciling theory, research, and practice: A models and modelling perspective. Educational Studies in Mathematics, 54(2–3), 225–248.

    Article  Google Scholar 

  • Frejd, P., & Ärlebäck, J. B. (2011). First results from a study investigating Swedish upper secondary students’ mathematical modelling competencies. In G. Kaiser, W. Blum, R. Borromeo Ferri, & G. Stillman (Eds.), Trends in teaching and learning of mathematical modelling (pp. 407–416). Dordrecht, The Netherlands: Springer.

    Chapter  Google Scholar 

  • Freudenthal, H. (1968). Why to teach mathematics so as to be useful. Educational Studies in Mathematics, 1(1), 3–8.

    Article  Google Scholar 

  • Gadanidis, G., & Hughes, J. (2011). Performing big math ideas across the grades. Teaching Children Mathematics, 17(8), 486–496.

    Article  Google Scholar 

  • Gainsburg, J. (2006). The mathematical modeling of structural engineers. Mathematical Thinking and Learning, 8(1), 3–36.

    Article  Google Scholar 

  • Galbraith, P., & Stillman, G. (2006). A framework for identifying student blockages during transitions in the modelling process. ZDM, 38(2), 143–162.

    Article  Google Scholar 

  • Hall, R. (1999). Following mathematical practices in design-oriented work. In C. Hoyles, C. Morgan, & G. Woodhouse (Eds.), Rethinking the mathematics curriculum (pp. 41–59). Philadelphia, PA: Falmer Press.

    Google Scholar 

  • Hjalmarson, M. A., Diefes-dux, H. A., & Moore, T. J. (2008). Designing model development sequences for engineering. In Models and modeling in engineering education (pp. 37–54). Leiden, The Netherlands: Brill Sense.

    Google Scholar 

  • Jankvist, U. T., & Niss, M. (2015). A framework for designing a research-based “maths counsellor” teacher programme. Educational Studies in Mathematics, 90(3), 259–284.

    Article  Google Scholar 

  • Kaiser, G., & Brand, S. (2015). Modelling competencies: Past development and further perspectives. In Mathematical modelling in education research and practice (pp. 129–149). Cham, Switzerland: Springer.

    Chapter  Google Scholar 

  • Kaput, J. (1994). Democratizing access to calculus: New routes to old roots. In Mathematical thinking and problem solving (pp. 77–156). Hillsdale, NJ: Erlbaum.

    Google Scholar 

  • Kaput, J. J. (1995, May). Long-term algebra reform: Democratizing access to big ideas. In The algebra initiative colloquium (pp. 37–53). Washington, DC: U. S. Department of Education, Office of Educational Technology.

    Google Scholar 

  • Konold, C., Higgins, T., Russell, S. J., & Khalil, K. (2015). Data seen through different lenses. Educational Studies in Mathematics, 88, 305–325.

    Article  Google Scholar 

  • Konold, C., & Miller, C. D. (2005). TinkerPlots: Dynamic data exploration. [Computer software]. Emeryville, CA: Key Curriculum Press.

    Google Scholar 

  • Lesh, R., Carmona, G., & Moore, T. (2009). Six sigma learning gains and long term retention of understandings and attitudes related to models & modeling. Mediterranean Journal for Research in Mathematics Education, 9(1), 19–54.

    Google Scholar 

  • Lesh, R., Cramer, K., Doerr, H. M., Post, T., & Zawojewski, J. S. (2003). Model development sequences. In R. Lesh & H. Doerr (Eds.), Beyond constructivism: Models and modeling perspectives on mathematics problem solving, learning and teaching (pp. 35–58). Mahwah, NJ: Lawrence Erlbaum Associates.

    Chapter  Google Scholar 

  • Lesh, R., & Doerr, H. (Eds.). (2003). Beyond constructivism: A models and modeling perspective. Mahwah, NJ: Lawrence Erlbaum Associates.

    Google Scholar 

  • Lesh, R., & Kelly, A. (2000). Multitiered teaching experiments. In A. Kelly & R. Lesh (Eds.), Handbook of research design in mathematics and science education (pp. 197–230). Mahwah, NJ: Lawrence Erlbaum Associates.

    Google Scholar 

  • Lesh, R., Hoover, M., Hole, B., Kelly, A., & Post, T. (2000). Principles for developing thought-revealing activities for students and teachers. In R. Lesh & A. Kelly (Eds.), Handbook of research design in mathematics and science education (pp. 591–646). Mahwah, NJ: Lawrence Erlbaum.

    Google Scholar 

  • 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.

    Article  Google Scholar 

  • Lesh, R., & Sriraman, B. (2005). Mathematics education as a design science. ZDM, 37(6), 490–505.

    Google Scholar 

  • Lesh, R., & Zawojewski, J. (2007). Problem solving and modeling. In F. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 763–804). Greenwich, CT: Information Age Publishing.

    Google Scholar 

  • Lesh, R. A., & Nibbelink, W. H. (1978). Mathematics around us: Kindergarten. Glenview, IL: Scott Foresman & Co.

    Google Scholar 

  • Lester, F., & Kehle, P. (2003). From problem solving to modeling: The evolution of thinking about research on complex mathematical activity. In R. Lesh & H. Doerr (Eds.), Beyond constructivism: Models and modeling perspectives on mathematical problem solving, learning, and teaching (pp. 501–517). New York, NY: Lawrence Erlbaum.

    Google Scholar 

  • Maass, K. (2006). What are modelling competencies? ZDM, 38(2), 113–142.

    Article  Google Scholar 

  • Makar, K., & Rubin, A. (2009). A framework for thinking about informal statistical inference. Statistics Education Research Journal, 8(1), 82–105.

    Article  Google Scholar 

  • Mousoulides, N., & English, L. D. (2008). Modeling with data in Cypriot and Australian primary classrooms. Proceedings of PME 32 and PME-NA 34, 3, 423–430.

    Google Scholar 

  • Mousoulides, N., Pittalis, M., & Christou, C. (2006). Improving mathematical knowledge through modelling in elementary schools. Proceedings of PME, 30(4), 201–208.

    Google Scholar 

  • 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). Singapore, Singapore: Mathematics and Mathematics Academic Group, National Institute of Education, Nanyang Technological University.

    Google Scholar 

  • Niss, M. (2003, January). Mathematical competencies and the learning of mathematics: The Danish KOM project. In 3rd Mediterranean conference on mathematical education (pp. 115–124). Athens, Greece: Hellenic Mathematical Society.

    Google Scholar 

  • Niss, M. (2019). Personal communication.

    Google Scholar 

  • Papert, S. (1988). A critique of technocentrism in thinking about the school of the future. In Children in the information age (pp. 3–18). Oxford, UK: Pergamon.

    Google Scholar 

  • Piaget, J. (1970). Genetic epistemology. (trans. Eleanor Duckworth). New York, NY: Columbia University Press.

    Book  Google Scholar 

  • Polya, G. (1945). How to solve it: A new aspect of mathematical method. Princeton, NJ: Princeton University Press.

    Book  Google Scholar 

  • Schoenfeld, A. H. (1992). Learning to think mathematically: Problem solving, metacognition, and sense making in mathematics. In, D. A. Grouws (Ed.) Handbook of research on mathematics learning and teaching (pp. 334–370). Reston, VA: NCTM.

    Google Scholar 

  • Schon, D. A. (1983). The reflective practitioner: How professionals think in action. New York: Basic Books.

    Google Scholar 

  • 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, 50(1–2), 5–18.

    Article  Google Scholar 

  • Schwarz, C., Reiser, B. J., Acher, A., Kenyon, L., & Fortus, D. (2012). MoDeLS: Challenges in defining a learning progression for scientific modeling. In A. Alonzo & A. W. Gotwals (Eds.), Learning progressions in science: Current challenges and future directions. Rotterdam, The Netherlands: Sense Publishers.

    Google Scholar 

  • Stillman, G. A. (2019). State of the art on modelling in mathematics education—Lines of inquiry. In Lines of inquiry in mathematical modelling research in education (pp. 1–20). Springer, Cham.

    Google Scholar 

  • Vorhölter, K. (2017). Measuring metacognitive modelling competencies. In G. A. Stillman, W. Blum, & G. Kaiser (Eds.), Mathematical modelling and applications: Crossing and researching boundaries in mathematics education (pp. 175–185). Cham, Switzerland: Springer.

    Chapter  Google Scholar 

  • Vorhölter, K. (2018). Conceptualization and measuring of metacognitive modelling competencies: Empirical verification of theoretical assumptions. ZDM Mathematics Education, 38(2), 113–142.

    Google Scholar 

  • Wittgenstein, L. (1958). Philosophical investigations. Malden, MA: Basil Blackwell Ltd.

    Google Scholar 

  • Zawojewski, J. S., Lesh, R. A., & English, L. D. (2003). A models and modeling perspective on the role of small group learning activities. In Beyond constructivism: Models and modeling perspectives on mathematics problem solving, learning, and teaching (pp. 337–358). Mahwah, NJ: Lawrence Erlbaum Associates.

    Google Scholar 

  • Zawojewski, J. S., Magiera, M., & Lesh, R. (2013). A proposal for a problem-driven mathematics curriculum framework. The Mathematics Enthusiast, 10(1&2), 469–506.

    Article  Google Scholar 

  • Zieffler, A., Garfield, J., delMas, R., & Reading, C. (2008). A framework to support research on informal inferential reasoning. Statistics Education Research Journal, 7(2), 40–58.

    Article  Google Scholar 

  • Zöttl, L., Ufer, S., & Reiss, K. (2011). Assessing modelling competencies using a multidimensional IRT-approach. In G. Kaiser, W. Blum, R. Borromeo Ferri, & G. Stillman (Eds.), Trends in teaching and learning of mathematical modelling (ICTMA 14) (pp. 427–437). Dordrecht, The Netherlands: Springer.

    Chapter  Google Scholar 

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Brady, C., Lesh, R. (2021). Development in Mathematical Modeling. In: Suh, J.M., Wickstrom, M.H., English, L.D. (eds) Exploring Mathematical Modeling with Young Learners. Early Mathematics Learning and Development. Springer, Cham. https://doi.org/10.1007/978-3-030-63900-6_5

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  • DOI: https://doi.org/10.1007/978-3-030-63900-6_5

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