Abstract
In modern societies in the 21st century, significant changes have been occurring in the kinds of “mathematical thinking” that are needed outside of school. Even in the case of primary school children (grades K-2), children not only encounter situations where numbers refer to sets of discrete objects that can be counted. Numbers also are used to describe situations that involve continuous quantities (inches, feet, pounds, etc.), signed quantities, quantities that have both magnitude and direction, locations (coordinates, or ordinal quantities), transformations (actions), accumulating quantities, continually changing quantities, and other kinds of mathematical objects. Furthermore, if we ask, what kind of situations can children use numbers to describe? rather than restricting attention to situations where children should be able to calculate correctly, then this study shows that average ability children in grades K-2 are (and need to be) able to productively mathematize situations that involve far more than simple counts. Similarly, whereas nearly the entire K-16 mathematics curriculum is restricted to situations that can be mathematized using a single input-output rule going in one direction, even the lives of primary school children are filled with situations that involve several interacting actions—and which involve feedback loops, second-order effects, and issues such as maximization, minimization, or stabilizations (which, many years ago, needed to be postponed until students had been introduced to calculus). …This brief paper demonstrates that, if children’s stories are used to introduce simulations of “real life” problem solving situations, then average ability primary school children are quite capable of dealing productively with 60-minute problems that involve (a) many kinds of quantities in addition to “counts,” (b) integrated collections of concepts associated with a variety of textbook topic areas, (c) interactions among several different actors, and (d) issues such as maximization, minimization, and stabilization.
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Notes
- 1.
Explicit principles for designing model-eliciting activities have been published in a variety of recent publications (e.g., English, 2009; English and Mousoulides, 2011; Lesh et al., 2000). And, these standards also have been specially adapted for the development of teacher-level MEAs (Zawojewski et al., 2009) or MEAs for older students in fields such as engineering (Haljmarson and Lesh, 2008).
References
Blanton, M., & Kaput, J. (2000). Generalizing and progressively formalizing in a third-grade mathematics classroom: conversations about even and odd numbers. In M. L. Fernandez (Ed.), Proceedings of the 22nd annual meeting of the North American chapter of the international group for the psychology of mathematics education (Vol. 1, pp. 115–119). Columbus: The ERIC Clearinghouse for Science, Mathematics, and Environmental Education.
Blanton, M., & Kaput, J. (2001a). Algebrafying the elementary mathematics experience. Part II: transforming practice on a district-wide scale. In H. Chick, K. Stacey, J. Vincent, & J. Vincent (Eds.), The future of teaching and learning of algebra: proceedings of the 12th ICMI study conference (pp. 87–95). Melbourne, Australia: The University of Melbourne.
Blanton, M., & Kaput, J. (2001b). Student achievement in an elementary classroom that promotes the development of algebraic thinking. In R. Speiser, C. Maher, & C. Walter (Eds.), The proceedings of the 23rd annual meeting of the North American chapter of the international group for the psychology of mathematics education (Vol. 1, pp. 99–108). Columbus: The ERIC Clearinghouse for Science, Mathematics, and Environmental Education.
Blanton, M., & Kaput, J. (2002). Design principles for tasks that support algebraic reasoning in elementary classrooms. In A. D. Cockburn & E. Nardi (Eds.), Proceedings of the 26th annual conference of the international group for the psychology of mathematics education (Vol. 2, pp. 105–112). Norwich, UK: University of East Anglia.
Blanton, M., & Kaput, J. (2004). Design principles for instructional contexts that support students’ transition from arithmetic to algebraic reasoning: elements of task and culture. In R. Nemirovsky, A. S. Rosebery, J. Solomon, & B. Warren (Eds.), Everyday matters in science and mathematics: studies of complex classroom events (pp. 211–234). Mahwah: Erlbaum.
Doerr, H., & Lesh, R. (2010). Model-eliciting activities: do they work in the education of students, teams or teachers? In G. Kaiser, W. Blum, & K. Maass (Eds.), Teaching mathematics modeling and applications (pp. 132–155). New York: Springer.
English, L. D. (2009). Promoting interdisciplinarity through mathematical modelling. ZDM: The International Journal on Mathematics Education, 41(1&2), 161–181.
English, L. D., & Mousoulides, N. (2011). Engineering-based modelling experiences in the elementary and middle classroom. In M. S. Khine & I. M. Saleh (Eds.), Models and modeling: cognitive tools for scientific enquiry (pp. 173–194). Dordrecht, The Netherlands: Springer.
Haljmarson, M., & Lesh, R. (2008). Six principles for developing model-eliciting activities. In L. English et al. (Eds.), Handbook of international research in mathematics education (pp. 241–268). New York: Routledge.
Hamilton, E., Lesh, R., Lester, F., & Brilleslyper, M. (2008). Model-eliciting activities (MEAs) and their assessment as a bridge between engineering education research and mathematics education research. Journal for Engineering Education, 1(2), 1–25.
Harel, G., & Kaput, J. (1991). The influence of notation in the development of mathematical ideas. In D. O. Tall (Ed.), Advanced mathematical thinking (pp. 82–94). Dordrecht, The Netherlands: Kluwer Academic.
Harel, G., & Lesh, R. (2003). Local conceptual development of proof schemes in a cooperative learning setting. In R. Lesh & H. Doerr (Eds.), Beyond constructivist: a models & modeling perspective on mathematics teaching, learning, and problems solving (pp. 359–382). Mahwah: Erlbaum.
Hegedus, S. (in press) Young children investigating mathematical concepts with haptic technologies: future design perspectives. In L. Moreno-Armella & M. Santos (Eds.), Mathematical problem-solving. Charlotte: Information Age Publishing. (To be published as part of the Montana Mathematics Enthusiast monograph series.)
Hegedus, S. & Roschelle, J. (Eds.) (2012). The SimCalc vision and contributions: democratizing access to important mathematics. Berlin: Springer.
Kaput, J. (1987). Representation and mathematics. In C. Janvier (Ed.), Problems of representation in the learning of mathematics (pp. 19–26). Hillsdale: Erlbaum.
Kaput, J. (1989a). Linking representations in the symbol system of algebra. In C. Kieran & S. Wagner (Eds.), A research agenda for the teaching and learning of algebra (pp. 167–194). Reston: National Council of Teachers of Mathematics.
Kaput, J. (1989b). The cognitive foundations of models that use intensive quantities. In W. Blum (Ed.), Applications and modeling in learning and teaching mathematics (pp. 389–397). London: Longman Higher Education.
Kaput, J. (1991). Notations and representations as mediators of constructive processes. In E. von Glaserfeld (Ed.), Constructivism and mathematics education (pp. 53–74). Dordrecht, The Netherlands: Kluwer Academic.
Kaput, J. (1995a). Creating cybernetic and psychological ramps from the concrete to the abstract: examples from multiplicative structures. In D. Perkins, J. Schwartz, M. West, & M. Wiske (Eds.), Software goes to school: teaching for understanding with new technologies (pp. 130–154). New York: Oxford University Press.
Kaput, J. (1995b). Long term algebra reform: democratizing access to big ideas. In C. Lacampagne, J. Kaput, & W. Blair (Eds.), The algebra colloquium (Vol. 1, pp. 33–49). Washington: Department of Education, Office of Research.
Kaput, J. (1998). Mixing new technologies, new curricula and new pedagogies to obtain extraordinary performance from ordinary people in the next century. In H. S. Park, Y. H. Choe, H. Shin, & S. H. Kim (Eds.), Proceedings of the 1st international commission on mathematical instruction East Asia regional conference on mathematics education (Vol. 1, pp. 141–156) Chungcheongbuk-do, South Korea: Korea National University of Education.
Kaput, J. (1999). Representations, inscriptions, descriptions and learning: a kaleidoscope of windows. Journal of Mathematical Behavior, 17(2), 265–281. G. Goldin & C. Janvier (Eds.), A special issue on “Representations and the psychology of mathematics education: Part II”.
Kaput, J., & Blanton, M. (2001). Algebrafying the elementary mathematics experience. Part I: transforming task structures. In H. Chick, K. Stacey, J. Vincent, & J. Vincent (Eds.), The future of teaching and learning of algebra: proceedings of the 12th ICMI study conference (Vol. 1, pp. 344–350). Melbourne, Australia: The University of Melbourne.
Kaput, J., & Blanton, M. (2005). Algebrafying elementary mathematics in a teacher-centered, systemic way. In T. Romberg, T. Carpenter, & F. Dremock (Eds.), Understanding mathematics and science matters (pp. 99–125). Mahwah: Erlbaum.
Kaput, J., & Goldin, G. (1996). A joint perspective on the idea of representation in learning and doing mathematics. In L. Steffe, P. Nesher, P. Cobb, G. Goldin, & B. Greer (Eds.), Theories of mathematical learning (pp. 397–430). Mahwah: Erlbaum.
Kaput, J., & Nemirovsky, R. (1995). Move to the next level: a mathematics of change theme throughout the K-16 curriculum. UME Trends, 6(6), 20–21.
Kelly, A. E., Lesh, R. A., & Baek, J. (2009). Handbook of design research methods in STEM education. New York: Routledge.
Lehrer, R., & Lesh, R. (2011). Mathematical learning. In W. Reynolds & G. Miller (Eds.), Comprehensive handbook of psychology (Vol. 7, pp. 357–391). New York: Wiley.
Lesh, R. (2007). Research design in mathematics education: focusing on design experiments. In L. English (Ed.), International handbook of research design in mathematics education (pp. 27–49). Hillsdale: Erlbaum.
Lesh, R. & Doerr, H. (Eds.) (2003). Beyond constructivism: a models and modeling perspective on mathematics teaching, learning, and problem solving. Hillsdale: Erlbaum.
Lesh, R., & Doerr, H. (2011). Alternatives to trajectories and pathways to describe development in modeling and problem solving. In W. Blum, R. Borromeo-Ferri, & K. Maaß(Eds.), Teaching mathematical modeling and applications (pp. 138–147). Wiesbaden, Germany: Springer.
Lesh, R., & Harel, G. (2007). Problem solving, modeling and local conceptual development. Mathematical Thinking & Learning, 5(2&3), 157–189.
Lesh, R., Hamilton, E., & Kaput, J. (Eds.) (2007). Models and modeling as foundations for the future in mathematics education. Mahwah: Erlbaum.
Lesh, R., & Kaput, J. (1998). Interpreting modeling as local conceptual development. In J. DeLange & M. Doorman (Eds.), Senior secondary mathematics education, Utrecht, The Netherlands: OW & OC.
Lesh, R., & Yoon, C. (2004). Evolving communities of mind—in which development involves several interacting and simultaneously developing strands. Mathematical Thinking and Learning, 6(2), 205–226.
Lesh, R., & Zawojewski, J. S. (2007). Problem solving and modeling. In F. K. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 763–804). Charlotte: Information Age Publishing.
Lesh, R., Galbraith, P., Haines, C., & Hurford, A. (Eds.) (2010). Modeling students’ mathematical modeling competencies: ICTMA 13. New York: Springer.
Lesh, R., Hoover, M., Hole, B., Kelly, A., & Post, T. (2000). Principles for developing thought-revealing activities for teachers and students. In A. Kelly & R. Lesh (Eds.), The handbook of research design in mathematics and science education (pp. 591–646). Mahwah: Erlbaum.
Lins, R., & Kaput, J. (2004). The early development of algebraic reasoning: the current state of the field. In H. Chick & K. Stacy (Eds.), The future of the teaching and learning of algebra. The 12th ICMI Study (Vol. 8, pp. 45–70). Boston: Kluwer Academic.
Zawojewski, J., Diefes-Dux, H., & Bowman, K. (Eds.) (2009). Models and modeling in engineering education: designing experiences for all students. Rotterdam, The Netherlands: Sense Publications.
Zawojewski, J., Chamberlin, M., Hjalmarson, M., & Lewis, C. (2008). Developing design studies in mathematics education professional development: studying teachers’ interpretive systems. In A. E. Kelly, R. A. Lesh, & J. Y. Baek (Eds.), Handbook of design research methods in education: innovations in science, technology, engineering, and mathematics learning and teaching (pp. 216–245). New York: Routledge.
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Lesh, R., English, L., Sevis, S., Riggs, C. (2013). Modeling as a Means for Making Powerful Ideas Accessible to Children at an Early Age. In: Hegedus, S., Roschelle, J. (eds) The SimCalc Vision and Contributions. Advances in Mathematics Education. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-5696-0_23
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