From problem solving to modeling – the emergence of models and modeling perspectives
DOI:
https://doi.org/10.7146/nomad.v12i1.148020Abstract
More than 25 years ago, a research project in the U.S investigated the question: ”What is needed by students, beyond having a mathematical idea, that enables students to use the mathematical idea in everyday problem solving situations? (Lesh, Landau & Hamilton, 1983). The answer to this question has begun to emerge after 25 years of systemic work in the domain of modeling. In this paper, we chronicle the emergence of models and modeling perspectives (MMP) from the genre of problem solving research via a synthesis of the major strands in the extant literature.
References
Battye, A. & Challis, M. (1997). Deriving learning outcomes for mathematical modeling units within an undergraduate programme. In S. Houston, W. Blum, I. Huntley & N. Neill (Eds.), Teaching and learning mathematical modeling - innovation, investigation and applications (Ch. 11). Chichester: Ellis Horwood.
Bell, A., Burkhardt, H. & Swan, M. (1992). Balanced assessment of mathematical performance. In R. Lesh & S. Lamon (Eds.), Assessment of authentic performance in school mathematics (pp. 119-144), Washington, DC: American Association for the Advancement of Science.
Blum, W. & Kaiser, G. (1997). Vergleichende empirische Untersuchungen zu mathematischen Anwendungsfähigkeiten von englischen und deutschen Lernenden. Unpublished application to Deutsche Forschungsgesellschaft.
Blum, W. & Niss, M. (1991). Applied mathematical problem solving, modeling, applications, and links to other subjects - state, trends, and issues in mathematics instruction. Educational Studies in Mathematics, 22 (1), 37-68. https://doi.org/10.1007/BF00302716
Bohl, J. (1998). Problems that matter: Teaching mathematics as critical engagement. Humanistic Mathematics Network Journal, 17, 23-31. https://doi.org/10.5642/hmnj.199801.17.14
Bonotto, C. & Basso, M. (2001). Is it possible to change the classroom activities in which we delegate the process of connecting mathematics with reality? International Journal of Mathematical Education in Science and Technology, 32 (3), 385-399. https://doi.org/10.1080/00207390110040201
Burkhardt, H. & Pollak, H (2006). Modeling in mathematics classrooms. Zentralblatt für Didaktik der Mathematik, 38 (2), 178-195. https://doi.org/10.1007/BF02655888
Choi, J. & Hannafin, M. (1997). The effects of instructional context and reasoning complexity on mathematics problem solving. Educational Technology Research and Development, 45 (3), 43-55. https://doi.org/10.1007/BF02299728
Christou, C., Mousoulides, N., Pittalis, M., Pitta-Pantazi, D. & Sriraman, B. (2005). An empirical taxonomy of problem posing processes. Zentralblatt für Didaktik der Mathematik, 37 (3), 149-158. https://doi.org/10.1007/s11858-005-0004-6
Confrey, J. & Doerr, H. (1994). Student modelers. Interactive Learning Environments, 4 (3), 199-217. https://doi.org/10.1080/1049482940040303
Crouch, R. & Haines, C. (2004). Mathematical modeling: transitions between the real world and the mathematical model. International Journal of Mathematics Education in Science and Technology,35 (2), 197-206. https://doi.org/10.1080/00207390310001638322
D'Ambrosio, U. (1998). Mathematics and peace: Our responsibilities. Zentralblatt für Didaktik der Mathematik. 98 (3), 67-73. https://doi.org/10.1007/BF02653170
DaPueto, C. & Parenti, L. (1999). Contributions and obstacles of contexts in the development of mathematical knowledge. Educational Studies in Mathematics, 39 (1), 1-21. https://doi.org/10.1023/A:1003702003886
De Lange, J. (1987). Mathematics, insight and meaning - teaching, learning and testing of mathematics for the life and social sciences. Utrecht University.
De Lange, J. (1992). Assessing mathematical skills, understanding, and thinking. In R. Lesh & S. Lamon (Eds.), Assessment of authentic performance in school mathematics (Ch. 8). Washington, DC: AAAS Press.
Design-Based Research Collective (2003). Design-based research: An emerging paradigm for educational inquiry. Educational Researcher, 32 (1), 5-8. https://doi.org/10.3102/0013189X032001005
Doerr, H. M. (2006). Examining the tasks of teaching when using students' mathematical thinking. Educational Studies in Mathematics, 62 (1). https://doi.org/10.1007/s10649-006-4437-9
Doerr, H. M & English, L. (2003). A Modeling perspective on students' mathematical reasoning about data. Journal of Research in Mathematics Education, 34 (2), 110-136. https://doi.org/10.2307/30034902
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. https://doi.org/10.1007/s10857-006-9004-x
Doerr, H. M. & Lesh, R. (2003). A modeling perspective on teacher development. In R. Lesh & H. M. Doerr (Eds.), Beyond constructivism: models and modeling perspectives on mathematics problem solving, learning and teaching (pp. 125-140). Hillsdale, NJ: Lawrence Erlbaum.
English, L.D. (2003). Reconciling theory, research, and practice: a models and modeling perspective. Educational Studies in Mathematics, 54 (2-3), 225-248. https://doi.org/10.1023/B:EDUC.0000006167.14146.7b
English, L. & Lesh, R. (2003). Ends-in-view problems. In R. Lesh & H. M. Doerr (Eds.), Beyond constructivism: models and modeling perspectives on mathematics problem solving, learning and teaching (pp. 297-316). Hillsdale, NJ: Lawrence Erlbaum. https://doi.org/10.4324/9781410607713
English, L. & Watters, J. (2004). Mathematical modeling in the early school years. Mathematics Education Research Journal, 16 (3), 59-80. https://doi.org/10.1007/BF03217401
Erickson, J. & Lehrer, R. (1998). The evolution of critical standards as students design hypermedia documents. The Journal of the Learning Sciences, 7 (3-4), 351-386. https://doi.org/10.1080/10508406.1998.9672058
Freudenthal, H. (1971). Geometry between the devil and the deep sea. Educational Studies in Mathematics, 3 (3-4), 413-435. https://doi.org/10.1007/BF00302305
Freudenthal, H. (1991). Revisiting mathematics education. China lectures. Dordrecht: Kluwer Academic Publishers.
Gravemeijer, K. (1997). Commentary. Solving word problems: a case of modeling? Learning and Instruction, 7 (4), 389-397. https://doi.org/10.1016/S0959-4752(97)00011-X
Gravemeijer, K., Cobb, P., Bowers, J. & Whitenack, J. (2000). Symbolizing, modeling and instructional design. In P. Cobb, E. Yackel & K. McClain (Eds.), Symbolizing and communicating in mathematics classrooms (pp. 225- 274). Mahwah, NJ: Lawrence Erlbaum.
Gravemeijer, K. & Doorman, M. (1999). Context problems in realistic mathematics education: a calculus course as an example. Educational Studies in Mathematics, 39 (1-3), 111-129. https://doi.org/10.1023/A:1003749919816
Greer, B. (1997). Modeling reality in mathematics classrooms: the case of word problems. Learning and Instruction, 7 (4), 293-307. https://doi.org/10.1016/S0959-4752(97)00006-6
Gutstein, E. (2006). Reading and writing the world with mathematics: toward a pedagogy for social justice. New York: Routledge.
Harel, G. & Lesh, R. (2003). Local conceptual development of proof schemes in a cooperative learning setting. In R. Lesh & H. M. Doerr (Eds.), Beyond constructivism: models and modeling perspectives on mathematics problem solving, learning and teaching (pp. 359-382). Mahwah, NJ: Lawrence Erlbaum.
Hiebert, J., Thomas P., Carpenter, E., Fennema, K., Fuson, P., et al. (1996). Problem solving as a basis for reform in curriculum and instruction: the case of mathematics. Educational Researcher, 25 (4), 12-21. https://doi.org/10.2307/1176776
Hjalmarson (2005). Designing presentation tools: a window into teacher practice. Unpublished doctoral dissertation. Purdue University.
Iversen, S., & Larson, C. (2006). Simple thinking using complex math vs. complex thinking using simple math. Zentralblatt für Didaktik der Mathematik, 38 (2), 281-292. https://doi.org/10.1007/BF02652811
Kaiser, G., Blomhøj, M. & Sriraman (2006). Towards a didactic theory for mathematical modeling. Zentralblatt für Didaktik der Mathematik, 38 (2), 82-85. https://doi.org/10.1007/BF02655882
Kaiser, G. & Sriraman, B. (2006). A global survey of international perspectives on modeling in mathematics education. Zentralblatt für Didaktik der Mathematik, 38 (3), 302-310. https://doi.org/10.1007/BF02652813
Kelly, A. & Lesh, R. (Eds.) (2000). Handbook of research design in mathematics and science education. Mahwah, NJ: Lawrence Erlbaum.
Kitchen, A. (1993). The 'Mechanics in Action Project'. In S. Houston (Ed.), Developments in curriculum and assessment in mathematics (p. 57-66). Coleraine, Northern Ireland: University of Ulster.
Koellner Clark, K. & Lesh, R. (2003). A modeling approach to describe teacher knowledge. In R. Lesh & H. M. Doerr (Eds.), Beyond constructivism: models and modeling perspectives on mathematics problem solving, learning and teaching (pp. 159-173). Mahwah, NJ: Lawrence Erlbaum.
Lesh, R., Cramer, K., Doerr, H. M., Post, T. & Zawojewski, J. (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). Hillsdale, NJ: Lawrence Erlbaum. https://doi.org/10.4324/9781410607713
Lesh, R. & Doerr, H. M. (2000). Symbolizing, communicating and mathematizing: key components of models and modeling. In P. Cobb, E. Yackel & K. McClain (Eds.), Symbolizing and communicating in mathematics classrooms: perspectives on discourse, tools and instructional design. Hillsdale, NJ: Lawrence Erlbaum.
Lesh, R. & Doerr, H.M. (2003a). Beyond constructivism: models and modeling perspectives on mathematics problem solving, learning and teaching. Mahwah, NJ: Lawrence Erlbaum. https://doi.org/10.4324/9781410607713
Lesh, R. & Doerr, H.M. (2003b). In what ways does a models and modeling perspective move beyond constructivism? In R. Lesh & H. M. Doerr (Eds.), Beyond constructivism: models and modeling perspectives on mathematics problem solving, learning and teaching (pp. 383-403). Hillsdale, NJ: Lawrence Erlbaum. https://doi.org/10.4324/9781410607713
Lesh, R., Doerr, H. M., Carmona, G. & Hjalmarson, M. (2003). Beyond constructivism. Mathematical Thinking and Learning, 5 (2), 211-234. https://doi.org/10.1080/10986065.2003.9680000
Lesh,R., Kaput, J. & Hamilton, E. (Eds.) (in press). Foundations for the future: the need for new mathematical understandings and abilities in the 21st century. Hillsdale, NJ: Lawrence Erlbaum.
Lesh, R. & Kelly, A. (2000). Multi-tiered teaching experiments. In R. Lesh & A. Kelly (Eds.), Handbook of research design in mathematics and science education (pp. 197-231). Mahwah, NJ: Lawrence Erlbaum.
Lesh, R., Landau, M. & Hamilton, E. (1983). Conceptual models in applied mathematical problem solving. In R. Lesh (Ed.), The acquisition of mathematical concepts and processes. New York: Academic Press.
Lesh, R. & Lehrer, R. (2003). Models and modeling perspectives on the development of students and teachers. Mathematical Thinking and Learning, 5 (2-3), 109-130. https://doi.org/10.1080/10986065.2003.9679996
Lesh, R. & Sriraman, B. (2005a). Mathematics education as a design science. Zentralblatt für Didaktik der Mathematik, 37 (6), 490-505. https://doi.org/10.1007/BF02655858
Lesh, R. & Sriraman, B. (2005b). John Dewey revisited - pragmatism and the models-modeling perspective on mathematical learning. In A. Beckmann, C. Michelsen & B.Sriraman (Eds.), Proceedings of the 1st International Symposium on Mathematics and its Connections to the Arts and Sciences (pp. 32-51). Hildesheim, Berlin: Franzbecker Verlag.
Lester, F. K. & Kehle, P. E. (2003). From problem solving to modeling: the evolution of thinking about research on complex mathematical activity. In R. Lesh & H. Doerr, H. (Eds.), Beyond constructivism: models and modeling perspectives on mathematics problem solving, learning and teaching (pp. 501- 518). Mahwah, NJ: Erlbaum.
McNair, R. (2000). Life outside the mathematics classroom: Implications for mathematics teaching reform. Urban Education, 34 (5), 550-570. https://doi.org/10.1177/0042085900345002
Michelsen, C. (2006). Commentary to Lesh & Sriraman: mathematics education as a design science. Zentralblatt für Didaktik der Mathematik, 38 (1), 73-76. https://doi.org/10.1007/BF02655909
NCTM (2000). Principles and standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics.
Niss, M. (1987). Applications and modeling in the mathematics curriculum - state and trends. International Journal of Mathematical Education in Science and Technology, 18 (4), 487-505. https://doi.org/10.1080/0020739870180401
Niss, M. (1993). Assessment of mathematical applications and modeling in mathematics teaching. In J. de Lange, C. Keitel, I. Huntley & M. Niss (Eds.), Innovation in mathematics education by modeling and applications. (p. 41-51) Chichester: Ellis Horwood.
OECD (2004). Problem solving for tomorrow's world - first measures of cross curricular competencies from PISA 2003. Retrieved September 29, 2005 from http://www.pisa.oecd.org/dataoecd/25/12/34009000.pdf
Pace, S. (2000). Teaching mathematical modeling in a design contest: a methodology based on the mechanical analysis of a domestic crusher. Teaching Mathematics and its Applications, 19(4), 158-165. https://doi.org/10.1093/teamat/19.4.158
Polya, G. (1962). Mathematical discovery. New York: Wiley.
Polya, G. (1973). How to solve it: a new aspect of mathematical methods (2nd Ed.). Princeton, NJ: Princeton University Press.
Resnick, L. (1988) Treating mathematics as an ill-structured discipline. In R. Charles & E. Silver (Eds.), The teaching and assessing of mathematical problem solving (pp. 32-60). Reston, VA: NCTM.
Reusser, K. & Stebler, R. (1997). Every word problem has a solution - the social rationality of mathematical modeling in schools. Learning and Instruction, 7 (4), 309-327. https://doi.org/10.1016/S0959-4752(97)00014-5
Schoenfeld, A. (1991). On mathematics as sense-making: an informal attack on the unfortunate divorce of formal and informal mathematics. In J. Voss, D. Perkins & J. Segal (Eds.), Informal reasoning and education (p. 311-343). Hillsdale, NJ: Lawrence Erlbaum.
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 (p. 334-370). New York: Simon & Schuster Macmillan.
Schorr, R. & Lesh, R. (2003). A modeling approach for providing teacher development. In R. Lesh & H. M. Doerr (Eds.), Beyond constructivism: Models and modeling perspectives on mathematics problem solving, learning, and teaching (pp. 141-158). Mahwah, NJ: Lawrence Erlbaum.
Skovsmose, O. (1994): Towards a philosophy of critical mathematics education. Dordrecht: Kluwer. https://doi.org/10.1007/978-94-017-3556-8
Skovsmose,O. (2000). Aporism and critical mathematics education. For the Learning of Mathematics, 20 (1), 2-8.
Smith, R. & Thatcher, D. (1989). An examination for mathematical modeling. International Journal of Mathematical Education in Science and Technology, 20 (4), 605-613. https://doi.org/10.1080/0020739890200416
Sriraman, B. & Lesh, R. (2006). Beyond Traditional conceptions of modeling. Zentralblatt für Didaktik der Mathematik, 38 (3), 247-254. https://doi.org/10.1007/BF02652808
Verschaffel, L. & De Corte, E. (1997). Teaching realistic mathematical modeling in the elementary school: a teaching experiment with fifth graders. Journal for Research in Mathematics Education, 28 (5), 577-601. https://doi.org/10.2307/749692
Verschaffel, L., De Corte, E. & Borghart, I. (1997). Pre-service teachers' conceptions and beliefs about the role of real-world knowledge in mathematical modeling of school word problems. Learning and Instruction, 7 (4), 339-359. https://doi.org/10.1016/S0959-4752(97)00008-X
Verschaffel, L., De Corte, E. & Lasure, S. (1994). Realistic considerations in mathematical modeling of school arithmetic word problems. Learning and Instruction, 4, 273-294. https://doi.org/10.1016/0959-4752(94)90002-7
Yoshida, H., Verschaffel, L. & De Corte, E. (1997). Realistic considerations in solving problematic word problems: do Japanese and Belgian children have the same difficulties? Learning and Instruction, 7 (4), 329-338. https://doi.org/10.1016/S0959-4752(97)00007-8
Zawojewski, J. S., Lesh, R. & English, L. (2003). A models and modeling perspective on the role of small group learning activities. In R. Lesh & H. M. Doerr (Eds.), Beyond constructivism: models and modeling perspectives on mathematics problem solving, learning, and teaching (pp. 337-358). Mahwah, NJ: Lawrence Erlbaum.
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