Theories of Mathematics Education pp 263-290 | Cite as

# Problem Solving for the 21^{st} Century

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## Abstract

Mathematical problem solving has been the subject of substantial and often controversial research for several decades. We use the term, *problem solving*, here in a broad sense to cover a range of activities that challenge and extend one’s thinking. In this chapter, we initially present a sketch of past decades of research on mathematical problem solving and its impact on the mathematics curriculum. We then consider some of the factors that have limited previous research on problem solving. In the remainder of the chapter we address some ways in which we might advance the fields of problem-solving research and curriculum development.

## Keywords

Word Problem National Council Mathematical Thinking Statistical Reasoning Mathematics Curriculum## Preview

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## References

- Anderson, J. R., Boyle, C. B., & Reiser, B. J. (1985). Intelligent tutoring systems.
*Science*,*228*, 456–462. CrossRefGoogle Scholar - Australian Association of Mathematics Teachers (AAMT) and Early Childhood Australia (ECA) (2009). http://www.aamt.edu.au (accessed 27.03.09).
- Baroody, A. J., Lai, M., & Mix, K. (2006). The development of young children’s early number and operation sense and its implications for early childhood education. In B. Spodek & O. Saracho (Eds.),
*Handbook of Research on the Education of Young Children*(2nd ed.). Mahwah: Lawrence Erlbaum. Google Scholar - Beckmann, A. (2009). A conceptual framework for cross-curricular teaching.
*The Montana Mathematics Enthusiast*,*6*(supplement 1), 1–58. Google Scholar - Begle, E. G. (1979).
*Critical Variables in Mathematics Education*. Washington D.C.: the Mathematics Association of America and the National Council of Teachers of Mathematics. Google Scholar - Brown, S. I., & Walter, M. I. (2005).
*The Art of Problem Posing*(3rd ed.). Mahwah, New Jersey: Lawrence Erlbaum. Google Scholar - Brownell, W. A. (1945). When is arithmetic meaningful?
*Journal of Educational Research*,*38*(3), 481–498. Google Scholar - Cai, J. (2003). What research tells us about teaching mathematics through problem solving. In F. Lester & R. Charles (Eds.),
*Teaching Mathematics Through Problem Solving*(pp. 241–253). Reston, Virginia: National Council of Teachers of Mathematics. Google Scholar - Campbell, S. (2006). Educational neuroscience: New horizons for research in mathematics education. In J. Novotna, H. Moraova, M. Kratka, & N. Stelikova (Eds.),
*Proceedings of the 30th Conference of the International Group for the Psychology of Mathematics Education*(Vol.*2*, pp. 257–264). Prague, Czech Republic: Charles University. Google Scholar - Charles, R., & Silver, E. (1988).
*The Teaching and Assessing of Mathematical Problem Solving*. Reston, VA: National Council of Teachers of Mathematics. Google Scholar - Charlesworth, R., & Lind, K. (2006).
*Math and Science Learning for Young Children*(6th ed.). NY: Delmar Publishers. Google Scholar - Davis, B., & Simmt, E. (2003). Understanding learning systems: Mathematics education and complexity science.
*Journal for Research in Mathematics Education*,*34*(2), 137–167. Google Scholar - De Abreu, G. (2008). From mathematics learning out-of-school to multicultural classrooms: A cultural psychology perspective. In L. D. English (Ed.),
*Handbook of International Research in Mathematics Education*. New York: Routledge. Google Scholar - Doerr, H. M., & English, L. D. (2001). A modelling perspective on students’ learning through data analysis. In M. van den Heuvel-Panhuizen (Ed.),
*Proceedings of the 25th Annual Conference of the International Group for the Psychology of Mathematics Education*(pp. 361–368). Utrecht University. Google Scholar - Doerr, H. M., & English, L. D. (2003). A modeling perspective on students’ mathematical reasoning about data.
*Journal for Research in Mathematics Education*,*34*(2), 110–137. CrossRefGoogle Scholar - Doerr, H., & English, L. D. (2006). Middle-grade teachers’ learning through students’ engagement with modelling tasks.
*Journal for Research in Mathematics Teacher Education*,*9*(1), 5–32. CrossRefGoogle Scholar - Doerr, H. M., & Tripp, J. S. (1999). Understanding how students develop mathematical models.
*Mathematical Thinking and Learning*,*1*(3), 231–254. CrossRefGoogle Scholar - English, L. D. (2003). Problem posing in the elementary curriculum. In F. Lester & R. Charles (Eds.),
*Teaching Mathematics Through Problem Solving*(pp. 187–198). Reston, Virginia: National Council of Teachers of Mathematics. Google Scholar - English, L. D. (2006). Mathematical modeling in the primary school: Children’s construction of a consumer guide.
*Educational Studies in Mathematics*,*62*(3), 303–329. CrossRefGoogle Scholar - English, L. D. (2007). Complex systems in the elementary and middle school mathematics curriculum: A focus on modeling. In B. Sriraman (Ed.),
*Festschrift in Honor of Gunter Torner. The Montana Mathematics Enthusiast*(pp. 139–156). Information Age Publishing. Google Scholar - English, L. D. (2008). Introducing complex systems into the mathematics curriculum.
*Teaching Children Mathematics*,*15*(1), 38–47. Google Scholar - English, L. D. (2009a). Promoting interdisciplinarity through mathematical modelling.
*ZDM: The International Journal on Mathematics Education*,*41*(1), 161–181. CrossRefGoogle Scholar - English, L. D. (2009b). Modeling with complex data in the primary school. In R. Lesh, P. Galbraith, W. Blum, & A. Hurford (Eds.),
*Modeling Students’ Mathematical Modeling Competencies: ICTMA 13*. Springer. Google Scholar - English, L. D., & Halford, G. S. (1995).
*Mathematics Education: Models and Processes*. Mahwah, New Jersey: Lawrence Erlbaum Associates. Google Scholar - English, L. D., & Watters, J. J. (2005). Mathematical modeling in the early school years.
*Mathematics Education Research Journal*,*16*(3), 58–79. Google Scholar - English, L. D., Lesh, R. A., & Fennewald, T. (2008).
*Future directions and perspectives for problem solving research and curriculum development*. Paper presented for TSG 19 at the International Congress on Mathematical Education. Monterrey, Mexico, July 6–13. Google Scholar *Enhancing the Teaching and Learning of Early Statistical Reasoning in European Schools*(2009). Project: http://www.earlystatistics.net/ (accessed 20 March, 2009).- Franklin, C. A., & Garfield, J. (2006). The GAISE project: Developing statistics education guidelines for grades pre-K-12 and college courses. In G. Burrill & P. Elliott (Eds.),
*Thinking and Reasoning with Data and Chance*(68th Yearbook, pp. 345–376). Reston, VA: National Council of Teachers of Mathematics. Google Scholar - Freudenthal, H. (1973).
*Didactical Phenomenology of Mathematical Structures*. Boston: Kluwer. Google Scholar - Gainsburg, J. (2006). The mathematical modeling of structural engineers.
*Mathematical Thinking and Learning*,*8*(1), 3–36. CrossRefGoogle Scholar - Ginsburg, H. P., Cannon, J., Eisenband, J. G., & Pappas, S. (2006). Mathematical thinking and learning. In K. McCartney & D. Phillips (Eds.),
*Handbook of Early Child Development*(pp. 208–230). Oxford, England: Blackwell. CrossRefGoogle Scholar - Gravemeijer, K. (1999). How emergent models may foster the construction of formal mathematics.
*Mathematical Thinking and Learning*,*1*, 155–177. CrossRefGoogle Scholar - Greer, B. (1997). Modeling reality in mathematics classroom: The case of word problems.
*Learning and Instruction*,*7*, 293–307. CrossRefGoogle Scholar - Greer, B., Verschaffel, L., & Mukhopadhyay, S. (2007). Modelling for life: Mathematics and children’s experience. In W. Blum, W. Henne, & M. Niss (Eds.),
*Applications and Modelling in Mathematics Education*(ICMI Study 14, pp. 89–98). Dordrecht: Kluwer. CrossRefGoogle Scholar - Hamilton, E. (2007). What changes are needed in the kind of problem solving situations where mathematical thinking is needed beyond school? In R. Lesh, E. Hamilton, & J. Kaput (Eds.),
*Foundations for the Future in Mathematics Education*(pp. 1–6). Mahwah, NJ: Lawrence Erlbaum. Google Scholar - Hamilton, E., Lesh, R., Lester, F., & Yoon, C. (2007). The use of reflection tools in building personal models of problem solving. In R. Lesh, E. Hamilton, & J. Kaput (Eds.),
*Foundations for the Future in Mathematics Education*(pp. 349–366). Mahwah, NJ: Lawrence Erlbaum. Google Scholar - Hutchins, E. (1995a).
*Cognition in the Wild*. Cambridge, MA: MIT Press. Google Scholar - Hutchins, E. (1995b). How a cockpit remembers its speeds.
*Cognitive Science*,*19*, 265–288. CrossRefGoogle Scholar - Kaiser, G., & Maass, K. (2007). Modelling in lower secondary mathematics classroom—problems and opportunities. In W. Blum, W. Henne, & M. Niss (Eds.),
*Applications and Modelling in Mathematics Education*(ICMI Study 14, pp. 99–108). Dordrecht: Kluwer. CrossRefGoogle Scholar - Kaiser, G., & Sriraman, B. (2006). A global survey of international perspectives on modelling in mathematics education.
*ZDM*,*38*(3), 302–310. CrossRefGoogle Scholar - Kaiser, G., Blomhoj, M., & Sriraman, B. (2006). Towards a didactical theory for mathematical modelling.
*ZDM*,*38*(2), 82–85. CrossRefGoogle Scholar - Langrall, C., Mooney, E., Nisbet, S., & Jones, G. (2008). Elementary students’ access to powerful mathematical ideas. In L. D. English (Ed.),
*Handbook of International Research in Mathematics Education*(2nd ed.). NY: Routledge. Google Scholar - Lehrer, R., & Schauble, L. (2004). Modeling natural variation through distribution.
*American Educational Research Journal*,*41*(3), 635–679. CrossRefGoogle Scholar - Lehrer, R., & Schauble, L. (2005). Developing modeling and argument in the elementary grades. In T. Romberg, T. Carpenter, & F. Dremock (Eds.),
*Understanding Mathematics and Science Matters*(pp. 29–53). NJ: Erlbaum. Google Scholar - Lehrer, R., Giles, N. D., & Schauble, L. (2002). Children’s work with data. In
*Investigating Real Data in the Classroom: Expanding Children’s Understanding of Math and Science*(pp. 1–26). Columbia Univ.: Teachers College. Google Scholar - Lesh, R. (2006). Modeling students modeling abilities: The teaching and learning of complex systems in education.
*The Journal of the Learning Sciences*,*15*(1), 45–52. CrossRefGoogle Scholar - Lesh, R. (2007). Foundations for the future in engineering and other fields that are heavy users of mathematics, science, and technology. In R. Lesh, E. Hamilton, & J. Kaput (Eds.),
*Foundations for the Future in Mathematics Education*(pp. vii–x). Mahwah, NJ: Lawrence Erlbaum. Google Scholar - Lesh, R. (2008). Directions for future research and development in engineering education. In J. Zawojewski, H. Diefes-Dux, & K. Bowman (Eds.),
*Models and Modeling in Engineering Education: Designing Experiences for All Students*. Rotterdam: Sense Publications. Google Scholar - Lesh, R., & Doerr, H. (2003). Foundation of a models and modeling perspective on mathematics teaching and learning. In R. A. Lesh & H. Doerr (Eds.),
*Beyond Constructivism: A Models and Modeling Perspective on Mathematics Teaching, Learning, and Problem Solving*(pp. 9–34). Mahwah, NJ: Erlbaum. Google Scholar - Lesh, R., & English, L. D. (2005). Trends in the evolution of models and modeling perspectives on mathematical learning and problem solving. In H. Chick & J. Vincent (Eds.),
*Proceedings of the 29th Annual Conference of the International Group for the Psychology of Mathematics Education*(pp. 192–196). University of Melbourne. Google Scholar - Lesh, R., & Sriraman, B. (2005). John Dewey revisited—pragmatism and the models-modeling perspective on mathematical learning. In A. Beckmann, C. Michelsen, & B. Sriraman (Eds.),
*Proceedings of the 1*^{st}*International Symposium of Mathematics and Its Connections to the Arts and Sciences*(pp. 7–31). Schwöbisch Gmund, Germany: The University of Education. Google Scholar - Lesh, R., & Zawojewski, J. S. (2007). Problem solving and modeling. In F. Lester (Ed.),
*The Second Handbook of Research on Mathematics Teaching and Learning*(pp. 763–804). Charlotte, NC: Information Age Publishing. Google Scholar - Lesh, R., Cramer, K., Doerr, H. M., Post, T., & Zawojewski, J. S. (2003a). Model development sequences. In R. A. Lesh & H. Doerr (Eds.),
*Beyond Constructivism: A Models and Modeling Perspective on Mathematics Teaching, Learning, and Problem Solving*(pp. 35–58). Mahwah, NJ: Erlbaum. Google Scholar - Lesh, R., Zawojewski, J. S., & Carmona, G. (2003b). What mathematical abilities are needed for success beyond school in a technology-based age of information? In R. Lesh & H. Doerr (Eds.),
*Beyond Constructivism: Models and Modeling Perspectives on Mathematic Problem Solving, Learning and Teaching*(pp. 205–222). Mahwah, NJ: Lawrence Erlbaum. Google Scholar - Lesh, R., Middleton, J., 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. CrossRefGoogle Scholar - Lester, F. K., & Charles, R. I. (Eds.) (2003).
*Teaching Mathematics Through Problem Solving: PreK-6*. Reston, VA: National Council of Teachers of Mathematics. Google Scholar - Lester, F. K., & Kehle, P. E. (2003). From problem solving to modeling: The evolution of thinking about research on complex mathematical activity. In R. A. Lesh & H. M. Doerr (Eds.),
*Beyond Constructivism: Models and Modeling Perspectives on Mathematics Problem Solving, Learning, and Teaching*(pp. 501–518). Mahwah, NJ: Lawrence Erlbaum Associates. Google Scholar - Lester, F. K., Garofalo, J., & Kroll, D. L. (1989). Self-confidence, interest, beliefs, and metacognition: Key influences on problem solving behavior. In D. B. McLeod & V. M. Adams (Eds.),
*Affect and Mathematical Problem Solving: A New Perspective*(pp. 75–88). New York: Springer-Verlag. Google Scholar - Lobato, J. (2003). How design experiments can inform a rethinking of transfer and vice versa.
*Educational Researcher*,*32*(1), 17–20. CrossRefGoogle Scholar - Maclean, R. (2001). Educational change in Asia: An overview.
*Journal of Educational Change*,*2*, 189–192. CrossRefGoogle Scholar - Meletiou-Mavrotheris, M., Paparistodemou, E., & Stylianou, D. (2009). Enhancing statistics instruction in elementary schools: Integrating technology in professional development.
*The Montana Mathematics Enthusiast*,*16*(1&2), 57–78. Google Scholar - National Council of Teachers of Mathematics (2000).
*Principles and Standards for School Mathematics*. Reston, VA: Author. Google Scholar - National Council of Teachers of Mathematics Standards (2008). http://standards.nctm.org/document/chapter3/index.htm (accessed: 23.03.09).
- Niss, M., Blum, W., & Galbraith, P. (2007). Introduction. In W. Blum, W. Henne, & M. Niss (Eds.),
*Applications and Modelling in Mathematics Education*(ICMI Study 14, pp. 3–33). Dordrecht: Kluwer. CrossRefGoogle Scholar - Nunes, T., & Bryant, P. (1996).
*Children Doing Mathematics*. Oxford: Blackwell. Google Scholar - Nunes, T., Schliemann, A. D., & Carraher, D. W. (1993).
*Street Mathematics and School Mathematics*. Cambridge, UK: Cambridge University Press. Google Scholar - PISA (2006). (Programme for International Student Assessment: http://www.pisa.oecd.org/; accessed 26.03.09).
- Polya, G. (1945).
*How to Solve It*. Princeton, NJ: Princeton University Press. Google Scholar - Romberg, T. A., Carpenter, T. P., & Kwako, J. (2005). Standards-based reform and teaching for understanding. In T. A. Romberg, T. P. Carpenter, & F. Dremock (Eds.),
*Understanding Mathematics and Science Matters*. Mahwah, NJ: Lawrence Erlbaum Associates. Google Scholar - Rubin, A. (2002). Interactive visualizations of statistical relationships: What do we gain? In
*Proceedings of the Sixth International Conference on Teaching Statistics*. Durban, South Africa. Google Scholar - Sabelli, N. H. (2006). Complexity, technology, science, and education.
*The Journal of the Learning Sciences*,*15*(1), 5–9. CrossRefGoogle Scholar - Sawyer, R. K. (2007).
*Group Genius: The Creative Power of Collaboration*. New York: Basic Books. Google Scholar - Saxe, G. (1991).
*Culture and Cognitive Development: Studies in Mathematical Understanding*. Hillsdale, NJ: Lawrence Erlbaum. Google Scholar - Schoen, & Charles (Eds.) (2003).
*Teaching Mathematics Through Problem Solving: Grades 6–12*. Reston, VA: National Council of Teachers of Mathematics. Google Scholar - Schoenfeld, A. (1992). Learning to think mathematically: Problem solving, metacognition, and sense making in mathematics. In D. A. Grouws (Ed.),
*Handbook of Research on Mathematics Teaching and Learning: A Project of the National Council of Teachers of Mathematics*(pp. 334–370). New York, NY: Macmillan Publishing Co. Google Scholar - Silver, E. A. (1985). Research on teaching mathematical problem solving: Some under represented themes and needed directions. In E. A. Silver (Ed.),
*Teaching and Learning Mathematical Problem Solving. Multiple Research Perspectives*(pp. 247–266). Hillsdale, NJ: Lawrence Erlbaum Associates. Google Scholar - Simon, H. (1978). Information-processing theory of human problem solving. In W. K. Estes (Ed.),
*Handbook of Learning and Cognitive Processes*(Vol.*5*, pp. 271–295). Hillsdale, NJ: Lawrence Erlbaum Associates. Google Scholar - Sriraman, B., & Adrian, H. (2008). A critique and response to multicultural visions of globalization.
*Interchange*,*39*(1), 119–130. CrossRefGoogle Scholar - Sriraman, B., & Dahl, B. (2009). On bringing interdisciplinary ideas to gifted education. In L.V. Shavinina (Ed.),
*The International Handbook of Giftedness*(pp. 1235–1254). Springer Science & Business. Google Scholar - Sriraman, B., & Steinthorsdottir, O. (2007). Research into practice: Implications of research on mathematics gifted education for the secondary curriculum. In C. Callahan & J. Plucker (Eds.),
*Critical Issues and Practices in Gifted Education: What the Research Says*(pp. 395–408). Prufrock Press. Google Scholar - Steen, L. A. (Ed.) (2001).
*Mathematics and Democracy: The Case for Quantitative Literacy*. USA: National Council on Education and the Disciplines. Google Scholar - Tan, J. (2002). Education in the twenty-first century: Challenges and dilemmas. In D. da Cunha (Ed.),
*Singapore in the New Millennium: Challenges Facing the Citystate*(pp. 154–186). Singapore: The Institute of Southeast Asian Studies. Google Scholar - Third International Mathematics and Science Study (TIMSS) (2003). http://timss.bc.edu/timss2003i/intl_reports.html; accessed 26.03.09).
- Van den Heuvel-Panhuzen, M. (2003). The didactical use of models in realistic mathematics education: An example from a longitudinal trajectory on percentage.
*Educational Studies in Mathematics*,*54*, 9–35. CrossRefGoogle Scholar - Van Engen, H. (1949). An analysis of meaning in arithmetic.
*Elementary School Journal*,*49*, 321–329, 395–400. CrossRefGoogle Scholar - Watson, J., & Moritz, J. B. (2000). Developing concepts of sampling.
*JRME*,*31*(1), 44–70. CrossRefGoogle Scholar - Zawojewski, J., & McCarthy, L. (2007). Numeracy in practice.
*Principal Leadership*,*7*(5), 32–38. Google Scholar - Zawojewski, J. S., Hjalmarson, M. A., Bowman, K. J., & Lesh, R. (2008). A modeling perspective on learning and teaching in engineering education. In J. S. Zawojewski, H. A. Diefes-Dux, & K. Bowman (Eds.),
*Models and Modeling in Engineering Education*. Rotterdam: Sense Publishers. Google Scholar