Abstract
It has been suggested that activities in discrete mathematics allow a kind of new beginning for students and teachers. Students who have been “turned off” by traditional school mathematics, and teachers who have long ago routinized their instruction, can find in the domain of discrete mathematics opportunities for mathematical discovery and interesting, nonroutine problem solving. Sometimes formerly low-achieving students demonstrate mathematical abilities their teachers did not know they had. To take maximum advantage of these possibilities, it is important to know what kinds of thinking during problem solving can be naturally evoked by discrete mathematical situations—so that in developing a curriculum, the objectives can include pathways to desired mathematical reasoning processes. This article discusses some of these ways of thinking, with special attention to the idea of “modeling the general on the particular.” Some comments are also offered about students’ possible affective pathways and structures.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
DeBellis, V. A., & Goldin, G. A. (1997). The affective domain in mathematical problem solving. In E. Pehkonen (Ed.), Proceedings of the 21st Conference of the International Group for the Psychology of Mathematics Education, Lahti, Finland (Vol. 2, pp. 209–216). Helsinki: University of Helsinki Department of Teacher Education.
DeBellis, V. A., & Rosenstein, J. G. (2004). Discrete mathematics in primary and secondary schools in the United States. ZDM, 36(2), 46–55.
Gardiner, A. D. (1991). A cautionary note. In M. J. Kenney & C. R. Hirsch (Eds.), Discrete Mathematics Across the Curriculum, K-12. Reston, VA: National Council of Teachers of Mathematics.
Goldin, G. A. (1984). Structure variables in problem solving. In G. A. Goldin & C. E. McClintock (Eds.), Task Variables in Mathematical Problem Solving (pp. 103–169). Philadelphia, PA: Franklin Institute Press (now Hillsdale, NJ: Erlbaum).
Goldin, G. A. (1998). Representational systems, learning, and problem solving in mathematics. Journal of Mathematical Behavior, 17(2), 137–165.
Goldin, G. A. (2000). Affective pathways and representation in mathematical problem solving. Mathematical Thinking and Learning, 2(3), 209–219.
Goldin, G. A., & Germain, Y. (1983). The analysis of a heuristic process: ‘Think of a simpler problem’. In J. C. Bergeron & N. Herscovics (Eds.), Proceedings of the 5th Annual Meeting of PME-NA [North American Chapter, International Group for the Psychology of Mathematics Education] (Vol. 2, pp. 121–128). Montreal: Univ. of Montreal Faculty of Educational Sciences.
Goldin, G. A., & Kaput, J. J. (1996). A joint perspective on the idea of representation in learning and doing mathematics. In L. Steffe, P. Nesher, P. Cobb, G. A. Goldin, & B. Greer (Eds.), Theories of Mathematical Learning (pp. 397–430). Hillsdale, NJ: Erlbaum.
Kenney, M. J., & Hirsch, C. R. (1991). Discrete Mathematics Across the Curriculum, K-12. Reston, VA: National Council of Teachers of Mathematics.
McLeod, D. B., & Adams, V. M. (Eds.) (1989). Affect and Mathematical Problem Solving: A New Perspective. New York: Springer Verlag.
Polya, G. (1962). Mathematical Discovery: On Understanding, Learning, and Teaching Problem Solving, Vol. I. New York: John Wiley & Sons.
Polya, G. (1965). Mathematical Discovery: On Understanding, Learning, and Teaching Problem Solving, Vol. II. New York: John Wiley & Sons.
Rosenstein, J. G., Franzblau, D., & Roberts, F. (Eds.) (1997). Discrete Mathematics in the Schools. Washington, DC: American Mathematical Society and Reston, VA: National Council of Teachers of Mathematics.
Schoenfeld, A. H. (1983). Episodes and executive decisions in mathematical problem-solving. In R. Lesh & M. Landau (Eds.), Acquisition of Mathematics Concepts and Processes (pp. 345–395). New York: Academic Press.
Schoenfeld, A. H. (1994). Mathematical Thinking and Problem Solving. Hillsdale, NJ: Erlbaum.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Goldin, G.A. (2010). Problem Solving Heuristics, Affect, and Discrete Mathematics: A Representational Discussion. In: Sriraman, B., English, L. (eds) Theories of Mathematics Education. Advances in Mathematics Education. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-00742-2_24
Download citation
DOI: https://doi.org/10.1007/978-3-642-00742-2_24
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-00741-5
Online ISBN: 978-3-642-00742-2
eBook Packages: Humanities, Social Sciences and LawEducation (R0)