So Let’s Prove It!

  • Arthur B. Powell


In previous chapters, we observed students throughout middle school and high school working on and making sense of two isomorphic problems in combinatorics – the towers problems and the pizza problems. In this chapter, we see how students just finishing high-school work on another isomorphic problem, demonstrating the application of techniques and ways of thinking that they developed throughout their previous years in the study. We further address the challenge that Davis (1992a) proposes to mathematics education researchers to investigate the emergence among learners of what lies at the core of mathematics: mathematical ideas. Here, a cohort of four high-school seniors – Brian, Jeff, Mike, and Romina – elaborates mathematical ideas and reasoning through work on the Taxicab Problem. They display criteria and techniques for justifying claims and an awareness of the power of generalizing, particularly as an aid to respond to special cases.


Short Route Mathematical Work Isomorphic Problem Efficient Route Problem Task 
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  1. Powell, A. B. (2006). Socially emergent cognition: Particular outcome of student-to-student discursive interaction during mathematical problem solving. Horizontes, 24(1), 33–42.Google Scholar
  2. Yackel, E., & Cobb, P. (1996). Sociomathematical norms, argumentation, and autonomy in mathematics. Journal for Research in Mathematics Education, 27(4), 458–477.CrossRefGoogle Scholar
  3. Davis, R. B. (1992a). Reflections on where mathematics education now stands and on where it may be going. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 724–734). New York: Macmillan.Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Department of Urban EducationRutgers UniversityNewarkUSA

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