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Mathematics Education Research Journal

, Volume 21, Issue 3, pp 7–35 | Cite as

Learning to reason in an informal math after-school program

  • Mary Mueller
  • Carolyn Maher
Articles

Abstract

This research was conducted during an after-school partnership between a University and school district in an economically depressed, urban area. The school population consists of 99% African American and Latino students. During an the informal after-school math program, a group of 24 6th-grade students from a low socioeconomic community worked collaboratively on open-ended problems involving fractions. The students, in their problem solving discussions, coconstructed arguments and provided justifications for their solutions. In the process, they questioned, corrected, and built on each other’s ideas. This paper describes the types of student reasoning that emerged in the process of justifying solutions to the problems posed. It illustrates how the students’ arguments developed over time. The findings of this study indicate that, within an environment that invites exploration and collaboration, students can be engaged in defending their reasoning in both their small groups and within the larger community. In the process of justifying, they naturally build arguments that take the form of proof.

Keywords

School Mathematics Mathematical Reasoning Counter Argument Mathematical Community Overhead Projector 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. Balacheff, N. (1988). Aspects of proof in pupils’ practice of school mathematics. In D. Pimm (Ed.),Mathematics, teachers and children (pp. 316–230). London: Hodder and Stoughton.Google Scholar
  2. Balacheff, N. (1991). The benefits and limits of social interaction: The case of a mathematical proof. In A. Bishop, S. Mellin-Olson, & J. van Doormolen (Eds.),Mathematical knowledge: Its growth through teaching (pp. 175–192). Dordrecht, The Netherlands: Kluwer Academic Publishers.Google Scholar
  3. Ball, D. L., & Bass, H. (2003). Making mathematics reasonable in school. In J. Kilpatrick, W.G. Martin, & D. Schifter (Eds.),A research companion to the principles and standards for school mathematics (pp. 27–44). Reston, VA: National Council of Teachers of Mathematics.Google Scholar
  4. Bulgar, S. (2002). Through a teacher’s lens: Children’s constructions of division of fractions.Dissertation Abstracts International, 63(5), 1754.Google Scholar
  5. Cobb, P. (2000). The importance of a situated view of learning to the design of research and instruction. In L. Burton (Series Ed.) & J. Boaler (Vol. Ed.),A series in international perspectives on mathematics learning: Multiple perspectives on mathematics teaching and learning (pp. 45–82). London: Ablex.Google Scholar
  6. Cobb, P., Stephan, M., McClain, K., & Gravemeijer, K. (2001). Participating in classroom mathematical practices.The Journal of the Learning Sciences, 10, 113–163.CrossRefGoogle Scholar
  7. Cobb, P., Wood, T., & Yackel, E. (1992). Learning and interaction in classroom situations.Educational Studies in Mathematics, 23, 99–122.CrossRefGoogle Scholar
  8. Cupillari, A. (2005).The nuts and bolts of proofs (3rd ed.). Oxford, UK: Elsevier.Google Scholar
  9. Doerr, H. M., & English, L. D. (2006). Middle grade teachers’ learning through students’ engagement with modeling tasks.Journal of Mathematics Teacher Education, 9, 5–32.CrossRefGoogle Scholar
  10. Fletcher, P., & Patty, C. W. (1995).Foundations of higher mathematics. Emeryville, CA: Brooks Cole.Google Scholar
  11. Francisco, J. M. (2005).Students’ epistemological ideas in mathematics: A 12-year longitudinal case study on the development of mathematical ideas. Unpublished doctoral dissertation, Rutgers, the State University of New Jersey, New Brunswick.Google Scholar
  12. Francisco, J. M. & Maher, C.A. (2005). Conditions for promoting reasoning in problem solving: Insights from a longitudinal study.Journal of Mathematical Behavior, 24, 361–372.CrossRefGoogle Scholar
  13. Goos, M. (2004). Learning mathematics in a classroom community of inquiry.Journal for Research in Mathematics Education, 35(4), 258–291.CrossRefGoogle Scholar
  14. Hanna, G. (1991). Mathematical proof. In D. Tall (Ed.),Advanced mathematical thinking (pp. 54–61). Dordrecht, The Netherlands: Kluwer.Google Scholar
  15. Hanna, G. (2000). Proof, explanation and exploration: An overview.Educational Studies in Mathematics, special issue on proof in dynamic geometry environments, 44(1–2), 5–23.Google Scholar
  16. Hanna, G., & Jahnke, H. N. (1996). Proof and proving. In A. Bishop, M. Clements, C. Keitel, J. Kilpatrick, & F. Leung (Eds.),International handbook of mathematics education (877–908). Dordrecht, The Netherlands: Kluwer Academic Publishers.Google Scholar
  17. Maher, C.A. (1995).Exploring the territory leading to proof. Paper presented to the research presession of the 73rd annual meeting of the National Council of Teachers of Mathematics, Boston, MA.Google Scholar
  18. Maher, C. A. (1998). Constructivism and constructivist teaching — Can they co-exist? In O. Bjorkqvist (Ed.),Mathematics teaching from a constructivist point of view (pp. 29–42). Finland: Abo Akeademi, Pedagogiska fakulteten.Google Scholar
  19. Maher, C. A. (2002). How students structure their own investigations and educate us: What we have learned from a fourteen-year study. In A. D. Cockburn & E. Nardi (Eds.),Proceedings of the 26th conference of the International Group for the Psychology of Mathematics Education (pp. 31–46). Norwich, England: University of East Anglia.Google Scholar
  20. Maher, C. A. (2005). How students structure their investigations and learn mathematics: Insights from a long-term study.The Journal of Mathematical Behavior, 24(1), 1–14.CrossRefGoogle Scholar
  21. Maher, C. A. (2008). Children’s reasoning: Discovering the idea of mathematical proof. In M. Blanton & D. Stylianou (Eds.),Teaching and learning proof across the grades (pp. 220–132). New Jersey: Taylor Francis — Routledge.Google Scholar
  22. Maher, C. A. & Davis, R. B. (1995). Children’s explorations leading to proof. In C. Hoyles & L. Healy (Eds.),Justifying and proving in school mathematics (pp. 87–105). London: Mathematical Sciences Group, Institute of Education, University of London.Google Scholar
  23. Maher, C.A. & Martino, A. M. (1996). The development of the idea of mathematical proof: A 5-year study. In F. Lester (Ed.),Journal for Research in Mathematics Education, 27(2), 194–214.CrossRefGoogle Scholar
  24. Maher, C. A., Powell, A. B., Weber, K., & Lee, H. S. (2006). Tracing middle school students’ arguments. In S. Alatorre, J. L. Cortina, M. Sáiz, & A. Méndez (Eds.),Proceedings of the 28th annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 403–410). Mérida, México: Universidad Pedagógica Nacional.Google Scholar
  25. Martin, L., Towers, J., & Pirie, S. (2006). Collective mathematical understanding as improvisation.Mathematical Thinking and Learning, 8(2), 149–183.CrossRefGoogle Scholar
  26. McCrone, S. S. (2005). The development of mathematical discussions: An investigation in a fifth-grade classroom.Mathematical Thinking and Learning, 7(2), 111–133.CrossRefGoogle Scholar
  27. Mueller, M. (2007).A study of the development of reasoning in sixth grade students. Unpublished doctoral dissertation. Rutgers, The State University of New Jersey, New Brunswick. University.Google Scholar
  28. National Council of Teachers of Mathematics (NCTM). (2000).Principles and standards for school mathematics (3rd ed.). Reston, VA: Author.Google Scholar
  29. Polya, G. (1981).Mathematical discovery. New York: Wiley.Google Scholar
  30. Powell, A. B. (2003). “So let’s prove it!”: Emergent and elaborated mathematical ideas and reasoning in the discourse and inscriptions of learners engaged in a combinatorial task. Unpublished doctoral dissertation, Rutgers, The State University of New Jersey, New Brunswick.Google Scholar
  31. Powell, A. B., Francisco, J. M., & Maher, C. A. (2003). An analytical model for studying the development of mathematical ideas and reasoning using videotape data.The Journal of Mathematical Behavior, 22(4), 405–435.CrossRefGoogle Scholar
  32. Reynolds, S. L. (2005).A study of fourth-grade students’ explorations into comparing fractions. Unpublished doctoral dissertation, Rutgers, The State University of New Jersey, New Brunswick.Google Scholar
  33. Skemp, R. R. (1979). Goals of learning and the qualities of understanding.Mathematics Teaching, 88, 44–49.Google Scholar
  34. Steencken, E. (2001).Tracing the growth in understanding of fraction ideas: A fourth grade case study. Unpublished doctoral dissertation, Rutgers, The State University of New Jersey, New Brunswick.Google Scholar
  35. Steencken, E. P., & Maher, C. A. (2002). Young children’s growing understanding of fraction ideas. In B. H. Littwiller & G. Bright (Eds.),2002 National Council of Teachers of Mathematics (NCTM) yearbook: Making sense of fractions, ratios, and proportions (pp. 49–60). Reston, VA: NCTM.Google Scholar
  36. Steencken, E. P., & Maher, C. A. (2003). Tracing fourth graders’ learning of fractions: Episodes from a yearlong teaching experiment.The Journal of Mathematical Behavior, 22(2), 113–132.CrossRefGoogle Scholar
  37. Stylianides, A. J. (2007). Proof and proving in school mathematics.Journal for Research in Mathematics Education, 38, 289–321.Google Scholar
  38. Thompson, P. W. (1996). Imagery and the development of mathematical reasoning. In L. P. Steffe, P. Nesher, P. Cobb, G. A. Goldin, & B. Greer (Eds.),Theories of mathematical learning (pp. 267–283). Mahwah, NJ: Erlbaum.Google Scholar
  39. Vygotsky, L. S. (1978).Mind in society: The development of higher psychological processes. (M. Cole, V. John-Steiner, S. Scribner, & E. Souberman, Trans.). Cambridge, MA: Harvard University Press. (Original work published 1930)Google Scholar
  40. Upper and lower bounds. (2008, February 6). InWikipedia, The Free Encyclopedia. Retrieved February 29, 2008, fromhttp://en.wikipedia.org/w/index.php?title=Upper_and_lower_boundsℓdid=189534718 Google Scholar
  41. Yackel, E., & Cobb, P. (1996). Sociomathematical norms, argumentation, and autonomy in mathematics.Journal for Research in Mathematics Education, 27(4), 458–477.CrossRefGoogle Scholar
  42. Yackel, E., & Hanna, G. (2003). Reasoning and proof. In J. W. Kilpatrick, G. Martin, á D. Schifter (Eds.),A research companion to principles and standards for school mathematics (pp. 227–236). Reston, VA: National Council of Teachers of Mathematics.Google Scholar
  43. Yankelewitz, D. (2009).The development of mathematical reasoning in elementary school students’ exploration of fraction ideas. Unpublished doctoral dissertation, Rutgers, the State University of New Jersey, New Brunswick.Google Scholar

Copyright information

© Mathematics Education Research Group of Australasia Inc. 2009

Authors and Affiliations

  • Mary Mueller
    • 1
  • Carolyn Maher
    • 2
  1. 1.Seton Hall UniversitySouth OrangeUSA
  2. 2.Department of Learning and Teaching, Graduate School of Education, RutgersThe State University of New JerseyNew BrunswickUSA

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