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Educational Studies in Mathematics

, Volume 82, Issue 3, pp 417–438 | Cite as

Learning in collaborative settings: students building on each other’s ideas to promote their mathematical understanding

  • John M. FranciscoEmail author
Article

Abstract

The purpose of this study is to contribute insights into how collaborative activity can help promote students’ mathematical understanding. A group of six high school students (15- to 16-year olds) worked together on a challenging probability task as part of a larger, after-school, longitudinal study on students’ development of mathematical ideas in problem-solving settings. The students solved the problem and produced a valid justification of their solution. This study shows that collaborative activity can help promote students’ mathematical understanding by providing opportunities for students to critically reexamine how they make claims from facts and also enable them to build on one another’s ideas to construct more sophisticated ways of reasoning. Implications for classroom teaching and ideas for future research are also discussed. The study helps address a documented need for a better understanding of how mathematical learning evolves in social settings.

Keywords

Collaborative work Students’ reasoning Probability Mathematical understanding 

References

  1. Balacheff, N. (1991). Benefits and limits of social interaction: The case of mathematical proof. In A. Bishop, S. Mellin-Olsen, & J. van Dormolen (Eds.), Mathematical knowledge: Its growth through teaching (pp. 175–192). Dordrecht: Kluwer.Google Scholar
  2. Bauersfeld, H. (1995). The structuring of structures: Development and function of mathematizing as a social practice. In L. P. Steffe & J. Galle (Eds.), Constructivism in education (pp. 137–158). Hillsdale: Lawrence Erlbaum Associates.Google Scholar
  3. Boero, P., & Guala, E. (2008). Development of mathematical knowledge and beliefs of teachers. In P. Sullivan & T. Wood (Eds.), International handbook of mathematics teacher education (pp. 223–244). Rotterdam: Sense Publishers.Google Scholar
  4. Bowers, J. S., & Nickerson, S. (1991). Identifying cyclic patterns of interaction to study individual and collective learning. Mathematical Thinking and Learning, 3, 1–28.CrossRefGoogle Scholar
  5. Cobb, P. (1999). Individual and collective mathematical development: The case of statistical data analysis. Mathematical Thinking and Learning, 1, 5–43.CrossRefGoogle Scholar
  6. Cobb, P., Boufi, A., McClain, K., & Whitenack, J. (1997). Reflective discourse and collective reflection. Journal for Research in Mathematics Education, 28(3), 258–277.CrossRefGoogle Scholar
  7. Cobb, P., & Yackel, E. (1996). Constructivist, emergent, and sociocultural perspectives in the context of developmental research. Educational Psychologist, 31(3/4), 175–190.Google Scholar
  8. Davis, B., & Simmt, E. (2003). Understanding learning systems: Mathematics education and complexity science. Journal for Research in Mathematics Education, 34, 137–167.CrossRefGoogle Scholar
  9. Davis, R. B., & Maher, C. A. (1990). The nature of mathematics: What do we do when we “do mathematics”? In R. B. Davis, C. A. Maher, & N. Noddings (Eds.), Constructivist views on the teaching and learning of mathematics (pp. 65–78). Reston: National Council of Teachers of Mathematics.Google Scholar
  10. Dörfler, W. (2000). Means for meaning. In P. Cobb, E. Yackel, & K. McClain (Eds.), Symbolizing and communicating in mathematics classrooms: Perspectives on discourse, tools, and instructional design (pp. 99–131). Mahwah: Lawrence Erlbaum Associates.Google Scholar
  11. Krummheuer, G. (1995). The ethnography of argumentation. In P. Cobb & H. Bauersfeld (Eds.), The emergence of mathematical meaning: Interaction in classroom cultures (pp. 229–269). Hillsdale: Lawrence Erlbaum Associates.Google Scholar
  12. Lampert, M., & Cobb, P. (2003). Communication and language. In J. Kilpatrick, G. Martin, & D. Schifter (Eds.), A research companion to principles and standards for school mathematics (pp. 237–248). Reston: National Council of Teachers of Mathematics.Google Scholar
  13. Lampert, M., Rittenhouse, P., & Crumbaugh, C. (1998). Agreeing to disagree: Developing sociable mathematical discourse. In D. R. Olson & N. Torrance (Eds.), Handbook of education and human development. Oxford: Blackwell.Google Scholar
  14. Lave, J. (1997). The culture of acquisition and the practice of understanding. In D. Kirshner & J. Whitson (Eds.), Situated cognition: Social, semiotic, and psychological perspectives (pp. 17–35). Mahwah: Lawrence Erlbaum Associates.Google Scholar
  15. Lave, J., & Wenger, E. (1991). Situated learning: Legitimate peripheral participation. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  16. Maher, C. A., & Martino, A. M. (1996). The development of the idea of mathematical proof: A 5-year case study. Journal for Research in Mathematics Education, 27(2), 194–214.CrossRefGoogle Scholar
  17. Manouchehri, A., & Enderson, M. (1999). Promoting mathematical discourse: Learning from classroom examples. Mathematics Teaching in the Middle School, 4, 216–222.Google Scholar
  18. Martin, L., Towers, J., & Pirie, S. (2006). Collective mathematical understanding as improvisation. Mathematical Thinking and Learning, 8(2), 149–183.CrossRefGoogle Scholar
  19. McCrone, S. S. (2005). The development of mathematical discussion: An investigation in a fifth-grade classroom. Mathematical Thinking and Learning, 7(2), 111–133.CrossRefGoogle Scholar
  20. NCTM. (2000). Principles and standards for school mathematics. Reston: National Council of Teachers of Mathematics.Google Scholar
  21. Pedemonte, B. (2007). How can the relationship between argumentation and proof be analysed? Educational Studies in Mathematics, 66, 23–41.CrossRefGoogle Scholar
  22. Powell, A. B., Francisco, J. M., & Maher, C. A. (2003). An evolving analytical model for understanding the development of mathematical thinking using videotape data. The Journal of Mathematical Behavior, 22(4), 405–435.CrossRefGoogle Scholar
  23. Rasmussen, C., & Stephan, M. (2008). A methodology for documenting collective activity. In A. E. Kelly & R. Lesh (Eds.), Design research in education (pp. 195–215). Mahwah: Lawrence Erlbaum Associates.Google Scholar
  24. Roschelle, J., & Teasley, S. D. (1995). The construction of shared knowledge in collaborative problem solving. In C. E. O’Malley (Ed.), Computer supported collaborative learning (pp. 88–110). Berlin: Springer.Google Scholar
  25. Salomon, G. (1993). No distribution without individuals’ cognition dynamic interactional view. In G. Salomon (Ed.), Distributed cognitions (pp. 111–138). Cambridge: Cambridge University Press.Google Scholar
  26. Sawyer, R. K. (2001). Creating conversations: Improvisation in everyday discourse. Cresskill: Hampton Press.Google Scholar
  27. Saxe, G. (2002). Children’s developing mathematics in collective practices: A framework for analysis. The Journal of the Learning Sciences, 11, 275–300.Google Scholar
  28. Schoenfeld, A. (1985). Mathematical problem solving. San Diego: Academic.Google Scholar
  29. Sfard, A. (2001). There is more to discourse than meets the ears: Looking at thinking as communicating to learn more about mathematical learning. Educational Studies in Mathematics, 46, 13–57.CrossRefGoogle Scholar
  30. Sfard, A., & Kieran, C. (2001). Cognition as communication: Rethinking learning by talking through multi-faceted analysis of students’ mathematical interactions. Mind, Culture, and Activity, 8, 42–76.CrossRefGoogle Scholar
  31. Shaughnessy, M. J. (2003). Research on students’ understanding of probability. In J. Kilpatrick, G. Martin, & D. Schifter (Eds.), A research companion to principles and standards for school mathematics (pp. 216–225). Reston: NCTM.Google Scholar
  32. Simon, M. (1995). Reconstructing mathematics pedagogy from a constructivist perspective. Journal for Research in Mathematics Education, 26, 114–145.CrossRefGoogle Scholar
  33. Stephan, M., Cobb, P., & Gravemeijer, K. (2003). Coordinating social and individual analyses: Learning as participation in mathematical practices. Supporting students’ development of measuring conceptions: Analyzing students’ learning in social context. Monograph of the Journal for Research in Mathematics Education, 12, 67–102.Google Scholar
  34. Stephan, M., & Rasmussen, C. (2002). Classroom mathematical practices in differential equations. The Journal of Mathematical Behavior, 21, 459–490.CrossRefGoogle Scholar
  35. Toulmin, S. (1969). The uses of arguments. Cambridge: Cambridge University Press.Google Scholar
  36. Vygotsky, L. S. (1978). Mind and society. Cambridge: Harvard University Press.Google Scholar
  37. Weber, K., & Alcock, L. (2005). Using warranted implications to understand and validate proofs. For the Learning of Mathematics, 25, 34–38.Google Scholar
  38. Weber, K., Maher, C., Powell, A., & Lee, H. (2008). Learning opportunities from group discussions: Warrants become the objects of debate. Educational Studies in Mathematics, 68(32), 247–261.CrossRefGoogle Scholar
  39. Yackel, E. (2001). Explanation, justification, and argumentation in mathematics classrooms. In M. van den Heuvel-Panhuizen (Ed.), Proceedings of the 25th Conference for the International Group of Psychology of Mathematics Education (1, 9–24). Utrecht, the Netherlands.Google Scholar
  40. Yackel, E., & Cobb, P. (1996). Sociomathematical norms, argumentation, and autonomy in mathematics. Journal for Research in Mathematics Education, 27, 458–477.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2012

Authors and Affiliations

  1. 1.School of EducationUniversity of MassachusettsAmherstUSA

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