The Longitudinal Study



Where do new ideas come from? Our view is that building new ideas is a process; new ideas come from old ideas that are revisited, reviewed, extended, and connected (Davis, 1984; Maher & Davis, 1995). Building new ideas also involves the retrieval and modification of representations of existing ideas. The representations that a learner builds for a mathematical idea or procedure can take different forms – physical objects or actions on objects, words, and symbols, for example. As the learner’s experience increases, old representations become elaborated, extended, and linked to new ones (Maher, 2008; Davis & Maher, 1997)


Mathematical Idea Teacher Development Elementary Grade Problem Task Classroom Session 


  1. Maher, C. A. (2002). How students structure heir own investigations and educate us: What we have learned from a fourteen year study. In A. D. Cockburn & E. Nardi (Eds.), Proceedings of the twenty-sixth annual meeting of the International Group for the Psychology of Mathematics Education (PME26) (Vol. 1, pp. 31–46). Norwich, England: School of Education and Professional Development, University of East Anglia.Google Scholar
  2. Maher, C. A. (2005). How students structure their investigations and learn mathematics: Insights from a long-term study. Journal of Mathematical Behavior, 24(1), 1–14.CrossRefGoogle Scholar
  3. Martino, A. M., & Maher, C. A. (1999). Teacher questioning to promote justification and generalization in mathematics: What research practice has taught us. Journal of Mathematical Behavior, 18(1), 53–78.CrossRefGoogle Scholar
  4. Landis, J. H., & Maher, C. A. (1989). Observations of Carrie, a fourth grade student, doing mathematics. Journal of Mathematical Behavior, 8(1), 3–12.Google Scholar
  5. Davis, R. B., & Maher, C. A. (Eds.). (1993). Schools, mathematics, and the world of reality. Needham, MA: Allyn & Bacon.Google Scholar
  6. Davis, R. B. (1984). Learning mathematics: The cognitive science approach to mathematics education. Hillsdale, NJ: Lawrence Erlbaum Associates.Google Scholar
  7. Davis, R. B., & Maher, C. A. (1997). How students think: The role of representations. In L. D. English (Ed.), Mathematical reasoning: Analogies, metaphors, and images (pp. 93–115). Hillsdale, NJ: Lawrence Erlbaum Associates.Google Scholar
  8. 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
  9. Bruner, J. (1960). The process of education. Cambridge, MA: Harvard University Press.Google Scholar
  10. Maher, C. A., & Martino, A. M. (1996a). 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
  11. Torkildsen, O. (2006). Mathematical archaeology on pupils’ mathematical texts. Un-earthing of mathematical structures. Unpublished doctoral dissertation, Oslo University, Oslo.Google Scholar
  12. Landis, J. H. (1990). Teachers’ prediction and identification of children’s mathematical behaviors: Two case studies. Unpublished doctoral dissertation, Rutgers, The State University of New Jersey, New Brunswick, NJ.Google Scholar
  13. Maher, C. A. (1988). The teacher as designer, implementer, and evaluator of children’s mathematical learning environments. The Journal of Mathematical Behavior, 6, 295–303.Google Scholar
  14. O’Brien, M. (1994). Changing a school mathematics program: A ten-year study. Unpublished doctoral dissertation, Rutgers, the State University of New Jersey, New Brunswick, NJ.Google Scholar
  15. Maher, C. A. (2008). The development of mathematical reasoning: A 16-year study (Invited Senior Lecture for the 10th International Congress on Mathematics Education, published in book with electronic CD). In M. Niss (Ed.), Proceedings of ICME 10 2004. Roskilde, DK: Roskilde University, IMFUFA, Department of Science, Systems and Models.Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Graduate School of Education, Rutgers UniversityNew BrunswickUSA

Personalised recommendations