# Influences on Students' Mathematical Reasoning and Patterns in its Development: Insights from a Longitudinal Study with Particular Reference to Geometry

- 304 Downloads
- 7 Citations

## Abstract

We report some findings of the Longitudinal Proof Project, which investigated patterns in high-attaining students' mathematical reasoning in algebra and in geometry and development in their reasoning, by analyses of students' responses to three annual proof tests. The paper focuses on students' responses to one non-standard geometry item. It reports how the distribution of responses to this item changed over time with some moderate progress that suggests a cognitive shift from perceptual to geometrical reasoning. However, we also note that many students made little or no progress and some regressed. Extracts from student interviews indicate that the source of this variation from the overall trend stems from the shift over the three years of the study to a more formal approach in the school geometry curriculum for high-attaining students, and the effects of this shift on what students interpreted as the didactical demands of the item.

## Key words

geometry curriculum geometrical reasoning longitudinal analysis proof## Preview

Unable to display preview. Download preview PDF.

## References

- Balacheff, N. (1988). Aspects of proof in pupils' practice of school mathematics. In D. Pimm (Ed.),
*Mathematics, teachers and children*(pp. 216–235). London: Hodder and Stoughton.Google Scholar - Ball, D.L. & Bass, H. (2000). Making believe: The collective construction of public mathematical knowledge in the elementary classroom. In D. Phillips (Ed.),
*Yearbook of the National Society for the Study of Education: Constructivism in education*(pp. 193–224). Chicago: Chicago University Press.Google Scholar - Brousseau, G. (1992). Didactique: What it can do for the teacher. In R. Douady & A. Mercier (Eds.),
*Research in the didactics of mathematics*(pp. 7–39). Paris: La Pensée Sauvage.Google Scholar - Cobb, P., Boufi, A., McClain, K., & Whitenack, J. (1997). Reflective discourse and collective reflection.
*Journal for Research in Mathematics Education, 28*, 258–277.CrossRefGoogle Scholar - Coe, R. & Ruthven, K. (1994). Proof practices and constructs of advanced mathematics students.
*British Educational Research Journal, 20*(1), 41–53.Google Scholar - Cuoco, A., Goldenberg, P., & Mark, J. (1995). Connecting geometry with the rest of mathematics. In P. House & A. Coxford (Eds.),
*Connecting mathematics across the curriculum: 1995 yearbook*. Reston, VA: NCTM.Google Scholar - de Villiers, M. (1990). The role and function of proof in mathematics.
*Pythagoras*,*24*, 17–24.Google Scholar - Duval, R. (1995). Geometrical pictures: Kinds of representation and specific processings. In R. Sutherland & J. Mason (Eds.),
*Exploiting mental imagery with computers in mathematics education*(pp. 142–157). Berlin: Springer.Google Scholar - Fischbein, E. (1982). Intuition and proof.
*For the Learning of Mathematics, 3*(2), 9–18, 24.Google Scholar - Fischbein, E. (1993). The theory of figural concepts.
*Educational Studies in Mathematics, 24*(2), 139–162.CrossRefGoogle Scholar - Godino, J.D. & Recio, A.M. (1997). Meaning of proofs in mathematics education. In E. Pehkonen (Ed.),
*Proceedings of the 21st Conference of the International Group for the Psychology of Mathematics Education, Vol. 2*, (pp. 313–320). Lahti, Finland.Google Scholar - Goldenberg, E.P., Cuoco, A.A., & Mark, J. (1998). A role for geometry in general education. In Lehrer, R. & Chazan, D. (Eds.),
*Designing learning environments for developing understanding of geometry and space*(pp. 3–44). Mahwah, NJ: Lawrence Erlbaum Associates.Google Scholar - Hanna, G. (1989). Proofs that prove and proofs that explain. In G. Vernaud, J. Rogalski & M. Artigue (Eds.),
*Proceedings of the 13th Conference of the International Group for the Psychology of Mathematics Education, Vol. 2*, (pp. 45–51). Paris, France.Google Scholar - Harel, G. & Sowder, L. (1998). Students' proof schemes: Results for exploratory studies. In A.H. Schoenfeld, J. Kaput & E. Dubinsky (Eds.),
*Research in collegiate mathematics education, III*(pp. 234–283). Providence, RI: American Mathematical Society and Washington, DC: Mathematical Association of America.Google Scholar - Heinze, A. & Reiss, K. (2003). Reasoning and proof: Methodological knowledge as a component of proof competence. In M.A. Mariotti (Ed.),
*Proceedings of the Third Conference of the European Society for Research in Mathematics Education, 2003*. Bellaria, Italy.Google Scholar - Hoyles, C. (1997). The curricular shaping of students' approaches to proof.
*For the Learning of Mathematics, 17*(1), 7–15.Google Scholar - Hoyles, C. & Küchemann, D. (2002). Students' understandings of logical implication.
*Educational Studies in Mathematics*,*51*(3), 193–223.CrossRefGoogle Scholar - Hoyles, C., Küchemann, D., Healy, L., & Yang, M. (2005). Students' developing knowledge in a subject discipline: Insights from combining quantitative and qualitative methods.
*International Journal of Social Research Methodology, 8*(3), 225–238.CrossRefGoogle Scholar - Jahnke, H.N. (2005). A genetic approach to proof. In
*Proceedings of the Fourth Conference of the European Society for Research in Mathematics Education, 2005*. Sant Feliu de Guíxols, Spain.Google Scholar - Laborde, C. (2005). The hidden role of diagrams. In J. Kilpatrick, C. Hoyles, O. Shovsmose & P. Valero (Eds.),
*Meaning in mathematics education*(pp. 159–179). New York: Springer.CrossRefGoogle Scholar - Lampert, M. & Cobb, P. (2003). Communication and language. In J. Kilpatrick, D. Shifter & G. Martin (Eds.),
*Principles and practices of school mathematics: Research companion volume*(pp. 237–249). Reston, VA: National Council of Teachers of Mathematics.Google Scholar - 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 - Mariotti, M.A. (1995). Images and concepts in geometrical reasoning. In R. Sutherland & J. Mason (Eds.),
*Exploiting mental imagery with computers in mathematics education*(pp. 99–116). Berlin: Springer.Google Scholar - Mariotti, M.A., Bartolini-Bussi, M.G., Boero, P., Ferri, F., & Garuti, R. (1997). Approaching geometry theorems in contexts: From history and epistemology to cognition. In E. Pehkonen (Ed.),
*Proceedings of the 21st Conference of the International Group for the Psychology of Mathematics Education, Vol. 2*, (pp. 180–195). Lahti, Finland.Google Scholar - Martino, A. & Maher, C. (1999). Teacher questioning to promote justification and generalization in mathematics: What research practice has taught us.
*The Journal of Mathematical Behaviour, 18*(1), 53–78.CrossRefGoogle Scholar - Morgan, C. (1998).
*Writing mathematically: The discourse of investigation.*London: Falmer.Google Scholar - Reid, D. (1999). Needing to explain: The mathematical emotional orientation. In O. Zaslavsky (Ed.),
*Proceedings of the 23rd Conference of the International Group for the Psychology of Mathematics Education, Vol. 4*, (pp. 105–112). Haifa, Israel.Google Scholar - Rodd, M. (2000). On mathematical warrants.
*Mathematical Thinking and Learning, 3*, 22–244.Google Scholar - Sfard, A. (2001). There is more to discourse than meets the ear: Looking at thinking as communicating to learn more about mathematical learning.
*Educational Studies in Mathematics, 46*(1–3), 13–57.CrossRefGoogle Scholar - Simon, M.A. (2000). Reconsidering mathematical validation in the classroom. In T. Nakahara & M. Koyama (Eds.),
*Proceedings of the 24th Conference of the International Group for the Psychology of Mathematics Education, Vol. 4*, (pp. 161–168). Hiroshima, Japan.Google Scholar - Vinner, S. (1983). The notion of proof—Some aspects of students' views at the senior high school level. In R. Hershkowitz (Ed.),
*Proceedings of the 7th Conference of the International Group for the Psychology of Mathematics Education*(pp. 289–294). Shoresh, Israel.Google Scholar - Yackel, E. (2002). What we can learn from analysing the teacher's role in collective argumentation.
*The Journal of Mathematical Behaviour, 21*(4), 423–440.CrossRefGoogle Scholar - Yackel, E. & Cobb, P. (1996). Socio-mathematical norms, argumentation, and autonomy in mathematics.
*Journal for Research in Mathematics Education, 27*, 458–477.CrossRefGoogle Scholar - Yackel, E. & Hanna, G. (2003). Reasoning and proof. In J. Kilpatrick, W.G. Martin & D. Schifter (Eds.),
*A research companion to NCTM*'*s principles and standards*(pp. 227–236). Reston, VA: NCTM.Google Scholar