Abstract
In contrast to existing research that has typically addressed the process from example generation to proof construction, this study aims at enhancing empirical examination after proof construction leading to revision of statements and proofs in secondary school geometry. The term “empirical examination” refers to the use of examples or diagrams to investigate whether a statement is true or a proof is valid. Although empirical examination after proof construction is significant in school mathematics in terms of cultivating students’ critical thinking and achieving authentic mathematical practice, how this activity can be fostered remains unclear. This paper shows the strength of a particular kind of mathematical task, proof problems with diagrams, and teachers’ roles in implementing the tasks, by analysing two classroom-based interventions with students in the eighth and ninth grades. In the interventions, the tasks and the teachers’ actions successfully prompted the students to discover a case rejecting a proof and a case refuting a statement, modify the proof, properly restrict the domain of the statement by disclosing its hidden condition, and invent a more general statement that was true even for the refutation of the original statement.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
Although I acknowledge the mathematical difference between lines and segments, segments BD and CE, not lines, are drawn in the diagram because this is enough to solve the initial problem.
I do not intend to undervalue this sequence. It would be worthwhile to investigate in future whether this sequence can lead to the activity aimed for in this study or another valuable activity.
References
Alcock, L., & Weber, K. (2005). Proof validation in real analysis: Inferring and checking warrants. The Journal of Mathematical Behavior, 24(2), 125–134.
Bieda, K. N. (2010). Enacting proof-related tasks in middle school mathematics: Challenges and opportunities. Journal for Research in Mathematics Education, 41(4), 351–382.
Buchbinder, O., & Zaslavsky, O. (2011). Is this a coincidence? The role of examples in fostering a need for proof. ZDM – The International Journal on Mathematics Education, 43(2), 269–281.
Common Core State Standards Initiative (2010). Common core state standards for mathematics. Retrieved August 27, 2015, from http://www.corestandards.org/wp-content/uploads/Math_Standards.pdf.
Henningsen, M., & Stein, M. K. (1997). Mathematical tasks and student cognition: Classroom-based factors that support and inhibit high-level mathematical thinking and reasoning. Journal for Research in Mathematics Education, 28(5), 524–549.
Herbst, P., & Arbor, A. (2004). Interactions with diagrams and the making of reasoned conjectures in geometry. ZDM – The International Journal on Mathematics Education, 36(5), 129–139.
Herbst, P., & Brach, C. (2006). Proving and doing proofs in high school geometry classes: What is it that is going on for students? Cognition and Instruction, 24(1), 73–122.
Iannone, P., Inglis, M., Mejia-Ramos, J. P., Simpson, A., & Weber, K. (2011). Does generating examples aid proof production? Educational Studies in Mathematics, 77(1), 1–14.
Ko, Y. Y., & Knuth, E. J. (2013). Validating proofs and counterexamples across content domains: Practices of importance for mathematics majors. The Journal of Mathematical Behavior, 32(1), 20–35.
Komatsu, K. (2010). Counter-examples for refinement of conjectures and proofs in primary school mathematics. The Journal of Mathematical Behavior, 29(1), 1–10.
Komatsu, K. (2016). A framework for proofs and refutations in school mathematics: Increasing content by deductive guessing. Educational Studies in Mathematics, 92(2), 147–162.
Komatsu, K., & Tsujiyama, Y. (2013). Principles of task design to foster proofs and refutations in mathematical learning: Proof problem with diagram. In C. Margolinas (Ed.), Task Design in Mathematics Education (pp. 471–479). Oxford: Proceedings of ICMI Study 22.
Komatsu, K., Tsujiyama, Y., Sakamaki, A., & Koike, N. (2014). Proof problems with diagrams: An opportunity for experiencing proofs and refutations. For the Learning of Mathematics, 34(1), 36–42.
Lakatos, I. (1976). Proofs and refutations: The logic of mathematical discovery. Cambridge: Cambridge University Press.
Lampert, M. (1992). Practices and problems in teaching authentic mathematics. In F. K. Oser, A. Dick, & J. L. Patry (Eds.), Effective and responsible teaching: The new synthesis (pp. 295–314). San Francisco: Jossey-Bass Publishers.
Larsen, S., & Zandieh, M. (2008). Proofs and refutations in the undergraduate mathematics classroom. Educational Studies in Mathematics, 67(3), 205–216.
Miyazaki, M., Nagata, J., Chino, K., Fujita, T., Ichikawa, D., Shimizu, S., & Iwanaga, Y. (2016). Developing a curriculum for explorative proving in lower secondary school geometry. Hamburg: Proceedings of the 13th International Congress on Mathematical Education.
Ohtani, M. (2002). Social formation of mathematical activity in mathematics classroom. Tokyo: Kazama Shobo [in Japanese].
Pedemonte, B. (2007). How can the relationship between argumentation and proof be analyzed? Educational Studies in Mathematics, 66(1), 23–41.
Sandefur, J., Mason, J., Stylianides, G. J., & Watson, A. (2013). Generating and using examples in the proving process. Educational Studies in Mathematics, 83(3), 323–340.
Sears, R., & Chávez, Ó. (2014). Opportunities to engage with proof: The nature of proof tasks in two geometry textbooks and its influence on enacted lessons. ZDM – The International Journal on Mathematics Education, 46(5), 767–780.
Stein, M. K., Engle, R. A., Smith, M. S., & Hughes, E. K. (2008). Orchestrating productive mathematical discussions: Five practices for helping teachers move beyond show and tell. Mathematical Thinking and Learning, 10(4), 313–340.
Stigler, J. W., & Hiebert, J. (1999). The teaching gap: Best ideas from the world’s teachers for improving education in the classroom. New York: Free Press.
Stylianides, A. J. (2007). Introducing young children to the role of assumptions in proving. Mathematical Thinking and Learning, 9(4), 361–385.
Stylianides, G. J., & Stylianides, A. J. (2009). Facilitating the transition from empirical arguments to proof. Journal for Research in Mathematics Education, 40(3), 314–352.
Stylianides, A. J., & Stylianides, G. J. (2013). Seeking research-grounded solutions to problems of practice: Classroom-based interventions in mathematics education. ZDM – The International Journal on Mathematics Education, 45(3), 333–341.
Stylianides, G. J., & Stylianides, A. J. (2014). The role of instructional engineering in reducing the uncertainties of ambitious teaching. Cognition and Instruction, 32(4), 374–415.
Acknowledgments
This paper was finalised during my visit to the University of Southampton as a visiting fellow. I would like to express my thanks to Keith Jones for his continuous support. I am grateful to Keisuke Makino and Isao Ohira for their cooperation in conducting classroom teaching experiments. I also thank the editors and the anonymous reviewers for their helpful comments on earlier versions of this paper. This study is supported by the Japan Society for the Promotion of Science (Nos. 15H05402, 16H02068, and 26282039).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Komatsu, K. Fostering empirical examination after proof construction in secondary school geometry. Educ Stud Math 96, 129–144 (2017). https://doi.org/10.1007/s10649-016-9731-6
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10649-016-9731-6