Learning to Prove
Proving is central to the practice of mathematics and plays an important role in learning mathematics; not surprisingly, both mathematics education scholars and reform initiatives have increasingly called for proof to play a more central role in the mathematics education of students at all grade levels. Yet, despite almost two decades of calls to elevate the status and role of proof in school mathematics, students of all ages continue to struggle to learn to prove, and teachers as well struggle to facilitate the development of students’ learning to prove. The Learning to Prove portfolio of projects focuses on examining students’ understanding of what counts as evidence in mathematics, and ways to foster the development of their learning to prove.
There has been a lack of systematic research focused on how students acquire and develop their understandings of what constitutes evidence and justification in mathematics and how such understandings can be extended and refined. The objectives of this project were to examine the longitudinal development of middle school students’ competencies in justifying and proving and to examine the instructional conditions and pedagogies necessary to promote the development of those competencies.
Principal Investigator: Eric Knuth
An enduring challenge in mathematics education is finding ways to help students understand the nature of evidence and justification in mathematics. Yet, cognitive science research has revealed surprising strengths in children’s abilities to reason inferentially in non-mathematical contexts. In particular, there is growing evidence that children are capable of developing sophisticated causal theories, and of using powerful strategies of inductive inference when reasoning about the natural world. This contrast raises a potential paradox: Why do children seem relatively adept at reasoning in non-mathematical contexts, yet seemingly poor at reasoning in mathematical contexts? The objective of this project was to examine this paradox. In particular, this project sought to understand both the strengths and weaknesses of students’ reasoning in and out of mathematics and, to examine ways in which students’ reasoning in non-mathematical domains may provide an important bridge to improving their ways of reasoning in mathematics.
Principal Investigators: Eric Knuth, Amy Ellis, & Charles Kalish
Mathematics education researchers have suggested that students’ overreliance on examples as a means of conviction and justification is a primary reason for their difficulties in learning to prove. Thus examples-based reasoning is typically viewed as a stumbling block to learning to prove and researchers have advocated that instructional approaches be designed to help students learn the limitations of examples. In contrast, we view examples-based reasoning as an important object of study and contend that examples play both a foundational and essential role in the development, exploration, and understanding of conjectures, as well as in subsequent attempts to develop proofs of those conjectures. The goals of the project were twofold: (a) investigate the nature of middle school and high school students’, undergraduate students’, and mathematicians’ thinking about the examples they use when developing, exploring, and proving conjectures; and (b) investigate ways in which thinking about and analyzing examples may facilitate the development of students’ learning to prove.
Principal Investigators: Eric Knuth, Orit Zaslavsky, & Amy Ellis
EXAMPLES II – Curriculum (2020-2021)
The objective of this project is to better understand the nature of the opportunities present in middle school curricular materials to engage students in productive uses of examples. The overarching goals of the project are to (a) examine the curricular treatment of examples in proving-related activities, (b) examine cross-cultural differences with regard to the curricular treatment of examples in proving-related activities given differences in students’ productive example use, and (c) provide insight into the design of curricular materials and associated teacher guidance with regard to example use in proving-related activities.
Principal Investigators: Eric Knuth, Orit Zaslavsky, & GwiSoo Na
Knuth, E., Kim, H., Zaslavsky, O., Vinsonhaler, R., Gaddis, D., & Fernandez, L. (Accepted). Teachers’ views about the role of examples in proving-related activities. To appear in Journal of Educational Research in Mathematics.
Ellis, A., Ozgur, Z., Vinsonhaler, R., Dogan, M., Carolan, T., Lockwood, E., Lynch, A., Sabouri, P., Knuth, E., & Zaslavsky, O. (2019). Student thinking with examples: The criteria-affordances-purposes-strategies framework. Journal of Mathematical Behavior, 53, 263-283.
Knuth, E., Zaslavsky, O., & Ellis, A. (2019). The role and use of examples in learning to prove. Journal of Mathematical Behavior, 53, 256-262.
Ozgur, Z., Ellis, A., Vinsonhaler, R., Dogan, M., & Knuth, E. (2019). From examples to proof: Purposes, strategies, and affordances of example use. Journal of Mathematical Behavior, 53, 284-303.
Lockwood, E., Ellis, A. B., & Lynch, A. G. (2016). Mathematicians’ example-related activity when exploring and proving conjectures. International Journal of Research in Undergraduate Mathematics Education, 2(2), 165-196.
Ellis, A.B., Bieda, K., & Knuth, E. (2012). Essential understandings project: Reasoning and Proving in High School Mathematics (Gr. 9-12). Reston, VA: National Council of Teachers of Mathematics.
Knuth, E., Kalish, C., Ellis, A., Williams, C., & Felton, M. (2011). Adolescent reasoning in mathematical and non-mathematical domains: Exploring the paradox. In V. Reyna, S. Chapman, M. Dougherty, & J. Confrey (Eds.), The adolescent brain: Learning, reasoning, and decision making (pp. 183-209). Washington, DC: American Psychological Association
Knuth, E., Choppin, J., & Bieda, K. (2009). Middle school students’ production of mathematical justifications. In D. Stylianou, M. Blanton, & E. Knuth (Eds.), Teaching and learning proof across the grades: A K-16 perspective (pp. 153-170). New York, NY: Routledge.
Knuth, E., Choppin, J., & Bieda, K. (2009). Proof in middle school: Moving beyond examples. Mathematics Teaching in the Middle School, 15(4), 206-211.
Stylianou, D., Blanton, M., & Knuth, E. (2009). Teaching and learning proof across the grades: A K-16 perspective. New York, NY: Routledge.
Knuth, E. (2002). Proof as a tool for learning mathematics. Mathematics Teacher, 95(7), 486-490.