A multi-faceted framework for identifying students’ understanding of the generality requirement of proof

Kimberly A. Conner

doi:10.29333/iejme/12270

Full Text (PDF)

Kimberly A. Conner ¹ ^*

More Detail

¹ Mathematics Department, University of Northern Iowa, Cedar Falls, IA, USA^* Corresponding Author

Abstract

The generality requirement, or the requirement that a proof must demonstrate a claim to be true for all cases within its domain, represents one of the most important, yet challenging aspects of proof for students to understand. This article presents a multi-faceted framework for identifying aspects of students’ work that have the potential to provide insight into their understanding of the generality requirement. I use a single student’s work across different snapshots in time to illustrate her understanding of the generality requirement as evidenced by the justifications, examples, variables, and diagrams present within her written and oral work when proving mathematical claims and her rationales when evaluating whether provided arguments were proofs. Analysis of the student’s work across multiple categories at a single moment in time shows that students can exhibit understanding of the generality requirement for some categories while demonstrating limited or unclear evidence of understanding for others. By looking across the framework dimensions and types of tasks, researchers can identify patterns in student understanding and specific aspects that could be addressed through future instruction.

Keywords

reasoning and proving
example-use
teaching experiment
case study
secondary mathematics

License

This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Article Type: Research Article

INT ELECT J MATH ED, Volume 17, Issue 4, November 2022, Article No: em0702

https://doi.org/10.29333/iejme/12270

Publication date: 26 Jul 2022

Article Views: 1282

Article Downloads: 879

Open Access References How to cite this article

References

Aricha-Metzer, I., & Zaslavsky, O. (2019). The nature of students’ productive and non-productive example-use for proving. Journal of Mathematical Behavior, 53, 304-322. https://doi.org/10.1016/j.jmathb.2017.09.002
Bieda, K. N., & Lepak, J. (2014). Are you convinced? Middle-grade students’ evaluations of mathematical arguments. School Science and Mathematics, 114(4), 166-177. https://doi.org/10.1111/ssm.12066
Boyle, J. D., Bleiler, S. K., Yee, S. P., & Ko, Y.-Y. (2015). Transforming perceptions of proof: A four-part instructional sequence. Mathematics Teacher Educator, 4(1), 32-70. https://doi.org/10.5951/mathteaceduc.4.1.0032
Buchbinder, O., & Zaslavsky, O. (2008). Uncertainty: A driving force in creating a need for proving. International Commission on Mathematical Instruction. http://www.orlybuchbinder.com/publications/ICMI_19_Buchbinder_Zaslavsky_Uncertainty%20as%20a%20driving%20force-final.pdf
Campbell, T. G., King, S., & Zelkowski, J. (2020). Comparing middle grade students’ oral and written arguments. Research in Mathematics Education, 23(1), 21-38. https://doi.org/10.1080/14794802.2020.1722960
Chazan, D. (1993). High school geometry students’ justification for their views of empirical evidence and mathematical proof. Educational Studies in Mathematics, 24(4), 359-387. https://doi.org/10.1007/BF01273371
Cirillo, M. (2011). “I’m like the sherpa guide”: On learning to teach proof in school mathematics. In B. Ubuz (Ed.), Proceedings of the 35^th Conference of the International Group for the Psychology of Mathematics Education (pp. 241-248). PME.
Cirillo, M. (2017). Engaging students with non-routine geometry proof tasks. In P. Herbst, U. Cheah, P. Richard, & K. Jones (Eds.), International perspectives on the teaching and learning of geometry in secondary schools (pp. 283-300). Springer. https://doi.org/10.1007/978-3-319-77476-3_16
Cirillo, M., & Herbst, P. G. (2011). Moving toward more authentic proof practices in geometry. The Mathematics Educator, 21(2), 11-33. http://math.coe.uga.edu/tme/issues/v21n2/v21n2_Cirillo&Herbst_Abs.html
Cobb, P., & Steffe, L. P. (1983). The constructivist researcher as teacher and model builder. Journal for Research in Mathematics Education, 14(2), 82-94. https://doi.org/10.5951/jresematheduc.14.2.0083
Conner, K. A., & Otten, S. (2021). General proof tasks provide opportunities for varied forms of reasoning about the domain of mathematical claims. Electronic Journal for Research in Science & Mathematics Education, 25(4), 42-70.
Dawkins, P. C., & Karunakaran, S. S. (2016). Why research on proof-oriented mathematical behavior should attend to the role of particular mathematical content. Journal of Mathematical Behavior, 44, 65-75. https://doi.org/10.1016/j.jmathb.2016.10.003
de Villiers, M. (1990). The role and function of proof in Mathematics. Pythagoras, 24, 17-23.
Ellis, A. B., Bieda, K., & Knuth, E. J. (2012). Developing essential understanding of proof and proving for teaching mathematics in grades 9-12. National Council of Teachers of Mathematics.
Ellis, A. B., Ozgur, Z., Vinsonhaler, R., Dogan, M. F., Carolan, T., Lockwood, E., Lynch, A., Sabouri, P., Knuth, E., & Zaslavsky, O. (2019). Student thinking with examples: The criteria-affordances-purposes-strategies framework. The Journal of Mathematical Behavior, 53, 263-283. https://doi.org/10.1016/j.jmathb.2017.06.003
Ely, R., & Adams, A. E. (2012). Unknown, placeholder, or variable: What is x? Mathematics Education Research Journal, 24(1), 19-38. https://doi.org/10.1007/s13394-011-0029-9
Fischbein, E. (1982). Intuition and proof. For the Learning of Mathematics, 3(3), 9-18.
Fischbein, E. (1993). The theory of figural concepts. Educational Studies in Mathematics, 24(2), 139-162. https://doi.org/10.1007/BF01273689
Hanna, G. (2000). Proof, explanation and exploration: An overview. Educational Studies in Mathematics, 44, 5-23. https://doi.org/10.1023/A:1012737223465
Hanna, G., & Jahnke, H. N. (1996). Proof and proving. In A. J. Bishop, K. Clements, C. Keitel, J. Kilpatrick, & C. Laborde (Eds.), International handbook of mathematics education (pp. 877-908). Springer. https://doi.org/10.1007/978-94-009-1465-0_27
Healy, L., & Hoyles, C. (2000). A study of proof conceptions in algebra. Journal for Research in Mathematics Education, 31(4), 396-428. https://doi.org/10.2307/749651
Herbst, P. G. (2002). Establishing a custom of proving in American school geometry: Evolution of the two-column proof in the early twentieth century. Educational Studies in Mathematics, 49, 283-312. https://doi.org/10.1023/A:1020264906740
Herbst, P. G., & 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. https://doi.org/10.1207/s1532690xci2401_2
Hoyles, C., & Healy, L. (2000). Curriculum change and geometrical reasoning. In P. Boero (Ed.), Theorems in school: From history, epistemology and cognition to classroom practice (pp. 81–115). Sense Publishers.
Knuth, E. J., & Sutherland, J. (2004). Student understanding of generality. In D. E. McDougall, & J. A. Ross (Eds.), Proceedings of the 26^th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 562-568). OISE/UT.
Knuth, E., Choppin, J., & Bieda, K. (2009). Middle school students’ production of mathematical justifications. In D. A. Stylianou, M. L. Blanton, & E. J. Knuth (Eds.), Teaching and learning proof across the grades: A K-16 perspective (pp. 153-170). Routledge. https://doi.org/10.4324/9780203882009-9
Knuth, E., Zaslavsky, O., & Ellis, A. B. (2019). The role and use of examples in learning to prove. Journal of Mathematical Behavior, 53, 256-262. https://doi.org/10.1016/j.jmathb.2017.06.002
Kobiela, M., & Lehrer, R. (2015). The codevelopment of mathematical concepts and the practice of defining. Journal for Research in Mathematics Education, 46(4), 423-454. https://doi.org/10.5951/jresematheduc.46.4.0423
Küchemann, D. (1978). Children’s understanding of numerical variables. Mathematics in School, 7(4), 23-26. https://doi.org/10.2307/30213397
Küchemann, D. E. (1981). Algebra. In K. M. Hart (Ed.), Children’s understanding of mathematics: 11-16 (pp. 102-119). John Murray.
Laborde, C. (1995). The hidden role of diagrams in students’ construction of meaning in geometry. In J. Kilpatrick, C. Hoyles, O. Skovsmose, & P. Valero (Eds.), Meaning in mathematics education (pp. 159-179). Springer. https://doi.org/10.1007/0-387-24040-3_11
Martin, T., McCrone, S., Bower, M., & Dindyal, J. (2005). The interplay of teacher and student actions in the teaching and learning of geometric proof. Educational Studies in Mathematics, 60(1), 95-124. https://doi.org/10.1007/s10649-005-6698-0
Martinez, M. V., & Superfine, A. C. (2012). Integrating algebra and proof in high school: Students’ work with multiple variables and a single parameter in a proof context. Mathematical Thinking and Learning, 14(2), 120-148. https://doi.org/10.1080/10986065.2012.657956
Mason, J. (1989). Mathematical abstraction as the result of a delicate shift of attention. For the Learning of Mathematics, 9(2), 2-8.
Mason, J. (2019). Relationships between proof and examples: Comments arising from the papers in this issue. The Journal of Mathematical Behavior, 53, 339-347. https://doi.org/10.1016/j.jmathb.2017.07.005
Mason, J., & Pimm, D. (1984). Generic examples: Seeing the general in the particular. Educational Studies in Mathematics, 15(3), 277-289. https://doi.org/10.1007/BF00312078
Otten, S., Bleiler-Baxter, S. K., & Engledowl, C. (2017). Authority and whole-class proving in high school geometry: The case of Ms. Finley. Journal of Mathematical Behavior, 46, 112–127. https://doi.org/10.1016/j.jmathb.2017.04.002
Otten, S., Gilbertson, N. J., Males, L. M., & Clark, D. L. (2014a). The mathematical nature of reasoning-and-proving opportunities in geometry textbooks. Mathematical Thinking and Learning, 16(1), 51-79. https://doi.org/10.1080/10986065.2014.857802
Otten, S., Males, L. M., & Gilbertson, N. J. (2014b). The introduction of proof in secondary geometry textbooks. International Journal of Educational Research, 64, 107-118. https://doi.org/10.1016/j.ijer.2013.08.006
Ozgur, Z., Ellis, A. B., Vinsonhaler, R., Dogan, M. F., & Knuth, E. (2019). From examples to proof: Purposes, strategies, and affordances of example use. The Journal of Mathematical Behavior, 53, 284-303. https://doi.org/10.1016/j.jmathb.2017.03.004
Philipp, R. (1992). The many uses of algebraic variables. The Mathematics Teacher, 85(7), 557-561. https://doi.org/10.5951/MT.85.7.0557
Roulston, K. (2010). Asking questions and individual interviews. In K. Roulston (Ed.), Reflective interviewing: A guide to theory and practice (pp. 9-32). SAGE. https://doi.org/10.4135/9781446288009.n2
Schoenfeld, A. H. (1989). Explorations of students’ mathematical beliefs and behavior. Journal for Research in Mathematics Education, 20(4), 338-355. https://doi.org/10.2307/749440
Schoenfeld, A., & Arcavi, A. (1988). On the meaning of variable. Mathematics Teacher, 81(6), 420-427. https://doi.org/10.5951/MT.81.6.0420
Senk, S. L. (1985). How well do students write geometry proofs? The Mathematics Teacher, 78(6), 448-456. https://doi.org/10.2307/27964580
Simon, M., Saldanha, L., McClintock, E., Akar, G. K., Watanabe, T., & Zembat, I. O. (2010). A developing approach to studying students’ learning through their mathematical activity. Cognition and Instruction, 28(1), 70-112. https://doi.org/10.1080/07370000903430566
Sommerhoff, D., & Ufer, S. (2019). Acceptance criteria for validating mathematical proofs used by school students, university students, and mathematicians in the context of teaching. ZDM–Mathematics Education, 51(5), 717-730. https://doi.org/10.1007/s11858-019-01039-7
Steffe, L. P., & Thompson, P. W. (2000). Teaching experiment methodology: Underlying principles and essential elements. In R. Lesh & A. E. Kelly (Eds.), Research design in mathematics and science education (pp. 267-307). Erlbaum.
Strauss, A., & Corbin, J. (1998). Basics of qualitative research: Techniques and procedures for developing grounded theory. SAGE.
Stylianides, A. J. (2007). Proof and proving in school mathematics. Journal for Research in Mathematics Education, 38(3), 289-321. https://doi.org/10.2307/30034869
Stylianides, A. J. (2019). Secondary students’ proof constructions in mathematics: The role of written versus oral mode of argument representation. Review of Education, 7(1), 156-182. https://doi.org/10.1002/rev3.3157
Stylianides, A. J., & Stylianides, G. J. (2009a). Proof constructions and evaluations. Educational Studies in Mathematics, 72(2), 237–253. https://doi.org/10.1007/s10649-009-9191-3
Stylianides, G. J., & Stylianides, A. J. (2009b). Facilitating the transition from empirical arguments to proof. Journal for Research in Mathematics Education, 40(3), 314-352. https://doi.org/10.5951/jresematheduc.40.3.0314
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. https://doi.org/10.1080/07370008.2014.948682
Usiskin, Z. (1999). Conceptions of school algebra and uses of variables. In B. Moses (Ed.), Algebraic thinking, grades K-12: Readings from NCTM’s school-based journals and other publications (pp. 7-13). National Council of Teachers of Mathematics.
Weber, K. (2002). Beyond proving and explaining: Proofs that justify the use of definitions and axiomatic structures and proofs that illustrate technique. For the Learning of Mathematics, 22, 14-17. https://www.jstor.org/stable/40248396
Weber, K., Lew, K., & Mejía-Ramos, J. P. (2020). Using expectancy value theory to account for individuals’ mathematical justifications. Cognition and Instruction, 38(1), 27-56. https://doi.org/10.1080/07370008.2019.1636796
Yin, R. K. (2014). Case study research: Design and methods (applied social research methods). SAGE.
Yopp, D. A., & Ely, R. (2016). When does an argument use a generic example? Educational Studies in Mathematics, 91(1), 37-53. https://doi.org/10.1007/s10649-015-9633-z

How to cite this article

APA

Conner, K. A. (2022). A multi-faceted framework for identifying students’ understanding of the generality requirement of proof. International Electronic Journal of Mathematics Education, 17(4), em0702. https://doi.org/10.29333/iejme/12270

Vancouver

Conner KA. A multi-faceted framework for identifying students’ understanding of the generality requirement of proof. INT ELECT J MATH ED. 2022;17(4):em0702. https://doi.org/10.29333/iejme/12270

AMA

Conner KA. A multi-faceted framework for identifying students’ understanding of the generality requirement of proof. INT ELECT J MATH ED. 2022;17(4), em0702. https://doi.org/10.29333/iejme/12270

Chicago

Conner, Kimberly A.. "A multi-faceted framework for identifying students’ understanding of the generality requirement of proof". International Electronic Journal of Mathematics Education 2022 17 no. 4 (2022): em0702. https://doi.org/10.29333/iejme/12270

Harvard

Conner, K. A. (2022). A multi-faceted framework for identifying students’ understanding of the generality requirement of proof. International Electronic Journal of Mathematics Education, 17(4), em0702. https://doi.org/10.29333/iejme/12270

MLA

Conner, Kimberly A. "A multi-faceted framework for identifying students’ understanding of the generality requirement of proof". International Electronic Journal of Mathematics Education, vol. 17, no. 4, 2022, em0702. https://doi.org/10.29333/iejme/12270