Browse > Article
http://dx.doi.org/10.7468/jksmed.2022.25.1.65

Secondary Teachers' Views about Proof and Judgements on Mathematical Arguments  

Kim, Hangil (STEM Education, University of Texas at Austin)
Publication Information
Research in Mathematical Education / v.25, no.1, 2022 , pp. 65-89 More about this Journal
Abstract
Despite its recognition in the field of mathematics education and mathematics, students' understanding about proof and performance on proof tasks have been far from promising. Research has documented that teachers tend to accept empirical arguments as proofs. In this study, an online survey was administered to examine how Korean secondary mathematic teachers make judgements on mathematical arguments varied along representations. The results indicate that, when asked to judge how convincing to their students the given arguments would be, the teachers tended to consider how likely students understand the given arguments and this surfaces as a controversial matter with the algebraic argument being both most and least convincing for their students. The teachers' judgements on the algebraic argument were shown to have statistically significant difference with respect to convincingness to them, convincingness to their students, and validity as mathematical proof.
Keywords
Proof; Reasoning; Secondary Teacher; Teacher Knowledge;
Citations & Related Records
Times Cited By KSCI : 1  (Citation Analysis)
연도 인용수 순위
1 Alcock, L., & Inglis, M. (2008). Doctoral students' use of examples in evaluating and proving conjectures. Educational Studies in Mathematics, 69(2), 111-129.   DOI
2 Australian Curriculum, Assessment and Reporting Authority. (2015). Australian Curriculum: Mathematics. Retrieved from https://australiancurriculum.edu.au/f10-curriculum/mathematics/
3 Balacheff, N. (1988). Aspects of proof in pupils' practice of school mathematics. In D. En Pimm (Ed.), Mathematics, teachers and children (pp. 216-235). London, England: Hodder & Stoughton.
4 Ball, D. L., Hoyles, C., Jahnke, H. N., & Movshovitz-Hadar, N. (2002). The teaching of proof. In L. I. Tatsien (Ed.), Proceedings of the international congress of mathematicians (Vol. III, pp. 907-920). Beijing, China: Higher Education Press.
5 Almeida, D. (2000). A survey of mathematics undergraduates' interaction with proof: some implications for mathematics education. International Journal of Mathematical Education in Science and Technology, 31(6), 869-890.   DOI
6 Balacheff, N. (2002). A researcher epistemology: A deadlock for educational research on proof. In Proceedings of the 2002 international conference on mathematics: Understanding proving and proving to understand (pp. 23-44). Department of Mathematics: National Taiwan Normal University.
7 Lockwood, E., Ellis, A., & Knuth, E. (2013). Mathematicians' example-related activity when proving conjectures. In Brown, S., Karakok, G., Roh, K., & Oehrtman, M. (Eds.). Proceedings for the Sixteenth Annual Conference on Research on Undergraduate Mathematics Education (Vol.1, pp. 16-28). Denver, CO: Northern Colorado University.
8 Knuth, E., Kim, H., Zaslavsky, O., Vinsonhaler, R., Gaddis, D., & Fernandez, L. (2020). Teachers' views about the role of examples in proving-related activities. Journal of Educational Research in Mathematics, 30(3), 115-134.   DOI
9 Knuth, E., Zaslavsky, O., & Kim, H. (2022). Argumentation, justification, and proof in middle grades: A rose by any other name. In Bieda, K., Conner, A., Kosko, K., & Staples, M. (Eds.). Conceptions and Consequences of Mathematical Argumentation, Justification, and Proof (pp. 129-136). Springer.
10 Kwon, O., Kim, A., & Cho, H. (2012). An analysis on the curriculum and teaching methods of Korean mathematics education departments. Mathematics Education, 51(3), 281-300.   DOI
11 Mason, J. & Pimm, D. (1984). Generic examples: Seeing the general in the particular. Educational Studies in Mathematics, 15(3), 277-289.   DOI
12 Ministry of Education. (2015). Revised Korean mathematics curriculum. Seoul, Korea: Ministry of Education
13 National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics.
14 Kim, H. (2020). Embracing proving into everyday lesson by problem posing. In Sacristan, A. I., Cortes-Zavala, J. C. & Ruiz-Arias, P. M. (Eds.). Mathematics education across cultures: Proceedings of the 42nd meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, Mexico.
15 Basturk, S. (2010). First-year secondary school mathematics students' conceptions of mathematical proofs and proving. Educational Studies, 36(3), 283-298.   DOI
16 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.   DOI
17 Hoyles, C. (1997). The Curricular Shaping of Students' Approaches to Proof. For the Learning of Mathematics, 17(1), 7-16.
18 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.   DOI
19 Lynch, A. G., & Lockwood, E. (2019). A comparison between mathematicians' and students' use of examples for conjecturing and proving. The Journal of Mathematical Behavior, 53, 323-338.   DOI
20 Hanna, G., & Jahnke, H. N. (1999). Using arguments from physics to promote understanding of mathematical proofs, In O. Zaslavsky (ed.), Proceedings of the twenty-third conference of the international group for the Psychology of Mathematics Education (Vol. 3, pp. 73-80). Haifa, Israel: PME.
21 Schoenfeld, A. (1986). On having and using geometric knowledge. In J. Hiebert (Ed.) Conceptual and procedural knowledge: The case of mathematics. (pp. 225-263). Hillsdale, NJ: Lawrence Erlbaum Associates.
22 Polya, G. (1954). Mathematics and plausible reasoning: Induction and analogy in mathematics (Vol. I). Princeton, NJ: Princeton University Press.
23 Reid, D. A., & Knipping, C. (2010). Proof in mathematics education: Research, learning, and teaching. The Netherlands: Sense Publishers.
24 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.   DOI
25 Schoenfeld, A. H. (1994). What do we know about mathematics curricula? The Journal of Mathematical Behavior, 13(1), 55-80.   DOI
26 Selden, A., & Selden, J. (2003). Validations of proofs considered as texts: Can undergraduates tell whether an argument proves a theorem? Journal for Research in Mathematics Education, 34(1), 4-36.   DOI
27 Council of Chief State School Officers. (2010). Common core state standards for mathematics. Washington DC: Council of Chief State School Officers.
28 Leron, U., & Zaslasvsky, O. (2013). Generic Proving: Reflections on scope and method. For the Learning of Mathematics, 33(3), 24-30.
29 Bell, A. (1976). A study of pupils' proof-explanations in mathematical situations, Educational Studies in Mathematics, 7, 23-40.   DOI
30 Blanton, M. L., & Stylianou, D. A. (2014). Understanding the role of transactive reasoning in classroom discourse as students learn to construct proofs. The Journal of Mathematical Behavior, 34, 76-98.   DOI
31 de Villiers, M. D. (1990). The role and function of proof in mathematics. Pythagoras, 24, 17-24.
32 Ellis, A. B., Lockwood, E., Dogan, M., Williams, C., & Knuth, E. J. (2013). Choosing And Using Examples: How Example Activity Can Support Proof Insight. In A. M. Lindmeier & A. Heinze (Eds.), Proceedings of the 37th conference of the international 2-265 group for the psychology of mathematics education (Vol. 2, pp. 265-272). Kiel, Germany: PME.
33 Simon, M. A., & Blume, G. W. (1996). Justification in the mathematics classroom: A study of prospective elementary teachers. The Journal of Mathematical Behavior, 15(1), 3-31.   DOI
34 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.   DOI
35 Stylianides, G. J., Stylianides, A. J., & Weber, K. (2016). Research on the teaching and learning of proof: Taking stock and moving forward. In Cai, J. (Ed.). Compendium for Research in Mathematics Education (pp. 237-266). Reston, VA: NCTM.
36 Healy, L., & Hoyles, C. (2000). A study of proof conceptions in algebra. Journal for Research in Mathematics Education, 31(4), 396-428.   DOI
37 Knuth, E. J. (2002b). Secondary school mathematics teachers' conceptions of proof. Journal for Research in Mathematics Education, 33(5), 379-405.   DOI
38 Knuth, E., Zaslavsky, O., & Ellis, A. (2019). The role and use of examples in learning to prove. The Journal of Mathematical Behavior, 53, 256-262.   DOI
39 Stylianides, G. J. (2007). Investigating the Guidance Offered to Teachers in Curriculum Materials: The Case of Proof in Mathematics. International Journal of Science and Mathematics Education, 6(1), 191-215.   DOI
40 Bieda, K. N., Ji, X., Drwencke, J., & Picard, A. (2014). Reasoning-and-proving opportunities in elementary mathematics textbooks. International Journal of Educational Research, 64, 71-80.   DOI
41 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.   DOI
42 Coe, R. & Ruthven, K. (1994). Proof practices and construct of advanced mathematics students. British Educational Research Journal, 20, 41-53.   DOI
43 Creswell, J. W., & Poth, C. N. (2016). Qualitative inquiry and research design: Choosing among five approaches. Thousand Oaks, CA: Sage publications.
44 Department for Education. (2014). Mathematics programmes of study: Key stage 4 (National curriculum in England). Retrieved from https://www.gov.uk/government/uploads
45 de Villiers, M. (1999). Rethinking proof with the Geometer's Sketchpad. Emeryville, CA: Key Curriculum Press.
46 Dickerson, D. S., & Doerr, H. M. (2014). High school mathematics teachers' perspectives on the purposes of mathematical proof in school mathematics. Mathematics Education Research Journal, 26(4), 711-733.   DOI
47 Epstein, D., & Levy, S. (1995). Experimentation and proof in mathematics. Notices of the AMS, 42(6), 670-674.
48 Hanna, G. (2000). Proof, explanation and exploration: an overview. Educational Studies in Mathematics, 44(1/2), 5-23.   DOI
49 Hersh, R. (1993). Proving is convincing and explaining. Educational Studies in Mathematics, 24, 389-399.   DOI
50 Harel, G., & Sowder, L. (1998). Students' Proof Schemes: Results from Exploratory Studies. Research in Collegiate Mathematics Education III, 7, 234-282.
51 Zack, V. (1999). Everyday and mathematical language in children's argumentation about proof, Educational Review, 51(2), 129-146.   DOI
52 Tall, D. (1999). The cognitive development of proof: Is mathematical proof for all or some? Developments in School Mathematics Education around the World, 4, 117-136.
53 Thompson, D. R., Senk, S. L., & Johnson, G. J. (2012). Opportunities to learn reasoning and proof in high school mathematics textbooks. Journal for Research in Mathematics Education, 43(3), 253-295.   DOI
54 Walther, G. (2006). Illuminating examples - an aspect of simplification in the teaching of mathematics. International Journal of Mathematical Education in Science and Technology, 17(3), 263-273.   DOI
55 Zaslavsky, O. (2018). Genericity, conviction, and conventions: examples that prove and examples that don't prove. In A. Stylianides, & G. Harel (Eds.). Advances in mathematics education research on proof and proving (pp. 283-298). Cham, Switzerland: Springer International Publishing AG.
56 Porteous, K. (1990). What do children really believe? Educational Studies in Mathematics, 21, 589-598.   DOI
57 Lakatos, I. (1976). Proofs and refutations: The logic of mathematical discovery. Cambridge, UK: Cambridge University Press.
58 Lockwood, E., Ellis, A., Knuth, E., Dogan, M. F., & Williams, C. (2013). Strategically chosen examples leading to proof insight: A case study of a mathematician's proving process. In Martinez, M. & Castro Superfne, A. (Eds.). Proceedings of the 35th annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 245-252). Chicago, IL: University of Illinois at Chicago.
59 Ministry of Education. (2011). Revised Korean mathematics curriculum. Seoul, Korea: Ministry of Education.
60 Rowland, T. (2002). Generic Proofs in Number Theory. Learning and Teaching Number Theory: Research in Cognition and Instruction, 157-183.
61 Schoenfeld, A. (1988). When good teaching leads to bad results: The disasters of "well-taught" mathematics courses. Educational Psychologist, 23(2), 145-166.   DOI
62 Senk, S. (1985). How well do students write geometry proofs? The Mathematics Teacher, 78(6), 448-456.   DOI
63 Sowder, L., & Harel, G. (1998). Types of students' justifications. Mathematics Teacher, 91, 670-675.   DOI
64 Sowder, L., & Harel, G. (2003). Case studies of mathematics majors' proof understanding, production, and appreciation. Canadian Journal of Math, Science & Technology Education, 3(2), 251-267.   DOI
65 Knuth, E. J. (2002a). Teachers' Conceptions of Proof in the Context of Secondary School Mathematics. Journal of Mathematics Teacher Education, 5, 61-88.   DOI
66 Stylianides, A. J. (2007). Proof and proving in school mathematics. Journal for Research in Mathematics Education, 38(3), 289-321.
67 Stylianides, A. J., & Stylianides, G. J. (2009). Proof constructions and evaluations. Educational Studies in Mathematics, 72(2), 237-253.   DOI
68 Kim, H. (2021). Problem posing in the instruction of proof: Bridging everyday lesson and proof. Research in Mathematical Education, 24(3), 255-278.   DOI
69 Knuth, E. J., 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). New York, NY: Routledge.