pp. 2373-2401 | Article Number: iejme.2016.203
Published Online: September 03, 2016
Article Views: 422 | Article Download: 433
Existing research have found that students’ creation and evaluation of mathematical proofs was inconsistent across content areas. Investigation into an explanation of the phenomena requires an analysis of students’ thinking processes when they conduct an evaluation of mathematical arguments. This study is conceptualized to contribute to this investigation. The analysis investigated the aspects and features of arguments that impacted students’ evaluation of the arguments. Eight 8th grade students participated in the interviews where they were asked to explain their rationale in evaluating arguments that justify conjectures from multiple strands of school mathematics. Interview data was coded using the Classification of Mathematical Argument (CMA) framework to identify the aspects and features of arguments that impacted students’ evaluation of the arguments. A detailed analysis of each subject’s interview response documented the complexity of each individual’s rationale and offered descriptions of the various differences among individuals. Despite such individual differences, the study also revealed a common theme among the subjects in their reasoning, i.e. the accepted statements in an argument, instead of its mode of presentation or mode of argumentation, had the largest impact on the subjects’ evaluation of an argument
Keywords: Argument evaluation, proof and reasoning, student thinking
Balacheff, N. (1988). Aspects of proof in pupils’ practice of school mathematics. In D. Pimm (Ed.), Mathematics, teachers and children (pp. 216-235). London, UK: Holdder & Stoughton.
Balacheff, N. (1991). The benefit and limits of social interaction: The case of mathematical proof. In A. Bishop, Mellin-Olsen, E. & van Dormolen, J. (Eds.), Mathematical knowledge: Its growth through teaching (pp. 175-192). Dordrecht, Netherlands: Kluwer.
Bell, A.W. (1976). A study of pupils’ proof-explanations in Mathematical situations. Educational Studies in Mathematics, 7(1-2), 23-40.
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.
Chazan, D., & Lueke, H. M. (2009). Relationships between disciplinary knowledge and school mathematics: Implications for understanding the place of reasoning and proof in school mathematics. In D. A. Stylianou, Blanton, M. L., & Knuth E. J. (Eds.), Teaching and Learning Proof Across the Grades: AK-16 Perspective (pp. 21-39). New York: Routledge.
Council of Chief State School Officers (2010). Common Core State Standards (Mathematics). National Governors Association Center for Best Practices. Washington, D. C.
de Villiers, M. (2003). Rethinking proof with the Geometer’s Sketchpad. Emeryville, CA: Key Curriculum Press.
de Villiers, M. (1998). An alternative approach to proof in dynamic geometry. In R. Lehrer & D. Chazan (Eds.), New directions in teaching and learning geometry (pp. 369-415). Lawrence Erlbaum.
de Villiers, M. (1990). The role and function of proof in mathematics. Pythagoras, 24, 17–24.
Dunning, D., Heath, C., & Suls, J. M. (2005). Flawed self-assessment: Implications for education, and the workplace. Psychological Science in the Public Interest, 5, 69-106.
Dreyfus, T. (1999). Why Johnny can’t prove. Educational Studies in Mathematics, 38(1/3), 85-109.
Freudenthal, H. (1971). Geometry between the devil and the deep sea. Educational Studies in Mathematics, 3(3-4), 413-435.
Freudenthal, H. (1973). Mathematics as an educational task. Dordrecht: Reidel.
González, G., & Herbst, P. (2006). Competing arguments for the geometry course: Why were American high school students to study geometry in the twentieth century? International Journal for the History of Mathematics Education, 1(1), 7-33.
Hanna, G. (2000a). A critical examination of three factors in the decline of proof. Interchange, 31(1), 21-33.
Hanna, G. (2000b). Proof, explanation and exploration: An overview. Educational Studies in Mathematics, 44, 5-23.
Hanna, G., & Jahnke, H. N. (Eds.). (1993). Aspects of proof [Special issue]. Educational Studies in Mathematics, 24(4).
Harel, G., & Sowder, L. (1998). Students’ proof schemes: Results from exploratory studies. In A. H. Schoenfeld, Kaput, J., & Dubinsky, E. (Eds.), Research in Collegiate Mathematics Education III (pp. 234-283). American Mathematical Society.
Harel, G., & Sowder, L. (2007). Toward comprehensive perspectives on the learning and teaching of proof. In F. Lester (Ed.), Second handbook of research in mathematics teaching and learning (pp. 805-842). Charlotte, NC: Information Age Publishing
Healy, L., & Hoyles, C. (2000). A study of proof conceptions in algebra. Journal for Research in Mathematics Education, 31(4), 396–428.
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.
Hersh, R. (2009). What I would like my students to already know about proof. In D. A. Stylianou, Blanton, M. L., & Knuth E. J. (Eds.), Teaching and Learning Proof Across the Grades: AK-16 Perspective (pp. 17-20). New York: Routledge.
Hoyles, C. (1997). The curricular shaping of students’ approaches to proof. For the Learning of Mathematics, 17(1),7-16.
Lakatos, I. (1976). Proofs and refutations: The logic of mathematical discovery. Cambridge, UK: Cambridge University Press.
Knuth, E. J., Choppin, J. M., & Bieda, K. N. (2009). Middle school students’ production of mathematical justifications. In D. A. Stylianou, Blanton, M. L., & Knuth, E. J. (Eds.), Teaching and Learning Proof Across the Grades: AK-16 Perspective (pp. 153-170). New York: Routledge.
Kuchemann, D., & Hoyles, C. (2009). From empirical to structural reasoning in mathematics: Tracking changes over time. In D. A. Stylianou, Blanton, M. L., & Knuth E. J. (Eds.), Teaching and Learning Proof Across the Grades: AK-16 Perspective (pp. 171-190). New York: Routledge.
Mejia-Ramos, J. P., & Inglis, M. (2009). Argumentative and proving activities in mathematics education research. In F.-L. Lin, F.-J. Hsieh, G. Hanna, and M. de Villiers (Eds.), Proceedings of the ICMI Study 19 conference: Proof and Proving in Mathematics Education (Vol. 2, pp. 88-93), Taipei, Taiwan.
Mejia-Ramos, J. P., Fuller, E., Weber, E.; Rhoads, K. & Samkoff, A. (2012). An assessment model for proof comprehension in undergraduate mathematics. Educational Studies in Mathematics, 79(1), 3-18.
McConaughy, S. H., & Achenbach, T. M. (2001). Manual for the Semistructured Clinical Interview for Children and Adolescents (2nd ed.). Burlington, VT: University of Vermont, Research Center for Children, Youth, and Families.
National Assessment of Educational Progress (NAEP). (2010). The Nation’s Report Card: Grade 12 Reading and Mathematics 2009 National and Pilot State Results. U.S. Department of Education, Institute of Education Sciences, National Center for Education Statistics. http://nces.ed.gov/nationsreportcard/pdf/main2009/2011455.pdf
National Council of Teachers of Mathematics (2000). Principles and standards for school mathematics. Reston, VA: NCTM.
Piaget, J. (1985). The Equilibration of Cognitive Structures: The Central Problem of Intellectual Development (T. Brown & K. J. Thampy, Trans.). Chicago, IL: University of Chicago Press.
Reid, D. A. (2011). Understanding proof and transforming teaching. In L. R. Wiest, & Lamberg, T. (Eds.), Proceedings of the 33rd Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 15-30). Reno, NV: University of Nevada, Reno.
Schoenfeld, A. H. (1988). When good teaching leads to bad results: The disasters of “well-taught” mathematics courses. Educational Psychologist, 23(2), 145–166.
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.
Simon, M. A. (1996). Beyond inductive and deductive reasoning: the search for a sense of knowing. Educational Studies in Mathematics, 90(2), 197-210.
Stylianides, G. J., & Stylianides, A. J. (2008a). Proof in school mathematics: Insights from psychological research into students’ ability for deductive reasoning. Mathematical thinking and learning, 10(2), 103-133.
Stylianides, G. J., & Stylianides, A. J. (2008b). Enhancing undergraduate students’ understanding of proof. In Electronic Proceedings of the 11th Conference on Research in Undergraduate Mathematics Education (http://sigmaa.maa.org/rume/crume2008/Proceedings/Stylianides&Stylianides_LONG(21).pdf), San Diego, CA.
Tall, D. et al. (2012). Cognitive development of proof. In G. Hanna, & de Villiers, M. (Eds.), Proof and proving in mathematics education (pp. 13-50). New York: Springer.
Van Hiele, P. M. (1980). Levels of thinking: How to meet them and how to avoid them. Paper presented at the Research Presession to the annual meeting of the National Council of Teachers of Mathematics. Seattle, WA.
van Hiele, P.M. (1986). Structure and insight: A theory of mathematics education. New York: Academic Press.
von Glasersfeld, E. (1994).A radical constructivist view of basic mathematical concepts. In Ernest, Paul (Ed.), Constructing mathematical knowledge: Epistemology and mathematics education (Studies in mathematics education Vol. 4, pp.5-7). Abingdon, Oxon: Routledge.
Usiskin, Z. (1982). Van Hiele Levels and achievement in secondary school geometry. The University of Chicago. Chicago, IL
Waring, S. (2000). Can you prove it? Developing concepts of proof in primary and secondary schools. Leicester, UK: The Mathematical Association.
Weber, K. (2001). Student difficulty in constructing proofs: The need for strategic knowledge. Educational Studies in Mathematics, 48(1), 101-119.
Yang, K., & Lin, F. (2008). A model of reading comprehension of geometry proof. Educational Studies in Mathematics, 67(1), 59-76.