ERG SES C 15, Pre-Service Teachers and Education
Many consider proof to be central to the discipline of mathematics; yet surprisingly, the role of proof in school mathematics has been peripheral at best (Harel & Sowder, 1998; Knuth, 2002a). Proof traditionally has been expected to play a role in high school geometry or advanced level math courses. Current reform efforts are calling for a transition in the role of proof to elevate proof beyond a topic of study in advanced mathematics courses to a tool for studying and learning mathematics at all levels (NGA/CCSSO 2010; NCTM, 2000).
Teachers' success in responding to this call, however, depends largely on their own conceptions of proof. In order to create such an environment for students, teachers must themselves have a deep understanding of proof. However, It was evident in many studies that implementing “proof for all” may be difficult for teachers; teachers viewed proof as appropriate for the mathematics majors only and as a topic of study rather than as a tool for communicating and studying mathematics (Knuth, 2002b).
Thus, one question remains to be asked: Are pre-service teachers who are required to successfully implement these current recommendations ready for the task? More specifically, this study investigated pre-service elementary teachers’ (PSTs’), who are a few steps away from being a classroom teacher, conceptions of proof which consists of their ability to construct as well as evaluate mathematical arguments and how their conceptions influenced their instructional decisions. In order to investigate whether PSTs are ready to respond to current recommendations, following two research questions guided the study:
- What are pre-service elementary teachers’ conceptions and/or misconceptions of proof and counterexamples in mathematics classrooms?
- Do pre-service elementary teachers’ conceptions and/or misconceptions of proof and counterexamples influence their teaching practices? If so, how?
In order to investigate PSTs’ conception of proof and counterexamples, the literature about proof schemes is reviewed. This study adopted the taxonomy of proof schemes, which consists of external, empirical, and analytical, proposed by Harel and Sowder (1998). However, in order to capture a broad spectrum of proof schemes, Level 0: No Justification in which students do not even need to provide a justification, Level 2: Naïve Reasoning in which students fail to produce a deductive argument even if they start with some deductions, and Level 4: Generic Example where students use a particular example to express their deductive reasoning were added to Harel and Sowder’s taxanomy. This taxanomy served as a theoretical framework to design the instruments as well to analyze the data collected in this study.
Participants were a sample of PSTs who enrolled in one section of geometry and measurement courses and one section of mathematics methods courses at a large Mid-western university. Data were collected in two phases. Two types of interviews—task-based and scenario-based interviews—which were conducted at the beginning and near the end of the semester, served as the main data. Secondary data sources including classroom observations and class work were also collected. Data were analyzed qualitatively using grounded theory techniques (Strauss& Corbin, 1990).
The results of this study exhibited mixed pictures of what constitutes a proof in the eyes of the PSTs who participated in this study. This study also showed that PSTs’ conceptions and/or misconceptions of proof and counterexamples played an important role in their instructional decisions. This study is in alignment with other studies such as Martin & Harel, 1989; Simon & Blume, 1996; Stylianides & Stylianides, 2009 to conclude that pre-service teachers should be well equipped to be able to convey all aspects of proof as suggested by current reform acts.
Balacheff, N. (1988). Aspects of proof in pupils' practice of school mathematics. In D. Pimm (Ed.), Mathematics, teachers, and children (pp.216-238). London: Hodder & Stoughton. Harel, G. (2007). Students’ proof schemes revisited: Historical and epistemological considerations. In P. Boera (Ed.), Theorems in school: From history, epistemology and cognition to classroom practice (pp. 65-78). Rotterdam: Sense Publishers. Harel, G., & Sowder, L. (1998). Students’ proof schemes: Results from exploratory studies. In A. Schoenfeld, J. Kaput, &E. Dubiensky (Eds.), Research in collegiate mathematics education III (pp. 234-283). Providence, R.I.: American Mathematical Society. Knuth, E. J. (2002a). Secondary school mathematics teachers’ conceptions of proof. Journal for Research in Mathematics Education, 33(5), 379-405. Knuth, E. J. (2002b). Proof as a tool for learning mathematics. Mathematics Teacher, 95(7), 486-490. Martin, W. G. & Harel, G. (1989). Proof frames of preservice elementary teachers. Journal for Research in Mathematics Education. 20 (1), 41-51. National Council of Teachers of Mathematics (NCTM): 2000, Principles and Standards for School Mathematics, Commission on Standards for School Mathematics, Reston, VA. National Governors Association Center for Best Practices & Council of Chief State School Officers (2010). Common Core State Standards for Mathematics, Washington, DC: Authors. Peressini, D., Borko, H., Romagnano, L., Knuth, E., & Willis-Yorker, C.(2004). A conceptual framework for learning to teach secondarymathematics: A situative perspective. Educational Studies in Mathematics, 56(1), 67-96. Quinn, A.L. (2009). Count on number theory to inspire proof. Mathematics Teachers, 103 (4), 298-304. Simon, M. A. & Blume, G. W. (1996). Justification in the mathematics classroom: A study of prospective elementary teachers. Journal of Mathematical Behavior, 15, 3-31. Strauss, A. & Corbin, J. (1990). Basics of qualitative research: Grounded theory procedures and techniques. Newbury Park, CA: Sage. Stylianides, A.J. & Stlyianides, G. J. (2009). Proof constructions and evaluations. Educational Studies in Mathematics, 72, 237-253.
- Search for keywords and phrases in "Text Search"
- Restrict in which part of the abstracts to search in "Where to search"
- Search for authors and in the respective field.
- For planning your conference attendance you may want to use the conference app, which will be issued some weeks before the conference
- If you are a session chair, best look up your chairing duties in the conference system (Conftool) or the app.