Are Pre-service Elementary Teachers Ready to Transition Proof From Being a Topic to a Tool of Learning?

Author(s):

Zulfiye Zeybek(presenting / submitting)

Conference:

ECER 2015

Network:

Emerging Researchers' Group (for presentation at Emerging Researchers' Conference)

Format:

Paper

Session Information

ERG SES C 15, Pre-Service Teachers and Education

Paper Session

Time:

2015-09-07

11:00-12:30

Room:

395. [Main]

Chair:

Jani Petri Ursin

Contribution

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.

Method

Participants To select participants representing a broad spectrum in terms of knowledge about proof, a proof questionnaire was developed and administered to all students in one section of geometry and measurement course and one section of mathematics methods course at the beginning of the semester. After administering the questionnaire, twelve PSTs who demonstrated wide range of proof schemes were selected. Data Collection The data was gathered in two distinct stages and the main sources of data were participants’ semi-structured interviews, classroom observations, and classroom work. Individual interviews took place at the beginning and near the end of the semester. All twelve participants were interviewed individually, and interviews lasted approximately 60 minutes and were audio-recorded. Task-based interviews focused on PSTs’ conceptions of proof, including their way of producing proofs/counterexamples, as well as validations of different types of arguments ranging from empirical to formal. Scenario-based interviews, on the other hand, focused on the participants’ usage of their conceptions that emerged from the analysis of task-based interviews. Theoretical Framework The literature in order to develop a taxonomy for teachers’ conception of proof was reviewed. The taxonomy of proof schemes, external, empirical, and analytical, proposed by Harel and Sowder (1998) and and later revisited by Harel (2007), is a fundamental framework for research on students’ conceptions of proof. It encapsulates the major categories included in other taxonomies and proposes further sub-categories. However, it is evidenced in the literature that some students may not even need to provide a justification, they may fail to produce a deductive argument even if they start with some deductions (Quinn, 2009), or they may use a particular example –generic example--to express their deductive reasoning (Balacheff, 1988; Simon & Blume, 1996). Since these students do not hold external, empirical, nor fully developed analytical proof schemes, it may be hard to classify these students’ proof schemes using Harel and Sowder’s taxonomy. I propose to add those levels to Harel and Sowder’s taxonomy in order to account for a broader spectrum of proof schemes. Data Analysis Each interview was transcribed, and the interview transcripts and participants’ written responses were carefully read. An inductive, grounded-theory approach (Strauss& Corbin, 1990) guided the analysis of data in this study. A team of experts, which consisted of three graduate students, was asked to code a sample of the interview transcripts. The coded samples were then compared and disagreements were discussed until the problems were resolved.

Expected Outcomes

Participants in this study demonstrated a wide range of reasoning abilities regarding mathematical proofs and counterexamples. It was clear that the PSTs had previous experience with constructing proofs and they were using those experiences to inform their judgments about what constituted a proof. However, despite their experience, it was also clear that the majority of the pre-service elementary teachers who participated in this study had a limited understanding of what constitutes proof in mathematics classrooms as well as demonstrated various misconceptions as described above. The fact that pre-service elementary teachers experience difficulty with proof and counterexamples in this study is not surprising considering the literature on pre-service teachers’ capabilities of validating arguments and constructing proofs and counterexamples (Martin & Harel, 1989; Simon & Blume, 1996). This study highlighted the importance of teachers’ conceptions in order to be able to engage students in tasks that require proofs or refutations. As Pressini et al. (2004) state “the extent to which mathematical ideas such as proof appear in classroom discourse will be influenced by both the teacher’s choice of task and the questions she asks during class, which are, in turn, influenced by the teacher’s knowledge of proof. In the scenario-based interviews, it was evident that the conceptions and/or misconceptions played an important role when the pre-service teachers evaluated the classroom scenarios. PSTs’ decisions as to whether an argument was a proof or a counterexample was appropriate to refute the statement were influenced heavily by their own conceptions of what constitutes proof and counterexamples. This study along with other studies documented the limited understanding of proof among PSTs and how their conceptions of proof played a key role in their instructional decisions. All these results highlight the need for empowering PSTs’ understanding of proof in order to help them meet the recent recommendations.

References

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.

Author Information

Zulfiye Zeybek (presenting / submitting)

Gazi Osman Pasa University

Mathematics education

Alasehir

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