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The meaning of mathematical proof for prospective middle school mathematics teachers
Date
2017-08-25
Author
Demiray, Esra
Işıksal Bostan, Mine
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Contribution To define mathematical proof is difficult since it may have different roles depending on the author, the community or the audience, and the methods of the proof (Cadwallader Olsker, 2011; Staples, Bartlo, & Thanheiser, 2012). One of the most simple and explicit definitions of mathematical proof was stated by Bruyn, Sidoli and Lomas (2004) as “a mathematical proof, by definition, can take a set of explicit givens (such as axioms, accepted principles or previously proven results), and use them, applying the principles of logic, to create a valid deductive argument” (p.82). Based on the related literature review, it can be seen that definitions of proof have different school of thoughts (VanSpronsen, 2008). For example, some definitions of proof focused on the characteristic of being a logical and deductive argument or the structure of proof (e.g., Griffiths, 2000; Hanna & Barbeau, 2002; Weber, 2005) and some of them mainly focused on the functions/roles of proof (e.g., Goetting, 1995; Hanna, 1989; Hersh, 1993). Yopp (2011) implied that there is not a clear and strict distinction between the terms role, purpose and function of mathematical proof and these terms were used interchangeably in the studies. While the roles of proof were examined in some studies in detail by considering many applications, there are also studies which examine the roles of proof under more general titles. According to Hersh (1993), there are two roles of proof which are convincing and explaining. Similarly, Volmink (1990) accepted conviction as the most important role of proof. Schoenfeld (1994) expressed the roles of proof as communicating ideas with others, thinking, exploring and understanding mathematical arguments. However, proof might involve some additional roles depending on the educational setting (Nordby, 2013). Students should have experience with activities involving justification, proof and reasoning processes in mathematics classrooms (Bieda, 2010; Bostic, 2016). To be able to integrate these processes into mathematics classrooms effectively, mathematics teachers should also have necessary knowledge and experience regarding proof and reasoning. In addition to the effects of mathematics teachers’ knowledge of proof on students’ understanding of proof, mathematics teachers’ proof definitions also affect students’ views, ideas, attitudes and perceptions of proof. In this respect, how mathematics teachers describe mathematical proof is an important issue. To state differently, since prospective middle school mathematics teachers are future teachers, their experiences, perceptions and ideas regarding proof might have effect on framing their future instructions (Bostic, 2016). Thus, one of the aims of this research is to investigate prospective middle school mathematics teachers’ definitions of mathematical proof. Moreover, students should be able to choose appropriate proof method in the proving process of the given statement (Rota, 1997) however they may have difficulties in naming the proof methods in some cases even though they know how to apply the method (Türker, Alkaş, Aylar, Güler, & İspir, 2010). Considering this issue, proof methods that prospective middle school teachers know while proving the given mathematical statements were also investigated in the present study. In this respect, this study was guided by two main research questions: 1. How do prospective middle school mathematics teachers define mathematical proof? 2. Which proof methods do prospective middle school mathematics teachers know while proving the given mathematical statement? Method In this study, a survey research design was employed since information is gathered from a sample instead of each member of the population in order to describe some characteristics of the population via asking questions (Fraenkel & Wallen, 2005). Moreover, convenience sampling method was used to determine the sample. The data was collected from 98 prospective teachers enrolled in the Elementary Mathematics Teacher Education program in a state university in Ankara, Turkey. There were 24 male (24.5%) and 74 female participants (75.5%) in this study. In terms of year level, 14 of participants (14.3%) were freshmen, 18 of them (18.4%) were sophomores, 39 of them (39.8%) were juniors and 27 of them (27.6%) were juniors. To answer the research questions, prospective middle school mathematics teachers were given a questionnaire and expected to answer two questions. In more detail, Question 1 asks students to define and explain the meaning of the concept of ‘mathematical proof” and Question 2 asks students to state proof methods that they know while proving the given mathematical statement. Pilot study was conducted with prospective middle school mathematics teachers in a different state university in Turkey. The analysis of prospective middle school mathematics teachers’ answers in the pilot study was also considered as a guide for the content analysis of the actual data. To analyze the data obtained from the participants, descriptive analysis and content analysis were applied. Based on the main phases of the content analysis which are coding the data, detecting themes, organizing codes and themes, and interpreting the findings (Yıldırım & Şimşek, 2008), prospective middle school mathematics teachers’ answers were coded and classified into relevant themes. Then, the codes and the themes obtained from the data were organized with frequencies and the findings were interpreted in accordance with the research questions. In this study, the list for roles of proof stated by VanSpronsen (2008) which was prepared as a summary of the related literature was used as framework in data analysis. This list involves seven roles of proof which are verification or justification (verifying the truth of a statement), explanation or illumination (asserting why a statement is true), conviction (removing doubts about statements), systematization (organizing a deductive system of axioms), communication (transmitting mathematical knowledge to others), discovery or construction (inventing new theorems/results/formulas), and enjoyment (dealing with an intellectual challenge properly). Expected Outcomes According to the results, 69 prospective teachers (70.4%) focused on the structure of proof while 82 of them (83.7%) wrote about the roles of proof in their definitions in Question 1. The definitions involving the structure of proof were categorized under four themes. In more detail, 53 participants (54.0%) defined proof by stating the application of mathematical facts/rules/theorems/knowledge/operations, 9 participants (9.2%) defined proof by mentioning particular proof methods, 8 participants (8.1%) defined proof by stating the use of assumptions explicitly, and 3 participants (3.1%) defined proof by writing some logical rules used in proof process. The definitions involving the roles of proof were coded based on the categorization of VanSpronsen (2008). In more detail, 56 participants (57.1%) defined proof by mentioning verification role; 26 participants (26.6%) defined by stating explanation role; 3 participants (3.1%) defined by stating discovery role, and 1 participant (1.0%) defined by describing communication role. This result is consistent with the study of Knuth (2002) which asserted that in-service secondary school mathematics teachers mentioned the primary role of proof as verification. To answer the second research question, proof methods that students emphasized were analyzed. According to the findings, 83 participants (84.7%) mentioned about proof by contradiction, 81 of them (82.7%) mentioned about mathematical induction, 47 of them (48.0%) stated counterexample, 40 of them (40.8%) stated direct proof, 18 of them (18.4%) stated proof by contrapositive, and 4 of them (4.1%) stated indirect proof. As seen, the percentages of participants who mentioned mathematical induction and proof by contradiction were near and considerably high. This result is consistent with study of Türker et al.(2010) who stated that induction and contradiction were common answers among prospective teachers. To analyze the answers of participants in-depth, the follow-up interviews might be conducted in the further studies.
URI
https://eera-ecer.de/ecer-programmes/conference/22/contribution/41789/
https://hdl.handle.net/11511/71274
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Department of Secondary Science and Mathematics Education, Conference / Seminar
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E. Demiray and M. Işıksal Bostan, “The meaning of mathematical proof for prospective middle school mathematics teachers,” presented at the The European Conference on Educational Research (ECER) (2017), Kopenhag, Danimarka, 2017, Accessed: 00, 2021. [Online]. Available: https://eera-ecer.de/ecer-programmes/conference/22/contribution/41789/.