The emphasis of undergraduate mathematics content is centered around abstract reasoning and proof, whereas students' pre-college mathematical experiences typically give them limited exposure to these concepts. Not surprisingly, many students struggle to make the transition to undergraduate mathematics in their first course on mathematical proof, known as a bridge course. In the process of this study, eight students of varied backgrounds were interviewed before during and after their bridge course at BYU. By combining the proof scheme frameworks of Harel and Sowder (1998) and Ko and Knuth (2009), I analyzed and categorized students’ initial proof schemes, observed their development throughout the semester, and their proof schemes upon completing the bridge course. It was found that the proof schemes used by the students improved only in avoiding empirical proofs after the initial interviews. Several instances of ritual proof schemes used to generate adequate proofs were found, calling into question the goals of the bridge course. Additionally, it was found that students’ proof understanding, production, and appreciation may not necessarily coincide with one another, calling into question this hypothesis from Harel and Sowder (1998).

Much research has been conducted in the past 25 years related to the teaching and learning of proof in Euclidean geometry. However, very little research has been done focused on preservice secondary school mathematics teachers’ notions of proof in Euclidean geometry. Thus, this qualitative study was exploratory in nature, consisting of four case studies focused on identifying preservice secondary school mathematics teachers’ current notions of proof in Euclidean geometry, a starting point for improving the teaching and learning of proof in Euclidean geometry. The unit of analysis (i.e., participant) in each case study was a preservice mathematics teacher. The case studies were parallel as each participant was presented with the same Euclidean geometry content in independent interview sessions. The content consisted of six Euclidean geometry statements and a Euclidean geometry problem appropriate for a secondary school Euclidean geometry course. For five of the six Euclidean geometry statements, three justifications for each statement were presented for discussion. For the sixth Euclidean geometry statement and the Euclidean geometry problem, participants constructed justifications for discussion. A case record for each case study was constructed from an analysis of data generated from interview sessions, including anecdotal notes from the playback of the recorded interviews, the review of the interview transcripts, document analyses of both previous geometry course documents and any documents generated by participants via assigned Euclidean geometry tasks, and participant emails. After the four case records were completed, a cross-case analysis was conducted to identify themes that traverse the individual cases. From the analyses, participants’ current notions of proof in Euclidean geometry were somewhat diverse, yet suggested that an integration of justifications consisting of empirical and deductive evidence for Euclidean geometry statements could improve both the teaching and learning of Euclidean geometry.

The purpose of the present study is to identify 10th grade students&rsquo / use of proof schemes in geometry questions and to investigate the differences in the use of proof schemes with respect to their cognitive style and gender. The sample of the study was 224 tenth grade students from four secondary schools. Of those, 126 participants were female and 98 participants were males. Data was collected at the end of the academic year 2005-2006 through uses of two data collection instruments: Geometry Proof Test (GPT) and Group Embedded Figure Test (GEFT). GPT, included eleven open-ended questions on triangle concept, was developed by researcher to investigate students&rsquo / use of proof schemes. The proof schemes reported by Harel and Sowder (1998) were used as a framework while categorizing the students&rsquo / responses. GEFT developed by Witkin, Oltman, Raskin and Karp (1971) was used to determine cognitive styles of the students as field dependent (FD), field independent (FI) and field mix (FM). To analyze data, descriptive analyses, repeated measure ANOVA with three proof schemes use scores as the dependent variables and a 2 (gender) x 3 (cognitive styles: FD, FM, FI) multivariate analysis of variance (MANOVA) with three proof schemes use scores as the dependent variables was employed. The results revealed that students used externally based proof schemes and empirical proof schemes significantly more than analytical proof schemes. Furthermore, females used empirical proof schemes significantly more than the males. Moreover, field dependent students used externally based proof schemes in GPT significantly more than field independent students. Also, field independent students use analytical proof schemes significantly more than field dependent mix students. There was no significant interaction between gender and cognitive style in the use of proof schemes. The significant differences in students&rsquo / use of proof schemes with respect to their gender and FDI cognitive style connote that gender and FDI cognitive styles are important individual differences and should be taken into consideration as instructional variables, while teaching and engaging in proof in geometry and in mathematics.

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