Investigation of instructors' proof presentation structures in mathematics lectures

Since proving is an act that allows mathematical thinking to be displayed both as a process and as a "product" (Tall, 1991), mathematicians, mathematics educators, and mathematics education researchers agree that proof is a very fundamental activity that should have an important role in mathematics lessons (Weber, 2010). In addition, mathematics teachers 'perceptions of mathematical proof affect not only the experiences they provide for their students but also their students' expectations for proof learning, so it is important for teachers to have a conceptual understanding of proof (Ko, 2010). Pre-service teachers are prepared both for such activities that will shape the understanding of proof of their future students and for advanced mathematical thinking in proof-based mathematics courses at the university. Within the framework of NCTM standards, curricula and teacher competencies, it can be said that the practices/actions and discourses towards proof are important in the university, since the pre-service teachers must have the competencies expected from a middle school/high school mathematics teacher in the context of proof teaching. In this context, understanding the practices of lecturers, who are the representatives of the mathematics discipline in the eyes of students, is beneficial in terms of helping us understand what students can do to be more successful and productive in their mathematical thinking and activities (Lynch & Lockwood, 2018). In other words, it is important to understand the opportunities that the lecturer provides to her/his students while explaining a proof in proof-based lessons and what kind of instructional tools such applications include. Considering these requirements, this study aimed to examine how the lecturers presented a proof in the course, as well as their views on the importance of proof in the course, the choice of proofs they will present in the course, their expectations from students about proof and their difficulties in proving. In this study, the term "proof presentation structure" refers to how the lecturers explain the proofs and their opinions about the proof presentation. The presentation of the proof includes the statement to be proved, the proof itself and all the definitions, comments, examples, questions etc. given during the proof. As a result, in this study, the proof presentation structures of the lecturers were explored by analyzing their practices and views about the importance of the proof, the textbook, the presentation of the proof in the lesson and students. In order to determine the proof presentation structures of the lecturers working in the department of mathematics education, grounded theory, one of the qualitative research methods, was used. Participants of this study consist of 7 lecturers who teach mathematics courses in primary or secondary mathematics teaching departments at the education faculty of a state university. After deciding what to do with the research and obtaining the necessary permissions, the courses of the lecturers were observed by video recording. Meanwhile, the questions to be asked about the proof expression structures were determined and the first interview questions were prepared. While the video observations of the lesson were continuing, the first interview was conducted, and this interview was analyzed. As a result of these analyzes and course video analysis, the second interview was planned and conducted for questions or observations that needed both explanation and confirmation of the instructors. At the same time, an interview form consisting of selection and ranking questions was created in order to determine the importance levels of the categories and themes that emerged as a result of the analysis of the questions selected from the first interview questions, and the third interview was held. Then, all interviews and course video observations were re-analyzed and interpreted together. As a result, two important theories have been reached as a result of the existence or frequency of important practices that are thought to not change from course to course, discourses that contain obvious differences in the course or interview, and the consensus or disagreement in opinions. One of the theories is the "ideal proof presentation structure" formed by the opinions of the instructors. This ideal proof presentation structure was formed as a result of the responses from the instructors to the questions pointing to what should be (ideal) for the statement of proof in the interview and the selection and ordering of these answers. The second theory, on the other hand, is the "focus proof presentation structure", which includes the characteristics that predominantly differ in the proof presentations or opinions of the lecturers and form a pattern (which is expected not to change from course to course and can be considered pedagogically important). The determination of these presentation structures was achieved by analyzing the patterns and differences in the practices of the instructors during the proof presentations. Other results of the study, which may be considered important, and how these results find their place in the literature are also discussed.