Proof and Reasoning Research Papers (original) (raw)

It can be said that proving activities are an important activity for both mathematicians and mathematics educators. Thus, developing the proving ability is among the aims of many courses in advanced mathematics. However, although the importance of proof is frequently emphasized, studies show that undergraduates have difficulty in this regard. To overcome these difficulties and to make sense of the complex structure of the proof, many theoretical frameworks have been presented. However, it can be said that very few of these frameworks deal with the proving activity from a process perspective. On the other hand, Kidron and Dreyfus (2014), who approaching proof from a knowledge construction perspective and analyzing affective-logical interaction in this process in the micro-analytical level, reached the concept of “proof image” in the context of the Abstraction in Context (AiC) framework. However, despite the elapsed time, it can be seen that the researches on this framework have not reached sufficient numbers. As stated by the authors, it is clear that more research is required to deepen this framework. Because, by making sense of the proof image, which constitutes a significant dimension of the proof process, the multidimensional structure of dynamics of the proof can be clarified, and thus difficulties experienced in this activity can be avoided. In this doctoral study designed with this perspective, it was aimed to deepen the understanding of "proof image" and to shed light on the spots left open in the theoretical framework. Taking into account both the views of Kidron & Dreyfus (2014) and the deficiencies emerged in the practices, the issues of "the relationship between the proof image and the formal proof", "the levels (or types) of the proof image" and "the socio-cultural dimension of the proof image" were examined in this study.
In the study, sophomore pre-service teachers in the primary school mathematics education department were selected by using the criterion sampling method, one of the purposive sampling methods. The reason for the selection of this class level is that the subjects related to Cantorian Set Theory and "infinity" take place at this level. The subject of infinity has been considered to be a suitable context for investigations. Because many studies in the literature indicate that pre-service teachers' understanding of infinity has developed from intuitive to formal just like the proof image. In the study, the case study (for the first two sub-problems) which is one of the qualitative research designs, and discourse analysis (for the third sub-problem) were used together. In addition, individual interview forms developed by Pala (2016) were used as a data collection tool in task-based interviews. On the other hand, while analyzing the data, descriptive analysis was used for the first two questions and, the discourse analysis technique with a socio-cultural interaction perspective was used for the third question.
As a result of the analysis related to the first sub-problem of the study, it was determined that the proof image would not provide a certainty to reach the formal proof. In many tasks, it has been observed that participants cannot make a transition to formal proof even though forming a proof image. Possible reasons for this situation are discussed in detail in the context of subcomponents. On the other hand, some of the prominent reasons for individuals, who formed a proof image, failed to reach the formal proof can be listed as "lack of conceptual knowledge", "frequent use of intuitive assumptions", and "limited perception of the theorem". In the context of the second sub-problem of the study, all these situations that arise between the proof image and the formal proof had been interpreted separately in the context of the role of formal knowledge. Thus, 4 different possibilities were determined for situations that include and do not include the image of proof. It was observed that situations involving a proof image may result in two cases that “result in formal proof” and “result in partial verification”. However, it was observed that situations, which do not contain a proof image, may result from either practice such as memorization and repetition that contain only inauthentic connections (Type 1) or insufficient justification (Type 2) that includes logical connections based on personal understanding. Lastly, in the context of the third sub-problem of the study, an interactive proof process was carried out and discourse analysis was made on the obtained data. In this context, 9 different discourse styles that affect the participants' proving processes were determined. These forms of discourse are "1. Questioning, 2. Making a definition of concepts, 3. Reminding the pre-knowledge, 4. Proposing a proving method, 5. Presenting a confirmatory or falsifying example, 6. Making comments or explanations, 7. Making inferences, 8. Expression of affective experiences, 9. Guessing or predicting”. Besides, with reference to the relationship on the transcription text, the discourse forms associated with each component of the proof image were determined. Consequently, the contributions of both discourses and formal knowledge to the formation of the proof image were explained and, suggestions for proving applications, which may involve a proof image, were presented.