Introduction to proof in the secondary school: The use of proofs without words within the framework of the realistic mathematics education

Proof, one of the basic building blocks of mathematics, is indispensable for the development, construction and transmission of mathematical knowledge. In many countries today, arithmetic concepts, calculations and algorithms are emphasized but a limited number of studies is done about the concept of proof in elementary mathematics teaching. However, it is expected that students will be able to understand some of the formal proofs and to do some basic proofs when secondary education starts. This can be termed as a "didactic void" created by curricula in transition from primary to secondary education.Proofs without words are images and diagrams that help the reader to understand why a certain mathematical statement might be right and how to get it started. It has been shown in various studies that mathematical process skills such as defining, representing, verifying and generalizing of an assumption have been developed by proofs without words. The purpose of this study is to investigate how proofs without words can be used as a means to facilitate the transition to formal proof and to fill the didactic void. In the study, the teaching experiment, which is a qualitative research method, was used within the theoretical framework provided by Realistic Mathematics Education. In this regard, six proofs without words suitable for the seventh-graders were selected and their introduction was planned in the form of word problems in the context of the activity design approach presented by the theory. The study was carried out in an optional mathematics course with the participation of 30 students. Approximately two lessons per week were reserved for each proof and the implementation lasted six weeks in total. The study's data were collected from student notebooks, researcher observation notes and video recording of one student group. The data were analyzed in accordance with the steps determined by the theory in a way that it reflects the work of both the whole class and the focus group. The results of the study show that students experienced many mathematical processes of proof, made connexions between mathematical areas and progressed in their own processes.