An investigation of mathematical thinking processes and mathematical language skills of secondary school students through the theoretical framework of 'three worlds of mathematics'

The three worlds of mathematics are a theoretical framework that examines the process of mathematical thinking in terms of the three worlds of thinking. Mathematical thinking is the process of interpreting knowledge of individuals. Hence mathematical thinking is a process. In this process, preliminary information about conceptual structures is revealed. Then new conceptual structures are formed with the existing information structures. According to the three worlds of mathematics, thinking begins with the conceptual embodied world. When the individual begins to think symbolically about a concept which is encountered in embodiment terms, he / she moves to the proceptual symbolic world dimension. This dimension also includes KİSS (procept) processes. KİSS processes include procedure, process and procept. When the individual creates and expresses information structures about the concept, he / she realizes the axiomatic formal world thinking process. The process of proof is expressed. Each thinking dimension includes different mathematical language skills within itself. These mathematical language skills focus on the ability to describe concepts in concrete terms in the conceptual embodied world. The skills of using oral, written and visual language are included in this dimension. Symbolic expression creation skills take place in the proceptual symbolic world. The definition of the concepts by individuals using the right language to make their own descriptions are in the axiomatic formal world dimension. The aim of this study is to examine mathematical thinking processes and mathematical language skills of secondary school students in terms of the theoretical framework of three worlds of mathematics. In addition, mathematical thinking processes of the students were examined in terms of KISS levels. The aim of this study is to provide students with the knowledge of the thinking dimensions and language skills of students with different levels of algebraic thinking and geometric thinking. Levels of 6th, 7th and 8th grades of secondary school were determined. The thesis study was carried out in Kocaeli. The thesis study has three stages. First of all, algebraic thinking and geometric thinking levels of secondary school students were determined. Then, semi-structured interviews were conducted with the students selected according to the levels of algebraic thinking and geometric thinking. The implementation process was carried out in the 2017-2018 academic year. The thesis study was designed in a mixed design. Algebraic Thinking Level Determination Test was developed in order to determine the level of algebraic thinking of secondary school students. This test is for the first three levels of algebraic thinking. Validity and reliability studies were conducted during the test development process. The reliability coefficient of the 27-item test which was developed as a result of all analyzes was calculated as 0,86. In order to determine the students' geometric thinking levels, Geometric Thinking Levels Scale developed by Alyeşil (2005) was used. This test is for the first four levels of geometric thinking. The thesis is based on the first three levels of geometric thinking. The reliability coefficient of this scale was calculated as 0,81. Algebraic Thinking Level Determination Test and Geometric Thinking Levels Scale were applied to 2808 students determined by cluster sampling method. Then, semi-structured interviews were conducted with 23 students with different levels of algebraic and geometric thinking, determined by purposeful sampling method. 8 of these students are in the 6th grade, 7 are in the 7th grade and 8 are in the 8th grade. The semi-structured interviews were held in three separate sessions with each student separately. In the first session, questions were asked about algebra. In this session, the knowledge structures and symbolic expression processes of the KİSS concept were examined. In the second session, questions were asked about geometry. The triangular inequality and the Pythagorean Theorem were presented to the students and they were asked to develop a proof process to explain the expressions. In the third session, the triangular inequality and the Pythagorean Theorem were expected to be formed in real life. As a result of the process of the thesis study, it was observed that the levels of algebraic thinking and geometric thinking of secondary school students were mostly at the first level. Levels of algebraic thinking and geometric thinking were found to be effective in developing a mathematical thinking process. A different view is presented in terms of the secondary school level regarding the theoretical framework of the three worlds of mathematics. It is determined that 6th and 7th grade students can develop thinking process in the axiomatic formal world dimension. The mathematical thinking processes of the 6th grade students were found to be more original. This situation is interpreted as the increase in the level of the class affects the mathematical thinking processes of students in developing the proof process. Therefore, it is thought that concept creation processes should be included in curriculum and course designs. It was found that mathematical language skills of secondary school students were mostly limited by the conceptual embodied world dimension. As the level of algebraic thinking increases, the ability to create symbolic expression also increases. However, students made mistakes in the use of written language. Visual language skills were insufficient. In conclusion, in the thesis study, a general framework is presented to examine the mathematical thinking processes and mathematical language skills of secondary school students in the theoretical framework of the three worlds of mathematics. The applicability of this framework has been demonstrated during the implementation process. It is thought that this general framework will guide the researchers.