Examination of 8th grade students' construction of the concept of slope based on realistic mathematics education in APOS framework

The concept of slope, which is frequently encountered in daily lives through terms such as 'steepness', 'inclined', 'ramp' and 'acclivity', plays a significant role in mathematics education because it provides the basis for other mathematical concepts, such as, derivative. This thesis explores the way middle school students, who have already acquired informal knowledge about slope in their daily lives, construct and mathematize the concept of slope. The construction of slope is analyzed according to APOS theoretical framework (Asiala et al.,1996). In Turkish educational system, the students encounter the mathematical concept of slope at 8th grade for the first time. The present study uses the instructional design perspective of RME and aims to enable students to appeal to their prior informal knowledge about slope. Data of this qualitative study, which is designed by using to the teaching experiment method, is collected by appealing to an open-ended test results, information in researcher's journal, work-sheets and clinical interviews with participating students. Before explaining the mathematical concept of slope, an open-ended test, which assesses students' prior knowledge of certain prerequisite concepts, such as the concepts of dependent-independent variable, proportion and equation of a line, was conducted. Having analyzed the results, students were divided into groups according to the difficulties they encounter and mistakes they make. Then, by using purposive sampling one student from each group was selected as a participant of that group. The results of the open-ended test were used to form heterogeneous groups and to ensure that each participant is a member of a different group. In order to monitor the construction and mathematization process of the concept of slope participants were interviewed three times throughout the teaching process. The genetic composition of the concept of slope is described according to these results. It has been observed that some students, who are considered to be at the first level, namely the 'action level' described by APOS teaching theory, memorize the formulation for finding the slope of a straight line as rise over run and use this formulation in their calculations. Students at the process level, on the other hand, were able to construct the concept of slope as a proportion, and they were able to grasp the fact that the slope of a straight line does not change based on the point that was taken on the line. One of the participants was able to attain the object level and complete the process level. She could use concept of slope to solve a problem which was not directly connected with it and relate the concept of slope to other mathematical concepts. Moreover, the results that have been acquired in relation to the mathematization process, shows that the self-developed right triangle model first appears to be domain specific. Then, it becomes a context independent cognitive tool which needs to be seen in physical form. Finally, it has been observed that the model becomes a cognitive tool and a mental entity that does not have to be seen in physical world. It has also been observed that discovering the relationship between the informal knowledge of the fact that slope of the same linear visual does not change and the knowledge of the fact that the proportion between rise and run remains constant, which is expected to be discovered in mathematization activities, is crucial for process conception of slope. Key Words: Slope, APOS, Realistic Mathematics Education, Mathematization