The Use of Variation Theory to Improve Secondary Three Students' Learning of the Mathematical Concept of Slope

This dissertation, "The Use of Variation Theory to Improve Secondary Three Students' Learning of the Mathematical Concept of Slope" by Chi-kit, Choy, 蔡志傑, was obtained from The University of Hong Kong (Pokfulam, Hong Kong) and is being sold pursuant to Creative Commons: Attribution 3.0 Hong Kong License. The content of this dissertation has not been altered in any way. We have altered the formatting in order to facilitate the ease of printing and reading of the dissertation. All rights not granted by the above license are retained by the author. Abstract: Abstract This study investigates how a learning study, which is grounded in the theory of variation, can improve student learning and teaching in mathematics in a secondary school in Hong Kong. Three teachers, 141 secondary three students and the researcher participated in the present study. The teachers in the study formed two groups, the learning study group and the comparison group. The learning study group includes two teachers, whereas the comparison group has one teacher. Both groups aim at achieving the agreed object of learning, which is to help the students to develop a capability of understanding the mathematical concept of slope. Based on the findings from the pilot study and the pre-test, both groups developed a lesson plan for teaching the mathematics topic by the teacher's own experience and intuition. The major difference between the learning study group and the comparison group was that the former made use of the theory of variation in planning their lessons whereas the latter did not have an explicit theory of learning to follow. All the lessons were videotaped and transcribed verbatim. After the lessons, a post-test was conducted for each class and three students from each class were selected for interviews by purposive sampling. Both inter-group and intra-group comparisons were performed in order to examine the differences between the intended objects of learning and the enacted objects of learning. The results of the study showed that four qualitatively different ways of experiencing the mathematical concept of slope were identified. The first two conceptions revealed that students could discern either horizontal distance or vertical distance as a critical aspect of slope. The third conception showed that students could discern both critical aspects, i.e. horizontal distance and vertical distance, simultaneously. Students who showed the fourth conception focused on negative slope of a straight line, taking into account the horizontal distance, the vertical distance and the angle of inclination with the x-axis at the same time. Based on the findings, students were found to be able to discern the critical aspects of slope after experiencing the variations in explicit or implicit way. However, although students belonging to the learning study group demonstrated a better understanding of the topic after the lessons than the comparison group, both groups were weak in handling mathematics problems with negative slopes. Differences in students' performance were found to be closely related to the differences in the enacted objects of learning. Two main conclusions were drawn for this study. Firstly, it concludes that experiencing variation is the key for making student learning possible. Secondly, the learning study approach may be an effective approach in bringing learning about. iii DOI: 10.5353/th_b3760836 Subjects: Experiential learning Learning, Psychology of Mathematics - Study and teaching (Secondary) - China - Hong Kong