Visual reasoning in solving mathematical problems on functions and their derivatives among Malaysian Pre-University Students / Haliza Abd Hamid

The main objective of this study was to develop a framework and assess the visual reasoning process adopted by pre-university students when integrating Cartesian graphs to solve mathematical problems on functions and derivatives. The study identified the usage levels of graphs, method of preference and graph-reasoning ability, and subsequently, the correlation among them. The study also investigated the misconceptions and difficulties faced by the students. The study employed a 3-phase descriptive quantitative method. The development of framework in Phase 1 involved a three-stage process: the document analysis on theories and models on visual reasoning and Cartesian graphs, focus group discussion among experts in the domain content and visual reasoning, and a 3-round Delphi method to confirm the framework. In Phase 2, three instruments were prepared; the Visual Representation Usage Level on four categories of using graphs, the Mathematical Visuality Test to measure the students’ preference method and the Graph-Reasoning Test measuring their graph-reasoning ability. Phase 3 involved the collection and analysis of data on 194 pre-university students. The developed framework consisted of seven categories of encoding and five categories of decoding processes. Results indicated between 41.75% to 84.02% of the students were very positive towards the use of graphs and diagrams in the teaching and learning of mathematics although between 56.70% and 78.87% said they faced difficulties in constructing and interpreting them. Students exhibited fluctuating patterns of visual reasoning ability. In the encoding process, the students were categorised into three types of mathematical visuality; 26.8% were visual, 16.5% were partially-visual and 56.7% were non-visual. This exhibits their reluctance to sketch graphs although they managed to obtain the correct solutions which indicates that they had a predominant preference for algebraic method as compared to visuals. Responses in the decoding process were based on the three levels of graph reasoning. At least 68% of the students managed to get correct solutions for all the read graph items. More than 70% of the students managed to obtain 75% and 43% of the items correct for the read between graph and read beyond graph respectively. These indicate that as the tasks get harder visually, more cognitive load is needed for the students to read and interpret graphs. Strong positive correlation values of at least 0.91 were obtained among the three decoding levels. They perform fundamental, operational and subjective types of errors and encountered the non-use of graph, generic and idiosyncratic difficulties when relating the algebraic forms of functions and derivatives to their visual representations on graphs and vice versa. The results of the study were significant in providing reliable and important ideas depicting the development of visual reasoning that is useful in the students’ thinking and understanding, to guide the development of instructional materials to improve students’ understanding and reasoning, for educators and curriculum developers to enhance the learning outcomes and teaching strategies to challenge students’ thinking and reasoning skills, and to reduce gap in the literature and knowledge on visual reasoning in the Malaysian educational system