| Summary: | 博士 === 國立交通大學 === 教育研究所 === 102 === Solving geometry problems which accompany with diagrams is considered a difficult task for most of students; even though the diagram is not complex. In this dissertation, the author aims to identify the sources of difficulties which solvers encountered in a geometry problem-solving scenario. To achieve the goal, four studies, which integrate self-report measures, eye movement, hand writing techniques, and statistical programs for analyzing eye movement data, were conducted to probe the cognitive processes during geometry problem-solving. In addition, the author developed programs that could help simultaneously trace eye movement as well as handwriting sequences.
In study one, the author attempted to identify the locus of difficulties in solving geometry problems based on the cognitive load theory. A series of problems of similar triangles were designed, and examined the validity for subsequent experiments. Five problems were selected and used to investigate source of difficulties in a geometry problem solving scenario.
The primary goal of study two was to evaluate whether the differences between the successful and unsuccessful solvers while solving the tasks with various difficulties could be observed with the help of an eye tracker. The eye tracking technique was used to observe the on-line processes of diagram comprehension for the successful versus unsuccessful problem solvers. The results indicated that eye movement is beneficial for observing the cognitive process in problem solving.
Anchoring on the findings derived from study one and study two, study three extensively investigated the usability and validity of applying eye tracker to explore the cognitive processes during the complete geometry problem solving (CPS) that involved simultaneously viewing (i.e., input) and writing (i.e., output) processes and the switching in between. The author examined whether the perceived difficulties and eye movements was correlated. In addition, the author searched for eye movement measures that are sensitive to the perceived difficulty of geometry problems. The results indicated that: (1) The perceived difficulties and eye movements were significantly correlated. (2) Unsuccessful solvers paid more attention on diagram comprehension than that in successful solvers, which was consistent to the finding of previous studies. (3) Three eye movement measures, including dwell time, fixation count, and run count significantly differed within specific areas of interest. The three eye movement measures were sensitive to the perceived difficulty when the problems were difficult.
The evidence from previous studies showed that diagram comprehension was one of the major sources of difficulties in solving geometry problems; therefore, study 4 sought to monitor and analyze the processes of diagram comprehension. The author examined potential explanations of problem solving difficulties by focusing on how the spatial relations of geometry diagram hindered the solvers from successfully solving the geometry problems. The most difficult problem (problem #5) was used as an example to conduct experiment. The author hypothesized that problem #5 was difficult because the two triangles were adjacent, which led to increase difficulties in recognizing diagrams that were required to correctly solve the problem. Another possible source of difficulties might be that the adjacent triangles could hinder solvers from identifying the length of specific sides. Therefore, separately presenting the adjacent triangles might help to solve the problem. This result suggested that separately presenting two triangles was helpful to reduce difficulties in integrating information in the diagram, though it might not be enough to successfully solving problems. The results partially supported the hypothesis that separately presented the pair of triangles was beneficial to diagram comprehension. Implications and suggestions for instructional design in mathematics education were discussed.
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