幾何問題解決的認知負荷與圖形理解: 以眼動追蹤及手寫板記錄歷程

Abstract

對許多學生來說，就算幾何題目中的圖形並不複雜，但是要解決仍然是個困難的任務。為了要探討這個現象，本論文的研究目的在探討問題解決者在問題解決的歷程中，難度的來源為何。為了完成此研究目的，本論文進行了四個子研究，整合了認知負荷自陳量表、眼動測量、手寫技術、認知負荷來源量表，並利用上述研究工具來探測幾何問題解決的認知歷程。此外，作者開發用於分析眼動資料的統計程式，以及同時觀察眼動與手寫軌跡的程式。 在研究一中，作者嘗試從認知負荷理論的角度出發，找出幾何問題解決時，發生困難的原因為何。首先，作者設計了一系列關於相似三角形的幾何題目，並且檢驗題目是否適用於後續的研究。最後選出了五個適用於後續研究的題目。 在研究二中，研究目的是評估問題解決者解決不同難度的幾何題目時，眼動儀是否能觀察到成功與不成功解題者的差異，如:答題正確度、認知負荷程度。研究結果顯示眼動技術適合用來觀察問題解決的認知歷程，正確解題者與不正確解題者的眼動軌跡有明顯不同。 基於研究一與研究二的結果，研究三更廣泛的調查眼動儀用於完全問題解決歷程的可用性與有效性。完全問題解決歷程指的是具有完整的閱讀資訊(輸入)以及寫出答案(輸出)的問題解決歷程。為了完成此研究目的，作者紀錄並且分析解題者在讀圖時的眼動歷程以及手寫結果，並試圖檢驗眼動測量是否與解題者感受到的難度有顯著相關，嘗試找出對於幾何問題的難度敏感的眼動測量。結果顯示:(1)眼動測量與主觀感受難度有顯著相關。(2)不成功解題者比成功解題者花費更多時間在理解圖形。(3)在特定興趣區的一些眼動測量，包含凝視總時間、凝視次數、與回視次數有顯著的不同。這三個眼動測量可偵測出作答較難的題目時，問題解決者感受到的難度。 基於上述三個研究，作者認為圖形理解可能是幾何問題解決歷程中主要的難度來源。因此在研究四中，作者嘗試找出在問題解決歷程中，幾何圖形理解是如何阻礙問題解決者，使其無法成功解題的原因。並以難度最高的題目(第5題)為例進行實驗。作者認為造成第5題困難的原因是由於圖形相連，造成視覺上看到的圖形與解題需要的圖形不同，因此產生困難。另一個可能原因是相鄰三角形會使研究者看不到變數訊息。因此將三角形分開將有助於解題。結果顯示雖然將圖形分開並無顯著的增加答題正確率的效果，然而有明顯證據顯示圖形分開可降低讀圖時的認知負荷，並且有助於整合圖形中的資訊。此結果部分支持分開呈現相似三角形有助於圖形理解(例如: 圖形旋轉)。論文最後探討此研究對於數學教育中教材設計的建議。

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.