高一學生在簡單多項式函數及其圖形單元的解題歷程分析-以新北市某高中為例

Abstract

本研究主要是利用放聲思考法針對高、中、低數學成就學生各兩人共六人進行六題非例行性題目的施測，探討不同數學成就的高一學生在簡單多項式函數及其圖形單元中解題歷程思路與解題策略運用的差異性，並了解影響解題成敗的因素，研究結果如下。 一、 解題歷程方面 (一) 所有的解題者幾乎都有歷經閱讀題意、分析題目、擬訂計畫與計畫執行四個階段，但部分解題者並沒有養成驗證檢討的習慣。 (二) 中、高成就的解題者閱讀完題意後都可以很快理解題意，但相較於中成就解題者分析、擬定與執行計畫的制式化，高成就解題者的處理較富靈活性，而在驗證檢討階段都會利用與初始運算不同面向的方法或概念執行。 (三) 低成就的解題者會花較多時間理解題意，且常常由於自身基模知識不完備導致會用猜測來分析、擬定與執行計畫，而驗證檢討方式往往都是重新計算一次。 二、 解題策略方面 (一) 幾乎所有的解題者在處理二次函數的題目都會下意識的先操作使用配方法來分析題目，其中以中低數學成就的解題者處理題目時比例偏高。 (二) 高數學成就的解題者在解題策略運用上，會利用自己最基礎的基模知識，靈活運用不同面向的解題方法進行思考。 (三) 中、低成就的解題者在解題策略運用會比較制式化的操作過去對類似題目做過的策略，但低成就的解題者往往由於基模知識不足導致最後會用猜測或放棄解題的比例偏高。 三、 影響解題成敗因素 (一) 數學知識 1. 解題者是否可以完整了解題意。 2. 解題者對於解題相關的基模知識是否可以與題目做適當的連結。 3. 解題者對於基模知識的理解是否混淆。 4. 解題者是否對於題目可以有適當的解題策略。 5. 解題者對於解題程序是否可以完全掌握。 6. 解題者是否可以多面向的驗證檢討答案。 (二) 後設認知 1. 解題者是否可以做答案基準的評估，並藉此檢查最後答案。 2. 解題者是否可以在一開始做解題策略的評估並預測解題結果。 3. 解題者是否可以在解題過程中有頓悟題意的歷程。 4. 解題者是否可以同步監控自己執行策略的過程。 (三) 情意態度 1. 解題者是否有保持解題企圖心。 2. 解題者是否可以在處理複雜的過程時可以兼具耐心與冷靜。 3. 解題者是否會有過度膨脹的自信。

This study is mainly based on the use of think-aloud protocol for 3 groups of students: high, medium and low mathematic achievers. Six students (two of each group) were selected to tackle six non-traditional problems to explore the difference in problem solving strategies between different high school students on the unit Simple Polynomial Functions and Its Graphic. The factors that impact the success or failure of solving a problem are also investigated. The results are as follows. A. Problem-solving process: 1. All the problem-solvers underwent the four stages of reading the problem, analyzing the problem, developing a plan, and implementing the plan. However, some problem-solvers don’t have the habit of checking their work and verifying their answers. 2. Medium and high mathematic achievers can both understand the problem after reading it. However, the high mathematic achievers show better problem-solving skills when analyzing the problem, developing and implementing a plan to find the solution compared to the medium mathematic achievers. In addition, high mathematic achievers use different method compared to their initial thinking when verifying and reviewing their answers. 3. Low mathematic achievers spend more time understanding the problem, and often guess the answers when analyzing, planning and implementing their plan to find the solution due to their limited knowledge. In addition, they always have more than one calculation when verifying and reviewing their answers. B. Problem solving strategies: 1. Almost every problem solver will subconsciously use completing the squares to analyze quadratic functions with more medium and low mathematic achievers using this method. 2. When solving problems, high mathematic achievers will use their basic knowledge in combination with other strategies to solve the problem. 3. Medium and low mathematic achievers’ methods for problem solving are very basic and standard when tackling problems that are similar to the ones they’ve done in the pass. However, low mathematic achievers showed more signs of guessing and giving up when problem solving due to their lack of knowledge. C. Factors that impact the success or failure of problem solving 1. Mathematical knowledge: a) Does the problem solver understand the meaning of the problem? b) Is the problem-solver able to make connections with prior knowledge when solving problems? c) Does the problem solver find the understanding of the knowledge of the basic model confusing? d) Is the problem-maker able to use the appropriate problem-solving strategies for the problems? e) Does the problem solver know exactly how to solve the problem? f) Is the problem solver able to use a multidimensional perspective to verify and review the problems? 2. Post-Cognition a) Is the problem solver able to assess his/her answer and check to the final answer? b) Is the problem solver able to strategize how to solve the problem and predict the results? c) Is the problem solver able to understand the course of the question when solving the problem? d) Is the problem solver able to synchronize his/her own problem solving process and monitor oneself when implementing strategies for problem solving? 3. Emotions and Attitude a) Do problem solvers solve problems with a clear goal? b) Is the problem solver able to tackle complicated problems calmly and patiently? c) Is the problem solver over confident?