以放聲思考法探討高二學生的數學解題思路與策略—以圓與直線為例

Abstract

本研究旨在探討普通高中二年級學生，對於圓與直線的關係，不同學習成就的學生，其解題的思路與策略有何不同之處。使用放聲思考法，針對高、中、低三個數學學習成就階層的學生，每階層各三人共九人，將其各自解題時的解題過程，以及所使用的解題思路及策略轉錄成原案，透過原案分析，探討影響解題者其解題成敗的各種因素。本研究主要發現如下： 一、解題歷程方面 (一)解題者在解題過程當中大致都經歷了閱讀題目、分析題目、解題計畫、解題執行此四個解題階段，驗證及檢討較少出現或者是沒有產生很好的效用。 (二)高數學成就組在解題歷程中較能注意到解題的關鍵條件，且在解題過程中有較多方面向的考量與監控解題的程序性過程。 (三) 中低數學成就組其解題歷程停頓較久時，常卡在基礎的數學基模概念，或者是誤解題意、誤判條件、較不合理的隨意假設數字代入運算。 二、解題策略方面 (一)成功的解題者在擬定解題策略的時後，較能正確使用高中所學的概念與知識，且能夠從使用不同面向的概念方法思考解題。 (二)中低成就組其解題策略上有時只使用了國中既有的概念，或者因題目條件無法有效的解讀，以及高中的相關公式不熟悉，而產生不了解題計畫。 三、影響解題成敗因素 (一)數學知識方面： 1.解題者是否充分了解題意。 2.解題者是否能使用解題相關的知識或公式。 3.解題者是否具有完備程序性知識。 4.解題者是否能發現題目所隱含的條件、限制。 (二)後設認知方面： 1.解題者是否能預測解題策略的可行性，評估接下來該如何修正或繼續。 2.解題者是否能評估自己目前的解題狀況。 3.解題者是否能檢查解題方法有誤。 4.解題者是否能在解題過程中有頓悟的歷程。 (三)情意態度方面： 1.解題者是否具有想要解題的企圖心。 2.解題者在解題遇到瓶頸時是否有足夠的耐心與堅持力。

This research aims to investigate the thinking and strategies employed by the 11th grade students with different mathematical achievements on solving math problems on lines and circles. This study utilized thinking-aloud protocols to analyze problem-solving processes, including the thinking and strategies used by the subjects who were students with high, average, and low mathematical achievements. They were grouped into three with three students in each group. Through protocol analysis, various factors for successful or failed math problem-solving were then discussed. The major findings of this study are summarized as follows: 1.Problem-solving processes a.The subjects substantially experienced the following four problem-solving stages: reading the given math questions, analyzing them, making plans and finally taking actions to solve the questions. Checking and proving their answers rarely occurred or did not show much effectiveness. b.The subjects with high math learning achievement were more able to notice the key conditions, think in a well-rounded way, and monitor the procedures of problem solving. c. The subjects with average and low mathematical achievement paused longer during problem-solving because they were often stuck in basic concepts of math schema, or they misunderstood the meanings of the questions, misjudged the conditions, and randomly picked an unreasonable number into operation.

2.Problem-solving strategies a.When devising problem-solving strategies, successful solvers were better able to accurately utilize the concepts and knowledge learned in senior high school, and to try various methods to answer the questions. b.Average and low achievers were sometimes confined to existing concepts acquired in junior high school, or they ineffectively interpreted the conditions of the questions, and they were unfamiliar with senior high school math formulas so that the problem-solving plans could not be made. 3.Factors for successful and failed problem-solving a.Mathematical knowledge: Whether the students. 1.fully understood the question meanings. 2.could utilize relevant knowledge or formulas. 3.have acquired complete procedural knowledge. 4.discovered the hidden conditions or restrictions of the questions. b.Metacognition: Whether the students could 1.predict the feasibility of the problem-solving strategies and evaluate how to further modify or continue. 2.evaluate their current problem-solving situations. 3.examine the mistakes in problem-solving methods. 4.have the epiphany during problem-solving. c.Affective attitudes: Whether the students 1.were ambitious to solve the problems. 2.had enough patience and persistence when encountering bottlenecks in problem-solving.