Please use this identifier to cite or link to this item: http://hdl.handle.net/11455/98476
標題: 台北市國中八年級學生在等差級數應用題之解題歷程分析
The Analysis of Problem Solving Process in Arithmetic Series of Eighth Grade Junior High School Students in Taipei City
作者: 陳妍儒
Yen-Ru Chen
關鍵字: 等差級數應用題;放聲思考法;學習策略;解題歷程;數學解題;後設認知;arithmetic series word problems;think-aloud protocols;learning strategy;problem-solving process;solving mathematics;metacognition
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摘要: 
本研究主要在分析台北市國中八年級學生在等差級數應用題的解題歷程,希望能從答題歷程中探討數學成就不同的學生所使用之解題策略,並針對錯誤題型分析,作為改進教學方法與實施課後輔導的依據。此研究使用放聲思考法與事後晤談方式,蒐集受測者的解題思考歷程。

此研究係以台北市松山某一國中八年級之班級為採樣對象,依106學年度第一學期八年級三次段考平均分數分層取樣。高分組為前27%,低分組為後27%,其餘為中間組。針對高分組、中間組和低分組,每組男學生及女學生各選取一位為研究對象。研究之進行係分別以一對一之方式讓六名受測學生透過放聲思考法進行六道等差級數應用題之解題,蒐集受測者的解題內部思考歷程加以分析,得到以下結論:

一、解題歷程
高分組及中間組的學生,均會經歷閱讀題目、分析題目、擬定計劃與執行計劃四個階段,也較常出現驗證解答的階段。由研究過程得知,受試者若較常驗證解答,其成功解題的比率也較高。然而,在低分組的學生的解題歷程中常出現隨機跳躍式的臆測性解題步驟。

二、解題策略
高分組的學生,能針對問題作出有系統的分析,提出適當的解題策略;中分組的學生較無系統地提出解題策略或使用單一解題策略,發現無法執行時,才會想另一個策略執行;低分組的學生,通常以直觀、臆測的方式,隨意使用解題策略獲得答案。

三、影響解題錯誤的原因
(一)解題知識方面:
對應用題目裡的詞彙理解程度。
對題目意義的理解程度。
能否正確且有效的觀察出其規律。
能否將待答問題轉換成數學思維的能力。
對級數與圖形觀察與歸納分類的能力。
新知識是否受舊經驗干擾。
是否精熟新學習到的數學概念。
提出可使用之解題策略。
熟練等差級數公式。
具先備知識足以正確運算並求出答案。

(二)後設認知方面:
能意識到自己提出的解題策略是否可行。
能知覺到錯誤的解題行為,及時修正彌補錯誤。
能正確判斷答案的合理性。

(三)情意態度方面:
是否具足夠的自我信心。
是否具足夠的專注力和毅力。
當解題遇到困難時,是否堅持並重新思考,持續完成解題。

最後,研究者根據研究結果,提出對於數學教學及未來研究議題之建議,盼能作為以後教學改善之參考,對國中階段的莘莘學子有所助益。

The study analyzes eighth grade students' problem solving processes for arithmetic series word problems. The solution strategies of students who have different achievement levels are studied and their error types are identified. This allows teachers to understand students' difficulty in solving arithmetic series word problems and could be used to improve teaching and for remedial instruction. The study is conducted in accordance with the think-aloud protocol and interview.

A sample of students with different achievement levels in Mathematics is taken from a junior high school in Songshan District, Taipei. Using the average scores for the three sectional examinations from the first semester of the academic year in 2017, the students are divided into three groups: high (first 27%), medium (28%-73%) and low (last 27 %) achievement levels. Two students, one female and one male, are chosen from each group to form a sample of six students for the study. The students each use the think-aloud strategy to solve six arithmetic series word problems. Their thinking processes from students' solutions are then acquired and analyzed. The main findings and results are as follows:

一、Problem solving process
While solving a problem, all the students who have a medium or high achievement level used all four steps of the problem solving process: reading, analysis, planning, and implementation. Justification is also frequently shown in their problem solving processes. It is also shown that the more frequently the students justify their answers, the higher is their success rate. The students with a low achievement level randomly use the four steps for problem solving.

二、Problem solving strategy
The students with a high achievement level systematically analyze the problem and use appropriate problem solving strategies. The students with a medium achievement level usually use problem solving strategies unsystematically or use only a single strategy and might use another strategy if they fail. The students with a low achievement level usually randomly select a problem solving strategy using intuition and assumption.

三、Decisive factors for the success or failure of problem solving
(一) Relevant Knowledge
1. The degree of identification of the vocabulary in the word problem.
2. The degree of comprehension of the meaning of the word problem.
3. Capability of observing the correct patterns.
4. Ability to transform a given problem into a mathematical thought process.
5. Capability to analyze the series.
6. New knowledge interferes with previous experience.
7. Use of applicable strategies.
8. Correctly calculating an answer.
9. Familiarity with the formulae for arithmetic series.
10. Use of prior knowledge to accomplish the computation.

(二) Metacognition
1. Being aware of whether the proposed strategy is feasible.
2. Being aware of a false solution and immediately correcting the mistakes.
3. Having good judgement on the rationality of a solution.

(三) Affective domain
1. Having self-confidence.
2. Paying sufficient attention to computation.
3. Willing to rethink and persist until the solution is calculated while having trouble with the solution.

Finally, I give some suggestions for mathematics teaching and future research, which will be beneficial in improving teaching and improving students' learning and achievement in mathematics.
URI: http://hdl.handle.net/11455/98476
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