臺灣博碩士論文加值系統

中　文　摘　要
問題解決一直是數學教育研究中的重要議題。在學校中，我們常以解決文字問題作為培養問題解決能力的重要方式。本研究據以建構主義的觀點，探討數學計算能力與不同語文理解的學習者，面對不同述敘方式的文字問題時，其解題表現、錯誤類型，及解題歷程。接以，加入合作協商之解題方式，初探合作解題之成效。
本研究探究問題解決的內涵，而發展出兩種文字問題－傳統文字問題、故事文字問題；五個解題歷程－理解題意、尋求模式、擬定解法、執行方法、判斷；以及解題評量方法。接以，本研究探討數學文字問題的相關研究，彙整影響解文字問題的因素，與錯誤類型。
本研究採用測驗、觀察方法，研究樣本為六班國二學生，共203人。首先將所有學生分為四組，依序分別為「數學計算與語文理解皆未發生困難的學生」(第一組)、「語文理解困難學生」(第二組)、「數學計算困難學生」(第三組)，與「數學計算與語文理解同時發生困難的學生」(第四組)。接著對其施以兩種數學文字問題，探究其個人與合作解題之得分、解題特徵與解題歷程。
研究結果分為四個部份。首先，在得分方面，傳統文字題與數學算術測驗結果相同，表示此類常見的學校數學文字問題，其簡短的敘述已去情境化，強調的是計算能力。而各組學生在故事文字題的得分皆變差，其中第二、三組學生已無顯著差異，各有不同的解題困難產生。
第二，在解題歷程方面，除第四組學生之外，餘三組學生在傳統文字題中大部份能達到高階段，但在故事文字題中，除了第一組學生之外，其餘學生大部份都停滯在低階段。而大部份學生沒有進行「判斷」的習慣，其憑直覺或歷程順利與否來決定是否驗證答案。另外，本研究亦嘗試建立一個解題歷程的模型。成功的解題必須先經歷「探索帶」，才能進入「解題核心」。
第三，在錯誤類型方面，本研究認為有故事背景的文字題對大部份學生而言，使其感到極為困擾，也表現得較差。尤其我們可以看到第二、三、四組學生，他們在「一片空白」的題數遠遠地超過傳統題。而本研究也指出，第二、四組學生有明顯的「語言知識錯誤」，需要接受閱讀理解的再訓練；第三、四組學生則需要加強自動化技能的加速技巧。
第四，各組學生合作解題後，皆能提昇兩種文字問題之得分與解題歷程階段。在合作的過程中，因和諧幽默的對話而產生情意上的支持，使得故事題之階段0(即沒有任何解題行為)的次數的大幅減少，顯示故事文字題可能較適合以合作的方式進行。第一組學生在各種組合中，通常擔任教導者的角色，其與其他三組學生的合作型式類似於專家－生手型態；而第二、三組學生的組合是由能力較為互補的兩方所形成，產生的口語互動較多，為在「相同平面」(equivalent planes)的合作關係。

最後，本研究認為在未來的數學問題解決研究上，合作解題可能是後續研究的重要方向之一。因此建議未來研究可以增加合作解題的配對模式及各組人數，以及讓合作的學生回頭再進行個別解題，以更深入地探究合作解題的功效。

Abstract
Problem solving has long been a crucial issue in mathematic education. In schools, providing students with word problems is an important way to help them become competent mathematics problem solvers. Based on the view of constructivism, this study mainly investigated the learners with different capabilities of math arithmetic and reading comprehension on their performance, error types, and the procedures for problem solving in dealing with different mathematical word problems. Moreover, the study explored the effectiveness on cooperative problem solving.
By reviewing the theoretical foundations of problem solving, there were two different mathematical word problems: traditional and narrative ones. Five steps were also proposed for problem solving: understanding the problem, matching the pattern, making a plan, carrying out the plan, and judgement. Afterward the study integrated the factors and error types in problem solving through surveying the researches on mathematical word problems.
Tests and observations were adopted in this study. The participants were 203 eighth-grade students, who were classified into four groups: having no difficulties in math arithmetic and reading comprehension (Group 1), having reading comprehension difficulties only (Group 2), having math arithmetic difficulties (Group 3), and having difficulties both in math arithmetic and reading comprehension (Group 4). All of the students were given traditional and narrative word problems individually and collaboratively. Their performance, features of solving behaviors, and procedures of problem solving were investigated.
Research findings were summarized as follows. First, students’ performance in traditional word problems was highly related to math arithmetic examination. It indicated that the traditional word problems were decontextualized and were highly coherent with their arithmetic abilities. However, students’ performance in narrative word problems was not as good as that in traditional word problems. Besides, though the students in Group 2 and Group 3 belonged to different difficulties, the performance of narrative word problems turned out no significant differences.
Second, most of the students attained high-level stages in solving traditional word problems except those in Group 4. However, except Group 1, most of the others stayed in low-level stages in solving narrative ones. Furthermore, depending on intuition or smooth working on the procedures of problem solving, most of the students did not judge their final answers. Based on the research findings, a model of problem solving was developed. Successful problem solving resulted from going through a ‘exploring belt’ and ‘the core of problem solving.’
Third, the findings also revealed that most of the students did not perform well in story-based narrative word problems. In particular, the unanswered situation of Group 2, Group3, and Group 4 students was much more frequent than that in traditional word problems. On the other hand, the students of Group 2 and Group 4 with obvious errors of linguistic knowledge may require interventions aimed at reading comprehension. The students of Group 3 and Group 4, on the other hand, may need instruction in automatic skills in mathematics.
Finally, solving problems cooperatively promoted both the scores and problem solving stages in traditional and narrative word problems. In the procedures of cooperation, the unanswered situations were greatly reduced in narrative word problems because of the affective supports from interactive conversations. Furthermore, the Group 1 students usually played a tutor role in cooperative activities with those in other groups, which were likely similar to an expert-novice relationship, while the complementary cooperative combination of Group 2 and Group 3 students was likely in a ‘equivalent plane,’ which revealed more verbal interaction.
The study indicated that cooperative problem solving may be an important research issue for mathematical problem solving. Further research was suggested to deeply investigate the effect on cooperative problem solving.

目 錄

頁數
中文摘要．．．．．．．．．．．．．．．．．．．．．．．．．．．．．． ⅰ
英文摘要．．．．．．．．．．．．．．．．．．．．．．．．．．．．．． ⅲ
目錄．．．．．．．．．．．．．．．．．．．．．．．．．．．．．． Ⅰ
表目錄．．．．．．．．．．．．．．．．．．．．．．．．．．．．． Ⅳ
圖目錄．．．．．．．．．．．．．．．．．．．．．．．．．．．．． Ⅵ
第一章 緒論．．．．．．．．．．．．．．．．．．．．．．．．． 1
第一節 研究動機與目的．．．．．．．．．．．．．．．．．．． 1
第二節 研究問題．．．．．．．．．．．．．．．．．．．．．． 3
第三節 名詞界定．．．．．．．．．．．．．．．．．．．．．． 5

第二章 文獻探討．．．．．．．．．．．．．．．．．．．．．． 7
第一節 問題解決．．．．．．．．．．．．．．．．．．．．．． 7
壹、問題的定義．．．．．．．．．．．．．．．．．．．．． 7
貳、問題與數學問題的類型．．．．．．．．．．．．．．．． 9
參、問題解決的歷程．．．．．．．．．．．．．．．．．．． 14
肆、解題的評量模式．．．．．．．．．．．．．．．．．．． 20
伍、建構學習觀對數學問題解決研究的影響．．．．．．．．． 24

第二節 數學文字問題的相關研究．．．．．．．．．．．．．．． 31
壹、影響解文字問題的因素．．．．．．．．．．．．．．．． 31
貳、文字問題錯誤類型的探討．．．．．．．．．．．．．．． 38

第三章 研究方法．．．．．．．．．．．．．．．．．．．．．．． 45
第一節 研究對象．．．．．．．．．．．．．．．．．．．．．．． 45
第二節 研究架構與設計．．．．．．．．．．．．．．．．．．．． 48
第三節 研究流程．．．．．．．．．．．．．．．．．．．．．．． 50
第四節 研究工具．．．．．．．．．．．．．．．．．．．．．．． 53
壹、「語文測驗」與「數學算術計算測驗」．．．．．．．．．． 53
貳、「傳統文字問題」、「故事文字問題」．．．．．．．．．． 55
參、解題歷程自評表．．．．．．．．．．．．．．．．．．．． 57
第五節 資料處理與分析．．．．．．．．．．．．．．．．．．．． 59
壹、資料處理方式．．．．．．．．．．．．．．．．．．．． 59
貳、資料分析方式-．．．．．．．．．．．．．．．．．．．． 62

第四章 研究結果與討論．．．．．．．．．．．．．．．．．．． 64
第一節 學生在背景能力測驗與兩種文字問題上的差異情形．．．．． 65
壹、學生在兩種文字問題得分的基本描述統計．．．．．．．． 65
貳、學生在背景能力測驗與兩種文字問題上的差異分析．．．． 66

第二節 學生之背景能力與兩種文字問題得分之相關情形．．．．．． 69
第三節 學生之背景能力對兩種文字問題得分的迴歸分析．．．．．． 70
壹、「語文測驗」與「數學算術計算測驗」對「傳統文字問題」的迴歸分析．．．．．．．．．．．．．．．．．．．．．．．
70
貳、「語文測驗」與「數學算術計算測驗」對「故事文字問題」的迴歸分析．．．．．．．．．．．．．．．．．．．．．．．
71

第四節 各組學生在兩種文字問題的解題歷程．．．．．．．．．．．． 73
壹、學生解題歷程階段自評分數的基本描述．．．．．．．．．． 73
貳、學生解題歷程階段的質化描述．．．．．．．．．．．．．． 77

第五節 各組學生在兩種文字問題的錯誤類型．．．．．．．．．．．． 84
壹、各組學生在傳統文字問題上錯誤類型分析．．．．．．．．． 84
貳、各組學生在故事文字問題上錯誤類型分析．．．．．．．．． 90
參、各組學生在兩種文字問題中的錯誤類型比較．．．．．．．．． 95

第六節 學生合作解題得分之差異．．．．．．．．．．．．．．．．．．． 100
壹、各種組合在兩種文字問題得分上的差異．．．．．．．．．． 102
貳、各組學生個人解題與合作解題得分的差異．．．．．．．．．． 103

第七節 學生合作解題的解題歷程．．．．．．．．．．．．．．．．．． 106
壹、各組學生在各種組合中合作解題之歷程的基本描述．．．．． 106
貳、各組學生「個人解題」與「合作解題」的解題歷程改變差異 109
參、各種組合學生解題情況的質化描述．．．．．．．．．．．． 113

第八節 歸納與討論．．．．．．．．．．．．．．．．．．．．．．． 125
壹、各組學生在兩種文字問題的表現．．．．．．．．．．．．． 125
貳、各組學生的解題歷程．．．．．．．．．．．．．．．．．． 127
參、各組學生的錯誤類型．．．．．．．．．．．．．．．．．． 130
肆、合作解題的成效．．．．．．．．．．．．．．．．．．．． 132

第五章 結論與建議．．．．．．．．．．．．．．．．．．．．．． 136
第一節 結論．．．．．．．．．．．．．．．．．．．．．．．．．．． 136
壹、各組學生解兩種文字問題的情況．．．．．．．．．．．．． 136
貳、各組學生合作解題的成效．．．．．．．．．．．．．．．． 138

第二節 在數學教育上的應用．．．．．．．．．．．．．．．．．． 140
第三節 研究限制與建議．．．．．．．．．．．．．．．．．．．． 142

參考文獻．．．．．．．．．．．．．．．．．．．．．．．．．．．． 144

附錄．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．． 150
附錄一、傳統文字問題測驗及解答．．．．．．．．．．．．．．．．．． 150
附錄二、故事文字問題測驗及解答．．．．．．．．．．．．．．．．．． 154
附錄三、文字問題的評量指標．．．．．．．．．．．．．．．．．． 160
附錄四、解題歷程自評表．．．．．．．．．．．．．．．．．．．． 161
附錄五、解題歷程評鑑的指標．．．．．．．．．．．．．．．．．． 162
附錄六、錯誤類型類目表．．．．．．．．．．．．．．．．．．．． 163
附錄七、傳統文字問題的錯誤類型範例．．．．．．．．．．．．．． 164
附錄八、故事文字問題的錯誤類型範例．．．．．．．．．．．．．． 172
附錄九、各組學生之個人解題與合作解題的各階段次數百分比(傳統文字題) ．．．．．．．．．．．．．．．．．．．．．．．．．．．
183
附錄十、各組學生之個人解題與合作解題的各階段次數百分比(故事文字題) ．．．．．．．．．．．．．．．．．．．．．．．．．．．
184
附錄十一、各組學生個人與合作解題之解題歷程列聯表(傳統文字題) 185
附錄十二、各組學生個人與合作解題之解題歷程列聯表(故事文字題) 187
附錄十三、使用學生資料同意書．．．．．．．．．．．．．．．． 189
附錄十四、各組學生「解題歷程階段」與「得分」之獨立性考驗 190





表 目 錄

頁數
表2－1－1 問題的分類．．．．．．．．．．．．．．．．．．．．． 11
表2－1－2 文字問題的分類舉例．．．．．．．．．．．．．．．．．． 13
表2－1－3 各學者對解題歷園的看法之整理，與本研究的解題歷程分類 20
表2－1－4 Charles & Lester (1982)的解題評量表．．．．．．．．． 21
表2－1－5 Schoenfeld (1985)的多重記分．．．．．．．．．．．． 22
表2－1－6 Mcaloon & robinson (1988)的解題評量表．．．．．．．． 22
表2－1－7 本研究文字問題的評量表．．．．．．．．．．．．．．． 23
表2－2－1 文字問題本身的因素．．．．．．．．．．．．．．．．．． 33
表2－2－2 影響解文字問題的因素．．．．．．．．．．．．．．．．．． 37
表2－2－3 學生在比較型問題的錯誤類型．．．．．．．．．．．．． 40
表2－2－4 解題的錯誤類型之分類．．．．．．．．．．．．．．．． 44
表3－1－1 203位學生在兩項測驗上的原始得分與T分數分佈摘要．．． 45
表3－1－2 四組學生的分類結構．．．．．．．．．．．．．．．．．． 46
表3－1－3 各組學生在兩項測驗的T分數摘要．．．．．．．．．．．． 47
表3－4－1 重測信度結果．．．．．．．．．．．．．．．．．．．． 54
表3－4－2 國民中學智力測驗與普通分類測驗之相關．．．．．．．． 55
表3－4－3 國民中學智力測驗與比西量表測驗之相關．．．．．．．． 55
表3－4－4 三位教師之基本資料．．．．．．．．．．．．．．．．．． 56
表3－4－5 兩種正式測驗之內容、題號、難易程度與適合程度．．．． 57
表3－4－6 解題歷程自評表之題號與配合的解題階段．．．．．．．． 58
表4－1－1 所有學生在兩種文字問題上的得分之摘要表．．．．．．． 65
表4－1－2 學生在兩種文字問題上的得分之相依t考驗結果．．．．．． 66
表4－1－3 各組學生在各測驗之差異摘要表．．．．．．．．．．．． 67
表4－2－1 203位學生在兩種背景能力測驗與兩種文字問題之相關係數矩陣．．．．．．．．．．．．．．．．．．．．．．．．
69
表4－3－1 「語文測驗」與「數學算術計算測驗」對「傳統文字問題」之多元迴歸分析摘要表(N＝203) ．．．．．．．．．．．．
70
表4－3－2 「語文測驗」與「數學算術計算測驗」對「故事文字問題」之多元迴歸分析摘要表(N＝203) ．．．．．．．．．．．．
72
表4－4－1 「傳統文字問題」的解題歷程階段自評分數之卡方考驗．．． 74
表4－4－2 「故事文字問題」的解題歷程階段自評分數之卡方考驗．．． 75
表4－4－3 對42位學生之解題歷程觀察統整．．．．．．．．．．．． 77
表4－5－1 各組學生在傳統文字問題的錯誤類型次數統計．．．．．． 85
表4－5－2 各組學生在「無法順利轉譯成代數式子」的次數統計(傳統文字題) ．．．．．．．．．．．．．．．．．．．．．．
86
表4－5－3 各組學生在故事文字問題的錯誤類型次數統計．．．．．． 91
表4－5－4 各組學生在「無法順利轉譯成代數式子」的次數統計(故事文字題) ．．．．．．．．．．．．．．．．．．．．．．
92
表4－5－5 各組的各項錯誤類型之題數比較．．．．．．．．．．．． 96
表4－5－6 各組「操作錯誤」之題數比較．．．．．．．．．．．．． 97
表4－5－7 各組學生在「無法順利轉譯成代數式子」之題數比較．．． 98
表4－5－8 各組「不適當的解題習慣」之題數比較．．．．．．．．． 99
表4－6－1 40位學生個人與合作解題得分之摘要表．．．．．．．．． 101
表4－6－2 各種組合學生在兩種文字問題得分之差異摘要表．．．．． 102
表4－6－3 各組學生在傳統文字問題上個人得分與合作得分之相依t考驗結果．．．．．．．．．．．．．．．．．．．．．．．
104
表4－6－4 各組學生在故事文字問題上個人得分與合作得分之相依t考驗結果．．．．．．．．．．．．．．．．．．．．．．．．
105
表4－7－1 各組學生合作解題之解題歷程的結果摘要(傳統文字問題)． 107
表4－7－2 各組學生合作解題之解題歷程的結果摘要(故事文字問題)． 108
表4－7－3 各組學生個人與合作解題的平均階段(傳統文字問題)．．．． 110
表4－7－4 各組學生個人與合作解題的平均階段(故事文字問題)．．．． 111
表4－7－5 各組「個人解題」與「合作解題」的平均階段差異之χ2值整理表．．．．．．．．．．．．．．．．．．．．．．．．
112
表4－7－6 對40位學生合作解題歷程之觀察摘要．．．．．．．．．． 114
表4－7－7 兩類組合模式的解題行為差異摘要．．．．．．．．．．．． 123
表4－8－1 背景能力對兩種文字問題的影響彙整表．．．．．．．．． 125
表4－8－2 各組的解題歷程與五種解題行為的比較．．．．．．．．． 130
表4－8－3 各組的錯誤類型與四種錯誤類型的比較．．．．．．．．． 131
表4－8－4 「個人解題」與「合作解題」之比較結果彙整表．．．．．． 134












圖 目 錄

頁數
圖2－1－1 個體解決問題的內在歷程(Newell & Simon, 1972, p.89)． 16
圖2－1－2 問題解決流程圖(Glass & Holyoak, 1986, p.67)．．．．． 17
圖3－2－1 研究架構圖．．．．．．．．．．．．．．．．．．．．． 48
圖3－2－2 研究設計圖．．．．．．．．．．．．．．．．．．．．．．．． 49
圖3－3－1 研究流程圖．．．．．．．．．．．．．．．．．．．．．．．． 50
圖3－3－2 正式施測流程圖．．．．．．．．．．．．．．．．．．． 52
圖4－1－1 第四章內容之架構圖．．．．．．．．．．．．．．．．． 64
圖4－6－1 各組合在故事文字題得分之差異解晰圖．．．．．．．．． 103
圖4－8－1 解題模型示意圖．．．．．．．．．．．．．．．．．．． 129

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