不同數學文字問題之解題與合作解題研究

Abstract

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.