本研究的目的主要有二：一是透過較長時期的後設認知教學，探討後設認知的技巧該如何教導才會有所成效，二是進一步探討後設認知相關的變數有哪些。本研究的研究樣本共23位，主要是國一與小六的數學資優生，他們大部份均有代表台灣參加國際性的小學數學競賽的經驗，之所以選擇他們當研究樣本的原因是因為一般學生可能不容易應付後設認知教學所帶來的心智負荷，故此，本研究希望藉由教導這群學生，初步探討後設認知技巧是否可以被年齡較低的學生習得。為此，本研究一方面綜合考量學生在數學解題與後設認知量表上的表現，從23人中挑選出5位學生進行教導後設認知技巧的教學實驗。另一方面，相對於後設認知相關因素的探討上，依據學生在尋找規律測驗、抽象符號測驗和選拔賽的成績，將他們分成高、低解題能力兩組，其中高解題能力組有12人，低解題能力組有11人，依此分類方式來比較他們在後設認知表現上的差異，藉以瞭解解題能力與後設認知之間的相關性。
在後設認知的教學方面，本研究以「教師引導式發問」和「相互教學法」為主軸，並輔以共同解題、分組討論、學生上台發表…等方式，為教學組學生進行10次、每次約2小時的後設認知導向的數學解題教學。結果顯示教學組的學生在經過後設認知的教學後，在解題表現與後設認知方面皆有略優於其他組的傾向，且教學組的學生，也大都認為經過後設認知的教學後，對他們在計畫解題步驟與檢查方面的能力上都有所增進，此外，他們對後設認知的教學也都持有正向的看法。
而在後設認知相關變數的探討上，高解題能力組無論是在後設認知量表、學習動機量表以及數學信念問卷的得分上，皆有優於低解題能力者的現象。再者，藉由兩次個別的晤談，也發現高解題能力者不論是在解題或是後設認知的表現上，皆比低解題能力者有較佳的表現。其中，在解題方面，高解題能力者比低解題能力者較具有數學思維、組織能力、解題動機…等。在後設認知方面，高解題能力者亦傾向比低解題能力者有更多的評估、計畫、監控以及檢查等行為出現。可見後設認知與解題能力兩者確有相關存在。
至於後設認知與學習動機及數學信念的關係，藉由典型相關分析(Canonical correlation analysis)的結果得知，在前測的後設認知量表中的「自我修正」與學習動機量表中的「成就動機」以及「追求成功」兩個次量表的表現有關；再者，前測的後設認知量表中的「策略運用」則與數學信念問卷中的「數學功用與數學學習的態度」以及「數學信念與做數學的態度」有關。可見後設認知與學習動機及數學信念亦其相關性存在。
根據上述的研究結果與發現，本研究建議進行後設認知的教學宜以長時期的方式來進行，才比較容易深入瞭解學生的學習成效以及教學實驗的效果。再者，後設認知的教學也應該同時考量學生在學習動機與數學信念的發展，才能達到預期的教學成效。此外，在後設認知的評量工具上，宜將一般的後設認知量表和即時的後設認知問卷併用，才能較真正衡量出學生後設認知能力。最後，本研究建議若有興趣進行後設認知教學，宜從「教師引導式的發問」開始進行。

There are two main purposes for this study. The first one is to explore the effectiveness of a teaching experiment of metacognition for a prolonged period. The other is to identify variables that are related to metacognition. There are 23 subjects in this study. Most of them are sixth or seventh graders who are gifted mathematically. The majority of them had participated in international mathematical contests at the primary level. The reason for choosing them as the subjects for this study is out of the consideration the normal students might have too much a cognitive load to learn metacognition as well as problem solving skills. Hence, this study focuses on studying if metacognitive skills could be learned by mathematically gifted students. For this purpose, five students, on one hand, were selected out of the twenty three students and were taught them metacognitive skills. The criteria of selection were based on multiple consideration, including their performance in mathematical problem solving as well as their rating on the metacognitive questionnaire. On the other hand, this study divided the 23 students into the high and low problem solving ability group ( with 12 students in the former and 11 students in the latter ) so as to investigate the relationship between problem-solving ability and metacognition.
" Teacher guided questioning " and " reciprocal teaching " represented the two major methods used to teach metacognition in this study. The instructional duration amounted to 10sessions, each one lasting for 2 hours. In addition, this study also adopted various instructional strategies, such as problem solving by the whole class, group discussion, and allowing the students to demonstrate their problem solving strategy to their classmates. The findings of this study were that students who were taught metacognitive skills not only had better performance, on the average, in mathematical problem solving and metacognition than others, but they also considered themselves as having improved with respect to their planning and checking ability. Besides, they also had positive attitude toward the kind of metacognitive instruction methods and strategies employed in this study.
As regards variables what variables are related to metacognition, it was found that the students with high problem solving ability had better ratings on the mathematical belief, learning motivation, and metacognition questionnaires than the students with low problem solving ability. Moreover, they also had showed better performance in problem solving and metacognition during the two follow-up interviews after all the instruction were completed. In relation to problem solving, it was found that the students with high problem solving ability possessed better mathematical thinking and organizing ability, they were also more motivation in problem solving etc. than the low ability group. They also had the tendency to exhibit more metacognitive behaviors in evaluating, planning, monitoring, and checking their solution process. Consequently, it is believed that there exists a relationship between metacognition and problem-solving ability.
Canonical correlation was need to explore the general relationship between metacognition questionnaire, and the instruments for mathematical belief and motivation. The result showed that the self connection subscale of the metacognition questionnaire was related to a weighted combination of the motivation for achievement and the success of the motivation questionnaire. Furthermore, the strategy subscale of the metacognition questionnaire was forced to the related to a weighted combination of the belief of students in mathematics and mathematical function subscales of the mathematical belief questionnaire. Hence, it is perceived that metacognition is relation and belief in mathematics in one for another.
Based on the above findings, it is suggested that metacognition should be taught for a longer duration to achieve better effect. As for the metacognitive assessment instruments, researchers should adopt both the general and the instantaneous metacognitive questionnaires in order to get a more comprehensive assessment of students' real metacognitive ability.
Finally, it is suggested that any teacher who is interested in metacognitive instruction should consider begining with the " teacher guided questioning " method.

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