[[abstract]]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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