抄録
We are often inclined to emphasize only the pure mathematical knowledge in its education. And we fail to make them active in students. Moreover, they fail to solve the mathematical problem and forget them soon after the paper and pencil test. Recently, metacognition has come to be noticed as an important function of human cognitive activities among researchers of mathematics education as well as among professional psychologists. Roughly speaking, we could regard "metacognition" as the knowledges and skills which make the objective knowledges active in one's thinking activities. Our unique conception is that this metacognition is thought to be originated and internalized from the teacher him/herself. So, we think this internalized metacognition is really a substitute or a copy of the teacher from whom the student learns. In this context, we refer to the subject of metacognition as "the inner teacher". In this paper, we developed a method of recording and analysing students' inner activity including cognitive-metacognitive aspect using paper and pencil as follows; [table] For this paper, student was indicated his/her cognitive activity at the left side of paper as usual answer in paper/pencil test and other thinking activity (including cognitive and metacognitive one) at the right side. After students answered, students are interviewed by turns after a junior school year. At first interviewer asks a question pointing at first line. "Is this a first statement occurred in your brain?" Similarly, the interviewer continues to ask a question, and adds some statements using red pen which students can't answer at his paper. Finally, the interviewer links students' statement using red pen like thinking a route map. In this paper, we implemented case study for 33 students in the university. Some findings in this study are as follows: 1. According to the quantitive aspect of "the inner teacher", students could not answer their full inner activity including cognitive and metacognitive one. So, the interview must be always made much of. 2. In analysing students' papers, we found the metacognition played a vital role. 3. Finally, we picked up and analysed three students' answers for getting their thinking route map qualitatively.
