Developmental Studies of Metacognition in Mathematics Education

We are often inclined to emphasize only pure mathematical knowledge in its education. And we fail to enact it in students. Consequently, they fail to solve mathematical problems and forget it soon after paper and pencil tests. 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 knowledge and skills which make the objective knowledge active in one's thinking activities. Our unique conception is that this "metacognition" is thought to be originated from and internalized by the teacher him/herself. We regard this internalized metacognition as a substitute to or a copy of the teacher from whom the students learn mathematics. In this context, we refer to "metacognition as "the inner teacher". In this paper, we investigate more deeply the concept of the inner teacher through the analysis of the responses to the questionnaire, specially designed for this research, from elemetary students in the 2nd, 3rd, 4th, 5th grades and those from junior high school students in their second year. We have obtained several findings as follows: 1. Analysing the number of responses to the questionnaire in each grade, we noticed two different conspicuous tendencies, one among the elementary students m the 2nd, 3rd and 4th grades, and the other among the elementary school students in the 4th, 5th and the second-year junior high school students. Elementary school teachers, especially at lower levels, repeatedly talk to the students in terms of metacognition at all stages of problem solving. As a result, students come to have a strong impression about their teacher's utterances. Responses from students in upper grades tend to become stable in terms of the number of items. This is probably because teachers at uppergrades are inclined to limit their talk of metacognition to particular stages of problem solving. 2. In the middle (3rd and 4th) grades, marty over-achievers had a strong impression about such teacher utterances as impressed under-achievers in lower grades. In upper grades, many over-achievers tend to lose interest in those teacher utterances. This seems to indicate that the inner teacher makes a weak impression upon upper grade students, but we assume that this is in fact because it has become automatic for them. 3. The teacher utterances which impressed many over-achievers in middle grades were mainly those on mathematical reasoning and problem solving's strategy