Van Hiele의 數學 學習水準 理論에 대한 小考

Abstract

In 1957 the van Hieles, Dutch mathematics teachers, reported on studies about a level theory of mathematical learning. Mathematics educators at the Soviet Academy of Pedagogical Sciences hastened to organize intensive research on the levels of development outlined by the van Hieles, verified the validity of their assertions, and accepted those as a basis for designing a new mathematics curriculum in 1960s. According to I. Wirszup, the new curriculum is the most radical change in Russian mathematics education in nearly a century. Recently, the levels of mathematical thinking approach has been the focus of increasing interest in the United States of America, and several recent studies have tested the hypotheses that the levels can be identified and form a hierarchy as described by the van Hieles. The van Hieles level theory has a great potentiality for uses in the mathematics curriculum development and classroom practices. In this paper, we found that. i) The basic principle of the van Hieles level theory emphasizes the process aspect of mathematics, the characteristics of mathematical thought; treating the means to put in order phenomena as themselves subjects of study, the inner order of thought as subject of study, and alternating patterns and subjects, forms and contents. This is supported also by C.S. Peirce, J. Piaget, H. Freudenthal, Z.P. Dienes and L. Burton. ii) The teaching and learning theories of Gestalt psychologists, }.S. Burner, H. Aebli and Z.P. Di enes, could not sufficiently account the kind of learning emphasizing the charac-teristics of mathematical thought. This very fact allows us recognize the value of the van Hieles theory in didactics of mathematics. iii)The criticism of P.M. van Hiele about Piagetian theory-Piaget distinguished only two levels, action and operational level, and did not distinguished more. Piaget used the term "logic" deviatingly and tried to deduce logical thinking directly from action. Piaget mistakenly thought that the concrete operational thoughts based on logic and natural number concept develops side by side with logic. piaget did not see the very important role of language in moving from one level to the next. Piaget did got into the absolute point of view about mathematics. Piaget did not clearly distinguished between development by biological maturation, confrontation with the cultural environment, and learning process, thus his theory has little value from the educational point of view. Piaget interpreted the subjects' reactions from his own perspective in order to verify his own assertions. These could be considered as misconceptions originating from his insufficient understanding about Piagetian theory, and the van Hieles level theory could be considered as an application of the operational constructivism to mathematics teaching and learning.