Study on learning process and its characteristics about interpreting  and using Euclidean algorithm:  On the basis of mathematical activities focused on spiral method

Abstract

　　 The aim of this paper is to study the learning process of interpreting and utilizing Euclidean algorithm and its characteristics. For this purpose, students will review what they learned in the process of learning common divisor and the greatest common divisor, or GCD when they were elementary students at first. Then they are introduced to Euclidean algorithm and will be able to understand it. The authors designed classes to develop learning which makes use of interpretations of Euclidean algorithm, and analyze qualitatively students’ activities and their descriptions made in classrooms. The findings are as follows: First, activities to revise the learning of GCD with an elementary school textbook of arithmetic not only increase the probability of getting different ideas from learning to interpret Euclidean algorithm, but also attach new meanings to the learning and lead to finding new values by connecting and overlooking the learning of GCD. Secondly, activities to understand Euclidean algorithm proves they have various ideas and expression styles. Discussion among the students advances the mutual understanding of the ideas as well as further interpretations, which also promotes motivation to learn and help students understand more deeply in the end. Finally, experiencing activities to learn Euclidean algorithm is applied to the indeterminate equation, which helps students deeply understand the solution to the indeterminate equation.