The flexible mathematical thinking - the ability to generate and connect various representations of concepts - is useful in understanding mathematical structure and variation in problem solving. In particular, the flexible mathematical thinking with the inventive mathematical thinking, the original mathematical problem solving ability and the mathematical invention is a core concept, which must be emphasized in all branches of mathematical education. In this paper, the author considered a case of flexible mathematical thinking with an inventive problem solving ability shown by his student via real analysis courses. The case is on the proofs of the equivalences of three different definitions on the concept of limit superior shown in three different real analysis books. Proving the equivalences of the three definitions, the student tried to keep the flexible mathematical thinking steadily.

Mathematics is to reflect on your or other people's psychological mathematic activities. Thus, learners need to reflect on their mathematical activities in order to cultivate mathematical thinking attitude and perceive learning contents. For this study, first of all, two classes of the fifth grade (29 students in experimental group and 31 students in control group) in 'Y' elementary school in Dae-gu city were selected as research targets and post-test of learning achievement and mathematical attitude examination were carried out in order to verify the differences of learning achievement and mathematical attitudes between experimental and control groups. The findings of this study mean that students' learning achievement and mathematical attitudes can be improved by applying mathematical reflective activities to the actual class.

This article deals with the teacher training for a minor in mathematics. We investigate the drawbacks of the teacher training which is enforced to resolve the overabundant teachers. And we claim that theachers must get a master's degree in a graduate school of education if they get a secondary teaching certificate.

이 논문은 "산학정의" (고종 4년 1867년, 남병길 편저, 이상혁 교정)의 하편 중 세 번째 부분의 내용 즉, 다원술(방정식의 표기법)에 대해 연구한 것이다. "산학정의" 하편에서의 다원절은 사원술을 가장 빨리 쉽게 이해시키고자 하였으며, 더 좋은 풀이법을 설명하고자 시도한 우수한 저술이다.

Arithmetic education is based not only on concept but also fundamentally on intuition. Pestalozzi understood time, a Kant's transcendental intuition, as numbers, a form of cognition, so that he considered intuition essential in arithmetic education. Pestalozzi and Herbart also recommended the intuitive arithmetic education. Significance of the arithmetic education based on intuition resides in the fact that arithmetic, an expression of nature and the world, is succeeded to modern arithmetic education because numbers, a cornerstone of mathematics, are symbolized as a law of mind reasoning.

The purpose of this study is to introduce the Common Core State Standards (CCSS) for mathematics which is released on June 2, 2010 in the U.S.A. The common core state standards are aligned with college and work expectations, and include rigorous content and application of knowledge through high-order skills. The most distinguishable differences between the CCSS standards and the NCTM standards are that the CCSS standards considers the mathematical modeling as one of the mathematical content domains such as algebra, geometry, ets, and the standards are designed by working together with school leaders of all states.

In this study we study Krutetskii and Krutetskii's monograph 'The Psychology of Mathematical Abilities in Schoolchidren' which is published in Russia(in Russian) and USA(in English). We describe Krutetskii's biography briefly, which was not published in Korea. We analyze the methods, procedures, and results which are presented in the Krutetskii' work, and systemize these. Our study will give a overview of Krutetskii' viewpoint about mathematical abilities.