In this study, we aim to improve an educational practice of "speed" that many children are difficult for understanding. In that case, we use the RTMaC lesson study with cooperation of elementary school’s teachers, junior high school’s teachers, high school’s teachers and university's teachers. In this paper, we examined children's cognition of "speed" by a teacher's made original cognition test, and we discussed improvement's teaching points of it. As a result, it has become clear that children’s cognition trend of "speed", and an improvement viewpoint for the teaching.

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In the present-day, it is generally important to teach the probability. However, it is said that both quantity and quality are not enough for teaching the probability in Japan. Especially, the probability is hardly taught in elementary schools. So the purpose of this study is to make the curriculumof teaching the probability in elementary schools. In this paper, we researched cognition about the probability to 4th, 5th and 6th graders who have not been taught the probability. A result of the research, we have found some characteristics of children’s cognition each school year.

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Generally speaking, junior high school students in Japan don’t value mathematics so much for how it can help them in their lives, compared with other countries. However their teachers value it. Educational experiment named ”Radioactivity and Mathematics” was conducted not only to change their evaluation of mathematics value, but also to introduce the exponential function, relating to the Fukushima nuclear accident. In the pretest and the posttest, we clarified the following. (1) The students changed their evaluation of mathematics value for how it could help them in their lives. (2) The students could find and understand as a function they had never learned. (3) Even in the public junior high school, it was possible for the students to learn a fundamental exponential function, if they were equal to surpass the average in scholastic ability. (4) It was not easy for the students to find two variables from the phenomenon in their lives.

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The purpose of this study is to show how students can enhance mathematical creativity. Flexibility is an important factor for creative problem solving. However, in the lesson of the present school mathematics, students’ flexible thinking is hardly used. We found that the inhibitory factor of students’ flexible idea. We analyzed the influence of fixation of a solution by using the problem of constructing a perpendicular bisector. We focused on the relationship between the intelligibility of learning contents and exertion of mathematical creativity. Students had the tendency to memorize a solution without understanding the mathematical background. We made a problem for overcoming students’ flexible thinking. When students used that problem, they were able to consider various solutions than the usual problem.

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The purpose of this study is to propose teaching method to activate the ability of creative problem solving. On the mathematics learning, it is important to acquire basic knowledge and skill, and it is necessary to grasp connection with the previous knowledge systematically. This method is based on students’ understanding about the relationship between method of problem solving and mathematical background. The lesson for carrying out representation of the relationship was practiced at three steps, "showing multiple expressions ", "comparing multiple expression" and "discovering common ground." As a result of using new method for "how to solve simultaneous equation", students were able to improve the ability of creative problem solving by using previous knowledge and skill.

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The new course of study was enforced in the high schools from 2012. In the new course of study for the 1st year high school students, the properties of integers are taught in an elective subject: Mathematics A. It was shown by Euclid in his Elements (volume 9, Proposition 20) that there exist an infinite number of primes. The purpose of this paper is to study the teaching materials about the Proposition 20 (volume 9) in Euclid’s Elements.

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The contents of the plane geometry in secondary education that can draw out student's various ideas are preferable. “Origamics” that the student actually works with the paper folding enables the student's trial and error in the proof problem. The purpose of this study is to propose the systematic plane geometry education using Origamics contents.

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