Modern society requires us to become statistically literate. The right and proper understanding and interpretation of statistical hypothesis testing is essential to statistical literacy, however it is frequently confused with mathematical proof by contradiction. This study explores how statistical hypothesis testing can be distinguished from mathematical proof by contradiction, that is, what is specifically different between them. To achieve the purpose in this study, their logical structures are compared using the analytical framework: argument. Consequently, four differences are found. In statistical hypothesis testing, unlike mathematical proof by contradiction, (a) its premise is mostly non-mathematical statement and not invariable, (b) contradiction in the strict sense of the word does not arise, (c) its conclusion/claim cannot always be supported and defended by its premise, and (d) defending its conclusion/claim is necessary. The analogical approach with proof by contradiction will work effectively when hypothesis testing is taught and learned, but using that approach alone hypothesis testing has the risk of being assimilated into proof by contradiction. To understand the essence of statistical hypothesis testing properly, it is necessary to compare intentionally hypothesis testing with proof by contradiction and characterize the former as not the same as the latter.

The objective of this paper is

(1) to revisit briefly the conception of mathematics education as a design science as it has been evolving alongside developmental research in the project "Mathe 2000" from 1987 to 2012

(2) to report in some detail on recent developments in the follow-up project "Mathe 2000+," as concerns both conceptual and practical issues.

The paper is a plea for appreciating and (re-)installing "well-understood mathematics" as the natural foundation for teaching and learning mathematics.

A fundamental problem in mathematics education is “What can we do for students to enhance their mathematical ability and achievement and how can we do it?” This paper searches for a promising solution to this problem. To that end, the author focuses on the lesson study approach, specifically adopting the same approach adopted by the mathematician George Polya, who reflected on his experience and described his methods of mathematical problem solving. As a mathematics educator, the author shares his experience of two types of lesson study of primary school mathematics, and proposes a framework of a dynamic cycle in the lesson study of school mathematics for promoting the professional development of mathematics teachers and educators. This professional development may improve the mathematical ability and achievement of students at schools.

The organization and procedures with lesson study are widely understood internationally; however, how Japanese teachers have made efforts to enhance the quality of mathematics lessons requires clarification if Japan is to make an international contribution in this regard. In this study, to approach the quality of mathematics lessons, we focused on narrative structures in mathematics lessons as beneficial viewpoints for understanding the cultural background of lessons in Japan. Especially, we considered sequentiality and the dual nature of acting and the complementary characteristic of the dual landscape as useful in clarifying the origins and development of the narrative for mathematics lessons. As an example showing the efforts of teacher study groups to develop quality lessons, we introduced a collaborative project “mathematics lessons incorporating students’ ‘questions’ as a main axis.” We examined the project in terms of how the teacher and students created the narrative of mathematics lessons based on the students’ continuous exploration of the subject. We conclude that a pioneering spirit on the part of teachers, which is diametrically opposed to the stereotyped approach to lesson study, should be introduced so that Japan can make a real international contribution in this area.