Introduction of Multiplication and Its Extension: How Does Japanese Introduce and Extend?

In Chap. 1, the Japanese approach was explained as developing students who learn mathematics by and for themselves (Isoda, 2015), and also as trying to cultivate human character, mathematical values, attitudes, and thinking as well as knowledge and skills (Isoda, 2012; Rasmussen and Isoda, Research in Mathematics Education 21:43-59, 2019). To achieve these aims, the approach is planned under the curriculum sequence to enable students to use their previous knowledge and reorganize it in preparation for future learning. By using their learned knowledge and reorganizing it, the students are able to challenge mathematics by and for themselves. In relation to multiplication, the Japanese curriculum and textbooks provide a consistent sequence for preparing future learning on the principle of extension and integration by using previous knowledge, up to proportions. (The extension and integration principle (MED, 1968) corresponds to mathematization by Freudenthal (1973) which reorganizes the experience in the our life (Freudenthal, 1991). Exemplars of the Japanese approach on this principle are explained in Chaps. 6 and 7 of this book.) This chapter is an overview of the Japanese curriculum sequence with terminology which distinguish conceptual deferences to make clear the curriculum sequence in relation to multiplication. First, the teaching sequence used for the introduction of multiplication, and the foundation for understanding multiplication in the second grade, are explained. Based on these, further study of multiplication is done and extended in relation to division up to proportionality. The Japanese approach to multiplication is explained with Japanese notation and terminology as subject specific theories for school mathematics teaching (Herbst and Chazan, 2016). The Japanese approach was developed by teachers through long-term lesson study for exploring ways on how to develop students who learn mathematics by and for themselves (Isoda, Lesson study: Challenges in mathematics education. World Scientific, New Jersey, 2015a; Isoda, Selected regular lectures from the 12th International Congress on Mathematical Education. Springer, Cham, Switzerland, 2015b). This can be done only through deep understanding of the curriculum sequence which produces a reasonable task sequence and a concrete objective for every class in the shared curriculum, such as in the Japanese textbooks (Isoda, Mathematical thinking: How to develop it in the classroom. Hackensack: World Scientific, 2012; Isoda, Pensamiento matemático: Cómo desarrollarlo en la sala de clases. CIAE, Universidad de Chile, Santiago, Chile, 2016) (This is also illustrated in Chap. 7 of this book.).