## Abstract

In Chap. 2, we posed questions about the differences in several national curricula, and some of them were related to the definition of multiplication. In Chap. 3, several problematics for defining multiplication were discussed, particularly the unique Japanese definition of multiplication, which is called definition of multiplication by measurement. It can be seen as a kind of definition by a group of groups, if we limit it to whole numbers. In Chap. 4, introduction of multiplication and its extensions in the Japanese curriculum terminology were illustrated to explain how this unique definition is related to further learning. Multiplicand and multiplier are necessary not only for understanding the meaning of multiplication but also for making sense the future learning. The curriculum sequence is established through the extension and integration process in relation to multiplication. In this chapter, two examples of lesson study illustrate how to introduce the definition of multiplication by measurement in a Japanese class. Additionally, how students develop and change their idea of units that any number can be a unit in multiplication beyond just counting by one is illustrated by a survey before and after the introduction of multiplication. After the illustration of the Japanese approach, its significance is discussed in comparison with the Chilean curriculum guidebook. Then, the conclusion illustrates the feature of the Japanese approach as being relatively sense making for students who learn mathematics by and for themselves by setting the unit for measurement (McCallum, W. (2018). Making sense of mathematics and making mathematics make sense. Proceedings of ICMI Study 24 School Mathematics Curriculum Reforms: challenges, changes and Opportunities (pp. 18). Tsukuba, Japan: University of Tsukuba.). A comparison with Chile is given in order to demonstrate the sense of it from the teachers side. In relation to lesson study, this is a good exemplar of how Japanese teachers develop mathematical thinking. It also illustrates the case for being able to see the situation based on the idea of multiplication (Isoda, M. and Katagiri, S. (2012). Mathematical thinking: How to develop it in the classroom. Singapore: World Scientific; Rasmussen and Isoda Research in Mathematics Education 21:4359, 2019), as seen in Figs. 4.2 and 4.3 in Chap. 4 of this book.

Original language | English |
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Title of host publication | Teaching Multiplication with Lesson Study |

Subtitle of host publication | Japanese and Ibero-American Theories for International Mathematics Education |

Publisher | Springer International Publishing |

Pages | 103-131 |

Number of pages | 29 |

ISBN (Electronic) | 9783030285616 |

ISBN (Print) | 9783030285609 |

DOIs | |

State | Published - 1 Jan 2020 |

## Keywords

- Make sense
- Meaning
- Multiplication
- Sense making