Building Opportunities for Learning Multiplication

This chapter seeks to illustrate a comprehension-based learning approach focused on multiplication. We use episodes to show the potential of a teaching design of multiplication focused on the development of mental calculation based on tasks with appropriate contexts and calculations anchored in the use of benchmark numbers and operation properties, with examples to explain the ideas that are introduced. They are based on a collaborative work experience between a third-grade primary teacher and a researcher (the first author of this chapter). Using these specific examples, we discuss mathematical ideas and didactic options that can guide the actions of teachers when teaching multiplication. Finally, we discuss some points of convergence between the approach introduced here and the lesson study approach.

“Necklaces”: A Didactic Sequence for Missing-Value Proportionality Problems

This research presents a sequence of didactic situations involving a proportionality relationship in which every value in a set (a number of necklaces) is mapped to a pair, a triad, or a quartet of values (numbers of blue beads, red beads, green beads, etc., required to make that number of necklaces) from another set. The sequence includes relatively simple multiplication and division problems, as well as more complex “missing-value” problems. This paper also presents the results of applying the sequence with a group of 4th grade students in a Mexican primary school (9 and 10 years old).

Can We Explain Students’ Failure in Learning Multiplication?

Teaching multiplication is a compulsory topic in elementary education mathematics programs. Much time is dedicated to teaching multiplication tables and algorithms, obtaining mediocre results. Recent investigations in neuroscience suggest that our way of teaching is not congruent with how the brain works and, as such, important changes should be made in teaching of numerical facts and algorithms. Additionally, other kinds of calculation, like mental calculation and calculations with a calculator, have not yet reached the level of importance of their use in education that citizens require in a contemporary society.

Multiplication of Whole Numbers in the Curriculum: Singapore, Japan, Portugal, the USA, Mexico, Brazil, and Chile

This chapter shows how the teaching of multiplication is structured in national curriculum standards (programs) around the world. (The documents are distributed by national governments via the web. Those documents are written in different formats and depths. For understanding the descriptions of the standards, we also refer to national authorized textbooks for confirmation of meanings.) The countries chosen for comparison in this case are two countries in Asia, one in Europe, two in North America, and two in South America: Singapore, Japan, Portugal, the USA (where the Common Core State Standards (2010) are not national but are agreed on by most of the states), Mexico, Brazil, and Chile, from the viewpoint of their influences on Ibero-American countries. (The National Council of Teachers of Mathematics (NCTM) standards (published in 2000) and the Japanese and Singapore textbooks have been influential in Latin America. Additionally, Portugal was selected to be compared with Brazil). To distinguish between each country’s standard and the general standards described here, the national curriculum standards are just called the “program.” The comparison shows the differences in the programs for multiplication in these countries in relation to the sequence of the description and the way of explanation. The role of this chapter in Part I of this book is to provide the introductory questions that will be discussed in Chaps. 3, 4, 5, 6, and 7 to explain the features of the Japanese approach. (As is discussed in Chap. 1, the Japanese approach includes the Japanese curriculum, textbooks, and methods of teaching which can be used for designing classes, as has been explored in Chile (see (Estrella, Mena, Olfos, Lesson Study in Chile: a very promising but still uncertain path. In Quaresma, Winsløw, Clivaz, da Ponte, Ní Shúilleabháin, Takahashi (eds), Mathematics lesson study around the world: Theoretical and methodological issues. Cham: Springer, pp. 105–122, 2018). The comparison focuses on multiplication of whole numbers. In multiplication, all of these countries seem to have similar goals—namely, for their students to grasp the meaning of multiplication and develop fluency in calculation. However, are they the same? By using the newest editions of each country’s curriculum standards, comparisons are done on the basis of the manner of writing, with assigned grades for the range of numbers, meanings, expression, tables, and multidigit multiplication. The relationship with other specific content such as division, the use of calculators, the treatment of multiples, and mixed arithmetic operations are beyond the scope of this comparison. Those are mentioned only if there is a need to show diversity.

The Teaching of Multidigit Multiplication in the Japanese Approach

Multidigit multiplication in vertical form uses the idea of the distributive law such as 27 × 3 = (20 + 7) × 3 = 20 × 3 + 7 × 3 for using a multiplication table under the base ten place value system. Multiplication in vertical form is not simply repeated addition such as 27 + 27 + 27. In this meaning, through the extension of multiplication from single digit to multidigit by use of vertical form with a multiplication table, students have to integrate their knowledge on the base ten system with the definition of multiplication by measurement (a group of groups; see Chaps. 10.1007/978-3-030-28561-6_3, 10.1007/978-3-030-28561-6_4, 10.1007/978-3-030-28561-6_5, and 10.1007/978-3-030-28561-6_6 of this book) and so on. How does the Japanese approach enable students to develop multiplication in vertical form by and for themselves based on their learned knowledge?This chapter illustrates this process as follows. Firstly, the diversity of multiplication in vertical form is explained in relation to the multiplier and multiplicand, and the Japanese approach in comparison with other countries such as Chile and the Netherlands is clearly illustrated. Secondly, how a Japanese teacher enables students to develop multiplication in vertical form beyond repeated addition is explained with an exemplar of lesson study. Thirdly, the exemplar illustrates a full-speck lesson plan under school-based lesson study which demonstrates how Japanese teachers try to develop students who learn mathematics by and for themselves including learning how to learn (see Chap. 1). Fourthly, it explains the process to extend multiplication in vertical form to multidigit numbers by referring to Gakko Tosho textbooks.

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

In Chap. 10.1007/978-3-030-28561-6_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. 10.1007/978-3-030-28561-6_6 and 10.1007/978-3-030-28561-6_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. 10.1007/978-3-030-28561-6_7 of this book.).

An Ethnomathematical Perspective on the Question of the Idea of Multiplication and Learning to Multiply: The Languages and Looks Involved

This chapter invites appreciation of the development of an ethnomathematical perspective on the question of the idea of multiplication. The teaching approach described here is grounded on miniprojects that integrate diverse areas of knowledge. It reveals a style of work being performed in the Waldorf Schools of São Paulo, Brazil, where the concept of multiplication is constructed together with the geometry of plane figures through the elaboration of mathematical thinking together with figures mounted on a circular wooden table. The sequence highlights ideas of context connected to the use of cellular phones by the students to introduce the concept of proportionality by taking photos of their bodies and faces, and then using them to study Leonardo da Vinci’s Vitruvian Man.

Teaching the Multiplication Table and Its Properties for Learning How to Learn

Why do the Japanese traditionally introduce multiplication up to the multiplication table in the second grade? There are four possible reasons. The first reason is that it is possible to teach. The second reason is that Japanese teachers plan the teaching sequence to teach the multiplication table as an opportunity to teach learning how to learn. The third reason is that memorizing the table itself has been recognized as a cultural practice. The fourth reason is to develop the sense of wonder with appreciation of its reasonableness. The second and the fourth reasons are discussed in Chap. 10.1007/978-3-030-28561-6_1 of this book as “learning how to learn” and “developing students who learn mathematics by and for themselves in relation to mathematical values, attitudes, ways of thinking, and ideas.” This chapter describes these four reasons in this order to illustrate the Japanese meaning of teaching content by explaining how the multiplication table and its properties are taught under the aims of mathematics education. In Chap. 10.1007/978-3-030-28561-6_1, these were described by the three pillars: human character formation for mathematical values and attitudes, mathematical thinking and ideas, and mathematical knowledge and skills.

Introduction: Japanese Theories and Overview of the Chapters in This Book

This introductory chapter explains the origin of this book and provides overviews of every chapter in Parts I and II of the book. Part I of the book is aimed at explaining what multiplication and lesson study are in relation to the Japanese approach. It provides an overview of Japanese theories on mathematics education for developing students who learn mathematics by and for themselves and provides necessary ideas to understand the Japanese approach and lesson study. Part II consists of contributions from leading researchers in Ibero-America. Through their contributions, this book provides various perspectives based on different theories of mathematics education which provide the opportunity to reconsider the teaching of multiplication and theories.

Japanese Lesson Study for Introduction of Multiplication

In Chap. 10.1007/978-3-030-28561-6_2, we posed questions about the differences in several national curricula, and some of them were related to the definition of multiplication. In Chap. 10.1007/978-3-030-28561-6_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. 10.1007/978-3-030-28561-6_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. 1–8). Tsukuba, Japan: University of Tsukuba.). A comparison with Chile is given in order to demonstrate the sense of it from the teacher’s 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 21:43–59, 2019), as seen in Figs. 10.1007/978-3-030-28561-6_4#Fig2 and 10.1007/978-3-030-28561-6_4#Fig3 in Chap. 10.1007/978-3-030-28561-6_4 of this book.