Abstract
Whole-number arithmetic is a core content area of primary mathematics, which lays the foundation for children’s later conceptual development. This paper focuses on teaching whole-number multiplication (WNM) to build a stepping stone for children’s proportional reasoning. Our intention in writing this paper is to obtain a practice-based perspective on the teaching of WNM, given the growing attention being given to significant connections between children’s understanding of multiplication and their proportional reasoning. To this end, we review studies on teaching WNM that were conducted in Japan. Our review highlights several aspects of teaching WNM that are integral to building an initial understanding of proportional relationships. Linear representations such as tape diagrams and number lines are also found to help make explicit the important properties of proportional relationships that can be used progressively across grade levels.
Highlights
Many researchers have suggested that multiplication differs significantly from addition and subtraction (e.g., Clark and Kamii 1996; Fuson 2003; Vergnaud 1983, 1994; Verschaffel et al 2007; Young-Loveridge and Bicknell 2015; Nunes et al 2016)
Drawing mainly on the work by Lamon (2005), we describe several aspects of proportional reasoning that we think are significant to the teaching of whole-number multiplication (WNM)
Three were ones concerning WNM in relation to proportional reasoning, which we describe
Summary
Many researchers have suggested that multiplication differs significantly from addition and subtraction (e.g., Clark and Kamii 1996; Fuson 2003; Vergnaud 1983, 1994; Verschaffel et al 2007; Young-Loveridge and Bicknell 2015; Nunes et al 2016). Multiplication and division involve quantities of different types (e.g., two boxes with three apples per box). This inherent difference makes multiplication not just a faster way of doing repeated addition; it is an operation that requires children to think at a higher level compared to thinking additively (Clark and Kamii 1996). Our strategy for teaching WNM is to attend to the aspect of proportional relationships at an early stage by foreseeing smooth progression of children’s proportional reasoning in later grades
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