These different ways of thinking may be characterized in terms of a hierarchy of four stages called number sequences (Steffe & Olive, 2010 Steff & Cobb, 1988 Olive, 2001 Ulrich, 2015, 2016). Examples of sixth graders’ solutions to the Cupcake Task You have baked 39 cupcakes and you will put the cupcakes in boxes of three. We will revisit these student solutions after presenting a hierarchy of how students work with units. Finally, the child in Figure 1f seemed to recognize the situation by reversing the context of the problem to one asking: “What multiplied by 3 would give me 39?” In each of these cases the child’s work represents a different way of thinking about and coordinating units. The child’s work in Figure 1d does not represent individual cupcakes, but instead uses skip counting by 3’s to 39, and then counts the number of “counts by 3” (notice the dots under each number). In Figure 1b, the child seems to create groups of 3 “cupcakes” in boxes until all 39 cupcakes are used up, and then counts the number of groups (notice the single dot in the boxes likely representing this counting act). In Figure 1a, the child seems to have first drawn 39 “cupcakes” without attending to groups of 3, then circled groups of 3 to make boxes until they were all used up, and then counted the boxes. All of these children produced the same correct answer, however, based on the solutions, these students all thought very differently about the problem. These solutions will be referred to throughout the article to highlight children’s ways of thinking and to make connections to instructional strategies for advancing children’s early number concepts. To motivate and facilitate our discussion of the ideas outlined above we first discuss several solutions to the task in Figure 1 produced by sixth graders. Moreover, we discuss the distinction between a child understanding, e.g., 5 × 3 as the result of their skip counting by 3s, from a child who has truly developed a multiplicative understanding of 5 × 3 as 5 times as much as 3. We also discuss the placement of skip counting within Virginia’s mathematics Standards of Learning and Curriculum Framework (Virginia Department of Education, 2009) and potential extensions and understandings that could be helpful for curriculum development for elementary-aged children. In particular, we discuss the role of skip counting (e.g., counting by 3’s: 3, 6, 9, 12, 15, 18, 21…) as both an indicator of, and as a way to foster, children’s construction and coordination of units throughout these stages of development. In this article we discuss the characteristic ways of thinking associated with these stages and instructional opportunities for moving children through these stages. Understanding these stages and the nature of children’s ways of thinking about units during each stage is important for teachers as they plan and prepare instructional activities for their students. An important part of this mathematical development is the construction and coordination of units, which does not occur all at once, but through several hierarchical stages (Ulrich, 2015, 2016). This developmental milestone affords children necessary tools for understanding more advanced mathematical concepts that is limited by additive thinking, such as fractions and proportional reasoning. Skip Counting and Figurative Material in Children’s Construction of Composite Units and Multiplicative ReasoningĪn important part of children’s mathematical development in the elementary and middle school years is the transition from additive to multiplicative thinking. The Role of Skip Counting in Children's Reasoning
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