Virginia Mathematics Teacher Spring 2017

instructional strategies for advancing children’s early number concepts. All of these children produced the same correct answer, however, based on the solutions, these students all thought very differently about the problem. 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. 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). 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). 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. We will revisit these student solutions after presenting a hierarchy of how students work with units. 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). These different stages describe how children work with units and coordinate them when working with the counting numbers. These four stages are referred to as the initial number sequence (INS), tacitly nested number sequence (TNS), explicitly nested number sequence (ENS) and the generalized number sequence (GNS). In this paper we focus on the first three sequences as they relate to skip counting and describe the extension of thinking required for the GNS which ultimately lays the groundwork for more advanced mathematical understanding. Here we briefly highlight the important characteristics of these stages; for a more detailed discussion, the interested reader should consult Ulrich (2015, 2016) and Olive (2001). The Number Sequences Prior to a child developing an INS, they are considered pre-numerical (Steffe & Olive, 2010; Olive, 2001). That is, for these students, numbers themselves do not represent cardinality, a quality of

a set, but instead are one part of their counting activity. At this point, children may be able to count a set of objects, but they would interpret the question, “How many?” as a request to say a sequence of numbers (e.g., 1, 2, 3, …, 7) while pointing to each object, not as a question about how “big” the set is. Furthermore, after counting a set of objects, if given additional objects and asked how many in all, this child would need to count all of the objects, first recounting the original set. A child who has constructed an INS (the first number sequence) recognizes that a number, such as 7, describes the cardinality of a set of objects and can stand in for counting them, that is, counting “1, 2, 3, …, 7” (Olive, 2001), an initial number sequence. In this case, the 7 is a numerical composite of units representing the result of counting the seven objects, and can serve as a starting point for additional counting. However, the 7 is not recognized as a unit that could be used to count with: “with an INS… [the number words] can only be used to symbolize the results of counting acts; they cannot yet be used as input for counting acts” (Olive, 2001, p. 6). The development of an INS affords a child with the ability to count-on , that is, if after counting a set of 7 objects, a child is given three more objects and asked, “How many altogether?” they would likely count: “7; 8, 9, 10,” while touching the three additional objects, or using their fingers to keep track of the additional three objects; and answer “10,” instead of having to count-all. A child with an INS will often rely on figurative materials to keep track of counting. Consider the child’s solution in Figure 1a. In order to solve this task, the child needs to represent all 39 “cupcakes,” making groups of three until they are all used up, and then count the number of groups. For a child with only an INS their counting acts are limited to using strings of the number sequence beginning at 1 (Ulrich, 2015). Later on, a child begins to recognize that there are subsequences embedded within larger sequences that can be used to aid them in their counting acts. For example, given 7 objects, a child is asked how many there would be if they added on 12 objects. In order to keep track this child might begin at 7 and count as follows: “8 is one more, 9 is two

Virginia Mathematics Teacher vol. 43, no. 2

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