Virginia Mathematics Teacher Spring 2017

more, 10 is three more, 11 is four more, …, 19 is twelve more, the answer is twelve.” This child has a tacit awareness of the subsequence 1 to 12 nested within the larger sequence from 1 to 19. This child has constructed a tacitly nested number sequence (Olive, 2001, Ulrich, 2015; Steffe & Olive, 2010). Double counting, as in the example, is a characteristic action of a child with a TNS. This awareness of the cardinality of the subsequence (1 to 12) irrespective of its location in the larger sequence suggests that the child is now able to work with a composite unit (Ulrich, 2015). Different from a numerical composite, in which the count stands in for the counting sequence 1 to the number, a composite unit is taken to represent the cardinality of the counting acts irrespective of where the subsequence occurs in the larger sequence. Also characteristic of a child with a TNS is the use of skip counting to solve tasks. For example, counting by 3’s, each count represents a composite unit of 3. With this ability, children can begin to answer questions such as, how many threes are in thirty-nine, by keeping track of their skip count: “3; 6; 9; 12, …”. Each of the numbers in the skip count represent a subsequence of cardinality 3, or a composite unit of 3. The child’s use of skip counting in Figure 1d is characteristic of this sort of thinking. The child uses skip counting by 3’s to 39, recognizing that each 3 represents a box of cupcakes (composite unit) and then counts the number of “counts by 3” and indicates “13 boxes” as their answer (notice the dots under each numeral likely representing the child’s count). Compare this to the solution in Figure 1e. Here, too, it is apparent that the child is working with composite units, but they no longer have to refer to their skip count but recognize that they are interested in the number of units and can use the sequence 1 to 13 to keep track of the units. With a TNS, the composite units remain tacit for a child. That is, these units are available to work with during counting activity, but children are not explicitly aware of the units prior to counting; they are reproduced through counting. Once children are able to reflect on units as a given, we say that a child has developed an explicitly nested number sequence (ENS). The defining characteristic of this stage is a child’s ability to construct an iterable unit of one (Ulrich, 2016;

Steffe and Olive, 2010; Olive, 2001). Here a child is able to recognize the activity of “adding one more” to the point that these additions are interchangeable units. That is, number words no longer only represent the result of counting, but represent a multiplicative relationship associated with the number of iterable units: for example, “7” no longer only represents “1, 2, 3, 4, 5, 6, 7,” but instead represents 7 ones, or 7 times as much as one unit. At this point the composite unit does not stand for a subsequence, but stands in for a multiplicative relationship. Olive (2001, p. 7) distinguishes a child with an ENS from a child with a TNS by comparing their activity for solving 1+1+1+1+1. For a child with only a TNS they would potentially need to solve this problem in steps by calculating the nested sums: 1+1 is 2, 2+1 is 3, …, 4+1 is 5. Whereas, for a child with an ENS the sums are taken as given, and they recognize 1+1+1+1+1 as simply 5 ones, and also recognize the reversibility of the relationship, that five ones are the same as one five. Children with an ENS can reflect on multiplicative situations (Ulrich, 2016; Olive, 2001) involving multiple levels of units. For example, combining 4 groups of 7 objects, can be viewed as making a composite unit of composite units. In other words, 4 groups of 7 objects is seen as a numerical composite of 4 composite units, each of which is a composite unit of 7 iterable units of one. Furthermore, a child can view this combination as a composite unit of 28 iterable units. However, a child with only an ENS has to create composite units of composite units—28 as 4 groups of 7 objects—in the moment. The child would have trouble operating on 28 without losing track of the 4 groups of 7. Although we will not elaborate greatly, for completeness, a child who has constructed a generalized nested number sequence (GNS) can work fluently with a composite of composite units because their composite units are now iterable in the same way units of 1 were iterable for students who have an ENS. Skip Counting Skip counting (e.g., counting by 3’s: 3, 6, 9, 12, 15, …) is often introduced to young children as a way to further develop their counting skills and build their knowledge of multiples. Much like they

Virginia Mathematics Teacher vol. 43, no. 2

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