Virginia Mathematics Teacher Spring 2017

this level of development does not guarantee that a child has developed multiplicative thinking (it does not imply that they have developed iterable units of 1). Many children like the one in Figure 1d are able to use their skip counting to solve multiplication tasks, but cannot work with units as countable objects and thus must depend on their additive strategies to solve such multiplicative tasks. Without further development, children will have trouble developing more sophisticated ways of thinking that require higher levels of units coordination (Ulrich, 2016). This understanding is necessary for multiplicative thinking, and thus many children reach middle school still working with their additive understandings. In addition to encouraging the continued use of skip counting, helping children develop notions of composite units can be aided through the intentional use of well-chosen mathematical manipulatives. For example, when students are working with tasks that ultimately involve multiples (e.g., Figure 1), making available single counters as well as counters clustered in different group sizes may help promote the use of composite units (see Figure 2). Consider a task similar to the Cupcake Task in which there are 15 cupcakes to be put into boxes of 3 cupcakes. Figure 2 shows four models of a solution to this task using different types of manipulatives. Children with an INS may use single objects to represent 15 cupcakes and then make groups of three cupcakes (see Figure 2a). For INS children, providing interlocking cubes (Figure 2b) could make it possible for them to build their own figurative composites, a first step in developing composite units. Making available objects pregrouped by threes (Figure 2c) with the single units still visible could stimulate children’s skip counting and use of a composite unit of 3. In addition, making available unpartitioned rods that represent 3 (Figure 2d) may help promote the notion of an iterable unit of one and a multiplicative unit relationship. Subtle changes in the manipulatives provided to children for solving tasks can afford or constrain their growth in understanding. For example, only having rods like those in Figure 2d for a child with only an INS may actually constrain their development. These children would likely use these “three” rods as single units and count out 15 of these rods and

make groups of 3, because they are not able to recognize that these composite units stand for 3 objects. However, for children with a TNS these rods could afford them the opportunity to develop the notion of an iterable unit of 1, thus helping them construct an ENS. Again, subtle changes in the use of manipulatives can provide opportunities for children to combine their skip counting and notions of composite units to make important growth in the construction of their number sequences. Conclusions Children’s transition from additive to multiplicative thinking represents an important shift in thinking that makes it possible to be successful with more advanced mathematical learning. Children who reach middle school without having made this transition are at a stark disadvantage because most of the middle school mathematics curriculum presumes multiplicative thinking. In this article we highlighted the characteristics of the different stages of number

Figure 2. Different models representing different types of unit structures for a Cupcake Task with 15 cup- cakes.

Virginia Mathematics Teacher vol. 43, no. 2

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