Virginia Mathematics Teacher Spring 2017

count these groups (or draw boxes while continually recounting the number of cupcakes they have used up). The solution in Figure 1b shows a child’s use of numerical composites to find a solution. Notice that each unit contains 3 visible objects that have been counted one by one, representative of a strategy by a child with an INS. The need to recount all the cupcakes after each “box” of cupcakes is drawn suggests a lack of a keeping track strategy, and the lack of a composite unit. In Figure 1c, the child has explicitly linked their figurative composite of 3 with their skip counting. This awareness of the link between skip counting and composite units is characteristic of a child with a TNS. In Figure 1d, we see a child’s solution using only skip counting and no representation of individual cupcakes. Here it is clear that the elements of the skip count are used to represent one composite unit of 3, in this case, a box of cupcakes. After reaching “39” notice that the child seems to have then counted the number of “boxes” indicated by a single dot associated with each numeral in the count. By writing down the skip count, children with a TNS are able to keep track of how many times they have applied their composite unit. In Figure 1e, we see a child who even more clearly interprets each composite of 3 as a single countable unit, indicated by the number sequence 1 to 13 (instead of the skip count) to find the number of boxes. As sophisticated as it may seem for a child to use skip counting to solve a task that seems multiplicative, we can not necessarily infer that the child is thinking multiplicatively. The children in 1d and 1e may still be using additive, not multiplicative, structures, as they are not representing a comparison between two numbers (Ulrich, 2016). Instead, they are describing their counting activity, e.g., “I had to count by 3 thirteen times to get to 39” (Ulrich, 2016). This is not representative of multiplicative thinking. Furthermore, children who can link their multiplication facts to their skip counting have not necessarily developed multiplicative thinking, but instead may be reinterpreting their multiplication facts in terms of additive skip counting. It is important not to de-emphasize children’s use of skip counting as implicitly suggested by the Curriculum Framework (VDOE, 2009) in favor of

a focus on multiplication facts alone. Continued use of skip counting for INS and TNS students focuses them on their use of composite structures and allows them to reflect on these composites. Focusing students on multiplication facts alone hides the iterations of composites involved in multiplication and may limit the ability of students to explicitly reflect on their number sequences, necessary for an ENS and GNS, further constraining their development of multiplicative structures (Ulrich, 2016). The solution presented in Figure 1f suggests that the child has constructed an ENS or GNS as they are able to repose the question as “Three of what makes thirty-nine?” Although they, too, may have counted by 3’s to reach to 39, they are able to reconceptualize 39 as 13 groups of 3, which represents a relationship between two numbers, or a multiplicative structure. A solution that many students gave for the Cupcake Task was to simply write “13.” Such an answer, without the apparent use of figurative materials, is only possible for children who have constructed an ENS; they are able to keep track of the numerical units of composite units that are necessary to solve this task. Finally, it is important to point out that many students were able to solve this task by recognizing it as a division problem, and carrying out the necessary division to reach an answer of 13. This particular solution strategy is not necessarily indicative of a higher stage of thinking (e.g., ENS, GNS), as many TNS children are able to strategically perform the necessary procedures associated with a given type of task without using multiplicative thinking. Implications for Teaching Recent research suggests that a significant number of children make it to middle school without having constructed an ENS (Ulrich & Wilkins, 2016a, 2016b). This suggests that many children in the sixth grade are still predominately additive thinkers and thus not poised for handling multiplicative thinking and relative thinking that is necessary for fractional and proportional thinking. It is thus important for teachers to provide continued opportunities for children to develop multiplicative thinking as early as possible. We have discussed the importance of skip counting to help children develop composite units, however,

Virginia Mathematics Teacher vol. 43, no. 2

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