Virginia Mathematics Teacher Spring 2017

begin their early counting as a sing-song, children also learn to skip count through repetition and song. Based on the Virginia Standards of Learning (SoLs; VDOE, 2009) children in Kindergarten are encouraged to count by fives and tens to 100 (see SoL K.4 in Table 1). Continuing in first grade, children are also encouraged to count by 2’s to 100 (see SoL 1.2 in Table 1). In second grade, children continue to skip count by 2’s, 5’s, and 10’s (see SoL 2.4 in Table 1). The Curriculum Framework (VDOE, 2009) highlights the role of skip counting for the general development of numerical patterns, as well as for use in very specific mathematical applications. For example, skip counting by 2’s lays the groundwork for understanding even and odd numbers (e.g., SoL 1.2); counting by 10’s lays the groundwork for place-value and money (e.g., SoLs 1.2 and 2.4); skip counting by fives lays the groundwork for telling time and counting money (SoL K.4). In all cases, skip counting is promoted for its relationship to learning multiplication facts. Beyond 2’s, 5’s, and 10’s, skip counting by other numbers is not explicitly highlighted in the Standards of Learning or Curriculum Framework. After second grade, skip counting is abandoned as an essential part of the SOLs and replaced with a focus on the learning of multiplication facts (SoL 3.5, Table 1). Beyond the specific role that counting by 2’s, 5’s, and 10’s has for developing particular mathematical concepts as prescribed in the VDOE Curriculum Framework, skip counting, in and of itself, can represent important developmental shifts in working with composite units and should be

encouraged for its own sake. Once children can relate their skip counting to their multiplication facts it should not then be assumed that children have developed multiplicative thinking. Skip counting alone does not imply the development of the multiplicative understanding inherent in an ENS or GNS. By encouraging children to use their skip counting for solving tasks involving multiplicative situations instead of relying on number facts, they may develop a more intentional and explicit awareness of the relationship between their skip count and the types of units with which they are working. This awareness affords children with increased opportunities to develop more powerful multiplicative understandings. Skip Counting and Number Sequences Here we discuss the solutions in Figure 1 as a way to exemplify different number sequences and the possible role of skip counting. In Figure 1a, the child represents all 39 “cupcakes” before making groups of three. This is a possible strategy that a child with an INS could use to solve this task. Interestingly, children with only an INS can successfully use skip counting to determine the cardinality of a set, but would be unable to keep track of their skip counting (Olive, 2001). For example, a child with an INS could have used skip counting to recount the cupcakes to make sure that they had created 39, but if asked “how many threes?” the question would not make sense since each “three” stands in for a numerical composite in which the 3’s represent three objects instead of one countable thing. As seen in the solution in Figure 1a, they would need to create groups of 3 and then

Table 1. Standards of Learning associated with skip counting.

K.4

The student will a) count forward to 100 and backward from 10; b) identify one more than a number and one less than a number; and c) count by fives and tens to 100.

1.2

The student will count forward by ones, twos, fives, and tens to 100 and backward by ones from 30.

2.4

The student will a) count forward by twos, fives, and tens to 100, starting at various multiples of 2, 5, or 10; b) count backward by tens from 100; and c) recognize even and odd numbers.

3.5 The student will recall multiplication facts through the twelves table, and the corresponding division facts.

Virginia Mathematics Teacher vol. 43, no. 2

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