Jesse L. M. Wilkins and Catherine Ulrich
Skip Counting and Figurative Material in Children’s Construction of Composite Units and Multiplicative Reasoning
An important part of children’s mathematical development in the elementary and middle school years is the transition from additive to multiplicative thinking. This developmental milestone affords children necessary tools for understanding more advanced mathematical concepts that is limited by additive thinking, such as fractions and proportional reasoning. An important part of this mathematical development is the construction and coordination of units, which does not occur all at once, but through several hierarchical stages (Ulrich, 2015, 2016). Understanding these stages and the nature of children’s ways of thinking about units during each stage is important for teachers as they plan and prepare instructional activities for their students. In this article we discuss the characteristic ways of thinking associated with these stages and instructional opportunities for moving children through these stages. In particular, we discuss the role of skip counting (e.g., counting by 3’s: 3, 6, 9, 12, 15, 18, 21…) as both an indicator of, and as a way to foster, children’s construction and coordination of units throughout these stages of development. We also discuss the placement of skip counting within Virginia’s mathematics Standards of Learning and Curriculum Framework (Virginia Department of Education [VDOE], 2009) and potential extensions and understandings that could be helpful for curriculum development for elementary-aged children. Moreover, we discuss the distinction between a child understanding, e.g., 5 × 3 as the result of their skip counting by 3s, from a child who has truly developed a multiplicative understanding of 5 × 3 as 5 times as much as 3.
To motivate and facilitate our discussion of the ideas outlined above we first discuss several solutions to the task in Figure 1 produced by sixth graders. These solutions will be referred to throughout the article to highlight children’s ways of thinking and to make connections to 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.
You have baked 39 cupcakes and you will put the cupcakes in boxes of three. How many boxes will you fill?
Figure 1. Examples of sixth graders’ solutions to the Cupcake Task
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 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 (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 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.
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.
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 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, 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.
Figure 2. Different models representing different types of unit structures for a Cupcake Task with 15 cupcakes.
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 sequences that children move through as they construct their notions of composite and iterable units on their way to developing multiplicative thinking. We highlighted the important role of skip counting as both an indicator of, and a way to foster, children’s development of composite units. We recommend that teachers continue to emphasize the use of skip counting as a way to develop students’ multiplicative thinking beyond just the connection with multiplication facts. Prematurely deemphasizing skip counting may cause children to focus only on the idea that 5 × 3 is the result of their additive counting by 3s, whereas continued use may afford them the opportunity to develop the more sophisticated notion that these 3s represent composite units, and that 5 × 3 represents 5 times as much as 3, a composite of composite units. At the same time, it is important for teachers to recognize that the proficient use of skip counting does not necessarily imply that children have developed multiplicative thinking. By being aware of the characteristic ways of thinking associated with each of the different number sequences teachers are better able to provide appropriate scaffolding to move children through the stages. As an example, we feel that the intentional and selective use of manipulatives, both with elementary school students and middle school students who have yet to develop composite or iterable units, could provide invaluable support for the students’ mathematical development.
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Ulrich, C. (2015). Stages in constructing and coordinating units additively and multiplicatively (Part 1). For the Learning of Mathematics, 35(3), 2–7.
Ulrich, C. (2016a). Stages in constructing and coordinating units additively and multiplicatively (Part 2). For the Learning of Mathematics, 36(1), 34–39.
Ulrich, C. & Wilkins, J. L. M. (2016a). Using written work to assess stages in sixth-grade students’ construction and coordination of arithmetic units. Manuscript submitted for publication.
Ulrich, C. & Wilkins, J. L. M. (2016b). Using student written work to investigate stages in sixth-grade students’ ways of operating with numbers. Paper presented at the American Educational Research Association, Washington, DC.
Virginia Department of Education (2009). Mathematics standards of learning for Virginia public schools. Richmond, VA: Commonwealth of Virginia Board of Education. Available online at: http://www.doe.virginia.gov/testing/sol/standards_docs/mathematics/index.shtml
Jesse L.M. Wilkins
Professor of Math