Children’s early understanding of number predicts their later problem-solving sophistication in addition

https://doi.org/10.1016/j.jecp.2017.12.010Get rights and content

Highlights

  • Preschool quantitative knowledge is related to first grade addition strategy choice.

  • Understanding of cardinality in preschool linked to more sophisticated strategy use.

  • Kindergarten quantitative skills mediate link between cardinality and strategy use.

Abstract

Previous studies suggest that the sophistication of the strategies children use to solve arithmetic problems is related to a more basic understanding of number, but they have not examined the relation between number knowledge in preschool and strategy choices at school entry. Accordingly, the symbolic and nonsymbolic quantitative knowledge of 134 children (65 boys) was assessed at the beginning of preschool and in kindergarten, and the sophistication of the strategies they used to solve addition problems was assessed at the beginning of first grade. Using a combination of Bayes and standard regression models, we found that children’s understanding of the cardinal value of number words at the beginning of preschool predicted the sophistication of their strategy choices 3 years later, controlling for other factors. The relation between children’s early understanding of cardinality and their strategy choices was mediated by their symbolic and nonsymbolic quantitative knowledge in kindergarten. The results suggest that sophisticated strategy choices emerge from children’s developing understanding of the relations among numbers, in keeping with the overlapping waves model.

Introduction

Siegler's (1996) overlapping waves model provides the theoretical foundation for understanding individual differences in the strategies children use for problem solving and developmental change in their strategy mix. Children have multiple ways to solve most problems, and the strategy used to solve any given problem is based on procedural and associative memories formed during prior problem solving and a conceptual understanding of the domain (e.g., Shrager and Siegler, 1998, Siegler and Shrager, 1984). The brain systems underlying developmental change in children’s strategy choices are now being studied (Qin et al., 2014), and the relation between strategy sophistication and concurrent and later academic achievement is well documented, especially for mathematics (Geary et al., 2017, Siegler, 1988). Despite the intensive study of children’s strategy choices, little is known about the precursor knowledge that puts young children on the path to sophisticated problem solving. An especially important issue is the sophistication of the strategies they use to solve arithmetic problems at school entry because this is predictive of their growth in mathematical competencies throughout the elementary school years (Geary, 2011).

We addressed this knowledge gap with a 3-year longitudinal study of the relation between children’s quantitative knowledge at the beginning of preschool and the sophistication of the strategies children used to solve addition problems in first grade, controlling for domain-general abilities (e.g., intelligence) and parental education. We begin with an overview of the mix of strategies that first graders use to solve addition problems and then describe the basic quantitative abilities that might provide the first steps toward sophisticated problem solving. The latter include children’s basic knowledge of numerals and number words, as well as their intuitive understanding of nonsymbolic quantity, because these often predict later mathematics achievement (Desoete et al., 2012, Gilmore et al., 2007, Jordan et al., 2007).

First graders solve the majority of simple addition problems using four basic strategies (Baroody, 1984, Carpenter and Moser, 1984, Geary and Brown, 1991, Groen and Parkman, 1972, Siegler and Robinson, 1982, Siegler and Shrager, 1984). Children’s initial problem solving is dominated by counting strategies (e.g., finger counting, verbal counting) that in turn are eventually replaced by more sophisticated retrieval-based strategies (e.g., decomposition, direct retrieval). Early on, children use their fingers to physically represent the addends and then count them to reach a sum. With verbal counting, children count audibly or move their lips as if counting implicitly. Whether counting fingers or counting verbally, children might count both addends starting from 1 (sum strategy), start with the smaller addend and count a number of times equal to the larger one (max strategy), or start with the larger addend and count a number of times equal to the smaller one (min strategy). The use of counting to solve these problems results in the formation of an associative relation between the problem and the generated answer.

The next time children see the same problem, memory representations of counting schema and the stored answer compete for expression. If the level of activation of the counting schema exceeds that of a candidate answer, then the counting procedure will be used. Repeated use of counting, however, builds the strength of the association between the problem and answer in long-term memory, and eventually results in use of retrieval-based problem solving (Siegler, 1996, Siegler and Shrager, 1984). One retrieval-based strategy involves decomposing the problem into more simple problems (Siegler, 1987). For example, the problem 5 + 7 might be solved by subtracting 2 from the 7, then retrieving the answer to 5 + 5, and finally adding back the 2. The use of decomposition is dependent on retrieval but also requires a conceptual understanding of number relations (Geary, Hoard, Byrd-Craven, & DeSoto, 2004). The other strategy is direct retrieval of the answer from long-term memory.

Children’s early abilities include an evolved intuitive sense of quantity that is supported by the approximate number system (ANS). The ANS allows children to compare and manipulate (e.g., add) representations of collections of items (Feigenson et al., 2004, Geary et al., 2015) and has been proposed as the foundation for children’s mathematical development (Dehaene, 2001). Individual differences in ANS acuity are correlated with concurrent and later mathematics achievement (Libertus et al., 2011, Mazzocco et al., 2011), but the magnitude and importance of this relation is debated (De Smedt et al., 2013, Fazio et al., 2014, Schneider et al., 2016).

For the current study, the key ANS feature is the proposed embodiment of an implicit understanding of basic arithmetic (Barth et al., 2005, Gallistel and Gelman, 1992). As an example, Barth et al. (2006) found that 5-year-olds were able to add the quantities associated with two collections of objects and determine whether the sum was more or less than the quantity of a third collection. Barth et al. also showed that 5-year-olds could sum visually (dot collections) and auditorily (tones) presented quantities and that performance varied with the ratio of the sum of the added quantities and the comparison quantity. The cross-modal representations of quantity- and ratio-dependent performance are signatures of the ANS (Barth et al., 2006, Gilmore et al., 2007). The sums generated in these contexts, however, are only approximate.

Using a different procedure, Huttenlocher and colleagues demonstrated that sometime between 2 and 3 years of age many children can add and subtract small collections of objects to produce the exact answer (Huttenlocher et al., 1994, Levine et al., 1992). Here, children watch as an experimenter places a set of disks on a mat. The mat is covered, and the experimenter then places additional disks under the cover or removes (subtracts) one or more of them. The child then generates the answer nonverbally by placing the correct number of disks on his or her mat (see also Starkey, 1992). It is possible that performance on these types of procedures is facilitated by ANS acuity, although Huttenlocher et al. (1994) proposed that performance was more dependent on domain-general abilities, especially intelligence.

Whatever the mechanisms, we included children’s beginning of preschool performance on this nonverbal calculation task as a predictor of their later strategy choices. Preschoolers’ performance on this task predicts their later fluency in processing the magnitudes associated with Arabic numerals (Moore, vanMarle, & Geary, 2016), and conceptually it seems to follow that an early implicit understanding of arithmetic might presage sophisticated strategy choices when solving symbolic arithmetic problems. We also included two measures that should be sensitive to ANS acuity and an analogous measure of children’s sensitivity to continuous quantity (i.e., area). The latter was included because it is debated whether children’s intuitive understanding of quantity is dependent on the ANS or on a more general system that represents discrete and continuous quantities (Lourenco, 2015). In a study of area discrimination, Odic, Libertus, Feigenson, and Halberda (2013) found that acuity for discrete and continuous quantities (area) both improved during childhood. Area acuity, however, was higher than number acuity, suggesting that the mechanisms encoding approximate area and number are different. Thus, we included measures of both.

We also included an ordinal choice task, performance on which is correlated with mathematics achievement (Chu et al., 2013, Geary and vanMarle, 2016). The task is based on a common procedure that has been used successfully with preverbal infants and nonhuman primates as an indicator of their sensitivity to “more” versus “less” (vanMarle, 2013, vanMarle et al., 2006). The child watches as collections of objects are sequentially placed into two opaque cups and then is asked to choose the cup with more objects. Preschoolers’ performance on this task is more strongly related to ANS acuity than to counting skills, suggesting that they make their choices based on an intuitive understanding of relative quantity (Geary & vanMarle, 2016). Use of min counting requires an explicit understanding of the smaller and larger of two addends; thus, we reasoned that children with a strong implicit understanding of more and less might later use more sophisticated counting strategies when solving addition problems.

Children’s learning of and fluency in processing number symbols is the leading alternative hypothesis to the ANS as the foundation for mathematical development (Bugden and Ansari, 2011, Chu et al., 2015, De Smedt et al., 2013, Iuculano et al., 2008, Schneider et al., 2016, vanMarle et al., 2014). Number words are the first mathematical symbols that children learn, followed by the count list (i.e., “one, two, three, …”) and its use in enumerating objects (Fuson, 1988, Gelman and Gallistel, 1978). Cardinality is children’s first mathematical concept—that each number word represents a unique quantity and that successive number words in the count list are exactly one more than the number before it (Carey, 2004, Le Corre and Carey, 2007, Wynn, 1992). Children’s understanding of this core mathematical concept unfolds slowly during the preschool years, as illustrated by the give-a-number task. Here, children are asked to provide x number of objects to an experimenter. One-knowers provide one object when asked to do so but cannot provide the correct amount for other number words. Two- to four-knowers learn the relation between these number words and quantity but do not generalize to larger counting words (Le Corre & Carey, 2007). Children who score above this generalize to larger number words in their count list and are considered cardinal principle (CP) knowers, although it takes several more years before they fully understand that the quantity of all successive number words is n + 1 (Cheung, Rubenson, & Barner, 2017).

Preschoolers’ performances on cardinality, verbal counting, and enumeration tasks, as well as their ability to identify Arabic numerals, are predictive of their later mathematics achievement (Geary and vanMarle, 2016, vanMarle et al., 2014), but these competencies have not been explicitly related to the sophistication of their later arithmetical problem solving. There are, nevertheless, relevant studies of older children. The most consistent finding is that children who are fast and accurate in comparing the magnitudes associated with numerals have better concurrent and later arithmetic skills generally and use a more sophisticated mix of strategies to solve arithmetic problems (e.g., Bartelet et al., 2014, Desoete et al., 2012, Vanbinst et al., 2012, Vanbinst et al., 2014, Vanbinst et al., 2015). Vanbinst et al. (2015), for instance, assessed first graders’ numerical processing (digit naming), performance on nonsymbolic and symbolic magnitude comparison tasks, and skill at solving single-digit arithmetic problems. Over time, children’s reliance on retrieval increased and was related in part to their symbolic and nonsymbolic numerical skills. In first grade, children’s knowledge of and fluency in processing numerical symbols predicted overall speed and accuracy of solving arithmetic problems and frequency of direct retrieval of answers.

Stepping back, preschoolers’ ability to compare the magnitudes of numerals is dependent on their understanding of cardinal value (Geary & vanMarle, 2017); thus, we would expect that cardinal knowledge at the beginning of preschool would be a good predictor of later strategy sophistication. Children’s verbal counting and enumeration (i.e., ability to count and adhere to one-to-one correspondence) may also be important because these support the use of counting procedures to solve arithmetic problems. Indeed, Muldoon, Towse, Simms, Perra, and Menzies (2013) found that 5-year-olds who were able to enumerate up to 20 performed better on a battery of mathematics tasks, which included early arithmetic.

At the same time, nearly all first graders will perform well on the give-a-number task; thus, it is not likely that it is cardinal knowledge per se but rather that children who have an early understanding of cardinality have more time to learn about the relations among numbers before school entry (Geary et al., 2018). With respect to their strategy choices, better number knowledge is associated with more frequent use of min counting and decomposition if the answer cannot be directly retrieved (Geary et al., 2004). In the context of this study, any relation between cardinal knowledge in preschool and strategy choices in first grade should be mediated by number knowledge in kindergarten. Unfortunately, we did not administer the same Arabic numeral comparison task in kindergarten as was used in these prior studies (e.g., Bartelet et al., 2014), but we did administer several tasks that should capture the same underlying skills.

To recap, much is known about the cognitive and brain mechanisms that support the mix of strategies used by elementary school children to solve arithmetic problems (Qin et al., 2014, Siegler, 1996). It is also clear that the sophistication of this strategy mix at school entry predicts concurrent and future mathematics achievement, controlling for other factors (Geary, 2011, Geary et al., 2017). Little is known, however, about the preschool quantitative knowledge that puts children on the path to a sophisticated strategy mix at school entry. On the basis of current theory and research, one candidate for this early precursor knowledge is children’s implicit understanding of simple arithmetic (Huttenlocher et al., 1994), potentially supported by the ANS (Barth et al., 2005). Another candidate is children’s emerging understanding of number symbols and the relations among them (Bartelet et al., 2014, Vanbinst et al., 2012). We assess these two hypotheses and assess whether the relation between the better of these two candidates and later strategy choices is mediated by children’s number knowledge in kindergarten.

Section snippets

Participants

Children were recruited from the Title I preschool program within the public school system in Columbia, Missouri. Title I preschool is a federally funded program that offers services to 3- to 5-year-olds with developmental needs and prepares them for successful school entry. Consent forms were sent to all entering 3-year-olds, and the sample consisted of those whose parents consented to their children’s participation. A total of 232 children were initially recruited across three cohorts. Of

Results

Descriptive information for the predictors and strategy score are provided in Table 3. Correlations for tasks and covariates are provided in Table 4.

Discussion

The cognitive and brain mechanisms that influence the mix of strategies used to solve addition problems are well understood in the context of Siegler’s (1996) model of cognitive development (Qin et al., 2014, Shrager and Siegler, 1998), and the sophistication of this mix at school entry predicts concurrent and later mathematics achievement (Geary, 2011, Siegler, 1988). Kindergarten and elementary school children’s strategy mix is influenced by knowledge of the relative magnitudes of numerals (

Acknowledgments

This study was supported by a grant from the University of Missouri Research Board and Grant DRL-1250359 from the National Science Foundation. We thank Mary Rook, Tina Sattler, and their staff for help in facilitating our assessments of the Title I preschool children. We are also grateful for the cooperation of Columbia Public Schools, and especially the children and parents involved in the study. We thank Mary Hoard, Lara Nugent, Tim Adams, Melissa Barton, Sarah Becktell, Samantha Belvin,

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