Sources of Individual Differences in Emerging Competence With Numeration Understanding Versus Multidigit Calculation Skill

Abstract

This study investigated contributions of general cognitive abilities and foundational mathematical competencies to numeration understanding (i.e., base-10 structure) versus multidigit calculation skill. Children (n = 394, M = 6.5 years) were assessed on general cognitive abilities and foundational numerical competencies at start of 1st grade; on the same numerical competencies, multidigit calculation skill, and numeration understanding at end of 2nd grade; and on multidigit calculation skill and numeration understanding at end of 3rd grade. Path-analytic mediation analysis revealed that general cognitive predictors exerted more direct and more substantial effects on numeration understanding than on multidigit calculations. Foundational mathematics competencies contributed to both outcomes, but largely via 2nd-grade mathematics achievement, and results suggest a mutually supportive role between numeration understanding and multidigit calculations.

The individual-differences literature suggests that mathematics learning during the elementary school years depends not only on the school’s instructional program but also on contributions from children’s early numerical competencies, such as understanding of magnitude, as well as their general cognitive abilities, including reasoning ability and working memory. This literature has focused on different forms of mathematics learning, including calculations (e.g., Andersson, 2008; DeStefano & LeFevre, 2004; Geary, 2011), word problems (e.g., Andersson, 2007; Fuchs et al., 2006; Fuchs, Geary, Compton, Fuchs, Hamlett, & Bryant, 2010; Fuchs, Geary, Compton, Fuchs, Hamlett, Seethaler, et al., 2010; H. L. Swanson & Beebe-Frankenberger, 2004), fractions (e.g., Hecht & Vagi 2010; Jordan et al., 2013; Vukovic et al., 2012), algebra (e.g., Fuchs et al., 2012; Lee, Ng, Bull, Pe, & Ho, 2011; Tolar, Lederberg, & Fletcher, 2009), and general mathematics achievement (e.g., De Smedt, Janssen, et al., 2009; De Smedt, Verschaffel, & Ghesquière, 2009).

Numeration understanding, as it develops across the primary school years, has not been the focus of such inquiry— despite its centrality to the development of mathematical competence. In the present study, we focused on the effects of foundational forms of numerical cognition and general cognitive abilities at the beginning of first grade on end-of-third-grade numeration understanding; considered whether the effects of these first-grade abilities and competencies are mediated via second-grade mathematics achievement; and examined whether these sources of individual differences differ for numeration understanding versus multidigit calculation skill.

Numeration Understanding and Multidigit Calculations and Their Role in Mathematical Development

Young children’s understanding of number involves the insight that collections of objects constitute magnitudes that can be represented by number words and Arabic numerals. These words and symbols, whose cardinal values denote quantities, can be systematically ordered from smaller to larger (Brainerd, 1979; Fuson, 1988; Sarnecka & Carey, 2008; Wynn, 1990). These forms of early mathematical competencies typically develop before first grade and are sometimes included as a component of number sense. With formal education, children’s understanding of number gradually expands to include multidigit representations of quantity and the base-10 structure of the Hindu–Arabic numerical system.

Such understanding of the multiplicative structure of the base-10 system (which we refer to as numeration) was the focus of the present study. It is a central form of children’s mathematics development generally and thought specifically to support the use of procedures in solving calculation problems represented in this system (referred to as multidigit calculations; National Mathematics Advisory Panel [NMAP], 2008). In the present study, we operationalized numeration understanding as identifying numerals from representations of base-10 concepts, identifying numerals that occur before and after a given a multidigit numeral, ordering strings of multidigit numerals, and identifying missing numerals on number lines. We operationalized multidigit calculation as accuracy in solving multidigit addition and subtraction problems with and without regrouping, although accuracy may reflect children’s understanding of the base-10 structure of multidigit numerals or rote reliance on algorithms (as discussed in Verschaffel, Greer, & De Corte, 2007, and in the hypotheses section below).

Although these forms of mathematics competence are expected to develop during the early years of formal mathematics education (see curricular context in Method section), many students do not achieve a solid conceptual understanding of the base-10 numeration system, and some fail to develop competence with multidigit calculations. By contrast, other children develop sophisticated understanding of the system’s multiplicative structure as well as strong multidigit calculation skill (Geary, Hoard, Nugent, & Bailey, 2013; Mulligan & Mitchelmore, 2009; Thomas, 2004).

Sources of Individual Differences in Multidigit Calculation Skill and Numeration Understanding

Prior research suggests that sources of such individual differences in multidigit calculation skill include children’s general cognitive abilities as well as their early numerical competencies. For example, H. L. Swanson and Beebe-Frankenberger (2004) documented that working memory was involved in multidigit calculation skill; Fuchs et al. (2005) found that working memory and attentive behavior were unique predictors; and Fuchs, Geary, Compton, Fuchs, Hamlett, Seethaler, et al. (2010) again identified a role for working memory, while demonstrating that two forms of early numerical competencies also made contributions. Despite this and other studies focused on multidigit calculation skill, development of numeration understanding has not been studied in the individual-differences literature.

On the basis of such prior work on multidigit calculation skills, as well as studies describing the nature of numeration understanding (e.g., Verschaffel et al., 2007) and individual-difference studies of other forms of mathematics competence, we expected four general cognitive abilities may be involved in determining individual differences in numeration: visuospatial reasoning, working memory, listening comprehension, and attentive behavior. In terms of visuospatial reasoning, structured interviews suggest that numeration understanding depends on children’s ability to recognize the structure of mathematical representations in visuospatial arrays (Gray, Pitta, & Tall, 2000; Mulligan & Mitchelmore, 2009; Pitta-Pantazzi, Gray, & Christou, 2004; Young-Loveridge, 2002). Also, pattern recognition at kindergarten predicts mathematics achievement through eighth grade (Claessens & Engel, 2013); visuospatial reasoning predicts other forms of mathematics learning, such as arithmetic, word problem learning, and general mathematics achievement (Fuchs, Geary, Compton, Fuchs, Hamlett, & Bryant, 2010; Fuchs, Geary, Compton, Fuchs, Hamlett, Seethaler, et al., 2010; Fuchs, Geary, et al., 2013; Li & Geary, 2013; H. L. Swanson & Beebe-Frankenberger, 2004); and visuospatial reasoning tasks, like the ones included in intelligence tests, are central to the kinds of highly symbolic learning reflected in numeration understanding (e.g., Fuchs et al., 2012). Further, a role for such reasoning makes sense because the recursive groupings that underlie our numeration system are typically depicted in terms of spatial configurations.

At the same time, working memory capacity likely represents an additional source of individual differences in numeration understanding. This possibility is based on Heirdsfield’s (2001, 2002) observations during child interviews; experimental studies of children’s and adults’ understanding of arithmetic concepts and skill at solving arithmetic problems (Geary, Hoard, & Nugent, 2012; Hitch, 1978; Klein & Bisanz, 2000; Lemaire, Abdi, & Fayol, 1996); and predictive studies of mathematics achievement (Bull, Espy, & Wiebe, 2008; De Smedt, Janssen, et al., 2009; Geary, 2011) and learning more generally (Deary, Strand, Smith & Fernandes, 2007). Numeration understanding, as reflected, for example, in assigning words to and recording numbers with the Hindu–Arabic notational number system, is a multicomponent processing task that taxes working memory by requiring students to sequentially formulate decisions about units of ones, tens, hundreds, and so on, as they hold intermediate results in mind. Potentially even more taxing on working memory are numeration understanding tasks that require combining or decomposing multidigit numbers by trading across columns, as Hitch (1978) found for adult problem solving.

The individual-differences literature also suggests potential roles for listening comprehension in numeration understanding. Listening comprehension predicts children’s development of competence with word problems (Fuchs et al., 2005; Fuchs, Geary, Compton, Fuchs, Hamlett, & Bryant, 2010; Fuchs, Geary, Compton, Fuchs, Hamlett, Seethaler, 2010), prealgebra (Fuchs et al., 2012), and fractions (Fuchs, Schumacher, et al., 2013; Jordan et al., 2013; Vukovic et al., 2012). Some studies show the influence of mathematical terminology, such as ones, tens, and hundreds, or multiple meanings for the same number name (two means two ones or two tens; ten means a collection of 10 objects or one unit of 10) on mental representations of mathematical concepts (Geary, 2006; Miura, Okamoto, Vlahovic-Stetic, Kim, & Han, 1999). Moreover, orally presented explanations of numeration concepts are lengthy and complex, thereby potentially taxing listening comprehension ability.

Attentive behavior in the classroom is a robust predictor across many forms of mathematical cognition and learning, including simple arithmetic, multidigit addition and subtraction, and word problems (Fuchs et al., 2005, 2006; Geary, Hoard, & Nugent, 2012), prealgebra (Fuchs et al., 2012), and fractions (Hecht & Vagi, 2010; Jordan et al., 2013; Vukovic et al., 2012). The multicomponent nature of operating in the numeration system and the need to attend in careful and sustained ways to lengthy explanations about numeration concepts suggest the importance of attentive behavior.

In addition to general cognitive abilities, prior studies further suggest that foundational mathematical competencies, specifically more basic knowledge of number and quantity, may contribute to children’s ease of learning numeration. This basic knowledge includes the ability to quickly apprehend the quantity of small collections of items (subitize), understand part–whole relationships, understand the cardinal value of single-digit Arabic numerals and their relative magnitude, and have a mental representation of the number line (e.g., Booth & Siegler, 2004; Fuchs et al., 2006; Fuchs, Geary, Compton, Fuchs, Hamlett, & Bryant, 2010; Fuchs, Geary, Compton, Fuchs, Hamlett, Seethaler, et al., 2010; Krajewski & Schneider, 2009; Kroesbergen, Van Luit, Van Lieshout, Van Loosbroek, & Van de Rijt, 2009; Vukovic et al., 2012). Also, these early forms of number sense are conceptually linked to and some are transparently embedded within base-10 numeration tasks.

In the present study, we assessed first graders’ foundational mathematical competencies with two measures that tap different early competencies. The Number Sets Test (Geary, Bailey, & Hoard, 2009) is a complex measure that assesses understanding of single-digit numerals, including part–whole relationships, and the fluency with which children transcode between Arabic numerals and the quantities they represented. We expected such early numerical competencies to contribute to individual differences in later numeration understanding based on von Aster and Shalev (2007), who hypothesized that number representational systems, combined with children’s mapping of counting words and Arabic numerals onto these representations, support learning of the base-10 number system. Also, fluency with these skills may free up working memory that can then be focused on the more challenging demands of operating in the base-10 system. The second measure, Number Line Estimation (Booth & Siegler, 2004), is designed to assess children’s understanding of the whole-number number line spanning 1 and 99. This form of early numerical cognition is transparently related to the base-10 number system. That is, children’s understanding of the meaning of tens versus the ones units will likely be reflected in number line performance.

Hypotheses

Numeration understanding

This provided the basis for three hypotheses concerning sources of individual differences in children’s development of numeration understanding. First, because numeration understanding is emerging across Grades 1–3 and because reasoning, listening comprehension, working memory, and attentive behavior contribute to the acquisition of new forms of symbolic learning, we expected these first-grade general cognitive abilities to have direct effects on third-grade numeration understanding. Second, because these general cognitive abilities also affect earlier forms of formal school mathematics learning, we also anticipated indirect effects of these first-grade general cognitive abilities on numeration understanding. We expected these indirect effects to occur via children’s second-grade mathematics achievement, as reflected in second graders’ accruing expertise with the multiple forms of number knowledge represented on the Number Sets Test, extended representations of the number line, and increased skill with multidigit calculations. Third, because first-grade numerical competencies are foundational across many forms of formal school mathematics learning (i.e., the kinds of learning reflected in performance on the second-grade Number Sets Test, second-grade mental representations of the number line, and second-grade multidigit calculations), we expected the effects of the first-grade numerical competencies to be entirely mediated by the three second-grade mathematics achievement measures.

Multidigit calculation skill

We also contrasted the direct and indirect effects of first-grade general cognitive abilities and foundational numerical competencies for third-grade numeration understanding to those involved in third-grade multidigit calculations. We might expect similar sources of individual differences based on research showing a connection between understanding of the base-10 system and use of appropriate calculation strategies for solving multidigit addition and subtraction problems (Blöte, Van der Burg, & Klein, 2001; Fuson & Kwon, 1992; Heirdsfield, 2001; Hiebert & Wearne, 1996). Children who understand the base-10 structure of multidigit written numerals or number words can rely on this system to solve multidigit calculations (NMAP, 2008). Even so, some children develop competence with multidigit addition and subtraction without adequate understanding of the base-10 numeration system (Thomas, 2004), which indicates reliance on algorithms without understanding their conceptual structure. This suggests that sources of individual differences in numeration understanding and multidigit calculations may differ. Moreover, because numeration understanding appears to involve greater symbolic complexity than multidigit calculations, we hypothesized that these general cognitive abilities are more central to acquiring numeration understanding than calculation skill and expected the effects of these general cognitive abilities on third-grade calculation skill to be mediated through second-grade numeration understanding.

Study Overview

At the start of first grade, we assessed children on the predictor variables: general cognitive abilities (visuospatial reasoning, working memory, listening comprehension, and attentive behavior) and foundational numerical competencies (the number sets and the number line estimation tasks). Approximately 2 school years later, to index the school mathematics learning that had occurred at the end of second grade, we tested the children on the same numerical competencies as well as on multidigit calculation skill and numeration understanding. One year later, at end of third grade, we examined their performance on the study’s outcomes: numeration understanding and multidigit calculation skill. We initiated the study at the start of first grade, when the range of individual differences on early numerical competencies is wide, based on informal learning experiences (before implementation of the formal mathematics curriculum). We chose third grade as the study’s endpoint based on curricular standards in the school district where the study took place (e.g., see curricular context in Method section). That is, multidigit representations of quantity and the base-10 structure of the Hindu–Arabic numerical system as well as multidigit addition and subtraction (the major study outcomes) were addressed instructionally by end of third grade. (In fourth grade, understanding about fractions, not a focus of the present study, becomes a dominant curricular focus).

To test the direct and indirect effects reflected in our hypotheses, we conducted path-analytic mediation analysis, for which the causal steps involve the following series of statistical relations. (In this explanation, the independent variable refers to the first-grade predictors; the mediator refers to the second-grade mathematics achievement variables; the dependent variable refers to the third-grade outcomes.) First, the independent variable must be associated with the dependent variable. This is the c path or the direct effect. It establishes there is an effect to mediate. Second, the independent variable must be associated with the mediator. This is the a path, which provides a test of the action theory. Third, the mediator must affect the dependent variable, when all independent variables are controlled. This is the b path. It substantiates that the mediator is related to the dependent variable. Fourth, the indirect (or mediated) effect, which is the product of the a and b paths (a × b), must be significant. This is equivalent to testing whether adding the mediator changes the relation between the independent and dependent variable (c is the relation before the mediator is added; c′ is the relation after the mediator is added). If so, showing that the direct effect is no longer significant in the face of a significant effect for a × b provides evidence for complete mediation; if the direct effect remains significant in the face of a significant effect for one or more mediators, the mediation effect is partial. Note that we conducted multiple mediation analysis, which simultaneously considered the effects of all second-grade mediators, in the presence of all seven first-grade predictors.

Identifying whether the direct effects of first-grade predictors are mediated by second-grade mathematics achievement helps researchers understand the process by which general cognitive abilities and early numerical competencies exert their effects on core third-grade mathematics outcomes. It also provides insight about which forms of second-grade mathematics achievement mitigate the effects of which types of general cognitive abilities and early numerical competencies. This has implications for what aspects of the second-grade curriculum are more and less important for promoting the types of third-grade outcomes we studied: numeration understanding and multidigit calculations. (We do, however, remind readers that because mediation analyses are correlational, causation should not be inferred.)

Method

Participants

Data were collected on three of four cohorts of a larger research project (e.g., Fuchs, Geary, Compton, Fuchs, Hamlett, & Bryant, 2010; Fuchs, Geary, Compton, Fuchs, Hamlett, Seethaler, et al., 2010; Fuchs, Geary, et al., 2013) conducted in Nashville, Tennessee, a southeastern metropolitan school district of 75,000 students in the United States. Participants were randomly sampled from first-grade classrooms, stratifying by risk status based on mathematics performance at the start of first grade. The study began with 506 at-risk and low-risk students (i.e., at-risk and not-at-risk students), with no more than eight students sampled from the same class. Of these students, 112 moved before the end of third grade to schools outside the district or to schools that chose not to participate. The sample of 394 children who did not move during three school years of data collection (start of Grade 1 to end of Grade 3) were in 102 first-grade classrooms in 26 schools. They dispersed to 167 second-grade classrooms and then to 219 third-grade classrooms across 54 schools (creating too many unique combinations of classroom sequences to make clustering relevant). The 112 exiters were not significantly different from the 394 students who remained on any first-grade predictor, as indicated by analyses of variance conducted on variables collected at the start of first grade. Data were complete for the 394 students who did not move, for whom the mean age was 6.5 years (SD = 4.26 months) in fall of first grade; 46% were female; 77% received subsidized lunch (i.e., they were entitled to participate in the federal program that provides lunch without charge or at reduced rates, if families met criteria for low income); 59% were African American, 26% non-Hispanic White, 9% White Hispanic, and 5% other. Thus, the sample is representative of urban centers in the United States.

Curricular Context

In the time frame for this study, the United States did not have a national curriculum. Instead, state standards guided instruction. Numeration standards were as follows:

• at Grade 1 (6–7 years), count and recognize number patterns by twos, fives, and tens up to 100 and start at any number to count forward or backward by ones to 18;

• at Grade 2 (7–8 years), start at any number to count forward or backward by ones to 1,000; count by tens from any number using a hundreds chart; identify odd and even whole numbers to 100; use a number line or hundreds grid to find one more or one less of a number lower than 50; use the number line to create visual representations of sequences; start at any number to count by twos, threes, fives, and tens to 1,000; demonstrate skip counting on a number line; use concrete objects to represent two-digit numbers in flexible ways to 99 (e.g., base-10 blocks, sticks, and straws); identify the position of a whole number to 100 on the number line; read and write numerals up to 1,000; write numbers in words to 10; read and write numerals to 999; order whole numbers to 100 using appropriate symbols; and identify and use ordinal numbers up to 12th;

• at Grade 3 (8–9 years), compare the magnitude of three numbers, build place value models, identify the value of a digit in a three-digit number, and convert place values in standard and expanded form between places in a three-digit number.

The state calculation standards were as follows:

• at Grade 1, add/subtract with pictures and Arabic numerals to 18; add one-digit numbers to two-digit numbers without regrouping; subtract one-digit numbers from two-digit numbers; add 0, doubles, three numbers, and tens; and subtract 0 and all, doubles, and tens;

• at Grade 2, add/subtract efficiently and accurately with one-digit numbers to 18 and add/subtract two-digit whole numbers with regrouping;

• at Grade 3, add two numbers with four or more digits with regrouping, subtract numbers up to three digits with regrouping, retrieve multiplication facts to 12, and retrieve division facts to 10.

First-Grade Predictor Measures

Visuospatial reasoning

Wechsler Abbreviated Scale of Intelligence–Matrix Reasoning (Wechsler, 1999) measures visuospatial reasoning (often referred to as nonverbal reasoning in the literature; e.g., Fuchs et al., 2006; Jordan et al., 2013). The tasks involve visuospatial pattern completion, classification, analogy, and serial reasoning tasks. With each item, students complete a matrix, from which one section is missing, from five response options. Reliability is .94; the correlation with the third-edition Wechsler Intelligence Scale for Children Full Scale IQ is .66. The maximum possible raw score is 32. The range of raw scores for this sample was 1–26.

Listening comprehension

Woodcock Diagnostic Reading Battery–Listening Comprehension (Woodcock, 1997) measures the ability to understand sentences or passages. With 38 items, students supply the word missing at the end of sentences or passages that progress from simple verbal analogies and associations to discerning implications. Reliability is .80 at ages 5–18. The maximum possible raw score is 38. The range of raw scores for this sample was 0–28.

Central executive working memory

We administered measures of central executive working memory from the Working Memory Test Battery for Children (Pickering & Gathercole, 2001): Counting Recall and Listening Recall. Each has proven a valuable predictor of mathematics development in previous studies. Each subtest has six items at span levels from 1–6 to 1–9. Passing four items at a level moves the child to the next level. At each span level, the number of items to be remembered increases by one. Failing three items terminates the subtest. We used the trials correct score. For Listening Recall, the child determines whether a sentence is true; after making true/false determinations for a series of sentences, the child recalls the last word of each sentence. For Counting Recall, the child counts a set of four, five, six, or seven dots on a card; after counting a series of cards, the child recalls the number of counted dots of each card. We considered these measures of the central executive separately, based on prior work showing their predictive value differs, depending on type of mathematics outcome (Fuchs, Geary, Compton, Fuchs, Hamlett, & Bryant, 2010). The maximum possible raw score is 36 for Listening Recall and 42 for Counting Recall. The range of trials correct scores for this sample was 0–16 for Listening Recall and 0–26 for Counting Recall.

Attentive behavior

The Strengths and Weaknesses of ADHD-Symptoms and Normal-Behavior (J. Swanson et al., 2004) samples items from the fourth-edition Diagnostic and Statistical Manual of Mental Disorders criteria for attention-deficit/hyperactivity disorder for inattention (nine items) and hyperactivity–impulsivity (nine items), but scores are normally distributed. Teachers rate items on a 1–7 scale. We report data for the inattentive subscale as the average rating across the nine items. The inattentive subscale correlates well with other dimensional assessments of behavior related to attention (http://www.adhd.net). In the present study, coefficient alpha on the inattentive subscale was .97. (We did not include the hyperactivity–impulsivity subscale because correlations with mathematics measures were low, as shown in prior work; e.g., Fuchs et al., 2005; Fuchs, Geary, Compton, Fuchs, Hamlett, & Bryant, 2010; Fuchs, Geary, Compton, Fuchs, Hamlett, Seethaler, et al., 2010.) The minimum possible attentive rating score is 9, and the maximum possible score is 63, which was the range of scores for this sample.

Foundational numerical competencies

We used two foundational numerical cognition tasks, each of which has proven a valuable predictor of individual differences in mathematics achievement and development. Number Sets Test (Geary et al., 2009) indexes understanding of number, numeral mapping, and part–whole relationships. It measures the speed and accuracy with which children understand and operate with small numerosities within and just beyond the subitizing range (<10), while partitioning sets and transcoding between quantities (sets of objects) and symbols (Arabic numerals). Two types of 1/2-in.-square stimuli are used: arrays of objects (circles, squares, diamonds, stars) and Arabic numerals (18-point font). Stimuli are joined in dominolike rectangles. The first five lines of the page show five 2-in.-square dominos; last two lines, three 3-in.-square dominos. At the top of the page, the target sum (5 or 9) is printed in larger font (36 point); 5 and 9 were chosen to represent smaller and larger values within the range of basic Arabic numerals. The first page for each target contains Objects/Same Objects and Objects/Different Objects; the second page, Objects/Arabic Numerals and Arabic Numerals/Arabic Numerals. On each page, 18 items match the target; 12 are larger, 6 are smaller, and 6 contain 0 or an empty square. The tester explains the task with a target of 3 and 4. The child is told to move across each line of the page from left to right, “circle any groups that can be put together to make the top number 5 [or 9],” and “work as fast as you can without making many mistakes.” The child has 60 s per page for the target 5; 90 s per page for 9 (time limits were chosen to avoid ceiling effects and to assess fluent recognition and processing of quantities).

Signal detection methods are applied to the number of hits and false alarms to generate a d′ variable representing sensitivity to quantities (Geary et al., 2009). Children who correctly identify many target quantities and commit few false alarms have high scores; as many hits as false alarms result in low scores (i.e., the high number of correct items is due to the bias to respond, not sensitivity to quantity). Individual differences in d′ are likely related to (a) acuity of the cognitive systems for representing exact small quantities or approximate larger ones (Koontz & Berch, 1996), (b) ease of mapping of Arabic numerals onto the underlying representation of magnitude (De Smedt & Gilmore, 2011; Rousselle & Noël, 2007), (c) implicit or explicit understanding of addition (Jordan, Huttenlocher, & Levine, 1992), and (d) understanding of cardinal value and that numbers are composed of sets of smaller-magnitude numbers (Butterworth, 2010). Controlling for general cognitive abilities, d′ predicts across-grade growth in mathematics but not reading achievement (Geary, 2011), and school-entry mathematics deficits of children with mathematics disability are fully mediated by d′, which in turn partially mediates slow growth in mathematics but not reading achievement through fifth grade (Geary, Hoard, Nugent, & Bailey, 2012). The range of d′ scores for this sample was −2.90 to 4.31 at first grade; −2.66 to 5.72 at second grade.

With Number Line Estimation (Siegler & Booth, 2004), children locate Arabic numerals on a number line marked with 0 and 100 as endpoints. A number line is presented, with a target number (3, 4, 6, 8, 12, 17, 21, 23, 25, 29, 33, 39, 43, 48, 52, 57, 61, 64, 72, 79, 81, 84, 90, 96) shown above the line. Children place the target number on the line, without a time limit. As in Siegler and Booth (2004, Experiment 2) and Geary, Hoard, Byrd-Craven, Nugent, and Numtee (2007), the tester explains, “The number 50 is half of 100, so we put it halfway in between 0 and 100 on the number line.” Use of these directions produces findings comparable to other work by Siegler and colleagues (e.g., Siegler & Booth, 2004; Geary et al., 2007).

Siegler and Opfer (2003) used group-level median placements fitted to linear and log models to make inferences about children’s modal representation for making placements, but used an accuracy measure for individual difference analyses. Although the extent to which the pattern of number line placements reflects the underlying way in which children mentally represent the line is debated (see Barth & Paladino, 2011; Cohen & Blanc-Goldhammer, 2011), the accuracy measure remains a good indicator of their overall understanding of the number line and predicts later mathematics achievement (Geary, 2011; Siegler & Booth, 2004). We correlated the following indices with numeration understanding and multidigit calculation skill: accuracy (the absolute difference between placement and correct position, which is automatically calculated by the computer with which the task is administered), percentage of trials consistent with a linear representation and percentage of errors for these trials, and percentage of trials consistent with a log representation and percentage of errors for these trials. For both outcomes, the best predictor was accuracy, as in Geary, Hoard, and Bailey (2012). The range of accuracy scores for this sample was 5.08 – 50.83 at first grade; 2.00 – 36.58 at second grade. Lower scores indicate stronger performance, but we multiplied scores by −1 before correlations and mediation analyses. Test-retest reliability on a subset of this sample (n = 83) was .87.

Second-Grade Mediation Measures and Third-Grade Outcome Measures

We assessed three forms of second-grade mathematics learning as potential mediators of the effects of first-grade predictors. When predicting each third-grade outcome, we considered second-grade number knowledge using Number Sets and Number Line, as described. When predicting third-grade numeration understanding, we also considered second-grade calculation skill as a mediator; for third-grade calculation skill, we also considered second-grade numeration understanding as a mediator.

To measure numeration understanding (as a second-grade mediator of the effects of the first-grade predictors on the calculation outcome and as the third-grade numeration outcome), we relied on KeyMath–Numeration (Connolly, 1998), with which children hold up fingers or say numerals to identify the number of objects in pictures, read numerals, identify ordinal numbers, identify numerals that occur before and after a given numeral, order strings of numerals, identify numerals from representations of base-10 concepts, and identify missing numerals on number lines. Second- and third-grade items address one- and two-digit numbers, but items also assess the ability to extend the numeration system beyond what has been explicitly taught in school by the end of third grade. Coefficient alpha on this sample was .84. The maximum possible raw score is 24. The range of scores for this sample was 4–23 at second grade and 3–23 at third grade.

To measure calculation skill (as a second-grade mediator of the effects of the first-grade predictors on the numeration outcome and as the third-grade calculation outcome), we relied on the Wide Range Achievement Test–Arithmetic (Wilkinson, 1993), with which children write answers to calculation problems of increasing difficulty. In this sample, at second and third grade, items address the four operations with whole numbers; the emphasis is on two-digit addition and subtraction with and without regrouping. Coefficient alpha on this sample was .88. The maximum possible raw score is 55. The range of scores for this sample was 9–28 at second grade and 12–38 at third grade.

Procedure

Testers were trained to criterion at each testing occasion and used standard directions for administration. In fall of first grade, children were assessed on the predictors (Wechsler Abbreviated Scale of Intelligence–Matrix Reasoning, Woodcock Diagnostic Reading Battery–Listening Comprehension, Working Memory Test Battery for Children subtests, Number Sets, and Number Line Estimation), and classroom teachers completed ratings of inattentive behavior. In spring of second grade, children were assessed on Number Sets, Number Line Estimation, KeyMath–Numeration, and Wide Range Achievement Test–Arithmetic. In spring of third grade, children were assessed on KeyMath–Numeration and Wide Range Achievement Test–Arithmetic. Testing occurred individually in a quiet room for all tasks except for Number Sets in first and second grade, which was group administered in classrooms or small groups in a quiet location outside the classroom. All individual sessions were audiotaped; 15% of tapes were selected randomly, stratifying by tester, for accuracy checks by an independent scorer. Agreement exceeded 99%.

Results

See Table 1 for means and standard deviations (raw scores and standard scores for nationally normed tests) and correlations (all p < .001). Raw scores, transformed to sample-based z scores, were used in analyses. We explored the distribution of each measure via statistical (skewness, kurtosis) and graphical (box plots, stem and leaf plots) methods. The variables entered into analyses (i.e., the standardized variables) were normally distributed except for reasoning (positively skewed and kurtotic), first-grade number sets (positively skewed), second-grade number sets (negatively skewed and kurtotic), and second-grade number line (positively skewed). On these variables, we completed square-root transformations. Because results for the original and the transformed variables were substantively the same, we present results for the original variables for ease of interpretation. Our analyses focused on examining total, direct, and indirect effects of the first- and second-grade variables on third-grade numeration understanding and on third-grade calculation skill. We used the Preacher and Hayes (2008) SPSS mediate macro to obtain estimates, with bootstrapping (5,000 draws to estimate standard errors) applied to construct 95% confidence intervals for indirect effects.

Table 1.

Means and Standard Deviations for Raw Scores and Nationally Norm-Referenced Standard Scores and Correlations (n = 394)

Variable	Scores
	Raw		Standard		Grade 1							Grade 2				Grade 3
	M	SD	M	SD	R	LC	CR	LR	AB	NL	NS	NL	NS	C	N	C
Grade 1 predictors
Reasoning (R)	9.63	5.76	50.26	9.82
Listen Comp (LC)	14.98	4.72	90.12	15.13	.39
Counting Recall (CR)	10.61	4.65			.39	.34
Listening Recall (LR)	4.64	3.94			.46	.54	.48
Attentive Behavior (AB)	36.59	13.02			.37	.44	.40	.51
Number Line (NL)	23.37	7.57			.44	.45	.35	.40	.38
Number Sets (NS)	0.00	1.00			.54	.46	.49	.54	.53	.53
Grade 2 potential mediators
Number Line (NL)	13.43	6.95			.38	.33	.27	.38	.35	.45	.49
Number Sets (NS)	0.00	1.00			.49	.44	.44	.47	.49	.47	.62	.50
Calculations (C)	21.88	3.46	96.23	14.16	.45	.40	.39	.46	.52	.44	.55	.53	.69
Numeration (N)	12.65	3.72	11.58	2.53	.58	.52	.45	.56	.55	.54	.69	.59	.69	.69
Grade 3 outcomes
Calculations (C)	26.90	4.29	100.47	15.74	.44	.42	.39	.52	.52	.46	.57	.56	.63	.69
Numeration (N)	15.13	3.85	11.21	3.85	.53	.51	.41	.54	.54	.50	.65	.58	.65	.65		.67

Note. Reasoning is Matrix Reasoning from Wechsler Abbreviated Scale of Intelligence (Wechsler, 1999). Listen Comp is Listening Comprehension from Woodcock Diagnostic Reading Battery (Woodcock, 1997). Listening Recall is from the Working Memory Test Battery for Children (WMTB; Pickering & Gathercole, 2001). Counting Recall is from WMTB. Attentive Behavior is Strengths and Weaknesses of ADHD-Symptoms and Normal-Behavior (Swanson et al., 2004). Number Line is Number Line Estimation error scores (Siegler & Booth, 2004), multiplied by −1 so higher scores indicate stronger performance. Number Sets is Number Sets Test (Geary et al., 2009), for which d′ scores are standardized for the sample at the grade level. Calculations is Wide Range Achievement Test-Arithmetic (Wilkinson, 1993; standard score: M = 100, SD = 15). Numeration is from KeyMath (Connolly, 1998; scale score: M = 10, SD = 3).

Direct and Indirect Effects on Third-Grade Numeration

Tables 2 and 3 summarize total, direct, and indirect effects of the first-grade predictors on third-grade numeration. As shown at the bottom of Table 2, the direct effects of first-grade reasoning, working memory listening recall, and attentive behavior on third-grade numeration were significant, while controlling for the direct and indirect effects of all first- and second-grade variables in the model. As in Table 3, the total effect (direct plus indirect effects) on third-grade numeration was significant for each of the first-grade predictors except counting recall, and the omnibus test of the total effect on third-grade numeration was significant, R² = .55, F(7, 386) = 67.84, p < .001.

Table 2.

Coefficients for Paths a, b, and c′ on Grade 3 Numeration

Path	Coefficient	SE	t	p
Paths b: Effect of Grade 2 skill on Grade 3 numeration
Grade 2 Number Line →Grade 3 Numeration	.18	.04	4.63	<.001
Grade 2 Number Sets →Grade 3 Numeration	.22	.05	4.58	<.001
Grade 2 Calculations →Grade 3 Numeration	.15	.05	3.39	<.001
Paths a: Effect of Grade 1 predictor on Grade 2 skill
Grade 1 Reasoning → Grade 2 Number Line	.10	.06	1.69	.092
Grade 1 Reasoning → Grade 2 Number Sets	.15	.05	3.01	.035
Grade 1 Reasoning → Grade 2 Calculations	.14	.05	2.72	.007
Grade 1 Listening Comprehension → Grade 2 Number Line	.01	.05	0.15	.882
Grade 1 Listening Comprehension → Grade 2 Number Sets	.07	.05	1.57	.118
Grade 1 Listening Comprehension → Grade 2 Calculations	.04	.05	0.89	.374
Grade 1 Counting Recall → Grade 2 Number Line	− .04	.05	− 0.75	.455
Grade 1 Counting Recall → Grade 2 Number Sets	.09	.05	2.11	.035
Grade 1 Counting Recall → Grade 2 Calculations	.06	.05	1.18	.240
Grade 1 Listening Recall → Grade 2 Number Line	.10	.06	1.67	.096
Grade 1 Listening Recall → Grade 2 Number Sets	.03	.05	0.65	.514
Grade 1 Listening Recall → Grade 2 Calculations	.08	.05	1.41	.157
Grade 1 Attentive Behavior → Grade 2 Number Line	.06	.05	1.10	.272
Grade 1 Attentive Behavior → Grade 2 Number Sets	.14	.05	3.07	.002
Grade 1 Attentive Behavior → Grade 2 Calculations	.24	.05	4.82	<.001
Grade 1 Number Line → Grade 2 Number Line	.23	.05	4.37	<.001
Grade 1 Number Line → Grade 2 Number Sets	.12	.05	2.60	.010
Grade 1 Number Line → Grade 2 Calculations	.12	.05	2.37	.018
Grade 1 Number Sets → Grade 2 Number Line	.25	.06	4.02	<.001
Grade 1 Number Sets → Grade 2 Number Sets	.30	.05	5.60	<.001
Grade 1 Number Sets → Grade 2 Calculations	.20	.06	3.54	<.001
Paths c′: Direct effect of Grade 1 predictor on Grade 3 numeration
Grade 1 Reasoning → Grade 3 Numeration	.09	.04	2.08	.038
Grade 1 Listening Comprehension → Grade 3 Numeration	.09	.04	2.35	.002
Grade 1 Counting Recall → Grade 3 Numeration	− .03	.04	− 0.82	.415
Grade 1 Listening Recall → Grade 3 Numeration	.13	.04	3.14	.002
Grade 1 Attentive Behavior → Grade 3 Numeration	.08	.04	2.01	.045
Grade 1 Number Line → Grade 3 Numeration	.02	.04	0.46	.643
Grade 1 Number Sets → Grade 3 Numeration	.15	.05	3.30	.001

Note. In all models, the effects of the other first-grade predictor variables were controlled. Bolded effects are potentially mediating effects, based on significant a and b paths.

Table 3.

Grade 1 Predictors’ Total and Indirect Effects (Via Grade 2 Mathematics Skills) on Grade 4 Numeration

Path	Indirect effect		ta	pa	95% CIb
	Coefficient	Path a × b
Reasoning
Total effect	.16	.05	3.43	.001
Indirect effect via Number Line	.02	.01			[.0226,.0368]
Indirect effect via Number Sets	.03	.01			[.0133,.0397]
Indirect effect via Calculations	.02	.01			[.0067,.0408]
Listening Comprehension
Total effect	.11	.04	2.64	.001
Indirect effect via Number Line	.00	.01			[−.0145,.0178]
Indirect effect via Number Sets	.02	.01			[−.0005,.0349]
Indirect effect via Calculations	.01	.01			[−.0057,.0211]
Counting Recall
Total effect	−.01	.04	−0.20	.845
Indirect effect via Number Line	−.01	.01			[−.0226,.0078]
Indirect effect via Number Sets	.02	.01			[.0039,.0397]
Indirect effect via Calculations	.01	.01			[−.0034,.0225]
Listening Recall
Total effect	.17	.05	3.57	<.001
Indirect effect via Number Line	.02	.01			[.0002,.0365]
Indirect effect via Number Sets	.01	.01			[−.0109,.0266]
Indirect effect via Calculations	.01	.01			[−.0115,.0279]
Attentive Behavior
Total effect	.16	.04	3.64	<.001
Indirect effect via Number Line	.01	.01			[−.0049,.0274]
Indirect effect via Number Sets	.03	.01			[.0125,.0532]
Indirect effect via Calculations	.04	.01			[.0164,.0603]
Number Line
Total effect	.10	.04	2.41	.016
Indirect effect via Number Line	.04	.01			[.0210,.0700]
Indirect effect via Number Sets	.03	.01			[.0086,.0464]
Indirect effect via Calculations	.02	.01			[.0041,.0345]
Number Sets
Total effect	.30	.05	5.89	<.001
Indirect effect via Number Line	.04	.01			[.0223,.0700]
Indirect effect via Number Sets	.06	.02			[.0366,.0976]
Indirect effect via Calculations	.03	.01			[.0119,.0532]

Note. In all models, the effects of the other first-grade predictors were controlled. Confidence intervals (CIs) that do not cover 0 are statistically significant. Bolded effects are significant mediating effects, based on significant a and b paths (see Table 2) as well as significant indirect effects (as per this table).

^aFor total effect.

^bFor indirect effects.

In considering whether a mediating effect is significant, it is necessary first to establish that the effect of the first-grade predictor on the potential mediator (i.e., path a) is significant and the effect of the potential mediator on the third-grade outcome (i.e., path b) is significant, while controlling for the effect of all first-grade predictors. As shown at the top of Table 2, the effect of each second-grade mathematics measure on third-grade numeration (i.e., path b) was significant. The middle panel of Table 2 (i.e., path a) shows that first-grade number line and number sets had significant effects on second-grade number line; first-grade reasoning, counting recall, attentive behavior, number line, and number sets had significant effects on second-grade number sets; and first-grade reasoning, attentive behavior, number line, and number sets had significant effects on second-grade number calculations. This left 11 potentially mediating effects (i.e., a and b paths were both significant). These are bolded in Table 2.

The final step in evaluating a mediation effect is to test whether the indirect effect of each potential mediator on the third-grade outcome is significant, while controlling for all first-grade predictors and all second-grade variables in the model. As bolded in Table 3, each of the 11 indirect effects was significant (95% confidence intervals that do not cover 0 are significant). Also, the omnibus test of direct effect was significant, R² = .08, F(7, 383) = 12.83, p < .001, indicating the first-grade predictors together made a significant contribution over the three second-grade variables considered as mediators. (Interactions between the set of first-grade predictors and each of the three mediators were not significant, so the homogeneity of regression assumption was met.)

See the top portion of Figure 1 for significant direct (solid lines), mediating effects (dotted lines), and total effects (gray boxes) on numeration understanding. First-grade general cognitive predictors and numerical competencies played a substantial role. The total effect of each cognitive predictor (except working memory counting recall) was significant; additional, mediated effects occurred for visuospatial reasoning, counting recall, and attentive behavior. Both foundational numerical competencies also contributed, with first-grade number sets exerting the largest total effect of all predictors. Much of this effect, however, was mediated via second-grade number sets, number line, and calculation scores.

Figure 1.

Significant direct effects for first-grade predictors (solid lines), significant indirect effects for first-grade predictors via second-grade mathematics learning (dotted lines), and significant total effects (shaded boxes) on third-grade numeration skill (top panel) and on third-grade calculation skill (bottom panel). Comp. = comprehension; WM = working memory.

Direct and Indirect Effects on Third-Grade Calculation Skill

See Tables 4 and 5 for total, direct, and indirect effects on third-grade calculation skill. As shown at the bottom of Table 4, the only significant direct effect for the first-grade predictors on third-grade calculation skill was attentive behavior. As in Table 5, the total effect was significant for three first-grade predictors: attentive behavior, number line, and number sets. Also, the omnibus test of the total effect on third-grade calculation skill was significant, R² = .43, F(7, 386) = 42.06, p < .001.

Table 4.

Coefficients for Paths a, b, and c′ on Grade 3 Calculations

Path	Coefficient	SE	t	p
Paths b: Effect of Grade 2 skill on Grade 3 calculations
Grade 2 Number Line → Grade 3 Calculations	.21	.04	4.70	<.001
Grade 2 Number Sets → Grade 3 Calculations	.23	.05	4.42	<.001
Grade 2 Numeration → Grade 3 Calculations	.21	.06	3.48	<.001
Paths a: Effect of Grade 1 predictor on Grade 2 skill
Grade 1 Reasoning → Grade 2 Number Line	.10	.06	1.69	.092
Grade 1 Reasoning → Grade 2 Number Sets	.15	.05	3.01	.003
Grade 1 Reasoning → Grade 2 Numeration	.22	.04	5.08	<.001
Grade 1 Listening Comprehension → Grade 2 Number Line	.01	.05	0.15	.882
Grade 1 Listening Comprehension → Grade 2 Number Sets	.07	.05	1.56	.118
Grade 1 Listening Comprehension → Grade 2 Numeration	.11	.04	2.72	.007
Grade 1 Counting Recall → Grade 2 Number Line	−.04	.05	−0.75	.455
Grade 1 Counting Recall → Grade 2 Number Sets	.09	.04	2.11	.035
Grade 1 Counting Recall → Grade 2 Numeration	.03	.04	0.87	.385
Grade 1 Listening Recall → Grade 2 Number Line	.10	.06	1.67	.100
Grade 1 Listening Recall → Grade 2 Number Sets	.03	.05	0.65	.514
Grade 1 Listening Recall → Grade 2 Numeration	.09	.04	2.12	.034
Grade 1 Attentive Behavior → Grade 2 Number Line	.06	.05	1.10	.272
Grade 1 Attentive Behavior → Grade 2 Number Sets	.14	.05	3.07	.002
Grade 1 Attentive Behavior → Grade 2 Numeration	.15	.04	3.68	<.001
Grade 1 Number Line → Grade 2 Number Line	.23	.05	4.37	<.001
Grade 1 Number Line → Grade 2 Number Sets	.12	.05	2.60	.010
Grade 1 Number Line → Grade 2 Numeration	.12	.04	3.17	.002
Grade 1 Number Sets → Grade 2 Number Line	.25	.06	4.02	<.001
Grade 1 Number Sets → Grade 2 Number Sets	.30	.05	5.59	<.001
Grade 1 Number Sets → Grade 2 Numeration	.32	.05	7.05	<.001
Paths c′: Direct effect of Grade 1 predictor on Grade 3 calculations
Grade 1 Reasoning → Grade 3 Calculations	−.01	.05	−0.11	.909
Grade 1 Listening Comprehension → Grade 3 Calculations	.01	.04	0.31	.755
Grade 1 Counting Recall → Grade 3 Calculations	.02	.04	0.50	.615
Grade 1 Listening Recall → Grade 3 Calculations	.01	.05	0.28	.780
Grade 1 Attentive Behavior → Grade 3 Calculations	.16	.04	3.54	<.001
Grade 1 Number Line → Grade 3 Calculations	.03	.04	0.72	.470
Grade 1 Number Sets → Grade 3 Calculations	.07	.05	1.23	.220

Note. In all models, the effects of the other first-grade predictor variables were controlled. Bolded effects are potentially mediating effects, based on significant a and b paths.

Table 5.

Grade 1 Predictors’ Total and Indirect Effects (Via Grade 2 Mathematics Skills) on Grade 3 Calculations

Path	Indirect effect		ta	pa	95% CIb
	Coefficient	Path a×b
Reasoning
Total effect	.09	.05	1.83	.069
Indirect effect via Number Line	.02	.01			[.0005,.0436]
Indirect effect via Number Sets	.03	.01			[.0131,.0586]
Indirect effect via Numeration	.05	.02			[.0217,.0753]
Listening Comprehension
Total effect	.06	.05	1.13	.257
Indirect effect via Number Line	.00	.01			[−.0168,.0206]
Indirect effect via Number Sets	.02	.01			[−.0010,.0360]
Indirect effect via Numeration	.02	.01			[.0073,.0434]
Counting Recall
Total effect	.04	.05	0.89	.371
Indirect effect via Number Line	− .01	.01			[−.0267,.0092]
Indirect effect via Number Sets	.02	.01			[.0043,.0413]
Indirect effect via Numeration	.01	.01			[−.0063,.0223]
Listening Recall
Total effect	.06	.05	1.15	.252
Indirect effect via Number Line	.02	.01			[.0003,.0430]
Indirect effect via Number Sets	.01	.01			[−.0112,.0274]
Indirect effect via Numeration	.02	.01			[.0034,.0401]
Attentive Behavior
Total effect	.23	.05	4.81	<.001
Indirect effect via Number Line	.01	.01			[−.0064,.0321]
Indirect effect via Number Sets	.03	.01			[.0127,.0556]
Indirect effect via Numeration	.03	.01			[.0131,.0548]
Number Line
Total effect	.13	.05	2.78	.006
Indirect effect via Number Line	.05	.02			[.0250,.0740]
Indirect effect via Number Sets	.03	.01			[.0089,.0483]
Indirect effect via Numeration	.03	.01			[.0096,.0479]
Number Sets
Total effect	.25	.06	4.55	<.001
Indirect effect via Number Line	.05	.02			[.0254,.0805]
Indirect effect via Number Sets	.07	.02			[.0375,.1021]
Indirect effect via Numeration	.07	.02			[.0345,.1076]

Note. In all models, the effects of the other first-grade predictors were controlled. Confidence intervals (CIs) that do not cover 0 are statistically significant. Bolded effects are significant mediating effects, based on significant a and b paths (see Table 4) as well as significant indirect effects (as per this table).

^aFor total effect.

^bFor indirect effects.

As shown at the top of Table 4, the effect of each second-grade mathematics measure on third-grade calculation skill (path b) was significant. The middle panel of Table 4 (path a) shows that first-grade number line and number sets had significant effects on second-grade number line; first-grade reasoning, counting recall, attentive behavior, number line, and number sets had significant effects on second-grade number sets; and first-grade reasoning, listening recall, listening comprehension, attentive behavior, number line, and number sets had significant effects on second-grade calculations. These 13 potentially mediating effects are bolded in Table 4. As shown in Table 5, each of these indirect effects was significant. The omnibus test of direct effect was significant, R² = .03, F(7, 383) = 3.29, p = .002, so the first-grade predictors made a significant contribution over the three second-grade mediators. (Interactions between the set of first-grade predictors and each of the three mediators were not significant, so the homogeneity of regression assumption was met.)

See Figure 1 (bottom panel) for significant direct, mediating, and total effects. General cognitive predictors played a smaller role in third-grade calculations than numeration understanding, with attentive behavior the only significant total effect for general cognitive abilities. Across all first-grade predictors, number sets had the largest total effect, most of which was mediated via second-grade number sets, number line, and numeration understanding. A significant total effect for number line was also mediated via the three second-grade forms of mathematics achievement.

Discussion

Understanding of our numeration system is a central but challenging component of children’s mathematical development. Wide variation in such understanding, even given the same classroom instruction (Mulligan & Mitchelmore, 2009; Thomas, 2004), suggests that sources of individual differences in numeration understanding extend beyond the effects of schooling to include students’ general cognitive abilities and their foundational numerical cognition. Development of numeration understanding has not, however, been studied in the individual-differences literature. In the present study, we examined the effects of first-grade general cognitive abilities and foundational forms of numerical competencies on third-grade numeration understanding, assessed whether these effects are mediated via second-grade mathematics achievement, and investigated whether sources of individual differences differ for numeration understanding versus multidigit calculation skill.

At the most general level, results indicated that both types of third-grade competence depend on a combination of general cognitive abilities and foundational mathematical competencies—as framed in Geary’s (2004) model of mathematics learning and as shown in the individual-differences literature for other forms of mathematics development (e.g., Andersson, 2008; De Smedt, Janssen, et al., 2009; De Smedt, Verschaffel, & Ghesquière, 2009; DeStefano & LeFevre, 2004; Fuchs, Geary, Compton, Fuchs, Hamlett, & Bryant, 2010; Fuchs, Geary, Compton, Fuchs, Hamlett, Seethaler, et al., 2010; Geary, 2011; H. L. Swanson & Beebe-Frankenberger, 2004). More specifically, on the basis of studies of other aspects of mathematical development and on the basis of interviews with children about their numeration understanding (Gray et al., 2000; Mulligan & Mitchelmore, 2009; Young-Loveridge, 2002), we expected important individual differences for third-grade numeration to involve first-grade visuospatial reasoning, central executive working memory, listening comprehension, attentive behavior, and basic numerical competencies. We further expected these first-grade sources of individual differences to exert a combination of direct and indirect effects on third-grade numeration understanding via second-grade mathematics achievement, and we expected effects to differ for third-grade numeration understanding versus multidigit calculations.

General Cognitive Abilities and Numeration Understanding

Four general cognitive abilities—visuospatial reasoning, listening comprehension, working memory in the form of listening recall, and attentive behavior—exerted direct effects on numeration understanding. Each of these direct effects was significant even though we controlled for (a) direct effects of the other cognitive predictors, (b) indirect effects of the six general cognitive abilities via each of the three types of second-grade mathematics achievement, and (c) direct and indirect effects of first-grade measures of foundational numerical competencies. At the same time, additional, indirect effects also occurred for three first-grade general cognitive abilities via second-grade mathematics achievement: for first-grade visuospatial reasoning via second-grade number sets performance and calculation skill, for working memory (this time counting recall) via second-grade number sets performance, and for attentive behavior via number sets performance and calculation skill. Across direct and indirect effects, the total effect was significant for visuospatial reasoning (β = .17), listening comprehension (β = .11), working memory-listening recall (β = .17), and attentive behavior (β = .16). In these ways, general cognitive abilities played a strong role in the development of third-grade numeration understanding, beyond what is attributable to foundational forms of number understanding.

Previous studies of the relation between visuospatial ability and performance on mathematical achievement and cognition measures has been mixed, once the central executive is controlled (Hegarty & Kozhevnikov, 1999; Holmes & Adams, 2006; Meyer, Salimpoor, Wu, Geary, & Menon, 2010; Rasmussen & Bisanz, 2005). It is likely that the importance of visuospatial reasoning varies with the content and complexity of mathematical tasks (Kyttälä & Lehto, 2008; Li & Geary, 2013); it has been found to contribute to skill at solving arithmetic and word problems (Fuchs, Geary, Compton, Fuchs, Hamlett, & Bryant, 2010; Fuchs, Geary, Compton, Fuchs, Hamlett, Seethaler, et al., 2010; Fuchs, Geary, et al. 2013; H. L. Swanson & Beebe-Frankenberger, 2004). Here we found a role for visuospatial reasoning for facilitating numeration understanding, which corroborates results from structured interview studies (Gray et al., 2000; Pitta-Pantazzi et al., 2004; Young-Loveridge, 2002). Our numeration system can be conceptualized, and within classrooms is often depicted, in terms of visuospatial configurations. Our results suggest that children with stronger visuospatial reasoning ability may more easily make sense of these arrays of recursive groupings to construe successive powers of 10, above and beyond the influence of the central executive, listening comprehension, attentive behavior, and foundational numerical competencies.

At the same time, using the Hindu–Arabic notational number system to assign numerals to such arrays involves multicomponent processing. It requires that students sequentially formulate decisions about units of ones, tens, hundreds, and so on, as they hold intermediate results in mind. So it makes sense that working memory, which is important when students are initially learning or consolidating mathematical information (e.g., Andersson, 2008; DeStefano & LeFevre, 2004; Fuchs, Geary, et al., 2013; Geary, 2011), is also involved in learning about numeration. In fact, on the basis of observations of children performing challenging numeration tasks, Heirdsfield (2001, 2002) inferred that children recruited working memory resources. Our results are also in keeping with previous studies of mathematical learning whereby working memory predicts competence in arithmetic (Andersson, 2008; DeStefano & LeFevre, 2004,Fuchs, Geary, Compton, Fuchs, Hamlett, & Bryant, 2010; Fuchs, Geary, Compton, Fuchs, Hamlett, Seethaler, et al., 2010; Geary, 2011), word problem solving (Andersson, 2007; Fuchs, Geary, Compton, Fuchs, Hamlett, & Bryant, 2010; Fuchs, Geary, Compton, Fuchs, Hamlett, Seethaler, et al., 2010; H. L. Swanson & Beebe-Frankenberger, 2004), fractions (Hecht & Vagi, 2010), and algebra (Lee et al., 2011; Tolar et al., 2009).

In the present study, working memory in the form of listening recall (with which children state the last word of a series of sentences after processing the veracity of each sentence) was a better predictor of numeration understanding than counting recall (in which students count a series of object arrays and then recall the counts). It is not clear whether this was due to the content of the tasks or whether the listening recall task is a better measure of the central executive than counting recall for the age range assessed in this study. In a large cross-sectional study, Gathercole, Pickering, Ambridge, and Wearing (2004) found that the strength of the relation between listening recall and counting recall and overall central executive skills varied with age; listening recall tended to be a better indicator than counting recall in school-age children. It is also possible that facility in maintaining verbal information in working memory is particularly important for numeration understanding. At the same time, it is important to note that effects for listening versus counting recall may vary across mathematical content. For example, Fuchs, Geary, Compton, Fuchs, Hamlett, Seethaler, et al. (2010) showed that counting recall was the predictor of calculation outcomes, whereas listening recall was the better predictor for mathematics word problem outcomes.

In this vein, it is important to note that more general listening comprehension ability at first grade had a unique and direct effect, above and beyond listening recall, on third-grade numeration understanding. Some studies show the influence of mathematical terminology on mental representations of mathematical concepts (Geary, 2006; Miura et al., 1999). In numeration understanding, terminology is highly confusing because the same word has multiple meanings. For example, in a single numeral, two can represent two ones and two tens. Even more confusing, ten can mean a collection of 10 objects or one unit of 10. Such terminology presses children’s listening comprehension ability, even as it recruits reasoning ability and working memory capacity. Additionally, however, instruction designed to foster numeration understanding is typically presented with lengthy and complex oral explanations that likely further tax children’s listening comprehension ability. The need to decipher complex, lengthy explanations about numeration concepts also requires attentiveness, as reflected in the predictive value on third-grade numeration understanding of teacher ratings of children’s first-grade attentive behavior nearly 4 years later. This corroborates previous findings for children’s learning of simple calculations and word problems (Fuchs et al., 2005, 2006; Geary, Hoard, & Bailey, 2012; Geary, Hoard, & Nugent, 2012) and prealgebra (Fuchs et al., 2012), as well as fractions (Jordan et al., 2013; Vukovic et al., 2012). On the one hand, finding that attentive behavior in the classroom predicts learning of academic content is not surprising. On the other hand, our finding here and in previous studies highlights the contributions of attentiveness to learning, above and beyond the contributions of traditionally assessed domain-general cognitive competencies.

In these ways, the present study indicates that visuospatial reasoning, working memory, listening comprehension, and attentive behavior contribute in important and unique ways to children’s development of numeration understanding. This echoes the individual-differences literature (e.g., Duncan et al., 2007; Jordan, Kaplan, Ramineni, & Locuniak, 2009), which, although not previously focused on numeration understanding, suggests a role for each of these abilities in other forms of mathematics development.

The Role of Foundational Mathematical Competencies in Numeration Understanding

This brings us to the role of foundational numerical competencies in numeration understanding. We considered two types of first-grade numerical cognition. The Number Sets Test is a complex measure that assesses understanding of single-digit numerals, including part-whole relationships, and the fluency with which children transcode between Arabic numerals and the quantities they represented (Geary et al., 2009). Individual differences in this form of number knowledge exerted a substantial direct effect on numeration understanding (β = .15), indicating that strong early fluency with processing and transcoding numerals and the quantities they represent support children’s development of numeration understanding. This makes sense in that understanding of numerical magnitudes of small magnitudes is embedded in base-10 numeration tasks, and fluent and flexible understanding of those quantities may free up working memory and other general cognitive abilities (e.g., attentive behavior) for addressing the more challenging demands of operating in the base-10 system. The finding is also in line with von Aster and Shalev (2007), who hypothesized that number representational systems, combined with children’s mapping of counting words and Arabic numerals onto these representations, support later learning of the base-10 number system.

This direct effect of number sets on numeration understanding combined with three significant indirect effects via second-grade mathematics achievement to produce an impressive total effect of .30. The first indirect effect, via second-grade number sets (.06), indicates that improvement across first and second grade on the foundational numerical competencies represented in the number sets task helps account for third-grade numeration understanding. The second significant indirect effect on third-grade numeration was mediated by second-grade calculation skill (.03). This is not surprising because the number sets task, in asking children to fluently partition numbers into constituent sets in flexible ways, should be foundational to calculation skill as it develops across first and second grade. Moreover, competence with second-grade calculation achievement, which involves multidigit addition and subtraction and trading across units in the numeration system, should support children’s understanding of the numeration system, a point we return to later. The final significant indirect effect of first-grade number sets performance on third-grade numeration was mediated by second-grade number line performance (β = .04). This is again in line with von Aster and Shalev (2007), who posited that understanding of small quantities provides a foundation not only for understanding the base-10 numeration system but also for learning the mathematical number line. Our findings support this. This indirect affect also indicates that understanding how numerals, including multidigit numerals, are ordered on the number line at the end of second grade contributes to their emerging numeration understanding.

What is more surprising is that the direct effect on numeration understanding of first-grade number line performance, which assesses understanding of numbers spanning 1 and 99, was not significant, and the total effect of number line, although significant, summed to .10—a considerably smaller total effect than was the case for first-grade number sets (.30). At the start of first grade, many students’ understanding of the number line may be restricted to smaller quantities. Moreover, the number line task is untimed, and the fluent appreciation of number required for the number sets task may discriminate more finely among first graders, thereby capturing variance that might otherwise be subsumed by the number line measure.

All three indirect effects for first-grade number line on third-grade numeration, via second-grade mathematics achievement, were however significant, corroborating previous studies that have found number line predictive of later mathematics achievement (e.g., Booth & Siegler, 2006; Geary, 2011; Siegler & Booth, 2004). The first indirect effect, via second-grade number line (.04), indicates that building understanding of the number line that spans one- and two-digit numbers across first and second grade supports third-grade numeration understanding. The second significant indirect effect, in which first-grade number line performance on third-grade numeration was mediated by second-grade number sets (.03), reemphasizes the contribution of fluent appreciation of small quantities for third-grade numeration understanding. The final significant indirect effect indicated that the effects of first-grade number line performance on third-grade numeration understanding are mediated by second-grade calculation skill. This makes sense to the extent that first graders who appreciate placement of two-digit numbers on the number line hold an advantage over first graders who lack such appreciation for mastering second-grade two-digit calculations, which demands trading units within the numeration system. This is turn should support third-grade numeration understanding.

Findings across the two forms of numerical cognition, as revealed by the direct and indirect effects of first-grade number sets, the indirect effects of first-grade number line, and the mediating effects of second-grade performance on these two forms of numerical cognition, support the conclusion that early understanding of numerals, the magnitudes they represent, and the relations between them supports the learning of more complicated types of number understanding. Findings also indicate that such early mathematical competencies and their development over first and second grade play an important role in numeration understanding at third grade. This echoes prior individual-differences studies examining other forms of mathematics learning (e.g., Fuchs et al., 2006; Fuchs, Geary, Compton, Fuchs, Hamlett, & Bryant, 2010; Fuchs, Geary, Compton, Fuchs, Hamlett, Seethaler, et al., 2010; Krajewski & Schneider, 2009; Kroesbergen et al., 2009; Vukovic et al., 2012). Results also suggest that developing understanding of the base-10 numeration system relies on a combination of domain-specific foundational competencies as well as general cognitive abilities, as Geary (2004) proposed.

Do Sources of Individual Differences Differ for Numeration Understanding Versus Multidigit Calculation Skill?

Our third hypothesis concerned similarities and differences between sources of individual differences for numeration understanding versus multidigit understanding. As NMAP (2008) suggested, only children who understand the base-10 structure of multidigit written numerals or number words can rely on that structure to solve multidigit calculation problems. And some studies show a connection between understanding the base-10 number system and use of appropriate strategies to solve multidigit addition and subtraction problems (Blöte et al., 2001; Fuson & Kwon, 1992; Hiebert & Wearne, 1996).

Yet, some children develop competence with multidigit addition and subtraction without adequate understanding of the base-10 numeration system (Thomas, 2004), which suggests strong reliance on procedural algorithms without understanding their underlying conceptual structure. If reliance on procedural algorithms dominates third-grade competence with multidigit calculations, without conceptual insight, then the sources of individual differences in numeration understanding and multidigit calculations may differ. Moreover, to the extent that numeration understanding involves greater symbolic complexity than calculation skill, numeration understanding may rely more than calculation skill on the type of mental flexibility, manipulation of symbolic associations, and maintenance of multiple representations supported by visuospatial reasoning, listening comprehension, and working memory. We thus hypothesized that (a) these general cognitive abilities operate directly for numeration understanding but not calculation skill and (b) the indirect effects of these general cognitive abilities on third-grade calculation skill are mediated through second-grade numeration understanding.

In line with these hypotheses, general cognitive abilities played a more central role in predicting third-grade numeration understanding than calculation skill. First-grade visuospatial reasoning ability, listening comprehension, and working memory as well as attentive behavior directly supported third-grade numeration, and the total effect of each of these first-grade predictors was significant. These general cognitive abilities also contributed to third-grade numeration skill indirectly via mathematics learning across first and second grade in multiple ways. By contrast, in determining third-grade multidigit calculation skill, attentive behavior was the only general cognitive ability for which the direct or total effect was significant. As noted earlier, teacher ratings of attentive behavior are a robust predictor across multiple types of mathematics learning. The present study extends this literature to include numeration understanding and to corroborate prior work on multidigit calculations. Yet, at this time, clarity is lacking about what these ratings indicate. Instead of or in addition to attention, they may reflect actual classroom mathematics learning or motivation to work hard in mathematics, and additional research is required to pinpoint what teachers’ ratings of inattentive behavior is measuring.

More striking is the lack of significant direct or total effects for visuospatial reasoning and working memory on multidigit calculations, given that such relations have been found in previous studies (e.g., Fuchs et al., 2005; Fuchs, Geary, Compton, Fuchs, Hamlett, & Bryant, 2010). Earlier studies did not, however, control for the possibility that such effects are mediated by interim numeration understanding. The present study indicates this may be the case. Significant indirect paths on third-grade calculation skill included the effects of four general cognitive abilities via second-grade numeration understanding: reasoning, working memory– listening recall, and attentive behavior. These indirect effects strengthen conclusions about the role of these abilities in numeration—beyond the direct effects of these first-grade abilities on third-grade numeration already discussed. They also indicate a role for listening comprehension in multidigit calculation skill, by suggesting that listening comprehension exerts its effect indirectly: Listening comprehension supports children’s second-grade numeration understanding, which in turn supports multidigit calculation skill. This, combined with indirect effects on third-grade multidigit calculation skill for reasoning, working memory, and attentive behavior via second-grade mathematics achievement, shows that the effects of these general cognitive abilities do operate on third-grade multidigit calculation skill, but primarily via second-grade numeration understanding.

Although results for the general cognitive ability predictors differed for third-grade multidigit calculation skill from those involved in numeration understanding, the pattern of effects for the first-grade general foundational numerical predictors was more similar. For both outcomes, many more indirect effects occurred via second-grade number sets performance than second-grade number line performance. Also, in predicting third-grade numeration understanding, a large proportion (36%) of the significant indirect effects occurred via second-grade multidigit calculation skill, and conversely, in predicting third-grade multidigit calculation skill, a large proportion (46%) occurred via second-grade numeration understanding. This suggests a reciprocal relation between numeration understanding and multidigit calculation skill at second and third grade, when calculation skill is dominated by two-digit addition and subtraction involving trading of tens and ones units. As per NMAP (2008), this suggests that understanding of the base-10 structure of multidigit written numerals or number words supports multidigit calculations. It also extends NMAP’s hypothesis to suggest that, in reciprocal fashion, skill with multidigit calculations supports understanding of the base-10 structure of multidigit written numerals or number words. It also supports research showing a connection between understanding of the base-10 number system and use of appropriate strategies in multidigit addition and subtraction (Blöte et al., 2001; Fuson & Kwon, 1992; Heirdsfield, 2001; Hiebert & Wearne, 1996).

Study Limitations and Conclusions

In interpreting findings, readers should consider study limitations. First, whereas our predictor variables explained 55% of the variance in third-grade numeration understanding, they accounted for only 43% of the variance in third-grade multidigit calculation skill. And for both outcomes, a substantial amount of variance remained unexplained, as is typically the case in the individual-differences literature. Considering additional or a different set of child characteristics may have explained additional variance or changed the pattern of effects. Future research might include other general cognitive resources, such as other forms of working memory (e.g., inhibition and updating; Miyake & Shah, 1999), an alternative type of visuospatial reasoning such as three-dimensional spatial visualization (Tolar et al., 2009), and analogical or inferential reasoning (e.g., Holyoak & Thagard, 1997). This may produce insights into additional processes that support development of numeration understanding. Second, because different measures of early numerical cognition may produce different findings (Ebersbach, Lowel, & Verschaffel, 2013), additional research with varying measures is required. Third, our measure of numeration understanding was limited largely to transcoding between numerals and conceptual representations of number, as well as extending the numeration system beyond what has been explicitly taught in school. Future studies should contrast alternative strategies for operationalizing numeration understanding. Fourth, we measured each construct with a particular measure. Including multiple measures to permit use of latent constructs should be pursued. Fifth, findings should be generalized only to settings with curricular contexts that resemble those in the present study and to children with similar demographics. Finally, conclusions about causality should be avoided because although longitudinal predictive relations were examined, our methods are at root correlational.

We caution readers to keep these limitations in mind when considering our major conclusions. First, first-grade general cognitive abilities play a substantial role in numeration understanding through end of third grade. These include visuospatial reasoning, listening comprehension, working memory in the form of listening recall, and attentive behavior. These sources of individual differences are direct and larger for numeration understanding than multidigit calculations, for which only attentive behavior had a significant direct or total effect. Second, first-grade foundational mathematics competencies support third-grade numeration understanding as well as multidigit calculation skill. This includes understanding of number, numeral mapping, and part–whole relationships, as reflected in the number sets measure, and understanding of the number line, and for both third-grade outcomes, these effects were largely mediated via second-grade mathematics achievement. Finally, results suggest a mutually supportive role between understanding of the numeration system and operating with multidigit calculations.

These findings, derived from an individual-differences perspective, supplement structured interviews of children, to contribute to a small literature on the development of numeration understanding. Given the important role for general cognitive abilities suggested in the present study, instruction designed to compensate for limitations in or strengthen such cognitive abilities, in the context of numeration instruction, may be warranted for at-risk children. And results indicate that intervention to build foundational mathematics competencies and strengthen the conceptual and operational connections between numeration and multidigit calculation may be valuable. Experimental studies should pursue these possibilities.