Preschool deficits in cardinal knowledge and executive function contribute to longer-term mathematical learning disability

Journal of Experimental Child Psychology 188 (2019) 104668

Contents lists available at ScienceDirect

Journal of Experimental Child Psychology journal homepage: www.elsevier.com/locate/jecp

Preschool deficits in cardinal knowledge and executive function contribute to longer-term mathematical learning disability Felicia W. Chu a, Kristy vanMarle a, Mary K. Hoard a, Lara Nugent a, John E. Scofield a, David C. Geary a,b,⇑ a b

Department of Psychological Sciences, University of Missouri, Columbia, MO 65211, USA Interdisciplinary Neuroscience Program, University of Missouri, Columbia, MO 65211, USA

a r t i c l e

i n f o

Article history: Received 29 March 2018 Revised 25 June 2019

Keywords: Mathematics achievement Learning disability Longitudinal study Preschool Number knowledge Executive function

a b s t r a c t In a preschool through first grade longitudinal study, we identified groups of children with persistently low mathematics achievement (n = 14) and children with low achievement in preschool but average achievement in first grade (n = 23). The preschool quantitative developments of these respective groups of children with mathematical learning disability (MLD) and recovered children and a group of typically achieving peers (n = 35) were contrasted, as were their intelligence, executive function, and parental education levels. The core characteristics of the children with MLD were poor executive function and delayed understanding of the cardinal value of number words throughout preschool. These compounded into even more substantive deficits in number and arithmetic at the beginning of first grade. The recovered group had poor executive function and cardinal knowledge during the first year of preschool but showed significant gains during the second year. Despite these gains and average mathematics achievement, the recovered children had subtle deficits with accessing magnitudes associated with numerals and addition combinations (e.g., 5 + 6 = ?) in first grade. The study provides unique insight into domain-general and quantitative deficits in preschool that increase risk for long-term mathematical difficulties. Ó 2019 Elsevier Inc. All rights reserved.

⇑ Corresponding author at: Department of Psychological Sciences, University of Missouri, Columbia, MO 65211, USA. E-mail address: [email protected] (D.C. Geary). https://doi.org/10.1016/j.jecp.2019.104668 0022-0965/Ó 2019 Elsevier Inc. All rights reserved.

2

F.W. Chu et al. / Journal of Experimental Child Psychology 188 (2019) 104668

Introduction It is now well recognized that children who start school with deficits in their understanding of number and basic arithmetic are very likely to lag behind their peers in mathematics throughout schooling and into adulthood (Duncan et al., 2007; Geary, Hoard, Nugent, & Bailey, 2013; Ritchie & Bates, 2013). Their slow development of mathematical competencies is not simply due to early number and arithmetic deficits—other factors contribute as well (e.g., working memory; Bailey, Watts, Littlefield, & Geary, 2014)—but nevertheless addressing these deficits is a necessary step toward improving the trajectory of their mathematical development. Interventions have in fact been developed to address these early deficits (Dyson, Jordan, & Glutting, 2013; Fuchs et al., 2013), but surprisingly little is known about why these deficits emerge in the first place. To be sure, there is a growing literature on the preschool quantitative abilities that predict later mathematics achievement (Geary et al., 2018; Libertus, Feigenson, & Halberda, 2011; Mazzocco, Feigenson, & Halberda, 2011), but few studies have focused on the early predictors of the lowest achieving children, those with a mathematical learning disability (MLD). In a unique 4-year longitudinal study, we identified a group of children whose average standardized mathematics achievement scores were consistently low (<25th percentile, means < 10th percentile) throughout preschool and into first grade (MLD) and another group with equally low preschool achievement but average achievement in first grade (recovered). We contrasted the quantitative and executive function (EF) development of children in these MLD and recovered groups and a group of typically achieving (TA) children across 2 years of preschool and assessed their knowledge of number and basic addition at the beginning of first grade. The results help to identify the domaingeneral and early quantitative deficits that may presage long-term MLD and the factors that contribute to the recovery of children who appear to be equally at risk based on their preschool mathematics achievement.

Early quantitative abilities It is unclear whether children’s initial foundation for mathematical development rests on an evolved number sense or their understanding of number words and numerals (De Smedt, Noël, Gilmore, & Ansari, 2013; Geary & vanMarle, 2016). The evolved number sense is supported by the approximate number system (ANS) that enables people and other animals to determine the relative quantity of two collections of objects and perhaps perform basic arithmetical operations over these representations (Feigenson, Dehaene, & Spelke, 2004; Gallistel & Gelman, 2000; Geary, Berch, & Mann Koepke, 2015). Performance on ANS measures is correlated with concurrent and prospective mathematics achievement (Bonny & Lourenco, 2013; Libertus, Feigenson, & Halberda, 2013; Starr, Libertus, & Brannon, 2013), and severe ANS deficits may underlie dyscalculia or MLD in some children (Piazza et al., 2010). Although a relation between ANS acuity and mathematics achievement has been confirmed in several meta-analyses (Chen & Li, 2014; Fazio, Bailey, Thompson, & Siegler, 2014; Schneider et al., 2017), the proposal that this is a causal relation is debated (Carey, Shusterman, Haward, & Distefano, 2017; De Smedt et al., 2013; Fuhs & McNeil, 2013; Rousselle & Noël, 2007). Alternative models focus on children’s early conceptual understanding of number words and numerals (Chu, vanMarle, & Geary, 2015; vanMarle, Chu, Li, & Geary, 2014) and their later fluency in accessing the corresponding magnitudes (Holloway & Ansari, 2009; Rousselle & Noël, 2007). Children’s memorization of number words, the standard counting sequence, and the use of this sequence to enumerate collections of objects provide the first foothold into symbolic mathematics, but in and of themselves do not indicate a conceptual understanding of number symbols (Fuson, 1988; Gelman & Gallistel, 1978). Children’s first major conceptual insight comes with their understanding of the quantities represented by number words, that is, their cardinal values (Carey, 2004; Le Corre & Carey, 2007; Wynn, 1992). Children’s progression on a standard measure of cardinality, give-a-number, illustrates the developmental unfolding of this insight (Wynn, 1992). In the give-a-number measure, children are

F.W. Chu et al. / Journal of Experimental Child Psychology 188 (2019) 104668

3

asked to provide x objects from a pile of objects. Most 3-year-olds can provide one object when asked to do so but provide random quantities for other number words and are called one-knowers. Over the next 1 or 2 years, children will learn the quantities associated with two, three, and four and are called two-, three-, and four-knowers, respectively (Le Corre & Carey, 2007). Sometime after this point, children come to understand that each number word represents a unique quantity and that each successive number is one more than the number before it. Children are then considered cardinal principle knowers (CPKs), although it will be several more years before they understand that each successive number is n + 1 ad infinitum (Cheung, Rubenson, & Barner, 2017; Le Corre & Carey, 2007; Wynn, 1990). Recent studies indicate that the age at which children achieve CPK status is predictive of their later number and arithmetic skills, controlling other factors (Chu, vanMarle, Rouder, & Geary, 2018; Geary et al., 2018; Spaepen, Gunderson, Gibson, Goldin-Meadow, & Levine, 2018). Thus, if there is a fundamental quantitative deficit that precedes the emergence of MLD, then this deficit most likely involves poor acuity of the ANS, delayed understanding of cardinality, or delayed learning of the basic number skills that precede children’s understanding of cardinality. The latter would include knowledge of the counting sequence and the use of this sequence to enumerate collections of items (Fuson, 1988; Gelman & Gallistel, 1978). Another skill to consider is children’s early recognition of numerals. This is because the delayed recognition of numerals will delay children’s learning of the associated cardinal values and because the slow or inaccurate processing of numeral magnitudes is related to individual differences in mathematics achievement and MLD during the elementary school years (Bugden & Ansari, 2011; Geary, Hoard, Byrd-Craven, Nugent, & Numtee, 2007; Iuculano, Tang, Hall, & Butterworth, 2008). School-entry number knowledge The relation between children’s competence with number and arithmetic during the early school years and their concurrent and later mathematics achievement is well established (Booth & Siegler, 2006; Geary, 2011; Siegler & Booth, 2004), as are deficits in these areas for children with MLD or persistently low mathematics achievement (Geary et al., 2007; Geary, Hoard, Nugent, & Byrd-Craven, 2008; Hanich, Jordan, Kaplan, & Dick, 2001). One well-established finding is that TA children use a more sophisticated mix of strategies to solve simple arithmetic problems (e.g., Geary, Brown, & Samaranayake, 1991). For 7 + 4, these children either retrieve the answer, decompose the problem into easier ones (e.g., 7 + 4 = 7 + 3 = 10 + 1), or use min counting (stating the larger number and counting the smaller one). Their low-achieving peers, in contrast, tend to commit many retrieval and counting errors, and when they count correctly they often use immature strategies (e.g., counting both addends). These students also have a poor understanding of the mathematical number line, as reflected in inaccurate placement of numerals on the line (Geary et al., 2008), and as noted they are slow at accessing the quantities associated with numerals (Koponen, Salmi, Eklund, & Aro, 2013; Rousselle & Noël, 2007). Performances on arithmetic, number line, and numeral processing tasks are highly correlated and in combination are indices of children’s knowledge of the quantities represented by number symbols and the relations among these quantities, sometimes called number system knowledge (NSK); in addition, school-entry performance on a composite of these measures predicts employment-relevant mathematical competencies during adolescence (Geary et al., 2013). The inclusion of these measures anchors the current study to the extant literature and enables a fuller understanding of the developmental trajectory of children with MLD from the preschool years to school entry. Domain-general cognitive abilities Along with early quantitative abilities and number knowledge, domain-general cognitive abilities contribute to individual differences in early mathematical development (Bull & Scerif, 2001; Espy et al., 2004; Geary, Nicholas, Li, & Sun, 2017; Lee & Bull, 2016). Executive function is one such domain-general ability and, depending on how it is assessed, is composed of several more specific abilities. These include working memory capacity (the ability to rehearse and update mentally represented information), inhibitory control, and shifting between tasks (Cragg & Gilmore, 2014; Diamond, 2013).

4

F.W. Chu et al. / Journal of Experimental Child Psychology 188 (2019) 104668

The inhibition component includes the top-down maintenance of attentional focus and the bottom-up suppression of task-irrelevant perceptions or memories. Task shifting or cognitive flexibility includes the ability to shift between perspectives or sets of information to meet changing task demands. One or several of these components of EF are consistently related to concurrent mathematics achievement and longitudinal gains in this achievement among school-age children (Bull & Lee, 2014; DeStefano & LeFevre, 2004; Geary, 2004) and have emerged as important contributors to individual differences in preschoolers’ mathematics achievement and academic development more broadly (Blair & Razza, 2007; Bull, Espy, & Wiebe, 2008; Clark, Pritchard, & Woodward, 2010). Moreover, many children with MLD have inhibitory and working memory deficits (Geary, Hoard, Nugent, & Bailey, 2012; Murphy, Mazzocco, Hanich, & Early, 2007). Resistance to proactive interference—confusing previously learned material with current to-belearned material—is one aspect of the inhibitory control component of EF (Diamond, 2013; Postle, Brush, & Nick, 2004). Proactive interference might be particularly important in mathematics learning because the core number symbols (i.e., Arabic numerals, basic number words) are finite and reference similar quantities (De Visscher & Noël, 2014; Geary, Hoard, & Bailey, 2012). Proactive interference contributes to arithmetic retrieval errors (e.g., retrieving 16 for 4 + 4) and, in theory, could also contribute to difficulties in learning the cardinal values of number words. During the learning of at least small values, children appear to associate number words with specific collection sizes (e.g., two = r r; Carey, 2004; Sullivan & Barner, 2014), and proactive (and retroactive) interference could result in confusions in relating specific number words to specific quantities, although this remains to be tested. The EF measure used here requires the inhibition of prepotent responses, and if our speculation is correct, then group differences on this measure should cluster with group differences in cardinal knowledge. The current study The study of groups of children with persistently low mathematics achievement has proven to be useful in the identification of underlying cognitive deficits (Geary et al., 2007; Murphy et al., 2007). We used a criterion of mathematics achievement below the 25th percentile for each of two assessments in preschool and our assessment at the end of first grade. We chose this cutoff because this is good estimate of the percentage of U.S. adults whose poor mathematical competencies compromise their economic prospects, and young children who score below this level are at risk for the same long-term difficulties with mathematics (Geary et al., 2013; Planty et al., 2010). The use of multiple assessments reduced the risk of false positives and identified children with average mathematics achievement scores less than the 10th percentile across assessments. This mean level of performance would typically be labeled as MLD even though the cutoff for any particular year was the 25th percentile. The procedure also led to the unexpected identification of children with low mathematics achievement in preschool (10th–12th percentile rank) but average achievement in first grade (49th percentile). Any differences in the preschool competencies of children in this recovered group and their peers with MLD will provide unique information on risk and recovery in young children with respect to their mathematical development. In addition to the EF measure, the contrasts of these groups and the TA children included intelligence and parental education because these are consistently related to mathematics achievement and development (Blair & Razza, 2007; Bull et al., 2008; Geary et al., 2017) and are often below average in children with MLD (Geary, Hoard, Nugent, et al., 2012). Method Participants Children were recruited from the Title I Preschool program within the public school system in Columbia, Missouri, in the midwestern United States. Title I Preschool is a U.S. federally funded program for children from low-income families and is designed to academically prepare them for successful school entry (https://www2.ed.gov/programs/titleiparta/index.html). Consent forms were sent to

5

F.W. Chu et al. / Journal of Experimental Child Psychology 188 (2019) 104668

all entering 3-year-olds across 3 academic years (i.e., three cohorts) and resulted in an initial sample of 232 children whose parents consented to their children’s participation. Of these students, 35 were dropped due to low IQ scores (IQ < 70; n = 14) or failure to complete the preschool tasks (e.g., due to poor attention; n = 21). Of the remaining 197 children, 63 did not complete all testing through the end of first grade (largely due to moving out of district), leaving 134 children (65 boys) who completed all or nearly all sessions in preschool, kindergarten, and first grade. After identifying students with consistently low mathematics achievement, students with recovered mathematics achievement, and TA controls, the final sample was 72 children (detailed below). All children received the Numbers Plus Preschool Mathematics Curriculum (Epstein, 2009) during their 2 years in preschool. Demographic information was obtained through a parent survey for a subset of the total sample. Not everyone answered all questions; thus, the ns varied for the demographic variables. Of this subsample (n = 50), 82% was non-Hispanic, 16% was Hispanic or Latino, and the remainder was unknown. The racial composition of this group (n = 56) was 57% White, 27% Black, 14% multiracial, and the remainder unknown. Self-reported annual household income (n = 52) was distributed as follows: $0–$24,999 (38%), $25,000–$49,999 (15%), $50,000–$74,999 (31%), $75,000–$99,999 (12%), $100,000–$149,999 (4%), and $150,000+ (0%). As a comparison, the median household income in the United States in 2017 was $58,820 (Guzman, 2018). Because mothers’ and fathers’ educational levels were highly correlated (r = .63, p < .001), we created a parental education variable based on the highest level attained of the two parents. This variable consisted of three categories (no information, high school diploma or less, and at least a college degree). Of these children, 46% had at least one parent with a high school diploma or less, 35% had one parent with at least a college degree, and the parental education of the remaining 19% was unknown (U.S. average education: 29% with a high school diploma and 41% with at least a college degree; U.S. Census, 2017). Overall, the sample is skewed toward somewhat lower income and educational levels than the United States as a whole. A further breakdown of the demographics by each group is provided in the online supplementary material (Table 1A). Mathematics achievement Children’s group status was based on performance on mathematics achievement tests administered at the end of both years of preschool (Test of Early Mathematics Ability [TEMA]; Ginsburg & Baroody, 2003) and at the end of first grade (Wechsler Individual Achievement Test-II–Abbreviated [WIAT-II-A Numerical Operations]; Wechsler, 2001). The TEMA consists of items that require producing finger displays to represent different quantities, counting, making numerical comparisons, and using some informal arithmetic. Children started on the first item of the test and continued until they failed on five consecutive items. The Numerical Operations subtest of the WIAT-II-A included items that assessed children’s knowledge of numbers, counting, and arithmetic. Simpler problems involved single-digit addition and subtraction, whereas more complex problems involved multi-digit addition and subtraction. As noted, three groups of children were identified based on these mathematics achievement scores. Children who scored below the 25th percentile in all three assessments were assigned to the persis-

Table 1 Standard mathematics achievement by group status. MLD (n = 14)

Y1 TEMA Y2 TEMA WIAT-II-A NO

Recovered (n = 23)

TA (n = 35)

M

SD

Mean percentile

M

SD

Mean percentile

M

SD

Mean percentile

76.56a 79.29a 78.07a

5.53 5.33 5.76

6.91 9.57 8.36

79.13a 81.30a 99.87b

6.07 5.47 7.68

9.83 11.96 48.96

98.46b 99.18b 99.97b

5.45 5.36 7.04

46.21 47.90 49.69

Note. MLD, mathematical learning disability; TA, typically achieving; Y1 and Y2, first and second years of preschool, respectively; TEMA, Test of Early Mathematics Ability; WIAT-II-A, Wechsler Individual Achievement Test-II–Abbreviated, administered in first grade; NO, Numerical Operations subtest of WIAT-II-A. Different superscripts in the same row indicate significant group differences. Achievement test scores are scaled scores based on national norms (M = 100, SD = 15).

6

F.W. Chu et al. / Journal of Experimental Child Psychology 188 (2019) 104668

tent low-achieving or MLD group (n = 14); as shown in Table 1, mean standard scores ranged between 76 and 79 (SDs = 5–6; 7th–10th percentile rank). Children in the recovered group (n = 23) were those who scored less than the 25th percentile in both years of preschool (Ms = 79 and 81, SDs = 6 and 5, respectively; 10th–12th percentile rank) but had average scores (M = 100, SD = 8; 49th percentile) in first grade. The TA group (n = 35) consisted of children with standard mathematics achievement scores between 90 and 110, inclusive, for all three assessments (Ms = 98–100, SDs = 5–7). We decided to restrict the sample to this narrow band so as to include only children with achievement scores similar to those of the recovered group in first grade. Thus, the final sample consisted of 72 children who had no or very little missing data. The average age across testing sessions was 4.05 years (SD = 0.32) during the first year of preschool, 5.03 years (SD = 0.33) during the second year of preschool, and 7.0 years (SD = 0.33) in first grade. Cognitive measures Intelligence Early in the spring semester in their first year of preschool, children completed the Receptive Vocabulary, Block Design, and Information subscales of the Wechsler Preschool and Primary Scale of Intelligence-III (WPPSI-III; Wechsler, 2002). Following standard procedures, scores were scaled and prorated to generate an estimate of full-scale IQ. The reliabilities of the composite score ranged from .89 to .95, depending on age. Executive function EF was assessed during the early spring semester of each year of preschool using the Conflict Executive Function scale developed for 2- to 6-year-olds (Beck, Schaefer, Pang, & Carlson, 2011; Carlson, 2012). Children were presented with two black plastic index card boxes, both of which had a label card affixed to the front and a slot cut into the top. Children were given a rule and asked to place a card in the box corresponding to the rule (e.g., in Level 2, big kitty in big kitty box, little kitty in little kitty box). For conflict trials, the rule was reversed (e.g., big kitty in little kitty box). In subsequent trials, children sorted cards according to shape or color, and more advanced trials required sorting by shape or color depending on whether a border was present or absent on the cards. There were 70 trials distributed among seven levels, with Levels 1–4 consisting of two 5-trial subsections and Levels 5–7 consisting of 10 trials each. For Levels 1–4 children needed to correctly answer 4 of 5 trials to advance to the next subsection or level, and for Levels 5–7 children needed to correctly complete 4 shape trials and 4 color trials (or 4 border trials and 4 nonborder trials) to advance. All children started at Level 2 and continued until they failed on a subsection or level. The score was the total number of correct trials (maximum score = 70). Intraclass correlations for test–retest reliability for this measure range from .75 to .80 (Beck et al., 2011). Preliteracy The Uppercase Alphabet Recognition subtest of the Phonological Awareness Literacy Screening– PreK (PALS; Invernizzi, Sullivan, Meier, & Swank, 2004) was administered during the early spring semester in both years of preschool and was used to index preliteracy skills. Children were presented with uppercase letters in the alphabet (a few at a time) and were asked to identify each one. The score was the total number of letters correctly identified. Preschool quantitative tasks Children completed a battery of 12 quantitative tasks four times (once in the fall and once in the spring) across the 2 years of preschool, but here we focus on 7 tasks that have previously been found to predict mathematics achievement (Geary & vanMarle, 2016; vanMarle et al., 2014): give-a-number, point-to-x, verbal counting, numeral recognition, enumeration, discrete quantity discrimination, and ordinal choice. The first 5 tasks assess early-emerging aspects of symbolic mathematics, whereas the latter 2 tasks are measures of ANS acuity. Details of the remaining 5 tasks are described in the online supplementary material.

F.W. Chu et al. / Journal of Experimental Child Psychology 188 (2019) 104668

7

Give-a-number and point-to-x The give-a-number and point-to-x tasks both assess children’s understanding of the cardinal value of number words (Wynn, 1990). For the give-a-number task, children were asked to ‘‘feed” a puppet exactly 1, 2, 3, 4, 5, or 6 cookies from a pile. Children placed the cookies on a plate and counted them to ensure that they had the intended number. Children began at set size 1 and advanced to the next set size after a correct response but went down a set size when incorrect. The highest number of objects correctly offered on at least two of three attempts was taken as children’s ‘‘knower level.” For the point-to-x task, children saw two sets of pictured objects displayed simultaneously on a laptop and identified which side of the screen contained x objects. Children received two blocks of 6 trials with ratios ranging from 0.5 to 0.67 (1 vs. 2, 5 vs. 10, 2 vs. 3, 6 vs. 9, 4 vs. 7, and 5 vs. 8). The smaller number was the target for half of the trials, and the side on which the smaller set was displayed was counterbalanced across trials. Children’s score was determined by multiplying the score of each trial (incorrect = 0, correct = 1) by the ratio of the comparison (e.g., 5 vs. 10 = 0.5). These products were then summed to generate a single score weighted for the difficulty of the comparison.

Verbal counting, numeral recognition, and enumeration For the verbal counting task, children counted as high as they could without an error or until they reached 100. Their score was the highest number they counted to without an error. In the numeral recognition task, children were presented with the Arabic numerals (one at a time) from 1 to 15 in random order and were asked to name each numeral. Their score was the total number of correctly named numerals. Children counted an array of 20 stickers while pointing to each sticker in the enumeration task, and their score was the highest number counted correctly.

Discrete quantity discrimination To assess ANS acuity, we used the Panamath program (Halberda, Mazzocco, & Feigenson, 2008). Children saw two separate arrays of blue and yellow dots presented simultaneously on a screen for 2533 ms and identified the set that ‘‘had more dots.” For each trial, there were 5–21 dots of each color, and the ratio between the blue and yellow dots was randomly determined. Children in the first cohort received 24 test trials, and based on low performance of some children in this cohort, 6 relatively easy trials were added for the second and third cohorts. For the 24 initial items the ratio ranged between 1.29 and 3.38, and the ratio of the additional 6 items ranged from 3.5 to 4.0. Arrays for individual trials were selected from ratio bins such that each child received the same number of trials per bin. Accuracy was used as the outcome for this task because it is a more reliable measure than the Weber fraction obtained from this task (Inglis & Gilmore, 2014). Total surface area was controlled for half of the items, and there was no difference in accuracy across trials with and without control of area (Geary & vanMarle, 2016).

Ordinal choice The ordinal choice task was 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, Aw, McCrink, & Santos, 2006). For preschool children, performance on this task is associated with ANS acuity (not counting; Geary & vanMarle, 2016). Children watched as an experimenter sequentially hid two different numbers of small objects (i.e., toy fish) in two opaque cups by dropping them into the cups one at a time. The experimenter filled one cup before filling the other. Children then selected the cup that contained more objects. There were six different comparisons (1 vs. 2, 2 vs. 3, 3 vs. 4, 4 vs. 5, 5 vs. 6, and 6 vs. 7). Because the comparisons varied in difficulty (i.e., ratios varied from 0.5 to 0.86), we generated a weighted score by first multiplying each trial’s score (incorrect = 0, correct = 1) by the ratio of the comparison (e.g., 3 vs. 4 = .75) and then summing the products. This score was then converted to a proportion correct by dividing by the highest possible score.

8

F.W. Chu et al. / Journal of Experimental Child Psychology 188 (2019) 104668

First-grade tasks Addition strategy Children completed 14 simple addition problems (e.g., 2 + 4) and 6 complex addition problems (e.g., 9 + 15) that were presented horizontally on flash cards one at a time. The simple problems consisted of the integers 2 through 9, with the constraint that the same two integers (e.g., 2 + 2) were never used in the same problem. For half of the problems, the smaller addend appeared in the first position, and half of the problems summed to 10 or less. For the current analyses, we focused on the strategies for solving the simple problems (i.e., two single-digit integers) because these have been more thoroughly studied than the strategies used for solving the more complex ones. Children were asked to solve each problem as quickly as possible without making too many mistakes using the strategy they found easiest to solve the problem. After children spoke each answer out loud, they were asked how they reached the answer. Based on their answer and the experimenter’s observations, the trials were classified into strategies, including counting fingers, fingers (use of fingers without counting, which appears to prompt retrieval of the answer), verbal counting, retrieval, decomposition (breaking down the problem into simpler problems or using known facts), or other/ mixed strategy (using a strategy in combination with another strategy). Counting trials (verbal and finger) were further broken down based on use of the min procedure (i.e., counting starting from the larger addend), sum procedure (i.e., counting both addends or starting with 1), or max procedure (i.e., counting starting from the smaller addend). Problems were then coded on a 6-point scale in which higher numbers reflected higher accuracy and greater sophistication of the strategy used: 1 = error in using retrieval, fingers, or decomposition; 2 = error in using a counting strategy (finger or verbal counting); 3 = correct use of max or sum counting; 4 = correct use of min counting; 5 = correct use of retrieval-related strategies (fingers or decomposition); and 6 = correct retrieval. A graded response model was then conducted using the ‘‘grm” procedure in the ltm package (Rizopoulos, 2006) in R (R Core Team, 2017), which is a type of item response theory model for ordinal polytomous categories. This model produces a z-score estimate of children’s problem-solving sophistication (see Chu et al., 2018).

Number line estimation task Children were presented with a series of 24 number lines ranging from 0 to 100 (measuring 25 cm) with a target number n printed in large font above the number line. This task was completed on paper, and children were asked to mark with a pencil where they thought the target should go on the number line. Absolute number line error has been found to be a good predictor of mathematics achievement (Siegler & Booth, 2004); thus, we used this as a measure of children’s understanding of the number line. The overall score was calculated by determining the absolute difference between the child’s placement and the correct position of the number for each trial (absolute error) and taking the mean of these differences across trials (a = .87).

Number sets sensitivity The stimuli for this task consisted of arrays of objects or Arabic numerals presented in half-inch squares. Pairs and triplets were joined in domino-like rectangles, and children were asked to determine as quickly and accurately as possible whether the pairs or triplets matched a target number, 5 or 9 (two pages each), in large font at the top of the page (Geary, Bailey, & Hoard, 2009). Children were instructed to move across each line of the page from left to right without skipping any items and to ‘‘circle any groups that can be put together to make the top number” and ‘‘work as fast as you can without making many mistakes.” Children were given 60 s per page for the target 5 and 90 s per page for the target 9. Children were scored on items correctly circled as matches (i.e., hits), correct matches that were not circled (i.e., misses), incorrect items that were not circled (i.e., correct rejections), and incorrect items that were circled as matches (i.e., false alarms). Following Geary et al. (2009), we calculated the signal detection measure for sensitivity by subtracting the standardized number of false alarms (a = .93) from the standardized number of hits (a = .83).

F.W. Chu et al. / Journal of Experimental Child Psychology 188 (2019) 104668

9

NSK composite The strategy choice, number line estimation, and number sets sensitivity scores all were highly correlated (rs > .50, p < .001); thus, these three variables all were standardized (M = 0, SD = 1) and the means were averaged to generate the NSK composite score (a = .79). Discrete quantity discrimination Children also completed the discrete quantity discrimination task in first grade. The procedure was the same as in preschool, but the task consisted of 36 trials and ratios of 1.18–3.10. Procedure For all sessions, children were tested individually in a quiet location at their school site. Preschool sessions typically lasted about 35 min, and first-grade sessions lasted about 45 min. In all, each preschool quantitative measure was administered four times: once in the spring semester and once in the fall semester of each year. The cognitive battery was administered in the early spring semester of each preschool year, the first-grade number and arithmetic tasks were administered in the early fall semester, and the achievement test was administered in the spring semester of first grade. A table showing each assessment, the tasks administered, the semester of the assessment, and group age breakdowns at each time point is available in the online supplementary material (Table 2A). The experimental procedure was reviewed and approved by the institutional review board of the University of Missouri. We obtained written consent from all parents and received verbal assent from all participants for all assessments. Analyses Two broad categories of analyses were conducted to examine how children’s achievement status was related to their early quantitative skills. The first category involved examining group differences on the covariates (Year 1 [Y1] and Year 2 [Y2] EF, IQ, and letter recognition), preschool quantitative skills, and the NSK composite. A series of multilevel analyses with time nested within group was conducted with the raw scores for the preschool quantitative tasks to determine whether the groups differed across any of the four time points, accounting for the covariates. For quantitative tasks with significant overall group differences, we conducted follow-up analyses to determine the time points at which the differences emerged. Group differences were also assessed for NSK because the measure captures variance in early number knowledge above and beyond that captured by standardized achievement tests (Geary et al., 2013, 2018). The second set of analyses involved logistic regressions to identify the best early quantitative predictors of group membership, specifically for discriminating students in the recovered and MLD groups and students in the TA and recovered groups. Group membership models were estimated using all measures at each year of preschool. Then, individual group membership classifications were predicted based on logistic regressions. These analyses allowed us to identify skill deficits and the timing of these deficits for children in the MLD and recovered groups. The data used in the analyses and associated R code have been archived in Open Science Framework (https://osf.io/phtxm/). Results Descriptive information by group is presented for mathematics achievement, covariates, quantitative tasks, and number knowledge composite in Table 2. Covariates A multivariate analysis of variance (MANOVA) examining the covariates (Y1 EF, Y2 EF, IQ, and letter recognition) indicated significant group differences, F(8, 134) = 4.80, p < .001, ɳ2p = .22. Box plots for each covariate can be seen in Fig. 1. Follow-up analyses of variance (ANOVAs) revealed that the groups

10

F.W. Chu et al. / Journal of Experimental Child Psychology 188 (2019) 104668

Table 2 Covariates and quantitative tasks by status. MLD (n = 14)

Recovered (n = 23)

TA (n = 35)

M

SD

M

SD

M

SD

Covariates Y1 EF Y2 EF IQ Letter recognition

21.57a 33.93a 87.40a 4.60a

9.57 12.55 11.08 6.26

26.00ab 44.22b 91.22ab 5.57a

11.50 9.51 13.17 6.58

31.54b 43.69b 99.57b 15.26b

13.37 14.34 15.35 8.34

Preschool quantitative tasks Year 1–Time 1 Give-a-number Point-to-x Verbal counting Numeral Recognition Enumeration Discrete quantity discrimination Ordinal choice

0.93 1.75 3.57 1.48 4.43 58.31 35.09

1.00 0.46 2.62 1.14 3.65 12.97 31.97

2.04 2.11 4.78 1.26 5.22 61.45 35.18

1.40 0.53 3.73 1.42 4.42 15.35 28.11

3.80 2.22 7.71 5.12 9.46 67.44 37.84

1.69 0.56 5.28 3.66 4.86 17.12 32.31

Year 1–Time 2 Give-a-number Point-to-x Verbal counting Numeral recognition Enumeration Discrete quantity discrimination Ordinal choice

1.43 2.13 6.83 2.95 7.79 59.15 32.74

0.85 0.69 5.04 2.60 4.68 17.83 32.00

2.43 2.20 5.90 2.22 7.13 62.32 43.76

1.24 0.48 4.18 2.39 4.85 16.26 32.52

4.88 2.25 12.93 7.60 12.20 70.93 38.68

1.29 0.51 7.59 3.72 5.34 19.67 32.70

Year 2–Time 3 Give-a-number Point-to-x Verbal counting Numeral recognition Enumeration Discrete quantity discrimination Ordinal choice

2.64 2.08 7.86 4.14 7.00 67.38 63.74

1.55 0.53 3.84 4.19 4.17 17.98 12.53

3.96 2.16 9.52 4.39 9.87 72.46 66.90

1.58 0.59 7.79 3.06 3.90 17.44 16.93

5.86 2.67 16.53 9.52 14.00 85.50 63.69

0.58 0.45 9.12 2.78 4.47 13.41 13.63

Year 2–Time 4 Give-a-number Point-to-x Verbal counting Numeral recognition Enumeration Discrete quantity discrimination Ordinal choice

4.00 2.19 9.00 4.29 10.64 61.90 53.72

1.75 0.39 5.04 3.50 3.41 18.75 23.74

5.13 2.40 13.57 5.99 11.78 77.97 62.01

1.25 0.49 9.69 2.99 4.90 19.61 17.37

5.95 2.84 21.01 10.59 15.54 92.43 67.39

0.21 0.46 10.44 2.85 4.29 9.27 19.56

0.30 0.96 0.89 0.43

0.01 0.31 0.63 0.31

0.46 0.99 1.21 0.76

0.39 0.28 0.17 0.28

0.30 0.68 0.72 0.38

First-grade quantitative tasks Addition strategy Number line Number sets sensitivity NSK composite

0.29 1.24 1.48 1.00

Note. MLD, mathematical learning disability; TA, typically achieving; Y1 and Y2, first and second years of preschool, respectively; EF, executive function; IQ, Wechsler Preschool and Primary Scale of Intelligence; NSK, number system knowledge. Different superscripts in the same row for the covariates indicate significant differences. Time refers to each of the four assessments across the 2 years of preschool. The first-grade tasks are standardized (M = 0, SD = 1), with higher scores meaning better performance.

differed on Y1 EF, F(2, 69) = 3.75, p = .028, ɳ2p = .10, and contrasts of group means revealed that the only significant difference was between the MLD and TA groups, t(69) = 2.60, p = .031, d = 0.80. The groups also differed on Y2 EF, F(2, 69) = 3.52, p = .035, ɳ2p = .09; the MLD group had significantly lower scores than the recovered group, t(69) = 2.40, p = .049, d = 0.96, and the TA group, t(69) = 2.44,

F.W. Chu et al. / Journal of Experimental Child Psychology 188 (2019) 104668

11

Fig. 1. Group differences in executive function (EF), intelligence (IQ), and letter recognition. Scores are averaged across the four preschool assessments. Filled circles () are the means, and horizontal lines are the medians. The top and bottom of the boxes are the 75th and 25th percentiles, respectively, and the end points of the top and bottom vertical lines are maximum and minimum scores, respectively. MLD, mathematical learning disability; R, recovered; TA, typically achieving.

p = .045, d = 0.70. There were also significant differences in IQ, F(2, 69) = 4.77, p = .012, ɳ2p = .12; the MLD group had lower IQ scores than the TA group, t(69) = 2.76, p = .020, d = 0.85, whereas there was a trend for the recovered group to have lower IQ scores than the TA group, t(69) = 2.23, p = .073, d = 0.57. Finally, the group differences for letter recognition were also significant, F(2, 69) = 16.49, p < .001, ɳ2p = .32, and indicated that children in both the MLD and recovered groups recognized significantly fewer letters, t(69) = 4.53, p < .001, d = 1.36 and t(69) = 4.85, p < .001, d = 1.26, respectively, than children in the TA group.

Preschool quantitative tasks The multilevel analyses revealed significant group differences (ps < .05) on all quantitative tasks with control of the covariates (including parent income and education status), with the exception of nonsignificant findings for the ordinal choice and enumeration tasks. Children with MLD had significantly lower scores than TA children on the give-a-number, point-to-x, verbal counting, and discrete quantity discrimination tasks across the four preschool time points, as summarized in Table 3. The recovered children, on average, also had lower scores than the TA children on the give-a-number, verbal counting, numerical recognition, and discrete quantity discrimination tasks. Box plots for the preschool quantitative tasks are shown in Fig. 2, collapsed across all four assessments. The full regression results are shown in the online supplementary material (Table 3A), and least squares mean contrasts for interaction effects are reported below for each task. In addition, significant time point differences

12

F.W. Chu et al. / Journal of Experimental Child Psychology 188 (2019) 104668

Table 3 Overall (across time) group differences on the preschool quantitative measures. Preschool task

Give-a-number Point-to-x Verbal counting Numeral recognition Enumeration Discrete quantity discrimination Ordinal choice

MLD vs. R

MLD vs. TA

t

p

6.06 1.92 0.65 0.65 0.27 0.58 1.03

<.001 .150 .795 .791 .960 .833 .561

t

R vs. TA p

10.94 3.65 2.59 2.35 1.80 2.64 1.49

<.001 .002 .034 .060 .181 .030 .306

t

p 6.23 2.20 2.46 3.79 1.94 2.62 0.59

<.001 .082 .046 .001 .141 .032 .827

Note. MLD, mathematical learning disability; R, recovered; TA, typically achieving.

were found on all measures (ps < .001), with the general trend showing performance improvements across time. Overall, the most consistent patterns emerged for the two cardinal knowledge tasks (give-anumber and point-to-x) and for numeral recognition. Children in the MLD and recovered groups lagged TA children on the give-a-number and numeral recognition tasks throughout preschool, with the exception that the give-a-number scores for the recovered and TA groups did not differ at the end of preschool (see Table 3A in online supplementary material). There were no group differences on the point-to-x task during the first year of preschool, but an advantage for the TA group emerged during the second year. Fig. 3 shows performance between groups at each time point for the give-anumber, verbal counting, and discrete quantity discrimination tasks where evidence for interaction effects was found. Give-a-number There was a significant interaction between group and time point, F(6, 147) = 2.95, p = .01, g2p = .11. The knower levels for children in the MLD and recovered groups were similar at the beginning of the first year of preschool, but there were more rapid gains in the recovered groups than in the MLD group across time points during the second year (ps < .001, ds = 0.84 and 0.77, respectively). Children in the TA group had knower-level advantages over both the recovered and MLD groups at all time points (ps < .01, average ds = 1.11 and 1.87, respectively) except for the contrast with the recovered group at the end of preschool (p = .300). The latter result needs to be interpreted with caution, however, given ceiling effects at this time point. The number of children who achieved CPK status at each time point is presented in Table 4, and Fisher’s exact test revealed significant group differences (p < .001). Post hoc pairwise tests revealed that the MLD and recovered groups did not differ in the proportions of children who achieved CPK status at each time point (p = .215). Compared with the TA group, both the MLD and recovered groups were significantly delayed in the development of CPK status (p < .001). Verbal counting There was marginal evidence for an interaction between group and time point, F(6, 147) = 1.90, p = .083, g2p = .07. There were very few group differences across the 2 years of preschool, and the two significant effects appeared only between the TA and MLD groups. Children in the TA group outperformed children in the MLD group at both time points during the second year of preschool (ps = .050 and .027, ds = 1.08 and 1.30, respectively). Discrete quantity discrimination There was also marginal evidence for an interaction in discrete quantity discrimination, F(6, 147) = 1.98, p = .072, g2p = .07. Children in the TA group outperformed their MLD peers at both time points during the second year of preschool (ps < .033, average d = 1.22). Children in the TA group also outperformed the recovered children at the first time point of the second year (p = .006, d = 0.86) and marginally at the final time point (p = .064, d = 1.01).

F.W. Chu et al. / Journal of Experimental Child Psychology 188 (2019) 104668

13

Fig. 2. Group differences in quantitative development during preschool. Scores are averaged across the four preschool assessments. Filled circles () are the means, and horizontal lines are the medians. The top and bottom of the boxes are the 75th and 25th percentiles, respectively, and the end points of the top and bottom vertical lines are maximum and minimum scores, respectively. MLD, mathematical learning disability; R, recovered; TA, typically achieving.

Point-to-x, numeral recognition, enumeration, and ordinal choice There were no significant interactions between group and time point for point-to-x, F(6, 147) = 0.64, p = .695, numeral recognition, F(6, 147) = 1.47, p = .193, enumeration, F(6, 147) = 0.97, p = .446, or ordinal choice, F(6, 147) = 0.09, p = .997. First-grade NSK and discrete quantity discrimination The full regression results for the NSK measures and discrete quantity discrimination are presented in the online supplementary material (Table 4A), and key group comparisons are presented in Table 5. The basic pattern is that children in the TA group performed better than children in one or both of the

14

F.W. Chu et al. / Journal of Experimental Child Psychology 188 (2019) 104668

Fig. 3. Group differences across time for core quantitative tasks. Filled circles () are the means, and horizontal lines are the medians. The top and bottom of the boxes are the 75th and 25th percentiles, respectively, and the end points of the top and bottom vertical lines are maximum and minimum scores, respectively. For the give-a-number task, the typically achieving (TA) children had higher scores than children in the mathematical learning disability (MLD) and recovered groups except for Time 4, where there was no difference between the TA and recovered groups. The verbal counting of children in the TA group exceeded that in the MLD group at Time 3 and Time 4. For discrete quantity discrimination, the TA group had advantages over the two other groups at Time 3 and Time 4.

two other groups on most measures. For the composite NSK measure and using mean scores, the gap between the MLD and recovered groups (d = 1.06) was about the same magnitude as the gap between the recovered and TA groups (d = 1.05). The gap between the TA and MLD groups was very large (d = 3.25) and was larger than the gap for mathematics achievement (d = 1.50).

15

F.W. Chu et al. / Journal of Experimental Child Psychology 188 (2019) 104668 Table 4 Achieving cardinal principle knower status.

MLD Recovered TA

Time 1

Time 2

Time 3

Time 4

Not by Time 4

0 (0.00%) 2 (8.70%) 12 (34.29%)

0 (0.00%) 1 (4.35%) 10 (28.57%)

2 (14.29%) 7 (30.43%) 12 (34.29%)

4 (28.57%) 8 (34.78%) 1 (2.86%)

8 (57.14%) 5 (21.74%) 0 (0.00%)

Note. MLD, mathematical learning disability; TA, typically achieving. Values indicate the number of students and percentage (noncumulative) of students (by achievement group) who achieved cardinal principle knower status by each time point.

Table 5 Contrasts for first-grade quantitative tasks. Estimate

SE

t

Cohen’s d

p

z-Strategy score MLD R MLD TA R TA

0.29 0.61 0.32

0.15 0.15 0.12

1.90 3.99 2.66

0.31 0.57 0.35

.151 .001 .029

Number line means MLD R MLD TA R TA

0.70 0.79 0.10

0.41 0.41 0.33

1.70 1.92 0.29

0.28 0.27 0.04

.216 .145 .955

Number sets sensitivity MLD R MLD TA R TA

0.84 1.64 0.80

0.43 0.43 0.34

1.96 3.79 2.34

0.32 0.54 0.31

.134 .001 .061

NSK composite MLD R MLD TA R TA

0.61 1.02 0.41

0.25 0.25 0.20

2.45 4.05 2.05

0.40 0.58 0.27

.047 .001 .113

Discrete quantity discrimination MLD R MLD TA R TA

0.61 2.60 1.99

2.61 2.63 2.08

0.23 0.99 0.96

0.04 0.14 0.13

.971 .589 .609

Note. MLD, mathematical learning disability; R, recovered; TA, typically achieving. Estimates for z-strategy score, number line means, number sets sensitivity, and number system knowledge (NSK) composite are based on standardized (M = 0, SD = 1) scores. The estimates for discrete quantity discrimination are differences in percentage correct.

As shown in Table 5, children in the TA group used more sophisticated strategies to solve addition problems and were more fluent in the processing of basic numeral information (number sets sensitivity) than children in the MLD and recovered groups; the effect sizes are smaller in Table 5 than those for mean scores due to control of the covariates (Table 4A). As shown in Table 6, the distribution of the children’s strategy choices revealed greater use of min counting, decomposition or finger use (advanced strategies), and more correct retrieval of answers by the TA children than by children in the two other groups. Although the overall z-strategy contrast was not significant, children in the

Table 6 Strategy distribution across achievement status.

MLD Recovered TA

N/A Missing

1 Retrieval error

2 Counting error

3 Sum/Max count

4 Min count

5 Advanced strategy

6 Correct retrieval

0.51 1.55 2.24

32.65 21.12 8.37

46.94 27.95 13.67

12.76 22.36 23.67

4.08 18.32 35.31

1.02 1.24 2.65

2.04 7.45 14.08

Note. N/A = not available; MLD, mathematical learning disability; TA, typically achieving. Values are percentages.

16

F.W. Chu et al. / Journal of Experimental Child Psychology 188 (2019) 104668

recovered group correctly retrieved more answers, used min counting more frequently, and committed fewer retrieval and counting errors than children in the MLD group. Controlling other factors (see Table 4A in online supplementary material), the group differences on the number line task were not significant, but the TA children were more fluent in processing numeral magnitudes relative to children in the MLD group (d = 0.54) and recovered group (d = 0.31). Children in the recovered and TA groups did not differ on the discrete quantity discrimination task (p = .609; Table 5). There was also similar performance between the MLD and TA groups (p = .589) as well as between the MLD and recovered groups (p = .971). Predicting group membership MLD versus recovered Logistic regressions were run separately for both years of preschool, discriminating between the MLD and recovered groups. The online supplementary material (Table 5A) shows analysis of deviance estimates for all predictors. Time 1 give-a-number (p = .014), point-to-x (p = .049), and ordinal choice (p < .001) scores were the best discriminators between the MLD and recovered groups by the end of the first year of preschool. A final logistic regression model using Time 1 give-a-number, point-to-x, and ordinal choice was significant, v2(7) = 18.90, p = .008, Nagelkerke’s R2 = .54. By the end of the second year of preschool, EF (p = .007), Time 2 enumeration (p = .001), and ordinal choice (p < .001) scores were the best predictors discriminating the MLD and recovered groups. A final logistic regression model using these predictors was significant, v2(3) = 9.25, p = .026, Nagelkerke’s R2 = .30. These two models from the first and second years were able to correctly classify group membership with accuracies of 70.27% and 62.16%, respectively, using a leave-one-out cross-validation (models were iteratively estimated with n 1 samples, leaving one sample out for prediction at each iteration). In sum, children in the recovered group had early advantages over their MLD peers in their understanding of the cardinal value of number words (give-a-number and point-to-x) and had a better intuitive understanding of more and less (ordinal choice). The latter advantage remained during the second year of preschool, and advantages in EF and in the practical use of counting (enumeration) emerged. Recovered versus TA Similarly, logistic regressions were run separately for both years of preschool, discriminating now between the recovered and TA groups. Table 5A in the SOM shows predictor estimates. By the end of the first year of preschool, the recovered and TA groups differed on PALS (p < .001) and TEMA (p < .001) scores. A final model including just these predictors (as well as the other covariates) was significant, X2(4) = 77.90, p < .001, Nagelkerke’s R2 = 1.00. By the end of the second year, the best predictors remained PALS (p < .001) and TEMA (p < .001) scores as well as full-scale IQ (p = .023), which was confirmed by a significant overall logistic model of these predictors, v2(4) = 77.90, p < .001, Nagelkerke’s R2 = 1.00. Both models at the first and second years were able to distinguish recovered versus TA groups with 100% accuracy with leave-one-out cross-validation. In sum, children in the recovered group showed general delays in preliteracy knowledge and general mathematics achievement. Discussion To our knowledge, this is the first study to track the quantitative development of children who exhibited persistently low mathematics achievement throughout the preschool years and through the end of first grade as well as the quantitative development of low-achieving preschoolers who recovered to have average mathematics achievement by the end of first grade. By identifying developmental trends in the domain-general abilities and preschool quantitative knowledge that distinguishes these two groups, we gain a better understanding of the early deficits that presage longterm risk for poor mathematics achievement as well as insight into what knowledge may be critical for children’s mathematical development more generally. Similarly, differences in the quantitative development of recovered children and their TA peers and group differences in specific facets of

F.W. Chu et al. / Journal of Experimental Child Psychology 188 (2019) 104668

17

NSK in first grade help to uncover the quantitative strengths and weaknesses of the recovered children. Mathematical learning disability Relative to their TA peers, the MLD children lagged in the acquisition of symbolic quantitative knowledge throughout the preschool years and continuing into first grade. These children’s broad quantitative deficits are not too surprising given the same pattern in older children (e.g., Geary et al., 2007; Geary, Hoard, Nugent, et al., 2012; Hanich et al., 2001; Landerl, Bevan, & Butterworth, 2004; Murphy et al., 2007), but they do extend the pattern to the preschool years. As is often found for older students with MLD, children with MLD in this study exhibited low average intelligence and EF scores; however, and again consistent with previous studies (Geary, Hoard, Nugent, et al., 2012), their quantitative deficits and mathematics achievement were often even worse than their domain-general deficits. As noted, a more unique contribution is the contrast of children in the MLD and recovered groups. The mathematics achievement of children in both of these groups was substantively below average throughout the preschool years, with no discernible group differences. Similarly, their intelligence and EF scores during the first year of preschool did not differ significantly, although the recovered children had significantly higher EF scores during the second year of preschool (see below). The best discriminators by the end of preschool of whether students would eventually recover or continue as MLD were EF, enumeration, and ordinal choice scores. These results suggest that a mix of early domaingeneral and symbolic and nonsymbolic (discriminating larger and smaller sets of objects) number deficits contribute to risk of long-term MLD. However, it is also important to note that give-a-number was the only preschool quantitative task in which children in the MLD group were consistently behind their TA peers across all preschool assessments (see Table 3A in online supplementary material). These deficits are in keeping with the importance of cardinal knowledge for later learning about number concepts and the relations among number words and numerals (Geary et al., 2018; Geary & vanMarle, 2018; Spaepen et al., 2018). The most consistently used measure of the ANS (i.e., nonsymbolic number knowledge) is the discrete quantity discrimination task (Halberda et al., 2008), and performance on this task is correlated with concurrent and later mathematics achievement (Chen & Li, 2014; Fazio et al., 2014; Schneider et al., 2017). We also found deficits for children with MLD on this task, but only during the second year of preschool. One possibility is that the normal development of the acuity of the ANS occurs more slowly in children with MLD than in their TA peers (Piazza et al., 2010). In this situation, any ANS deficit is more of a developmental delay in this system than an early fundamental deficit. The other possibility is that their poor performance on the discrete quantity discrimination task is due to their poor understanding of cardinality. Shusterman, Slusser, Halberda, and Odic (2016) and Geary and vanMarle (2018) found that performance on this task accelerated after children became CPKs. From this perspective, the apparent ANS deficit of children with MLD is actually secondary to their especially poor understanding of the cardinal value of number words. Indeed, 8 of the 14 children with MLD had not achieved CPK status by the end of preschool. In contrast, 18 of the 23 recovered children and all 35 of the TA children had achieved CPK status by the end of preschool. It is potentially important that the TA children had mean knower levels of four at the beginning of preschool, that is, levels indicating that many of them had become, or were on the cusp of becoming, CPKs (Carey, 2004; Le Corre & Carey, 2007; Wynn, 1992); when examining the time points at which children became CPKs, 34% of the TA children had achieved CPK status at the beginning of preschool and an additional 29% had achieved it by the end of the first year of preschool. Children in the recovered group did not achieve the same level of performance until the beginning of the second year of preschool, when 43% of the recovered children had achieved CPK status, and an additional 34% had achieved this status by the end of preschool. In other words, children in the recovered group showed a 1-year delay in the acquisition of cardinal knowledge and children in the MLD group showed an even more substantive delay. The symbolic number deficits of children with MLD continued into first grade. Relative to their TA peers, their deficits were evident on measures of addition strategy choices, fluency of numeral

18

F.W. Chu et al. / Journal of Experimental Child Psychology 188 (2019) 104668

processing, and overall NSK. Their overall deficit on the latter was very large (>3 standard deviations) and double the magnitude of the deficits suggested by their performance on the mathematics achievement test (d = 1.50). The difference indicates that mathematics achievement tests may substantively underestimate the magnitude of the numerical and arithmetical deficits of children with MLD and, in doing so, substantively underestimate their long-term risk of poor employment-relevant mathematical competencies at school completion (Geary et al., 2013). The performance of children with MLD on specific first-grade tasks is consistent with many prior studies of school-age children with MLD (Andersson, 2010; Geary et al., 2008, 2009; Holloway & Ansari, 2009) and may provide insight into the nature of their underlying deficits. For instance, they committed retrieval or counting errors on approximately 80% of the addition problems they attempted to solve (Geary, 1990, 1993). Retrieval errors are very common among children with MLD and appear to reflect, at least in part, proactive interference, whereby errors occur because answers to similar problems are simultaneously retrieved during problem solving (De Visscher & Noël, 2014; Geary, Hoard, & Bailey, 2012). This interference may be related to abnormalities in the functional connectivity between several prefrontal regions (e.g., dorsolateral cortex, inferior frontal gyrus) and the intraparietal sulcus (De Smedt, Holloway, & Ansari, 2011; De Visscher et al., 2018; Jonides & Nee, 2006; Rosenberg-Lee et al., 2015) as well as the hippocampus during initial learning of number relations (Qin et al., 2014). A deficit in this network and potential difficulties with proactive interference are in keeping with the poor EF performance of children with MLD during preschool. Relative to TA children, their performance suggested about a 1-year delay in the development of at least the inhibitory component of EF, but given that retrieval, inhibitory, and working memory deficits tend to persist in children with MLD (Geary, Hoard, Nugent, et al., 2012; Murphy et al., 2007; Passolunghi & Siegel, 2004), it is not likely to be a simple developmental delay but rather a long-term deficit. The EF measure used in our study requires a combination of inhibitory control—inhibition of prepotent responses and rules—and working memory (maintaining rules), but the inhibitory processes appear to be more important for overall performance (Best & Miller, 2010; Carlson, 2005). Interference would occur during conflict trials, where children must determine whether the previously learned rule or the new rule applies in the current context. In other words, proactive interference resulting from poor inhibition of prepotent responses may have contributed to the poor EF performance of the children with MLD. Moreover, a relation between inhibitory control and conflict resolution and preschoolers’ understanding of cardinality and mathematics achievement more generally has been found in previous studies (Fuhs & McNeil, 2013; Mou, Berteletti, & Hyde, 2018). As we suggested in the Introduction, inhibitory deficits due to proactive (and retroactive) interference could contribute to delays in learning the cardinal values of number words, specifically confusions in relating specific number words to specific quantities. Whether this is a factor that contributes to the delayed acquisition of cardinal knowledge in children with MLD cannot, however, be determined from our results. A more direct test would require assessment of susceptibility to interference and performance on estimation tasks whereby children are asked to state the number word associated with briefly presented (to prevent counting) collections of objects. We hypothesize that susceptibility to proactive (and presumably retroactive) interference will be associated with inconsistency in associating number words with object sets (e.g., sometimes stating ‘‘three” to three-object sets and other times stating ‘‘two” or ‘‘one”; Libertus, Odic, Feigenson, & Halberda, 2016). Although we speculate that susceptibility to interference might have the biggest effects on inhibitory control as measured in our study, it also constrains other aspects of EF such as cognitive flexibility and working memory (Diamond, 2013). In other words, inhibitory control need not be the only explanatory mechanism underlying EF deficits (Blakey, Visser, & Carroll, 2016; Munakata, 2001). Whatever the specific underlying cognitive mechanisms, the poor early number knowledge and domain-general deficits of children with MLD likely combined to result in the very large NSK deficits found at the beginning of first grade. We base this conclusion on the finding that domain-specific knowledge and domain-general abilities independently contribute to subsequent mathematical learning (Geary et al., 2017; Lee & Bull, 2016). For these children, deficits in both domain-specific knowledge and domain-general abilities will likely compound over time, resulting in a widening gap between their mathematical competencies and those of TA children. The combined deficits indicate

F.W. Chu et al. / Journal of Experimental Child Psychology 188 (2019) 104668

19

that any interventions focused on improving their domain-specific knowledge (e.g., cardinal knowledge) will need to be designed in ways that provide additional supports that compensate for their domain-general deficits, as is being done by Fuchs and colleagues for older children (e.g., Fuchs et al., 2014). Any such supports and domain-specific interventions will almost certainly need to be implemented for key topics (e.g., fractions) throughout their schooling (Bailey, Duncan, Odgers, & Yu, 2017). Recovery from early deficits Prior studies have identified older children with low mathematics achievement scores in one grade and at least average scores in the next (e.g., Geary, 1990; Geary et al., 1991), with recovered children showing no or only subtle underlying cognitive (e.g., working memory) deficits. To the best of our knowledge, the current study is unique in the identification of children who had substantively below average mathematics achievement during the preschool years but average achievement at the end of first grade. We suspect that the recovered children’s rapid increase in EF from the first year to the second year of preschool is the critical domain-general competence that enabled them to eventually obtain average mathematics achievement scores. We base this conclusion on the importance of EF for preschoolers’ mathematical development (Blair & Razza, 2007; Bull et al., 2008; Clark et al., 2010) and the consistent finding of related deficits in older children with persistent low mathematics achievement and the importance of EF for mathematical development more generally (Bull & Lee, 2014; DeStefano & LeFevre, 2004; Geary, 2004; Geary, Hoard, Nugent, et al., 2012; Mazzocco & Kover, 2007; Swanson, Jerman, & Zheng, 2008). Children’s performance at Time 2 and Time 3 on the give-a-number task was a significant predictor of classification in the recovered or TA groups. Although children in the recovered group lagged significantly behind their peers at Time 2 and Time 3, they also showed significant improvements in cardinal knower levels during the second year of preschool and closed the gap with their TA peers by the end of preschool. The rapid gains in both cardinal knower level and EF could be a coincidence but is also consistent with the importance of EF for coming to understand the cardinal value of number words. As noted above, one potential common factor may be susceptibility to proactive interference, but this remains to be determined. In any case, the closing of the gap between the recovered and TA groups on the give-a-number task was likely influenced in part by ceiling effects. In fact, TA children outperformed recovered children on the more difficult cardinality task, point-to-x, due to larger gains in the TA group than in the recovered group during the second year of preschool. Thus, despite rapid gains, recovered children still lagged behind their TA peers in their conceptual understanding of number. There were other differences as well. Recovered children’s recognition of numerals lagged behind that of TA children throughout the preschool years. The average child in the recovered group recognized only a single numeral (M = 1.26) at the beginning of preschool, somewhat less than the average child with MLD (M = 1.48). It is possible that these children received little exposure to numerals (or letters) at home before entering preschool (LeFevre et al., 2009). Once in preschool, their gain of about 5 additional recognized numerals over the next 2 years was similar to that of TA children’s 5.5 numeral gain. If this were the only factor, however, we might then have expected a larger difference between the recovered and MLD groups by the second year of preschool given the recovered group’s advantage in EF at that time and its modest (but not significant) advantage in IQ. There was in fact a nonsignificant trend for recovered children’s recognition of numerals to grow more rapidly than that of children with MLD during the second year. It is also possible that some of these children—and children with MLD—have delays or deficits in the perceptual and brain systems that support the learning of visual symbols and the ease of discriminating between them (Holloway, Battista, Vogel, & Ansari, 2013; Yeo, Wilkey, & Price, 2017), but this cannot be determined from the current study. Whatever the cause, children in the recovered group showed a mix of strengths and weaknesses in first grade. Their performance on the number line and discrete quantity discrimination tasks did not differ from that of TA children. The former suggests that recovered children’s understanding of numerical magnitude was age appropriate, and it is possible that this contributed to their age-appropriate

20

F.W. Chu et al. / Journal of Experimental Child Psychology 188 (2019) 104668

performance on the discrete quantity discrimination task or that they influenced one another (Feigenson, Libertus, & Halberda, 2013; Geary & vanMarle, 2018). Modest deficits were evident in the sophistication of the strategies they used to solve addition problems (d = 0.41) and in the fluency of accessing the quantities associated with numerals (d = 0.34); these competencies are associated with mathematics achievement generally and with mathematics difficulties in particular (Bugden & Ansari, 2011; Geary, 1993; Holloway & Ansari, 2009; Rousselle & Noël, 2007). Looking more closely at how these children solved addition problems (Table 6), we see a mix of mature (min) and immature (sum/max) counting and a high percentage of retrieval errors (21% vs. 7% correct retrieval). Children who are delayed in the use of mature counting generally catch up, but retrieval deficits can be more persistent (Geary, Hoard, Nugent, et al., 2012; Jordan, Hanich, & Kaplan, 2003). The pattern of poor fluency in numeral processing and retrieval errors suggests that these children may have subtle deficits in accessing numerical information—numeral magnitudes and problem–answer associations—from long-term memory (De Smedt & Gilmore, 2011; Rousselle & Noël, 2007) despite their average mathematics achievement scores.

Limitations and conclusion There are some important limitations to this study. Although the results are based on longitudinal relations, they should be interpreted with caution because the data are still correlational. More definitive conclusions must await follow-up experimental or intervention studies to test our proposed deficits associated with the quantitative competencies of the MLD and recovered groups (e.g., regarding heightened susceptibility to proactive interference). Moreover, we do not know how attrition may have influenced our results. Many of the students who were dropped from the study during the preschool years had low IQ scores or persistent difficulties attending to the tasks; thus, there is a distinct possibility that we have underestimated the number of young children with MLD and the severity of their deficits. Our EF measure largely tapped inhibitory abilities; thus, it cannot be determined whether the difficulties of the children with MLD were largely due to inhibitory deficits or to EF deficits more broadly. In addition, although we focused on specific quantitative abilities during the preschool years and in first grade based on previous research, it is still possible that other quantitative abilities may better differentiate children in the achievement groups we studied. Even with these caveats, our results support the importance of the understanding of the cardinal values of number words to children’s early mathematics development and provide a link between this initial conceptual understanding of number and later learning disabilities in mathematics. The results also provide direction for future studies regarding the potential antecedents (e.g., proactive interference) of later MLDs and have implications for the early identification of children at risk for longterm difficulties with mathematics.

Acknowledgments This study was supported by grants from the University of Missouri Research Board and DRL1250359 from the National Science Foundation, United States. We thank Mary Rook and Tina Sattler and her 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 Tim Adams, Melissa Barton, Sarah Becktell, Samantha Belvin, Erica Bizub, Kaitlyn Bumberry, Lex Clarkson, Stephen Cobb, Danielle Cooper, Alexis Currie, Amanda Evans, Dillon Falk, James Farley, Lauren Johnson-Hafenscher, Jared Kester, Morgan Kotva, Bradley Lance, Kate Leach, Kayla Legow, Natalie Miller, Lexi Mok, Yi Mou, Molly O’Byrne, Rebecca Peick, Kelly Regan, Nicole Reimer, Laura Roider, Brandon Ryffe, Sara Schroeder, Jin Seok, Jonathan Thacker, Claudia Tran, Hannah Weise, Melissa Willoughby, Grace Woessner, and Samantha York for help with data collection and other aspects of the project.

F.W. Chu et al. / Journal of Experimental Child Psychology 188 (2019) 104668

21

Appendix A. Supplementary material Supplementary data to this article can be found online at https://doi.org/10.1016/j.jecp.2019. 104668. References Andersson, U. (2010). Skill development in different components of arithmetic and basic cognitive functions: Findings from a 3year longitudinal study of children with different types of learning difficulties. Journal of Educational Psychology, 102, 115–134. Bailey, D., Duncan, G. J., Odgers, C. L., & Yu, W. (2017). Persistence and fadeout in the impacts of child and adolescent interventions. Journal of Research on Educational Effectiveness, 10, 7–39. Bailey, D. H., Watts, T. W., Littlefield, A. K., & Geary, D. C. (2014). State and trait effects on individual differences in children’s mathematical development. Psychological Science, 25, 2017–2026. Beck, D. M., Schaefer, C., Pang, K., & Carlson, S. M. (2011). Executive function in preschool children: Test–retest reliability. Journal of Cognition and Development, 12, 169–193. Best, J. R., & Miller, P. H. (2010). A developmental perspective on executive function. Child Development, 81, 1641–1660. Blair, C., & Razza, R. P. (2007). Relating effortful control, executive function, and false belief understanding to emerging math and literacy ability in kindergarten. Child Development, 78, 647–663. Blakey, E., Visser, I., & Carroll, D. J. (2016). Different executive functions support different kinds of cognitive flexibility: Evidence from 2-, 3-, and 4-year-olds. Child Development, 87, 513–526. Bonny, J. W., & Lourenco, S. F. (2013). The approximate number system and its relation to early math achievement: Evidence from the preschool years. Journal of Experimental Child Psychology, 114, 375–388. Booth, J. L., & Siegler, R. S. (2006). Developmental and individual differences in pure numerical estimation. Developmental Psychology, 42, 189–201. Bugden, S., & Ansari, D. (2011). Individual differences in children’s mathematical competence are related to the intentional but not automatic processing of Arabic numerals. Cognition, 118, 35–47. Bull, R., Espy, K. A., & Wiebe, S. A. (2008). Short-term memory, working memory, and executive functioning in preschoolers: Longitudinal predictors of mathematical achievement at age 7 years. Developmental Neuropsychology, 33, 205–228. Bull, R., & Lee, K. (2014). Executive functioning and mathematics achievement. Child Development Perspectives, 8, 36–41. Bull, R., & Scerif, G. (2001). Executive functioning as a predictor of children’s mathematics ability: Inhibition, switching, and working memory. Developmental Neuropsychology, 19, 273–293. Carey, S. (2004). Bootstrapping and the origin of concepts. Daedalus, 133, 59–68. Carey, S., Shusterman, A., Haward, P., & Distefano, R. (2017). Do analog number representations underlie the meanings of young children’s verbal numerals?. Cognition, 168, 243–255. Carlson, S. M. (2005). Developmentally sensitive measures of executive function in preschool children. Developmental Neuropsychology, 28, 595–616. Carlson, S. M. (2012). Executive Function Scale for Preschoolers [test manual]. Minneapolis: Institute of Child Development, University of Minnesota. Chen, Q., & Li, J. (2014). Association between individual differences in non-symbolic number acuity and math performance: A meta-analysis. Acta Psychologica, 148, 163–172. Cheung, P., Rubenson, M., & Barner, D. (2017). To infinity and beyond: Children generalize the successor function to all possible numbers years after learning to count. Cognitive Psychology, 92, 22–36. Chu, F. W., vanMarle, K., & Geary, D. C. (2015). Early numerical foundations of young children’s mathematical development. Journal of Experimental Child Psychology, 132, 205–212. Chu, F. W., vanMarle, K., Rouder, J., & Geary, D. C. (2018). Children’s early understanding of number predicts their later problemsolving sophistication in addition. Journal of Experimental Child Psychology, 169, 73–92. Clark, C. A. C., Pritchard, V. E., & Woodward, L. J. (2010). Preschool executive functioning abilities predict early mathematics achievement. Developmental Psychology, 46, 1176–1191. Cragg, L., & Gilmore, C. (2014). Skills underlying mathematics: The role of executive function in the development of mathematics proficiency. Trends in Neuroscience and Education, 3(2), 63–68. De Smedt, B., & Gilmore, C. K. (2011). Defective number module or impaired access? Numerical magnitude processing in first graders with mathematical difficulties. Journal of Experimental Child Psychology, 108, 278–292. De Smedt, B., Holloway, I. D., & Ansari, D. (2011). Effects of problem size and arithmetic operation on brain activation during calculation in children with varying levels of arithmetical fluency. NeuroImage, 57, 771–781. De Smedt, B., Noël, M.-P., Gilmore, C., & Ansari, D. (2013). How do symbolic and non-symbolic numerical magnitude processing skills relate to individual differences in children’s mathematical skills? A review of evidence from brain and behavior. Trends in Neuroscience and Education, 2, 48–55. De Visscher, A., & Noël, M.-P. (2014). Arithmetic facts storage deficit: The hypersensitivity-to-interference in memory hypothesis. Developmental Science, 17, 434–442. De Visscher, A., Vogel, S. E., Reishofer, G., Hassler, E., Koschutnig, K., De Smedt, B., & Grabner, R. H. (2018). Interference and problem size effect in multiplication fact solving: Individual differences in brain activations and arithmetic performance. NeuroImage, 172, 718–727. DeStefano, D., & LeFevre, J. A. (2004). The role of working memory in mental arithmetic. European Journal of Cognitive Psychology, 16, 353–386. Diamond, A. (2013). Executive functions. Annual Review of Psychology, 64, 135–168. Duncan, G. J., Dowsett, C. J., Claessens, A., Magnuson, K., Huston, A. C., Klebanov, P., ... Japel, C. (2007). School readiness and later achievement. Developmental Psychology, 43, 1428–1446.

22

F.W. Chu et al. / Journal of Experimental Child Psychology 188 (2019) 104668

Dyson, N. I., Jordan, N. C., & Glutting, J. (2013). A number sense intervention for low-income kindergartners at risk for mathematics difficulties. Journal of Learning Disabilities, 46, 166–181. Epstein, A. S. (2009). Numbers plus preschool mathematics curriculum. Ypsilanti, MI: HighScope. Espy, K. A., McDiarmid, M. M., Cwik, M. F., Stalets, M. M., Hamby, A., & Senn, T. E. (2004). The contribution of executive functions to emergent mathematic skills in preschool children. Developmental Neuropsychology, 26, 465–486. Fazio, L. K., Bailey, D. H., Thompson, C. A., & Siegler, R. S. (2014). Relations of different types of numerical magnitude representations to each other and to mathematics achievement. Journal of Experimental Child Psychology, 123, 53–72. Feigenson, L., Dehaene, S., & Spelke, E. (2004). Core systems of number. Trends in Cognitive Sciences, 8, 307–314. Feigenson, L., Libertus, M. E., & Halberda, J. (2013). Links between the intuitive sense of number and formal mathematics ability. Child Development Perspectives, 7, 74–79. Fuchs, L. S., Geary, D. C., Compton, D. L., Fuchs, D., Schatschneider, C., Hamlett, C. L., ... Changas, P. (2013). Effects of first-grade number knowledge tutoring with contrasting forms of practice. Journal of Educational Psychology, 105, 58–77. Fuchs, L. S., Schumacher, R. F., Sterba, S. K., Long, J., Namkung, J., Malone, A., ... Changas, P. (2014). Does working memory moderate the effects of fraction intervention? An aptitude–treatment interaction. Journal of Educational Psychology, 106, 499–514. Fuhs, M. W., & McNeil, N. M. (2013). ANS acuity and mathematics ability in preschoolers from low-income homes: Contributions of inhibitory control. Developmental Science, 16, 136–148. Fuson, K. C. (1988). Children’s counting and concepts of number (Springer Series in Cognitive Development). New York: SpringerVerlag. Gallistel, C. R., & Gelman, R. (2000). Non-verbal numerical cognition: From reals to integers. Trends in Cognitive Sciences, 4, 59–65. Geary, D. C. (1990). A componential analysis of an early learning deficit in mathematics. Journal of Experimental Child Psychology, 49, 363–383. Geary, D. C. (1993). Mathematical disabilities: Cognitive, neuropsychological, and genetic components. Psychological Bulletin, 114, 345–362. Geary, D. C. (2004). Mathematics and learning disabilities. Journal of Learning Disabilities, 37, 4–15. Geary, D. C. (2011). Cognitive predictors of achievement growth in mathematics: A 5-year longitudinal study. Developmental Psychology, 47, 1539–1552. Geary, D. C., Bailey, D. H., & Hoard, M. K. (2009). Predicting mathematical achievement and mathematical learning disability with a simple screening tool: The number sets test. Journal of Psychoeducational Assessment, 27, 265–279. Geary, D. C., Berch, D. B., & Mann Koepke, K. (Eds.). (2015). Evolutionary origins and early development of number processing (Mathematical Cognition and Learning, Vol. 1). San Diego: Elsevier Academic Press. Geary, D. C., Brown, S. C., & Samaranayake, V. A. (1991). Cognitive addition: A short longitudinal study of strategy choice and speed-of-processing differences in normal and mathematically disabled children. Developmental Psychology, 27, 787–797. Geary, D. C., Hoard, M. K., & Bailey, D. H. (2012). Fact retrieval deficits in low achieving children and children with mathematical learning disability. Journal of Learning Disabilities, 45, 291–307. Geary, D. C., Hoard, M. K., Byrd-Craven, J., Nugent, L., & Numtee, C. (2007). Cognitive mechanisms underlying achievement deficits in children with mathematical learning disability. Child Development, 78, 1343–1359. Geary, D. C., Hoard, M. K., Nugent, L., & Bailey, D. H. (2012). Mathematical cognition deficits in children with learning disabilities and persistent low achievement: A five-year prospective study. Journal of Educational Psychology, 104, 206–223. Geary, D. C., Hoard, M. K., Nugent, L., & Bailey, D. H. (2013). Adolescents’ functional numeracy is predicted by their school entry number system knowledge. PLoS One, 8(1), e54651. Geary, D. C., Hoard, M. K., Nugent, L., & Byrd-Craven, J. (2008). Development of number line representations in children with mathematical learning disability. Developmental Neuropsychology, 33, 277–299. Geary, D. C., Nicholas, A., Li, Y., & Sun, J. (2017). Developmental change in the influence of domain-general abilities and domainspecific knowledge on mathematics achievement: An eight-year longitudinal study. Journal of Educational Psychology, 109, 680–693. Geary, D. C., & vanMarle, K. (2016). Young children’s core symbolic and nonsymbolic quantitative knowledge in the prediction of later mathematics achievement. Developmental Psychology, 52, 2130–2144. Geary, D. C., & vanMarle, K. (2018). Growth of symbolic number knowledge accelerates after children understand cardinality. Cognition, 177, 69–78. Geary, D. C., vanMarle, K., Chu, F. W., Rouder, J., Hoard, M. K., & Nugent, L. (2018). Early conceptual understanding of cardinality predicts superior school-entry number-system knowledge. Psychological Science, 29, 191–205. Gelman, R., & Gallistel, C. R. (1978). The child’s understanding of number. Cambridge, MA: Harvard University Press. Ginsburg, H. P., & Baroody, A. J. (2003). Test of Early Mathematics Ability (3rd ed.). Austin, TX: Pro-Ed. Guzman, G. G. (2018). Household income: 2017. Retrieved from: . Halberda, J., Mazzocco, M. M. M., & Feigenson, L. (2008). Individual differences in non-verbal number acuity correlate with maths achievement. Nature, 455, 665–668. Hanich, L. B., Jordan, N. C., Kaplan, D., & Dick, J. (2001). Performance across different areas of mathematical cognition in children with learning difficulties. Journal of Educational Psychology, 93, 615–626. Holloway, I. D., & Ansari, D. (2009). Mapping numerical magnitudes onto symbols: The numerical distance effect and individual differences in children’s mathematics achievement. Journal of Experimental Child Psychology, 103, 17–29. Holloway, I. D., Battista, C., Vogel, S. E., & Ansari, D. (2013). Semantic and perceptual processing of number symbols: Evidence from a cross-linguistic fMRI adaptation study. Journal of Cognitive Neuroscience, 25, 388–400. Inglis, M., & Gilmore, C. (2014). Indexing the approximate number system. Acta Psychologica, 145, 147–155. Invernizzi, M., Sullivan, A., Meier, J., & Swank, L. (2004). PreK teacher’s manual: Phonological awareness and literacy screening. Charlottesville: University of Virginia. Iuculano, T., Tang, J., Hall, C. W. B., & Butterworth, B. (2008). Core information processing deficits in developmental dyscalculia and low numeracy. Developmental Science, 11, 669–680.

F.W. Chu et al. / Journal of Experimental Child Psychology 188 (2019) 104668

23

Jonides, J., & Nee, D. E. (2006). Brain mechanisms of proactive interference in working memory. Neuroscience, 139, 181–193. Jordan, N. C., Hanich, L. B., & Kaplan, D. (2003). Arithmetic fact mastery in young children: A longitudinal investigation. Journal of Experimental Child Psychology, 85, 103–119. Koponen, T., Salmi, P., Eklund, K., & Aro, T. (2013). Counting and RAN: Predictors of arithmetic calculation and reading fluency. Journal of Educational Psychology, 105, 162–175. Landerl, K., Bevan, A., & Butterworth, B. (2004). Developmental dyscalculia and basic numerical capacities: A study of 8–9-yearold students. Cognition, 93, 99–125. Le Corre, M., & Carey, S. (2007). One, two, three, four, nothing more: An investigation of the conceptual sources of the verbal counting principles. Cognition, 105, 395–438. Lee, K., & Bull, R. (2016). Developmental changes in working memory, updating, and math achievement. Journal of Educational Psychology, 108, 869–882. LeFevre, J.-A., Skwarchuk, S.-L., Smith-Chant, B. L., Fast, L., Kamawar, D., & Bisanz, J. (2009). Home numeracy experiences and children’s math performance in the early school years. Canadian Journal of Behavioural Science/Revue canadienne des sciences du comportement, 41, 55–66. Libertus, M. E., Feigenson, L., & Halberda, J. (2011). Preschool acuity of the approximate number system correlates with school math ability. Developmental Science, 14, 1292–1300. Libertus, M. E., Feigenson, L., & Halberda, J. (2013). Numerical approximation abilities correlate with and predict informal but not formal mathematics abilities. Journal of Experimental Child Psychology, 116, 829–838. Libertus, M. E., Odic, D., Feigenson, L., & Halberda, J. (2016). The precision of mapping between number words and the approximate number system predicts children’s formal math abilities. Journal of Experimental Child Psychology, 150, 207–226. Mazzocco, M. M. M., Feigenson, L., & Halberda, J. (2011). Preschoolers’ precision of the approximate number system predicts later school mathematics performance. PLoS One, 6(9), e23749. Mazzocco, M. M. M., & Kover, S. T. (2007). A longitudinal assessment of executive function skills and their association with math performance. Child Neuropsychology, 13, 18–45. Mou, Y., Berteletti, I., & Hyde, D. C. (2018). What counts in preschool number knowledge? A Bayes factor analytic approach toward theoretical model development. Journal of Experimental Child Psychology, 166, 116–133. Munakata, Y. (2001). Graded representations in behavioral dissociations. Trends in Cognitive Sciences, 5, 309–315. Murphy, M. M., Mazzocco, M. M. M., Hanich, L. B., & Early, M. C. (2007). Cognitive characteristics of children with mathematics learning disability (MLD) vary as a function of the cutoff criterion used to define MLD. Journal of Learning Disabilities, 40, 458–478. Passolunghi, M. C., & Siegel, L. S. (2004). Working memory and access to numerical information in children with disability in mathematics. Journal of Experimental Child Psychology, 88, 348–367. Piazza, M., Facoetti, A., Trussardi, A. N., Berteletti, I., Conte, S., Lucangeli, D., ... Zorzi, M. (2010). Developmental trajectory of number acuity reveals a severe impairment in developmental dyscalculia. Cognition, 116, 33–41. Planty, M., Provasnik, S., Hussar, W., Snyder, T., Kena, G., Hampden-Thompson, G., ... Choy, S. (2010). The condition of education 2010 (NCES 2010–028). Washington, DC: National Center for Education Statistics. Postle, B. R., Brush, L. N., & Nick, A. M. (2004). Prefrontal cortex and the mediation of proactive interference in working memory. Cognitive, Affective, & Behavioral Neuroscience, 4, 600–608. Qin, S., Cho, S., Chen, T., Rosenberg-Lee, M., Geary, D. C., & Menon, V. (2014). Hippocampal-neocortical functional reorganization underlies children’s cognitive development. Nature Neuroscience, 17, 1263–1269. R Core Team. (2017). R: A language and environment for statistical computing. Vienna, Austria: R Foundation for Statistical Computing. Retrieved from: . Ritchie, S. J., & Bates, T. C. (2013). Enduring links from childhood mathematics and reading achievement to adult socioeconomic status. Psychological Science, 24, 1301–1308. Rizopoulos, D. (2006). ltm: An R package for latent variable modelling and item response theory analyses. Journal of Statistical Software, 17, 1–25. Rosenberg-Lee, M., Ashkenazi, S., Chen, T., Young, C. B., Geary, D. C., & Menon, V. (2015). Brain hyper-connectivity and operationspecific deficits during arithmetic problem solving in children with developmental dyscalculia. Developmental Science, 18, 351–372. Rousselle, L., & Noël, M.-P. (2007). Basic numerical skills in children with mathematics learning disabilities: A comparison of symbolic vs non-symbolic number magnitude. Cognition, 102, 361–395. Schneider, M., Beeres, K., Coban, L., Merz, S., Schmidt, S. S., Stricker, J., ... De Smedt, B. (2017). Associations of non-symbolic and symbolic numerical magnitude processing with mathematical competence: A meta-analysis. Developmental Science, 20(3), e12372. Shusterman, A., Slusser, E., Halberda, J., & Odic, D. (2016). Acquisition of the cardinal principle coincides with improvement in approximate number system acuity in preschoolers. PLoS One, 11(4), e153072. Siegler, R. S., & Booth, J. L. (2004). Development of numerical estimation in young children. Child Development, 75, 428–444. Spaepen, E., Gunderson, E. A., Gibson, D., Goldin-Meadow, S., & Levine, S. C. (2018). Meaning before order: Cardinal principle knowledge predicts improvement in understanding the successor principle and exact ordering. Cognition, 180, 59–81. Starr, A., Libertus, M. E., & Brannon, E. M. (2013). Number sense in infancy predicts mathematical abilities in childhood. Proceedings of the National Academy of Sciences of the United States of America, 110, 18116–18120. Sullivan, J., & Barner, D. (2014). Inference and association in children’s early numerical estimation. Child Development, 85, 1740–1755. Swanson, H. L., Jerman, O., & Zheng, X. (2008). Growth in working memory and mathematical problem solving in children at risk and not at risk for serious math difficulties. Journal of Educational Psychology, 100, 343–379. U.S. Census. (2017). Educational attainment in the United States: 2017. Retrieved from: . vanMarle, K. (2013). Infants use different mechanisms to make small and large number ordinal judgments. Journal of Experimental Child Psychology, 114, 102–110.

24

F.W. Chu et al. / Journal of Experimental Child Psychology 188 (2019) 104668

vanMarle, K., Aw, J., McCrink, K., & Santos, L. R. (2006). How capuchin monkeys (Cebus apella) quantify objects and substances. Journal of Comparative Psychology, 120, 416–426. vanMarle, K., Chu, F. W., Li, Y., & Geary, D. C. (2014). Acuity of the approximate number system and preschoolers’ quantitative development. Developmental Science, 17, 492–505. Wechsler, D. (2001). Wechsler Individual Achievement Test-II–Abbreviated. San Antonio, TX: Psychological Corporation, Harcourt Brace. Wechsler, D. (2002). Wechsler Preschool and Primary Scale of Intelligence-III. San Antonio, TX: Psychological Corporation, Harcourt Brace. Wynn, K. (1990). Children’s understanding of counting. Cognition, 36, 155–193. Wynn, K. (1992). Children’s acquisition of the number words and the counting system. Cognitive Psychology, 24, 220–251. Yeo, D. J., Wilkey, E. D., & Price, G. R. (2017). The search for the number form area: A functional neuroimaging meta-analysis. Neuroscience and Biobehavioral Reviews, 78, 145–160.