Mathematical knowledge lays the foundation for success in science, technology, engineering, and mathematics (STEM) disciplines, and thus plays a critical role in the education and training of the next generation of STEM professionals (PCAST, 2012; Sadler & Tai, 2007). Yet even before the start of formal schooling, children from lowincome households underperform on a variety of basic number skills compared to middleincome children, including counting, recognizing numerals, and comparing numbers (Jordan, Kaplan, Olah, & Locuniak, 2006; Starkey, Klein, & Wakeley, 2004). Moreover, math performance at the start of school predicts longterm math achievement in later grades (Duncan et al., 2007; Watts, Duncan, Siegler, & DavisKean, 2014), making early gaps especially concerning. Data from older students suggest that the incomebased math achievement gap may widen over time. Among a nationally representative sample of U.S. eighth graders, only 18% of students from lowincome households performed at or above the proficient level in math, compared to 48% of students from middle and upperincome households (U.S. Department of Education, Institute of Education Sciences, National Center for Education Statistics, 2015). The gaps in math achievement are attributed to children from lowincome households having fewer and less supportive experiences with math, both at home and in school (Clements & Sarama, 2007; Dyson, Jordan, & Glutting, 2013; Siegler, 2009).
The present study investigates two hypotheses related to the early math achievement gap between children from lowincome and middleincome families. The first is that consistent, reliable gaps between low and middleincome preschoolers’ symbolic numerical knowledge, defined as culturally specific verbal and visual labels for numbers (i.e., verbally stated or written numerals), may not be replicated in parallel measures of nonsymbolic numerical knowledge, defined as nonverbal representations of approximate quantities (i.e., sets of objects). The second is that exposure to dual representations of symbolic and nonsymbolic number can help lowincome children build their numerical knowledge and improve their foundational math skills. Both hypotheses are informed by a theory of numerical development, which proposes that children’s understanding of symbolic and nonsymbolic numerical magnitudes underlie their later math achievement (Siegler, 2016). Investigating these two hypotheses will help elucidate the nature of the differences in numerical knowledge between children from lower and higherincome backgrounds as well as ways to narrow this SESrelated gap in young children.
Theoretical Background [TOP]
According to Siegler and colleagues (Siegler, 2016; Siegler & LortieForgues, 2014; Siegler, Thompson, & Schneider, 2011), numerical development consists of increased understanding of the numerical magnitudes of a continually expanding range of numbers. They propose that children’s symbolic magnitude representations can be mapped onto their nonsymbolic number representations. Nonsymbolic magnitude representations are evident in infancy (e.g., Lipton & Spelke, 2003; Xu & Spelke, 2000), and have been collectively termed the Approximate Number System (ANS; Libertus, Feigenson, & Halberda, 2011). In general, the precision of the ANS to reliably differentiate between nonsymbolic sets increases throughout infancy and early childhood (Libertus & Brannon, 2009). As young children continue to refine their nonsymbolic magnitude representations, they simultaneously acquire symbolic number skills, including knowledge of number words (e.g., one, two, three) and numeral symbols (e.g., 1, 2, 3) (Siegler, 2016). These early forms of numerical knowledge underlie later mathematical skills, such as arithmetic (Geary, Hoard, & Hamson, 1999).
Both symbolic and nonsymbolic numerical knowledge are related to mathematics learning. Studies have demonstrated that the ability to compare the magnitude of two numerals (a form of symbolic number representation) and two sets of objects (a form of nonsymbolic number representation) is associated with mathematics achievement among children and adults (Chen & Li, 2014; De Smedt, Noël, Gilmore, & Ansari, 2013; Fazio, Bailey, Thompson, & Siegler, 2014; Schneider et al., 2016). However, based on their integrative review of the literature, De Smedt and colleagues (2013) conclude that the data on nonsymbolic magnitude comparison tasks predicting general mathematics achievement are mixed, with some studies demonstrating significant effects and others not. In contrast, the data on measures of symbolic magnitude comparison tasks showed a reliable relation to math achievement across studies: participants with poor performance on symbolic magnitude comparison tasks tended to have lower mathematics achievement both concurrently and up to 2 years later (De Smedt et al., 2013). Similarly, recent metaanalyses report a significant, albeit weak, relation between nonsymbolic numerical knowledge and mathematics achievement (Chen & Li, 2014; Fazio et al., 2014; Schneider et al., 2016). The relation was maintained while controlling for general cognitive abilities (Chen & Li, 2014), but may decrease as children enter formal schooling environments (Fazio et al., 2014). When both symbolic and nonsymbolic magnitude knowledge were included as predictors of children’s math achievement, they each significantly explained a portion of the variance, however the effect size for measures of symbolic magnitude knowledge was larger (Fazio et al., 2014). Across studies of children and adults, the effect size is consistently higher for symbolic than nonsymbolic comparison tasks (Schneider et al., 2016). In sum, preschoolers’ symbolic magnitude comparison skills predict their later mathematics success (Geary et al., 1999), and tend to be more predictive of mathematics achievement than measures of nonsymbolic magnitude comparison (Kolkman, Kroesbergen, & Leseman, 2013; van Marle, Chu, Li, & Geary, 2014).
Given the significance of symbolic and nonsymbolic magnitude knowledge during early childhood, it is important to understand whether there are incomebased gaps present in both types of numerical skills. Although there are documented gaps between children from lowincome and middleincome families on measures of symbolic magnitude comparison (e.g., determining which is the larger number; Jordan et al., 2006; Ramani & Siegler, 2011), no studies to date have measured children’s performance on a parallel measure of nonsymbolic magnitude comparison. Previous work demonstrated that children from lowincome backgrounds performed equivalently to children from middleincome backgrounds on nonsymbolic arithmetic problems, but were less accurate on symbolic (verbal) arithmetic problems (Gilmore, McCarthy, & Spelke, 2010; Jordan, Huttenlocher, & Levine, 1992). A second study found that children’s socioeconomic status does not impact performance on nonsymbolic estimation tasks (Mejias & Schiltz, 2013). These findings suggest that children from lowincome backgrounds may also be equally proficient on nonverbal, nonsymbolic magnitude comparison tasks that do not include additional calculation or estimation demands. If so, lowincome children’s nonsymbolic magnitude understanding could potentially be used to promote their symbolic magnitude understanding.
Traditional numerical playing cards provide simultaneous exposure to symbolic and nonsymbolic magnitude representations, in the form of Arabic numerals and sets of objects (e.g., hearts, diamonds, spades, clubs). Exposing children to materials with multiple redundant cues to numbers and their magnitudes can help build their numerical knowledge (Ramani & Siegler, 2008; Siegler & Booth, 2004). In creating an intervention for lowincome preschoolers using numerical cards, we sought to capitalize on participants’ nonsymbolic magnitude knowledge to support their symbolic magnitude knowledge, as well as engender the motivational and learning benefits of playful, gamebased activities.
PlayBased Early Mathematics Interventions [TOP]
A number of recent early mathematics interventions have incorporated playful, informal learning activities (Clements & Sarama, 2007; Ramani & Siegler, 2008; Wilson, Dehaene, Dubois, & Fayol, 2009). Playful learning activities and games broadly represent a developmentally appropriate mechanism for teaching young children in an engaging and supportive way (HassingerDas et al., 2017; Singer, Golinkoff, & HirshPasek, 2006; Weisberg, HirshPasek, & Golinkoff, 2013). These activities can also give children opportunities to practice their early numeracy skills (Ramani & Eason, 2015). When offered an engaging learning context and the opportunity for active participation, children tend to benefit more than those who are directly taught or allowed to freely explore without guidance (HassingerDas et al., 2017; Weisberg et al., 2013). For example, Siegler and Ramani (2009) randomly assigned preschoolers to play a linear numerical board game, a circular numerical board game, or a set of nonplayful, typical numerical activities (counting aloud, counting objects, identifying numerals) for four 1520 min sessions over a 3week period. They found that children who played the numerical board games outperformed children in the other two conditions on tasks measuring their numerical magnitude knowledge. Similarly, Ferrara and colleagues (2011) found that parentchild dyads who engaged in a playful, interactive block building activity to assemble a target structure used significantly more spatial language than dyads who were given preassembled block structures to interact with for the same amount of time. These patterns of findings underscore that playful learning activities may lead to greater learning than engaging in nonplayful activities, particularly for preschoolaged children.
Children similarly appear to benefit from playful, informal mathematics activities in the home (LeFevre et al., 2009; Melhuish et al., 2008; Ramani, Rowe, Eason, & Leech, 2015). Previous research has demonstrated that parental reports of engaging in informal mathematics activities at home (e.g., playing board games with dice or spinners) predict children’s math achievement (LeFevre et al., 2009; Melhuish et al., 2008; Ramani et al., 2015). Furthermore, selfreport and observational data suggest that lowincome families engage in fewer informal mathematical learning activities in the home than middleincome families (Saxe, Guberman, & Gearhart, 1987; Starkey et al., 2004), perhaps indicating one type of experience that contributes to the incomebased early math achievement gap.
Given the engagement and motivational benefits of playful learning, combining play with a game that integrates a theory of numerical development may be a particularly effective method to promote specific skills. Numerical card games can provide parallel symbolic and nonsymbolic magnitude representations, which allow children to rely on either type of information to engage with the materials (i.e., identifying a numeral, making a comparison between two quantities). Further, specific card games can vary in the type and amount of numerical knowledge they rely on to complete the game. For example, a common numerical card game for young children, War, involves two players dividing a deck of cards between them. Each player then turns over one of their cards and together they determine which has the larger quantity. The player whose card has the larger quantity keeps both cards and the player with the most cards at the end of the game wins. The game offers children the opportunity to practice counting, identifying numerals, and importantly, making magnitude comparisons between two numbers to determine which of the numbers represented on the cards is largest. Other card games for young children, such as Memory, involve players turning over pairs of facedown cards to try to find a match. The player who finds the most pairs wins. When played with numerical cards, games like Memory may offer similar practice with counting and recognizing numerals, but focus on children’s ability to remember the location of cards as opposed to making magnitude comparisons between numbers. Thus, cards can provide children with multiple representations of numerical magnitudes, and card games can vary on the type of practice with numbers that children have while playing them.
The Current Paper [TOP]
In the current paper, we pursued two goals. The first goal was to determine whether preschoolers from lowincome households were less accurate than preschoolers from middleincome households on parallel tasks of symbolic and nonsymbolic magnitude comparison. In Experiment 1, we hypothesized that we would find incomebased performance gaps on a symbolic numerical magnitude comparison task during which participants are shown two Arabic numerals and then asked, “Which is more: X or Y?” In contrast, we predicted that lowincome children would show similar accuracy to their middleincome peers on a parallel measure of nonsymbolic magnitude comparison with quantities less than ten. In this task, children are shown two sets of dots and similarly asked to indicate which set has more. These hypotheses are based on previous research, which has found SESrelated gaps in verbally stated, symbolic numerical tasks, but not on nonverbal numerical tasks (Gilmore et al., 2010; Jordan et al., 1992; Ramani & Siegler, 2011).
The second goal was to assess whether approximately one hour of playing numerical card games with symbolic and nonsymbolic representations would lead to improvements in participants’ early symbolic and nonsymbolic numerical knowledge. Expanding on the results of Experiment 1, Experiment 2 examined whether a magnitude comparison card game, based on the card game War, could improve lowincome children’s numerical knowledge. The game involves players comparing two cards with both symbolic number representations (Arabic numerals) and nonsymbolic number representations (circles). The dual representations allow children to practice comparing symbolic magnitudes with nonsymbolic quantities to use for additional support in making their decision. For example, although a child may not know if 7 is more or less than 2, they are likely able to tell that a card with seven circles has more circles than a card with only two. Other components of the game involve counting nonsymbolic representations and identifying numerals, which could also contribute to improvements to symbolic numerical knowledge. Likewise, the repeated practice comparing nonsymbolic quantities may improve children’s nonsymbolic magnitude skills, which have been shown to malleable with targeted practice (e.g., Wang, Odic, Halberda, & Feigenson, 2016). As a comparison group, half of the children were randomly assigned to play a memory and matching game (Memory) with the same numerical cards. Since the game focuses on children’s ability to remember the location of cards, it allowed for the effects of magnitude comparison training to be isolated from any effects of practice counting and identifying numerals.
We examined whether playing the two games would improve children’s symbolic numerical knowledge, specifically their symbolic numerical magnitude comparison skills, verbal counting from 110, and identifying written numerals from 110, as well as their nonsymbolic numerical magnitude knowledge. We hypothesized that children in both conditions would improve their counting and numeral identification skills because all children had similar exposure in playing with the numerical cards. However, we hypothesized that only children who played the magnitude comparison card game would improve their symbolic magnitude comparison skills because of their unique experiences making comparisons during the game. We did not have specific predictions regarding children’s performance on the nonsymbolic numerical tasks, given that we expected children’s nonsymbolic numerical knowledge to be proficient for small quantities less than 10. Finally, we investigated whether lowincome children’s magnitude comparison skills postintervention would be comparable to the numerical knowledge of sameage peers from middleincome backgrounds.
Experiment 1 [TOP]
The goal of Experiment 1 was to determine whether SESrelated differences in numerical magnitude knowledge are specific to symbolic representations, or whether lowincome preschoolers would show similar performance gaps on nonsymbolic magnitude measures compared to middleincome preschoolers. To equate the nonsymbolic and symbolic measures of numerical magnitude, both used the same number range of 1 through 9. If children performed equivalently on small number comparisons using nonsymbolic representations, it might be possible to capitalize on their nonsymbolic comparison skills to foster their symbolic magnitude comparison skills with similar quantities.
Method [TOP]
Participants [TOP]
Participants were 46 preschoolers. Informed consent was obtained for all participants from their parent or guardian. Twentythree preschoolers were recruited from one Head Start center in a midAtlantic state, ranging in age from 3 years, 4 months to 5 years, 6 months (M = 4 years, 8 months, SD = 0.60; 30% female; 57% African American/Black, 17% Caucasian, 9% Asian, 17% Multiracial; 87% not Hispanic/Latino, 13% Hispanic/Latino). Head Start is an American federally funded early childhood education program targeted at families living at or below the poverty line. The program seeks to promote young children’s school readiness skills (social, emotional, and academic), healthy development (perceptual, motor, and physical), and family wellbeing (Administration for Children and Families [ACF], 2016). For enrollment in the 20142015 school year, the U.S. federal poverty guideline for a family of four was $23,850.
The remaining 23 preschoolers were recruited from middleincome households. These children were part of a larger study on mathematics and memory development that assessed their knowledge in those areas during two brief sessions. The preschools were located in the same geographic area as the Head Start center, and all participants were recruited during the same academic year. The middleincome preschoolers were selected as an age and gendermatched sample to the lowincome participants, ranging in age from 3 years, 8 months to 5 years, 9 months (M = 4 years, 9 months, SD = 0.57; 30% female; 61% Caucasian, 26% African American/Black, 4% Asian, 9% Multiracial; 96% not Hispanic/Latino, 4% Hispanic/Latino). Selfreported annual household incomes ranged from $76,000 to over $151,000 (23% of incomes were between $76,000 and $100,000; 9% of incomes were between $101,000 and $150,000; and 68% of incomes were $151,000 or more). Annual household incomes for a family of four ranging from $58,000 to $172,000 were considered middleincome for the metropolitan area where participants were recruited (Pew Research Center, 2016).
Measures of Numerical Knowledge [TOP]
Two measures of magnitude comparison skills were used to assess the numerical understanding of participants.
Symbolic magnitude comparison [TOP]
Participants were asked to compare 20 pairs of symbolic numbers ranging from 19 presented in a paper booklet (Ramani & Siegler, 2008). After two practice trials with experimenter feedback, participants were shown 18 test pairs of numbers in the booklet and asked to indicate which number is larger. The test pairs were read aloud by the experimenter, without any accuracy feedback. Each number was counterbalanced for side of presentation (i.e., 38, 83). The ratio between pairs ranged from 1.1 (e.g., 89) to 9.0 (e.g., 91). The dependent measure was percentage of correct comparisons. The interitem reliability was α = .80.
Nonsymbolic magnitude comparison [TOP]
Participants were asked to compare 20 pairs of magnitudes (dot arrays) displayed on a laptop computer. Using the Panamath software (e.g., Libertus, Feigenson, & Halberda, 2013), the children saw a set of yellow dots presented on the left side of the screen and a set of blue dots simultaneously presented on the right side of the screen for 2.8 seconds and were asked to press either a yellow or blue button to indicate which side had more dots. The dot quantities ranged from 19, and included numerical comparisons with ratios ranging from 1.3 (e.g., 79) to 3.5 (e.g., 27). The Panamath program counterbalanced each magnitude for side of presentation (i.e., 36, 63) and controlled for dot area and density. Neither the Panamath program nor the experimenter provided accuracy feedback on the test trials. The dependent measure was the percentage of correct comparisons. Although other researchers using the Panamath program have also included Weber fractions as a dependent measure, which indicate the most difficult ratios between two sets to which the child can respond accurately, there is some evidence that Weber fractions are noisy and possibly inaccurate measures for such young children (Libertus et al., 2011). The interitem reliability was α = .69.
Procedure [TOP]
All participants completed a series of math assessments individually with an experimenter within the child’s preschool in a quiet area of the hallway or room nearby their classroom. The middleincome participants also completed noncomputerized assessments of working memory. However, because the focus of the current study was preschoolers’ symbolic and nonsymbolic number knowledge, only the identical assessments of symbolic and nonsymbolic magnitude comparison administered to both low and middleincome participants were analyzed for this experiment. To sustain children’s motivation, the symbolic magnitude comparison task was presented first followed by the nonsymbolic magnitude comparison task, which participants considered more exciting because it was computerbased. During test trials, experimenters limited their feedback to general encouragement (e.g., “I can tell you’re thinking hard about these”) and progress (e.g., “We’re almost done!”). Each assessment session lasted approximately 15 minutes, and children were given a sticker to thank them for their participation. The experimenter who interacted with the lowincome participants was a Caucasian graduate student; the experimenters who interacted with the middleincome participants were one Caucasian graduate student, one Caucasian and one Asian American undergraduate student. All four experimenters were female.
Results [TOP]
Independent samples ttests showed that there were no accuracy differences between the lower and middleincome samples on the nonsymbolic magnitude comparison task (M = 84% vs. M = 89%; t(44) = 1.14, p = .261, d = 0.34). Although the mean difference in accuracy between the two groups was not statistically significant, the average accuracy of the middleincome preschoolers was 0.34 standard deviations higher than the accuracy of the lowincome preschoolers. However, lowincome preschoolers were significantly less accurate than the middleincome preschoolers on symbolic magnitude comparisons of the same numerical range and similar ratios (M = 77% vs. M = 93%; t(44) = 3.52, p = .001, d = 1.04). This gap in symbolic magnitude comparison skills represents a one standard deviation difference in mean accuracy between lowincome and middleincome participants. The effect size for the symbolic numerical knowledge task was three times as large as the effect size for the symbolic magnitude comparison task (d = 1.04 vs. d = 0.34), and considered a large effect size by Cohen’s benchmarks (Cohen, 1988). These results suggest the performance gap on symbolic magnitude comparison skills does not extend to less verbal, nonsymbolic measures of magnitude comparison.
Discussion [TOP]
In the present experiment, we found that the SESrelated gaps in children’s symbolic magnitude knowledge were statistically significant, however, the differences between lowincome and middleincome participants on the parallel nonsymbolic magnitude comparison task were not statistically significant. This suggests that with these measures of symbolic and nonsymbolic magnitude comparison, the gaps between middle and lowerincome children’s numerical knowledge were specific to children’s performance on a symbolic magnitude comparison measure. These findings are consistent with previous research examining SESrelated differences on symbolic and nonsymbolic measures of early arithmetic (e.g., Gilmore et al., 2010; Jordan et al., 2006). In general, symbolic numerical knowledge is thought to be highly dependent on the type of input and instruction that children receive (Jordan & Levine, 2009), and children from lowerincome households experience less frequent and high quality math experiences at home and in school than middleincome children (Clements & Sarama, 2007; Siegler, 2009).
The pattern of effects found in this experiment may vary under different task specifications. The symbolic and nonsymbolic magnitude comparison tasks used in Experiment 1 included slightly different ratios (1.1 to 9.0 versus 1.3 to 3.5) and numbers of test trials (18 versus 20 trials). However, the two tasks were equivalent on the range of quantities presented (1 – 9). In addition, previous work using the Panamath program with preschool children used shorter presentation times, larger quantities of dots, and more difficult comparison ratios than the ones used in the present study (e.g., Libertus et al., 2011, 2013). It is possible that the relative ease of the nonsymbolic magnitude comparison task in the present study may be part of the reason why incomebased performance gaps were not observed in the present study.
Nonetheless, our intention was to match the nonsymbolic magnitude comparison task parameters closely to the demands of the symbolic magnitude comparison task to investigate the SESrelated gaps in young children’s numerical magnitude knowledge on quantities less than ten. The results of Experiment 1 demonstrate that lowincome children are considerably more accurate (equivalent to middleincome children) in making nonsymbolic comparisons than they are making symbolic comparisons between the same numbers. This implies that lowincome children may benefit from magnitude comparison experiences that allow them to leverage their nonsymbolic magnitude comparison skills to promote their symbolic magnitude comparison skills, a premise tested in Experiment 2.
Experiment 2 [TOP]
Building from the findings of Experiment 1, Experiment 2 tested an experimental, gamebased intervention using numerical cards with both nonsymbolic and symbolic numerical representations. Children from lowincome households were randomly assigned to play one of two games using the same numerical cards. The first game is a magnitude comparison card game, based on the card game War, which involves players comparing the magnitude of two cards. The second game, based on the card game Memory, involves players finding matching pairs of cards. Thus, the two games offered similar exposure to numerical cards, but differed in the opportunities to make magnitude comparison judgments between two quantities. We hypothesized that the targeted practice with magnitude comparison while playing the magnitude comparison game (War) may support improvements in lowincome preschoolers’ symbolic magnitude comparison skills. Given that the games were otherwise similar (i.e., same numerical cards, practice counting, identifying numerals), we predicted that children in both the experimental magnitude comparison condition (War) and the control numerical card game condition (Memory) would experience similar gains on other foundational symbolic numerical skills, specifically their counting and numeral identification skills. Further, we hypothesized that the lowincome preschoolers in the experimental condition would improve their symbolic magnitude comparison skills to a point of statistical equivalence with a group of age and gendermatched middleincome preschoolers.
In addition to assessing whether playing the card games improved children’s symbolic numerical knowledge, we were also interested in whether playing the games improved their nonsymbolic magnitude knowledge. Therefore, we included two measures of children’s nonsymbolic magnitude knowledge, specifically a nonsymbolic magnitude comparison task, similar to the one used in Experiment 1, and an additional nonsymbolic comparison task, which gauged children’s discrimination between ordinal quantities of objects (e.g., N and N+1). Previous research has shown that the performance of lowincome children on computerized nonsymbolic magnitude comparison tasks may be affected by their inhibitory control (Fuhs & McNeil, 2013), hence our second measure of nonsymbolic magnitude knowledge was an noncomputerized task developed for use with lowincome preschoolers (Chu, Van Marle, & Geary, 2013).
Given that the lowincome children had high accuracy on the nonsymbolic measure of numerical magnitude knowledge in Experiment 1, no specific predictions were made about whether playing the magnitude comparison card game would lead to improved nonsymbolic magnitude understanding. It is possible that the experience children have comparing numerical magnitudes using cards with both nonsymbolic and symbolic representations could increase performance on nonsymbolic measures of numerical knowledge. However, the highly accurate performance of lowincome children on a measure of nonsymbolic magnitude comparison in Experiment 1 suggests that it is also possible that no further gains would be observed from additional experience. To improve the sensitivity of the measure to detect improvements in nonsymbolic magnitude comparison skills and provide a better match to the parameters in previously published studies with preschoolers, we increased the difficulty of the nonsymbolic magnitude comparison measure by adjusting the quantities of the numbers and the presentation time.
Method [TOP]
We next describe the participants, measures of numerical knowledge, and procedures. We report all data exclusions and justifications for their removal, all experimental manipulations, and all measures collected in this study.
Participants [TOP]
Participants were 70 preschoolers. Informed consent was collected for all participants from their parent or guardian. Fortysix lowincome preschoolers were recruited from four Head Start centers in a midAtlantic state, comprising the main experimental sample, none of which participated in Experiment 1. Lowincome preschoolers ranged in age from 3 years, 6 months to 5 years, 7 months (M = 4 years, 9 months, SD = 0.62; 48% female; 54% Caucasian, 35% African American/Black, 7% Asian, 4% Multiracial; 54% Hispanic/Latino, 46% not Hispanic/Latino). Three additional lowincome children were recruited but were not included in the final analyses because the children did not complete the game sessions or the posttest tasks (n = 2), or the child had prolonged absences from school (n = 1).
Lowincome participants were randomly assigned to either a numerical magnitude comparison card game condition (War) or to a numerical matching card game condition (Memory). The War card game condition included 24 children (M = 4 years, 8 months, SD = 0.67; 50% female). The Memory card game condition included 22 children (M = 4 years, 10 months, SD = 0.57; 46% female).
The remaining 24 preschoolers were from middleincome backgrounds (M = 4 years, 8 months, SD = 0.68; 50% female; 46% Caucasian, 25% African American/Black, 21% Multiracial, 8% Asian; 100% not Hispanic/Latino) who were recruited from the larger study on math and memory development described in Experiment 1. The middleincome preschoolers were selected as an age (within 4 months) and gendermatched sample to the lowincome participants in the numerical magnitude comparison (War) condition. In order to provide the closest age and gendermatched sample for comparison, 10 of the middleincome children served as matchedcomparison participants in both Experiments 1 and 2. Selfreported annual household incomes ranged from $76,000 to over $151,000 (12% of incomes were between $76,000 and $100,000; 17% of incomes were between $101,000 and $150,000; and 71% of incomes were $151,000 or more). The matched sample of middleincome children did not participate in the intervention, but instead provided a comparison point for the symbolic magnitude knowledge of the lowincome participants before and after the magnitude comparison card game intervention training.
Measures of Numerical Knowledge [TOP]
Five measures were used to assess the lowincome participants’ numerical knowledge, described in detail below.
Rote counting [TOP]
Participants were asked to count aloud from 1 through 10 (Ramani & Siegler, 2008). The dependent measure was the highest number counted to without errors divided by the highest possible score (i.e., 10). The interitem reliability was α = .93 at the first administration.
Numeral identification [TOP]
Children were presented with 10 randomly ordered cards, each with an Arabic numeral from 1 to 10, and asked to identify the numeral (Ramani & Siegler, 2008). The dependent measure was the percentage of numerals correctly labeled. The interitem reliability was α = .91 at the first administration.
Nonsymbolic ordinality [TOP]
Children watched an experimenter sequentially hide two different numbers of objects (i.e., fuzzy pompoms) in two opaque cups as a measure of nonsymbolic magnitude discrimination (Chu et al., 2013). The experimenter dropped the objects into the cups one at a time and then asked the children to point to the cup that contained more objects. The task included the presentation of two practice trials and six test trials. The number of objects in each cup ranged from 1 to 7, and each test comparison involved a number and the next largest number (e.g., 5 vs. 6). The cup containing more objects was randomized and counterbalanced across comparisons. The dependent measure was the number of accurate comparisons (excluding two practice trials) divided by the highest possible score (i.e., 6). The interitem reliability was markedly low at the first administration, α = .24 (see Discussion for potential explanations).
Symbolic magnitude comparison [TOP]
Participants were asked to complete the same measure of symbolic magnitude comparison described previously in Experiment 1. The dependent measure was percentage of correct comparisons. The interitem reliability was α = .84 at the first administration.
Nonsymbolic magnitude comparison [TOP]
Participants were asked to complete a series of comparisons between magnitudes (dot arrays) on a laptop computer, using the Panamath program described in Experiment 1. The settings were adjusted from those used in Experiment 1 to increase the task difficulty and better replicate those used in other studies of preschoolaged children, such that participants were asked to compare a greater number of pairs (32), and the presentation time decreased (stimuli presented for 2.3 seconds). The dot quantities ranged from 415 and included numerical comparisons with ratios of 2.0 (e.g., 48, 25% of trials); 1.5 (e.g., 46, 25% of trials); 1.3 – 1.4 (e.g., 811, 25% of trials); and 1.1 – 1.2 (e.g., 89, 25% of trials). The dependent measure was the percentage of accurate comparisons. As in Experiment 1, Weber fractions were not used as a dependent measure. The interitem reliability was α = .69 at the first administration.
Numerical Card Games [TOP]
Two numerical card games were used to provide lowincome participants with experience related to magnitude comparison and memory of numerical information. The numerical magnitude comparison card game (War) and the numerical matching card game (Memory) are described in turn.
Numerical magnitude comparison card game (War) [TOP]
The materials for the magnitude comparison card game were a set of 40 cards in the dimensions of standard playing cards (3.5 inches X 2.5 inches). The set included four subsets of cards representing quantities 1 through 10. Each card had both red Arabic numerals (0.5 inches in height) in the upper left and lower right corners and red dots (0.5 inches in diameter) representing the quantity (Figure 1).
For each game, the experimenter first divided the cards equally so that the child and the experimenter each had 20 cards stacked in a pile face down. To play, each person turned over his/her top card, said the number on the card, and the child was asked to say which number was greater. If children struggled to identify the numeral on the card, the experimenter encouraged them to count the circles on the card to determine the number, having previously established that the numeral and quantity of circles were always the same. If children could not compare the two quantities, or did so incorrectly, the experimenter encouraged them to look at the quantities of circles visually presented on each card to help them decide. The experimenter always corrected inaccurate counting, numeral identification, and magnitude comparison decisions, typically by asking the child to try again and scaffolding their arrival at the correct answer. The player with the card of greater magnitude (“the card with more”) took both cards. If two cards were the same, both players put three additional cards down, then turned a fourth card to determine who took all of the cards. At the end of the game, the players counted their cards to see who had more. During each session, the experimenter and child played as many rounds as possible until 15 minutes elapsed.
Numerical matching card game (Memory) [TOP]
Children who played the numerical memory and matching card game used the same playing cards described above, organized into two sets of 10 cards each. Each set of 10 cards contained five pairs of matching numbers between one and ten, and the experimenter alternated the use of each set so that children were exposed to all pairs equally. This allowed for similar exposure to all of the numbers from 1 through 10 for children in both numerical card games.
For each game, the 10 cards were placed face down in two rows of five columns. Each player took turns flipping over two cards and saying the number on each card, trying to find two cards with the same number. As in the game War, if children struggled to identify the numeral on the card, the experimenter encouraged them to count the circles on the card to determine the number. The experimenter always corrected inaccurate counting, numeral identification, and matching decisions. If the two cards matched, the player kept both, but if they did not match, the player returned the cards to their face down position. The players counted the cards at the end of the game to determine who had more. During each session, the experimenter and child played as many rounds as possible until 15 minutes elapsed.
Procedure [TOP]
Lowincome participants met individually with an experimenter in a quiet area of the hallway or room nearby their classroom for six 1520 minute sessions over a 3week period on average. During Sessions 1 and 6, the children completed five pretest and posttest assessments of their numerical knowledge in the order listed above, including multiple measures assessing nonsymbolic and symbolic representations. Following Session 1, children were randomly assigned at the sample level stratified by gender to play either the numerical magnitude comparison card game condition (War; n = 24) or to the numerical matching card game condition (Memory; n = 22) during Sessions 25. Previous intervention research targeting lowincome children’s numerical skills found significant improvements to children’s knowledge after four 15minute training sessions (e.g., Ramani & Siegler, 2008, 2011; Siegler & Ramani, 2008, 2009). Each child played the same game with the same experimenter for all of the card game training (Sessions 25), while another experimenter blind to their condition conducted 48% of the posttest assessments. After each session, children were given a sticker to thank them for their participation. The experimenters were four females: two Caucasian graduate students, one Caucasian and one Asian American undergraduate student.
The middleincome children completed one session of assessments of numerical knowledge and working memory abilities individually with an experimenter, including the symbolic magnitude comparison assessment described in Experiment 1. The experimenters were three females: one Caucasian graduate student, one Caucasian undergraduate student, and one Asian American undergraduate student. These experimenters were blind to the specific hypotheses of this study.
Results [TOP]
To address our first set of hypotheses, we began by investigating the effectiveness of the two intervention conditions (magnitude comparison and matching) on lowincome preschoolers’ numerical knowledge. Given the related nature of the five measures of numerical knowledge, we chose to conduct a doubly multivariate repeated measures analysis to examine the effects of participant age, condition, and session, and then conduct univariate analyses to examine the effects on each individual task. We report Pillai’s Trace as an Fstatistic because it is more robust with smaller sample sizes. For each of the numerical knowledge tasks, a single measure of accuracy was used. To account for the age range of lowincome participants in the study, a median split (59 months) was applied to create a group of younger children (n = 20, M = 4 years, 2 months, SD = 0.42, range = 3 years, 6 months to 4 years, 8 months) and a group of older children (n = 26, M = 5 years, 2 months, SD = 0.25, range = 4 years, 11 months to 5 years, 7 months). The median split served as a proxy for participants’ number of years in preschool (i.e., one versus two years).
Preliminary Analyses [TOP]
To ensure that lowincome children in the two conditions were equivalent at pretest, and to check for potential effects of classrooms on pretest knowledge, a 2 (condition: War or Memory card game) x 4 (classroom: Head Start classroom membership) MANOVA was first conducted on the five pretest measures of numerical knowledge. There were no significant main effects on the pretest numerical knowledge measures of condition, F(5, 34) = 1.35, p = .267, ω^{2} = .04, or class, F(15, 108) = 1.46, p = .135, ω^{2} = .05. Given that there were no differences, classroom was not controlled for in the following analyses. Similarly, as there was no main effect of condition, the two groups are considered equivalent on pretest numerical knowledge measures.
The difference in the amount of time children spent playing the card games in the two conditions fell just short of conventional significance levels (M_{War} = 59 minutes vs. M_{Memory} = 57 minutes, t(44) = 1.98, p = .054). On average, participants spent 58 minutes total playing the games across sessions. However, given the nature of the two card games, there was a difference in the number of games played across sessions between the two conditions. Children in the War condition on average played a total of 8.5 games, while children in the Memory condition on average played a total of 22.3 games. This reflects the fact that across participants, completing a single game of War took approximately 8 minutes, while a single game of Memory took approximately 3 minutes. As described previously, games of Memory alternated between two decks of cards that each included half of the quantities between 1 and 10, thus playing approximately twice as many games as children who played War resulted in a similar amount of overall exposure to the quantities 1 to 10.
Multivariate Analyses [TOP]
A 2 (age: above or below median) x 2 (condition: War or Memory numerical card game) x 2 (session: pretest or posttest) repeated measures MANOVA was conducted on the five measures of numerical knowledge: rote counting, numeral identification, nonsymbolic ordinality, symbolic magnitude comparison, and nonsymbolic magnitude comparison. There were significant main effects for age, F(5, 38) = 2.52, p = .046, ω^{2} = .15; session, F(5, 38) = 3.47, p = .011, ω^{2} = .22; and the condition x session interaction, F(5, 38) = 2.81, p = .029, ω^{2} = .17. Table 1 presents descriptive statistics and comparisons for the pretest and posttest measures. Additional univariate analyses for each task are presented below to establish the sources of the abovementioned differences.
Table 1
Measure  Pretest

Posttest

Pre vs. Post



M  SD  Min  Max  M  SD  Min  Max  p  r  ES  
Numerical magnitude comparison card game, War (n = 24)  
Rote counting  0.88  0.26  0.20  1.00  0.93  0.17  0.40  1.00  †  .82  0.39 
Numeral identification  0.80  0.31  0.10  1.00  0.82  0.32  0.00  1.00  .92  0.16  
Nonsymbolic ordinality  0.63  0.22  0.17  1.00  0.67  0.26  0.00  1.00  .47  0.16  
Symbolic magnitude comparison  0.72  0.23  0.28  1.00  0.80  0.20  0.39  1.00  ***  .91  0.88 
Nonsymbolic magnitude comparison  0.63  0.13  0.41  0.88  0.64  0.15  0.41  0.88  .61  0.08  
Numerical matching card game, Memory (n = 22)  
Rote counting  0.95  0.18  0.30  1.00  0.98  0.11  0.50  1.00  .56  0.22  
Numeral identification  0.88  0.23  0.30  1.00  0.91  0.20  0.30  1.00  †  .96  0.49 
Nonsymbolic ordinality  0.71  0.21  0.33  1.00  0.61  0.17  0.33  1.00  †  .07  0.39 
Symbolic magnitude comparison  0.82  0.18  0.44  1.00  0.83  0.16  0.56  1.00  .72  0.08  
Nonsymbolic magnitude comparison  0.71  0.14  0.50  0.94  0.66  0.14  0.44  0.94  †  .46  0.34 
Note. ^{†}p < .10. ***p < .001.
Symbolic magnitude comparison [TOP]
There were significant main effects on the symbolic magnitude comparison measure for age, F(1, 42) = 6.88, p = .012, ω^{2} =.11; session, F(1, 42) = 5.84, p = .020, ω^{2} = .10; and the condition x session interaction, F(1, 42) = 5.70, p = .022, ω^{2} = .10. Across sessions and conditions, older children tended to be more accurate in their symbolic magnitude comparisons than younger children (average of 85% comparisons answered correctly versus 72% comparisons). Across conditions and age groups, children tended to improve on symbolic magnitude comparison between pretest and posttest sessions (an average of 76% of comparisons answered correctly at pretest versus an average of 81% of comparisons at posttest).
As predicted, the significant condition x session interaction revealed that the children who played the War card game improved more on the symbolic magnitude comparison measure than their peers who played the Memory card game. Children in the War condition improved from correctly comparing 72% of the pairs at pretest to 80% of the pairs at posttest, t(23) = 4.27, p < .001, d = 0.88, whereas children in the Memory condition showed no significant improvement in performance, answering 82% correct at pretest and 83% correct at posttest.
Rote counting [TOP]
Children’s ability to count correctly from 110 varied by session, F(1, 42) = 4.68, p = .036, ω^{2} = .08; though there was not a significant condition x session interaction, F(1, 42) = .05, p = .819, ω^{2} =  .02. The main effect of session indicated that children in both conditions improved on average in their counting accuracy between the pre and posttest sessions, with accuracy improving from an average of 91% at pretest to an average of 95% at posttest.
Numeral identification [TOP]
Children’s ability to correctly identify each Arabic numeral from 1 to 10 showed no statistically significant effects of age or the condition x session interaction, however the main effect of session approached significance, F(1, 42) = 2.93, p = .094, ω^{2} = .04. This suggested that some children in both card game conditions showed improvement in their numeral identification skills between the pretest and posttest sessions. Indeed, the percentage of the sample that correctly identified all ten Arabic numerals was 63% at the pretest and 67% at the posttest.
Nonsymbolic ordinality [TOP]
Performance on the nonsymbolic ordinality measure approached statistically significant effects for age, F(1, 42) = 3.19, p = .081, ω^{2} = .05, and the condition x session interaction, F(1, 42) = 3.76, p = .059, ω^{2} = .06, however there were no effects that were statistically significant at the α = .05 level.
Nonsymbolic magnitude comparison [TOP]
On the nonsymbolic magnitude comparison measure, there was a significant effect of age, F(1, 42) = 9.51, p = .004, ω^{2} = .16, but no effects of session nor a condition x session interaction. Older children across sessions and conditions were more accurate in their nonsymbolic magnitude judgments than younger children, with 70% of comparisons made correctly by older children compared to 60% of comparisons made correctly by younger children.
Low and MiddleIncome Comparison [TOP]
To address our second set of hypotheses, we examined the extent to which playing the magnitude comparison card game improved the lowincome children’s symbolic magnitude skills relative to a matched sample of middleincome participants, which was the area of numerical knowledge with the greatest improvements among lowincome preschoolers. The lowincome children’s symbolic magnitude comparison performance at pretest and posttest was compared to the performance of the matched middleincome children. Independent samples ttests showed that there were differences between the lower and middleincome samples at pretest (M = 72% vs. M = 88%; t(46) = 2.97, p = .005, d = 0.86). However, after playing the War game the posttest scores of the lowincome children were no longer significantly different than those of the middleincome sample (M = 80% vs. M = 88%; t(46) = 1.67, p = .103, d = 0.48). Although the symbolic magnitude comparison accuracy of lowincome children after playing the War game was not significantly less than the accuracy of the middleincome children, the estimated effect size of the difference decreased from a 0.86 standard deviation gap (large) to a 0.48 standard deviation gap (moderate).
Discussion [TOP]
In Experiment 2, we found that playing both card games improved children’s counting skills. However, as predicted, only playing War improved children’s symbolic magnitude comparison performance. There were no gains found for children on either of the nonsymbolic measures of numerical knowledge, which was somewhat surprising in light of the improvements in symbolic magnitude comparison demonstrated by children in the magnitude comparison card game condition. Possible explanations for the lack of improvement could be the competing salience of symbolic number representations on the cards (stressed by the experimenter), and generally poor performance on the ordinal choice nonsymbolic magnitude comparison task. In particular, the ordinal choice task required a nontrivial working memory load in order to represent the quantities obscured by the opaque cups, as well as a fairly sophisticated tracking strategy (e.g., counting the objects placed into one cup, then remembering the first count while simultaneously keeping an accurate count of the objects in the second cup, then comparing the two counts). These potential explanations are discussed in more detail in the General Discussion.
Early gaps in numerical knowledge tend to widen over the course of schooling (Alexander & Entwisle, 1988; Geary, 1994, 2006), therefore, it is important to examine whether the intervention helped children to “catchup” to their peers from higherincome backgrounds, ideally preventing small gaps in numerical knowledge from growing into larger ones. The results of the lowincome and middleincome comparison analysis suggest that playing the numerical magnitude comparison card game improved the lowincome children’s performance to a point of statistical equivalence with the middleincome children, thus narrowing the observed achievement gap in early numerical knowledge.
General Discussion [TOP]
In the present study, we examined the specificity of SESrelated performance gaps on measures of magnitude representation, designed and implemented a numerical card game intervention for lowincome preschoolers, and compared children’s improved performance to a matched sample of middleincome preschoolers. We found that lowincome preschoolers did not show a significant performance gap on a measure of nonsymbolic magnitude comparison, although they did significantly lag behind middleincome preschoolers on a matched measure of symbolic magnitude comparison. We then examined whether we could capitalize on these differences by providing experience playing numerical card games with both symbolic and nonsymbolic representations to improve lowincome preschoolers’ early number skills. We found that playing the numerical magnitude comparison card game improved the lowincome children’s symbolic numerical magnitude knowledge.
Card Games and Numerical Magnitude Knowledge [TOP]
Given that magnitude comparison skill predicts broader mathematics achievement (Jordan et al., 2006; Mazzocco, Feigenson, & Halberda, 2011), we were particularly interested in whether targeted practice playing a magnitude comparison card game would boost preschoolers’ numerical magnitude understanding. As hypothesized, children who played the numerical magnitude comparison card game for approximately one hour also showed improvements in their symbolic magnitude comparison skills. We found that their skills improved to the point of statistical equivalence with a matched sample of middleincome preschoolers. Additionally, we found that children in both conditions, having similar opportunities to count circles and label Arabic numerals, improved similarly on those skills as hypothesized.
However, as discussed in Experiment 2, playing the numerical magnitude comparison (War) card game did not improve children’s performance on either of the two measures of nonsymbolic magnitude comparison. One explanation may be that including both numerals and nonsymbolic quantities (circles) on the cards led children who played the War card game to focus more on the symbolic numeral information while making comparisons, largely ignoring the nonsymbolic information. Theories of numerical development suggest that between 4 and 5years old children begin to fully integrate their symbolic number knowledge with nonsymbolic magnitude information (Siegler, 2016; Siegler & LortieForgues, 2014; Siegler, Thompson, & Schneider, 2011), which may mean the symbolic magnitude information was particularly salient to our sample of predominately 4 and 5year old children. In addition, the nonsymbolic ordinality task showed only marginally significant pretestposttest reliability and low interitem reliability (Pearson’s r(44) = .29, p = .051; interitem reliability at first administration α = .24), which may be an indication that the children had trouble focusing on or understanding that specific task.
Nevertheless, at pretest some of the children in our experimental sample were unable to recognize written numerals, and many children performed poorly when asked to make symbolic magnitude comparisons, which suggests they may have needed to rely on the nonsymbolic magnitude information to make accurate comparisons during the card game. Although the present study suggests that children may scaffold symbolic magnitude understanding from their nonsymbolic magnitude comparison skills, future work could further tease apart this hypothesis by contrasting the effectiveness of intervention conditions with purely nonsymbolic, symbolic, and dual (nonsymbolic and symbolic) magnitude representations.
These results add additional support to the growing literature of playbased mathematics interventions for lowincome and struggling early learners (Laski & Siegler, 2014; Ramani & Siegler, 2008; Ramani, Siegler, & Hitti, 2012; Räsänen, Salminen, Wilson, Aunio, & Dehaene, 2009; Whyte & Bull, 2008). Although many previous studies with lowincome preschoolers have focused on linear number board game interventions designed to promote children’s mental number line (e.g., Ramani & Siegler, 2008; Siegler & Ramani, 2008; Siegler & Ramani, 2009), a representation of their magnitude understanding, the current study suggests that numerical card games may lead to similar improvements in children’s early math skills. There are several existing curricula targeting early childhood education that similarly incorporate informal mathematics activities such as games, both commercially available and experimental (e.g., Building Blocks; Big Math, Little Kids; Number Sense Interventions). Understanding the impact of playing card games on children’s numerical knowledge can help to determine when and how numerical card games could be used as an activity independently in classrooms or coupled with existing curricular programs in preschools.
Limitations [TOP]
There are several limitations of the current study that should be noted. First, although participants were randomly assigned to the card game condition and preliminary analyses confirmed there were no statistically significant effects of condition at pretest, the sample average symbolic magnitude comparison performance of children in the Memory condition at pretest was greater than that of the children in the War condition. Thus, while children in the War condition improved their symbolic magnitude comparison performance to the point of equivalence at posttest to an age and gendermatched middleincome sample (Experiment 2), the scores of lowincome children in the Memory condition were statistically equivalent to the same middleincome sample at pretest. Since children were randomly assigned to the two groups, it is likely that this was simply due to variation in the magnitude comparison skills of the greater population of lowincome preschoolers. Future work will seek to replicate these effects with larger sample sizes, which should help reduce the variability in estimating children’s abilities.
Second, the relatively high performance at pretest of many preschoolers on the rote counting and numeral identification meant there was limited room for improvement. All measures were selected based on previous studies that found sufficient variability in samples of lowincome preschoolers (e.g., Chu et al., 2013; Jordan et al., 2006; Ramani & Siegler, 2008; Siegler & Ramani, 2008), however our results suggest that they may no longer be appropriate in today’s academically focused preschool settings. Future studies could adapt tasks to be more challenging for the children (e.g., have children count as high as they can as opposed to stopping at 10), or target children with lower initial knowledge for the training.
Third, all participants were involved in an active condition with exposure to numerical card games. Including a passive control condition or card game condition that did not involve numerical cards may provide a more accurate picture of the improvements in children’s early number skills that can be attributed to their experience playing the numerical magnitude comparison card game. In a similar vein, incorporating more distal measures of magnitude understanding such as the number line task (Siegler & Booth, 2004), which assesses magnitude understanding without directly referencing magnitude comparison, may help explain if the training effects from the magnitude comparison card game are specific to magnitude knowledge.
Fourth, although the two card games were matched in terms of total duration of playing time and exposure to nonsymbolic and symbolic representations of quantities from 110, the two games did differ in average duration per card game. This may in turn have led children to tire more quickly of playing the memory and matching game (as opposed to the longer numerical magnitude comparison game). Anecdotally, children appeared to be equally engaged in both types of games despite the variation in game duration. Children in both conditions rarely asked to end the gameplaying sessions prior to the 15minute time limit (less than 5 percent of intervention training sessions ended early at the child’s request). However, the present study did not collect a formal measure of children’s attention to or engagement in the card games, which could be added to future studies to control for potential variability between the two conditions.
Finally, the current study assessed learning by means of a posttest assessment conducted within several days of the final training condition. It remains an open question whether or not the improvements observed using a shortterm posttest assessment are maintained in the longterm, several weeks or months after the intervention training.
Conclusions [TOP]
The current study expands the early mathematics intervention literature to include numerical card games: a readily accessible and affordable resource. The results revealed significant effects of playing numerical card games on improving lowincome children’s basic numerical skills, an area in which many lowincome children are behind relative to middle and upperincome peers (Jordan et al., 2006; Starkey et al., 2004). Although the theoretical links between nonsymbolic and symbolic representations are debated, our study suggests exposure to both types of representation simultaneously may support lowincome children’s symbolic number skills by capitalizing on their more accurate nonsymbolic magnitude discriminations among quantities less than ten. Numerical card games provide the aforementioned opportunity for dual numerical representations, and future work should further explore their potential as intervention materials in classrooms and homes.
Promoting early mathematical understanding prior to the start of formal schooling has the potential to boost lowincome students’ longterm academic performance and combat perpetual STEM performance gaps. Basic numerical skills such as counting and the ability to make comparisons between numbers are foundational to the development of later mathematics skills such as addition and subtraction (Geary et al., 1999; Siegler & LortieForgues, 2014). Thus, improvements in these skills may support children’s success in elementary school and later mathematics experiences, narrowing the early mathematics gap.