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Preschoolers from low-income households lag behind preschoolers from middle-income households on numerical skills that underlie later mathematics achievement. However, it is unknown whether these gaps exist on parallel measures of symbolic and non-symbolic numerical skills. Experiment 1 indicated preschoolers from low-income backgrounds were less accurate than peers from middle-income backgrounds on a measure of symbolic magnitude comparison, but they performed equivalently on a measure of non-symbolic magnitude comparison. This suggests activities linking non-symbolic and symbolic number representations may be used to support children’s numerical knowledge. Experiment 2 randomly assigned low-income preschoolers (Mean Age = 4.7 years) to play either a numerical magnitude comparison or a numerical matching card game across four 15 min sessions over a 3-week period. The magnitude comparison card game led to significant improvements in participants’ symbolic magnitude comparison skills in an immediate posttest assessment. Following the intervention, low-income participants performed equivalently to an age- and gender-matched sample of middle-income preschoolers in symbolic magnitude comparison. These results suggest a brief intervention that combines non-symbolic and symbolic magnitude representations can support low-income preschoolers’ early numerical knowledge.

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 (

The present study investigates two hypotheses related to the early math achievement gap between children from low-income and middle-income families. The first is that consistent, reliable gaps between low- and middle-income 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 non-symbolic 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 non-symbolic number can help low-income 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 non-symbolic numerical magnitudes underlie their later math achievement (

According to Siegler and colleagues (

Both symbolic and non-symbolic 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 non-symbolic number representation) is associated with mathematics achievement among children and adults (

Given the significance of symbolic and non-symbolic magnitude knowledge during early childhood, it is important to understand whether there are income-based gaps present in both types of numerical skills. Although there are documented gaps between children from low-income and middle-income families on measures of symbolic magnitude comparison (e.g., determining which is the larger number;

Traditional numerical playing cards provide simultaneous exposure to symbolic and non-symbolic 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 (

A number of recent early mathematics interventions have incorporated playful, informal learning activities (

Children similarly appear to benefit from playful, informal mathematics activities in the home (

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 non-symbolic 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,

In the current paper, we pursued two goals. The first goal was to determine whether preschoolers from low-income households were less accurate than preschoolers from middle-income households on parallel tasks of symbolic and non-symbolic magnitude comparison. In Experiment 1, we hypothesized that we would find income-based 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 low-income children would show similar accuracy to their middle-income peers on a parallel measure of non-symbolic 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 SES-related gaps in verbally stated, symbolic numerical tasks, but not on nonverbal numerical tasks (

The second goal was to assess whether approximately one hour of playing numerical card games with symbolic and non-symbolic representations would lead to improvements in participants’ early symbolic and non-symbolic numerical knowledge. Expanding on the results of Experiment 1, Experiment 2 examined whether a magnitude comparison card game, based on the card game

We examined whether playing the two games would improve children’s symbolic numerical knowledge, specifically their symbolic numerical magnitude comparison skills, verbal counting from 1-10, and identifying written numerals from 1-10, as well as their non-symbolic 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 non-symbolic numerical tasks, given that we expected children’s non-symbolic numerical knowledge to be proficient for small quantities less than 10. Finally, we investigated whether low-income children’s magnitude comparison skills post-intervention would be comparable to the numerical knowledge of same-age peers from middle-income backgrounds.

The goal of Experiment 1 was to determine whether SES-related differences in numerical magnitude knowledge are specific to symbolic representations, or whether low-income preschoolers would show similar performance gaps on non-symbolic magnitude measures compared to middle-income preschoolers. To equate the non-symbolic 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 non-symbolic representations, it might be possible to capitalize on their non-symbolic comparison skills to foster their symbolic magnitude comparison skills with similar quantities.

Participants were 46 preschoolers. Informed consent was obtained for all participants from their parent or guardian. Twenty-three preschoolers were recruited from one Head Start center in a mid-Atlantic state, ranging in age from 3 years, 4 months to 5 years, 6 months (

The remaining 23 preschoolers were recruited from middle-income 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 middle-income preschoolers were selected as an age- and gender-matched sample to the low-income participants, ranging in age from 3 years, 8 months to 5 years, 9 months (

Two measures of magnitude comparison skills were used to assess the numerical understanding of participants.

Participants were asked to compare 20 pairs of symbolic numbers ranging from 1-9 presented in a paper booklet (

Participants were asked to compare 20 pairs of magnitudes (dot arrays) displayed on a laptop computer. Using the Panamath software (e.g.,

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 middle-income participants also completed non-computerized assessments of working memory. However, because the focus of the current study was preschoolers’ symbolic and non-symbolic number knowledge, only the identical assessments of symbolic and non-symbolic magnitude comparison administered to both low- and middle-income participants were analyzed for this experiment. To sustain children’s motivation, the symbolic magnitude comparison task was presented first followed by the non-symbolic magnitude comparison task, which participants considered more exciting because it was computer-based. 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 low-income participants was a Caucasian graduate student; the experimenters who interacted with the middle-income participants were one Caucasian graduate student, one Caucasian and one Asian American undergraduate student. All four experimenters were female.

Independent samples

In the present experiment, we found that the SES-related gaps in children’s symbolic magnitude knowledge were statistically significant, however, the differences between low-income and middle-income participants on the parallel non-symbolic magnitude comparison task were not statistically significant. This suggests that with these measures of symbolic and non-symbolic magnitude comparison, the gaps between middle- and lower-income children’s numerical knowledge were specific to children’s performance on a symbolic magnitude comparison measure. These findings are consistent with previous research examining SES-related differences on symbolic and non-symbolic measures of early arithmetic (e.g.,

The pattern of effects found in this experiment may vary under different task specifications. The symbolic and non-symbolic 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.,

Nonetheless, our intention was to match the non-symbolic magnitude comparison task parameters closely to the demands of the symbolic magnitude comparison task to investigate the SES-related gaps in young children’s numerical magnitude knowledge on quantities less than ten. The results of Experiment 1 demonstrate that low-income children are considerably more accurate (equivalent to middle-income children) in making non-symbolic comparisons than they are making symbolic comparisons between the same numbers. This implies that low-income children may benefit from magnitude comparison experiences that allow them to leverage their non-symbolic magnitude comparison skills to promote their symbolic magnitude comparison skills, a premise tested in Experiment 2.

Building from the findings of Experiment 1, Experiment 2 tested an experimental, game-based intervention using numerical cards with both non-symbolic and symbolic numerical representations. Children from low-income 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

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 non-symbolic magnitude knowledge. Therefore, we included two measures of children’s non-symbolic magnitude knowledge, specifically a non-symbolic magnitude comparison task, similar to the one used in Experiment 1, and an additional non-symbolic 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 low-income children on computerized non-symbolic magnitude comparison tasks may be affected by their inhibitory control (

Given that the low-income children had high accuracy on the non-symbolic 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 non-symbolic magnitude understanding. It is possible that the experience children have comparing numerical magnitudes using cards with both non-symbolic and symbolic representations could increase performance on non-symbolic measures of numerical knowledge. However, the highly accurate performance of low-income children on a measure of non-symbolic 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 non-symbolic magnitude comparison skills and provide a better match to the parameters in previously published studies with preschoolers, we increased the difficulty of the non-symbolic magnitude comparison measure by adjusting the quantities of the numbers and the presentation time.

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 were 70 preschoolers. Informed consent was collected for all participants from their parent or guardian. Forty-six low-income preschoolers were recruited from four Head Start centers in a mid-Atlantic state, comprising the main experimental sample, none of which participated in Experiment 1. Low-income preschoolers ranged in age from 3 years, 6 months to 5 years, 7 months (

Low-income participants were randomly assigned to either a numerical magnitude comparison card game condition (

The remaining 24 preschoolers were from middle-income backgrounds (

Five measures were used to assess the low-income participants’ numerical knowledge, described in detail below.

Participants were asked to count aloud from 1 through 10 (

Children were presented with 10 randomly ordered cards, each with an Arabic numeral from 1 to 10, and asked to identify the numeral (

Children watched an experimenter sequentially hide two different numbers of objects (i.e., fuzzy pom-poms) in two opaque cups as a measure of non-symbolic magnitude discrimination (

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 inter-item reliability was α = .84 at the first administration.

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 preschool-aged 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 4-15 and included numerical comparisons with ratios of 2.0 (e.g., 4|8, 25% of trials); 1.5 (e.g., 4|6, 25% of trials); 1.3 – 1.4 (e.g., 8|11, 25% of trials); and 1.1 – 1.2 (e.g., 8|9, 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 inter-item reliability was α = .69 at the first administration.

Two numerical card games were used to provide low-income participants with experience related to magnitude comparison and memory of numerical information. The numerical magnitude comparison card game (

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 (

Numerical cards used in

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.

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

Low-income participants met individually with an experimenter in a quiet area of the hallway or room nearby their classroom for six 15-20 minute sessions over a 3-week 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 non-symbolic 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 (

The middle-income 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.

To address our first set of hypotheses, we began by investigating the effectiveness of the two intervention conditions (magnitude comparison and matching) on low-income 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 F-statistic 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 low-income participants in the study, a median split (59 months) was applied to create a group of younger children (

To ensure that low-income children in the two conditions were equivalent at pretest, and to check for potential effects of classrooms on pretest knowledge, a 2 (condition: ^{2} = .04, or class, ^{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 (_{War}_{Memory}

A 2 (age: above or below median) x 2 (condition: ^{2} = .15; session, ^{2} = .22; and the condition x session interaction, ^{2} = .17.

Measure | Pretest |
Posttest |
Pre vs. Post |
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Min | Max | Min | Max | ES | |||||||

Numerical magnitude comparison card game, |
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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 | |

Non-symbolic 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 |

Non-symbolic 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, |
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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 |

Non-symbolic 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 | |

Non-symbolic magnitude comparison | 0.71 | 0.14 | 0.50 | 0.94 | 0.66 | 0.14 | 0.44 | 0.94 | † | .46 | -0.34 |

^{†}

There were significant main effects on the symbolic magnitude comparison measure for age, ^{2} =.11; session, ^{2} = .10; and the condition x session interaction, ^{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

Children’s ability to count correctly from 1-10 varied by session, ^{2} = .08; though there was not a significant condition x session interaction, ^{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.

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, ^{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.

Performance on the non-symbolic ordinality measure approached statistically significant effects for age, ^{2} = .05, and the condition x session interaction, ^{2} = .06, however there were no effects that were statistically significant at the α = .05 level.

On the non-symbolic magnitude comparison measure, there was a significant effect of age, ^{2} = .16, but no effects of session nor a condition x session interaction. Older children across sessions and conditions were more accurate in their non-symbolic magnitude judgments than younger children, with 70% of comparisons made correctly by older children compared to 60% of comparisons made correctly by younger children.

To address our second set of hypotheses, we examined the extent to which playing the magnitude comparison card game improved the low-income children’s symbolic magnitude skills relative to a matched sample of middle-income participants, which was the area of numerical knowledge with the greatest improvements among low-income preschoolers. The low-income children’s symbolic magnitude comparison performance at pretest and posttest was compared to the performance of the matched middle-income children. Independent samples

In Experiment 2, we found that playing both card games improved children’s counting skills. However, as predicted, only playing

Early gaps in numerical knowledge tend to widen over the course of schooling (

In the present study, we examined the specificity of SES-related performance gaps on measures of magnitude representation, designed and implemented a numerical card game intervention for low-income preschoolers, and compared children’s improved performance to a matched sample of middle-income preschoolers. We found that low-income preschoolers did not show a significant performance gap on a measure of non-symbolic magnitude comparison, although they did significantly lag behind middle-income 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 non-symbolic representations to improve low-income preschoolers’ early number skills. We found that playing the numerical magnitude comparison card game improved the low-income children’s symbolic numerical magnitude knowledge.

Given that magnitude comparison skill predicts broader mathematics achievement (

However, as discussed in Experiment 2, playing the numerical magnitude comparison (

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 non-symbolic magnitude information to make accurate comparisons during the card game. Although the present study suggests that children may scaffold symbolic magnitude understanding from their non-symbolic magnitude comparison skills, future work could further tease apart this hypothesis by contrasting the effectiveness of intervention conditions with purely non-symbolic, symbolic, and dual (non-symbolic and symbolic) magnitude representations.

These results add additional support to the growing literature of play-based mathematics interventions for low-income and struggling early learners (

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

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 low-income preschoolers (e.g.,

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 (

Fourth, although the two card games were matched in terms of total duration of playing time and exposure to non-symbolic and symbolic representations of quantities from 1-10, 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 game-playing sessions prior to the 15-minute 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 short-term posttest assessment are maintained in the long-term, several weeks or months after the intervention training.

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 low-income children’s basic numerical skills, an area in which many low-income children are behind relative to middle- and upper-income peers (

Promoting early mathematical understanding prior to the start of formal schooling has the potential to boost low-income students’ long-term 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 (

This work was supported by the Heising-Simons Foundation [Grant number 2014-142] and the Spencer Foundation [Grant number 201600002] to Geetha B. Ramani; the National Science Foundation [DGE 1322106] to Nicole R. Scalise; and the National Institutes of Health [NICHD Training Grant 1T32 HD07542-9] to Emily N. Daubert.

The authors have declared that no competing interests exist.

The authors give special thanks to the students and teachers at the Howard County and Prince George’s County Head Start classrooms for their participation in this research. The authors also thank Talia Rosenstrauch, Naomi Silverman, and Joanne Vu for their assistance with data collection and entry.