Associations Between Young Children's Flexible Attention to Numerical and Spatial Magnitudes and Early Math Skills.

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Published before final editing as: J Cogn Dev. 2025 Apr 15:10.1080/15248372.2025.2480072. doi: 10.1080/15248372.2025.2480072 Search in PMC Search in PubMed View in NLM Catalog Add to search Associations Between Young Children’s Flexible Attention to Numerical and Spatial Magnitudes and Early Math Skills Mary C Wagner Mary C Wagner 1 University of Dayton Find articles by Mary C Wagner 1 , Marissa Brown Marissa Brown 1 University of Dayton Find articles by Marissa Brown 1 , Molly K Griffin Molly K Griffin 1 University of Dayton Find articles by Molly K Griffin 1 , Mitchell Hanson Mitchell Hanson 1 University of Dayton Find articles by Mitchell Hanson 1 , Danielle A Barrett Danielle A Barrett 1 University of Dayton Find articles by Danielle A Barrett 1 , Madelyn H Hales Madelyn H Hales 1 University of Dayton Find articles by Madelyn H Hales 1 , Julia R Fabian Julia R Fabian 1 University of Dayton Find articles by Julia R Fabian 1 Author information Copyright and License information 1 University of Dayton PMC Copyright notice PMCID: PMC12338338  NIHMSID: NIHMS2070073  PMID: 40857435 The publisher's version of this article is available at J Cogn Dev Abstract Attending to numerical and spatial magnitude information is important for many math skills (e.g., measurement, proportional reasoning). The flexible attention to magnitudes (FAM) account proposes that preschool-aged children’s early ability to disentangle numerical and spatial magnitude information and flexibly shift attention between the two is a significant predictor of early math achievement. We recruited 226 children from diverse racial/ethnic and family income backgrounds in the Midwestern U.S. (51% female; Mage = 55 months; SDage = 8 months) to examine the associations between their FAM skill and math achievement. We first tested the hypothesis that children’s FAM skill is specific to their ability to flexibly shift between dimensions of both numerical and spatial magnitude. We did not find evidence that children were using a single-dimension strategy to complete the FAM task. We did find that children’s performance depended on which dimension they had previously attended to in prior trial levels, suggesting that the task indeed assesses children’s flexibly shifting between different dimensions of magnitudes. Second, we tested the association between FAM skill and math achievement while also taking into account proportional reasoning, number line estimation, subitizing, and non-symbolic numerical magnitude comparison. We found that children’s FAM task performance was significantly related to their math achievement after controlling for demographic covariates, executive function (EF) skills, and specific math skills. Implications of these findings for understanding the development of early math skills for diverse preschool-aged children in the U.S. is discussed. Strong math skills are crucial for children’s later academic success and for creating a well-prepared workforce in an increasingly technological society ( Hanushek & Woessmann, 2010 ). However, difficulties with math achievement are often evident early in children’s math skills development ( Jordan & Levine, 2009 ). The flexible attention to magnitudes (FAM) account proposes that children’s early ability to disentangle numerical and spatial magnitude information and flexibly shift attention between the two is a significant predictor of children’s early math achievement ( Fuhs et al., 2021 ) and that targeting this skill for early instruction could potentially facilitate improvement in early math skills. In the current study, we tested two novel hypotheses of the FAM account. First, we tested the hypothesis that children’s FAM skill is specific to their ability to flexibly shift between dimensions of numerical and spatial magnitude, rather than their ability to attend to differing elements within a single magnitude dimension. Additionally, we ran further analyses to better understand how the FAM task measures children’s flexible attention shifting rather than their ability to simply attend to numerical magnitudes. Second, we tested the hypothesis that children’s FAM ability is significantly related to math achievement above and beyond other specific math skills that have been shown to predict math achievement (number line estimation, proportional reasoning, subitizing, non-symbolic numerical magnitude discrimination). Math Achievement in Early Childhood Siegler (2016) argues in his integrated theory of numerical development that young children’s development of math skills is driven by an increasing ability to accurately represent numerical magnitudes, starting with representing non-symbolic numerosities (e.g., discrete quantities of objects) early in development to eventually representing all rational numbers. Correlational and longitudinal evidence supports a consistent association between children’s symbolic numerical magnitude skills (i.e., skills involving Arabic numerals or number words) and their math achievement (e.g., De Smedt et al., 2009 ; Geary, 2011 ; Geary et al., 2008 ). There are also several examples of successful interventions for symbolic numerical magnitude understanding or a combination of symbolic and non-symbolic magnitudes, particularly using number line activities (e.g., Laski & Siegler, 2014 ; Ramani & Siegler, 2008 ; Scalise et al., 2018 ). The research evidence is clear that children’s symbolic numerical magnitude processing facilitates their math achievement in early childhood. What is much less clear is if and how non-symbolic numerical magnitude processing abilities facilitate children’s math achievement. Many studies have found such a correlation (e.g., Halberda et al., 2008 ; Libertus et al., 2011 ; Starr et al., 2017 ), while others have not (e.g., Sasanguie et al., 2012 ; Sasanguie et al., 2014 ). Core number knowledge theorists propose that children have an innate sense of non-symbolic numerical magnitude that manifests in above-chance accuracy in representing numerical magnitudes via two distinct numerical systems ( Feigenson et al., 2004 ). However, children’s non-symbolic numerical magnitude skills may be affected by incongruencies that children notice in numerical and spatial magnitudes in tasks that are commonly used to assess this skill. For example, conditions of salient incongruency between numerical and spatial magnitudes – asking which set has more stars, 5 large stars or 10 small stars - results in the lowest performance for young children ( Fuhs & McNeil, 2013 ; Gilmore et al., 2013 ; Szűcs et al., 2013 ). Incongruency may actually tap into children’s executive function (EF) skills because children must ignore salient conflicting spatial magnitude information. According to the sense of magnitude hypothesis, spatial magnitude processing develops prior to numerical magnitude processing skills, and numerical magnitude processing is made possible by the development of EF skills ( Leibovich et al., 2017 , although see Victorsson et al., 2023 for an example of contradictory empirical evidence). More recently, Aulet and Lourenco (2023) have argued for the interdependence of visual processing of numerical and non-numerical magnitudes throughout development and suggest that only through the development of selective attention processes do children start to differentiate among numerical and non-numerical magnitude information. When examining factors that influence performance on non-symbolic numerical magnitude comparison tasks, empirical evidence suggests that EF skills are particularly associated with incongruent trials on non-symbolic numerical comparison tasks in early childhood ( Fuhs & McNeil, 2013 ), but this link may not be evident in older children ( Wilkey et al., 2021 ). FAM Account Current debates around the role of non-symbolic numerical magnitude processing in children’s early math skills are limited in several ways that prevent the field from moving forward with more effective theories and interventions. First, debates tend to focus on the question of whether numerical magnitude processing is innate or learned, and although very important to understand basic developmental processes in infancy, this perspective has limited practical or clinical utility in addressing how both numerical and spatial magnitude processing abilities are connected to math achievement. Second, these debates are often limited to the discussion of children’s non-symbolic numerical magnitude skills, or approximating large discrete quantities without counting. However, tangible math manipulatives and number books involving non-symbolic numerical magnitudes (e.g., sets of objects) are used widely in early math activities and typically do not require that children estimate quantities while being prevented from counting. Therefore, it is difficult to make theoretical and empirical connections between these tasks and everyday activities for young children in early math. Third, in debates about the role of EF skills in children’s non-symbolic number comparison skills, theoretical accounts have often focused solely on instances in which children are asked to pay attention to numerical magnitude and ignore spatial magnitudes, rather than flexibly shift between magnitudes. Finally, much of the prior research in this area has been conducted primarily with White, higher-income families. The FAM account addresses the limitations of prior theoretical and empirical work in early childhood by proposing that an important skill for young children is the ability to disentangle and flexibly shift between attending to numerical and spatial magnitudes in early mathematical tasks. An important component of this account is that it focuses on explicit, directed shifting between numerical and spatial magnitudes, rather than spontaneous attention to one dimension in an unguided context. A similar, yet notably distinct, skill presented in existing literature on young children’s math development is spontaneous focusing on numerosity (SFON) ( McMullen et al., 2019 ). SFON broadly refers to children’s tendency to focus on numerosity when presented with an ambiguous task and no external prompting to attend to mathematical attributes ( McMullen et al., 2019 ). Significant relations have been identified between SFON tendencies and math skills, and these relations held after children’s age and a variety of other cognitive abilities were controlled for ( McMullen et al., 2019 ). Measures of SFON tendency and FAM skill are similar in that both use small item quantities to eliminate the possibility that results are dependent on children’s ability to recognize large quantities, but they differ in the construct that each attempts to measure. While SFON tendency is defined as a child’s ability to spontaneously recognize and apply mathematical concepts to vague tasks, FAM ability is defined as intentional, explicit attention to a specific magnitude and a child’s ability to make flexible cognitive shifts between magnitudes ( Fuhs et al., 2021 ; McMullen et al., 2019 ). Similarly, spontaneous orientation towards different dimensions of magnitude (SOMAG) measures children’s tendency to focus on numerical or non-numerical dimensions of magnitude when presented with both in a single task without external prompting (Viarouge et al., 2018). Although SOMAG and FAM accounts are similar in that both involve multiple dimensions of magnitude, SOMAG tendency examines children’s proclivity for attending to numerical or non-numerical dimensions, while FAM account measures their ability to flexibly shift their focus intentionally between dimensions ( Fuhs et al., 2021 ; Viarouge et al., 2018). The FAM task was created to align with everyday mathematical encounters while testing outside of the parameters of estimating and comparing large quantities without counting ( Fuhs et al., 2021 ). In this task, children were asked to attend to and discriminate between two incongruent object sets with varying prompts asking them to either focus on numerical or spatial magnitudes. All object sets were incongruent with respect to numerical and spatial magnitudes, such that one object set always had more numerous objects while the other set had larger objects. Conceptually, the primary skill involved in the FAM task is cognitive flexibility, but we know from latent variable analyses that children’s EF skills in early childhood are best represented by a unitary latent EF factor (e.g., Hughes, Ensor, Wilson, & Graham, 2009 ). Researchers found that the FAM task significantly correlated with children’s EF skills and math achievement longitudinally across a six-month period ( Fuhs et al., 2021 ). In a second study, researchers found that even when controlling for children’s EF skills and non-symbolic numerical discrimination ability, FAM skill was uniquely and significantly associated with children’s math achievement ( Fuhs et al., 2021 ). Prior FAM studies have addressed if FAM is a significant predictor of math achievement, but several open questions remain. First, it is not yet clear if children’s FAM task performance truly represents their ability to flexibly shift between numerical and spatial magnitudes or if it simply represents their ability to: a) attend to different aspects of a single magnitude dimension, or b) attend to numerosity and ignore spatial magnitudes or understand the meaning of numerical magnitude language (e.g., more). For example, because all FAM test trials in prior work were incongruent with respect to spatial and numerical magnitudes, we cannot rule out the possibility that young children may simply be using a single-dimension strategy to complete the task. For example, if a child attended to “big” objects in the first level of the task when being asked to attend to spatial magnitude, they would score very well on these trials. When switching to the next level of the task where they are asked to attend to number, the child could potentially still focus solely on spatial magnitudes and attend to the “small” objects and perform quite well on the task. Whether and how often children use a single dimension strategy to solve early math problems is an important theoretical question to address because it will allow for a better understanding of the developmental precursors to children’s FAM skill and will provide insights about the saliency of different dimensions of magnitude in early childhood. Second, how is FAM task performance related to other specific math skills that predict math achievement? To answer this, we need to address correlations between FAM and math achievement controlling for other known predictors of math achievement to determine how FAM relates to them and if FAM provides unique predictive validity above and beyond these other factors. These are essential questions to ask prior to investigating FAM instruction as an intervention target for improving young children’s early math achievement. FAM and Math Skills Development There are several specific math skills that presumably require children to flexibly shift between numerical and spatial magnitudes. For example, number line estimation requires that children attend to both the numerical quantity they are asked to place on a number line as well as the line length and distance from zero. Proportional reasoning also requires children to attend to both numerical quantities as well as spatial proportions, for example comparing bar graphs with and without discrete units (e.g., Boyer & Levine, 2015 ). These skills have been found to relate to children’s general math achievement ( Ramani & Siegler, 2008 ; Boyer & Levine, 2015 ) but also to children’s EF skills ( Fuhs et al., 2016 ; Kolkman et al., 2013 ). These skills should then, theoretically, be significantly related to but distinct from children’s FAM skill, while other skills that do not require as much attention to both numerical and spatial abilities, such as subitizing, would not be significantly related (although still correlated with math achievement, Yun et al., 2011 ). It is important, then, to examine these skills in conjunction with FAM skill to understand the extent to which overlap occurs between these related skills and if FAM skill makes a unique contribution to math achievement above and beyond other specific math skills that have been correlated with math achievement in prior work. Current Study In the current study, we extended existing theoretical and empirical literature in two important ways. First, we tested the hypothesis that children’s FAM skill is specific to their ability to flexibly shift between dimensions of both numerical and spatial magnitude, rather than their ability to attend to different elements of a single dimension of magnitude or their ability to simply attend to numerosity and ignore spatial dimensions of magnitude. To investigate this, we incorporated “check” trials into the FAM assessment that were congruent with respect to both the size and quantity of the items presented, unlike the test trials, which were incongruent. Building on our previous work, the addition of the check trials allowed us to determine if children were employing a single dimension decision-making strategy rather than flexibly shifting their attention between the two magnitudes. We also conducted an in-depth trial-by-trial analysis of FAM performance to determine the extent to which children’s performance related primarily to their shifting ability between magnitudes versus their ability to attend to numerosity and ignore spatial magnitudes when necessary. Second, we tested the hypothesis that children’s FAM ability is significantly related to math achievement above and beyond other specific math skills that have been shown to predict math achievement (number line estimation, proportional reasoning, subitizing, non-symbolic numerical magnitude discrimination) to determine if and how FAM ability aligns with accounts of specific math skills and their relation to math achievement. Method Participants Children (N = 226) were recruited from childcare providers, public pre-k programs, and Head Start programs in the Midwestern U.S.. This sample size needed to achieve .80 power using an effect size of f 2 = .04, or a small effect, for the estimated association between children’s FAM ability and their math achievement in a multiple regression analysis with 11 predictors was N = 199, with additional children recruited to account for attrition and missingness. Children ranged in age from 36 – 70 months ( M = 55 months; SD = 8 months). The sample was 51% female and 49% male. The sample was 67% Black or African American, 23% White, 8% Multiple Races, and 2% Asian or Asian American (parents/guardians of 15% of participants did not respond). Most parents had a high school diploma or less as their highest level of education (61%), and most families had annual household incomes of less than $42,000 USD (65%). Missingness was higher for family income (41%) and parent education (39%) due to parents and guardians choosing not to respond to those particular demographic questions. Measures FAM Skill. The FAM assessment was created with the intention to represent everyday mathematical activities, without requiring children to estimate and compare large quantities while being prohibited from counting. There are three trial levels (pre-switch, post-switch, and mixed) and two conditions (number first and size first) in this assessment. In the first pre-switch level, children are randomly assigned to select either the object set with more numerous stars (number first condition) or the object set with larger stars (size first condition). In the post-switch level, children are asked to shift their attention to the dimension that they did not attend to in the pre-switch trials. In the final mixed trial level, children are asked to shift back and forth between attending to numerical magnitudes and spatial magnitudes across trials in a pseudo-random order while making judgments about which object set has more (or bigger) stars. These pre-switch and post-switch trial levels each include a demonstration trial, two practice trials, and six test trials. The mixed trial level includes two demonstration trials, two practice trials, and twelve test trials. In this task, children are provided with feedback on their correctness for the practice trials. If a child answers correctly, the researcher provides positive feedback and repeats the rule. If the child answers incorrectly, the researcher corrects the child by explaining the differences in features of the stars in both boxes (e.g., “Well, the stars in that box are bigger, but there are more stars in this box. They are smaller but there are more of them” ( Negen & Sarnecka, 2015 )). This feedback provided in practice trials has been shown to help young children better understand the numerosity term meaning in previous studies ( Negen & Sarnecka, 2015 ). Children are not provided with feedback on the test trials. The number of trials in each level were selected to align with already existing flexible attention paradigms (e.g., Dimensional Change Card Sort, Zelazo, 2006 ). Each object set includes set sizes of 1 – 10 items, which is within most preschoolers’ experience level with numerosity. FAM Task Modification. In the standard FAM task, all object set comparisons are incongruent with respect to numerical and spatial magnitudes, such that one object set always has more items, and the other always has larger items. However, one might wonder if children could potentially adopt a single-dimension strategy to solve these problems that would allow them to score artificially high on the test trials. For example, if a child is asked to first select the set with bigger objects on the pre-switch trials, and then asked to switch to selecting the set with more numerous objects on the post-switch trials, a child could get all post-switch answers correct by simply switching from bigger to smaller objects within the spatial dimension and not actually switching between spatial and numerical magnitudes (see Figure 1 ). Figure 1. Open in a new tab To explore this possibility, we modified the original FAM task to include two “check” trials in the pre- and post-switch trials and four “check” trials in the mixed trials. For the “check” trials, if an object set had more numerous objects, it also had larger objects. If a child was using a single-dimension strategy, they would score high on the test trials but low on the “check” trials despite the fact that the “check” trials are generally considered easier than the test trials to solve because they do not involve incongruency. Accuracy on the FAM task was calculated as the proportion correct on post-switch and mixed test trials as these are the two trial levels that explicitly require flexible attention to magnitudes, and check trials were examined separately to test for the possibility that children used a single-dimension strategy. Mathematics Achievement. Children’s overall math achievement was evaluated using the WJ-IV ECAD Number Sense subtest ( Schrank & Dailey, 2014, 2015 ). This standardized measure assesses children’s numerical skills including recognizing and identifying numbers, counting, and quantity estimation, and has verified reliability and validity ( Schrank et al., 2015 ). Standard scores were used in analyses. Language Covariate. Children’s language ability was assessed using the WJ-IV ECAD Picture Vocabulary subtest ( Schrank et al., 2015 ). This subtest appraises children’s vocabulary skills by asking them to verbally identify pictures of items presented in increasing difficulty. Standard scores were used in analyses. Executive Functioning Skills. The Minnesota Executive Function Scale (MEFS) was used to examine children’s executive functioning skills ( Casey et al., 2014 ). This computer-based iPad app was derived from the Dimensional Change Card Sort Task (DCCS; Zelazo, 2006 ), and includes 7 levels, increasing in difficulty, that are adaptive to the child’s age and performance. Depending on the level of difficulty, children are prompted to sort cards into one of two boxes, based on components like color, shape and size. Lower levels only require children to attend to one characteristic at a time, while higher levels ask them to shift focus between traits to make their sorting decision. The MEFS software uses an algorithm that incorporates both response time and accuracy and produces standardized scores that were used in analyses. Number Line Estimation. Children completed the number-to-position task as a measure of number line estimation as it requires attention to both numerical and spatial magnitudes and has been shown to predict a variety of math skills in pre-k and elementary-school-aged students ( Booth et al., 2014 ; Booth & Siegler, 2006 , 2008 ; Ramani & Siegler, 2008 ; Siegler & Booth, 2004 ; Siegler & Booth, 2005 ; Siegler & Opfer, 2003 ). Children completed a 0–10 number line and were asked to “point to #”, with a total of 20 trials (2 trials for each number). For each trial, experimenters then viewed a set of boxes below the line and scored the item correct if the child pointed either exactly to the correct spot or within one box to the right or left of the correct box, resulting in total scores of 0 – 20. Proportional Reasoning. Children’s proportional reasoning was assessed using a task developed by Spinillo and Bryant (1991) in which children must match a small picture of a box in which a portion is shaded blue against a white background (continuous representation) to one of two response choices that show larger versions of the box filled with a number of blue and white blocks that either represent the same proportion of blue shaded space as the picture or a different proportion (discrete match) (16 trials). Children’s scores were calculated as the total number of items correct (0 – 16). Subitizing. Children’s subitizing ability was assessed using the fast counting task ( Schleifer & Landerl, 2011 ). In this task, children were presented with 12 trials showing a random number of dots in small quantities (1 – 6) and are asked to say how many dots there are quickly without counting. Total scores were used in analyses (0 – 12). Non-symbolic Numerical Comparison Skills. Children’s non-symbolic numerical comparison skills were assessed using the Panamath task ( Halberda et al., 2012 ). In this computer task, children are presented with two object sets and are asked to choose, without counting, which object set has more dots. Trials are only displayed for a short time period (600 ms.) to prevent counting. This task was chosen as it is the most commonly used assessment of young children’s non-symbolic numerical magnitude comparison skills. Percent correct for this assessment was used in analyses. Procedure Children were assessed at their preschool/childcare facility at a time determined by the teachers and in a quiet area of the classroom or center. Children completed the assessments in a fixed order across two separate sessions, each lasting 20 – 30 minutes. Children received stickers and a gift card for participating. The order of assessments in session 1 was MEFS, WJ-IV ECAD Picture Vocabulary, WJ-IV ECAD Number Sense, and the FAM task. The order of assessments in session 2 was Number Line Estimation, Proportional Reasoning, Subitizing, and Panamath. Analytic Approach Descriptives were calculated in SPSS v. 28. Correlation and regression analyses were conducted in Mplus v. 8.8. First, we compared children’s performance on specific trials within and across conditions and trial levels to determine if children’s performance was influenced by their flexible shifting ability between numerical and spatial magnitudes or their ability to attend to numerosity alone. Second, we examined the possibility of children using a single-dimension strategy to perform well on the FAM task to determine if it occurred, and if so, how often. To do so, we performed several analyses. First, we examined zero-order correlations between check trial performance and test trial performance because a significant negative correlation would suggest that children overall were using a single dimension strategy during the FAM task. Then, we computed the percentage of children who scored both below chance on check trials and above chance on test trials to identify how many individual children may have used a single-dimension strategy during the FAM task. Finally, we compared children who met criteria for using a single-dimension strategy to those who did not on a number of outcomes to understand how these two groups may differ. We then examined bivariate correlations between FAM ability and specific math skills. Third, we computed a multiple regression model (Model 1) that included these specific math skills as additional predictors of math achievement. In a second model, we examined an exploratory research question by computing an additional model without children who used a single-dimension strategy on the FAM task (Model 2). In both regression models, we allowed the residuals of all predictors to correlate, which resulted in a fully saturated model. We used Full Information Maximum Likelihood (FIML) estimates to utilize all available data and reduce bias due to missingness. Results Descriptive Statistics and Missingness Descriptive statistics are presented in Table 1 . On average, children scored below national norms on language skills, EF skills, and math achievement. Missingness on assessments was primarily due to children choosing to stop participating at some point during the study, and there were a number of children who withdrew from their school or were unavailable at assessing times to participate in the study. We found it particularly challenging to complete Panamath assessments as children were challenged by the repetitive nature of the task and the large number of trials. Therefore, missingness was higher on this assessment compared to the other assessments. We conducted independent samples t-tests to compare the demographics and assessments scores of children who completed the Panamath versus children who did not complete the assessment. A number of differences were noted. Children who did not complete the Panamath were somewhat younger ( Mcompleted = 55.15 months, Mmissing = 52.56 months, t (221) = 1.85, p = .065, d = .34) and significantly more likely to be male ( Female percent completed = 89%, Male percent completed = 79%, t (220) = 2.01, p = .046, d = .37). Children who were missing Panamath scores also scored significantly lower on the MEFS ( Mcompleted = 97.35, Mmissing = 93.40, t (219) = 2.28, p = .023, d = .42), Subitizing ( Mcompleted = 6.94, Mmissing = 3.07, t (215) = 6.07, p < .001, d = 1.21), Number Line Estimation ( Mcompleted = 3.91, Mmissing = 1.66, t (215) = 3.48, p < .001, d = .69), and Proportional Reasoning ( Mcompleted = 10.51, Mmissing = 2.55, t (215) = 10.81, p < .001, d = 2.16). Table 1. Descriptive Statistics N Min Max M SD Number Sense Standard Score 221 47 146 96.42 17.63 Picture Vocab Standard Score 223 60 146 101.27 13.66 MEFS Standard Score 221 61 118 96.73 9.49 FAM Post + Mixed Trial Accuracy 209 0.22 1.00 0.69 0.18 Subitizing Total Score 217 0 12 6.42 3.45 Number Line Estimation Total Score 217 0 16 3.61 3.33 Proportional Reasoning Total Score 217 0 16 9.45 4.58 Panamath Percent Correct 188 35 100 64.54 15.82 Open in a new tab Note: MEFS refers to the Minnesota Executive Function Scale. FAM refers to the Flexible Attention to Magnitudes task. FAM Task Descriptive Analyses We first assessed children’s performance on test trials by FAM condition (size first, number first) and FAM trial level (pre-switch, post-switch, mixed). Results of a 2 × 3 ANOVA revealed a significant interaction between FAM condition and FAM trial level (Greenhouse-Geisser corrected F (1.95, 404.52) = 10.55, p < .001, η p 2 = .05). We ran post-hoc contrasts using estimated marginal means (EMM) to explore this interaction. We first compared performance of FAM conditions within each trial level, and then we compared performance across trial levels within each FAM condition (see Figure 2 ). Figure 2. Open in a new tab FAM Performance by FAM Condition within FAM Trail Levels Children assigned to the size first FAM condition scored significantly higher on pre-switch trials compared to the number first FAM condition, F (1,207) = 10.54, p = .001, η p 2 = .05), and children assigned to the number first FAM condition scored significantly higher on post-switch trials compared to the size first FAM condition, ( F (1,207) = 7.88, p = .001, η p 2 = .04). There was not an overall significant difference between FAM condition performance in the mixed trial level ( M size first = .59, M number first = .61, F (1,207) = .42, p = .516, η p 2 = .002). Note that Figure 2 further breaks down the FAM mixed trial level performance by trial type (size or number trials within the mixed condition). Those results are discussed below in the Hypothesis 1 results section. We also found significant effects of performance across trial levels within the size first condition ( Wilks’ Lambda F (2,206) = 81.33, p < .001, η p 2 = .44) and the number first condition ( Wilks’ Lambda F (2,206) = 85.09, p < .001, η p 2 = .45). Specifically, children in the size first condition significantly decreased their performance from pre-switch to post-switch trials ( EMM difference = .13, SD = .03, p < .001) and mixed trials ( EMM difference = .34, SD = .03, p < .001) and from post-switch to mixed trials ( EMM difference = .21, SD = .03, p < .001). Within the number first condition, children marginally increased their performance from pre-switch to post-switch trials ( EMM difference = .05, SD = .03, p = .064), significantly decreased their performance from post-switch to mixed trials ( EMM difference = .33, SD = .03, p < .001), and significantly decreased their performance from pre-switch to mixed trials ( EMM difference = .28, SD = .03, p < .001). Overall, the findings indicate that the FAM condition affected performance such that children scored lower on number trials regardless of if they were pre- or post-switch, but that FAM condition did not affect mixed trials performance. Because children scored lower on number trials regardless of condition, we explored if children who completed number trials as the pre-switch level scored higher on number trials than those who completed number trials as the post-switch level, suggesting that the FAM task measures a switching effect rather than simply a numerosity effect. Children who completed the number trials as pre-switch trials performed better than children who completed the number trials as post-switch trials, t (210) = 1.50, p = .068, d = .21). We also explored if children who completed the size trials as pre-switch trials also performed better than children who completed the size trials as post-switch trials, t (212) = .87, p = .193, d = .12), but this finding was non-significant. It should be noted that children generally scored well on these trials overall, leaving little variability. Hypothesis 1: Children’s FAM skill is specific to their ability to flexibly shift between dimensions of numerical and spatial magnitude, rather than their ability to attend to different elements of a single dimension of magnitude or their ability to attend to numerical magnitude alone. Analyses testing for evidence of single-dimension responding on the FAM task. Within the pre- and post-switch levels, there were two additional check trials added to the six test trials, and within the mixed trial levels, there were four additional check trials. Check trials were congruent in spatial and numerical magnitude such that the box that had more stars also had bigger stars (See Figure 1 for an example of a congruent check trial and an incongruent test trial within the number game). The purpose of the check trials was to assess if children used a single-dimension strategy, which would mean that they would score lower on the check trials but do well on the pre- and post-switch test trials. In other words, evidence of children using a single-dimension strategy would include a significant negative correlation between check trials and test trials within each of the three levels of the task. Therefore, we examined correlations between check trials in each level and the test trial accuracy. We found significant positive correlations across levels, pre-switch ( r = .38, p < .001), post-switch ( r = .18, p = .011), and mixed ( r = .20, p = .003). This suggests that on average, there was not evidence that children were using a single-dimension strategy to complete the assessment. Even though we did not find evidence of children using a single-dimension strategy when averaging across the entire sample, it was still possible that at least a small subset of the children within the sample were using a single-dimension strategy. To test this possibility, we examined the percentage of children who appeared to use a single-dimension strategy within each trial level. We made the assumption that individual children who were using a single-dimension strategy would score below chance on check trials (< 50% accuracy or scoring 0/2 on check trials, and/or <2/4 on mixed trials) and also score above chance on performance trials (> 50% accuracy or equal to or greater than 4/6 on pre- and post-switch trials, and/or greater to or equal to 9/12 on mixed trials). For pre-switch trials, 6% (13 out of 214 children) met this criteria. For post-switch trials, 9% (18 out of 212 children) met this criteria, and 1% (2 out of 209 children) met this criteria on mixed trials. Further examination of only the pre-switch and post-switch trials across FAM conditions (and not the mixed trials) revealed that children only met this criteria when they were completing trials in the number trial level, regardless of whether it was the pre-switch or post-switch trials. This suggests that there was a small subset of children who used a spatial strategy on numerical magnitude comparison test trials where they answered test trials using only spatial magnitude information. Children who used a single-dimension strategy on one or more trial levels (n = 32, note that one child met criteria on both the post-switch trial level and the mixed trial level) differed from their peers on a few variables. Children who used a single-dimension strategy on any of the trial levels were significantly younger ( M = 51.81 months) than children who did not ( M = 55.54 months, t (211) = 2.58, p = .011, d = .49), they scored significantly lower on non-symbolic numerical magnitude comparison skills ( M = 53.05% accuracy) than children who did not ( M = 66.53% accuracy, t (181) = 4.31, p < .001, d = .89). Children who used a single-dimension strategy also scored lower on language ( M = 96.19 vs M = 102.32, t (211) = 2.34, p = .020, d = .45) and math achievement ( M = 89.44 vs M = 98.48, t (209) = 2.74, p = .007, d = .53). To summarize, we did not find a general pattern of using a single-dimension strategy overall, given that check trials and test trials were positively significantly correlated in all three levels of the FAM task. However, when examining individual differences, we found that a subset of 32 children (of 214 who completed some or all of the FAM task) showed possible evidence of using a single-dimension strategy, as evidenced by scoring below chance on check trials and above chance on test trials. This subset of children differed from their peers in age, non-symbolic numerical magnitude processing, language achievement, and math achievement. Analyses testing if the FAM task assesses numerical magnitude understanding alone and not switching. An additional concern of the FAM measure is the extent to which it measures children’s difficulty flexibly shifting between dimensions of magnitude as opposed to just their difficulty understanding numerical magnitude alone. To achieve a better understanding of exactly what the FAM task measures, we conducted a trial-by-trial analysis of the mixed FAM trial level to determine if children’s performance was based on their ability to switch to number trials alone rather than their ability to flexibly shift between magnitudes 1 . Overall, children scored lower on number trials in the mixed level compared to size trials in the mixed level, t (150) = 2.83, p = .005, d = .23. However, this finding varied by which level children completed at pre-switch and post-switch, prior to the mixed level. Children who completed the number game at pre-switch and the size game at post-switch, directly before the mixed level, scored significantly worse on mixed number trials compared to children who completed the size game at pre-switch and number game at post-switch, t (151) = 2.62, p < .010, d = .43 (see Figure 2 ), suggesting that their number trials performance in the mixed level depended on their switching condition in prior trial levels. Conversely, children who completed the size game at pre-switch scored significantly lower on mixed size trials compared to children who completed the number game at pre-switch, t (149) = 3.00, p = .022, d = .49 (see Figure 2 ). Examining the switch trials in the mixed level, similar patterns were found. Switch trials within the FAM mixed trial level can be defined as a trial where the prior trial was asking for attention to a different dimension. For example, a trial asking for size comparison following a trial asking for number comparison would be considered a “switch” trial. First, we compared switch trials to non-switch trials within the mixed condition regardless of FAM condition to test if there was a difference between these two types of trials overall. Children scored lower on switch trials within the mixed trial level ( M = .59, SD = .23) than non-switch trials ( M = .62, SD = .26), but this difference was not significant, t (150) = 1.42, p = .157, d = .12. Within the switch trials in the mixed trial level, children had more difficulty switching to number if the prior trial was a size trial ( M = .54, SD = .37) than switching to size if the prior trial was a number trial ( M = .64, SD = 37), t (150) = 2.05, p = .042, d = .17. When examining mixed trials without considering FAM condition, the evidence appears to support a numerical magnitude attention alone argument for what the FAM task is measuring rather than a switching account. However, when FAM condition is considered, children in the number first condition who completed the number game at pre-switch (and size game at post-switch) scored significantly lower on size-to-number switch trials compared to children in the size first condition who completed the size game at pre-switch (and number game at post-switch), t (152) = 2.39, p = .018, d = .39 (see Figure 3 ). Similarly, children in the size first condition who completed the size game at pre-switch (and number game at post-switch) scored significantly lower on number-to-size switch trials than children in the number first condition who completed the number game at pre-switch (and size game at post-switch, t (149) = 2.95, p = .002, d = .48 (see Figure 3 ). Figure 3. Open in a new tab FAM Mixed Trail Performance within Switch Trail Types for Each Condition To summarize, these additional exploratory analyses suggest that the FAM task does not simply measure children’s numerical magnitude comparison skills alone but also importantly captures their ability to switch attention between numerical and spatial magnitude dimensions. Hypothesis 2: Children’s FAM ability is significantly related to math achievement above and beyond other specific math skills that have been shown to predict math achievement (number line estimation, proportional reasoning, subitizing, non-symbolic numerical magnitude discrimination). Bivariate correlations between FAM skill and other math skills ( Table 2 ) provided evidence for construct validity for the FAM task. As predicted, children’s FAM skill was significantly correlated with their proportional reasoning skills and their non-symbolic numerical magnitude comparison skills and marginally correlated with their number line estimation skills. Also as predicted, there was not a significant correlation between FAM skill and children’s subitizing abilities. Table 2. Correlations 1 2 3 4 5 6 7 8 9 10 11 1. Number Sense Standard Score 1 2. Age −0.19 ** 1 3. Gender 0.17 * 0.003 1 4. Parent Education 0.41 ** −0.05 0.09 1 5. Family Income 0.39 ** −0.05 0.10 0.63 ** 1 6. Picture Vocab Standard Score 0.69 ** −0.31 ** 0.12 0.33 ** 0.40 ** 1 7. MEFS Standard Score 0.36 ** −0.05 0.09 0.30 ** 0.30 ** 0.33 ** 1 8. FAM Post + Mixed Trial Accuracy 0.34 ** 0.28 ** 0.04 0.09 0.07 0.25 ** 0.29 ** 1 9. Subitizing Total Score −0.02 −0.10 0.09 0.08 0.04 0.09 −0.05 −0.05 1 10. Number Line Estimation Total Score 0.41 ** 0.17 * 0.10 0.30 ** 0.42 * 0.30 ** 0.29 ** 0.33 ** −0.004 1 11. Proportional Reasoning Total Score 0.19 * 0.18 * −0.10 −0.04 0.13 0.14 * 0.24 ** 0.19 ** −0.01 0.40 ** 1 12. Panamath Percent Correct 0.45 ** 0.29 ** −0.01 0.20 * 0.16 0.25 * 0.19 ** 0.38 ** −0.40 ** 0.42 ** 0.33 ** Open in a new tab Note: * p < .05. ** p < .01. MEFS refers to the Minnesota Executive Function Scale. FAM refers to the Flexible Attention to Magnitudes task. Results of multiple regression analyses are presented in Table 3 . Model 1 examined the associations between FAM ability and math achievement controlling for specific math skills, EF, language, and demographics. In this model, FAM skill remained a significant predictor of math achievement, above and beyond the effect of the other specific math skills. Given that there was a subset of children who employed a single-dimension strategy on the FAM task, suggesting that their performance may not fully represent their FAM skill but potentially a precursor to FAM skill, we computed a second model, replicating Model 1 but without the children who employed a single-dimension strategy. There are several unique findings of note. First, the effect size for parent education was larger than in the previous model. Second, the subitizing assessment emerged as a significant predictor of math achievement. Finally, the Panamath effect size was smaller, half the size of the effect for the other model with Panamath as a predictor. However, FAM skill remained a significant predictor of math achievement. Table 3. Associations between FAM Ability and Math Achievement Model 1 Model 2 Age −.17(.05) ** −.22(.06) ** Gender .09(.05) † .12(.05) * Parent Education .13(.08) † .30(.09) ** Family Income .01(.08) −.07(.09) Picture Vocab Standard Score .44(.06) ** .34(.07) ** MEFS Standard Score .05(.05) .01(.06) FAM Post + Mixed Trial Accuracy .11(.05) * .12(.06) * Subitizing Total Score .04(.06) .20(.07) ** Number Line Estimation Total Score .08(.06) .06(.07) Proportional Reasoning Total Score .01(.06) −.02(.06) Panamath Percent Correct .30(.07) ** .14(.07) * Open in a new tab Notes: Estimates are standardized betas, with standard error in parentheses. * p < .05. ** p < .01. † p < .10. Models 1 and 2 include the same predictors, but Model 1 includes all children (n = 226), while Model 2 only includes children who did not exhibit evidence of using a single-dimension strategy on the FAM task (n = 194). We also examined a model additionally controlling for FAM form (size first or number first) and found the same results, but the model output produced a warning and is therefore not presented here. In this table, MEFS refers to the Minnesota Executive Function Scale. FAM refers to the Flexible Attention to Magnitudes task. Discussion Understanding how a diverse sample of preschool aged children attend to and flexibly shift between both numerical and spatial magnitudes has theoretical and practical importance for understanding the early development of math skills. We modified an existing FAM task ( Fuhs et al., 2021 ) in this study to better understand how children’s FAM task performance represents their ability to flexibly shift between dimensions of magnitude versus capturing only children’s ability to shift between different values within the same magnitude or their ability to simply attend to numerical magnitude alone. We accomplished this by incorporating check trials into each trial level and conducting trial-by-trial analyses of the mixed trial level. On check trials, numerical and spatial magnitude were congruent across object sets. We found that children’s performance on check trials was positively significantly correlated with their performance on test trials, suggesting that children who got more check trials correct also got more test items correct. If children were overall using a single-dimension strategy, we would have seen a negative correlation whereby children were choosing, for example, “big” items on pre-switch trials and “small” items on post-switch trials, which would result in higher scores on test trials and lower scores on check trials, but the evidence did not support a negative correlation. However, we did find that a small subset of children showed evidence of using a spatial-only strategy on numerical magnitude test trials. Understanding the circumstances under which children use a single-dimension strategy could be an important indicator of the developmental progression of FAM skill. It could be that using a single-dimension spatial strategy is a precursor to FAM skill that could be leveraged to help children understand the differences between spatial and numerical magnitudes. That this subset of children appeared to use only a spatial-focused single-dimension strategy is in line with recent theories of magnitude understanding development suggesting that children are more attuned to spatial magnitudes over numerical magnitudes in their environment in early childhood, prior to formal math instruction ( Aulet & Lourenco, 2023 ; Leibovich et al., 2017 ). It is possible that the extent to which children are able to use inhibitory control to handle interferences between numerical and non-numerical dimensions of magnitude depends on children’s individual differences in their spontaneous tendency to attend to one dimension of magnitude over another, as is suggested by research on SOMAG (Viarouge et al., 2018). This spontaneous tendency could help explain the propensity of some children to exclusively attend to a single dimension. Importantly, we found that FAM task performance was still a significant predictor of math achievement even when removing the children who used a single-dimension strategy from the prediction model (Model 2). This suggests that prior findings of a significant association between FAM task performance and children’s math achievement were not driven primarily by children who employed a single-dimension strategy. Evidence that children who used a single-dimension strategy may represent a different point on the developmental trajectory of math skills was found in Model 2 where these children were removed from the exploratory model. In Model 2, several of the predictors in the model yielded differing effect sizes compared to previous models. For example, the effect sizes for parental education and subitizing were larger, while the effect size for non-symbolic numerical magnitude comparison skill was noticeably smaller. Future longitudinal research using the FAM task with children across their early childhood years could help delineate the pattern of development to understand why associations differ and could help inform competing theories of early magnitude processing development. Another question one might ask is if the FAM task only captures children’s attention to number and not their ability to shift between different dimensions of magnitude. Indeed, children performed worse on number trials than on size trials regardless of if they completed number trials at pre-switch or post-switch. Theoretically, however, that getting worse from pre- to post-switch regardless of condition is necessary evidence of difficulty switching rests on the premise that the two conditions were equally difficult to start with, which may not be the case. There are studies suggesting that children are generally more attentive to spatial magnitudes early in their development ( Leibovichet al., 2017 , although see Victorsson et al., 2023 for an example of contradictory empirical evidence). Also, by controlling total surface area, the FAM task ends up with even larger ratios for surface area per item (e.g., on some trials with a 1:3 total surface area, the ratio of surface area per item is 1:9), which may contribute to the size trials being “easier” overall. Nonetheless, comparing the mean performance on pre- and post-switch trials of children from each condition suggested that switching to attend to number was more difficult for children than attending to number pre-switch. This finding aligns with the hypothesis that flexibly switching between dimensions of magnitude is uniquely challenging for young children ( Fuhs et al., 2021 ). To further understand the use of the FAM task as a measure of children’s ability to flexibly switch between dimensions of magnitude, a trial-by-trial analysis of the mixed FAM level across FAM conditions was conducted. The results of these analyses suggest that children’s performance on size and number trials within the mixed level is dependent on the order in which they completed size and number trials at pre-switch and post-switch. For instance, children scored significantly better on the mixed level trials that tested the same dimensional focus that they had focused on in their post-switch level when compared to children who focused on that dimension at the pre-switch level. This effect held for both number and size trials (see Figure 3 ). Similarly, when looking only at switch trials in the mixed level, children in both conditions scored higher on trials that switched to the dimension tested in their post-switch trials as compared to switch trials that switched to the dimension tested in their pre-switch trials (see Figure 4). These findings all suggest that it is more challenging for children to flexibly shift to the magnitude they attended to in the pre-switch level than to flexibly shift to the magnitude they most recently attended to in the post-switch trial level. In other words, rather than simply assessing children’s ability to switch between values within a dimension or to simply attend to numerical magnitude dimensions alone, our results suggest that children’s FAM task performance is related specifically to their ability to flexibly shift between numerical and spatial magnitudes. The results of correlations with the FAM task were consistent with our hypothesis that FAM skill will be related to other math assessments that require children to attend flexibly to both numerical and spatial magnitudes. Finding that FAM skill was not significantly correlated with subitizing skill, which requires simply identifying the numerical total of a set of dots that are identical, was thus expected. Also consistent with our expectations, we found that correlations between FAM skill and EF skills, number line estimation, proportional reasoning, and non-symbolic numerical magnitude comparison skills were significant with a small to medium effect size. Overall, these correlational results help to establish construct validity for the FAM task as a measure of FAM skill that is unique from both EF and other math skills, but significantly correlated with related constructs. Overall, results revealed a significant association between children’s FAM task performance and their math achievement, controlling for specific early math skills. One of the unexpected findings in the regression analyses was that children’s performance on non-symbolic numerical magnitude comparison was especially strongly correlated with children’s math achievement in this study compared to other studies recruiting heterogeneous populations, including children from low-income and minoritized communities ( Fuhs & McNeil, 2013 ; Fuhs et al., 2021 ; Gilmore et al., 2013 ). There are several possible explanations for this. First, it could be related to the measure used. We found differences between children who completed the entire Panamath measure versus children who could not complete the measure, as we found that children who did not complete the measure were younger, more likely to be male, and scored significantly lower on EF and math skills generally. We used FIML to utilize all available data in our primary analyses to reduce bias due to missingness, but it is still possible that we were not capturing information relevant to this particular subgroup of children who could not complete the measure. Had we been able to get complete data from those children, the findings could potentially have looked more similar to prior studies with heterogeneous populations. Interestingly, for the children who did complete the non-symbolic numerical magnitude comparison task, their scores did not significantly differ across different congruent and incongruent trial types, which is also different from prior findings with heterogeneous preschool populations using other measures (e.g., Fuhs & McNeil, 2013 , Gilmore et al., 2013 ). Another potential reason for the differing findings is that there was a cohort effect, such that children who participated in this study experienced the significant impacts of the COVID-19 pandemic while prior cohorts did not. It is possible that differing prior experiences at preschool and home for this cohort could have played a role in the findings. Nevertheless, the results indicated that both FAM skill and non-symbolic numerical magnitude comparison skill were significantly correlated, but that each provided unique predictive value for math achievement above and beyond the impact of the other variables. These findings also suggest particular promise for FAM skill instruction to impact both children’s non-symbolic numerical comparison skills as well as FAM skill and math achievement. Limitations and Future Directions There are still several questions about FAM skill and its relationship to math achievement. One of these questions relates to the limited information on the efficacy of FAM skill interventions and their possible effects on early math development in children. This study does not answer this question, instead it adds to the research base on the correlation between FAM skill and math achievement, creating a foundation for future research to explore this question. Second, because this study was correlational, we were unable to determine if there is a causal effect between FAM skill and math achievement. Future research experiments should be designed to test the causality of the correlation between FAM and math achievement. Because these data were collected as part of a longitudinal study of preschool experiences, we administered the assessments in a fixed order, which may have impacted the missingness for the non-symbolic numerical magnitude comparison task as it was administered at the end of a short battery of assessments. Testing for order effects in future counterbalanced designs will help to better understand how children’s non-symbolic numerical magnitude comparison skills relate to math achievement in children from heterogeneous backgrounds. As a cross-sectional study, we were not able to address the development of children’s FAM ability over time. The current study suggests that at ages 3 to 5 in this particular population, children may have more familiarity with spatial magnitudes than numerical magnitudes. However, it is possible that the ease with which children attend to each dimension of magnitude may change over time as young children gain more exposure to and familiarity with numerical magnitudes in a more formal schooling setting. Future studies should consider a longitudinal design that can measure children’s performance on size and number questions as children gain more consistent exposure to numerical magnitudes in school to address how children’s familiarity with and understanding of different dimensions affects their ability flexibly shift between dimensions in more complex math contexts. This study, and others like it, specifically focused on understanding the development of children from diverse backgrounds, both racially/ethnically and in terms of family income, add to the heterogeneity of populations that are sampled for research on the development of early math skills. Researchers have recently published several important calls for diversifying early childhood math and cognitive development research (e.g., Miller-Cotto et al., 2021 ; Prather, 2022 ; Prather et al., 2022 ). Moving away from convenience samples and towards samples that reflect the real communities in which we operate as researchers offers important benefits for translational research. It is certainly the case that this type of research involves increased efforts on the part of researchers to develop ongoing, reciprocal partnerships with children and families in the local community, to develop research agendas that align with community priorities, and it also often requires researchers to meet families where they spend their time rather than having them come to a laboratory setting. However, the benefits of increasing the diversity of samples in cognitive development research and understanding the particular developmental patterns and assets of children and families from low-income and/or minoritized communities is a crucial part of ensuring that cognitive development research is situated in context and has relevance for practice across diverse populations. Summary Math skills are critical for children’s academic success ( Hanushek & Woessmann, 2010 ). However, difficulties in math are often evident early in children’s math skills development ( Jordan & Levine, 2009 ). The FAM account suggests that FAM skill is an important early math skill that is correlated with math achievement, but, for young children, flexibly shifting of attention between spatial and numerical magnitudes is uniquely difficult. We found FAM skill was predictive of math achievement controlling for other early math skills (EF, number line estimation, proportional reasoning, subitizing, non-symbolic numerical magnitude discrimination) as well as demographic and language covariates. Interestingly, we also found that a small group of children showed evidence of using a single-dimension spatial-only strategy on numerical magnitude test trials. Understanding why this subset of children used a single-dimension strategy could aid in the understanding of the developmental progression of FAM skill in future studies. Examining the development of FAM skill using instructional interventions is an avenue of future research that could provide practical applications for early childhood education. Acknowledgements We are grateful for the children, families, and teachers who made this work possible. This research was supported by NIH R15HD100936 awarded to Mary Wagner (formerly Fuhs). 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