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Do Magnitude Knowledge Gains Mediate Outcomes of a Kindergarten Mathematics Intervention?

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Title: Do Magnitude Knowledge Gains Mediate Outcomes of a Kindergarten Mathematics Intervention?
Language: English
Authors: Taylor Lesner (ORCID 0000-0003-2797-1491), Marah Sutherland (ORCID 0000-0002-6108-8515), Madison Cook, Emily Wilke (ORCID 0009-0003-5985-8490), Keith Smolkowski (ORCID 0000-0003-2565-3297), Ben Clarke (ORCID 0000-0002-1021-1886)
Source: Grantee Submission. 2025.
Peer Reviewed: Y
Page Count: 27
Publication Date: 2025
Sponsoring Agency: Institute of Education Sciences (ED)
Contract Number: R324A120304
Document Type: Reports - Research
Education Level: Early Childhood Education
Elementary Education
Kindergarten
Primary Education
Descriptors: Kindergarten, Mathematics Education, Knowledge Level, Outcomes of Education, Number Concepts, Numeracy, Intervention, Achievement Gains, Mathematics Achievement, Elementary School Students, Numbers, Student Attitudes
DOI: 10.1177/09388982251321536
Abstract: Understanding numerical magnitude is critical to the development of mathematics proficiency, and math interventions targeting magnitude knowledge have been shown to improve outcomes for students with math learning difficulties across grade levels. While recent studies have found that growth in magnitude knowledge mediates fractions intervention outcomes, similar analyses have not been conducted in the context of early numeracy interventions. This secondary analysis examined whether and to what degree gains in magnitude understanding explained overall intervention gains for kindergarten students (n = 1,251) receiving a validated Tier 2 early math intervention, ROOTS. Results of an indirect-effects mediation analysis were consistent with a partial mediation effect across proximal and distal measures of mathematics proficiency. Findings add to a growing literature base highlighting the importance of magnitude understanding to overall mathematics proficiency in the early grades, and indicate that whole number magnitude understanding may be an "active ingredient" driving early mathematics intervention outcomes. [This paper will be published in "Learning Disabilities Research & Practice."]
Abstractor: As Provided
IES Funded: Yes
Entry Date: 2025
Accession Number: ED670981
Database: ERIC
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  Value: <anid>AN0184489649;z2z01may.25;2025Apr17.01:46;v2.2.500</anid> <title id="AN0184489649-1">Do Magnitude Knowledge Gains Mediate the Outcomes of a Kindergarten Mathematics Intervention? </title> <p>Understanding numerical magnitude is critical to the development of mathematics proficiency. Math interventions targeting magnitude knowledge have been shown to improve outcomes for students with math learning difficulties across grade levels. However, while recent studies have found that growth in magnitude knowledge mediates fractions intervention outcomes, similar analyses have not been conducted in the context of early numeracy interventions. This secondary analysis examined whether, and to what degree, gains in magnitude understanding explained overall intervention gains for kindergarten students (N = 1,251) receiving a validated Tier 2 early math intervention, ROOTS. Results of an indirect-effects mediation analysis were consistent with a partial mediation effect across proximal and distal measures of mathematics proficiency. Findings add to a growing literature base highlighting the importance of magnitude understanding to overall mathematics proficiency in the early grades, and indicate that whole number magnitude understanding may be an "active ingredient" driving early mathematics intervention outcomes.</p> <p>Keywords: magnitude knowledge; mediation; early mathematics intervention</p> <hd id="AN0184489649-2">Introduction</hd> <p>Magnitude understanding refers to the ability to comprehend the size of numbers represented both symbolically (i.e., with numerals) and non-symbolically (e.g., with dots or objects). Students with strong magnitude understanding demonstrate knowledge of the exact magnitude of numbers, and are able to estimate approximate magnitude and compare the magnitudes of two or more numbers ([<reflink idref="bib14" id="ref1">14</reflink>]; [<reflink idref="bib25" id="ref2">25</reflink>]). Magnitude knowledge is typically assessed using either number line tasks, in which children estimate the approximate location of a given quantity on the number line, or magnitude comparison tasks, in which children are asked to select the greater (or lesser) of two quantities ([<reflink idref="bib1" id="ref3">1</reflink>]; [<reflink idref="bib10" id="ref4">10</reflink>]; [<reflink idref="bib32" id="ref5">32</reflink>]).</p> <p>Magnitude understanding is one of several domain-specific skills critical to the development of overall mathematics proficiency, with prior research linking magnitude processing skills to early numeracy skill development, whole number and fractions arithmetic, numerical working memory, and broad math achievement ([<reflink idref="bib14" id="ref6">14</reflink>]; [<reflink idref="bib24" id="ref7">24</reflink>]; [<reflink idref="bib31" id="ref8">31</reflink>]). Siegler and colleagues ([<reflink idref="bib35" id="ref9">35</reflink>]; [<reflink idref="bib38" id="ref10">38</reflink>]) propose an integrated theory of numerical development in which understanding of numerical magnitude in relation to a mental number line is the key driver of overall mathematical development across the lifespan, such that representations of numerical magnitude become more precise and are applied to a broader range of numbers (e.g., larger whole numbers, fractions, negative numbers) over time. Thus, magnitude understanding plays an important role in understanding number systems, from the whole number system in early numeracy to more complex number systems in the later grades.</p> <p>Critically, prior research indicates that students with or at risk for mathematics learning difficulties or disabilities (MLD) tend to show deficits in understanding and representing numerical magnitude compared to their typically achieving peers ([<reflink idref="bib17" id="ref11">17</reflink>]; [<reflink idref="bib18" id="ref12">18</reflink>]). Specifically, a literature review ([<reflink idref="bib11" id="ref13">11</reflink>]) and meta-analysis ([<reflink idref="bib33" id="ref14">33</reflink>]) both indicate that children with MLD demonstrate consistently weaker performance than their grade-level peers on magnitude comparison tasks, particularly when tasks represent quantities symbolically (i.e., with numerals). Further, in longitudinal studies examining kindergarten early numeracy measures and later math outcomes, poor performance on symbolic magnitude comparison tasks has emerged as one of the strongest, most consistent predictors of mathematics difficulties in later years ([<reflink idref="bib4" id="ref15">4</reflink>]; [<reflink idref="bib45" id="ref16">45</reflink>]).</p> <p>Together, these findings suggest that difficulty linking symbolic representations of number to numerical meaning is a hallmark deficit of MLD and may be an essential early indicator of mathematics difficulties (e.g., [<reflink idref="bib29" id="ref17">29</reflink>]; [<reflink idref="bib33" id="ref18">33</reflink>]). Therefore, building magnitude knowledge is also likely to be an important instructional target for children with or at risk for MLD.</p> <hd id="AN0184489649-3">Importance of Magnitude Understanding to Whole Number Proficiency</hd> <p>Understanding of numerical magnitude is a critical construct in the early elementary grades as it relates to proficiency with whole number skills ([<reflink idref="bib31" id="ref19">31</reflink>], [<reflink idref="bib32" id="ref20">32</reflink>]). [<reflink idref="bib31" id="ref21">31</reflink>] conducted a meta-analysis examining magnitude comparison tasks, synthesizing 284 effect sizes across 17,201 participants. They found that correlations between magnitude comparison tasks and students' overall mathematics competence ranged from <emph>r</emph> =.30 for symbolic magnitude comparison tasks (i.e., those using numerals) to <emph>r</emph> =.24 for non-symbolic tasks (i.e., those using dots or drawings of objects). Mathematics competence was assessed using a variety of measures, including diagnostic measures such as the Test of Early Mathematics Achievement-Third Edition (TEMA-3; [<reflink idref="bib19" id="ref22">19</reflink>]), curriculum-based measures (CBMs), and written or mental arithmetic tasks. In light of their findings, the authors concluded that future research is needed to better understand how magnitude knowledge relates to intervention response, for example, by exploring whether gains in magnitude comparison performance explain broader intervention-related gains.</p> <p>The early elementary school years are thought to be a critical period for development of whole number magnitude understanding. [<reflink idref="bib36" id="ref23">36</reflink>] found that children's accuracy in placing numbers on a number line improved substantially from kindergarten to second grade, and that first- and second-grade students were more accurate in their magnitude estimates when placing multiple numbers on the same number line. Based on these findings, the authors concluded that magnitude understanding appears to develop in a predictable way across the early grades and can be improved through experience. For example, having opportunities to interact with number line representations, such as during board game play, may lead to improved magnitude knowledge and more accurate magnitude estimates.</p> <p>In line with this hypothesis, researchers have noted that interventions that teach whole number magnitude skills are effective in improving overall mathematics outcomes among students at risk for MLD. For example, [<reflink idref="bib37" id="ref24">37</reflink>] found that playing a board game with numerals represented sequentially on the board led to gains in magnitude estimation accuracy in a sample of preschool students at risk for later learning difficulties (also see [<reflink idref="bib43" id="ref25">43</reflink>]). Broader early mathematics intervention programs designed to build number sense – of which magnitude understanding is a critical component – have also been shown to improve outcomes across a wide range of early numeracy outcomes for students at risk for MLD (e.g., [<reflink idref="bib6" id="ref26">6</reflink>]; [<reflink idref="bib13" id="ref27">13</reflink>]; [<reflink idref="bib39" id="ref28">39</reflink>]). These findings support the importance of magnitude understanding as a focus of early mathematics intervention for students with MLD, and suggest that growth in magnitude understanding may account for broader math intervention gains in the early elementary school years.</p> <hd id="AN0184489649-4">Magnitude Understanding as a Mediator of Math Intervention Outcomes</hd> <p>Given the growing emphasis on magnitude understanding as a critical construct and intervention target, researchers have begun to investigate whether growth in magnitude understanding mediates the outcomes of mathematics intervention among students with MLD. To date, investigations in this area have focused on the later elementary grades. For example, Fuchs and colleagues (2013, 2014) found that for fourth-grade students receiving supplemental fractions intervention, gains on a measure of overall fraction knowledge compared to the control group were partially mediated by gains in magnitude understanding, as measured by a number line estimation task. Jayanthi and colleagues (2021) observed similar results with a sample of fifth-grade students, finding that pre- to post-intervention growth on a general fractions outcome measure was mediated by gains on a number line estimation task.</p> <p>Together, these results lend support to the theory that improving magnitude understanding is essential to overall mathematics proficiency for students with MLD in the upper-elementary grades (i.e., fractions). However, despite the strong literature base suggesting the importance of early magnitude understanding to the development of whole number skills, the extent to which magnitude understanding mediates whole number intervention effects for younger students with or at risk for MLD is largely unknown. Thus, to our knowledge, prior research has not examined gains in magnitude knowledge as a mediator of response to early mathematics interventions focused on whole number skills.</p> <hd id="AN0184489649-5">Present Study</hd> <p>Determining whether, and to what degree, gains in magnitude understanding account for gains in early mathematics skills is necessary to provide further insight into the mechanisms by which validated intervention programs impact mathematics achievement outcomes among students with MLD. The present study sought to fill this gap in the literature by examining whether gains in magnitude understanding, as measured by the Assessing Student Proficiency in Early Number Sense – Magnitude Comparison CBM ([<reflink idref="bib7" id="ref29">7</reflink>]), explain overall intervention gains for kindergarten students with or at risk for MLD in the context of a validated Tier 2 early math intervention, ROOTS ([<reflink idref="bib5" id="ref30">5</reflink>]).</p> <p>This study addressed the following research question:</p> <p></p> <ulist> <item> For kindergarten students receiving the ROOTS intervention, are overall mathematics gains on proximal (i.e., ROOTS Assessment of Early Numeracy Skills; [<reflink idref="bib12" id="ref31">12</reflink>]) and/or distal (i.e., Test of Early Mathematics Achievement-Third Edition; [<reflink idref="bib19" id="ref32">19</reflink>]) outcome measures explained by gains in magnitude comparison skills?</item> </ulist> <hd id="AN0184489649-6">Method</hd> <p>This study was a secondary analysis of data from the ROOTS Efficacy Project (2012-2015), a partially nested, randomized controlled trial to evaluate the impact of ROOTS on kindergarten mathematics outcomes. The sample and procedures for the parent study were reported in detail by [<reflink idref="bib6" id="ref33">6</reflink>] and described briefly here.</p> <hd id="AN0184489649-7">Participants</hd> <p>The sample consisted of students across 23 elementary schools and 138 classes in Oregon and Massachusetts. Kindergarten students with parental consent (<emph>N</emph> = 3,130) were screened in the fall with the Assessing Student Proficiency in Early Number Sense (ASPENS; [<reflink idref="bib7" id="ref34">7</reflink>]) and the Number Sense Brief (NSB; [<reflink idref="bib22" id="ref35">22</reflink>]). Students at risk for MLD (i.e., with ASPENS composite scores in the Strategic or Intensive range and NSB scores of 20 or lower) were eligible to participate in the study. The final sample consisted of 1,251 students randomly assigned to receive ROOTS small-group intervention (<emph>n</emph> = 880) or business-as-usual control (<emph>n</emph> = 371). Table 1 reports district-provided demographic characteristics for the final sample.</p> <p>Table 1. Descriptive Statistics for Student Demographic Characteristics.</p> <p>Graph</p> <p> <ephtml> <table><colgroup><col align="left" /><col align="char" char="(" /><col align="char" char="(" /><col align="char" char="(" /></colgroup><thead><tr><th align="left" /><th align="left">Treatment<italic>N</italic> (%)</th><th align="left">Control<italic>N</italic> (%)</th><th align="left">Total <italic>N</italic> (%)</th></tr></thead><tbody><tr><td>Total sample (<italic>N</italic>)</td><td>880</td><td>371</td><td>1,251</td></tr><tr><td>Sex</td><td /><td /><td /></tr><tr><td> Female</td><td>441 (51)</td><td>186 (50)</td><td>627 (51)</td></tr><tr><td> Male</td><td>425 (49)</td><td>184 (50)</td><td>609 (49)</td></tr><tr><td>Race</td><td /><td /><td /></tr><tr><td> White</td><td>500 (64)</td><td>214 (64)</td><td>714 (64)</td></tr><tr><td> Hispanic</td><td>185 (24)</td><td>79 (24)</td><td>264 (24)</td></tr><tr><td> Black</td><td>45 (6)</td><td>15 (4)</td><td>60 (5)</td></tr><tr><td> Asian</td><td>20 (3)</td><td>12 (4)</td><td>32 (3)</td></tr><tr><td> Multiracial</td><td>19 (2)</td><td>7 (2)</td><td>26 (2)</td></tr><tr><td> American Indian/ Alaskan Native</td><td>11 (1)</td><td>7 (2)</td><td>18 (2)</td></tr><tr><td> Hawaiian/Pacific Islander</td><td>3 (<1)</td><td>0 (0)</td><td>3 (<1)</td></tr><tr><td> First language: Spanish</td><td>258 (34)</td><td>117 (36)</td><td>375 (35)</td></tr></tbody></table> </ephtml> </p> <p>1 <emph>Note.</emph> Sample <emph>n'</emph>s for each variable do not sum to the total sample <emph>n</emph> reported in the top row due to missing responses. Percentages represent the proportion of students for whom each variable was reported. Race and ethnicity categories were mutually exclusive.</p> <hd id="AN0184489649-8">ROOTS Intervention</hd> <p>ROOTS ([<reflink idref="bib5" id="ref36">5</reflink>]) is a small-group, Tier 2 kindergarten mathematics intervention consisting of 50 scripted, 20-min lessons designed to build proficiency with whole number content (i.e., standards from the Counting and Cardinality, Operations and Algebraic Thinking, and Number and Operations in Base Ten strands of the kindergarten Common Core State Standards [CCSS] for Mathematics; CCSS Initiative, 2010). Lessons employ an explicit, systematic instructional architecture ([<reflink idref="bib2" id="ref37">2</reflink>]) with increasingly complex mathematical content building across sessions. Each lesson consists of an opening routine, as well as several activities covering different instructional objectives. Throughout ROOTS activities, students work with a variety of mathematical representations (e.g., ten frames, counters, tally marks, number lines) to build a deep understanding of key concepts.</p> <p>A primary goal of ROOTS is to build students' understanding of number, including numerical magnitude. Given the focus of the present study, the authoring team coded all ROOTS activities to provide an estimate of instructional dosage of magnitude-focused activities. Across the 50 ROOTS lessons, 44% of activities were found to target some aspect of magnitude understanding. Examples of common magnitude-focused activities in ROOTS, including number comparison, sequencing and identifying missing numerals on the number line, and intentional vocabulary instruction, are described below.</p> <p>As part of the opening routine in lessons 16-50, students build magnitude understanding through comparison activities. Initially, each student is given a set of objects to count before being asked to compare the quantity in their set with that of a partner to determine who has more and who has less. In later lessons, this routine becomes more challenging as students are given numeral cards and asked to engage in magnitude comparison using the numerals themselves, rather than sets of objects.</p> <p>Outside of the opening routine, students engage in number comparison within 20 in a variety of ways. When students are first learning the concepts of more and less, they use concrete models to compare quantities. For example, students visually compare cups of teddy bear counters to identify which has more and use linking cubes to build "trains" that are then compared based on length and number of cubes. As the program progresses, concrete models are gradually paired and replaced with numerals. For example, on a guided worksheet, students are asked to circle the greater numeral in each pair, and then use a printed number line to check their work.</p> <p>Students also build magnitude understanding through frequent exposure to and activities involving number line representations. Throughout the ROOTS program, students are often tasked with sequencing numbers they have learned on a number line. In these activities, the teacher guides students to work together to place numeral cards in order on the number line, then check their work by counting along the number line. The teacher then removes one of the numerals, and students must draw upon their knowledge of magnitude and the counting sequence to identify the missing numeral.</p> <p>Finally, the ROOTS intervention addresses magnitude understanding through intentional vocabulary instruction. In early ROOTS lessons, students are introduced to the concepts of more and less. These terms are introduced by the teacher and used by students during comparison activities (e.g., comparing groups of teddy bear counters to determine which has more and which has less). Over time and with teacher support, this evolves to the more formal mathematical language of "greater than" and <emph>"</emph>less than." For example, when given a numeral card, students are asked to verbally compare their numeral to a target numeral following a teacher model (e.g., "Eight is greater than three."). These terms are incorporated throughout subsequent activities, including those that do not specifically target magnitude understanding.</p> <hd id="AN0184489649-9">Intervention Procedures and Interventionists</hd> <p>During the efficacy project, the intervention group received ROOTS from late fall to spring, in addition to core mathematics instruction. Students receiving ROOTS showed significantly greater growth compared to peers in the control condition on a range of proximal and distal math measures (Hedges' <emph>g</emph> effect sizes range from.18-.81; [<reflink idref="bib6" id="ref38">6</reflink>]).</p> <p>ROOTS interventionists were either employees of participating schools or hired by the research team. The majority identified as female (93.5%) and White (76.1%) and reported previous experience delivering small-group instruction (92.3%), with 22.0% holding a teaching license or certification. Interventionists received training and coaching support to ensure high-quality implementation. Fidelity of ROOTS implementation was assessed via direct observation, and data indicated that ROOTS was implemented with fidelity ([<reflink idref="bib6" id="ref39">6</reflink>]).</p> <hd id="AN0184489649-10">Measures</hd> <p>Trained project staff administered three measures of mathematics achievement at pretest and posttest.</p> <hd id="AN0184489649-11">Assessing Student Proficiency in Early Number Sense (ASPENS): Magnitude Comparison</hd> <p>ASPENS ([<reflink idref="bib7" id="ref40">7</reflink>]) is a set of standardized, individually administered, timed CBMs focused on early numeracy skills. Test-retest reliabilities of kindergarten ASPENS measures are in the moderate to high range (.74 to.85, or.74 to.80 for the Magnitude Comparison subtest). The predictive validity of ASPENS scores in the fall of kindergarten with spring scores on the TerraNova 3 ([<reflink idref="bib41" id="ref41">41</reflink>]) ranges from.45–.52 (.45 for the Magnitude Comparison subtest). In the present study, scores on the Magnitude Comparison subtest were utilized as a measure of whole number magnitude knowledge. For this 1-min measure, students were presented with pairs of numerals from 1 to 20 and asked to identify the greater number in each pair. Raw scores representing the total number of correct comparisons are reported in the current study.</p> <hd id="AN0184489649-12">ROOTS Assessment of Early Numeracy Skills (RAENS)</hd> <p>RAENS ([<reflink idref="bib12" id="ref42">12</reflink>]) is a researcher-developed, individually administered, untimed measure of early numeracy skills covered in ROOTS (i.e., items assess knowledge of counting and cardinality, number operations, and the base ten number system). RAENS' predictive validity with commonly used math achievement measures ranges from.68-.83. High interrater agreement (100%) and internal consistency (Cronbach's alpha =.91) have been reported for RAENS ([<reflink idref="bib5" id="ref43">5</reflink>]). In the present study, RAENS was conceptualized as a proximal outcome measure, as it was specifically designed to measure skills taught in the ROOTS intervention using similar language, representations, and problem types.</p> <hd id="AN0184489649-13">Test of Early Mathematics Ability-Third Edition (TEMA-3)</hd> <p>The TEMA-3 ([<reflink idref="bib19" id="ref44">19</reflink>]) is a standardized, norm-referenced, individually administered measure of early mathematics proficiency, assessing skills in counting, comparison, simple problem-solving, and addition/subtraction. The authors report test-retest and alternate-form reliabilities at or above.93. Concurrent validity with other mathematics achievement measures ranges from.54-.91. In the present study, TEMA-3 was conceptualized as a distal outcome measure because it requires greater application and generalization of knowledge and skills targeted by ROOTS to novel problem types and situations.</p> <hd id="AN0184489649-14">Statistical Analysis</hd> <p>We tested whether the effect of condition on RAENS or TEMA gains may have been explained by an indirect (mediated) effect through gains in Magnitude Comparison. [<reflink idref="bib3" id="ref45">3</reflink>] causal steps approach to indirect effects has considerable intuitive appeal but produces biased estimates when variables are measured at only one time point (e.g., [<reflink idref="bib23" id="ref46">23</reflink>]; [<reflink idref="bib26" id="ref47">26</reflink>]). Due to concerns with this model, therefore, we followed the approach recommended by [<reflink idref="bib42" id="ref48">42</reflink>] examining mediation with growth curve models. With only two assessment time points, we estimated pre-post gains over time rather than growth. The model does not use latent variables for growth, so it does not benefit from increased power. However, growth has been shown to be a "natural extension of the observed difference score" ([<reflink idref="bib44" id="ref49">44</reflink>], p. 414), and the analysis of gains is preferred for mediation "when the focus is on within-individual change" ([<reflink idref="bib34" id="ref50">34</reflink>], p. 158). The models, therefore, allow a test of whether the associations are consistent with the hypothesis of mediation. The correlational analyses, however, do not allow temporal or causal inferences. Additionally, we modeled effects for the RAENS and TEMA separately because, in the context of our research question, these measures represent theoretically different constructs (proximal and distal mathematics outcomes, respectively), and gain scores across these measures were only moderately correlated (<emph>r</emph> = 0.43, representing 19% shared variance).</p> <p>The indirect-effects models were estimated in M<emph>plus</emph> 8.4 ([<reflink idref="bib27" id="ref51">27</reflink>]). To address the nonnormality of the sampling distribution for the test of the indirect path from condition through magnitude comparison to RAENS or TEMA, we used bias-corrected bootstrapped confidence intervals ([<reflink idref="bib28" id="ref52">28</reflink>]) based on 5,000 samples. The analyses used full-information maximum likelihood estimation with all available data, which produces potentially unbiased results even in the face of substantial missing data, provided the missing data were missing at random ([<reflink idref="bib30" id="ref53">30</reflink>]), although nonrandom missingness "is often not sufficient to affect the internal validity of an experimental study to any practical extent" ([<reflink idref="bib20" id="ref54">20</reflink>], p. 568). M<emph>plus</emph> reported missing data for each pair of variables in the analysis for each dependent variable (independent variable, mediator baseline and posttest, and dependent variable baseline and posttest); each pair included 90% to 99% of cases. Missing data were primarily the result of student absences during testing sessions and are not believed to violate the missing-at-random assumption, and attrition did not differ by study condition (see [<reflink idref="bib6" id="ref55">6</reflink>], for additional information). The analysis did not nest intervention students within small groups or include covariates. Our sensitivity analyses of impact estimates produced similar intervention effects and standard errors when nested. We report effect sizes using Hedges' <emph>g</emph>.</p> <hd id="AN0184489649-15">Results</hd> <p>We tested whether the effects of condition on gains on the RAENS and TEMA were potentially mediated by gains on the Magnitude Comparison subtest of ASPENS. The direct effect of condition on gains in the RAENS (4.93, 95% CI [4.28, 5.60]) and the TEMA (1.76 [1.08, 2.43]) were statistically significant. These estimates closely match the main effects results from [<reflink idref="bib6" id="ref56">6</reflink>]. Support for mediation hinges on reductions in the direct effects when adding a putative mediator and associated indirect path to the mathematics outcomes into the model.</p> <p>Figure 1 depicts indirect-path model results consistent with Magnitude Comparison as a mediator of intervention gains. When added to the model of the effect of condition on RAENS growth, the indirect path a·b through Magnitude Comparison was statistically significant, 0.63 [0.42, 0.88] (a × b = 6.95 × 0.09 = 0.63). The effect of condition on gains in the RAENS dropped from 4.93 [4.28, 5.60] to 4.28 [3.62, 4.96] and remained statistically significant. The total effect of <emph>g</emph> = 0.81 [0.69, 0.93] decreased, in the mediation model, to a direct effect of 0.71 [0.60, 0.82]. The indirect path represents an effect size of 0.10 [0.07, 0.14].</p> <p>Graph: Figure 1. Indirect-Path Model to Test Mediation of Mathematics Outcomes by Magnitude Comparison.</p> <p>Similarly, the effect of condition on TEMA gains dropped from a total effect of 1.76 [1.08, 2.43] to 0.81 [0.15, 1.48] with the indirect a·b path in the model. Both the direct and indirect path a·b (0.94 [0.68, 1.24]) were statistically significant. The total effect of <emph>g</emph> = 0.23 [0.14, 0.32] decreased to 0.10 [0.02, 0.19] in the mediation model. The effect size for the indirect path was 0.12 [0.09, 0.16].</p> <p>These findings support the partial mediation of intervention gains through gains on Magnitude Comparison. In the mediation model for the RAENS, the indirect effect through Magnitude Comparison accounted for only 13% of the total effect. Support for mediation was stronger for the TEMA, for which the indirect effect accounted for 53% of the total effect. Importantly, mediation models are correlational and cannot support a directional or causal interpretation ([<reflink idref="bib23" id="ref57">23</reflink>]).</p> <hd id="AN0184489649-16">Discussion</hd> <p>The purpose of this study was to investigate whether, and to what degree, gains in magnitude understanding mediate whole number intervention gains for kindergarteners with or at risk for MLD. Results indicated that magnitude understanding partially mediated ROOTS intervention gains on both a proximal and distal outcome measure (the RAENS and TEMA-3, respectively). The study expands upon previous literature supporting magnitude knowledge as a mediator of math intervention outcomes in the later elementary grades (e.g., [<reflink idref="bib15" id="ref58">15</reflink>], [<reflink idref="bib16" id="ref59">16</reflink>]; [<reflink idref="bib21" id="ref60">21</reflink>]), highlighting the critical relationship between magnitude understanding and early mathematical development. Together, these findings indicate that magnitude understanding plays a key role in developing mathematics proficiency through intervention across grade levels and math content areas (e.g., whole number and rational number understanding) for students with MLD.</p> <p>Results indicated a stronger mediation effect on intervention gains on the more distal measure, TEMA-3. Mediation through Magnitude Comparison appeared to reduce the direct effect of condition more for the TEMA-3 than for the proximal measure, RAENS. This finding aligns with results from a meta-analysis indicating that correlations between magnitude comparison skills and the TEMA-3, a broad measure of early numeracy skills (<emph>r</emph> =.41), were stronger than those between magnitude comparison and more proximal outcome measures (e.g., curriculum-based measures, <emph>r</emph> =.21; arithmetic tasks, <emph>r</emph> =.28; [<reflink idref="bib31" id="ref61">31</reflink>]).</p> <p>In interpreting this finding, it is important to consider the nature of the proximal and distal measures utilized in the present study, which assess early numeracy content in different ways. While RAENS items use wording and representations similar to what students encounter in the ROOTS program, TEMA-3 items require students to generalize whole number knowledge to different types of problems (e.g., those involving time or money). It is possible that strong magnitude understanding, as an essential component of early number sense (e.g., [<reflink idref="bib13" id="ref62">13</reflink>]), facilitates students' ability to flexibly apply their whole number knowledge to novel problem situations encountered on the TEMA-3, resulting in a stronger mediation effect on this distal outcome measure.</p> <hd id="AN0184489649-17">Practical Implications</hd> <p>Our results have several implications for intervention research and school practice. The finding that magnitude understanding partially mediated student intervention gains across outcome measures suggests that students' understanding of whole number magnitude may be a relatively robust and "active ingredient" of student response to early numeracy intervention. In the early math intervention literature, the extent to which evidence-based programs target magnitude understanding varies. As described above, nearly half of ROOTS intervention activities target some aspect of magnitude knowledge, such as sequencing numerals on a number line and making greater than/less than comparisons between sets or numbers. Results of the present study suggest that the explicit focus on magnitude understanding in ROOTS may partially account for student gains from pre- to post-intervention. Given the established relationship between magnitude understanding and mathematics outcomes, further exploration related to the development of magnitude understanding and delivery of related instructional supports is warranted.</p> <p>In considering the essential early numeracy skills that students must master in kindergarten, we would be remiss not to acknowledge the role that other whole number skills, such as counting, simple addition and subtraction, and place value knowledge, play in influencing student outcomes. Given the interrelated nature of whole number skills, future research should investigate the extent to which these areas of mathematics are distinct mediators of student intervention gains, or are captured under the broader umbrella of magnitude understanding skills. For example, counting and knowledge of the number list from 1-10 may be a prerequisite skill to understanding number magnitude, whereas place value understanding within 20 may add the distinct skill of composing and decomposing units of 10, and therefore contribute uniquely to whole number intervention gains. Investigating such questions may lead to a better understanding of how to focus early numeracy intervention programs to yield the greatest impact for students with MLD.</p> <p>Our findings also further support the foundational nature of magnitude understanding to mathematics proficiency in the early grades. Previous research examining magnitude knowledge as a mediator of math intervention outcomes for students with MLD has focused on fractions interventions in the upper-elementary grades, with results supporting partial to full mediation ([<reflink idref="bib15" id="ref63">15</reflink>], [<reflink idref="bib16" id="ref64">16</reflink>]; [<reflink idref="bib21" id="ref65">21</reflink>]). Our results expand on this literature, positioning magnitude understanding as a critical factor in early mathematics development that may set students up for later mathematics success (or failure), in line with the integrated theory of numerical development ([<reflink idref="bib38" id="ref66">38</reflink>]).</p> <p>It is believed that the role of magnitude knowledge as a driver of overall mathematics proficiency may grow stronger over time and as mathematics content becomes increasingly complex. As students build whole number knowledge in the early elementary years, those who lack strong magnitude understanding may be able to compensate via memorization of the counting number list. Regardless of conceptual understanding of the numerical magnitude of the counting numbers, students may utilize a memorized number list to aid them in completing a wide range of early numeracy tasks, from comparing numbers to simple addition and subtraction. However, the centrality of magnitude understanding to overall mathematics proficiency may become more apparent as math content expands beyond whole numbers in later years, when compensation strategies like these become ineffective. For example, when fractions are introduced in the upper-elementary grades, students who lack magnitude understanding may struggle with the lack of a meaningful counting sequence and the fact that all fraction skills (i.e., comparing, arithmetic) require reasoning about magnitude based on the numerator/denominator relationship. The present findings support the importance of targeting magnitude understanding during early mathematics intervention to lay the foundation for students with MLD to be successful with more advanced math content in later years.</p> <hd id="AN0184489649-18">Limitations and Future Directions</hd> <p>When interpreting our findings, it is important to consider the limitations of the present study. First, the model used to test for mediation effects does not allow for true causal conclusions about the relation between growth in magnitude understanding and overall math outcomes. Therefore, while our results suggest that the strong focus on building magnitude understanding in the ROOTS intervention may be linked to student outcomes, future research utilizing designs that allow for causal analysis is needed to corroborate this finding.</p> <p>Second, we utilized a magnitude comparison task to measure magnitude knowledge, whereas previous research on magnitude understanding as a mediator of upper-elementary math intervention outcomes used a number line estimation task. While both measures are commonly used to assess magnitude knowledge (e.g., [<reflink idref="bib31" id="ref67">31</reflink>], [<reflink idref="bib32" id="ref68">32</reflink>]), it is likely that the tasks relate to various math outcome measures to different degrees. Our decision to utilize the magnitude comparison task was informed by research indicating that early numeracy CBMs outperform the number line assessment as screening tools during kindergarten ([<reflink idref="bib40" id="ref69">40</reflink>]). However, future research should explore, compare, and contrast the relation between magnitude understanding and whole number intervention gains using a range of magnitude knowledge and outcome measures.</p> <p>Additionally, with the measures available, we were not able to isolate other key early numeracy skills (e.g., base ten knowledge) that could also contribute to gains on general outcome measures. This makes it difficult to determine the relative importance of magnitude knowledge vs. other potential mediators of ROOTS intervention gains, and points to the necessity of thinking intentionally about how to measure the hypothesized "active ingredients" of programs in intervention research to isolate individual skills.</p> <p>Relatedly, additional research seeking to understand the mechanisms that drive change in whole number-focused early math interventions is needed. While mathematics intervention research has resulted in a number of effective early numeracy intervention programs based on sound instructional design principles and best practice guidelines for teaching students with or at risk for MLD, far less is known about what specific activities and content within a program are most strongly linked to growth. Additionally, school systems have limited resources to provide mathematics intervention, and the mathematics content covered in both the early grades curricular standards and many effective early math intervention programs is broad. Considering these factors, research that builds on the present study to offer additional insight into how effective interventions contribute to gains in mathematics proficiency for students with MLD (e.g., by experimentally manipulating hypothesized intervention active ingredients; testing components or sequences within effective intervention programs; [<reflink idref="bib8" id="ref70">8</reflink>]) may have high practical utility, allowing curriculum developers to streamline intervention programs, and schools to focus their implementation resources on the components of programs that are essential for positive outcomes, while modifying or adapting other components to improve contextual fit.</p> <ref id="AN0184489649-19"> <title> References </title> <blist> <bibl id="bib1" idref="ref3" type="bt">1</bibl> <bibtext> Ansari D. (2008). Effects of development and enculturation on number representation in the brain. 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Learning and Instruction, 60, 29–40. https://doi.org/10.1016/j.learninstruc.2018.11.006</bibtext> </blist> </ref> <ref id="AN0184489649-20"> <title> Footnotes </title> <blist> <bibtext> The authors declared the following potential conflicts of interest with respect to the research, authorship, and/or publication of this article: Ben Clarke is eligible to receive a portion of royalties from the University of Oregon's distribution and licensing of certain ROOTS-based works. Potential conflicts of interest are managed through the University of Oregon's Research Compliance Services. An independent external evaluator and coauthor of this publication completed the research analysis described in the article.</bibtext> </blist> <blist> <bibtext> The author(s) disclosed receipt of the following financial support forThe author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research was supported by the Institute of Education Sciences, U.S. Department of Education, under Grant R324A120304 to the Center on Teaching and Learning at the University of Oregon. The opinions expressed are those of the authors and do not represent views of the institute or the U.S. Department of Education.</bibtext> </blist> <blist> <bibtext> Taylor Lesner https://orcid.org/0000-0003-2797-1491 Marah Sutherland https://orcid.org/0000-0002-6108-8515 Madison Cook https://orcid.org/0000-0003-0195-7587 Emily Wilke https://orcid.org/0009-0003-5985-8490 Keith Smolkowski https://orcid.org/0000-0003-2565-3297</bibtext> </blist> </ref> <aug> <p>By Taylor Lesner; Marah Sutherland; Madison Cook; Emily Wilke; Keith Smolkowski and Ben Clarke</p> <p>Reported by Author; Author; Author; Author; Author; Author</p> </aug> <nolink nlid="nl1" bibid="bib14" firstref="ref1"></nolink> <nolink nlid="nl2" bibid="bib25" firstref="ref2"></nolink> <nolink nlid="nl3" bibid="bib10" firstref="ref4"></nolink> <nolink nlid="nl4" bibid="bib32" firstref="ref5"></nolink> <nolink nlid="nl5" bibid="bib24" firstref="ref7"></nolink> <nolink nlid="nl6" bibid="bib31" firstref="ref8"></nolink> <nolink nlid="nl7" bibid="bib35" firstref="ref9"></nolink> <nolink nlid="nl8" bibid="bib38" firstref="ref10"></nolink> <nolink nlid="nl9" bibid="bib17" firstref="ref11"></nolink> <nolink nlid="nl10" bibid="bib18" firstref="ref12"></nolink> <nolink nlid="nl11" bibid="bib11" firstref="ref13"></nolink> <nolink nlid="nl12" bibid="bib33" firstref="ref14"></nolink> <nolink nlid="nl13" bibid="bib45" firstref="ref16"></nolink> <nolink nlid="nl14" bibid="bib29" firstref="ref17"></nolink> <nolink nlid="nl15" bibid="bib19" firstref="ref22"></nolink> <nolink nlid="nl16" bibid="bib36" firstref="ref23"></nolink> <nolink nlid="nl17" bibid="bib37" firstref="ref24"></nolink> <nolink nlid="nl18" bibid="bib43" firstref="ref25"></nolink> <nolink nlid="nl19" bibid="bib13" firstref="ref27"></nolink> <nolink nlid="nl20" bibid="bib39" firstref="ref28"></nolink> <nolink nlid="nl21" bibid="bib12" firstref="ref31"></nolink> <nolink nlid="nl22" bibid="bib22" firstref="ref35"></nolink> <nolink nlid="nl23" bibid="bib41" firstref="ref41"></nolink> <nolink nlid="nl24" bibid="bib23" firstref="ref46"></nolink> <nolink nlid="nl25" bibid="bib26" firstref="ref47"></nolink> <nolink nlid="nl26" bibid="bib42" firstref="ref48"></nolink> <nolink nlid="nl27" bibid="bib44" firstref="ref49"></nolink> <nolink nlid="nl28" bibid="bib34" firstref="ref50"></nolink> <nolink nlid="nl29" bibid="bib27" firstref="ref51"></nolink> <nolink nlid="nl30" bibid="bib28" firstref="ref52"></nolink> <nolink nlid="nl31" bibid="bib30" firstref="ref53"></nolink> <nolink nlid="nl32" bibid="bib20" firstref="ref54"></nolink> <nolink nlid="nl33" bibid="bib15" firstref="ref58"></nolink> <nolink nlid="nl34" bibid="bib16" firstref="ref59"></nolink> <nolink nlid="nl35" bibid="bib21" firstref="ref60"></nolink> <nolink nlid="nl36" bibid="bib40" firstref="ref69"></nolink>
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  Data: <searchLink fieldCode="DE" term="%22Kindergarten%22">Kindergarten</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematics+Education%22">Mathematics Education</searchLink><br /><searchLink fieldCode="DE" term="%22Knowledge+Level%22">Knowledge Level</searchLink><br /><searchLink fieldCode="DE" term="%22Outcomes+of+Education%22">Outcomes of Education</searchLink><br /><searchLink fieldCode="DE" term="%22Number+Concepts%22">Number Concepts</searchLink><br /><searchLink fieldCode="DE" term="%22Numeracy%22">Numeracy</searchLink><br /><searchLink fieldCode="DE" term="%22Intervention%22">Intervention</searchLink><br /><searchLink fieldCode="DE" term="%22Achievement+Gains%22">Achievement Gains</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematics+Achievement%22">Mathematics Achievement</searchLink><br /><searchLink fieldCode="DE" term="%22Elementary+School+Students%22">Elementary School Students</searchLink><br /><searchLink fieldCode="DE" term="%22Numbers%22">Numbers</searchLink><br /><searchLink fieldCode="DE" term="%22Student+Attitudes%22">Student Attitudes</searchLink>
– Name: DOI
  Label: DOI
  Group: ID
  Data: 10.1177/09388982251321536
– Name: Abstract
  Label: Abstract
  Group: Ab
  Data: Understanding numerical magnitude is critical to the development of mathematics proficiency, and math interventions targeting magnitude knowledge have been shown to improve outcomes for students with math learning difficulties across grade levels. While recent studies have found that growth in magnitude knowledge mediates fractions intervention outcomes, similar analyses have not been conducted in the context of early numeracy interventions. This secondary analysis examined whether and to what degree gains in magnitude understanding explained overall intervention gains for kindergarten students (n = 1,251) receiving a validated Tier 2 early math intervention, ROOTS. Results of an indirect-effects mediation analysis were consistent with a partial mediation effect across proximal and distal measures of mathematics proficiency. Findings add to a growing literature base highlighting the importance of magnitude understanding to overall mathematics proficiency in the early grades, and indicate that whole number magnitude understanding may be an "active ingredient" driving early mathematics intervention outcomes. [This paper will be published in "Learning Disabilities Research & Practice."]
– Name: AbstractInfo
  Label: Abstractor
  Group: Ab
  Data: As Provided
– Name: CodeSource
  Label: IES Funded
  Group: SrcInfo
  Data: Yes
– Name: DateEntry
  Label: Entry Date
  Group: Date
  Data: 2025
– Name: AN
  Label: Accession Number
  Group: ID
  Data: ED670981
PLink https://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=eric&AN=ED670981
RecordInfo BibRecord:
  BibEntity:
    Identifiers:
      – Type: doi
        Value: 10.1177/09388982251321536
    Languages:
      – Text: English
    PhysicalDescription:
      Pagination:
        PageCount: 27
    Subjects:
      – SubjectFull: Kindergarten
        Type: general
      – SubjectFull: Mathematics Education
        Type: general
      – SubjectFull: Knowledge Level
        Type: general
      – SubjectFull: Outcomes of Education
        Type: general
      – SubjectFull: Number Concepts
        Type: general
      – SubjectFull: Numeracy
        Type: general
      – SubjectFull: Intervention
        Type: general
      – SubjectFull: Achievement Gains
        Type: general
      – SubjectFull: Mathematics Achievement
        Type: general
      – SubjectFull: Elementary School Students
        Type: general
      – SubjectFull: Numbers
        Type: general
      – SubjectFull: Student Attitudes
        Type: general
    Titles:
      – TitleFull: Do Magnitude Knowledge Gains Mediate Outcomes of a Kindergarten Mathematics Intervention?
        Type: main
  BibRelationships:
    HasContributorRelationships:
      – PersonEntity:
          Name:
            NameFull: Taylor Lesner
      – PersonEntity:
          Name:
            NameFull: Marah Sutherland
      – PersonEntity:
          Name:
            NameFull: Madison Cook
      – PersonEntity:
          Name:
            NameFull: Emily Wilke
      – PersonEntity:
          Name:
            NameFull: Keith Smolkowski
      – PersonEntity:
          Name:
            NameFull: Ben Clarke
    IsPartOfRelationships:
      – BibEntity:
          Dates:
            – D: 01
              M: 02
              Type: published
              Y: 2025
          Titles:
            – TitleFull: Grantee Submission
              Type: main
ResultId 1