Modeling variability in treatment effects for cluster randomized controlled trials using by-variable smooth functions in a generalized additive mixed model.

Variability in treatment effects is common in intervention studies using cluster randomized controlled trial (C-RCT) designs. Such variability is often examined in multilevel modeling (MLM) to understand how treatment effects (TRT) differ based on the level of a covariate (COV), called TRT × COV. In detecting TRT × COV effects using MLM, relationships between covariates and outcomes are assumed to vary across clusters linearly. However, this linearity assumption may not hold in all applications and an incorrect assumption may lead to biased statistical inference about TRT × COV effects. In this study, we present generalized additive mixed model (GAMM) specifications in which cluster-specific functional relationships between covariates and outcomes can be modeled using by-variable smooth functions. In addition, the implementation for GAMM specifications is explained using the mgcv R package (Wood, 2021). The usefulness of the GAMM specifications is illustrated using intervention data from a C-RCT. Results of simulation studies showed that parameters and by-variable smooth functions were recovered well in various multilevel designs and the misspecification of the relationship between covariates and outcomes led to biased estimates of TRT × COV effects. Furthermore, this study evaluated the extent to which the GAMM can be treated as an alternative model to MLM in the presence of a linear relationship.

Connections between Mathematics and Reading Development: Numerical Cognition Mediates Relations between Foundational Competencies and Later Academic Outcomes.

We examined longitudinal relations between 1st-grade cognitive predictors (early nonverbal reasoning, processing speed, listening comprehension, working memory, calculation skill, word-problem solving, word-reading fluency, attentive behavior, and numerical cognition) and 2nd-grade academic outcomes (calculations, word-problem solving, and word reading) in 370 children (M age = 6.55 years, SD age = 0.33 years at the start of the study) who were identified as at-risk or not-at-risk for mathematics disability. Path analysis mediation models revealed that numerical cognition, assessed at an intermediary timepoint, mediated the effects of processing speed, working memory, calculation skill, word-problem solving, and attentive behavior on all 3 outcomes. Findings indicate that multiple early domain-general cognitive abilities are related to later mathematics and reading outcomes and that numerical cognition processes, which may track ease of forming symbol-concept associations, predict later performance across both academic domains.

Closing the Word-Problem Achievement Gap in First Grade: Schema-Based Word-Problem Intervention with Embedded Language Comprehension Instruction.

The main purpose of this study was to test the effects of word-problem intervention, with versus without embedded language comprehension instruction, on at-risk 1st graders' word-problem performance. We also isolated the need for a structured approach to word-problem intervention and tested the efficacy of schema-based instruction at 1st grade. Children (n=391; mean age = 6.53, SD = 0.32) were randomly assigned to 4 conditions: schema-based word-problem intervention with embedded language instruction, the same word-problem intervention but without language comprehension instruction, structured number knowledge intervention without a structured word-problem component, and a control group. Each intervention included 45 30-min sessions. Multilevel models, accounting for classroom and school effects, revealed the efficacy of schema-based word-problem intervention at 1st grade, with both word-problem conditions outperforming the number knowledge condition and the control group. Yet, word-problem performance was significantly stronger for the schema-based condition with embedded language comprehension instruction compared to the schema-based condition without language comprehension instruction. Number knowledge intervention conveyed no word-problem advantage over the control group, even though all 3 intervention conditions outperformed the control group on arithmetic. Results demonstrate the importance of a structured approach to word-problem intervention; the efficacy of schema-based instruction at 1st grade; and the added value of language comprehension instruction within word-problem intervention. Results also provide causal evidence on the role of language comprehension in word-problem solving.

Keys to the Gate? Equal Sign Knowledge at Second Grade Predicts Fourth-Grade Algebra Competence.

Algebraic competence is a major determinant of college access and career prospects, and equal sign knowledge is taken to be foundational to algebra knowledge. However, few studies have documented a causal effect of early equal sign knowledge on later algebra skill. This study assessed whether second-grade students' equal sign knowledge prospectively predicts their fourth-grade algebra knowledge, when controlling for demographic and individual difference factors. Children (N = 177; Mage  = 7.61) were assessed on a battery of tests in Grade 2 and on algebraic knowledge in Grade 4. Second-grade equal sign knowledge was a powerful predictor of these algebraic skills. Findings are discussed in terms of the importance of foregrounding equal sign knowledge to promote effective pedagogy and educational equity.

Prevention: Necessary But Insufficient? A 2-Year Follow-Up of an Effective First-Grade Mathematics Intervention.

We present first-grade, second-grade, and third-grade impacts for a first-grade intervention targeting the conceptual and procedural bases that support arithmetic. At-risk students (average age at pretest = 6.5) were randomly assigned to three conditions: a control group (n = 224) and two variants of the intervention (same conceptual instruction but different forms of practice: speeded [n = 211] vs. nonspeeded [n = 204]). Impacts on all first-grade content outcomes were significant and positive, but no follow-up impacts were significant. Many intervention children achieved average mathematics achievement at the end of third grade, and prior math and reading assessment performance predicted which students will require sustained intervention. Finally, projecting impacts 2 years later based on nonexperimental estimates of effects of first-grade math skills overestimates long-term intervention effects.

Does the Severity of Students' Pre-Intervention Math Deficits Affect Responsiveness to Generally Effective First-Grade Intervention?

The purpose of this analysis was to assess whether effects of 1st-grade mathematics intervention apply across the range of at-risk learners' initial skill levels. Students were randomly assigned to control (n=213) and 2 variants of intervention (n=385) designed to improve arithmetic. Of each 30-minute intervention session (48 over 16 weeks), 25 minutes were identical in the 2 variants, focused on number knowledge that provides the conceptual bases for arithmetic. The other 5 minutes provided non-speeded conceptual practice (n=196) or speeded strategic practice (n=199). Contrasts tested effects of intervention (combined across variants) versus control and effects between the variants. Moderation analysis indicated no significant interactions between at-risk children's pre-intervention mathematics skill and either contrast on any outcome. Across pre-intervention math skill, effects favored intervention over control on arithmetic and transfer to double-digit calculations and number knowledge and favored speeded over non-speeded practice on arithmetic.

Using Moderator Analysis to Identify the First-Grade Children Who Benefit More and Less from a Reading Comprehension Program: A Step Toward Aptitude-by-Treatment Interaction.

Because of the importance of teaching reading comprehension to struggling young readers and the infrequency with which it has been implemented and evaluated, we designed a comprehensive first-grade reading comprehension program. We conducted a component analysis of the program's decoding/fluency (DF) and reading comprehension (COMP) dimensions, creating DF and DF+COMP treatments to parse the value of COMP. Students (N = 125) were randomly assigned to the 2 active treatments and controls. Treatment children were tutored 3 times per week for 21 weeks in 45-min sessions. Children in DF and DF+COMP together performed more strongly than controls on word reading and comprehension. However, pretreatment word reading appeared to moderate these results such that children with weaker beginning word reading across the treatments outperformed similarly low-performing controls to a significantly greater extent than treatment children with stronger beginning word reading outperformed comparable controls. DF+COMP children did not perform better than DF children. Study limitations and implications for research and practice are discussed.

Connections Between Reading Comprehension and Word-Problem Solving via Oral Language Comprehension: Implications for Comorbid Learning Disabilities.

In this article, we discuss the approach adopted within the Vanderbilt University Learning Disabilities Innovation Hub, which focuses on students with higher-order comorbidity: students with concurrent difficulty with reading comprehension and word-problem solving. The aim of the Hub's Research Project is to test what we refer to as the higher-order comorbidity hypothesis: that language comprehension plays a critical role in reading comprehension and word-problem solving. In the Hub's study, we test the hypothesize that language comprehension offers a coordinated approach for improving both outcomes and that this approach thus provides direction for understanding higher-order comorbidity and support for the validity of reading comprehension and word-problem solving comorbidity as a learning disabilities subtyping framework. In the first segment of this article, we describe a model that connects reading comprehension and word-problem solving development via oral language comprehension, and we provide a brief overview of prior related research on these connections. This first section provides the basis for the second segment of this article, in which we discuss the Vanderbilt Hub's innovative approach for investigating these connections. This study tests a theoretically-coordinated framework on students' performance in both high-priority domains of academic development, while exploring effects for boys versus girls and for linguistically diverse learners.

The Mathematical Performance of At-Risk First Graders as a Function of Limited English Proficiency Status.

The purpose of this study was to explore interactions between limited English proficiency (LEP) status, as a function of risk status (low math performance at the start of the school year), on computation and word-problem solving performance. Among 260 1st-grade students, classified as at-risk (AR) or not at-risk (NAR) for math disability, we compared the performance of LEP students to native English-speaking peers. A series of 2-way ANOVAs were conducted on computation and word-problem solving skill at 2 time points, fall and spring of 1st-grade. On fall computation measures, there was no main effect for LEP status and no interaction between LEP and risk status. On spring computation, a main effect for LEP status had emerged, but again no interaction. By contrast, on fall word-problem solving, there was an interaction between LEP and risk status; however, this interaction was no longer significant by spring. Results suggest that language proficiency is an important factor in the development of computation and word-problem solving skill. Implications for future research are discussed.

Text Comprehension and Oral Language as Predictors of Word-Problem Solving: Insights into Word-Problem Solving as a Form of Text Comprehension.

This study was designed to deepen insights on whether word-problem (WP) solving is a form of text comprehension (TC) and on the role of language in WPs. A sample of 325 second graders, representing high, average, and low reading and math performance, was assessed on (a) start-of-year TC, WP skill, language, nonlinguistic reasoning, working memory, and foundational skill (word identification, arithmetic) and (b) year-end WP solving, WP-language processing (understanding WP statements, without calculation demands), and calculations. Multivariate, multilevel path analysis, accounting for classroom and school effects, indicated that TC was a significant and comparably strong predictor of all outcomes. Start-of-year language was a significantly stronger predictor of both year-end WP outcomes than of calculations, whereas start-of-year arithmetic was a significantly stronger predictor of calculations than of either WP measure. Implications are discussed in terms of WP solving as a form of TC and a theoretically coordinated approach, focused on language, for addressing TC and WP-solving instruction.

Does Initial Learning about the Meaning of Fractions Present Similar Challenges for Students with and without Adequate Whole-Number Skill?

The purposes of this study were to (a) explore whether early fractions understanding at 4th grade is differentially challenging for students with versus without adequate whole-number competence and (b) identify specific whole-number skill associated with difficulty in fractions understanding. Based on initial whole-number competence, 1,108 4th graders were classified as having (a) adequate whole-number competence (n = 775), (b) less severe whole-number difficulty (n = 201), and (c) severe whole-number difficulty (n = 132). At the end of 4th grade, they were assessed on fractions understanding and further classified as with versus without difficulty in fractions understanding. Multi-level logistic regression indicated that compared to students with adequate whole-number competence, those with less severe whole-number difficulty were almost 5 times as likely to experience difficulty with fractions understanding whereas those with severe whole-number difficulty were about 32 times as likely to experience difficulty with fractions understanding. Students with severe whole-number difficulty were about 7 times as likely to experience difficulty with fractions understanding compared to those with less severe whole-number difficulty. Among students with adequate whole-number competence, the pretest whole-number skill distinguishing those with versus without difficulty in fractions understanding was basic division facts (i.e., 2-digit dividend ÷ 1-digit divisor) and simple multiplication (i.e., 3-digit × 1-digit without regrouping). The role of whole-number competence in developing initial fractions understanding and implications for instruction are discussed.

Measuring arithmetic: A psychometric approach to understanding formatting effects and domain specificity.

Using multitrait, multimethod data, and confirmatory factor analysis, the current study examined the effects of arithmetic item formatting and the possibility that across formats, abilities other than arithmetic may contribute to children's answers. Measurement hypotheses were guided by several leading theories of arithmetic cognition. With a sample of 1314 3rd grade students (age M=103.24 months, SD=5.41 months), Abstract Code Theory, Encoding Complex Theory, Triple Code Theory, and the Exact versus Approximate Calculations Hypothesis were evaluated, using 11 measures of arithmetic with symbolic problem formats (e.g., Arabic numeral and language-based formats) and various problem demands (e.g., requiring both exact and approximate calculations). In general, results provided support for both Triple Code Theory and Encoding Complex Theory. As predicted by Triple Code Theory, arithmetic outcomes with language formatting, Arabic numeral formatting, and estimation demands (across formats) were related but distinct from one another. As predicted by Encoding Complex Theory, executive attention was a direct predictor of all arithmetic outcomes. Language was no longer a direct predictor of arithmetic outcomes when executive attention was accounted for in the model; however, a strong and enduring relationship between language and executive attention suggested that language may play a facilitative role in reasoning during numeric processing. These findings have important implications for assessing arithmetic in educational settings and suggest that in addition to arithmetic-focused interventions, interventions targeting executive attention, language, and/or the interplay between them (i.e., internal speech during problem-solving) may be a promising avenues of mathematical problem-solving intervention.

The Role of Domain-General Cognitive Abilities and Decimal Labels in At-Risk Fourth-Grade Students' Decimal Magnitude Understanding.

The purpose of the study was to determine whether individual differences in at-risk 4th graders' language comprehension, nonverbal reasoning, concept formation, working memory, and use of decimal labels (i.e., place value, point, incorrect place value, incorrect fraction, or whole number) are related to their decimal magnitude understanding. Students (n = 127) completed 6 cognitive assessments, a decimal labeling assessment, and 3 measures of decimal magnitude understanding (i.e., comparing decimals to the fraction [Formula: see text] benchmark task, estimating where decimals belong on a 0-1 number line, and identifying fraction and decimal equivalencies). Each of the domain-general cognitive abilities predicted students' decimal magnitude understanding. Using place value labels was positively correlated with students' decimal magnitude understanding, whereas using whole-number labels was negatively correlated with students' decimal magnitude understanding. Language comprehension, nonverbal reasoning, and concept formation were positively correlated with students' use of place value labels. By contrast, language comprehension and nonverbal reasoning were negatively correlated with students' use of whole number labels. Implications for the development of decimal magnitude understanding and design of effective instruction for at-risk students are discussed.

Pathways to Third-Grade Calculation Versus Word-Reading Competence: Are They More Alike or Different?

Children (n = 747; 6.5 years) were assessed on domain-general processes and mathematics and reading-related competencies (start of first grade), addition retrieval (end of second grade), and calculations and word reading (end of third grade). Attentive behavior, reasoning, visuospatial memory, and rapid automatized naming (RAN) indirectly contributed to both outcomes, via retrieval. However, there was no overlap in domain-general direct effects on calculations (attentive behavior, reasoning, working memory) versus word reading (language, phonological memory, RAN). Results suggest ease of forming associative relations and abilities engaged during the formation of these long-term memories are common to both outcomes and can be indexed by addition-fact retrieval, but further growth in calculations and word reading is driven by different constellations of domain-general abilities.

Cognitive and numerosity predictors of mathematical skills in middle school.

There is a strong research base on the underlying concomitants of early developing math skills. Fewer studies have focused on later developing skills. Here, we focused on direct and indirect contributions of cognitive measures (e.g., language, spatial skills, working memory) and numerosity measures, as well as arithmetic proficiency, on key outcomes of fraction performance, proportional reasoning, and broad mathematics achievement at sixth grade (N=162) via path analysis. We expected a hierarchy of skill development, with predominantly indirect effects of cognitive factors via number and arithmetic. Results controlling for age showed that the combination of cognitive, number, and arithmetic variables cumulatively accounted for 38% to 44% of the variance in fractions, proportional reasoning, and broad mathematics. There was consistency across outcomes, with more proximal skills providing direct effects and with the effects of cognitive skills being mediated by number and by more proximal skills. Results support a hierarchical progression from domain-general cognitive processes through numerosity and arithmetic skills to proportional reasoning to broad mathematics achievement.

A longitudinal study on predictors of early calculation development among young children at risk for learning difficulties.

The purpose of this study was to explore domain-general cognitive skills, domain-specific academic skills, and demographic characteristics that are associated with calculation development from first grade to third grade among young children with learning difficulties. Participants were 176 children identified with reading and mathematics difficulties at the beginning of first grade. Data were collected on working memory, language, nonverbal reasoning, processing speed, decoding, numerical competence, incoming calculations, socioeconomic status, and gender at the beginning of first grade and on calculation performance at four time points: the beginning of first grade, the end of first grade, the end of second grade, and the end of third grade. Latent growth modeling analysis showed that numerical competence, incoming calculation, processing speed, and decoding skills significantly explained the variance in calculation performance at the beginning of first grade. Numerical competence and processing speed significantly explained the variance in calculation performance at the end of third grade. However, numerical competence was the only significant predictor of calculation development from the beginning of first grade to the end of third grade. Implications of these findings for early calculation instructions among young at-risk children are discussed.

Cognitive Predictors of Calculations and Number Line Estimation with Whole Numbers and Fractions among At-Risk Students.

The purpose of this study was to examine the cognitive predictors of calculations and number line estimation with whole numbers and fractions. At-risk 4th-grade students (N = 139) were assessed on 7 domain-general abilities (i.e., working memory, processing speed, concept formation, language, attentive behavior, and nonverbal reasoning) and incoming calculation skill at the start of 4th grade. Then, they were assessed on whole-number and fraction calculation and number line estimation measures at the end of 4th grade. Structural equation modeling and path analysis indicated that processing speed, attentive behavior, and incoming calculation skill were significant predictors of whole-number calculations whereas language, in addition to processing speed and attentive behavior, significantly predicted fraction calculations. In terms of number line estimation, nonverbal reasoning significantly predicted both whole-number and fraction outcome, with numerical working memory predicting whole-number number line estimation and language predicting fraction number line estimation. Findings are discussed in terms of distinctions between whole-number and fraction development and between calculations and number line learning.

Cognitive and Linguistic Predictors of Mathematical Word Problems With and Without Irrelevant Information.

The purpose of this study was to identify cognitive and linguistic predictors of word problems with versus without irrelevant information. The sample was 701 2nd-grade students who received no specialized intervention on word problems. In the fall, they were assessed on initial arithmetic and word-problem skill as well as language ability, working memory capacity, and processing speed; in the spring, they were tested on a word-problem measure that included items with versus without irrelevant information. Significant predictors common to both forms of word problems were initial arithmetic and word problem-solving skill as well as language and working memory. Nonverbal reasoning predicted word problems with irrelevant information, but not word problems without irrelevant information. Findings are discussed in terms of implications for intervention and future research.

Does the Value of Dynamic Assessment in Predicting End-of-First-Grade Mathematics Performance Differ as a Function of English Language Proficiency?

The purpose of this study was to assess the added value of dynamic assessment (DA) beyond more conventional static measures for predicting individual differences in year-end 1st-grade calculation (CA) and word-problem (WP) performance, as a function of limited English proficiency (LEP) status. At the start of 1st grade, students (129 LEP; 163 non-LEP) were assessed on a brief static mathematics test, an extended static mathematics test, static tests of domain-general abilities associated with CAs and WPs (vocabulary; reasoning), and DA. Near end of 1st grade, they were assessed on CA and WP. Regression analyses indicated that the value of the predictor depends on the predicted outcome and LEP status. In predicting CAs, the extended mathematics test and DA uniquely explained variance for LEP children, with stronger predictive value for the extended mathematics test; for non-LEP children, the extended mathematics test was the only significant predictor. However, in predicting WPs, only DA and vocabulary were uniquely predictive for LEP children, with stronger value for DA; for non-LEP children, the extended mathematics test and DA were comparably uniquely predictive. Neither the brief static mathematics test nor reasoning was significant in predicting either outcome. The potential value of a gated screening process, using an extended mathematics assessment to predict CAs and using DA to predict WPs, is discussed.

Development of fraction concepts and procedures in U.S. and Chinese children.

We compared knowledge of fraction concepts and procedures among sixth and eighth graders in China and the United States. As anticipated, Chinese middle school children had higher knowledge of fraction concepts and procedures than U.S. children in the same grades, and the difference in procedural knowledge was much larger than the difference in conceptual knowledge. Of particular interest, national differences in knowledge of fraction concepts were fully mediated by differences in knowledge of fraction procedures, and differences between the knowledge of Chinese and U.S. children were most pronounced among the lowest achieving children within each country. Based on these and previous results, a theoretical model of the mutually facilitative interaction between conceptual and procedural knowledge of fractions is proposed and discussed.