Comorbid Word Reading and Mathematics Computation Difficulty at Start of First Grade.

The purpose of this analysis was to describe cognitive processes associated with comorbid difficulty between word reading (WR) and mathematics computation (MC) at the start of first grade among children selected for WR and MC delays. A sample of 234 children (mean age 6.50 years, SD = 0.31) was assessed on WR, MC, core cognitive processes (phonological processing, rapid automatized naming, verbal counting [VC]), and domain-general cognitive processes (working memory, oral language, nonverbal reasoning, attentive behavior). Structural equation modeling was used to predict a latent Comorbidity factor, which modeled shared variance between WR and MC, and to identify processes associated with that Comorbidity factor. Results identified each of the core cognitive processes, especially VC, and each of the domain-general cognitive processes, especially working memory, as explaining shared variance between WR and MC. Implications for understanding comorbid difficulty at the start of first grade and designing coordinated first-grade interventions are discussed.

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.

Severe Pandemic Learning Loss and the Promise of Remotely Delivered Intervention in Students With Comorbid Reading and Mathematics Learning Difficulty.

Analyses were conducted with second graders, drawn from an ongoing multi-cohort randomized controlled trial (RCT), who had been identified for RCT entry based on comorbid reading comprehension and word-problem solving difficulty. To estimate pandemic learning loss, we contrasted fall performance for 3 cohorts: fall of 2019 (pre-pandemic; n = 47), 2020 (early pandemic, when performance was affected by the truncated preceding school year; n = 35), and 2021 (later pandemic, when performance was affected by the truncated 2019 to 2020 school year plus the subsequent year's ongoing interruptions; n = 75). Across the 2 years, declines (standard deviations below expected growth) were approximately 3 times larger than those reported for the general population and for students in high-poverty schools. To estimate the promise of structured remote intervention for addressing such learning loss during extended school closures, we contrasted effects in the RCT's 2018 to 2019 cohort (entirely in-person intervention delivery; n = 66) against the same intervention's effects in the 2020 to 2021 cohort (alternating periods of remote and in-person delivery; n = 29). Large intervention effects were not moderated by pandemic status, suggesting potential for structured remote intervention to address student needs during extended school closures.

Modelling multilevel nonlinear treatment-by-covariate interactions in cluster randomized controlled trials using a generalized additive mixed model.

A cluster randomized controlled trial (C-RCT) is common in educational intervention studies. Multilevel modelling (MLM) is a dominant analytic method to evaluate treatment effects in a C-RCT. In most MLM applications intended to detect an interaction effect, a single interaction effect (called a conflated effect) is considered instead of level-specific interaction effects in a multilevel design (called unconflated multilevel interaction effects), and the linear interaction effect is modelled. In this paper we present a generalized additive mixed model (GAMM) that allows an unconflated multilevel interaction to be estimated without assuming a prespecified form of the interaction. R code is provided to estimate the model parameters using maximum likelihood estimation and to visualize the nonlinear treatment-by-covariate interaction. The usefulness of the model is illustrated using instructional intervention data from a C-RCT. Results of simulation studies showed that the GAMM outperformed an alternative approach to recover an unconflated logistic multilevel interaction. In addition, the parameter recovery of the GAMM was relatively satisfactory in multilevel designs found in educational intervention studies, except when the number of clusters, cluster sizes, and intraclass correlations were small. When modelling a linear multilevel treatment-by-covariate interaction in the presence of a nonlinear effect, biased estimates (such as overestimated standard errors and overestimated random effect variances) and incorrect predictions of the unconflated multilevel interaction were found.

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.

Comorbidity in Reading Comprehension and Word-Problem Solving Difficulties: Exploring Shared Risk Factors and Their Impact on Language Minority Learners.

The purpose of this study was threefold: to examine unique and shared risk factors of comorbidity for reading comprehension and word-problem solving difficulties, to explore whether language minority (LM) learners are at increased risk of what we refer to as higher order comorbidity (reading comprehension and word-problem solving difficulties), and to examine the profiles of at-risk LM learners compared with at-risk non-LM learners. At-risk (LM n = 70; non-LM n = 89) and not-at-risk (LM n = 44; non-LM n = 114) students were evaluated on foundational academic (word reading, calculation), behavioral (behavioral attention), cognitive (working memory, processing speed, nonverbal reasoning), and language (vocabulary, listening comprehension) measures in English. Results indicated listening comprehension was the only shared risk factor for higher order comorbidity. Furthermore, LM learners were 3 times more likely to be identified as at risk compared with non-LM learners. Finally, among at-risk learners, no differences were found on cognitive dimensions by language status, but LM learners had lower reading and listening comprehension skills than non-LM learners, with a relative advantage in behavioral attention. Results have implications for understanding higher order comorbidity and for developing methods to identify and intervene with higher order comorbidity among the growing population of LM learners.

Bringing Data-Based Individualization to Scale: A Call for the Next-Generation Technology of Teacher Supports.

The purpose of this narrative synthesis of the curriculum-based measure (CBM) instructional utility literature is to deepen insight into the supports required to enrich teachers' instructional decision-making within curriculum-based measure-data-based individualization (CBM-DBI) in ways that enhance the learning outcomes of students with intensive intervention needs, including students with learning disabilities. We begin by summarizing a recent meta-analysis of CBM-DBI studies focused on academic outcomes. We then reconsider studies from that meta-analysis to further explore the supports required to enrich teachers' instructional decision-making within CBM-DBI and improve student learning. We next draw conclusions and propose a renewed program of instructional utility CBM-DBI research for capitalizing on technology's potential to enhance productive instructional decision-making for students who require intensive intervention, fulfill DBI's potential, and bring CBM-DBI to scale.

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.

Language-Related Longitudinal Predictors of Arithmetic Word Problem Solving: A Structural Equation Modeling Approach.

We investigated the longitudinal relations between cognitive skills, specifically language-related skills, and word-problem solving in 340 children (6.10 to 9.02 years). We used structural equation modeling to examine whether word-problem solving, computation skill, working memory, nonverbal reasoning, oral language, and word reading fluency measured at second grade were associated with performance on measures of word-problem solving in fourth grade. Results indicated that prior word-problem solving, computation skill, nonverbal reasoning, and oral language were significantly associated with children's later word-problem solving. Multi-group modeling suggested that these relations were not significantly different for boys versus girls. Implications of these findings are discussed.

Is "Response/No Response" Too Simple a Notion for RTI Frameworks? Exploring Multiple Response Types With Latent Profile Analysis.

We conducted a secondary analysis of data from a randomized control trial to explore this question: Does "response/no response" best characterize students' reactions to a generally efficacious first-grade reading program, or is a more nuanced characterization necessary? Data were collected on 265 at-risk readers' word reading prior to and immediately following program implementation in first grade and in spring of second grade. Pretreatment data were also obtained on domain-specific skills (letter knowledge, decoding, passage comprehension, language) and domain-general skills (working memory, non-verbal reasoning). Latent profile analysis of word reading across the three time points with controls as a local norm revealed a strongly responsive group (n = 45) with mean word-reading z scores of 0.25, 1.64, and 1.26 at the three time points, respectively; a mildly responsive group (n = 109), z scores = 0.30, 0.47, and 0.55; a mildly non-responsive group (n = 90), z scores = -0.11, -0.15, and -0.55; and a strongly non-responsive group (n = 21), z scores = -1.24, -1.26, and -1.57. The two responsive groups had stronger pretreatment letter knowledge and passage comprehension than the two non-responsive groups. The mildly non-responsive group demonstrated better pretreatment passage comprehension than the strongly non-responsive group. No domain-general skill distinguished the four groups. Findings suggest response to early reading intervention was more complicated than response/no response, and pretreatment reading comprehension was an important predictor of response even with pretreatment word reading controlled.

Sight word acquisition in first grade students at risk for reading disabilities: an item-level exploration of the number of exposures required for mastery.

The purpose of this study was to examine word learning efficiency in at-risk first grade students (N = 93) participating in a yearlong study evaluating a multicomponent intervention targeting word reading and decoding skills. As part of each intervention lesson, students participated in a 1 to 3-min sight word reading activity in which high-frequency words were read from a list until mastered, at which point the word dropped off the list. This study explored factors predicting the number of exposures required for item reading mastery (N = 145 words). Specifically, we explored how the number of word exposures required to reach mastery varied as a function of linguistic features of the words and cognitive characteristics of the students. Using item-level crossed-random effects models, we found students required an average of 5.65 exposures for mastery, with word features representing word length, vocabulary grade, and imageability being significant predictors of learning efficiency. We also found a significant interaction between pretest word reading skill and imageability of a word, with this semantic feature being especially important for the poorest readers. Results indicate that in the absence of typical word recognition skills, poor readers tend to rely on other sources of information to learn words, which tend to be related to the semantic features of words.

Do the Processes Engaged During Mathematical Word-Problem Solving Differ Along the Distribution of Word-Problem Competence?

Conventional research methods for understanding sources of individual differences in word-problem solving (WPS) only permit estimation of average relations between component processes and outcomes. The purpose of the present study was instead to examine whether and if so how the component processes engaged in WPS differ along the spectrum of WPS performance. Second graders (N = 1,130) from 126 classrooms in 17 schools were assessed on component processes (reasoning, in-class attentive behavior, working memory, language comprehension, calculation fluency, word reading) and WPS. Multilevel, unconditional quantile multiple regression indicated that 3 component processes, calculation fluency, language comprehension, and working memory, are engaged in WPS differentially depending on students' overall word-problem skill. The role of calculation fluency and language comprehension was stronger with more competent word-problem solving ability. By contrast, the role of working memory was stronger with intermediate-level than for strong problem solving. Results deepen insight into the role of these processes in WPS and provide the basis for hypothesizing how instructional strategies may be differentiated depending on students' overall level of WPS competence.

Improving Language Comprehension to Enhance Word-Problem Solving.

The focus of this article is the role of language comprehension within word-problem solving (WPS). The role of the language comprehension in WPS is explained, and an overview of research illustrating language comprehension's contribution to WPS is described. Next, an innovative intervention that embeds WP-specific language comprehension instruction within a validated form of schema-based WP intervention is described, and the methods and results of a randomized controlled trial assessing the added value of embedding WP-specific language comprehension instruction are outlined. Implications for practice and future research are drawn.

Are Individual Differences in Response to Intervention Influenced by the Methods and Measures Used to Define Response? Implications for Identifying Children With Learning Disabilities.

Response to intervention (RTI) has been promoted for nearly 20 years as a valid supplement to or alternative method of learning disability (LD) identification. Nevertheless, important unresolved questions remain about its role in disability identification. We had two purposes when conducting this study of 229 economically and racially diverse poor readers in Grades 4 and 5 in 28 public elementary and middle schools in Nashville. First, we examined predictors of the children's response to a reading comprehension tutoring program. Second, we explored the utility of different methods (growth vs final status) and measures (near- and mid-transfer vs far-transfer) in operationalizing "response," and whether these contrasting methods and measures identified similar children. Findings indicated students with higher pretreatment scores on expressive vocabulary, nonverbal IQ, teacher ratings of attention, and reading comprehension measures were more likely classified as responsive with final status methods. Students with lower pretreatment comprehension scores were more likely identified as responsive with growth methods. These and other findings suggest "response" is strongly context dependent, raising questions about the validity of RTI as a means of disability identification.

Dynamic Assessment for Identifying Spanish-Speaking English Learners' Risk for Mathematics Disabilities: Does Language of Administration Matter?

We examined dynamic assessment's (DA's) added value over traditional assessments for identifying Spanish-speaking English learners' (ELs) risk for developing mathematics disabilities, as a function of the language of test administration (English vs. Spanish), type of math outcome, and EL's language dominance. At the start of first grade, ELs (N = 368) were randomly assigned to English-DA or Spanish-DA conditions, were assessed on static mathematics measures and domain-general (language, reasoning) measures in English, and completed DA in their assigned language condition. At year's end, they were assessed on calculation and word-problem solving outcomes in English. Results from multigroup path models indicated that Spanish-DA mitigates the impact of ELs' language dominance on DA performance. Moreover, ELs' language dominance moderated DA's predictive validity differentially depending on DA language and type of outcome. Spanish-DA showed higher predictive validity in Spanish-dominant ELs than English-dominant ELs when predicting calculations but not word-problem solving. English-DA was predictive for both outcomes, regardless of ELs' language dominance.

Cognitive Correlates of the Covariance in Reading and Arithmetic Fluency: Importance of Serial Retrieval Fluency.

This study examines the core predictors of the covariance in reading and arithmetic fluency and the domain-general cognitive skills that explain the core predictors and covariance. Seven-year-old Finnish children (N = 200) were assessed on rapid automatized naming (RAN), phonological awareness, letter knowledge, verbal counting, number writing, number comparison, memory skills, and processing and articulation speed in the spring of Grade 1 and on reading and arithmetic fluency in the fall of Grade 2. RAN and verbal counting were strongly associated, and a constructed latent factor, serial retrieval fluency (SRF), was the strongest unique predictor of the shared variance. Other unique predictors were phonological awareness, number comparison, and processing speed. Findings highlight the importance of SRF in clarifying the relation between reading and arithmetic fluency.

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.

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.

Embedding Self-Regulation Instruction Within Fractions Intervention for Third Graders With Mathematics Difficulties.

The purpose of this study was to explore the efficacy of fractions intervention with and without an embedded self-regulation (SR) component for third-grade students at risk for mathematics disabilities. Fractions intervention focused on magnitude understanding and word problems. Embedded SR was designed to support a growth mindset (fostering belief that intellectual and academic abilities can be developed) along with SR processes in which students set goals, self-monitor, and use strategies to engage motivationally, metacognitively, and behaviorally through challenging tasks. Students (n = 69) were randomly assigned to business-as-usual control and the two versions of fractions intervention. Multilevel models, accounting for the nested structure of the data, identified a moderation effect on fraction word problems: For students receiving fractions intervention with embedded SR, response to intervention was robust across the continuum of students' pretest word problem skill; by contrast, without SR, response to fractions intervention depended on students' pretest word problem skill. On the remaining outcomes, results reflected stronger outcomes when fractions intervention embedded SR instruction without moderation.

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.