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This study compared three types of personal-space preference (PSP)--Initial PSP, Approach PSP, and Stop-Distance PSP--of male elementary and high school students in talking with an adult female conversation partner in a natural conversation setting. The boys were divided into three groups: boys with severe visual impairments, sighted boys who were blindfolded, and sighted boys with no visual restrictions. The visually impaired boys and the sighted, blindfolded boys chose smaller Initial PSPs, but on Approach PSP and Stop-Distance PSP, all three groups performed comparably. [ABSTRACT FROM AUTHOR]

In this article, we introduce the term vulnerability to achievement stressors , which refers to differentially low achievement when shifts in the educational environment "stress" or threaten the capacity of an individual or a group of individuals to make academic progress. We also introduce a methodological framework for assessing vulnerability to achievement stressors. Vulnerability to achievement stressors in students with learning disabilities (LD), relative to students without disabilities, is illustrated with two achievement stressors: (a) the shift in learning standards codified in Common Core State Standards, specifically the increase in complexity of the fourth-grade fractions curriculum, and (b) the COVID-19 pandemic, which decreased instructional structure by disrupting in-person teaching. Because these illustrations were embedded within randomized controlled trials, each with an inclusive instruction condition and an intensive intervention condition, they also provide the basis for concluding that intensive intervention is more effective than inclusive instruction for addressing students with LD's vulnerability to achievement stressors and for narrowing their persistently severe achievement gaps., Competing Interests: Declaration of Conflicting InterestsThe author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

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., Competing Interests: Declaration of Conflicting InterestsThe author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

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., (© 2023. The Psychonomic Society, Inc.)

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.

Despite policy relevance, longer-term evaluations of educational interventions are relatively rare. A common approach to this problem has been to rely on longitudinal research to determine targets for intervention by looking at the correlation between children's early skills (e.g., preschool numeracy) and medium-term outcomes (e.g., first-grade math achievement). However, this approach has sometimes over-or under-predicted the long-term effects (e.g., 5th-grade math achievement) of successfully improving early math skills. Using a within-study comparison design, we assess various approaches to forecasting medium-term impacts of early math skill-building interventions. The most accurate forecasts were obtained when including comprehensive baseline controls and using a combination of conceptually proximal and distal short-term outcomes (in the nonexperimental longitudinal data). Researchers can use our approach to establish a set of designs and analyses to predict the impacts of their interventions up to two years post-treatment. The approach can also be applied to power analyses, model checking, and theory revisions to understand mechanisms contributing to medium-term outcomes.

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.

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., (© 2022 The British Psychological Society.)

How does language shape mathematical development? In this article, we consider this question by reviewing findings from cross-sectional and longitudinal research. In this literature, we find that differences in the structures of languages and individual variation in language ability are associated with mathematical performance in both obvious and unexpected ways. We then consider the causal nature of these relations, with a focus on experimental studies that have tested the effects of language instruction on mathematical outcomes. Findings from this work show that certain forms of language instruction meaningfully improve performance in several mathematical domains, providing strong evidence of a linguistic pathway in mathematical development. However, much additional research is needed to understand how language instruction may be integrated optimally into math education. We conclude with recommendations for research.

The first purpose of this study was to examine the prevalence of risk for comorbid reading and mathematics disabilities (RMD) at start of first grade, when measured in a representative sample of 3,062 students with first-grade fluency measures (word reading; computation). The second purpose was to examine the utility of these measures for predicting RMD status within a sample of 577 students when RMD status was assessed at the end of second grade in terms of reading and math accuracy. When set at or below the 16 th percentile, first-grade risk for RMD was two times more common than chance; at or below the 7 th percentile, it was five times more common. Logistic regression showed that the two first-grade fluency measures accurately distinguished students with and without RMD in second grade; however, when cut scores were set to capture 85% of students with RMD, false positives were high. Overall, the results provide support for the use of fluency measures as an initial gating procedure in first grade, but additional gating steps appear necessary in the screening process to reduce false positives.

We examined longitudinal relations between 1 st -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 2 nd -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.

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.

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 1 st 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 1 st 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 1 st 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 1 st 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.

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.

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.

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., (© 2018 Society for Research in Child Development.)

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.

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.

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., Competing Interests: Declarations of interest: none

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; M age  = 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., (© 2018 Society for Research in Child Development.)

Specific learning disabilities (SLDs) are highly relevant to the science and practice of psychology, both historically and currently, exemplifying the integration of interdisciplinary approaches to human conditions. They can be manifested as primary conditions-as difficulties in acquiring specific academic skills-or as secondary conditions, comorbid to other developmental disorders such as attention-deficit hyperactivity disorder. In this synthesis of historical and contemporary trends in research and practice, we mark the 50th anniversary of the recognition of SLDs as a disability in the United States. Specifically, we address the manifestations, occurrence, identification, comorbidity, etiology, and treatment of SLDs, emphasizing the integration of information from the interdisciplinary fields of psychology, education, psychiatry, genetics, and cognitive neuroscience. SLDs, exemplified here by specific word reading, reading comprehension, mathematics, and written expression disabilities, represent spectrum disorders, each occurring in approximately 5% to 15% of the school-aged population. In addition to risk for academic deficiencies and related functional social, emotional, and behavioral difficulties, those with SLDs often have poorer long-term social and vocational outcomes. Given the high rate of occurrence of SLDs and their lifelong negative impact on functioning if not treated, it is important to establish and maintain effective prevention, surveillance, and treatment systems involving professionals from various disciplines trained to minimize the risk and maximize the protective factors for SLDs. (PsycINFO Database Record (c) 2020 APA, all rights reserved).

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.

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.

This study explored the developmental trajectories and predictors of word reading and reading comprehension among young at-risk readers. In fall of first grade, 185 students identified as at-risk for reading difficulties were assessed on measures of domain-specific skills (phonological awareness, letter knowledge, and vocabulary), domain-general skills (working memory, nonverbal reasoning, and processing speed), and word reading and reading comprehension. Word reading and reading comprehension were assessed again in spring of grades 1-4. Individual growth curve modeling showed that the children demonstrated decelerated growth on word reading and linear growth on reading comprehension, although their performance on both word reading and reading comprehension were consistently below average on national norms. After controlling for word reading and reading comprehension in first grade, letter knowledge predicted growth in word reading; vocabulary and nonverbal reasoning predicted growth in reading comprehension. That is, we found different developmental trajectories and different predictors for word reading and reading comprehension among our at-risk sample. Implications are discussed for theory and early reading instruction for at-risk children.

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., (© 2019 Wiley Periodicals, Inc.)

Students' explanations of their mathematical thinking and conclusions have become a greater part of the assessment landscape in recent years. With a sample of 71 fourth-grade students at risk for mathematics learning disabilities, we investigated the relation between student accuracy in comparing the magnitude of fractions and the quality of students' explanations of those comparisons, as well as the relation between those measures and scores on a criterion test: released fraction items from the National Assessment of Educational Progress. We also considered the extent to which reasoning and language contribute to the prediction. Results indicated a significant moderate correlation between accuracy and explanation quality. Commonality analyses indicated that explanation quality accounts for little variance in National Assessment of Educational Progress scores beyond what is accounted for by traditional measures of magnitude understanding. Implications for instruction and assessment are discussed.

Disorders of reading, math, and attention frequently co-occur in children. However, it is not yet clear which cognitive factors contribute to comorbidities among multiple disorders and which uniquely relate to one, especially because they have rarely been studied as a triad. Thus, the present study considers how reading, math, and attention relate to phonological awareness, numerosity, working memory, and processing speed, all implicated as either unique or shared correlates of these disorders. In response to findings that the attributes of all three disorders exist on a continuum rather than representing qualitatively different groups, this study employed a dimensional approach. Furthermore, we used both timed and untimed academic variables in addition to attention and activity level variables. The results supported the role of working memory and phonological awareness in the overlap among reading, math, and attention, with a limited role of processing speed. Numerosity was related to the comorbidity between math and attention. The results from timed variables and activity level were similar to those from untimed and attention variables, although activity level was less strongly related to cognitive and academic/attention variables. These findings have implications for understanding cognitive deficits that contribute to comorbid reading disability, math disability, and/or attention-deficit/hyperactivity disorder.

The purpose of this analysis was to assess whether effects of 1 st -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.

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.

Response to Intervention (RTI) was accepted in the early 2000s as a new framework for identifying learning difficulties (LD) in the U.S. In Finland, a similar multi-tiered framework has existed since 2010. In the present study, these frameworks are presented from the viewpoint of the role of assessment and instruction as expressed in documents that describe the frameworks, as it seems that these two components of RTI are the most disparate between the U.S. and Finland. We present a suggestion for the Finnish framework as an example of support in mathematics learning that incorporates principles of RTI (such as systematized assessment and instruction, cyclic support, and modifiable instruction). Finally, recommendations are presented for further refining and developing assessment and instruction policies in the two countries.

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.

The purposes of this study were to (a) explore whether early fractions understanding at 4 th 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 4 th 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 4 th 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.

This study examined differences in cognitive processing between 4 th -grade students who respond adequately, as opposed to inadequately, to intervention on 3 fraction outcomes: number-line estimation, calculation, and word problems. Students were assessed on 7 cognitive processes and on the 3 fraction outcomes. Students were grouped as adequate or inadequate responders, using as the cut-point the control-group mean on pre-to-post improvement on the relevant measure. Between-group differences identified reasoning, concept formation, and listening comprehension related to all 3 fraction outcomes. On the number-line outcome, within-group profile analysis indicated that inadequate responders experienced low reasoning ability relative to their other forms of cognitive processing.

In this article, the authors summarize results from 5 randomized controlled trials assessing the effects of intervention to improve the fraction performance of fourth-grade students at risk for difficulty in learning about fractions. The authors begin by explaining the importance of competence with fractions and why an instructional focus on fractions magnitude understanding may improve learning. They then describe an intervention that relies strongly on this type of understanding about fractions instruction, and they provide an overview of the intervention's overall effects. This is followed by an overview of 5 intervention components for which the authors isolated effects. They conclude by discussing some of the lessons learned from this research program.

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 relation between 2 forms of mathematical cognition, calculations and word problems, was examined. Across grades 2-3, performance of 328 children (mean starting age 7.63 [ SD =0.43]) was assessed 3 times. Comparison of a priori latent change score models indicated a dual change model, with consistently positive but slowing growth, described development in each domain better than a constant or proportional change model. The bivariate model including change models for both calculations and word problems indicated prior calculation performance and change were not predictors of subsequent word-problem change, and prior word-problem performance and change were not predictors of subsequent calculation change. Results were comparable for boys versus girls. The bivariate model, along with correlations among intercepts and slopes, suggest calculation and word-problem development are related, but through an external set of overlapping factors. Exploratory supplemental analyses corroborate findings and provide direction for future study.

The purpose of the study was to determine whether individual differences in at-risk 4 th 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.

The three purposes of this study were to (a) describe fraction ordering errors among at-risk fourth grade students, (b) assess the effect of part-whole understanding and accuracy of fraction magnitude estimation on the probability of committing errors, and (c) examine the effect of students' ability to explain comparing problems on the probability of committing errors. Students ( N = 227) completed a nine-item ordering test. A high proportion (81%) of problems were completed incorrectly. Most (65%) errors were due to students misapplying whole number logic to fractions. Fraction-magnitude estimation skill, but not part-whole understanding, significantly predicted the probability of committing this type of error. Implications for practice are discussed.

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., (Copyright © 2016 Elsevier Inc. All rights reserved.)

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.

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 1 st -grade calculation (CA) and word-problem (WP) performance, as a function of limited English proficiency (LEP) status. At the start of 1 st 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 1 st 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.

The purpose of this study was to examine child-level pathways in development of prealgebraic knowledge versus word-problem solving, while evaluating the contribution of calculation accuracy and fluency as mediators of foundational skills/processes. Children (n = 962; mean 7.60 years) were assessed on general cognitive processes and early calculation, word-problem, and number knowledge at start of Grade 2; calculation accuracy and calculation fluency at end of Grade 2; and prealgebraic knowledge and word-problem solving at end of Grade 4. Important similarities in pathways were identified, but path analysis also indicated that language comprehension is more critical for later word-problem solving than prealgebraic knowledge. We conclude that pathways in development of these forms of 4th-grade mathematics performance are more alike than different, but demonstrate the need to fine-tune instruction for strands of the mathematics curriculum in ways that address individual students' foundational mathematics skills or cognitive processes. (PsycINFO Database Record, ((c) 2016 APA, all rights reserved).)

Development of fraction number line estimation was assessed longitudinally over 5 time points between 4th and 6th grades. Although students showed positive linear growth overall, latent class growth analyses revealed 3 distinct growth trajectory classes: Students who were highly accurate from the start and became even more accurate (n = 154); students who started inaccurate but showed steep growth (n = 121); and students who started inaccurate and showed minimal growth (n = 197). Younger and minimal growth students typically estimated both proper and improper fractions as being less than 1, failing to base estimates on the relation between the numerator and denominator. Class membership was highly predictive of performance on a statewide-standardized mathematics test as well as on a general fraction knowledge measure at the end of 6th grade, even after controlling for mathematic-specific abilities, domain-general cognitive abilities, and demographic variables. Multiplication fluency, classroom attention, and whole number line estimation acuity at the start of the study predicted class membership. The findings reveal that fraction magnitude understanding is central to mathematical development. (PsycINFO Database Record, ((c) 2016 APA, all rights reserved).)

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., (Copyright © 2016 Elsevier Inc. All rights reserved.)

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., (© 2015 The Authors. Child Development © 2015 Society for Research in Child Development, Inc.)

This study investigated the relation between symbolic and nonsymbolic magnitude processing abilities with 2 standardized measures of math competence (WRAT Arithmetic and KeyMath Numeration) in 150 3rd-grade children (mean age 9.01 years). Participants compared sets of dots and pairs of Arabic digits with numerosities 1-9 for relative numerical magnitude. In line with previous studies, performance on both symbolic and nonsymbolic magnitude processing was related to math ability. Performance metrics combining reaction and accuracy, as well as weber fractions, were entered into mediation models with standardized math test scores. Results showed that symbolic magnitude processing ability fully mediates the relation between nonsymbolic magnitude processing and math ability, regardless of the performance metric or standardized test.

The purpose of this study was to examine the cognitive predictors of calculations and number line estimation with whole numbers and fractions. At-risk 4 th -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 4 th grade. Then, they were assessed on whole-number and fraction calculation and number line estimation measures at the end of 4 th 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.