Search

The purposes of this study were to assess the effects of fractions intervention for students who are at risk for poor outcomes and to examine whether a component that combines self-regulated learning with growth-mindset instruction (SR-GM) provides added value for improving outcomes. At-risk students (N = 84) were randomly assigned to three conditions: fractions intervention, fractions intervention with embedded SR-GM, and a control group. Intervention was conducted three times per week for 35 min per session for 13 weeks. Multilevel models indicated both fractions intervention conditions produced strong effects, with no added value for SR-GM. Posttest fractions achievement gaps for both intervention conditions held steady, narrowed, or closed, whereas the control group's gaps remained sizeable or grew. Results suggest that intervention can address challenging mathematics standards for at-risk learners and that SR-GM instruction may not be necessary in the context of strong intervention. [For the corresponding grantee submission, see ED604271.]

This study's first purpose was to investigate effects of a fourth- and fifth-grade "next-generation" fraction intervention, which included six enhancements over a previously validated fraction intervention, designed to address career- and college-readiness standards. The study's second purpose was to assess effects of the next-generation fraction intervention at follow-up, 1 year after intervention ended. The third purpose was to isolate the effects of one of the six intervention enhancements: interleaved fraction calculations instruction. Students with intensive intervention needs were randomized to next-generation fraction intervention (Super Solvers [SSINT]) with blocked calculations instruction (SSINT_B), SSINT with interleaved calculations instruction (SSINT_I), and control. On a mix of proximal and transfer outcomes, SSINT (across conditions) produced strong, significant effects over control at posttest. At follow-up, effect sizes were weaker but remained significant on calculations: g = 1.22. On other measures, follow-up g was 0.39 to 0.58. The effect of SSINT_I over SSINT_B, although not significant at posttest (g = 0.28), was statistically significant and large at follow-up (g = 0.65), in line with the cognitive science literature showing long-term advantages for interleaved instruction. Results suggest next-generation fraction intervention efficacy for intensive-needs students and the importance of interleaved instruction. [For the corresponding grantee submission, see ED623746.]

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 2 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.

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. [For the corresponding grantee submission, see ED603185.]

Background/Context: The Individuals with Disabilities Education Act (IDEA, 2004) requires that all students with disabilities (SWD) receive a free, appropriate public education designed to meet their unique needs to prepare them for post-school education and employment (American Psychological Association, 2018). In the past two decades, momentum has grown for a supplementary idea: that schools be held accountable for SWD achieving grade-level standards. Thus standards-based reform for SWD is often caught between ideals of standardization and principles of differentiation. Purpose and Research Questions: The purpose of this study was to examine the extent to which district administrators, principals, general educators, and special educators differ in their policy perceptions of the newest college- and career-readiness standards (CCRS) for SWD versus other learners. Our research questions were: To what extent do teachers of SWD report delivering similar or different instructional content compared to general education teachers? What types of instructional supports do teachers provide, what types of professional development do teachers receive, and how do these differ by teacher type? How do policy perceptions differ between teachers of SWD and general education teachers? How do district administrators, principals, and teachers differ in their policy perceptions of the CCRS as they relate to SWD? Research Design: In three states (Texas, Ohio, and Kentucky), we surveyed a stratified sample of teachers, principals, and district administrators on the implementation of their state's standards and directed them to respond for SWD who participate in the regular accountability system. Conclusions/Recommendations: Results indicate an environment where SWD continue to receive less grade-level content and, in Texas and Ohio, are served by personnel who do not believe that the standards are appropriate. Kentucky demonstrated greater consistencies between general education and SWD instruction and policy environments. Findings raise questions about whether CCRS are being implemented for all students.

This study's purpose was to investigate effects of 3 intervention approaches for building working memory (WM) and improving word-problem solving (WPS). Children with mathematics difficulties (n = 240; 7.51 years [SD = 0.33]) were randomized to 4 conditions: a control group, general WM training with contiguous math practice, WPS intervention without WM training, and WPS intervention with domain-specific WM training. WM, WPS, and arithmetic were assessed before and 1-3 weeks after intervention; delayed WPS and arithmetic posttesting occurred 4-6 weeks later. Multilevel modeling of main effects and mediation effects were employed. Compared to control, general WM training with contiguous math practice and WPS intervention without WM training increased WM and WPS. The 3rd training condition, WPS intervention with domain-specific WM training, which minimized WM training time, improved WPS but without effects on WM. Both WPS intervention conditions outperformed general WM training on WPS. Conclusions are as follows. (a) General WM training with contiguous math practice improves WM and WPS. (b) WM training is not a substitute for WPS intervention when the goal is to strengthen WPS. (c) WPS intervention without WM training improves WM but is not a substitute for WM training when the goal is to strengthen WM. (d) For WM effects to accrue, WM training needs to occur with sufficient intensity. (e) WM plays a causal role in WPS, but not in arithmetic. Implications are drawn for research and practice, including assessing instructional supports in future research to build cognitive-academic bidirectionality. [For the grantee submission, see ED620371.]

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 variations 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 16th percentile, first-grade risk for RMD was two times more common than chance; at or below the 7th 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.

Reading nonfiction texts with understanding is important to school success, yet many students struggle to do so. This randomized controlled trial extends previous research by contrasting an earlier iteration of a comprehension tutoring program (Comp) against a variant with strategies for transferring learning (Comp+Transfer). Participants were 189 fourth and fifth graders with weak reading comprehension. To evaluate their efficacy, we used commercially developed far-transfer measures and experimenter-made near- and mid-transfer measures of reading comprehension. In contrast to controls, students in both programs significantly improved their understanding of near-transfer passages. Additionally, students in Comp+Transfer improved performance on mid-transfer passages. These findings suggest the value of teaching for transfer and the importance of measuring program efficacy with researcher-made tests alongside commercial tests.

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. [For the corresponding grantee submission, see ED581313.]

The purpose of this analysis was to assess whether effects of first-grade mathematics intervention apply across the range of at-risk learners' initial skill levels. Students were randomly assigned to control (n = 213) and two 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 two variants, focused on number knowledge that provides the conceptual bases for arithmetic. The other five minutes provided nonspeeded 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 nonspeeded practice on arithmetic.

Background: The previous body of research literature has reported several separate cognitive processes relevant in solving mathematics wps. Therefore, it is of the essence to seek for effective intervention and instruction for students in need for support in learning. Aim: This article reports the outcome of an intervention targeted at mathematics word problem (wp) skills. Setting: This study included three data collection points: (1) Pre-measurements, (2) post-measurements and (3) follow-up measurements. Pre-measurements were performed in August, post-measurements immediately after the intervention period in October and followup measurements in December. Methods: A programme, which included face-to-face support in mathematics wp strategies with the think-aloud protocol, was applied. The participants were 28 Finnish third-graders (14 training group students and 14 control students). Their mathematics wp skills were tested three times (pre-, post- and follow-up assessments). The groups were matched by gender, family type and the mathematics wp pre-measurement score level. The groups differed neither by literacy skills (i.e. technical reading, reading comprehension) nor by task orientation at baseline. Results: Some acceleration of mathematics wp skills among the training group students was found but the growth dramatically declined as soon as the face-to-face support stopped. The results further showed improvement in the efficacy of correct answers or attempted mathematics wp items among training group students. Conclusion: The results suggested that training consisting of face-to-face support is crucial for accelerating mathematics wp strategies among students struggling with mathematics. Repeated, cyclic periods of support are suggested for sustained effect.

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 (Mage = 6.55 years, SDage = 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 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.

We conducted a randomized control trial assessing the effects of two variants of word-problem intervention with third graders (n = 304) experiencing mathematics difficulty. Students were assigned to a business-as-usual condition (BaU) or one of two variants of word-problem intervention. One variant included a pre-algebraic reasoning component (relational understanding of the equal sign as well as standard and nonstandard equation solving); the other included word-problem intervention without pre-algebraic reasoning. Students in both interventions significantly outperformed students in the BaU with large effect sizes (2.66 and 2.44), but there were no significant differences between the two intervention conditions. The purpose of the present analysis was to assess maintenance of effects 6-12 months after intervention with the students (n = 229), now in fourth grade. At follow-up, only students in the word-problem intervention with pre-algebraic reasoning significantly outperformed the BaU on a measure of word-problem solving. This finding suggests an advantage for the pre-algebraic reasoning component, yet the follow-up effect between intervention conditions was not significant. The ESs of 0.43 and 0.31 and persistence rates of 16% and 13% reveal substantial forgetting for both conditions, which suggests the dose of intervention may not be adequate for many students.

Many general and special education teachers teach mathematics word problems by defining problems as a single operation and linking key words to specific operations. Unfortunately, teaching students to approach word problems in these ways discourages mathematical reasoning and frequently produces incorrect answers. This article lists eight common key words, identifies the operation typically associated with each, and provides word problems that illustrate how reliance on key words can result in incorrect answers. Neither of these approaches--defining problems in terms of a single operation or linking key words to specific operations--has evidence to support its use. In contrast, other approaches do promote mathematical reasoning and substantially boost word-problem performance among students with learning disabilities. Two practices that have emerged from research studies as particularly effective for word-problem instruction are: (1) Attack strategies, which provide students with a general plan for processing and solving word problems (Montague, 2008; Xin & Zhang, 2009); and (2) Schema instruction, in which students learn to categorize word problems within problem types (i.e., schemas based on the word problem's mathematical structure), apply an efficient solution strategy for each word-problem schema, and understand the meaning of word problem language (Fuchs et al., 2014; Jitendra & Star, 2012). These strategies are discussed in detail.

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.

This study examined differences in cognitive processing between 4th-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. [This article was published in "Learning Disabilities Research & Practice" (EJ1159515).]

This article describes the Taxonomy of Intervention Intensity, which articulates seven principles for evaluating and building intervention intensity based upon research. The Taxonomy's seven dimensions of intensity are strength, dosage, alignment, attention to transfer, comprehensiveness, behavioral support, and individualization (see Table 1). In explaining the Taxonomy, the authors present a case study illustrating how the Taxonomy can be used to systematize the process by which special educators and related personnel (a) set up the intensive intervention process and (b) monitor the student's response and systematically improve the program to match the target student's individual needs. The goal is to increase the quality of intensive intervention, improve student outcomes, and help schools design intensive intervention programs that are clearly distinguishable from less intensive (Tier 2) intervention programs. [This article can also be accessed in ERIC as Grantee Submission ED571640.]

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.

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 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 U.S. 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. [ABSTRACT FROM AUTHOR]

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. [This article was published in "Journal of Experimental Child Psychology," v145 p95-119 May 2016.]

Development of fraction number line estimation was assessed longitudinally over five time points between fourth and sixth grades. Although students showed positive linear growth overall, latent class growth analyses revealed three 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 one, 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 sixth 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. [This paper was published in "Developmental Psychology" (EJ1098744).]

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.

We attempted to strengthen an evidence-based, peer-mediated, first-grade reading program (First Grade Peer-Assisted Learning Strategies [PALS]) by modestly revising its content and adding a repeated-reading (RR) component. In a cluster-randomized trial, we conducted a component analysis of the revised program by creating two versions of it. "PALS+Fluency" represented the modified program with an RR component, whereas "PALS-Only" represented the same program without RR. We tested the efficacy of the two PALS versions together against controls and against each other to determine if peer-mediated RR had "value added." With moderator analyses, we further explored whether the PALS programs benefited weaker and stronger readers alike. Thirty-three first-grade classroom teachers (and 491 students) from eight urban schools were randomly assigned to the three study groups within their schools. The PALS-Only and PALS+Fluency programs ran for 22 weeks. Multilevel modeling showed that the combined effects of the two PALS programs on phonological awareness (PA), word reading, and reading fluency were superior to those of controls. Students' pretreatment PA moderated combined program effects on PA and word reading, indicating stronger effects for students with weaker PA. PALS-Only students improved PA skills (over PALS+Fluency students) with stronger effects for those with weaker pretreatment PA.

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

The 3 purposes of this study were to: (a) describe fraction ordering errors among at-risk 4th-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 9-item ordering test. A high proportion (81%) of problems were completed incorrectly. Most (65% of) 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. [At time of submission to ERIC this article was in press with "Journal of Learning Disabilities."]

The purpose of this analysis was to examine achievement gaps on fractions for very-low-performing students as a function of whether they receive inclusive fraction instruction or specialized fraction intervention and with the shift to Common Core State Standards (CCSS). In three randomized control trials conducted in 3 consecutive years, 203 students who scored at or below the 10th percentile in mathematics (mean standard score ~75) at the start of fourth grade were randomly assigned at the individual level to 12 weeks of inclusive fraction instruction or specialized fraction intervention. In Year 1, the fourth-grade mathematics curriculum was guided by initial state standards; in Years 2 and 3, the state was transitioning to CCSS. In each of the 3 years on each measure, results indicated significantly stronger learning and markedly smaller post-intervention achievement gaps for specialized fraction intervention than for inclusive fraction instruction. Yet, the size of achievement gaps grew over the years in both conditions, as CCSS increased the depth and challenge of the fraction curriculum and produced differentially stronger learning in not-at-risk classmates. Implications are discussed in terms of the provision of services for students with learning disabilities in the era of CCSS and the meaning of "access to the general education curriculum". [This paper was published in "Exceptional Children" (EJ1049289).]

Students who demonstrate persistent mathematics difficulties and whose performance is severely below grade level require "intensive intervention". Intensive intervention is an individualized approach to instruction that is more demanding and concentrated than Tier 2 intervention efforts. We present the elements of intensive intervention that teachers should consider when planning for, implementing, and monitoring intensive intervention in mathematics. Each of these elements is based on evidence from validated interventions. We also highlight strategies for intensifying instruction. We provide two examples of intensive intervention, one of which launches from a Tier 2 intervention platform and the other which is completely generated by a teacher. We conclude with considerations for intensive intervention in mathematics. [This paper was published in: "Learning Disabilities Research & Practice" v30 n4 p182-192 2015 (EJ1079645)].