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The purpose of this study was to investigate the effects of the concrete-representational-abstract–integrated (CRA-I) sequence on students with learning disabilities' performance when learning fraction and decimal concepts. Three elementary-level students in Grades 4 and 6 participated in a single-case multiple probe across behaviors study. The intervention involved explicit use of fraction blocks, coins, base ten blocks, number lines, pictures, and abstract symbols to teach unit fractions, fraction and decimal equivalence, addition of fractions with unlike denominators, and writing fractions as decimals. The researchers demonstrated a functional relation between CRA-I and three different behaviors related to fraction concepts: decreased error in estimating fraction magnitude, addition of unlike fractions, and writing fractions as decimals. The findings show promise in the use of CRA-I for teaching fraction concepts to students with learning disabilities. [ABSTRACT FROM AUTHOR]

State standards include fluent use of standard computational algorithms. However, learning and using them require conceptual understanding of numbers and operations. Previous research using the concrete–representational–abstract (CRA) sequence has been effective in teaching computational algorithms to students at risk of and students with learning disabilities by emphasizing conceptual understanding. However, all the research was face-to-face and few captured the impact of the intervention on students' conceptual understanding. The current study occurred during the COVID-19 pandemic, so researchers modified CRA for remote instruction. This study investigated the effects of modified CRA on the fluency and accuracy of three sixth-grade students with learning disabilities. They also included an assessment of conceptual understanding. The researchers used a multiple probe across participants design, demonstrating a functional relation between CRA and students' fluency and accuracy. Researchers assessed conceptual understanding with informal measures that required application of their conceptual understanding. The results and implications are discussed. [ABSTRACT FROM AUTHOR]

This pilot study examined the promise of a Tier 2 Grade 6 intervention program for students at risk for difficulties in mathematics. The study utilized a quasi-experimental design. The final sample included 112 students in treatment (Promoting Algebra Readiness) and 86 students in control (standard district practice) conditions. The Promoting Algebra Readiness program consisted of 93 lessons across four strands focused on key concepts and applications of fractions. Measures of mathematics achievement were collected at pretest and posttest. Feasibility and usability data indicated favorable impressions by users and strong levels of implementation fidelity. Gain scores of treatment students were significantly greater than those of control peers on two of four proximal measures of mathematics achievement. Positive nonsignificant effects were found on additional proximal and distal measures. Implications for educators delivering instruction for at-risk students in multitier service delivery models are discussed. [ABSTRACT FROM AUTHOR]

This meta-analysis assessed the effect of word-problem-solving interventions on the word-problem-solving accuracy of students identified as having a learning disability (LD) or at risk for an LD in kindergarten to the sixth grade. Eighteen randomized control group designed studies met the inclusion criteria. Overall, word-problem-solving interventions yielded a significant positive effect on the word-problem-solving accuracy of students in elementary grades with LD (effect size [ES] = 1.08). Instructional components that underlie effective studies were also identified. Results suggest that peer interaction and transfer instructions yielded large effects on treatment outcomes. Results also suggested that intensive interventions (50-min sessions, 34 total sessions) in Grade 3 regardless of instructional setting yielded the largest ESs. These findings support the need to develop and implement quality evidence-based instruction in classroom settings (Tier 1 instruction) prior to utilizing additional resources for more intensive and individualized intervention. [ABSTRACT FROM AUTHOR]

The purpose of the study was to compare the deficit profiles of two important types of mathematics difficulties. Three cognitive measures (working memory, processing speed, and reasoning), two mathematics measures (numerical facts retrieval and mathematics vocabulary), and reading comprehension were assessed among 237 Chinese fourth-grade students, among whom 28 were classified as students with only computational difficulties (CD), 34 were classified as having only word problem-solving difficulties (WPD), 20 were classified as students with computational and word problem-solving difficulties (CD + WPD), and 43 typically developing (TD) peers. Multivariate analysis showed that, compared with TD, CD was associated with weakness in numerical working memory; WPD was associated with weakness in reading comprehension; both CD and WPD were associated with weakness in mathematics vocabulary. However, CD and WPD did not differ from each other on any of those profiling measures. Implications for understanding mathematics competence and identification of mathematics difficulties are discussed. [ABSTRACT FROM AUTHOR]

Instructional interactions that occur between teachers and students around foundational mathematics topics are critical for supporting mathematical proficiency among students with mathematics learning disabilities (MLD). This study investigated whether the initial mathematics skill of 880 kindergarten students at risk of MLD predicted the quantity and quality of explicit instructional interactions (i.e., overt teacher modeling, student practice opportunities, and academic feedback) experienced during an empirically validated Tier 2 kindergarten mathematics intervention. It also examined whether the quantity and quality of such instructional interactions predicted gains in student mathematics achievement. Researchers conducted 740 direct observations of 255 intervention groups within a multiyear, randomized controlled trial. Results suggested that intervention groups with lower initial mathematics skill received higher rates of academic feedback and made more frequent errors. In addition, more frequent and higher quality academic feedback and group practice opportunities predicted increased mathematics achievement. Implications for investigating the active ingredients of mathematics interventions are discussed. [ABSTRACT FROM AUTHOR]

This study examined the impact of linguistic and cultural enhancements to evidence-based mathematics instruction within a multitiered support system for English learners. The study employed a single subject changing criterion design for four fourth-grade students who were English learners with or at risk of a learning disability diagnosis in mathematics. Dual dependent variables were mathematics vocabulary acquisition and application in story problems. Student performance on identifying the correct mathematics vocabulary words when given the definition was measured across baseline and four phases of intervention. Students were also assessed on their ability to complete story problems that contained the target vocabulary. At each phase of the intervention, students showed an increase in performance on both vocabulary words correctly identified and story problems correctly completed. Implications for practice and future directions for research are discussed. [ABSTRACT FROM AUTHOR]

English learners (ELs) experience difficulty with mathematical problem solving because word problems require complex processes beyond basic math skills, such as the use of linguistic information, identifying relevant information, and constructing the appropriate problem statement. This study used a combined multiple baseline design and criterion changing design to assess the effectiveness of a paraphrasing intervention on the problem-solving performance for nine third-grade students who are ELs and at risk of mathematical disabilities (MD). Although the magnitude of the Tau-U effect sizes was in the small range, the visual analysis indicated that all students displayed increasing trends in problem-solving accuracy as a function of the paraphrasing intervention. The results were discussed in terms of providing continual support toward maintaining intervention outcomes. [ABSTRACT FROM AUTHOR]

Accumulating research has established explicit mathematics instruction as an evidence-based teaching practice. This study utilized observation data from a multi-year efficacy trial to examine the longitudinal effects of a core kindergarten mathematics program on the use of explicit mathematics instruction among two distinct groups of teachers: one group that used standard practices in Year 1 of the efficacy trial and the core program in Year 2, and a second group that used the core program in both years. Targeted teaching practices consisted of teacher models, student practice opportunities, and teacher-provided academic feedback. Implementation of the program in Year 2 was found to increase the mean rates of teaching practices of teachers who used standard teaching practices in Year 1. Effect sizes are also suggestive of a positive impact of a second year of implementation with the core program. Implications for designing explicit mathematics programs and investigating evidence-based practices in future research are discussed. [ABSTRACT FROM AUTHOR]

Mathematical reasoning is important in conceptual understanding and problem solving. In current reform-based, discourse-oriented mathematics classrooms, students with learning disabilities (LD) encounter challenges articulating or explaining their reasoning processes. Enlightened by the concept of conversational repair borrowed from the field of linguistics, this study designed an intervention program to facilitate mathematical reasoning of students with LD. Conversational repair, an ability to repair communicative breakdowns or inaccuracies, was designed in an implicit–explicit continuum to elicit self-explanation from students with LD in the context of mathematics word problem solving. Using a multiple-baseline across participants design, the study found that the intervention was effective for improving students’ mathematical reasoning and problem-solving ability measured by their self-explanation and word problem–solving performance. It provided implications for future studies concerning the use of conversational repair in mathematics classroom discourse for individuals with LD. [ABSTRACT FROM AUTHOR]

For students with mathematics difficulties (MD), math word problem solving is especially challenging. The purpose of this study was to examine the effects of a problem-solving strategy, bar model drawing, on the mathematical problem-solving skills of students with MD. The study extended previous research that suggested that schematic-based instruction (SBI) and cognitive strategy instruction (CSI) delivered within an explicit instruction framework can be effective in teaching various math skills related to word problem solving. A multiple-baseline design replicated across groups was used to evaluate the effects of the intervention of bar model drawing on math problem-solving performance of students with MD. Student achievement was measured in terms of increased correct use of cognitive strategies and overall accuracy of math word problem solving. Results showed that bar modeling drawing is an effective strategy for increasing elementary students' accuracy in solving math word problems and their ability to use cognitive strategies to solve the problems. [ABSTRACT FROM AUTHOR]

The Common Core Mathematics Standards have raised expectations for schools and students in the United States. These standards demand much deeper content knowledge from teachers of mathematics and their students. Given the increasingly diverse student population in today’s classrooms and shortage of qualified special education teachers, computer-assisted instruction may provide supplementary support, in conjunction with the core mathematics instruction, for meeting the needs of students with different learning profiles. The purpose of this study was to explore the potential effects of the Please Go Bring Me-Conceptual Model-Based Problem Solving (PGBM-COMPS) intelligent tutor program on enhancing the multiplicative problem-solving skills of students with learning disabilities or difficulties in mathematics. [ABSTRACT FROM AUTHOR]

Word problems are prevalent on high-stakes assessments, and success on word problems has implications for grade promotion and graduation. Unfortunately, English Language Learners (ELLs) continue to perform significantly below their native English-speaking peers on mathematics assessments featuring word problems. Little is known about the instructional needs and performance of ELLs at risk of mathematics difficulty (MD). In the present study, an exploratory quasi-experimental design was used to investigate word-problem instruction for ELLs in a culturally and linguistically diverse public elementary school. Specifically, we studied the efficacy of a word-problem intervention for ELLs with MD (N = 9) that combined culturally and linguistically responsive practices with schema instruction (CLR-SI). The study is unique in that it combines research on effective instruction for ELLs and students with MD; CLR-SI has not been investigated for either ELLs or students with MD. Results have implications for teachers, administrators, and researchers of ELLs with MD. [ABSTRACT FROM AUTHOR]

Documenting how students with learning disabilities (LD) initially conceive of fractional quantities, and how their understandings may align with or differ from students with mathematics difficulties, is necessary to guide development of assessments and interventions that attach to unique ways of thinking or inherent difficulties these students may face understanding fraction concepts. One way to characterize such conceptions is through the creation of a framework that depicts key understandings evidenced as students work with problematic situations. The present study extends current literature by presenting key understandings of fractions, documented through problem-solving activity, language, representations, and operations, evidenced by students with LD and mathematics difficulties as they engaged with equal sharing problems. Clinical interviews were conducted with 43 students across the second, third, fourth, and fifth grades. Results of the study suggest that students with LD hold similar informal notions of key understandings of fractions as students with mathematics difficulties and that many of the students evidenced rudimentary understandings of fractional quantities. Researchers discuss implications of the findings in relation to considerations for designing interventions to support and extend students’ initial conceptions of fractional quantity. [ABSTRACT FROM AUTHOR]

Minority students at risk for math difficulties (MD) struggle with word problems for various reasons beyond procedural or calculation challenges. As a result, these students require support in reading and language development in addition to math. The purpose of this study was to assess the effectiveness of a math comprehension strategy based on a dynamic assessment (DA) framework, which provided scaffolding support based on the student's reading and language levels. A multiple baseline design was used to assess eight third-grade minority students at risk for MD. As compared with the baseline phase, the strategy increased problem-solving ability for all participants. All students maintained performance levels during follow-up sessions. Results suggest that the intervention facilitated word-problem-solving performance. [ABSTRACT FROM AUTHOR]

This study used a multiple probe across participants design, replicated across two interventions and counterbalanced across participant groups to examine the effects of number sense intervention and extensive practice intervention on strategy transformation when students with mathematics learning disabilities (MLD) solved basic fact problems. In addition, the effects of both interventions on students’ fact retrieval performance and generalization performance were also tested. To examine the effects of each intervention, the participants were divided into two groups and the order of the interventions was varied between the two groups. The results showed that, in general, students demonstrated a greater extent of strategy transformation and generalization after receiving the number sense intervention. However, the extensive practice intervention promoted students’ use of retrieval-based strategies to solve basic facts. Findings suggested that both interventions had positive impacts on students with MLD and should be used in tandem. [ABSTRACT FROM AUTHOR]

Systematically observing instruction for students with learning disabilities (LD) provides information regarding the quality of school-based interventions, allocation of instructional time, and other implementation variables associated with student outcomes. In this synthesis, observation studies of reading and mathematics instruction from 2000 to 2013 were identified through a systematic literature search. The purpose was to understand findings from observations of typical practice for students with LD during an era of increased exposure to evidence-based practices, educational policy reforms, and reports on reading and mathematics designed to influence instruction, and amid a period of increased academic expectations. Eleven studies focusing on academic instruction for students with LD were identified and reviewed. Cognitive strategy instruction, differentiated instruction, and opportunities for independent application (i.e., text reading, problem solving) were reported infrequently across the published observational research. However, findings provide some evidence of improved teacher use of effective instructional practices in several areas compared with earlier observation study syntheses. Implications for practice and areas for future research are discussed. [ABSTRACT FROM AUTHOR]

Little to no information exists explaining the nature of conceptual gaps in understanding fractions for students with learning disabilities (LD); such information is vital to practitioners seeking to develop instruction or interventions. Many researchers argue such knowledge can be revealed through student’s problem-solving strategies. Despite qualitative differences in thinking and representation use in students with LD that may exist, existing frameworks of student’s strategies for solving fraction problems are not inclusive of students with LD. This exploratory study extends existing literature by documenting the strategies students with LD use when solving fraction problems. Clinical interviews were conducted with 10 students across the third, fourth, and fifth grades (N = 10). Results indicate students with LD used similar strategies as previously reported in research involving non-LD students, although the dominant strategy utilized was less advanced and the range of strategy use was relatively compact. Researchers suggest the nature of conceptual gaps students with LD display in their understanding of fractions originates from a malleable source. Implications for instruction and assessment are presented. [ABSTRACT FROM AUTHOR]

Researchers and practitioners indicate students require explicit instruction on mathematics vocabulary terms, yet no study has examined the effects of an embedded vocabulary component within mathematics tutoring for early elementary students. First-grade students with mathematics difficulty (MD; n = 98) were randomly assigned to addition tutoring with an embedded vocabulary component, addition tutoring without the embedded vocabulary component, or business-as-usual control. At posttest, students who received addition tutoring without vocabulary demonstrated greater gains than control students on addition fluency. On a measure of mathematics vocabulary, students in the active tutoring conditions demonstrated improved performance on mathematics vocabulary over control students. Results indicate that exposure to addition tutoring with or without an embedded vocabulary component positively improves mathematics vocabulary performance. [ABSTRACT FROM AUTHOR]

This article introduces the thematic special series on intersection of mathematics education and special education. This special series will include four research papers and one commentary. The four research papers rooted in a working group from the annual conferences of the International Group of the Psychology of Mathematics Education (PME) as well as its North American Chapter (PME-NA). [ABSTRACT FROM AUTHOR]

The purpose of this study was to investigate the effectiveness of explicit, direct, and generative strategy training and working memory capacity (WMC) on mathematical word problem-solving accuracy in elementary schoolchildren. In this study, children in third grade (N = 82) identified as at risk for math difficulties (MD) were randomly assigned (within classrooms) to one of three treatment conditions that explicitly directed students’ attention to different propositions within word problems—paraphrase question propositions (Restate), paraphrase relevant propositions (Relevant), and paraphrase all propositions (Complete)—or an untreated control condition. A significant treatment by covariate design indicated that generative strategy outcomes were conditional on the level of pretest WMC. A clear advantage in posttest problem-solving accuracy and solution planning was found for the complete generative condition relative to the control condition, but this advantage was conditional on setting WMC to a high level. Although no significant treatment advantages were found for solution accuracy when WMC was set to a low level, treatment advantages relative to the control condition were found for measures of schema activation. The results indicated that the effectiveness of generative strategies among children at risk for MD was directly dependent on the level of WMC. [ABSTRACT FROM PUBLISHER]

The purpose of the present study was to examine the effectiveness of a strategic training program for improving students’ performance in solving multiplication problems. The participants were 3 third graders with math difficulties. In this teaching experiment, microgenetic analysis was used to analyze improvement in students’ strategic development and problem-solving accuracy. The instructional components included selective task assignment and OFSD (encouraging students to use Own strategy to solve problems, providing Feedback, asking for students’ Self-explanations, and explicit Demonstration of strategies when needed). The results showed that the three participants began the intervention at different strategic developmental levels and consequently were given differentiated tasks to promote their strategic development during the intervention. In response to the intervention, the three participants improved their problem-solving accuracy, use of advanced strategies, and flexibility in choosing backup strategies. [ABSTRACT FROM AUTHOR]

An introduction is presented in which the author discusses various issues including errors of students with mathematics learning disabilities, effects of 30-min reading intensive interventions on English learners, and effects of using concrete and virtual manipulative during algebra instruction.