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Mathematics learning is sequential and builds in complexity as children learn more advanced skills. The transition that many schools are making to the Common Core State Standards may require that some mathematics skills are introduced at earlier grades. Current focus on high-stakes assessments to understand student achievement may lead to external pressures to pass grade-level standards without consideration about the sequential continuum of mathematics. This paper addresses commonly used mathematics language that may disjoint the natural progression of mathematics learning and proposes alternatives that may allow for more seamless expanding of conceptual mathematics learning.

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... Mathematics language is comprised terms and vocabulary that are unique to mathematical use (i.e., technical vocabulary; Monroe & Panchyshyn, 1995), such as parallelogram, while others have both general-English and mathematics-specific meanings (i.e., subtechnical vocabulary; Monroe & Panchyshyn, 1995), such as product (Hughes, Powell, & Stevens, 2016) while still others have more than one mathematical meaning, for example, square. Similarly, the writing structure of mathematical word problems differs from that of narrative, informational, or persuasive writing. ...

... Many students often face barriers when they encounter questions that require open-ended responses ( Baxter et al., 2005). They usually leave an answer sheet blank or write only one sentence with an answer because they are not proficient at mathematical written communication with mathematical language ( Hughes et al., 2016) and writing strategies (Authors, under review). A long response does not always guarantee a more logical sequence of student-written responses, but we need to guide students to generate more mathematical ideas and organize them in a coherent way within a limited time for a state assessment. ...

... This result may be explained by the fact that our participants were in the middle-grade level (i.e., sixth grade) and the selected MW prompts focused more on mathematical communication for their mathematical reasoning than basic math facts. In addition, because the language of mathematics is different from the one of writing in other domains ( Hughes et al., 2016), sentence writing fluency may not be a unique contributor for quality of student-written response for mathematical reasoning on its own. Considering that the regression model including all of the variables was statistically significant, we can conclude that all of them in a combined model played a role as predictors for the state assessment score of mathematical reasoning, even if individual factors did not. ...

The benefits of content writing are widely recognized, yet there are still few studies that evaluate the benefits and challenges of mathematical writing (MW) for students. The National Council of Teachers of Mathematics and the Common Core State Standards for Mathematics have an emphasis on written communication of mathematical reasoning as an integral part of curriculum, peer-reviewed publications on this topic are lacking. This study evaluates the effects of self-regulated strategy development (SRSD) intervention on sixth-grade students’ MW. Findings indicate that the intervention had significant impact on MW outcomes. Additionally, the five predictors assessed prior to the intervention accounted for approximately 73.6% of variance in MW quality. Implications for the field and future research are discussed.

... Since mathematical writing requires both mathematics and writing proficiencies , requirements for written expression of mathematical reasoning increases cognitive loads for students with LD who have challenges with mathematics fact fluency, mathematics conceptual knowledge, and application mathematics in context (Jitendra, Sczesniak, & Deatline-Buchman, 2005;Moran et al., 2014), and mathematics language (Hughes, Powell, & Stevens, 2016). ...

... The absence of explicit instruction for mathematical vocabulary and students' lack of conceptual understanding about fractions and decimals resulted in inaccurate and infrequent mathematical language use. Especially for students with mathematical disabilities and difficulties, teachers need to allocate the instructional time for mathematical language (Hughes et al., 2016;Powell, Stevens, & Hughes, 2019) and teach students to make connections between new and previously understood terms (Riccomini, Smith, Hughes, & Fries, 2015). ...

... First, students with LD need strategic instruction to support development and understanding of mathematics. Using clear and concise mathematical language (Hughes et al., 2016) to explicitly teach mathematical vocabulary terms (Riccomini et al., 2015) and mathematical writing structures may not only allow better access to the meaning of word problems, but also to increase accurate expressions of mathematics. Frequent and dynamic discourse also benefits student to correct misconception in the process of mathematical reasoning (Xin et al., 2016. ...

Assessment results from two open-construction response mathematical tasks involving fractions and decimals were used to investigate written expression of mathematical reasoning for students with learning disabilities. The solutions and written responses of 51 students with learning disabilities in fourth and fifth grade were analyzed on four primary dimensions: (a) accuracy, (b) five elements of mathematical reasoning, (c) five elements of mathematical writing, and (d) vocabulary use. Results indicate most students were not accurate in their problem solution and communicated minimal mathematical reasoning in their written expression. In addition, students tended to use general vocabulary rather than academic precise math vocabulary and students who provided a visual representation were more likely to answer accurately. To further clarify the students struggles with mathematical reasoning, error analysis indicated a variety of error patterns existed and tended to vary widely by problem type. Our findings call for more instruction and intervention focused on supporting students mathematical reasoning through written expression. Implications for research and practice are presented.

... This model estimated random effects for classroom intercepts. We conducted analyses using Mplus (Mplus 7.31; Muthén & Muthén, 1998-2016. ...

... In turn, educators who teach courses in teacher-preparation programs should understand the importance of mathematics vocabulary as it relates to understanding mathematics, and provide mathematics methods courses with adequate attention paid to the language of mathematics. Finally, all educators should focus on the precision of mathematics vocabulary used during instruction and understand when formal vocabulary should replace informal vocabulary (e.g., denominator instead of bottom number; Hughes, Powell, & Stevens, 2016). With educators providing more precision and focusing on mathematics vocabulary, students may develop a deeper connection to mathematics and demonstrate robust mathematics performance. ...

To read mathematics textbooks, answer questions on mathematics assessments, and understand educator and student communication, students must develop an understanding of the academic language of mathematics. A primary aspect of academic language is vocabulary. In this study, we focused on the mathematics-vocabulary performance of students in 3rd and 5th grade. We designed and implemented a measure of key mathematics vocabulary in the late elementary grades, and we compared performance on this measure to scores from general vocabulary and mathematics computation measures. Student performance at both grades was variable, with a 62-point range at 3rd grade and a 95-point range at 5th grade. General vocabulary and mathematics computation were significant predictors of mathematics vocabulary, but the influence of these predictors differed by mathematics-vocabulary performance levels.

... • When referring to equations, such as 5 + 2 = 7, aloud it is helpful to say, "Five plus two is equal to seven" or "Five plus two is the same amount as seven," instead of "Five plus two equals seven" or "Five plus two makes seven" and to encourage children to use this more precise language as well. The use of precise and accurate mathematical language is key to supporting children's conceptual understanding of mathematics (Hughes, Powell, and Stevens 2016). Similarly, refer to these arithmetic problems as equations or mathematics problems rather than number sentences. ...

Modify arithmetic problem formats to make the relational equation structure more transparent. We describe this practice and three additional evidence-based practices: (1) introducing the equal sign outside of arithmetic, (2) concreteness fading activities, and (3) comparing and explaining different problem formats and problem-solving strategies.

... These findings support that concrete manipulatives may help students develop the conceptual understanding of abstract concepts in mathematics (Maccini & Gagnon, 2000;Root, Browder, Saunders, & Lo, 2017). Furthermore, VM is particularly effective in teaching a diverse range of skills to students with ASD as it offers instruction in a systematic and visual manner using clear and consistent language, while helping students pay focused attention on the task (Hughes, Powell, & Stevens, 2016;Hughes & Yakubova, 2019;Yakubova et al., 2015). The structure of the intervention allows for various dosage, based on need, and perhaps Robert would have benefitted from more sessions and opportunities to practice the skill to mastery. ...

Background
With the surge of intervention research examining ways of supporting students with autism spectrum disorder (ASD) in inclusive settings, there remains a need to examine how technology supports could enhance students’ learning by offering one size fits one instruction. Furthermore, intervention studies focused on teaching students with ASD how to solve fractions are scarce.
Aims
The purpose of this research study was to examine the effects of providing instruction via video modeling (VM), concrete manipulatives, a self-monitoring checklist, and practice for comprehension check on the accuracy of fraction problem solving of three middle school students with ASD.
Methods and procedures
Through the use of single-case multiple probe across students experimental design, we examined whether a functional relation existed between the intervention and students’ improved accuracy of solving simple proper fraction problems.
Outcomes and results
All three students improved the accuracy of solving simple proper fraction problems from baseline to intervention sessions and two students generalized the skill to solving whole proper fraction problems.
Conclusions and implications
The intervention consisting of VM and concrete manipulatives along with additional behavioral strategies offers an option for teachers to accommodate diverse learning needs of students with ASD in a variety of settings.

... Research also points to the importance of using precise mathematics vocabulary to engage in meaningful discussion and discourse about mathematics (Hughes, Powell, & Stevens, 2016). For example, the phrase "5 is greater than 3" is more precise than saying, "5 is bigger than 3." While the mathematics practice standards do not directly prescribe the use of precise language, the "Attend to Precision" mathematics practice assumes that teachers and students use correct vocabulary. ...

Within a multitiered system of support (MTSS), students who struggle to learn mathematics often receive core instruction and supplemental intervention in different settings, with different teachers and different sets of curriculum materials, all of which can result in poor alignment. This curriculum crosswalk describes how three sets of materials commonly used to provide core instruction and intervention differ with regard to mathematics practices and vocabulary. The results indicate that there is little overlap among all three programs for the majority (n = 6) of the mathematics practices, and very little overlap in mathematics vocabulary (ranging from 6.3 to 24 percent). We also provide a set of research‐based instructional recommendations intended to help teachers address gaps and improve alignment of core instruction and intervention.

... Multiple factors contribute to these foundational skills, including the acquisition of a mathematics vocabulary. Indeed, a relation between students' mathematics vocabulary and their mathematics achievement has been demonstrated in elementary-school students (Hughes et al., 2016;, but much less is known about this link for older students or students outside of the United States (US). In this study, we address this gap by examining the association between eighth graders' (ages 13-14) mathematics vocabulary and their mathematics achievement for students from the US and Turkey, controlling for general vocabulary. ...

This study explored the relationship between mathematics vocabulary and mathematics achievement, controlling general vocabulary, for eighth graders (ages 13 to 14) from the United States (US; n = 89) and Turkey (n = 188). The mathematics achievement of Turkish students fell into higher- and lower-achieving groups, with students in the higher-achieving group showing achievement levels like their US peers. For US students and the corresponding higher-achieving Turkish students, mathematics vocabulary predicted mathematics achievement and mediated the relation between general vocabulary and mathematics achievement. For the lower-achieving Turkish students, only general vocabulary predicted mathematics achievement. The pattern of results extends findings from studies of younger students and is interpreted in terms of the domain-general and domain-specific contributions to mathematics achievement.

... In addition, most SETs are responsible for HLP 20, and they will require curricular resources that support intensive instruction in their particular content area. For example, SETs providing intensive instruction in math will need curricula that provides them with support for skills specific to intensive math instruction, including task analyzing math skills, consistently defining key vocabulary, using a gradual transition from concrete to abstract representations of math concepts, teaching strategies to solve word problems, and providing ample opportunities to practice with feedback (e.g., Hughes, Powell, & Stevens, 2016;Powell & Driver, 2015). Curricula that support intensive instruction in particular content areas are essential for SETs to be able to learn how to provide intensive instruction and enact it well. ...

... In addition, most SETs are responsible for HLP 20, and they will require curricular resources that support intensive instruction in their particular content area. For example, SETs providing intensive instruction in math will need curricula that provides them with support for skills specific to intensive math instruction, including task analyzing math skills, consistently defining key vocabulary, using a gradual transition from concrete to abstract representations of math concepts, teaching strategies to solve word problems, and providing ample opportunities to practice with feedback (e.g., Hughes, Powell, & Stevens, 2016;Powell & Driver, 2015). Curricula that support intensive instruction in particular content areas are essential for SETs to be able to learn how to provide intensive instruction and enact it well. ...

Improving educational outcomes for students with disabilities and others who struggle in school largely depends on teachers who can deliver effective instruction. Although many effective practices have been identified to address the academic and behavioral needs of students who struggle in school, including those with disabilities, these practices are not used extensively in classrooms. This article provides a rationale for and description of major changes that are occurring in teacher preparation programs that are designed to improve the practice of beginning teachers. This is followed by a description of a set of high-leverage practices that was recently approved by the Council for Exceptional Children. These practices represent an initial attempt to delineate a core curriculum for special education teacher preparation to support the changes that are occurring in teacher education.

... In addition, most SETs are responsible for HLP 20, and they will require curricular resources that support intensive instruction in their particular content area. For example, SETs providing intensive instruction in math will need curricula that provides them with support for skills specific to intensive math instruction, including task analyzing math skills, consistently defining key vocabulary, using a gradual transition from concrete to abstract representations of math concepts, teaching strategies to solve word problems, and providing ample opportunities to practice with feedback (e.g., Hughes, Powell, & Stevens, 2016;Powell & Driver, 2015). Curricula that support intensive instruction in particular content areas are essential for SETs to be able to learn how to provide intensive instruction and enact it well. ...

Induction is designed to support teachers’ effectiveness, improve their students’ learning, and foster their retention. We consider how high-leverage practices (HLPs) might provide an instructional framework for special education teacher (SET) induction. With sensemaking theory as a conceptual foundation, we posit that, by structuring induction experiences and instructional conditions around HLPs, schools and districts can send more coherent messages about effective instruction, thereby easing new SETs’ efforts to make sense of their roles. We first provide a brief review of research on new SETs’ experiences. Next, we consider how specific induction components (i.e., professional development and mentoring, teacher evaluation, and collaboration) and instructional conditions (i.e., collaboration, instructional curricula and resources, and schedules) might be structured to support SETs’ learning of and use of these HLPs. We conclude with considerations for researchers and practitioners.

... repetition contribute to students' understanding and mastery. Consistent language affords students the opportunity to make connections and scaffolds to independent practice (Hughes, Powell, & Stevens, 2016;Yakubova et al., 2016). ...

With the increasing attention and surge of empirical research in providing academic instruction for students with autism spectrum disorder (ASD) comes the need to provide teachers with research-supported strategies. Using one evidence-based strategy for teaching mathematics to students with high incidence disabilities, and another for teaching primarily nonacademic skills to students with ASD, this article offers practical tips for implementing a unique and innovative approach to providing mathematics instruction to students with ASD across a variety of instructional contexts. Guidelines to develop and implement the concrete-representational-abstract (CRA) sequencing instruction with video-based instruction (VBI) in teaching mathematics to students with ASD are provided.

... As such, educators must be mindful and purposeful about the language used to teach math. In 2016, we provided recommendations for elementary educators related to improving math language for elementary students (Hughes, Powell, & Stevens, 2016). Just as math concepts and procedures build in complexity through school years, so does the language necessary to access math and distinguish between or among concepts. ...

... Similarly, understanding verbal instructions that use words such as "add," "sum," or "digit" depends upon students' knowledge of those math-specific words. Further, some researchers contend that there are implicit difficulties associated with learning math vocabulary that may not be associated with general vocabulary (Hughes, Powell, & Stevens, 2016;Thomas et al., 2015;Thompson & Rubenstein, 2000). This combination of math-specific vocabulary's complexity with its omnipresence in math instruction has led researchers to posit that it is more important than general vocabulary in predicting math performance. ...

The purpose of this study was to examine the associations between child language ability and mathematics performance. First- and second-grade children (N = 365) were assessed on language ability and mathematics performance. Structural equation models revealed that receptive syntax and a broad screening tool significantly predicted math performance, while vocabulary did not. Path analyses corroborate these findings, with receptive syntax emerging as the only significant predictor of all indicators of mathematics performance. We conclude that syntax is a strong predictor of mathematics performance while vocabulary is not. Further, although many studies use receptive vocabulary to index language, it may not be the most predictive of or the best proxy for language ability in young children in the context of mathematics learning.

... Similarly, understanding verbal instructions that use words such as "add," "sum," or "digit" depends upon students' knowledge of those math-specific words. Further, some researchers contend that there are implicit difficulties associated with learning math vocabulary that may not be associated with general vocabulary (Hughes, Powell, & Stevens, 2016;Thomas et al., 2015;Thompson & Rubenstein, 2000). This combination of math-specific vocabulary's complexity with its omnipresence in math instruction has led researchers to posit that it is more important than general vocabulary in predicting math performance. ...

The purpose of this study was to examine the associations between child language ability and mathematics performance. First- and second-grade children (N = 365) were assessed on language ability and mathematics performance. Structural equation models revealed that receptive syntax and a broad screening tool significantly predicted math performance, while vocabulary did not. Path analyses corroborate these findings, with receptive syntax emerging as the only significant predictor of all indicators of mathematics performance. We conclude that syntax is a strong predictor of mathematics performance while vocabulary is not. Further, although many studies use receptive vocabulary to index language, it may not be the most predictive of or the best proxy for language ability in young children in the context of mathematics learning.

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.

This article presents a checklist of 10 evidence-based practices for educators to apply in mathematics instruction for students with learning disabilities. The checklist is “actionable,” meaning the items on the checklist can be put into action immediately. It provides practical strategies teachers can adopt to fit their lessons regardless of their specific mathematical domain areas or student grade level. The focus of this article is translating research of evidence-based strategies into practice for mathematics instruction.

Number talks are increasingly used in general education mathematics classes to engage students. Yet, and despite the potential benefits, number talks are given limited attention for students with high-incidence disabilities in special education settings. This article presents special education teachers with both the why and, more importantly, the how for implementing number talks to support students with high-incidence disabilities in special education settings. Specifically, the authors address how number talks can serve as both a formative assessment and an intervention for fluency and activating students’ background knowledge to be successful in general education settings. The article also provides suggestions for implementing number talks with fidelity and flexibility (e.g., use of manipulatives, pictorial representations, and teacher explicit instruction of numerical strategies).

This study examines four of the most commonly-used core mathematics curricula in the USA for evidence of support for research-based instructional strategies for mathematics vocabulary in first and second grade. Content analyses of the teachers’ editions of two units for each grade level were analyzed per curriculum (n = 16). Statistically significant differences among curricula were found for number of target words (range 6–51 per unit), level of difficulty of terms (basic to technical), and number of support strategies per word. Multiple means of representation varied in terms of verbal and non-verbal strategies for target terms. These differences indicate children are experiencing substantially different mathematics vocabulary learning opportunities, which may impact later mathematics achievement. Implications for practice, curriculum development, and future research are addressed.

In recent years, the idea of language influencing the cognitive development of an understanding of place value has received increasing attention. This study explored the influence of using explicit number names on prekindergarten and kindergarten students’ ability to rote count, read two-digit numerals, model two-digit numbers, and identify the place value of individual digits in two-digit numerals. Through individual student interviews, pre and post assessments were administered to evaluate rote counting, reading five two-digit numerals, modeling five two-digit numbers, and identifying place value in two two-digit numerals. Chi-square tests for independence showed two significant relations: (a) the relationship between the traditional and explicit group membership on the post assessment of modeling twodigit numbers, and (b) the relationship between place value identifications and group membership. Analysis of the children’s performance and error patterns revealed interesting differences between children taught with explicit number names and children taught with traditional number names. The improvement of the explicit group overall exceeded the improvement of the traditional group. This study indicates that teaching children to use explicit number names can, indeed, have a positive influence on their understanding of place value.

Vocabulary understanding is a major contributor to overall comprehension in many content areas including mathematics. Effective methods for teaching vocabulary in all content areas are diverse and long standing. The importance of teaching and learning the language of mathematics is vital for the development of mathematical proficiency. Students’ mathematical vocabulary learning is a very important part of their language development and ultimately mathematical proficiency. This article draws upon currently available research-based evidence to provide a: (a) rationale for teaching vocabulary, (b) review of research that supports the importance of teaching mathematics vocabulary, and (c) description of specific strategies to teach mathematics vocabulary. Implications and the need for future research are also addressed.

Geometry is a course that is increasingly required for students to graduate from high school. However, geometric concepts can pose a great challenge to high school youth with math difficulties. Although research on teaching geometry to students with mathematics difficulties is limited, teachers are challenged daily with providing support for youth who do not make adequate progress through high-quality core mathematics instruction. This article provides practical and promising practices for providing additional small group support (i.e., Tier II interventions) to youth that promote understanding of finding the area of a trapezoid. Specifically, the authors focus on the use of a graduated instructional sequence and peer-mediated instruction within an explicit instruction model. Recommendations and sample lessons are provided.

Despite the current theoretical momentum for the importance of academic English and the acknowledgment that academic materials increase in complexity through the grades, little empirical attention has been devoted to the role of academic English in academic achievement. This study examined the amount of variance in academic achievement explained by academic word knowledge for diverse middle school students. A linguistically and socioeconomically diverse sample of grade 7 and 8 students (N�=339) was administered measures of overall breadth of vocabulary knowledge and general (i.e., cross-discipline) academic word knowledge, and the explanation of variance in standardized academic achievement tests across 4 disciplines was explored. For the entire sample, knowledge of general academic words explained a considerable and significant amount of variance in academic achievement across 4 disciplines. Findings lend empirical support to current calls for providing academic language support for early adolescents from non-native English speaking and low socioeconomic backgrounds.

Students often misinterpret the equal sign (=) as operational instead of relational. Research indicates misinterpretation of the equal sign occurs because students receive relatively little exposure to equations that promote relational understanding of the equal sign. No study, however, has examined effects of nonstandard equations on the equation solving and equal-sign understanding of students with mathematics difficulty (MD). In the present study, second-grade students with MD (n = 51) were randomly assigned to standard equations tutoring, combined tutoring (standard and nonstandard equations), and no-tutoring control. Combined tutoring students demonstrated greater gains on equation-solving assessments and equal-sign tasks compared to the other two conditions. Standard tutoring students demonstrated improved skill on equation solving over control students, but combined tutoring students' performance gains were significantly larger. Results indicate that exposure to and practice with nonstandard equations positively influence student understanding of the equal sign.

This study investigates the oral academic language used by English as a second language prepared teachers during content area instruction in five upper elementary classrooms in the United States. Using ethnographic and sociolinguistic perspectives the authors examine the oral, academic language exposure students received from their teachers during mathematics, social studies, and language arts instruction in mainstream classrooms.Findings suggest that English language learners in these classrooms (1) had limited opportunities to hear the specialized language of the content areas, and (2) encountered a variety of opaque terms (e.g., homophones, idiomatic expressions), which can potentially hinder understanding. These findings have important implications for: understanding the subtle and overt aspects of the language of school, increasing our understanding of teacher talk during content area instruction, and preparing educators to teach the unique linguistic demands of each academic content area.

An important issue in understanding mathematical cognition involves the similarities and differences between the magnitude representations associated with various types of rational numbers. For single-digit integers, evidence indicates that magnitudes are represented as analog values on a mental number line, such that magnitude comparisons are made more quickly and accurately as the numerical distance between numbers increases (the distance effect). Evidence concerning a distance effect for compositional numbers (e.g., multidigit whole numbers, fractions and decimals) is mixed. We compared the patterns of response times and errors for college students in magnitude comparison tasks across closely matched sets of rational numbers (e.g., 22/37, 0.595, 595). In Experiment 1, a distance effect was found for both fractions and decimals, but response times were dramatically slower for fractions than for decimals. Experiments 2 and 3 compared performance across fractions, decimals, and 3-digit integers. Response patterns for decimals and integers were extremely similar but, as in Experiment 1, magnitude comparisons based on fractions were dramatically slower, even when the decimals varied in precision (i.e., number of place digits) and could not be compared in the same way as multidigit integers (Experiment 3). Our findings indicate that comparisons of all three types of numbers exhibit a distance effect, but that processing often involves strategic focus on components of numbers. Fractions impose an especially high processing burden due to their bipartite (a/b) structure. In contrast to the other number types, the magnitude values associated with fractions appear to be less precise, and more dependent on explicit calculation. (PsycINFO Database Record (c) 2013 APA, all rights reserved).

In keeping with the early childhood chapter of Principles and Standards for School Mathematics , this department examines activities and children's thinking in geometry and, in the next issue, number. From prekindergarten to grade 12, the Geometry Standard addresses four main areas: properties of shapes, location and spatial relationships, transformations and symmetry, and visualization. For each area, we quote the goal of the Standard and the associated early-childhood expectations. We then present snippets of research and sample activities to develop ideas within each area with students.

This study examined math growth trajectories by disability category, gender, race, and socioeconomic status using a nationally representative sample of students ages 7 to 17. The students represented 11 federal disability categories. Compared with the national norming sample, students in all 11 disability categories had lower math achievement levels and slower growth in elementary school. In secondary school, however, the math growth rate slowed down and was similar for all students. Among students with disabilities, those with speech or visual impairments had the highest math achievement, and those with multiple disabilities or intellectual disability had the lowest. Relative to students with learning
disabilities on calculation, growth rates for students with autism were significantly slower and those for students with speech impairments decelerated significantly faster. For students with disabilities, gender, White–Black, and socioeconomic status math achievement gaps were significant and stable over time, whereas White–Hispanic math achievement gaps widened over time.

The purposes of this study were to assess the efficacy of remedial tutoring for 3rd graders with mathematics difficulty, to investigate whether tutoring is differentially efficacious depending on students' math difficulty status (mathematics difficulty alone vs. mathematics plus reading difficulty), to explore transfer from number combination (NC) remediation, and to examine the transportability of the tutoring protocols. At 2 sites, 133 students were stratified on mathematics difficulty status and site and then randomly assigned to 3 conditions: control (no tutoring), tutoring on automatic retrieval of NCs (i.e., Math Flash), or tutoring on word problems with attention to the foundational skills of NCs, procedural calculations, and algebra (i.e., Pirate Math). Tutoring occurred for 16 weeks, 3 sessions per week and 20-30 min per session. Math Flash enhanced fluency with NCs with transfer to procedural computation but without transfer to algebra or word problems. Pirate Math enhanced word problem skill as well as fluency with NCs, procedural computation, and algebra. Tutoring was not differentially efficacious as a function of students' mathematics difficulty status. The tutoring protocols proved transportable across sites.

Imagine a teacher running her hands across her desk as she tells her students, “A plane is a perfectly flat surface.” The students listen quietly, but one of them is thinking, “I thought a plane was something that flies.”

Overgeneralizing commonly accepted practices, using imprecise vocabulary, and relying on tips and tricks that do not promote conceptual mathematical understanding can lead to misunderstanding later in students' math careers.

ELLs need to practice using the language in their speech. Teachers can ask students to restate the definition in their own words and provide opportunities for students to use academic vocabulary in discussions. Chunking (instead of teaching inch in isolation, also teach foot, centimeter, and yard) helps students develop their schema and mentally organize their new vocabulary. Gestures and movements will help students remember what words mean. They'll be able to associate the meaning of a new word to a movement or gesture. Requiring students to keep a journal for math terms that contains definitions or nonlinguistic representations of words helps them learn better, too.

Teachers and parents often use trade books to introduce or reinforce mathematics concepts. To date, an analysis of the early numeracy content of trade books has not been conducted. Consequently, this study evaluated the properties of numbers and counting within trade books. We coded 160 trade books targeted at establishing early numeracy skill in children to determine the numbers included, the representations of number presented, and how books used representations of number to inform children about numbers, including counting. The main findings included limited opportunity to learn the number 0 and numbers beyond 10 as well as limited exposure to multiple representations of number deemed necessary to build strong number understanding and counting skills. We discuss practical implications for the selection and use of trade books about number with young children.

The article discusses the importance of mathematical vocabulary, difficulties students encounter in learning this vocabulary, and some strategies for teachers to use in instruction. Two general methods for teaching vocabulary are discussed: context and explicit vocabulary instruction. The article summarizes these two methods as they apply to mathematical vocabulary instruction and describes one example of a combined approach.

This article synthesizes research by applied linguists and mathematics educators to highlight the linguistic challenges of mathematics and suggest pedagogical practices to help learners in mathematics classrooms. The linguistic challenges include the multi-semiotic formations of mathematics, its dense noun phrases that participate in relational processes, and the precise meanings of conjunctions and implicit logical relationships that link elements in mathematics discourse. Research on pedagogical practices supports developing mathematics knowledge through attention to the way language is used, suggesting strategies for moving students from informal, everyday ways of talking about mathematics into the registers that construe more technical and precise meanings.

Competence with fractions predicts later mathematics achievement, but the codevelopmental pattern between fractions knowledge and mathematics achievement is not well understood. We assessed this codevelopment through examination of the cross-lagged relation between a measure of conceptual knowledge of fractions and mathematics achievement in sixth and seventh grades (N=212). The cross-lagged effects indicated that performance on the sixth grade fractions concepts measure predicted 1-year gains in mathematics achievement (ß=.14, p<.01), controlling for the central executive component of working memory and intelligence, but sixth grade mathematics achievement did not predict gains on the fractions concepts measure (ß=.03, p>.50). In a follow-up assessment, we demonstrated that measures of fluency with computational fractions significantly predicted seventh grade mathematics achievement above and beyond the influence of fluency in computational whole number arithmetic, performance on number fluency and number line tasks, central executive span, and intelligence. Results provide empirical support for the hypothesis that competence with fractions underlies, in part, subsequent gains in mathematics achievement.

The goal of this first major report from the Western Reserve Reading Project Math component is to explore the etiology of the relationship among tester-administered measures of mathematics ability, reading ability, and general cognitive ability. Data are available on 314 pairs of monozygotic and same-sex dizygotic twins analyzed across 5 waves of assessment. Univariate analyses provide a range of estimates of genetic (h(2) = .00 -.63) and shared (c(2) = .15-.52) environmental influences across math calculation, fluency, and problem solving measures. Multivariate analyses indicate genetic overlap between math problem solving with general cognitive ability and reading decoding, whereas math fluency shares significant genetic overlap with reading fluency and general cognitive ability. Further, math fluency has unique genetic influences. In general, math ability has shared environmental overlap with general cognitive ability and decoding. These results indicate that aspects of math that include problem solving have different genetic and environmental influences than math calculation. Moreover, math fluency, a timed measure of calculation, is the only measured math ability with unique genetic influences.

In this study, the author aimed at measuring how much limited working memory capacity constrains early numerical development before any formal mathematics instruction. To that end, 4- and 5-year-old children were tested for their memory skills in the phonological loop (PL), visuo-spatial sketchpad (VSSP), and central executive (CE); they also completed a series of tasks tapping their addition and counting skills. A general vocabulary test was given to examine the difference between the children's numerical and general vocabulary. The results indicated that measures of the PL and the CE, but not those of the VSSP, were correlated with children's performance in counting, addition and general vocabulary. However, the predictive power of the CE capacity was significantly stronger than that of the PL capacity. Poor CE capacity should thus be taken into consideration when identifying children at risk of experiencing learning disabilities.

Developmental and school-related changes in basic number, counting, and arithmetic skills from infancy to old age are reviewed. Nearly all of the quantitative competencies that emerge during infancy and the preschool years appear to reflect the operation of a biological primary, or inherent, cognitive system, and appear to be universal in their expression and development. In contrast, most of the basic quantitative competencies acquired in school and that are of importance in industrial societies do not have a direct inherent foundation. As a result, the development of these secondary quantitative abilities varies considerably with educational practices and can, and often does, vary from one country or generation to the next. Variability in the development of secondary quantitative abilities greatly complicates the study of the relation between pathological (e.g., dyscalculia due to stroke) and age-related processes and these abilities.

To determine the natural history of developmental dyscalculia (DC) and factors impacting on its prognosis, we performed a prospective six-year longitudinal study. One hundred and forty children of normal intelligence diagnosed with DC in the fifth grade of elementary school were re-examined for dyscalculia three and six years later, in eighth (n=123) and eleventh (n=104; 41 males, 63 females) grades respectively. Mean age of the children in fifth grade was 11 years 1 month (SD 4 months), in eighth grade 14 years 2 months (SD 1 month), and in eleventh grade 17 years 2 months (SD 5 months). The assessment included standardized arithmetic, reading and writing tests, behavioural rating scales, information on socioeconomic status, educational interventions, and familial learning problems. Participants in eleventh grade were recategorized as having DC if their score on the arithmetic test was not more than the fifth centile for grade. At the six-year follow-up, 99/104 (95%) children diagnosed with dyscalculia in fifth grade were still performing poorly in arithmetic, scoring within the lowest quartile for their grade, and 42/104 (40%) were recategorized with DC. Chronicity of DC was associated with severity of the dyscalculia in fifth grade (p<0.05), lower IQ (p<0.01), inattention (p<0.01), and writing problems (p<0.01). Thus, DC is an enduring specific learning difficulty, persisting into late adolescence in almost half of affected individuals.

Influence of general vocabulary and mathematics knowledge on mathematics vocabulary

- S R Powell
- G Nelson

Powell, S. R., & Nelson, G. (2016). Influence
of general vocabulary and mathematics
knowledge on mathematics vocabulary.
Manuscript submitted for publication.

Magnitude comparison with different types of rational numbers Journal of Experimental Psychology: Human Perception and Performance Teaching geometry to students with math difficulties using a graduated and peer-mediated instruction in a response-to-intervention model. Preventing School Failure

- M Dewolf
- M A Grounds
- M Bassok
- K J Holyoak
- A Dobbins
- J C Gagnon
- T Ulrich

DeWolf, M., Grounds, M. A., Bassok, M.,
& Holyoak, K. J. (2014). Magnitude
comparison with different types
of rational numbers. Journal of
Experimental Psychology: Human
Perception and Performance, 40, 71-82.
doi:10.1037/a0032916
Dobbins, A., Gagnon, J. C., & Ulrich, T.
(2014). Teaching geometry to students
with math difficulties using a graduated
and peer-mediated instruction in
a response-to-intervention model.
Preventing School Failure, 58, 17-25. doi:
10.1080/1045988x.2012.743454

The connections among general vocabulary, mathematics computation, and mathematics vocabulary knowledge at Grades 3 and 5

- S R Powell
- M K Driver
- G Roberts

Powell, S. R., Driver, M. K., & Roberts, G.
(2016). The connections among general
vocabulary, mathematics computation,
and mathematics vocabulary knowledge
at Grades 3 and 5. Manuscript submitted
for publication.

Developmental dyscalculia: A prospective six-year follow-up. Developmental Medicine and Child Neurology Evidence for the importance of academic word knowledge for the academic achievement of diverse middle school students

- R S Shalev
- O Manor
- V Grosstsur

Shalev, R. S., Manor, O., & GrossTsur, V. (2005). Developmental
dyscalculia: A prospective six-year
follow-up. Developmental Medicine
and Child Neurology, 47, 121-125.
doi:10.1111/j.1469-8749.2005.tb01100.x
Townsend, D., Filippini, A., Collins, P., &
Biancarosa, G. (2012). Evidence for the
importance of academic word knowledge
for the academic achievement of diverse
middle school students. Elementary
School Journal, 112, 497-518.
doi:10.1086/663301

Math growth trajectories of students with disabilities: Disability category, gender, racial, and socioeconomic status differences from ages 7 to 17

- X Wei
- K B Lenz
- J Blackorby

Wei, X., Lenz, K. B., & Blackorby, J.
(2013). Math growth trajectories of
students with disabilities: Disability
category, gender, racial, and
socioeconomic status differences
from ages 7 to 17. Remedial and
Special Education, 34, 154-165.
doi:10.1177/0741932512448253

Remediating number combination and word problem deficits among students with mathematics difficulties: A randomized control trial From infancy to adulthood: The development of numerical abilities

- L S Fuchs
- S R Powell
- P M Seethaler
- P T Cirino
- J M Fletcher
- D Fuchs
- R O Zumeta
- D C Geary

Fuchs, L. S., Powell, S. R., Seethaler, P. M.,
Cirino, P. T., Fletcher, J. M., Fuchs, D.,
... Zumeta, R. O. (2009). Remediating
number combination and word problem
deficits among students with mathematics
difficulties: A randomized control trial.
Journal of Educational Psychology, 101,
561-576. doi:10.1037/a0014701
Geary, D. C. (2000). From infancy to
adulthood: The development of
numerical abilities. European Child
and Adolescent Psychiatry, 9(Suppl. 2),
11-16. doi:10.1007/s007870070004

! A systematic analysis of number representations in children's books

! A systematic analysis of number
representations in children's books. Early
Education and Development, 26, 377-398.
doi:10.1080/10409289.2015.994466