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Abstract

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. ...
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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. ...
Article
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. ...
Article
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. ...
Article
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. ...
Article
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. ...
Article
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. ...
Article
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. ...
Article
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. ...
Article
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). ...
Article
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. ...
Article
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. ...
Article
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.
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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.
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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.
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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.
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