LSHSS

Research Article

The Relationship Between Mathematics and Language: Academic Implications for Children With Specific Language Impairment and English Language Learners Mary Alt,a Genesis D. Arizmendi,a and Carole R. Beala

Purpose: The present study examined the relationship between mathematics and language to better understand the nature of the deficit and the academic implications associated with specific language impairment (SLI) and academic implications for English language learners (ELLs). Method: School-age children (N = 61; 20 SLI, 20 ELL, 21 native monolingual English [NE]) were assessed using a norm-referenced mathematics instrument and 3 experimental computer-based mathematics games that varied in language demands. Group means were compared with analyses of variance. Results: The ELL group was less accurate than the NE group only when tasks were language heavy. In contrast, the group with SLI was less accurate than the groups with NE and ELLs on language-heavy tasks and some language-light

tasks. Specifically, the group with SLI was less accurate on tasks that involved comparing numerical symbols and using visual working memory for patterns. However, there were no group differences between children with SLI and peers without SLI on language-light mathematics tasks that involved visual working memory for numerical symbols. Conclusion: Mathematical difficulties of children who are ELLs appear to be related to the language demands of mathematics tasks. In contrast, children with SLI appear to have difficulty with mathematics tasks because of linguistic as well as nonlinguistic processing constraints.

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of learning the language of instruction. A language other than English is spoken in the home of nearly 20% of the U.S. population, and of that group, nearly a quarter reports that they are not able to speak English well (Shin & Kominski, 2010). In addition, some children who are English language learners (ELLs) may also have a language impairment that may be difficult to recognize. That is, they may have problems learning a second language because of their language impairment. Both children with SLI and who are ELLs are at risk for difficulties in learning mathematical skills (Shaftel, BeltonKocher, Glasnapp, & Poggio, 2006). Clearly, the instructional recommendations for children should vary with the nature of their language challenge. However, the specific nature of the mathematics difficulties experienced by children with SLI and by children who are ELLs is not well understood. The present study was designed to compare the performance of children with SLI and those who are ELLs on mathematics tasks specifically designed to vary in terms of the demands

uccess in school depends heavily on children’s ability to understand and use the language of instruction to learn. However, there is much to learn about how, precisely, language proficiency can affect academic performance, particularly mathematics achievement. One clear prediction is that children who face challenges with language will be at a disadvantage compared with their peers who are proficient with language. However, children might have difficulty with language for different reasons. Some children have specific language impairment (SLI). About 7% of kindergarteners (Tomblin et al., 1997) have difficulty learning their native language, despite having normal hearing, cognition, and behavioral development. Others are in the process

a

University of Arizona, Tucson

Correspondence to Mary Alt: [email protected] Editor: Marilyn Nippold Associate Editor: Sonja Pruitt-Lord Received January 11, 2013 Revision received May 21, 2013 Accepted February 12, 2014 DOI: 10.1044/2014_LSHSS-13-0003

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Key Words: children, English language learners, language, education, specific language impairment, mathematics

Disclosure: The authors have declared that no competing interests existed at the time of publication.

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on language, symbolic processes, and visuospatial working memory. As outlined next, these candidate factors were suggested by previous research on mathematics learning by children with SLI and by those who are ELLs.

Children With SLI There is clear evidence that children with early diagnosed language impairment are at risk for later academic difficulties (e.g., Dockrell, Lindsay, & Palikara, 2011; Harrison, McLeod, Berthelsen, & Walker, 2009; Morgan, Farkas, & Wu, 2011). However, there is much to be learned about how, precisely, a language impairment can affect the academic performance of a child with SLI. In particular, the role of language impairment on children’s mathematics learning is not yet well understood. One straightforward possibility is that children with SLI have difficulties with mathematical tasks that are so-called language-heavy—for example, tasks that require verbal counting (Arvedson, 2002) or naming skills such as those for arithmetic problems, where children need to recall number names or facts (Arvedson, 2002; Cowan, Donlan, Newton, & Lloyd, 2005; Donlan, Cowan, Newton, & Lloyd, 2007; Fazio, 1996; 1999; Hall & Segarra, 2007; Koponen, Mononen, Räsänen, & Ahonen, 2006; MainelaArnold, Alibali, Ryan, & Evans, 2011). Other research indicates that children with SLI have deficits related to retrieving number names (e.g., Fazio, 1996, 1999; Hall & Segarra, 2007). These studies suggest that if the language demands of a mathematics task were reduced, children with SLI might show performance comparable to children without language challenges. To address this prediction, tasks designed to vary in language demands were included in the present study. Alternatively, there is evidence to suggest that the mathematics difficulties of children with SLI may involve a more general difficulty with symbols. Recall that language is defined as “a complex and dynamic system of conventional symbols” (American Speech-Language-Hearing Association, 1982). Like language, mathematics involves symbols (e.g., –, 6, %), many of which are context-specific and with meanings that are dependent upon explicit classroom instruction, arguably making them difficult to learn (Pierce & Fontaine, 2009). Literature on children with typical development shows a relationship between symbolic (e.g., specific knowledge of a number) and nonsymbolic mathematical knowledge (e.g., approximate number representations; Gilmore, McCarthy & Spelke, 2007; McNeil & Alibali, 2005; Mundy & Gilmore, 2009; Rittle-Johnson, Siegler & Alibali, 2001; Sherman & Bisanz, 2009). This research suggests that children who have challenges learning linguistic symbols in their native language might also struggle with the representation of mathematical symbols. Consistent with this hypothesis, some studies suggest that children with SLI have difficulty with tasks involving mathematical symbols, including answering written problems (e.g., 2 + 3) in either the spoken or written modality (e.g., Fazio, 1996). Koponen et al. (2006) found that second graders with SLI demonstrated lower performance than age-matched peers on nonverbal mathematics tasks that involved symbols.

However, Donlan and Gourlay (1999) found that 8-year-olds with SLI performed at a level comparable to age-matched peers when asked to make judgments about which of two single-digit or double-digit numbers was larger. Thus, the nature of a specific nonverbal mathematical symbolic deficit in children with SLI is unclear. In the present study, mathematics tasks designed to vary in the use of symbols were included to learn whether the performance of children with SLI would be affected. A third possibility is that problems with counting and number names in children with SLI may be linked to poor working memory skills. In particular, visuospatial working memory seems to play a critical role in mathematics performance. Kyttälä, Aunio, and Hautamäki (2010) provided evidence that children rely on visuospatial skills when solving mathematics tasks, for example, by holding information in mind while solving mental calculations or for representing a number line. Visuospatial representations also appear to be involved in the understanding of place value (Donlan & Gourlay, 1999; Donlan et al., 2007). However, there is no direct indication that children with SLI perform less well on mathematics tasks that involve visuospatial working memory. Cowan et al. (2005) found that children with SLI have lower scores on the Corsi span task than age-matched peers, and performance on that task predicted performance on addition combinations, story problems, and relative magnitude tasks. However, the Corsi span task, in which participants are asked to tap blocks from an array in the same order that an examiner does, is more a measure of spatial skills than it is of visual skills. Although Kyttälä et al. (2010) directly measured visual and spatial working memory, their group did not specifically categorize children in terms of having impaired language or not. Thus, in the present study, tasks designed to vary in the demands on visuospatial working memory were included. Clarifying the nature of the difficulty with mathematics learning (i.e., language-specific, symbol-specific, or related to visuospatial-working-memory) is important for developing appropriate instruction for children with SLI. For example, a language-specific problem might lead to a child answering a mathematics question incorrectly because he does not understand what the word divide means when he hears or reads it. The student’s difficulty is not mathematics-based; it is driven by lack of vocabulary knowledge. A symbolspecific problem might manifest as a child providing an incorrect answer because she does not recognize the symbol for divide (e.g., ) in a problem. She might know the word divide, but if she cannot recognize the corresponding symbol, she will be at a loss. A child with cognitively based problems might recognize both the word and symbol for divide. Unfortunately, the word and symbol alone may not be sufficient to allow him to perform the appropriate solution steps, which would involve retaining the problem information in visuospatial working memory. Effective instruction for each of the cases presented above would most likely look quite different. Without precise knowledge of the nature of the mathematics difficulty, it is nearly impossible to design the most effective treatment plan.

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ELLs There is considerable evidence that children who are not proficient in the language of instruction tend to have lower academic achievement than children whose primary language is also the one that they encounter in school. In the United States, children who are ELLs do not perform as well in school as native English-speaking students (August & Shanahan, 2008). Not only do children who are ELLs score less well in reading and related literacy subjects, but they also perform much less well in mathematics than students whose primary language is English (National Clearinghouse for English Language Acquisition, 2007). Beal, Adams, and Cohen (2010) found that both children who are ELLs and English primary students improved after working with a mathematics instructional software program, but the gap between the two groups remained; that is, the children who were ELLs improved but never “caught up” with their peers. Similar findings have been reported in Europe where children of immigrant families have lower mathematics achievement than children whose primary language is also the language of instruction in school (Kempert, Saalbach, & Hardy, 2011). The achievement gap remains even after factors such as differences in socioeconomic status have been considered (Kempert et al., 2011). The precise reason children who are ELLs have more trouble than English primary students with mathematics is not clearly established. Some studies suggest that at least some early numerical concepts are acquired by children relatively independently of language, such as an understanding of relative quantity and conservation of number (Arvedson, 2013). Also, there are preliminary indications from neuroscience research to suggest that mathematical and language processing may involve relatively distinct brain networks (Cappelletti, Butterworth, & Kopelman, 2001; Salillas & Wicha, 2012). This work implies that the use of languageheavy assessments may mask the actual numerical knowledge of children who are ELLs. Consistent with this prediction, some studies show that English learners perform less well on mathematics problems that involve a high degree of linguistic complexity, compared with problems with equivalent mathematics but with simpler language (Abedi & Lord, 2001; Abella, Urrutia, & Shneyderman, 2005; Arvedson, 2002; Beal & Barbu, 2010; Martinello, 2008; Shaftel et al., 2006). Similarly, Alt, Arizmendi, Beal, and Hurtado (2013) found that the performance of children who are ELLs on a standardized mathematics assessment improved when the children were offered the opportunity to respond in Spanish to items that they had missed when presented in English. An alternate possibility is that the conceptual understanding of children who are ELLs might actually be weaker than that of their English primary peers. For one thing, learning in a nonprimary language involves additional cognitive demands. Emerging evidence from research on adult bilingualism indicates that there is a cognitive cost associated with shifting from one language to another (Saalbach, Eckstein, Andri, Hobi, & Grabner, 2013; Spelke & Tsivkin, 2001). For example, the person might translate the word problem

into his or her primary language in order to form a problem representation, but this “cognitive switching” may increase the chances of errors. Children who are ELLs may also be impeded by a lack of familiarity with mathematics vocabulary, including both “technical” words that have a specific mathematical meaning (e.g., triangle) and “subtechnical” words that may be confusing because they have a familiar meaning as well as a mathematical interpretation (e.g., yard as a unit of measurement and a reference to an area outside a house; Pierce & Fontaine, 2009). Without explicit instruction, there are relatively few opportunities to acquire mathematical vocabulary, but teachers do not typically allocate much instructional time to the language of mathematics (Monroe & Orme, 2002). Children who are ELLs may also experience more limited access to rich and meaningful instruction in content areas (Callahan, 2005). If children who are ELLs have fewer “opportunities to learn” in the classroom, we might expect that they would perform less well than English primary students even if language-light assessments are used (Abedi & Herman, 2010). With the goal of better understanding the nature of mathematics learning in children who are ELLs, we included tasks designed to vary in linguistic complexity as well as mathematics difficulty in the present study.

Value of Contrasting These Groups Despite the fact that using English is the main challenge for both groups of children, it is clear that the underlying source of their difficulties should be different. Children with SLI have brain-based differences that make language difficult for them to use, whereas children who are ELLs have typical language learning skills but a lack of experience with English. Combining both groups in a single study can strengthen the findings and help with interpretations. Students who are ELLs should have difficulties only with language, not with learning. Therefore, any between-group differences for children with SLI and children who are ELLs will help highlight the nature of the learning difficulties of children with SLI. Specifically, it will give us insight into how large a role language alone plays in the mathematics performance of children with SLI. Additionally, this type of comparison sets the stage for the next phase of research that looks at mathematics challenges faced by children who both are ELLs and have SLI. Without a direct comparison, it would be difficult to interpret future findings.

Current Study The present study included three groups: children with SLI, children who are ELLs, and, for comparison, monolingual native English-speaking (NE) peers with typical development. Our overarching research question was: Would children with different sources of language challenges perform differently on mathematics tasks designed to vary language and mathematical demands? (See Table 1.) Predictions for each task are listed under the following descriptions: 1.

Language-heavy, symbol-heavy task: Standardized mathematics test. This task is considered language and symbol heavy because it involves mathematical

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Table 1. Demands by task. Task

Language processing

Symbol processing

heavy

heavy

light

light light light light

heavy heavy light light

light light light heavy

KeyMath3, unmodified KeyMath3, Spanish modified (ELL group only) Number comparison game Quantity comparison game Concept mapping game

Visual working memory

Note. ELL = English language learner.

symbols as well as language without any accommodations in terms of syntactic complexity, vocabulary, and so forth. It is also language-heavy for the students who are ELLs, given that it is in English. Prediction: NE > ELL = SLI 2.

Language-light (comparatively), symbol-heavy task: Modified mathematics test. This task is considered comparatively language light for those in the ELL group because, although it still involves a significant amount of language, it now removes the load of processing language in the child’s nonnative language and allows children access to vocabulary in both of their languages. Prediction: NE = ELL > SLI

3.

Language-light, symbol-heavy task: Lab-made computer assessment. This task is language-light in that linguistic instructions are designed to limit complexity and are supplemented with a visual context for any language that is presented. It is considered symbol-heavy in that children are expected to use mathematical symbols to make decisions. Prediction: NE = ELL > SLI

4.

Language-light, symbol-light task: Lab-made computer assessment. This task is language-light in that linguistic instructions are designed to limit complexity and are supplemented with a visual context for any language that is presented. It is considered symbol-light in that no mathematical symbols are needed to make decisions. Prediction: NE = ELL = SLI unless conceptual learning is problematic. Then, SLI < NE = ELL

5.

Language-light, visual working memory heavy: Labmade computer assessment. This task is language-light in that linguistic instructions are designed to limit complexity and are supplemented with a visual context for any language that is presented. The task is heavy on visual working memory in that children must remember what they see and then make decisions about visual information while inhibiting irrelevant information. Prediction: NE = ELL > SLI

We did not expect those in the ELL group to have difficulty with any mathematics tasks when the language load is lightened, be it through providing input in the native language, providing a visual context for language, or decreasing linguistic complexity. However, we do allow for different possible problem sources for the group with SLI, including problems with symbols, concepts, and visual working memory.

Method Participants Sixty-one school-aged children, primarily second graders (age range 6;9–9;1 [years;months]), were recruited to participate in this study.1 Among these children, 21 formed the native monolingual English group (NE; 13 female, 8 male), 20 formed the group with SLI (9 female, 11 male), and 20 formed the ELL group (10 female, 10 male). The racial and ethnic composition of each group is reported in Table 2. To recruit participants, we created a database of schools in Tucson, Arizona (from www10.ade.az.gov/ReportCard/), that (a) had similar performance scores on state mathematics assessments, (b) showed similar indicators of economic disadvantage, and (c) met the criteria for having a reasonable percentage of students who were classified as ELLs. After receiving permission from the district administration, we invited schools that were similar in these dimensions to participate in the study. Ten schools and one district summer program accepted our invitation. At each school, we asked the second grade teacher, ELL teacher, and speech-language pathologist (SLP) to distribute packets to their students. Additional students were recruited through advertisements placed in the community (e.g., Boys & Girls Clubs). In order to qualify as having SLI, children needed to achieve a standard score of 85 or less on the Clinical Evaluation of Language Fundamentals, Fourth Edition (CELF–4), which has a sensitivity of 100% and a specificity of 82% using a cutoff of –1 SD (Semel, Wiig, & Secord, 2003). Of the children in the group with SLI, 11 were receiving services, two were being evaluated for services, and two had parents who had expressed concerns about their child’s language or academic performance but had not previously pursued evaluation. Only five participants had never received services and had no reported parental concerns. All children in the group with SLI had language skills that were judged as impaired by the certified SLP or graduate student clinician performing the assessment. Given that more than 70% of parents of kindergarten-aged children with SLI have not been told that their child has a language problem (Tomblin et al., 1997), it is not surprising to find a subset of previously unidentified children in any given sample. 1

Three children in the SLI group were in third grade, two children in the NE group had completed first grade but had not yet entered second grade, and one was in third grade.

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Table 2. Reporting of race and ethnicity by group. Group Race/ethnicity

NE

SLI

ELL

African American American Indian Mixed race Not reported White Hispanic Not Hispanic Not reported

3 0 2 6 10 8 9 4

4 2 2 6 6 10 6 4

0 0 0 15 5 20 0 0

Note. NE = native speakers of English; SLI = specific language impairment.

To be included in the typically developing (NE) group, children needed to earn a score of better than 85 on the CELF–4. They also had to have no reported parental concerns about language or academics. Six children were excluded due to parental concerns. For this study, all students classified as ELLs were native Spanish speakers. In order to qualify as an ELL, participants were classified as ELLs by their school, according to state criteria (n = 12) and were in a special class for ELLs. The Arizona Department of Education website provides details about ELL classification (www.azed.gov/englishlanguage-learners/overviewmore/). Alternatively, participants in the ELL group were classified by their parents via parent questionnaire2 (n = 8). Parents reported that their children’s native and dominant language was Spanish. Parental report has been found to be a valid and useful tool when assessing bilingual children’s language skills (GutiérrezClellen & Kreiter, 2003; Restrepo, 1998). Children who were identified as native Spanish speakers but who had a history of special education services or parental concerns about language were excluded from the study (n = 5). In order to rule out SLI, participants in the ELL group also had to demonstrate intact language skills on a story retell task. The children in the ELL group were asked to retell a wordless storybook in both English and Spanish. Their stories were recorded, transcribed, and analyzed using the Systematic Analysis of Language Transcription (SALT) software (Miller & Iglesias, 2006). With SALT, we compared their performance to a normative database of children performing the same task. The stories were analyzed for mean length of utterance in words and total number of grammatical errors. Children were excluded from the study if their scores were more than 1 SD below the normative mean (n = 2). The average mean length of utterance in words for our group was 7.79 (SD = .74) with a range of 5.69–8.68. The average percentage of utterances in a sample with errors was 2.2 (SD = 0.8) with a range of 0.5–3.8. All children were able to provide a language sample in Spanish. Two children declined to retell the story in English. Eleven children’s scores were based on 2

The parent questionnaire was available in both English and Spanish.

their Spanish samples, and nine children’s scores were based on their English samples. Given the substantial heterogeneity of children who are ELLs, it is important to describe the language characteristics of our sample. Children were native Spanish speakers who were deemed dominant in Spanish as reported by school, parents, or both. In Arizona, the law requires that all children are taught in English. Thus, even students who are native Spanish-speakers often begin a process of Spanish language attrition. We created a scale that allowed us to characterize Spanish language abilities using the following pieces of information: what the child reported as his/her preferred language; which language the child reported speaking with family; which language the child reported speaking with friends; which language the child actually spoke with the native-Spanishspeaking examiner; whether the language sample was within 1 SD of the mean for English; whether the language sample was within 1 SD of the mean for Spanish. Each of these six areas was worth one point, so a child who spoke only Spanish would score 6.0, and a child with significant Spanish attrition would score 0. We were missing data to fully complete this scale for two of our participants. The average score was 2.41 (SD = 1.53) with a range of 0.75–5.75. Other inclusionary criteria for all children included achieving standard scores of better than 75 (70 + standard error of the mean) on the Kaufman Assessment Battery for Children, Second Edition (K–ABC II; Kaufman & Kaufman, 2004) nonverbal index in order to rule out intellectual disability, which is typically diagnosed when an individual scores 70 or less on an IQ test.3 One child was excluded on the basis of this criterion. All participants also passed a near vision acuity screening, a color vision screening, and a hearing screening. One child was excluded for failing the color vision screening, and one was excluded for failing the hearing screening. All participants’ parents filled out a questionnaire. Children whose parents reported that they had a history of seizures (n = 1), brain injury, attention deficit hyperactivity disorder (n = 1), or other type of speech-language issue (e.g., stuttering n = 1) were not enrolled in the study. See Table 3.

Measures Standardized assessment: Language-heavy, symbolheavy task. In order to have a baseline measure of children’s mathematics skills on a traditional, language-heavy, symbolheavy task, participants completed the KeyMath3 (Connolly, 2007), a developmental assessment of mathematical concepts that has been used frequently in research, including studies with exceptional populations (Baylor, 1998; Eaves, 1992; Flores, 2009; Kratochwill & Demuth, 1976; Tinney, 1975).4 3 There is some disagreement in the literature about the appropriate cutoff for nonverbal IQ for children with SLI. For a more detailed justification of the inclusion of children with a nonverbal IQ below 85, please see Alt and Spaulding (2011), p. 644. 4

The results of the groups who are ELL and NE on the KeyMath3 have been published previously (Alt, Arizmendi, Beal, & Hurtado, 2013) in a discussion of test translation. All other results presented in this article are novel.

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Table 3. Participants’ demographic features on descriptive and inclusionary measures. NE Variable Age in years;months MLEa CELF–4b K–ABCc

ELL

SLI

M (SD)

Range

M (SD)

Range

M (SD)

Range

7;10 (0;8) 14.35 (2.34) 101.66 (7.81) 107.38 (9.12)

6;9–9;4 10–18 87–114 92–127

8;2 (0;3) 11.94 (2.27) N/A 97.73 (10.10)

7;6–8;7 8–16 N/A 83–119

8;3 (0;5) 13.68 (1.73) 75.10 (7.26) 91.1 (10.70)

7;6–9;1 10–18 54–85 78–118

Note. MLE = maternal level of education in years; CELF–4 = Clinical Evaluation of Language Fundamentals, Fourth Edition (Semel et al., 2003); K–ABC = Kaufman Assessment Battery for Children, Second Edition (Kaufman & Kaufman, 2004). a

NE > ELL group at p = .004 bNE > SLI group at p < .001 cNE > ELL = SLI groups at p < .05

The KeyMath3 includes subtests that are organized into three general areas: Basic Concepts (conceptual knowledge), Operational Knowledge (computational skills) and Applications (problem solving). The complete test includes 372 items and can be used with participants from age 4;6 through 21. The item content is aligned with the standards of the National Council of Teachers of Mathematics (2000). Each item includes a picture or graphic, and responses include short answers as well as pointing. Children are presented with items in sequence until they fail a specific number of items in a row, defined by the test. We selected KeyMath3 for several reasons: First, it has well-established psychometric properties, and includes alternate equivalent forms (Connolly, 2007; Williams, Fall, Eaves, Darch, & Woods-Groves, 2007). Second, it is appropriate for children in the early grades, with subtests that assess mathematical concepts in addition to those that involve computation and applied problem solving. We administered eight subtests of the KeyMath3, including all subtests of the Basic Concepts (Numeration, Algebra, Geometry, Measurement, Data Analysis, and Probability) and Applications (Foundations of Problem Solving, Applied Problem Solving) areas. We only administered the Mental Computation and Estimation subtest in the Operational Knowledge area and not the Addition and Subtraction and Multiplication and Division subtests. It has been clearly established in the literature that children with SLI lag behind peers on basic number skills (e.g., Cowan et al., 2005), and these subtests involve writing, making them difficult to compare directly to the other tasks that require only oral language. Norm-referenced assessment: Language-light (comparatively), symbol-heavy measure. To reduce the linguistic demands on the English Learner group, we modified the KeyMath3 to accommodate for the participants’ native language.5 Two native Spanish speakers translated the KeyMath3 from English to Spanish. To ensure appropriate outcomes, the translators focused on establishing linguistic equivalence, functional equivalence, and cultural equivalence (Peña, 2007). A back-translation was performed to ensure equivalence of

5 For more information on the differences between the English-only and Spanish-modified versions, please see Alt et al. (2013).

the testing materials in English and Spanish, and 90.22% of the items were considered to have fully functional equivalence.

Experimental Tasks We created stimuli for three experimental tasks. Each task was built using animation created with Adobe Flash and programmed using Adobe ActionScript 3.0. The tasks were presented on a Planar touch screen that was attached to a Gateway laptop computer. By using computer administration and scoring, we were able to ensure 100% fidelity for presentation and scoring. Number comparison game: Language-light, symbolheavy. For the number comparison game, we created nine dinosaur races in which children had to choose which of three novel dinosaurs had the largest numeral on its racing bib. The dinosaur with the largest numeral would win the race. Koponen et al. (2006) pointed out that number comparison tasks are the “most commonly used indicator of nonverbal number skill” (p. 65). We wanted to be certain the children with SLI understood the directions for the game. Thus, children were trained on the game using computer animations that used easily understandable numerals (i.e., 1–5) and that explained the task using language and animation. For example, children were not just told that the number on the racing bib equaled a specific quantity. They also saw an animation where the bib was lifted up to reveal the corresponding number of dots on the dinosaur’s body. To play the game, children first had to pass the training, demonstrating that they understood the task and could choose the largest number. All but three passed the training on the first try. Two children from the NE group and one from the ELL group required two tries to pass the training. In the actual races, the numerals ranged from 2 to 37 and were deliberately assembled to systematically create foils. For example, if the set was 21, 12, and 15, there could be confusion in terms of place value with 21 and 12, either through confusion of order or the idea that the 15 would be the highest because it contains the largest single digit (i.e., 5). For each of the nine races, the dinosaurs were positioned in one of three starting places. Starting places for the winning dinosaur were varied across the nine races (see Supplemental Figure 1). After a child touched a dinosaur, it would light up to indicate that a selection had been made. Children were given the

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opportunity to change their minds. Once they made a final decision, they pressed the “I’m done” button, and the race began. Children were given feedback about their selection at the end of each race via an animation showing the chosen dinosaur winning a ribbon (correct) or looking sad with the crowd making sympathetic noises (incorrect). Quantity comparison game: Language-light, symbollight. The quantity comparison game was set up in exactly the same way as the number comparison game, with one exception. As in the number comparison, there were nine races. However, in this game, dinosaurs did not wear racing bibs. Instead, they had dots on their bodies, and children were instructed to choose the dinosaur with the most dots. In this game, the number of dots ranged from six to 16.6 The use of dots removed the need for understanding symbols. Dots were randomly placed, and children simply had to count the dots and not rely on knowledge of numerical symbols. Counting was encouraged by using quantities that were not easy to group (e.g., greater than five) or recognize perceptually and by keeping the range of numbers in each race small (e.g., 8, 9, 11; see Supplemental Figure 2). This game also had a training in which the dots on the dinosaurs were equated with power; thus the more power the dinosaur had, the faster it would run. This concept was explained with language but demonstrated with animation that showed the dots fueling a power bar, much like the ones that are found in video games. All but one child passed the training for this game on the first try. One child from the NE group required two trials to pass the training. Attribute memory game: Language-light, heavy visual working memory. The attribute memory game exposed children to a dinosaur who walked across a screen, ate something, and burped out a cloud with a numeral in it. The animation included four concepts that the child would need to recognize. Two of the concepts were nonmathematical: the dinosaur’s color and what it ate. Two of the concepts were mathematical. One involved symbols (i.e., the number in the burp cloud), whereas the other involved quantity. Each dinosaur’s body had some sort of marking (e.g., an asterisk, a jagged line) that appeared between one and nine times on the body. The child’s task was to recreate the four concepts from the animation. After the children saw the animation, they saw a screen with an outline of the dinosaur in the middle of the screen. Each side of the screen showed the four alternatives for each attribute along with an icon that corresponded to that attribute. For example, for color, we created an icon of a crayon, and provided the child with four boxes, each of which was a different color (see Supplemental Figure 3). This task was designed to rely heavily on visual working memory. To recreate the attributes, the child had to attend to the four concepts during the animation, remember them, and then select each correct target from the distractors while

6

This range is more limited than the range of the number comparison game due to space constraints. Having up to 37 dots would have made it difficult for participants to discriminate.

holding the correct answer for the remaining concepts in memory. The potential for interference from the distractors, the other attributes, and previous trials taxed working memory. This task was language light in two ways. First, there were no words presented throughout the task. Second, we intentionally made the choices difficult to lexicalize in terms of the stimuli we chose and the foils we provided. For example, it would be difficult for a child to simply remember “green” because we used different shades of each color and it was possible for two versions of a particular color to be provided as a choice (see Supplemental Figure 3). We created 15 unique dinosaur animations and 15 unique choice screens to provide the same degree of difficulty in terms of the foils that were presented for choices. Foil choices for each dinosaur were preselected and were the same for each participant, but foil/dinosaurs combinations were presented in random order. After children made their selections, they could change their minds. Their responses were scored after they pressed the “I’m done” button. Children had up to four chances to choose the correct answer for each attribute and received visual feedback (a picture of electronic coins) after each “I’m done” button press. If they were correct on the first try, they earned four coins. If they were correct on the second try, they earned three coins, and so on. Their cumulative earnings were displayed across the game. If an answer was correct on the first try, they did not have to respond about that attribute on subsequent trials. This game also included a training to make sure the children understood what each icon signified, how the game worked, and what the reinforcement scheme was. The dinosaur animation was replayed if a child got something wrong during the training, so memory would not be an issue while learning the concept of the game. The training was set up so that children had to show they understood how to mark each of the four concepts for two different animals (a dinosaur and a bunny). Like the actual game, they had four chances for each animal to demonstrate they understood what to do. Thus, a passing score on the training would be eight or less, with a perfect score being two. All children were able to demonstrate that they understood the concept of the game by passing the training (x = 3.39, range = 2–6).

Analytic Strategies Standard scores on KeyMath3 subtests served as dependent variables. For the number comparison and quantity comparison games, the dependent variable was accuracy, or number of correct responses, which was calculated automatically by the computer. The dependent measures in the attribute memory game were overall accuracy as well as a ratio of number of attempts to number correct. Recall that children had four attempts to choose the correct response for each attribute. Therefore, the most desirable ratio was 1.0, indicating that a child made the correct selection on the first try. We included the ratio measurement to accommodate the fact that children with SLI

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often can learn the same information as peers, but they often need more trials (e.g., Rice, Oetting, Marquis, Bode, & Pae, 1994). We were interested in each attribute individually as well as in comparing performance on the mathematical to the nonmathematical semantic attributes. We addressed our research question with a mixed-design repeated-measures analysis of variance (ANOVA), with group as the betweensubject variable and attribute as the within-subject variable.

Procedure Completing the experiment typically took three to four visits of roughly 1 hr each that were completed within a 2-week span. All children completed the testing and were enthusiastic about the computer tasks. Testing occurred in a place chosen by participants’ parents: their child’s school, afterschool program, or home. After receiving parental consent and assent from the participants, we screened the participants’ vision and hearing. If they passed the screenings, we presented the other norm-referenced measures (K–ABC interrater reliability = 98.25%; CELF–4 interrater reliability = 96.47%; KeyMath3 interrater reliability = 97.18%) in random order. The participants in the ELL group were administered the Spanish-modified version of the KeyMath3. The KeyMath3 was administered in English, but the standard administration was modified so that if a child failed to answer an item correctly, the item was readministered in Spanish before proceeding to the next item. We started with the English version because (a) it allowed us to see what performance would be like without modifications, as it is typically done in schools, and (b) all students were educated in English. Thus, it was assumed that their mathematics vocabulary would be in English. However, the Spanish-modified version is considered to be a comparatively language-light measure because it eliminates the linguistic processing demands in English for all nonmathematical terms for those in the ELL group. Earlier examination of the translation verified that improved performance was unlikely to be due to guessability (see Alt et al., 2013). After the norm-referenced measures, children played the experimental computer games, which were presented in random order. There was a version of each game made with instructions in English (for groups with NE and SLI) and in Spanish (for those in the ELL group). All tests were administered by teams of research assistants who had been trained by the first author to administer the normreferenced tests, the experimental tasks, or both. All research assistants passed fidelity training checks. Testing teams always included native Spanish speakers for the participants in the ELL group.

F(2, 58) = 17.01, p < .001, hp2 = .36. Unequal-n post hoc testing with p < .05 showed that the children in the NE group performed better than those in the ELL group, who performed better than the children with SLI (see Table 4).

Language-Light Measures The first hypothesis was that reducing the language load would improve performance for those in the ELL group. This hypothesis was supported. There was a main effect for group with the Spanish-modified version of the KeyMath3, for Basic Concepts, F(2, 58) = 19.04, p < .001, hp2 = .39, and Applications, F(2, 58) = 20.49, p < .001, hp2 = .41. Unequal-n post hoc testing with p < .01 showed that the performance of those in the ELL group was not significantly different from the performance of those in the NE group and was different from the group with SLI (see Table 4). As reported in Alt et al. (2013), the amount of gain a child received from the Spanishmodified version of the Basic Concepts area was predicted by Spanish dominance score, with children with higher dominance showing greater gains.

Number Comparison Game: Language-Light, Symbol-Heavy To determine whether reducing the language load would help the language-challenged groups with a task that involved symbols, we performed an ANOVA with accuracy as the dependent variable. Unequal-n post hoc testing was used with p = .01. Both groups without SLI had equivalent performance and performed better than the group with SLI, F(2, 58) = 6.42, p = .003, hp2 = .18 (see Table 5).

Quantity Comparison Game: Language-Light, Symbol-Light We ran the same statistical tests for the quantity comparison game. There were no significant differences between groups, F(2, 58) = 2.42, p = .09, hp2 = .07 (see Table 5).

Attribute Memory Game: Language-Light, Visual Working Memory Heavy To determine how language-challenged groups would perform on a language-light but visual working memory– heavy task, we ran a mixed-design ANOVA with group (NE, ELL, SLI) as the between-group variable and attribute (color, eat, burp, pattern) as the within-group variable. We ran Table 4. Standard scores for participants by group on KeyMath3 areas.

Results KeyMath3: Language Heavy, Symbol Heavy Our baseline measure was children’s performance on the KeyMath3, a language-heavy, symbol-heavy normreferenced test. We compared children’s standard scores using a one-way ANOVA. There was a main effect for Basic Concepts, F(2, 58) = 15.46, p < .001, hp2 = .34, and Applications,

Basic Concepts Group NE SLI ELL ELL, Spanish modified

Applications

M

SD

M

SD

99.47 80.40 91.40 97.00

9.86 9.47 13.26 12.52

100.33 78.15 90.25 97.60

10.89 10.40 14.81 14.40

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Table 5. Number of races correct by group. Number comparison racea M (SD)

Range

M (SD)

Range

8.71 (0.46) 7.85 (1.56) 8.85 (0.36)

8–9 4–8 8–9

7.23 (1.37) 6.35 (1.26) 6.85 (1.22)

5–9 4–9 5–9

Group NE SLI ELL

Quantity comparison raceb

a

NE = ELL > SLI at p < . 05 bNE = SLI = ELL.

two separate ANOVAs: one for accuracy and one for ratio (attempts: correct response). We used unequal-n post hoc testing to explore main effects and interactions for betweengroup measures, and t tests with Bonferroni corrections to examine within-group differences. There were main effects of accuracy for group, F(2, 58) = 5.54, p = .006, hp2 = .16, and attribute, F(3, 6, 174) = 12.56, p < .001, hp2 = .17, that were tempered by an interaction between group and attribute, F(3, 6, 174) = 2.50, p = .02, hp2 = .07. Post hoc testing using Bonferroni corrections ( p < .008) revealed that children with SLI had particular trouble with pattern. Their performance on pattern was significantly less accurate than their performance on any other semantic attribute. In contrast, performance on pattern was equivalent to performance on all other semantic attributes for the groups with NE and ELL. In terms of direct comparisons across groups, post hoc unequal-n Tukey testing ( p < .05) showed that the group with SLI was less accurate on pattern than the group with ELL (see Table 6). We expected high performance (close to 15) for all groups, given that children were given four chances to get the correct answer. Therefore, we also examined the ratio of tries to correct answers, which provides an estimate of learning efficiency. The findings were amplified when we examined ratio. A perfect score would be 1.0 (first attempt yielded the correct answer). Again, there were main effects for group, F(2, 58) =

Table 6. Means (SDs) for raw scores and for ratios by group and attribute for the attribute memory game. Attribute Pattern Raw score Ratio Burp cloud Raw score Ratio Color Raw score Ratio Eat Raw score Ratio

NE

SLI

ELL

13.67 (1.64) 1.86 (0.66)c

12.75 (2.12)a,b 2.50 (1.53)c,d

14.35 (0.74)a 1.54 (0.24)c

14.28 (0.90) 1.47 (0.32)

14.30 (0.80) 1.44 (0.38)

14.48 (0.88) 1.45 (0.44)

14.28 (0.71) 1.46 (.025)

14.00 (0.91) 1.65 (0.30)

14.40 (0.75) 1.52 (0.31)

14.76 (0.43) 1.25 (0.15)e

14.25 (0.91) 1.40 (0.34)

14.85 (0.36) 1.22 (0.24)f

Note. a,cSLI group significantly different ( p < .001) from other group(s). b,d,e,fAttribute significantly different ( p ≤ .008) from other attributes within a group.

4.50, p = .01, hp2 = .13, and attribute, F(3, 6, 174) = 19.57, p < .001, hp2 = .25, that were tempered by an interaction between group and attribute, F(3, 6, 174) = 3.95, p < .001, hp2 = .11. The post hoc testing revealed similar patterns to those found in the accuracy analysis, but in this case, the children with SLI were less accurate than both the groups with ELL and NE on pattern, and the groups with NE and ELL showed some within-group differences related to their relative efficiency on the eat attribute (see Supplemental Figure 4 and Table 6).

Explanatory Analyses Participants were chosen to represent their respective populations and were not matched. Thus, it is possible that other factors might have contributed to between-group differences. To rule out this possibility, we performed t tests on the relevant outcome measures (KeyMath3, Number Comparison, Quantity Comparison, Attribute Memory– patterns) for each group by gender (see Else-Quest, Hyde, & Linn, 2010 for a meta-analysis of gender differences in mathematics performance and attitudes). With p < .05, there were no significant differences between boys and girls for any outcome measure for any group. We know that the groups differed by socioeconomic status as indexed by maternal level of education and on nonverbal intelligence as measured by the K–ABC. Although there were no statistically significant differences between groups in terms of age, there was still a fairly large range. Therefore, we checked to see whether these measures, and the Spanish dominance scores, were correlated with relevant outcome measures using Pearson’s product–moment correlations with p < .05. There were no significant correlations between any of these variables and the relevant outcome measures.

Discussion The goal of the study was to investigate the relationship of language and mathematical proficiency in children who face language challenges. Performance on multiple mathematical tasks was compared for three groups of children: native English speakers with typical development, children who are ELL with typical development, and children with SLI. The tasks were selected to vary in the demands associated with language processing, symbolic processing and visual working memory.

Children With SLI In accordance with our predictions for Tasks 1 through 5, children with SLI demonstrated lower performance than the children in the NE group on every task except one: Quantity Comparison (Task 4). The effect sizes (Cohen’s d ) for these differences were large for the KeyMath3 (d = 1.97 for Basic Concepts and 2.08 for Applications), Number Comparison (d = 0.74 NE and 0.88 ELL) and medium to large for the pattern portion of fast mapping (d = 0.54 NE and 0.87 ELL). The results may help to clarify the nature of mathematical

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deficits in students with SLI. One possibility is that poor mathematics performance results directly from poor language skills. The largest effect sizes for the most language-heavy task support this idea. Language is the primary mode of human communication and is pervasive throughout the school curriculum. Although there are hands-on components to mathematics instruction, there is still a lot of language involved in the instruction of most mathematical concepts. Children must learn new vocabulary, but children with SLI are poor word learners (e.g., Alt & Plante, 2006; Rice et al., 1994). We did find that children with SLI performed poorly on the KeyMath3 mathematics test, which includes a heavy language component through the use of complex sentence structure and sometimes-challenging academic vocabulary. This is in line with the findings from the literature that show decreased academic performance for children with SLI in mathematics (e.g., Harrison et al, 2009). However, this task also involved mathematical symbols. We might expect mathematical symbols to be difficult for a group with a demonstrated deficit in using linguistic symbols to master. Indeed, as predicted with Task 3, children with SLI also did not perform well on the dinosaur race task that involved the use of symbols (numerals), despite the fact that the groups without SLI were close to ceiling performance on this task. However, we cannot conclude that children with SLI find symbolic tasks challenging across the board. In the visual working memory task (Task 5), they recognized numerical symbols as well as peers with typical development. Recall that in our task, the symbols were somewhat decontextualized. Numerals appeared in burp clouds from dinosaurs, but the information was not connected to any specific meaning (e.g., a 6 did not mean the dinosaur had eaten six flowers). Thus, although children with SLI may have adequate visual working memory for symbols for a recognition task, they may not have the ability to manipulate those symbols as efficiently or meaningfully as peers. The finding that different types of tasks might be differentially affected by symbolic knowledge is in line with the findings of Powell, Fuchs, Fuchs, Cirino, and Fletcher (2009). They found that different types of word problems were differentially difficult for children with mathematical difficulties. The types of problems that were difficult were not the same for children who had mathematical and reading disabilities. The point is that mathematical knowledge, symbolic or otherwise, is not an all-or-nothing proposition (e.g., McNeil & Alibali, 2005). Our findings are most in line with those of Koponen et al. (2006), who found deficits in children with SLI compared with age-matched peers for nonverbal mathematical measures that included a number comparison task. Their task was more difficult than ours (i.e., children were asked to compare numbers up to five digits, whereas we used only one- and two-digit numbers), yet we still found a difference between groups. Why, then, did Donlan and Gourlay (1999) find no group differences on their numeral comparison task? One likely explanation is differences in sample size. They had 13 children in their group with SLI compared with the 20 in our study and the 29 in the Koponen et al. study. Certainly, Donlan and Gourlay found trends toward differences.

For example, they set a criterion of 85% accuracy on the tasks. Only 61% of the children with SLI met the criterion compared with 83% of the peers without SLI. Finally, Donlan and Gourlay matched their groups for nonverbal reasoning skills. If nonverbal reasoning contributes to mathematics performance, as several researchers (including Donlan and Gourlay) have suggested, matching on this cognitive component might have the effect of masking real underlying differences between groups. It seems likely that at least a subset of children with SLI is likely to struggle with mathematics tasks that involve mathematical symbols, including numerals. An additional hypothesis was that children with SLI would perform as well as peers without SLI when presented with a task with a light language load and no symbols (Task 4). In our study, children with SLI did not show a performance difference compared with the other groups only on the task that involved both reduced linguistic demands and relatively little involvement of symbols. However, given the relatively poor performance of the children with typical development on this task, we are hesitant to place much stock in the results without replication. Specifically, if children with typical development were actually counting the dots on the dinosaurs, their scores should have been closer to nine. They were not. Therefore, it is likely that our findings are confounded by impulsivity. The task was a race, and perhaps the motivating nature of the task actually backfired in this case. Thus, we are not able to make conclusions about how children with SLI perform on mathematics tasks that are language and symbol light. Our final hypothesis (Task 5) was that children with SLI would show difficulty with visual working memory, which might be a source of difficulty for understanding concepts. We did find a deficit for children with SLI in comparison with peers with typical development on the task designed to place heavy demands on visual working memory, particularly for memory for pattern. This confirms that visual working memory, not just spatial working memory (Cowan et al., 2005), is related to lower mathematics performance in children with SLI. Again, we need to be careful in terms of interpretation because our findings showed that this was not an across-the-board deficit. Children with SLI showed ageappropriate visual working memory for numerals and other nonmathematical features. However, it is important to consider the role of working memory for patterns in mathematical conceptual development, especially considering how prevalent patterns are in mathematics. Other findings are emerging to support the view that the deficits exhibited in children with SLI may be more general and cognitive in nature rather than specific to language. For one thing, there are indications that children with SLI do not learn new information as quickly as peers with typical development. For example, they may have problems with semantic information (McGregor, Newman, Reilly, & Capone, 2002). In other words, they might know a word like cat but not know as much about a cat as a typical peer (e.g., is an animal, called a feline, meows). When learning a new word, children with SLI do not as learn as many semantic features (e.g., color, shape, pattern,) about novel nouns

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(Alt & Plante, 2006) or verbs (Alt, Plante, & Creusere, 2004) as do peers with typical development in a fast-mapping situation. They often also need more examples or input than do peers with typical development (Gray, 2003; Rice et al., 1994). These findings are consistent with the view that children with SLI may have an underlying cognitive weakness, perhaps associated with working memory, as suggested by our results in the present study. They are in line with Johnston and Smith’s (1989) findings that preschoolers with SLI could solve verbal but not nonverbal problems related to size as well as peers. They interpreted these findings as being in line with processing limitations for children with SLI. Although children with SLI, by definition, have typical nonverbal intelligence, they also routinely score lower than age-matched peers on tests of nonverbal intelligence (Plante, 1998). Some theorists posit that SLI is a consequence of a more general cognitive processing deficit, often referred to as a limited processing capacity (e.g., Just & Carpenter, 1992) that in turn contributes to poor language skills and possibly poor mathematics skills. If these children simply have more limited working memory resources to efficiently process complex information quickly, performance should be more limited on a range of tasks, including those that involve language but also on other types of tasks. One implication of this view is that the performance of children with SLI may be affected more by the overall demands of the task than simply by the involvement of language or symbols. In fact, children with SLI performed quite well on symbol-heavy tasks that required only recognition of relatively simple features. For example, they could remember the symbol represented in the “burp cloud” as well as peers with typical development (Task 5). However, when the task required them to hold symbols in working memory and compare them, as in the dinosaur race game, their performance faltered (Task 3; see Table 5). Overall, the results of this study point to multiple possible problem sources for mathematical difficulties for children with SLI. These include the manipulation of mathematic symbols, the use of working memory for patterns, and the combination of complex linguistic syntax plus mathematical symbols. These are not the only possible sources of mathematical problems. For example, our study did not consider amount of instruction, which has been found to be important to mathematics performance in work done in England (e.g., Cowan et al., 2005). Given the already high testing load for the participants in this study, we did not add any additional tests of working memory. However, it would be helpful in the future to further examine the role of working memory in mathematics performance. If future work provides additional evidence to support the sources we have identified, then the results may help to support the development of instructional approaches that might benefit children with SLI in addition to language support. For example, cognitive-strategy instruction for mathematics problem solving has been shown to be successful with children with learning and attention deficits and other cognitive challenges (Montague & Dietz, 2009). Similar approaches might also be useful to explore for use with children with SLI.

ELLs Consistent with prior research and our predictions (Task 1), performance for children who are ELLs was less accurate than for those in the NE group for language-heavy, symbol-heavy tasks. Effect sizes (Cohen’s d = 0.69 for Basic Concepts and d = 0.77 for Applications) for the differences between groups on the English-only administration of the KeyMath3 can be considered medium to large. However, consistent with the predictions for Tasks 2 through 5, the performance of children in the ELL group was not statistically different from the children in the NE group on tasks with light language demands; there was no indication that children in the ELL group had difficulty with symbolic processing or with tasks that placed high demands on visual working memory. The academic implications of these findings are that educators need to be aware that language-heavy assessments may not provide an accurate picture of a child’s mathematical knowledge or proficiency. The results will likely be confounded with English language knowledge. In this study, we found multiple ways to lighten the language load and let the students who are ELLs reveal their mathematical competence. When we provided access to the assessment materials in Spanish, the performance of students who are ELLs did not differ significantly from children who were assessed in their native language. We also used technology to minimize the language load. First, we were able to provide instructions using animations and visual examples that did not rely upon language. Second, by using touch-screen technology to allow children to respond, the children did not have the additional cognitive load of trying to come up with the appropriate linguistic responses. On all tasks using technology, children who are ELLs did not differ significantly in their performance compared with monolingual peers. Granted, both group performed close to ceiling on these measures, thus we cannot say with confidence that they have equivalent number concepts. However, these data suggest that lightening the language load will lead to improved mathematics performance for students with typical development who are ELLs. This finding is consistent with other research showing that children who are ELLs perform better when the linguistic complexity of the assessment materials is reduced (Abedi & Lord, 2001; Abella, Urrutia, & Shneyderman, 2005; Arvedson, 2002, 2013; Beal & Barbu, 2010; Martinello, 2008; Shaftel et al., 2006).

Clinical Implications There are academic implications for children with different types of language challenges when educators do not consider the role that language plays in assessment and instruction in mathematics. Specifically, poor performance in mathematics may be confounded by poor English language skills. If these differences are not disambiguated, it is unlikely that children will receive effective instruction. Although students who are ELLs are likely to be able to demonstrate equivalent mathematics skills to peers who are NE when the language load is lightened, children with SLI do not benefit as readily from these types of modifications. They may also

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need explicit language intervention to be most successful in mathematics. Although further research needs to be conducted to determine the most effective way of delivering this intervention, the current work speaks to a role for SLPs in working with teachers who are teaching and assessing the mathematics skills of children with language challenges. The main role would be to alert teachers to the pitfalls of assuming that a standardized assessment will provide the best measure of the mathematics skills of a child with language challenges. Particularly for students who are ELLs, actual mathematics skills might be underestimated and the ability for a child to hear questions in both English and their heritage language should be encouraged. In addition to thinking about including mathematics vocabulary in a child’s individualized education program, school-based SLPs could also work with teachers to provide instruction that is more visually based, thus alleviating some of the language load for students with language challenges.

Acknowledgments This material is based on work supported by National Science Foundation Grant No. SBE-0548130. Preliminary findings were presented at the Symposium on Research in Child Language Disorders in 2012. Disclaimer: Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation. We would like to thank all the participants and their families and schools for taking part in this project. We also greatly appreciate the work of all contributing members of the L4 Lab.

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