The language of mathematics: Investigating the ways language counts for children’s mathematical development

Abstract

This longitudinal study examined how language ability relates to mathematical development in a linguistically and ethnically diverse sample of children from 6 to 9 years of age. Study participants were 75 native English speakers and 92 language minority learners followed from first to fourth grades. Autoregression in a structural equation modeling (SEM) framework was used to evaluate the relation between children’s language ability and gains in different domains of mathematical cognition (i.e., arithmetic, data analysis/probability, algebra, and geometry). The results showed that language ability predicts gains in data analysis/probability and geometry, but not in arithmetic or algebra, after controlling for visual–spatial working memory, reading ability, and sex. The effect of language on gains in mathematical cognition did not differ between language minority learners and native English speakers. These findings suggest that language influences how children make meaning of mathematics but is not involved in complex arithmetical procedures whether presented with Arabic symbols as in arithmetic or with abstract symbols as in algebraic reasoning. The findings further indicate that early language experiences are important for later mathematical development regardless of language background, denoting the need for intensive and targeted language opportunities for language minority and native English learners to develop mathematical concepts and representations.

Introduction

There is growing recognition that language ability is important for children’s mathematical development (e.g., Carey, 2004, Kleemans et al., 2011, LeFevre et al., 2010). Indeed, Moses and Cobb (2001) argued that language underlies children’s ability to gain the conceptual understanding necessary to make sense of the otherwise abstract symbols inherent in mathematics. It has even been suggested that children’s mathematical difficulties may reflect deficient linguistic processes as opposed to deficits in nonverbal numerical processes (Lager, 2006, LeFevre et al., 2010, Vukovic, 2012). These suppositions are in part supported by neuropsychological evidence showing that the left angular gyrus supports the manipulation of numbers in verbal form (Dehaene, Piazza, Pinel, & Cohen, 2003). Yet very few studies have systematically examined the relation between linguistic processes and mathematical cognition beyond number and arithmetic.

To begin to increase the specificity of our understanding of the linguistic basis of mathematics, this study examined how language ability relates to children’s development across different domains of mathematical cognition from first to fourth grades. We focused specifically on linguistically and ethnically diverse children attending high-poverty urban schools, and this represents a significant step toward advancing the science of children’s mathematical development in light of trends in the demographics of school-age populations. There is a particular need for research with language minority learners—students who come from homes where the primary language spoken is not the societal language. In industrialized countries worldwide, the population of children growing up in linguistically diverse homes is on the rise (UNICEF Innocenti Research Centre, 2009). In the United States, for example, the past several decades have seen a dramatic increase in the number of school-age children coming from homes where English is not the primary language spoken; between 1980 and 2009, this population of children rose from 10% to 21% of school-age children (Aud et al., 2011). The research that has been conducted with language minority students focuses predominantly on their literacy development and its instruction. Investigating the linguistic basis of mathematics with language minority learners not only provides an important window into the relation between language and mathematics but also has implications for identifying sources of individual differences in mathematical development in this understudied population.

The bulk of studies investigating the linguistic basis of mathematics have focused specifically on understanding whether numerical cognition represents an invented construction based on language (e.g., Le Corre and Carey, 2007, Sarnecka and Carey, 2008, Sarnecka and Lee, 2009) or whether concept of number exists independent of language (Ansari et al., 2003, Frank et al., 2008, Gelman and Butterworth, 2005, Libertus and Brannon, 2010, Libertus et al., 2011). It turns out that language—having unique words for exact quantities specifically—plays a role in some, but not all, aspects of numerical cognition. In particular, having number words appears to be involved in the uniquely human ability to cognitively represent large numbers (i.e., ⩾5) with precision (e.g., Dehaene et al., 1999, Gordon, 2004, Spaepen et al., 2011). This is in contrast to the preverbal number system not unique to humans that is specialized for representing only small numerosities (i.e., 1–4) precisely and large quantities approximately (see Gelman and Butterworth, 2005, Landerl et al., 2009). Thus, this body of research indicates that although concepts of quantity exist independent of language, exact representations of number—which is foundational for formal mathematics—are dependent on a language system.

There is burgeoning evidence that language processes are indeed involved in solving basic mathematical problems, in particular arithmetical addition and subtraction. For instance, Russian–English bilingual adults have been shown to retrieve newly learned addition and subtraction facts more efficiently in the language of training, whether Russian or English, compared with the untrained language, suggesting that arithmetic facts are stored in language-specific ways (Dehaene et al., 1999, Spelke and Tsivkin, 2001). Indeed, Spelke and Tsivkin (2001) showed that even incidental exposure to exact numbers (e.g., learning a date in history) is stored in language-specific ways, such that language of training affects how numerical information is stored and subsequently retrieved.

It is hypothesized that phonological processes specifically underlie this relation, presumably because completing arithmetic problems requires the retrieval of phonological codes (Fuchs et al., 2005, Hecht et al., 2001, Koponen et al., 2007, Simmons and Singleton, 2008). Indeed, the well-documented relation between phonological processing and arithmetic performance helps to explain the finding that many children with reading difficulties also have difficulty with arithmetic (e.g., Dirks et al., 2008, Rubinsten, 2009, Simmons and Singleton, 2008). However, there exist some children with mathematical difficulties who are nonetheless good readers (and therefore tend to have age-appropriate phonological skills) and vice versa (e.g., Landerl et al., 2009, Vukovic, 2012), suggesting that phonological skills are not the sole influential factor for some children. Moreover, Jordan and colleagues have found that children can compensate for arithmetical difficulties by using verbal strategies to complete arithmetic problems, suggesting that language skills beyond phonological processing are involved in arithmetic performance (e.g., Jordan, 2007, Jordan and Hanich, 2003). There is further evidence that the language used in arithmetic problems influences how children symbolically represent and solve such problems (e.g., Abedi and Lord, 2001, Brissiaud and Sander, 2010, Lager, 2006). Together, these findings suggest that language ability more broadly may play a unique role in children’s mathematical cognition.

Yet the role of language in more abstract mathematical concepts has been relatively unexplored, especially in more diverse samples of children. In a relevant study, LeFevre and colleagues (2010) proposed that the linguistic circuit advanced by Dehaene and colleagues (2003) is engaged when children perform mathematical tasks that depend on the formal number system. The authors found that a linguistic composite (i.e., vocabulary, phonological awareness, and number identification) measured in a sample of 182 preschool children explained unique variance in second-grade arithmetic, numeration, geometry, and measurement. The linguistic composite was consistently the most important predictor across different domains of mathematical cognition—more so than either nonverbal subitizing or nonverbal visual–spatial working memory—suggesting that language skills play a critical role in children’s understanding of not only arithmetic but also higher order mathematical domains.

That the linguistic composite used by LeFevre and colleagues (2010) included vocabulary, elision, and number identification makes it difficult to definitively conclude that language per se drives the relation between the linguistic pathway and mathematical development. Specifically, the relation between the linguistic composite and mathematical outcomes could reflect a relation between mathematical outcomes and phonological skills and/or domain-specific skills, consistent with previous research (e.g., Fuchs et al., 2005, Jordan and Levine, 2009, Locuniak and Jordan, 2008, Swanson and Beebe-Frankenberger, 2004). Indeed, Dehaene and colleagues (1999) found that whereas adults store exact arithmetic sums as language-based representations, approximate calculations—including advanced mathematical facts such as cube roots—may be completed independently of language. Therefore, the authors speculated that higher order forms of mathematics might not be as dependent on language as is arithmetic. One purpose of the current study, thus, was to build on and advance previous research (e.g., Dehaene et al., 1999, LeFevre et al., 2010, Spaepen et al., 2011) by focusing specifically on whether general language ability is related to the development of both arithmetic and higher order mathematical domains. A second purpose was to examine the role language plays in the mathematical development of language minority learners.

Most research examining children’s mathematical development has been conducted with native English speakers, primarily in middle-income settings, resulting in little knowledge about language minority learners’ mathematical development. Studying mathematical development in language minority learners provides an important alternative angle by which to examine questions surrounding the relation between language and mathematics. Children from language minority backgrounds—especially those in high-poverty urban settings—disproportionately struggle with mathematics (Kieffer, Lesaux, Rivera, & Francis, 2009; National Center for Education Statistics [NCES], 2009), suggesting that language proficiency plays a role in mathematical development. However, these findings do not shed light on whether language plays a similar role in mathematical development in language minority learners and their native English-speaking peers or whether language proficiency serves as a barrier to mathematical performance in language minority learners simply because these children do not understand the task requirements. Indeed, in mathematics classrooms and curricula across the United States, language minority learners do not understand much of the language that is used, and most learners are not explicitly taught to read, write, or speak mathematically (Lager, 2006).

At the same time, language minority learners in the United States tend to be concentrated in high-poverty urban schools (e.g., Gándara, Rumberger, Maxwell-Jolly, & Callahan, 2003), and there is a well-established relation between poverty and underdeveloped language and vocabulary skills, at least for monolinguals (e.g., Hart & Risley, 1995). Thus, it is possible that poverty—manifested as fewer opportunities to learn in and out of school than in middle-class neighborhoods—detrimentally affects the language development of both language minority learners and their native English-speaking peers, which in turn affects the formation of concepts—including mathematical concepts—that are necessary for school success (Carey, 2004). Indeed, Lager (2006) found that many of the language obstacles that affected language minority learners’ ability to successfully complete algebra word problems were also barriers for native speakers. For instance, both groups struggled to understand specialized mathematical language (e.g., pattern, show, figure), symbols and notation (e.g., parentheses), and the use of variables (e.g., “if x = 10, y = ?”). Thus, in the context of research with a diverse sample, comparing language minority learners with their native English-speaking classmates, who attend the same schools and often share many educational experiences as well as socioeconomic characteristics, offers additional insight into the role language plays in mathematical development.

This study sought to contribute to an understanding of the linguistic basis of mathematics by using a developmental lens to examine the relation between general language ability and mathematical cognition in a linguistically and ethnically diverse sample of children. Building on previous research (e.g., Dehaene et al., 2003, Kleemans et al., 2011, Le Corre and Carey, 2007, LeFevre et al., 2010), we examined whether language ability predicted gains in children’s arithmetic, data analysis/probability, algebra, and geometry. Given that longitudinal studies of school-age children are largely absent from the research base, we studied the period from early to middle childhood, following a sample of children from first to fourth grades. Two research questions guided this study:

• Does children’s early language ability predict later gains in arithmetic, data analysis/probability, algebra, and geometry?

• Do the relations between early language ability and gains in mathematical cognition differ between language minority learners and native English speakers?