Introduction: Language and Culture in Mathematical Cognitive Development
Daniel B. Berch*; David C. Geary†; Kathleen Mann Koepke‡ * University of Virginia, Charlottesville, VA, United States
† University of Missouri, Columbia, MO, United States
‡ NICHD/CDBB, Rockville, MD, United States
Abstract
In this introductory chapter, we present a brief historical review of research on the roles of language and culture in mathematical cognition, and then provide readers with a useful framework for categorizing specific linguistic levels of influence on numerical processing (e.g., lexical, semantic, and syntactic). We then describe other considerations in studying linguistic influences on mathematical cognition, such as the neural substrates of verbal, language-based representations of numerical information and the impact of bilingualism on numerical processing. With respect to the role of cultural factors that influence mathematical cognition, we briefly review many of the classic studies on this topic, and then present some purported myths of cultural psychology as a framework for discussing specific kinds of cultural factors that could potentially influence mathematical learning and achievement. We conclude by discussing the complexities associated with disentangling linguistic and cultural influences on numerical processing and development.
Keywords
Lexical; Semantic; Syntactic; Phonological; SNARC effect; MARC effect; Approximate number system; Object tracking system; Analogical mapping; Linguistic transparency; Inversion; Transcoding; Intraparietal sulcus; Dyscalculia; Bilingualism
Introduction
The contributions of linguistic and cultural factors to the development and learning of numerical, arithmetic, and other mathematical concepts and skills are broad and varied, ranging from different types of linguistic influences to contrasting cultural beliefs about the relative importance of ability and effort in learning mathematics, to a traditional society whose language possesses only a limited numerical vocabulary. Indeed, the expansive collection of studies reviewed by the authors in this edited volume is a testament to the diversity of themes, critical issues, theoretical perspectives, research methods, and even the participants (e.g., dual-language learners, children with specific language impairments, and members of a remote indigenous population in New Guinea) that exemplify the fascinating and important spheres of linguistic and cultural influences on mathematical cognition.
In this introductory chapter, we begin by presenting a brief historical review of research on the roles of language and culture in mathematical cognition. Next, we provide readers with a useful framework for categorizing specific linguistic levels of influence on numerical processing (e.g., lexical, semantic, and syntactic), followed by a brief discussion of one of the major questions that has guided research on the role of language in numerical development. We then describe other considerations in studying linguistic influences on mathematical cognition, including the neural substrates of verbal, language-based representations of numerical information and the impact of bilingualism on mathematical thinking and learning.
With respect to the role of cultural factors that influence mathematical cognition, we begin by briefly reviewing many of the classic studies on this topic. Next, we present some purported myths of cultural psychology as a framework for discussing specific kinds of cultural factors that could potentially influence mathematical learning and achievement. We then discuss the complexities associated with disentangling linguistic and cultural influences on numerical processing and development and conclude by acknowledging how studying the intersection between language, culture, and mathematics has flourished since the early groundbreaking research on these topics, exemplified by the work described in this volume that has certainly produced a deeper understanding of this intersection.
A Brief Historical Review of Research on the Roles of Language and Culture in Mathematical Cognition
The systematic, large-scale study of cross-national differences in students' mathematical development began in 1964 with the International Project for the Evaluation of Educational Achievement (IEA; Husén, 1967). The goal was to assess a broad range of mathematical skills of 13- and 17-year olds from developed nations and to examine potential influences on these skills, such as the amount of homework and family background, as we elaborate in “The Role of Culture in Mathematics Achievement”. The results revealed substantive cross-national differences in mathematics achievement at both ages. The second IEA study was conducted about two decades later and confirmed substantive cross-national variation in mathematical competencies (Crosswhite, Dossey, Swafford, McKnight, & Cooney, 1985). These cross-national studies have expanded over the years and are now conducted on a regular basis through the Program for International Student Assessment (PISA; https://www.oecd.org/pisa/aboutpisa/) and Trends in International Mathematics and Science Study (TIMSS; http://timssandpirls.bc.edu/). The story remains the same; there is substantive cross-national variation in mathematical competencies.
The systematic study of language influences on these cross-national differences began several decades after the first IEA and focused largely on the transparency of the base-10 structure of Arabic numerals in the corresponding number words (e.g., Miller & Stigler, 1987; Miura, 1987; Miura, Okamoto, Kim, Steere, & Fayol, 1993). For example, number words in most Asian languages are a direct reflection of the base-10 system, whereas number words in most European-derived languages are more arbitrary, at least until the 100s. For instance, the Chinese number word for 27 is translated as two ten seven, which makes it obvious to children that 27 is composed of 2 tens and 7 ones. This structure is less obvious in English or in most other European-based languages, especially for number words in the teens. These differences in turn can influence students' learning of some aspects of number and arithmetic (Fuson & Kwon, 1992). Studies of this type continue today, as illustrated in Chapter 4 by Okamoto and Chapter 10 by Göbel.
More recently, the study of number words has shifted to those that emerge in traditional populations (e.g., Butterworth, Reeve, & Reynolds, 2011; Gordon, 2004). These studies have revealed that people in many of these groups have a limited set of number words (e.g., the equivalent of “one, two, and three”), which allows researchers to separate the influence of number words from children's and adults' inherent understanding of relative quantity, as noted in Section “How Do Children Learn the Meaning of Number Words?” Cultures with more developed trading systems generally have an expanded set of number words that are often tailored to aid in specific functional tasks (Beller & Bender, 2008), as contrasted with the systematic mathematical system that children now learn with formal education. Within economically developed nations with multilingual populations, there is now a focus on how learning number words and other aspects of mathematics in two different languages influence children's mathematical development, as illustrated in Chapter 7 by Wicha et al., Chapter 8 by Salillas and Martínez, and Chapter 9 by Sarnecka et al.
Specific Linguistic Levels of Influence on Numerical Processing
In a recent editorial for a special issue of Frontiers in Psychology concerning linguistic influences on mathematical cognition, Dowker and Nuerk (2016) contend that the preponderance of research in this area has not gone beyond simplistic demonstrations that language influences number, thus failing to explore the specific level at which particular language effects operate. As a consequence, these authors devote their editorial to differentiating seven linguistic levels. We adopt these here as a framework both for introducing the various ways in which language can influence mathematical processing and for previewing the topics to be covered in the present volume. The specific levels identified by Dowker and Nuerk include lexical, semantic, syntactic, phonological, visuospatial orthographic, conceptual, and other language-related skills.
Lexical: The Composition of Number Words
The study of lexical factors in mathematical cognition has focused primarily on the effects of number words. As Dowker and Nuerk (2016) point out, the extent to which a language's number word structure is transparent (i.e., compatible with its written/digital number system) can influence numerical performance, even when number words are not involved in a mathematical problem. One such lexical property is what Dowker and Nuerk refer to as power transparency—the degree to which the construction of number words provides a clear and consistent representation of the base-10 system in the language. For example, in East Asian languages, the power of any number beyond 10 can be derived directly from the number word (e.g., “ten-one” for 11). And since such correspondences are more direct (i.e., more transparent) than in languages such as English where many number words are irregular (e.g., “eleven”), some researchers have maintained that the comparatively superior counting and arithmetic skills of Asians may be attributable, at least in part, to the greater transparency of their number words (see Chapter 13 by Bender and Beller, for a linguistic typological analysis of regularity in different numeration systems). However, as Dowker and Nuerk contend, findings based upon such comparisons are often confounded by numerous other cultural and educational differences between countries. (See Section “The Role of Culture in Mathematical Cognition” below for a delineation of these potentially confounding factors, along with the section on “Disentangling Linguistic and Cultural Influences”.)
Dowker and Nuerk (2016) also discuss another type of lexical influence known as the inversion property. For example, as Chapter 10 by Göbel describes it, in some languages, the number word starts with the unit followed by the decade (e.g., drei-und-zwanzig, three and twenty), which conflicts with the order of the digits in the Arabic form (e.g., 23). However, in languages such as English that contain noninverted number words, the number word sequence for two-digit numbers is decade first followed by unit (e.g., twenty-three), consistent with the Arabic notation (23). As Göbel describes it, the evidence suggests that children whose language contains decade-unit number word inversions make frequent inversion errors in numerical transcoding—writing numbers in response to spoken number words (e.g., writing 32 for three and twenty)—which seldom occurs for children whose language does not possess number word inversion (see also Chapter 4 by Okamoto, for a discussion of the role of inversion in number-line estimation).
Despite the strong evidence demonstrating that number word inversions can lead to numerical transcoding errors in children, as Dowker and Nuerk (2016) summarize, other research has shown that the inversion property does not influence all numerical and arithmetic skills. That being said, Göbel (Chapter 10 by Göbel) found that for 7- to 9-year olds, number word inversion can impact symbolic arithmetic when addition problems require carry operations (e.g., 27 + 6), in contrast to those that do not (e.g., 31 + 6). Moreover, Lonnemann and Yan (2015) recently demonstrated that inverted number words can complicate arithmetic processing even for adults in countries where arithmetic processing is an accomplished skill.
Before leaving the topic of inversion to review other lexical influences on numerical and arithmetic processing treated in the present volume, it is worth noting that Dowker and Nuerk (2016) discuss the results of an interesting study in their special issue. Here, Prior, Katz, Mahajna, and Rubinsten (2015) examined inversion in Arabic-Hebrew bilinguals, since Arabic words possess the inversion property while Hebrew words do not. Administering arithmetic problems orally, the authors found that the participants were better at solving these problems when the language structure paralleled that of the arithmetic operation, regardless of whether the problem was presented in their native language or the one they acquired later on. As a consequence, the authors concluded that the language in which math facts are originally learned does not entirely govern arithmetic processing.
As a final example of the role of lexical factors in mathematical cognition, Chapter 4 by Okamoto discusses how differences between the English and Chinese terms for geometric shapes may influence the ease with which young children learn the shape names, depending on how directly they refer to the visuospatial properties of such shapes. More specifically, with respect to the etymology of English terms for geometric shapes, Okamoto points out that these shape names, like “rectangle” and “pentagon,” are composed of Greek and Latin root words and are comparatively lengthy and difficult for young children to learn. She then contrasts such words with Chinese geometric terms (written in the form of logograms—characters that stand for words or phrases), describing how these not only are much less complicated but also have embedded in them the meanings of the shapes to which they refer (e.g., “three-angle shape” for triangle and “five-angle shape” for pentagon). And although Okamoto speculates that children who learn the simple terms may find it easier to associate such words with visual shapes and their properties as compared with those who have to memorize the longer and more complex words, she acknowledges that no such cross-cultural investigations have been carried out to date.
Semantic: The Meaning of Words
Semantics is the branch of linguistics concerned with the study of meaning in language. As Brysbaert (2005) has pointed out, there are several reasons for distinguishing between the lexical system, the word-form level, and the semantic system, the word-meaning level. Included among these are some cases in which one-to-one mappings between words and their meanings are lacking, where, for example, the meaning of a given word may vary depending in part on the context in which it is presented—as in the meaning of “big” within the phrase the “big mouse” as compared with its meaning in the phrase the “big building.” Furthermore, as many words possess multiple meanings (i.e., homonyms), such as point, right, change, and watch, it is challenging to explain how the ambiguities in the mappings from word-form to word-meaning can be resolved within one and the same level of representations (Brysbaert, 2005). In discussing how the semantic level or word meanings may influence mathematical processing, Dowker and Nuerk (2016) introduce a study by Daroczy, Wolska, Meurers, and Nuerk (2015) who reviewed evidence showing that semantic and numerical properties frequently interact in word problems. More specifically, when the requisite mathematical operation is lexically consistent with the words in the text (e.g., having to carry out addition for a problem in which the words “more” or “buy” appear or having to subtract is associated with the words “less” or “sell”), the word problem is easier to solve than when the words and the required operations are inconsistent.
The words “more” and “less,” along with others such as “few,” “most,” “many,” and “some,” are examples of essentially quantitative terms. Purpura and Reid (2016) have argued that although there has been a good deal of research on the role of language in the early development of numerical knowledge, most of this work has concentrated on the use of domain-general language measures—such as vocabulary or phonological awareness—rather than on domain-specific measures of quantitative or mathematical language (Harmon, Hedrick, & Wood, 2005). Furthermore, they point out that even though general language skills are consistently related to numeracy performance, enhancing these skills has not been shown to improve children's numeracy knowledge (Jordan, Glutting, Dyson, Hassinger-Das, & Irwin, 2012). Consequently, Purpura and Reid suggested that as mathematical language is a more proximal (i.e., to math content) type of linguistic skill, individual differences in young children's mathematical language may be uniquely predictive of their numeracy performance (i.e., independent of their differences in general language skills).
To test this hypothesis, Purpura and Reid (2016) administered both a general language measure (a definitional vocabulary test) to preschoolers and kindergartners along with an author-developed, content-specific test of mathematical language that included items consisting of quantitative terms, such as “take away” and “more” and spatial terms, such as “nearest” and “middle.” Fifteen of the 16 items were receptive and were tested with pictures, and one of the quantitative items was expressive and involved the use of manipulatives. As predicted, mathematical language accounted for more unique variance in numeracy skills than did general language. As a consequence, these authors reasoned that these more content-specific language skills are probably more modifiable than domain-general factors and thus may constitute a more viable target for instructional interventions.
Following up on this supposition, Purpura, Napoli, Wehrspann, and Gold (2017) randomly assigned 3- to 5-year-old children in a Head Start program to either a mathematical language intervention group or a comparison (business as usual) condition. A so-called dialogic storybook reading intervention was administered throughout an eight-week period, in which the children became the storytellers while an adult interventionist prompted and questioned them by placing emphasis exclusively on the book's mathematical language—the quantitative and spatial terms and pictures (e.g., more and near)—by repeating, expanding on, etc. the children's descriptions. Students in the intervention group subsequently performed significantly better on a mathematical language test than students in the comparison group, and more importantly, were also superior on a test of mathematical knowledge.
Although several authors in the present volume discuss some aspect of semantic properties in numerical cognition, the reader is directed to Chapter 6 by Donlan for an interesting discussion regarding the kinds of quantitative modifiers discussed above and differences between their logical and pragmatic interpretations in children and adults, where the latter is concerned with how context influences meaning.
Syntactic: The Grammatical Structure of Languages Beyond Word-Level Influences
The terms for cardinal values used to count quantities, such as “one” and “two,” are known as lexical numbers. However, as Dowker and Nuerk (2016) point out, numerical information can also be communicated through the use of so-called grammatical numbers—morphological influences on numerical processing that are principally characterized by singular/plural language markings (e.g., the “s” for plural in English) that differentiate sets of one from sets of two or more (Sarnecka, 2014; see Chapter 6 by Donlan, for other kinds morphological influences on number processing). Based upon a review by Sarnecka (2014) in their special issue, Dowker and Nuerk conclude that the development of numerical understanding occurs later for children whose language does not distinguish between singular and plural (see also Chapter 9 by Sarnecka et al.).
Almoammer et al. (2013) provided a particularly compelling outcome in support of the importance of grammatical number. These researchers investigated the acquisition of number words in young children (2-, 3-, and 4-year olds) who were learning Slovenian and Saudi Arabic—two unrelated languages that possess not only singular/plural marking but also dual marking (i.e., for sets of two). These researchers found that learning a dual marking was associated with more rapid acquisition of the cardinal meaning of the number word “two” than in English-speaking children (or in any other previously studied language), even when having had less experience with counting. As the authors assert, these findings are particularly striking given that children learn the Slovenian and Saudi Arabic languages in completely different cultural contexts (see Chapter 6 by Donlan, for a more detailed description of this study).
Finally, drawing on the intriguing study by Overmann (2015), Chapter 13 by Bender and Beller affirms that although lexical numbers are almost universal in the world's languages, grammatical number is less prevalent and is neither necessary nor sufficient for the (evolutionary) emergence of linguistic numbers—despite its role in facilitating children's acquisition of cardinal numbers when it exists as a feature of their language.
Phonological: The Structure and Sequencing of Speech Sounds
Phonological awareness refers to the conscious awareness of the sound structure of words, including phonemes, the smallest unit of sounds. These skills are tested in a variety of ways, including the ability to rhyme, to match words with beginning sounds, and to blend sounds into words. Based on the articles in their special issue, Dowker and Nuerk (2016) claim that although phonological skills are not directly related to general mathematical functioning, they are specifically associated with verbal representations of and operations on numbers. For example, Jordan, Wylie, and Mulhern (2015) found that the impairments in phonological processing experienced by children with reading disabilities and co-occurring reading and mathematical difficulties played a negligible role in mediating performance on a battery of seven mathematics tasks. However, Pixner, Leyrer, and Moeller (2014) demonstrated that deaf and profoundly hearing-impaired children with cochlear implants who had experienced atypical language development exhibited specific difficulties with place-value understanding, which appears to be directly mediated by language. Chapter 6 by Donlan describes how phonological processing difficulties in children with specific language impairment can adversely impact numerical transcoding—the formation of associations between spoken number words and Arabic numerals. Finally, Chapter 13 by Bender and Beller discusses how phonological codes can mediate the effect of one of our numeration systems on the acquisition of number words. Namely, they explain that learning to count on one's fingers actually helps the learner distinguish between number words by associating their different phonological patterns with unique finger patterns.
Visuo-Spatial-Orthographic: Conventions for Writing and Reading Directions of Languages
As Dowker and Nuerk (2016) characterize it, visual-spatial-orthographic influences on numerical processing largely entail the predominant reading and/or writing direction of a given culture's script or its complexity. They point out that these kinds of factors are most frequently associated with space-number relations, such as the spatial-numerical association of response codes (SNARC) effect, where participants tend to associate relatively small numbers with the left side of space and relatively larger numbers with the right side. Dowker and Nuerk also make reference to Rodic et al.'s (2015) study in their special issue. These researchers reported that 6- to 9-year-old Russian children who were learning a second language that makes use of a spatially complex, character-based writing system—Chinese or Japanese—over one school year did not exhibit a greater improvement in arithmetic than Russian children who spent the same amount of time learning English and/or Spanish. Based upon these findings, Dowker and Nuerk conclude that while visuospatial orthographic skills appear to influence the direction of space-number relations, they do not have an effect on the learning of arithmetic skills per se.
More detailed reviews of work concerning spatial-numerical associations in general and the SNARC effect in particular can be found in Chapter 10 by Göbel and Chapter 11 by Shaki and Fischer. For the most part, the work reviewed here not only corroborates in part the conclusions arrived at by Dowker and Nuerk (2016) but also expands upon and in some cases qualifies them. For example, Shaki and Fischer review evidence suggesting that the acknowledged influence of culturally-based reading direction on the SNARC effect is not as persistent and long-lasting as once believed. Furthermore, these authors contend that the effects that reading direction has on spatial-numerical associations often occur in combination with the influences of other factors and are also dependent on the situational context.
Likewise, Chapter 10 by Göbel reports that despite strong evidence that one's culturally dominant reading direction indeed shapes number processing, this outcome does not necessarily mean that reading direction influences individual differences in mathematical performance and development. Furthermore, she reports data demonstrating that neither children's nor adults' overall performance on standardized mathematics tests appears to be related to the degree to which they have been influenced by their culture's dominant reading direction.
Conceptual: Conceptual Properties of Language
Dowker and Nuerk (2016) also call attention to another albeit comparatively unusual category of linguistic influences on mathematical cognition, specifically, conceptual properties of language—attributes that go beyond basic linguistic structures such as phonemes and words, by operating at what they see as the highest linguistic level of abstract verbal concepts (see Nuerk, Iversen, & Willmes, 2004, for a more detailed account of why they consider the conceptual level to be the highest linguistic level, based in large part on the production model of Levelt, Roelofs, & Meyer, 1999). A primary example is known as linguistic markedness: the condition of being conspicuously unusual as compared with a regular or more common form. With respect to an unmarked-marked oppositional pair of adjective terms, the unmarked one is considered the more comprehensive, dominant, and default form; in contrast, the marked one is irregular and is sometimes marked or derived by a negative prefix—as in the pair “happy-unhappy.” More generally, the first or unmarked term of such pairs is the one that is posed in questions, such as “how successful” or “how clean” are you? Alternatively, the use of marked terms in such queries—“how unsuccessful” or “how dirty” are you?—would presume that the person being questioned is in fact unsuccessful or dirty. In other words, unmarked-marked adjectives pairs often possess an evaluative nonequivalence.
In the area of mathematical cognition, linguistic markedness has primarily been studied with respect to the SNARC effect, focusing on parity tasks in which the relevant adjective pairs are “even-odd” and “right-left,” where even and right are considered unmarked and odd and left marked. As has been demonstrated, when the linguistic markedness of the numerical stimuli and the response side codes are both congruent, that is, the correct response to an even digit is to the right side and to an odd digit the left side, performance is facilitated, whereas incongruent codes (even-left and odd-right) produce interference (Nuerk et al., 2004). The evidence also suggests that this so-called markedness of response codes (MARC) effect can attenuate the SNARC effect (Berch, Foley, Hill, & Ryan, 1999).
Although linguistic markedness is not discussed in the present volume, Shaki and Fischer's (Chapter 11 by Shaki and Fischer) “harmony” model of the SNARC effect can explain a recent finding pertaining to the role of linguistic markedness in the parity task for native speakers of Hebrew. First, according to Shaki and Fischer, for the SNARC effect to emerge in a given task, participants' directional habits for reading text (from left to right or right to left) must be consistent or in harmony with their directional habits for reading numbers (i.e., from left to right or right to left, respectively). Until recently, there was no evidence of a SNARC effect for parity (odd-even) judgments in native speakers of Hebrew—who read and write words from right to left, but numbers from left to right—presumably due in large part to the conflict that arises between their alphabetic and numeric-writing systems. As Zohar-Shai, Tzelgov, Karni, and Rubinsten (2017) recently reasoned, because the Hebrew word for odd, ecpzugi, means “not even” (zugi is even), the morphological prefix negates the unmarked word root, making this linguistic markedness factor more salient than for their corresponding English-term equivalents (even vs odd), thereby enhancing the MARC effect with Hebrew participants. Based upon this analysis, Zohar-Shai et al. (2017) designed a series of studies that revealed the first evidence of a reliable SNARC effect in Hebrew participants after managing to reduce the MARC effect.
To understand how these researchers accomplished this feat, it is important to point out first that under standard parity task conditions, a within-participants design is employed, wherein the two types of parity-to-response-side mapping blocks—congruent (i.e., even-right/odd-left) and incongruent (even-left/odd- right) with respect to markedness—are run within the same single session. In contrast, Zohar-Shai et al. (2017) ran these blocks on two different days separated by a one week interval, reasoning that this approach might reduce the coactivation of the two mappings, thereby attenuating the MARC effect, which in turn could counteract its typical overriding of the SNARC effect, permitting it to emerge. This is indeed what happened, presumably because the time interval allowed the activation of the first mapping to decay, thereby effectively eliminating the standard conflict (i.e., lack of harmony) that arises between the two parity-to-response-side mappings in the standard, single-session judgment task.
Other Language-Related Skills: Verbal Working Memory and Other Cognitive Skills Related to Language Representations
Working memory is generally considered to be a limited capacity system responsible for temporarily storing, maintaining, and mentally manipulating information over brief time periods to serve other ongoing cognitive activities and operations. As Dowker and Nuerk (2016) acknowledge, it has been known for at least 35 years that verbal working memory is associated with mental arithmetic (Ashcraft & Stazyk, 1981). Nevertheless, Soltanlou, Pixner, and Nuerk (2015) demonstrated that although verbal working memory is related to multiplication skills in Grade 3, visual-spatial working memory plays a more important role in this kind of mathematical processing in Grade 4. Readers should also note that even beyond this developmental change, the association between working memory and mathematical processing is not always straightforward. For example, as Raghubar, Barnes, and Hecht (2010) have noted, language of instruction is one of the numerous factors that may influence and therefore complicate the interpretation of findings pertaining to the relations between working memory and math performance.
Some of the chapters in the current volume corroborate the importance of verbal working memory and other cognitive skills associated with language representations. For example, Chapter 6 by Donlan concludes that for children with specific language impairment, the increased cognitive load resulting from the combination of verbal memory and phonological processing needed to learn the number word sequence can be overwhelming, limiting how effectively associations can be formed between spoken words and Arabic numerals. Chapter 7 by Wicha et al. discusses how, in the context of bilingualism, theories differ with respect to whether arithmetic facts are encoded in a verbal memory store limited to the language in which they were learned or whether these facts exist in verbal memory stores that are distinct for each language—even though the facts were predominantly learned solely in one of the languages. Finally, Chapter 13 by Bender and Beller discusses how numeration systems that differ by modality (verbal, notational, body-based, and material) may differentially impact working memory load, which in turn can influence their use as cognitive tools for enabling the representation and development of numerical and arithmetic processing.
Finally, we think that another example of a language-related cognitive skill that can influence numerical cognition is that of analogical mapping—the construction of correspondences between two representations that share similar structures—as discussed in Chapter 2 by Marchand and Barner. First, however, as Ferry, Hespos, and Gentner (2015) have acknowledged, even though possessing relational language is not necessary for analogical ability, language (e.g., the use of common labels) has been shown to enhance children's relational capacity by strengthening attention to specific relations and by inviting comparison processes (Namy & Gentner, 2002). In their chapter, Marchand and Barner review evidence supporting their contention that learning the meaning of number words is essentially reliant on the construction of analogical mappings between language and nonverbal representations. They also provide compelling evidence that learning to rapidly assign labels to large arrays of objects when estimating is likewise dependent on a process of analogical mapping.
A Key Question Driving Research on Numerical Development
How Do Children Learn the Meaning of Number Words?
Number words are different from other types of words. They refer to the abstract property of sets of individuals, such as five dogs, rather than attributes of the individuals themselves, such as large dogs. Number words are also children's first foray into symbolic mathematics and as such set the foundation for their long-term mathematics achievement. Given their unique linguistic properties and their importance for symbolic mathematics, it is not surprising that children's learning of the meaning of number words—their cardinal value (e.g., that “five” represents a set of five things)—has been the subject of theoretical debate and empirical study for many decades (Carey, 2009; Gelman & Gallistel, 1978; Le Corre & Carey, 2007; Lipton & Spelke, 2005; Wynn, 1992), and continues to this day (e.g., Spelke, 2017; vanMarle et al., in press). At the center of the debate are the processes that support children's learning of the cardinal value of number words and reasons for the protracted development of this knowledge. This insight is thought to be based, at least in part, on the acquisition of the successor principle, that each successive number word in the count list is exactly one more than the word before it.
It is now well established that many 2-year olds can recite at least part of the verbal count list (e.g., “one, two, three, four…”) but may take another 2 years before they understand the cardinal value of these number words (Gelman, 1972; Wynn, 1990, 1992). The development of this knowledge is revealed in the give-N task, whereby children are asked to give an experimenter a particular number of objects. Children who understand the cardinal value of one—“one knowers”—will give the experimenter one object when asked to do so and a random number of objects for all other number words. Over the next 3–6 months, these children will become “two knowers” and, several months later, “three knowers.” Once children understand the meaning of “four,” they transition to “cardinal principle knowers” (CP-knowers); that is, they provide to the experimenter the exact number of requested objects up to the limit of their count list (Carey, 2004; Le Corre & Carey, 2007; Wynn, 1992). Children's transition to CP-knowers is quite remarkable, because even our closest relatives do not achieve this conceptual insight, even with extensive training (Beran, Parish, & Evans, 2015).
One key and still debated question is the extent to which this insight is dependent on the evolutionarily ancient approximate number system (ANS). The ANS supports the representation and comparison of the relative quantity of collections of discrete objects and for performing simple arithmetic operations on these representations (Feigenson, Dehaene, & Spelke, 2004; Gallistel & Gelman, 2005; for recent reviews, see Geary, Berch, & Mann Koepke, 2015). In other words, the ANS is thought to provide an intuitive understanding of aspects of number, counting, and arithmetic (Gelman & Gallistel, 1978), but this implicit knowledge must still be linked to the child's count list and number words. One possibility is that young children first form associative mappings between numerals and ANS representations as they learn the meaning of number words and the ordinal nature of the counting system (Lipton & Spelke, 2005). And, children and adults do appear to have a mapping between number words and corresponding ANS representations of quantity, but these mappings may only be for the first five or six number words, at least in children (Sullivan & Barner, 2014). Moreover, ANS representations of quantity are approximate, whereas the cardinal values of number words are exact. vanMarle and colleagues (in press) proposed that this mismatch can be bridged if children's learning of the cardinal value of the first few number words is supported by both the ANS and the object tracking system (OTS). The latter supports the individuation and tracking of up to three or four objects as they move through space. The OTS is not a quantitative system per se but does provide an exact representation that, in combination with the ANS and number words, may contribute to children's understanding of the cardinal value of the first three to four number words (see also Spelke, 2017).
It has also been proposed that perhaps children can deduce the cardinal value of number words through the same language mechanisms that support the learning of other words, without reliance on the ANS or OTS (Bloom & Wynn, 1997); specifically, the syntax of a sentence and the semantics of other words in the sentence allow children to narrow the range of possible meanings of an unknown word used in that sentence. Bloom and Wynn argued there are some linguistic features that are consistently associated with use of number words and that these features, when averaged across many spoken examples, allow children to deduce the cardinal value of specific number words. Number words are generally used with count (e.g., three dogs) not mass (e.g., half a glass) nouns; they do not appear with modifiers (e.g., very two), suggesting absolute properties; they precede but never follow adjectives (two big dogs vs big dogs two), and they can occur in partitive contexts (X of the Ys). Syrett, Musolino, and Gelman (2012) evaluated a corpus of parent-child utterances to determine if partitive phrases occurred frequently enough and in contexts that unambiguously allowed children to infer the exact quantities of number words. They found that most of the partitive phrases that conveyed quantitative information referred to general amounts (get some of the cookies), and less than 14% included exact amounts (get three of the cookies). Two follow-up experiments also showed that children use this syntactic feature of language to make inferences about specific amounts, only under restrictive conditions. They did not assess the other aspects of language proposed by Bloom and Wynn, but their results and other factors suggest that aspects of natural language may aide children's learning the cardinal value of number words but is probably not sufficient in and of itself (Barner, 2012; Butterworth, 2012; Spelke, 2017).
At this point, the developmental pattern of children's learning of the cardinal value of number words is well established (Wynn, 1992), and there is recent evidence suggesting that the ease with which children learn number words is critically important to their early mathematics achievement (Geary & vanMarle, 2016), but exactly how this learning occurs is not fully understood. Most likely, some combination of the ANS, OTS, and the system for natural language learning are key to this learning, but the relative timing and mix of these processes to learning the meaning of number words remains to be determined.
Other Considerations in Studying Linguistic Influences on Numerical Processing
Language and Number in the Brain
Much is now known about the representation of numerical information in the brain, including involvement of the right and left intraparietal sulcus and areas of the prefrontal cortex (for reviews, see Berch, Geary, & Mann Koepke, 2016; Nieder & Dehaene, 2009). The former are not classic language areas and, as noted earlier, people in traditional cultures without extensive number words have an intuitive sense of approximate number (e.g., Gordon, 2004). Clearly, the basic aspects of people's inherent number sense are independent of language, but language is critical for aspects of children's learning about symbolic number and arithmetic. On top of the earlier described relation between language and children's learning of the cardinal value of number words, children rely on counting when first learning to solve arithmetic problems (Siegler & Shrager, 1984). Use of fingers to represent cardinal value and use of verbal counting using fingers may provide an early bridge between spatial representations of quantity, language, and numerical representations in the brain (Bertelleti & Booth, 2016). Indeed, early studies of brain injury involving areas of the parietal cortex around the angular gyrus revealed a link between dyscalculia and finger agnosia (Gerstmann, 1940); these studies were with educated adults and involved symbolic arithmetic, not an intuitive sense of number. Over the course of development, the repeated use of finger representations and verbal counting in quantitative contexts appears to result in an integration of brain regions involved in hand movements, speech, and those involved in number representation, including the intraparietal sulcus.
Dehaene and Cohen (1995) argued that some aspects of arithmetic, especially addition and multiplication facts, eventually become stored in long-term memory as a verbal, language-based code, which will be dependent in part on the left angular gyrus and left temporal cortex associated with storage of word meanings. The storage as a verbal code follows from the use of verbal counting and other language-related strategies to solve these problems (Siegler & Shrager, 1984); the formation of these types of long-term memories does not occur as easily for subtraction and division because they are not commutative (9−5 does not equal 5−9). There are however some differences between addition and multiplication, with long-term memory representations of multiplication facts showing more involvement of the temporal cortex associated with memory for words than addition facts (Bertelleti & Booth, 2016). In any case, these studies indicate that children's use of verbal counting, with and without fingers, eventually results in an integration of the intraparietal sulcus with other areas of the brain, including some language areas, at least for a few aspects of symbolic arithmetic.
Domain-General and Domain-Specific Influences of Bilingualism on Cognition
A Brief History of the Study of Bilingualism and Its Effects on Cognitive Development
The impacts of bilingualism on cognitive development and capacities have been of interest at least since the late 1800s and been studied for a corresponding length of time by psychologists and educators. Investigators initially compared the IQs or other standardized general cognitive measures of monolinguals with bi- or multilingual individuals and typically found an advantage for monolinguals (Jones & Stewart, 1951; Laurie, 1890; Saer, 1923), generally hypothesizing that multilingual individuals or “polyglots” are constantly battling the interference of competing languages (Weinreich, 1953) and greater isolation from the majority language and experiences (Lambert, Hodgson, Gardner, & Fillenbaum, 1960). Researchers and educators alike believed that the bilingual's “confusion” and presumed need to inhibit competing words and potentially competing representations would inhibit cognitive development (Saer, 1923).
But as neutral or positive data on the benefits of bilingualism emerged along with awareness of appropriate control variables (Peal & Lambert, 1962) such as socioeconomic status (McCarthy, 1954) and degree of bilingualism (Saer, 1931), and the advent of multiple factor analysis, new research suggested the possibility of distinct cognitive factors (Guilford, 1956) and a possible role for cognitive organization and/or cognitive inhibition (Bialystok & Ryan, 1985; Green, 1998; Peal & Lambert, 1962) in the polyglot intellect. In particular, evidence of changes in interference and cross language facilitation during simple language tasks in some polyglots has been attributed to a variety of cognitive capacities including levels of processing (Garrett, 1975), working memory (Morales, Calvo, & Bialystok, 2012; Soliman, 2014), attention (Astheimer, Berkes, & Bialystok, 2016; Chung-Fat-Yim, Sorge, & Bialystok, 2017), and executive function (Bialystok, 2017; Wiseheart, Viswanathan, & Bialystok, 2016).
In contrast to early conjectures, bilingualism has more recently been demonstrated to produce positive benefits in each domain identified above. Despite this mounting evidence for a “bilingual advantage,” many still debate the unique contributions of bilingualism, the consistency of these effects, and decry the publication bias in favor of finding such an effect (Paap & Greenberg, 2013; Valian, 2015). Even one of the earliest and most vocal proponents of a bi-/multilingual advantage admits that a diverse array of experiences have the power to positively impact behavior and brain structure and function (Bialystok, 2016; Bialystok & Werker, 2017). It is nevertheless noteworthy that the above list of likely impacted cognitive domains is largely domain-general; it would therefore be prudent for students and investigators of any cognitive capacity to consider the potential role that bi-/polylingualism might have on their both verbal and nonverbal, symbolic and nonsymbolic cognitive domains of interest.
Further, the prevalence of multilingual environments and the complexities associated with trying to accurately characterize these and the participants sampled should give additional pause to all cognitive and behavioral scientists; it is estimated that more than half the world is at least bilingual (Grosjean, 2010 as quoted by Ellen Bialystok & Werker, 2017) to some degree. Even countries that, for political or social reasons, have declared a single “official language” and resisted learning other languages typically have high rates of bilingualism. This list includes the United States, where although English is the official language, the US Census Bureau reported that 382 individual languages and language groups were most commonly spoken in U.S. homes in 2013, and more than 20% of the population (over the age of 5 years) spoke a language other than English at home (Ryan, 2013), with some urban areas as high as 70% (see www.lep.gov).
The complexity of defining “bilingualism” was recognized early on, as not all bilinguals are equally proficient in both languages. Hywela Saer (1931) attempted to create a standardized measure of the degree of bilingualism by creating a ratio of proficiency in the two spoken languages where equivalent proficiency (or a ratio = 1) was considered true bilingualism. But bilingualism is seldom perfectly balanced; it often changes over the course of a lifetime, and it is sensitive to environmental factors such as the density of other speakers. And rarely are any but so-called simultaneous bilinguals (i.e., who learn two languages from birth) “true bilinguals.” Thus our capacity to study, as well as to accurately account for and statistically adjust our analyses for, the potential impact of bilingualism on cognitive factors is most likely more challenging than is typically recognized. Finally, it should be noted that any domain-general benefits of bilingualism may be dependent on factors such as cross lingual lexical similarity (Duyck & Brysbaert, 2008), articulatory speeds (Elliott, 1991), or other domain-specific factors. Here again, the complexity of defining “bilingualism” and understanding its effects demands additional study both in breadth and depth.
Bilingualism and Mathematical Cognition
As Chapter 8 by Salillas and Martínez discusses, the study of bilingualism can serve as a window to various forms of language influences on numerical and arithmetic processing in several ways, for example, for testing the distinction between approximate and exact calculation; by investigating the effect of having two accessible, bilingual linguistic contexts on calculation; and to determine the relative dominance of two languages for mathematics learning (see their chapter for a review of such studies). Along these lines, by recording event-related potentials (a time-sensitive measure of neural electrical activity) with Spanish-English bilingual children and adults, Chapter 7 by Wicha et al. demonstrates that both the language in which math facts were initially learned and the frequency with which these facts are used in the second language can affect the manner in which bilinguals access them in each of their languages—and the efficiency with which they do so. Also employing the event-related potentials technique, Chapter 8 by Salillas and Martínez reports new data suggesting that brain differences in the management of numerical judgments between bilingual dyscalculic children and matched bilingual controls without dyscalculia are regulated by the input language of the number words—that is, either the language in which they originally learned math or their other language.
The Role of Culture in Mathematical Cognition
The earlier noted cross-national differences in mathematics achievement are important findings, because the mathematical competencies of children and adolescents will have a long-term impact on their later educational and employment opportunities (Ritchie & Bates, 2013; Rivera-Batiz, 1992) and influence their ability to deal with now routine quantitative tasks (Reyna, Nelson, Han, & Dieckmann, 2009). The sources of national differences in success in fostering children's mathematical development are not fully understood but are related at least in part to the opportunity to learn. The latter is related to national differences in the cultural valuation of mathematics that in turn contributes to differences in the quantity and quality of children's exposure to mathematics at home and in school (Geary, 1994). These differences are nicely illustrated by Harold Stevenson's and colleagues' studies of the mathematics achievement of American (USA), Japanese, and Chinese (Taiwan) students (Stevenson, Chen, & Lee, 1993, 1986, Stevenson, Lee, & Stigler, 1986, Stevenson et al., 1990; Stevenson & Stigler, 1992). The gist of these studies is that Chinese and Japanese students have an advantage over their American peers by first grade, and the gap widens across grades and perhaps even across generations (Geary, Salthouse,Chen, & Fan, 1996, Geary et al., 1997).
Among other things, these differences and the differences noted in the large-scale international studies are related to curriculum. Based on the first IEA, Husén concluded “thus, national differences [in mathematical achievement] can in part be explained by differences in emphasis in curriculum” (Husén, 1967, Vol. 2, p. 300). Generally, achievement in different mathematical areas (e.g., algebra) varied directly with the relative degree of emphasis on that area in the mathematics curriculum. Stevenson and colleagues provided a fine-grained analysis of these differences, with extensive classroom observations. In one study of classroom behavior, American first graders were engaged in academic activities about 70% of the time, compared with 85% and 79% of the time for their Chinese and Japanese peers, respectively (Stevenson et al., 1986). The gap widened by the fifth grade. In the first grade, American teachers spent 1–2 hours less per week on mathematics than did teachers in Taiwan and Japan (Stevenson et al., 1990). Again, the gap widened by fifth grade. In addition to time, there are also substantive differences in teaching approaches across nations (Stigler, Gonzales, Kwanaka, Knoll, & Serrano, 1999) and substantial differences in the amount of time spent on homework per week. American first graders spent very little time on homework, whereas their peers in Japan and China spent, on average, 4–8 hours per week on homework. The gap narrowed a bit by fifth grade but was still significant.
The differences in mathematics classrooms and homework are compounded by national differences in the rigor of the mathematics curriculum. Stevenson and Bartsch (1992), for instance, found that Japanese and American elementary and secondary school mathematics textbooks generally presented on the same topics, but the material was conceptually more difficult and presented in an earlier grade in Japanese, in relation to American, textbooks. American textbooks were also cumbersome and included too much unnecessary, nonmathematical material, a concern echoed 16 years later by the National Mathematics Advisory Panel (2008). In a related study, Fuson, Stigler, and Bartsch (1988) found that arithmetic topics and problems presented in American fifth–sixth grade textbooks were presented two–three grades earlier in East Asian textbooks and in textbooks from the former Soviet Union. The common core initiative in the United States was, in part, designed to help to address these gaps (http://www.corestandards.org/), but its success at doing so is uncertain due to resistance to the standards.
There are also wider cultural beliefs that may influence children's investment in learning mathematics. For instance, American students and parents generally believe that success in mathematics is more strongly influenced by some type of inherent ability or propensity to learn mathematics, whereas parents from East Asian nations emphasize the importance of hard work over ability (Stevenson et al., 1990). Interventions that shift students' emphasis from ability to hard work are associated with gains in mathematics achievement (Blackwell, Trzesniewski, & Dweck, 2007), suggesting these attributions about what it takes to become skilled in mathematics matter. These differences in beliefs and a high valuation of mathematics in Asian culture (Hatano, 1990) translate into differences in parental expectations for their children's mathematical development and associated differences in emphasis on informal learning of mathematics in the home (Crystal & Stevenson, 1991).
In all, there are clearly myriad factors that influence students' opportunity to learning mathematics in one nation or another, and these influences translate into large and very persistent cross-national differences in students' mathematics achievement. These influences range from national or local policy for the mathematics curriculum to the rigor of the mathematics presented in textbooks to students' and teachers' engagement with mathematics in the classroom to students' and parents' attitudes about what it takes to excel in mathematics. These differences in turn may reflect deeper cultural differences in the valuation of mathematics, the relative focus on individual agency (e.g., resulting in more or less emphasis on student-directed learning), and perceived opportunities for second chances, that is, the extent to which the early learning trajectory influences later educational opportunities (Geary, 1996; Hatano, 1990).
Five Myths of Cultural Psychology
In an effort to extol the virtues of studying cultural influences in psychological research, Wang (2017) recently acknowledged that there are a number of myths regarding cultural psychology that frequently diminish the likelihood that psychologists will incorporate such factors in their research. More specifically, Wang offers what she considers to be five false assumptions that she then critiques, largely by providing counterexamples from her own seminal research on the role of culture in the development of autobiographical memory, autonomous and relational self-goals, knowledge of emotions, and future thinking. Here, we briefly describe each of these myths and then point readers toward the chapters in this volume that further demonstrate how, in direct opposition to these erroneous assertions, investigating the impact of cultural factors on mathematical cognitive development can not only increase the generalizability of findings obtained from conventionally studied participants in Western societies but also provide unique methodological and theoretical contributions.
Assumption 1. Cultural Psychological Science Focuses Only on Finding Group Differences
Wang (2017) argues that although studying the kinds of differences that exist between cultures is important, cultural psychology can go beyond such basic outcomes by investigating why and how those differences occur. She points out that one way this can be achieved is by manipulating variables in experimental settings that can make individuals from one culture behave more like those from another culture. Readers can find such examples pertaining to mathematical performance in the chapter by Göbel. Other authors in this volume also discuss evidence from different cultures that goes beyond foundational findings of group differences, thereby making progress toward uncovering the mechanisms responsible for such differences and the factors responsible for giving rise to such outcomes. These include the chapters by LeFevre, Cankaya, Xu, and Lira; Okamoto; Opfer, Kim, and Qin; and Shaki and Fischer.
Assumption 2. Cultural Psychology Disregards Group Similarities
With respect to the second false assumption, Wang (2017) contends that far from being a failure, a cross-cultural psychological study that yields no cultural differences can reveal that a psychological construct or process may be closely coupled with shared sociocultural experiences, neurobiological constraints, or an interaction between these factors. We would add that developmental changes can also be informative, as children from different cultures may, for example, exhibit similarities at an earlier age but differences at a later one. Taken together, several of the chapters in the present volume reveal how both group similarities and differences can inform our understanding of mathematical cognitive development, including the contributions by Göbel, LeFevre et al., Okamoto, Opfer et al., and Shaki & Fischer. Basically, as Wang states, “cultural similarities may suggest universality in the underlying biological cognitive mechanisms on the one hand, and shared human conditions and life circumstances on the other” (p. 22).
Assumption 3. Cultural Psychology Concerns Only Group-Level Analysis
According to Wang (2017), although it is true that cultural psychological research frequently entails comparisons of cultural groups, this should not be interpreted as diminishing the importance of individual differences. Indeed, as she cogently points out, investigating individual differences is often essential for revealing the factors that can explain the group-level cultural differences that have been observed (see Chapter 5 by Opfer et al. and Chapter 10 by Göbel for examples derived from the study of mathematical cognitive development).
Assumption 4. Cultural Psychology Is Irrelevant to Basic Psychological Processes
Wang (2017) summarizes evidence demonstrating that contrary to this generally accepted assumption, various basic psychological processes, including neuronal functioning, sensation, visual illusions, face processing, color perception, and event segmentation, are indeed sensitive to cultural influences. Several chapters in this volume demonstrate further how basic cognitive processes underlying numerical representations and processing may likewise be responsive to cultural influences (see Chapter 3 by LeFevre et al., Chapter 4 by Okamoto, Chapter 5 by Opfer et al., Chapter 10 by Göbel, Chapter 11 by Shaki and Fischer, Chapter 12 by Saxe, and Chapter 13 by Bender and Beller).
Assumption 5. Cultural Psychological Research Only Confirms the Generalizability of Theories
According to Wang (2017), one of the principal contributions that cultural psychology can make to psychological science is to enable researchers to test their theories and hypotheses beyond their customary pool of participants—specifically, those sampled from western, educated, industrialized, rich, and democratic societies (Henrich, Heine, & Norenzayan, 2010). However, she also argues that in to addition to confirming the generalizability of theories, cross-cultural research can play an important role in modifying and augmenting extant theories, and possibly even permit the discovery of mechanisms that may only be found in non-Western samples. An illustration of the former with respect to numerical processing can be found in Chapter 11 by Shaki and Fischer, while a prime example of the latter is described in Chapter 12 by Saxe concerning his work on the 27-body part counting system used by the Oksapmin of Papua New Guinea. We would add that the study of cultural influences on the acquisition of numerical and arithmetic skills can also yield unique findings that can inform the design of instructional interventions (e.g., see Chapter 5 by Opfer et al. and Chapter 10 by Göbel).
Before moving on to the next section, it is important to point out that both the cross-cultural and within-culture research methods discussed by Wang (2017) fall under the rubric of what Chapter 12 by Saxe refers to as dichotomous approaches to the study of culture and cognition. As he cogently points out, these constitute both an epistemological stance and a set of empirical techniques wherein researchers treat cognitive processes as occurring within the heads of individuals and as dependent variables that are influenced by a distinct and separate set of surrounding cultural elements. These cultural influences are regarded as independent variables, consisting of factors such as early numeracy experiences, numerical language characteristics, and parental attitudes toward mathematics. In contrast, Saxe adopts a methodological framework that he refers to as an intrinsic relations approach where the emphasis is on studying the dynamic and complex interleaving of cultural factors (especially “collective practices” that are cultural microcosms) and the development of numerical cognition in the daily activities of individuals as “processes in motion” (see the Saxe chapter for more details).
Disentangling Linguistic and Cultural Influences on Numerical Processing and Mathematics Achievement
Dowker and Nuerk (2016) point out that attributing the superiority of East Asians' numerical and arithmetic skills over that of students who speak European languages to linguistic (power) transparency may be moderated by the confounding of many other differences between countries with respect to their cultural and educational practices. They also contend that one approach for gathering more evidence of language effects per se is to compare children who are schooled in different languages, albeit in the same country and educational system. The example they provide is Wales, where both the English (irregular) and the Welsh (regular/transparent) counting systems are in use. By comparing children receiving these different counting systems within the same country, cultural environment, educational system, and curriculum, Dowker, Bala, and Lloyd (2008) found that children in Welsh-medium primary schools exhibited specific advantages in reading and comparing two-digit numbers, even though they did not perform better in arithmetic overall. Dowker and Nuerk discuss an extension of this study using the same kind of design and another study employing a related design using additional controls with children in Hong Kong who can learn mathematics using both the Chinese (regular/transparent) and English (irregular) languages (Mark & Dowker, 2015). Taken together, these results show that in spite of some selective numerical processing benefits associated with a transparent counting system, this linguistic property does not have an all-encompassing impact on cross-national differences in arithmetic (Dowker & Nuerk, 2016).
Chapter 3 by LeFevre et al. discusses in further detail the studies just described, acknowledging that cultural, experiential, and other demographic factors have frequently been confounded with linguistic differences in studies that have compared the influences of language on disparities in the acquisition of early numeracy skills. Such factors have included, among others, the amount of practice children have had in their respective languages, the amount of direct instruction in math at home, and the level of encouragement to engage in math-related activities. Furthermore, LeFevre et al. provide a comprehensive review and analysis of these kinds of factors with respect to her (and her colleagues') own studies of young children's counting across five different languages, taking into account the role of language, country, parents' education, and home numeracy factors. Likewise, Chapter 5 by Opfer et al. examines numerous contrasts in attitudes toward and educational practices pertaining to the learning of math that may contribute to the cross-national gap between East Asian and American students' mathematics proficiency. As we described earlier, these factors include beliefs concerning whether ability or effort is more important in learning mathematics, both the quantity and quality of mathematics instruction in school, and the quality of teacher knowledge, among others. In addition, Chapter 4 by Okamoto discusses the contribution of qualitatively different problem-solving experiences between children in different cultures to cross-national gaps in mathematics achievement.
As Chapter 3 by LeFevre et al. concludes, researchers who investigate differences in the acquisition of early numeracy skills among languages should also measure other prospective causal factors and then control for as many of them as is feasible, either by developing other procedures for testing language-related differences or by means of statistical methods. Interestingly, Chapter 5 by Opfer et al. discusses how they have applied both of these recommended approaches to the exploration of cross-national differences in mathematics achievement.
Conclusions
The concepts and procedures that define modern mathematics as an academic discipline may be universal, but the language, educational, and broader cultural supports (e.g., beliefs about the factors that facilitate mathematical learning, such as hard work or native ability) that influence people's access to this discipline and the ease with which they learn it are not. Many of these language and cultural supports are invisible in the sense that the ways in which their structure facilitates (or does not) children's early learning of symbolic number and arithmetic are not obvious to the casual observer. Few parents and for that matter not many teachers understand that something as simple as the names of number words, such as eleven versus ten-one, can provide cues to children about the underlying structure of the number system, the base-10 structure in this case (Miura, 1987). And, teachers can use these words to teach this structure, at least in languages in which it is transparently reflected in number words (e.g., as in ten-one for 11 or two-ten-one for 21). Research on the mapping between number words and children's learning of the mathematical number system was among the first studies to highlight how language can influence children's access to and understanding of symbolic mathematics. Since this early pioneering research, the study of the intersection between language, culture, and mathematics has blossomed and, as illustrated by the chapters in this volume, has yielded a richer and more nuanced understanding of this intersection.
Included among these advances and building on Saxe's (1981) seminal work with the Oksapmin are ever-increasing studies of number words and concepts among people living in traditional cultures who are not exposed to formal education and analyses of the cultural evolution of number systems other than the Hindu-Arabic system that is the basis for academic mathematics as we know it. This burgeoning cross-cultural research is providing insights into similarities and differences in how human's construct mathematical systems, depending on needs (e.g., trade) within the culture, and how this knowledge can be transmitted across cultures. These types of studies add to the more familiar comparisons of the relative academic achievement of students receiving formal education in economically developed societies (e.g., Stevenson et al., 1986). Not surprisingly, the latter differences are related in part not only to differences in the content and rigor of mathematics instruction but also to broader social beliefs about mathematics learning (e.g., that it takes hard work and practice) and the valuation (or not) of mathematics achievement (Stevenson & Stigler, 1992). A fuller understanding of all of these factors—language, education, and cultural—will provide insights into how the human mind creates and learns mathematics, and this in turn will provide guidance on how to better structure (e.g., sequence grade-level content) and teach mathematics.