Cognitive Benefits and Costs of Bilingualism in Elementary ... - UGent

Journal of Educational Psychology 2011, Vol. 103, No. 3, 547–561

© 2011 American Psychological Association 0022-0663/11/$12.00 DOI: 10.1037/a0023619

Cognitive Benefits and Costs of Bilingualism in Elementary School Students: The Case of Mathematical Word Problems Sebastian Kempert

Henrik Saalbach

University of Frankfurt

ETH Zurich

Ilonca Hardy University of Frankfurt Previous research has emphasized the importance of language for learning mathematics. This is especially true when mathematical problems have to be extracted from a meaningful context, as in arithmetic word problems. Bilingual learners with a low command of the instructional language thus may face challenges when dealing with mathematical concepts. At the same time, speaking two languages can be associated with cognitive benefits with regard to attentional control processes, although such benefits have only been found in highly proficient bilinguals. In the present study, we attempted to disentangle the effects of bilingual proficiency on mathematical problem solving in Turkish–German bilingual elementary school students. We examined whether the positive cognitive effects of bilingualism could be found not only in highly proficient bilinguals but also in students with an immigrant background and a low command of the instructional or native language. Our findings emphasize the importance of language proficiency for mathematics problem solving, as shown by the predictive value of students’ proficiency in the language of testing (German/Turkish) for their performance on mathematical word problems. No additional effect of the language of instruction (German) was found for problem solving in the bilingual students’ native language (Turkish). Furthermore, bilinguals gained scores comparable to those of their monolingual peers on word problems that required attentional control skills although performing significantly below their monolingual classmates on ordinary word problems, suggesting that bilinguals have an advantage when it comes to attentional control. Finally, bilingual students with a relatively high command of the instructional language performed better on word problems presented in German than on those presented in Turkish, thus facing cognitive costs when transferring knowledge from one language to the other. Implications of our findings for bilingual education are discussed. Keywords: bilingualism, executive control, mathematical word problems, instructional language, immigrant children

(Baumert & Schu¨mer, 2001; Schwippert, Bos, & Lankes, 2004; Schwippert, Hornberg, & Goy, 2008; Walter & Taskinen, 2007). Throughout their school careers, bilingual students’ risk of being retained is two to three times higher than that of their monolingual classmates (Baumert & Schu¨mer, 2001). Some analyses have shown the contribution of general measures of socioeconomical and cultural backgrounds to these striking differences in school careers (Mu¨ller & Stanat, 2006). Most specifically, however, mastering the instructional language is the greatest barrier for students from immigrant families (Baumert & Schu¨mer, 2001; Organisation for Economic Co-Operation and Development [OECD], 2006): Almost 50 % of all 15-year-olds with an immigrant background failed to reach more than the first reading competence level in PISA 2000. In other words, these students’ reading abilities barely go beyond the localization of explicitly stated information and the interpretation of and the reflection on very simple statements in a given text. Similar results have been reported from PISA 2003, 2006, and 2009 (Stanat, Rauch, & Segeritz, 2010; Walter, 2008). On the basis of these findings, it has been suggested that the deficits of students with an immigrant background in mastering the language of instruction, as reflected by low reading scores, will

The importance of students’ competence in the instructional language for school achievement has been an issue of educational policy ever since international comparative studies revealed that students with an immigrant background face serious academic problems both in elementary school and secondary school. For countries like Germany or Switzerland, studies on school achievement such as the Program for International Student Assessment (PISA) or the Progress in International Reading Literacy Study (PIRLS) have shown that in all three domains of assessment— reading, mathematics, and science—students with an immigrant background performed significantly below their native-born peers

This article was published Online First June 20, 2011. Sebastian Kempert and Ilonca Hardy, Department of Educational Sciences, University of Frankfurt, Frankfurt, Germany; Henrik Saalbach, Institute of Behavioral Sciences, Eidgenössische Technische Hochschule (ETH) Zu¨rich, Zu¨rich, Switzerland. Correspondence concerning this article should be addressed to Sebastian Kempert, Department of Educational Sciences, University of Frankfurt, Frankfurt am Main 60325, Germany. E-mail: kempert@em .uni-frankfurt.de 547

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KEMPERT, SAALBACH, AND HARDY

have a cumulative effect in most academic domains, thus contributing to an impaired development of competencies in other academic fields such as mathematics and science (Baumert & Schu¨mer, 2001; OECD, 2006). However, these disadvantages of bilingual students with an immigrant background sharply contrast with research findings that highlight a number of specific positive cognitive consequences of speaking two or more languages (for reviews, see Adesope, Lavin, Thompson, & Ungerleider, 2010; Bialystok, 2009; Bialystok, Craik, Green, & Gollan, 2009). Numerous studies with highly proficient bilingual speakers revealed that bilinguals, due to their daily practice of two different language systems, tend to develop an increased ability to perform executive control processes (Bialystok, 1988; 1999; 2009; Bialystok, Craik, & Luk, 2008; Bialystok, Craik, & Ryan, 2006; Cromdal, 1999; Martin-Rhee & Bialystok, 2008; Rodriguez-Fornells, de Diego Balaguer, & Mu¨nte, 2006). In the present study, we attempted to disentangle the effects of bilingualism on academic achievement in students with an immigrant background. In particular, we examined whether and to what extent bilingual (Turkish–German) third graders’ mathematical achievement is affected by aspects of bilingual proficiency. Turkish–German bilinguals, who constitute one of the largest groups of primary and secondary students with an immigrant background in Germany (Ramm, Prenzel, Heidemeier, & Walter, 2005), have been found to be the most strongly disadvantaged group in school assessments as compared with other immigrant groups (Stanat, 2008). In the following, we will first discuss the role of students’ proficiency in the instructional language and native language for mathematical learning. We will then provide arguments for why the positive cognitive effects of bilingualism found in highly proficient bilinguals may be related to mathematics achievement and show why it is likely that these cognitive advantages will also extend to bilingual students with an immigrant background.

Language and Mathematics Learning In the present study, we explored the effects of bilingualism on arithmetic word problem solving. This context was chosen not only because of the great importance attributed to mathematics in the school curriculum but also because the learning of mathematics is more strongly related to language processes than has been assumed. According to Bialystok (2001), mathematics is a domain where cognitive effects in bilinguals are likely to occur as language and mathematics share common critical features such as abstract mental representation, conventional notations, and interpretive function. Furthermore, cross-cultural differences in children’s mathematical performance have been partly traced back to differences in the way languages code numerosity (e.g., Gordon, 2004; Miller, 1996; Miller, Smith, Zhu, & Zhang, 1995). Indeed, both cognitive and educational research has produced ample evidence that language processing and mathematical problem solving are closely connected. The importance of language for the acquisition of mathematical concepts and procedures has also been revealed by experimental research. For example, in a training study, Spelke and Tsivkin (2001) had highly proficient adult Russian–English bilinguals practice different mathematical problems in their respective native

languages in a training course. A posttraining test, carried out in both languages, revealed that the retrieval of information was significantly more accurate and faster when the language relied on for problem solving was the language of instruction rather than participants’ other language (in which they were proficient as well). Similar effects were also reported with respect to mathematical word problems (Bernardo, 1998) and in domains other than mathematics (e. g., Marian & Fausey, 2006). These effects have been attributed to the specific cognitive costs (so-called switching costs) that arise when participants are required to switch languages for information retrieval (Saalbach, Eckstein, & Grabner, 2010; Spelke & Tsivkin, 2001). For example, in a recent neuropsychological study on Italian–German bilinguals, Saalbach et al. (2010) found that the cognitive costs associated with language switching in the context of mathematical problem solving were due to additional calculation processes required when transferring knowledge from the language of instruction to the language of retrieval. These findings suggest that the encoding and retrieval of content taught in school are closely connected to the language of instruction. Students with a low command of the instructional language may thus face the risk of insufficient mental representation of information presented at school, which may eventually result in poor performance in different subjects. In fact, Marian and Fausey (2006) found that their dominant Spanish–English bilingual participants (i.e., participants whose proficiency was significantly higher in Spanish than in English) performed significantly below balanced bilinguals (i.e., participants whose proficiency was the same in Spanish and in English) in the retrieval of information when English was the language of both instruction and retrieval. In school programs like Structured Immersion or Transition, instructors often cope with this problem by using the students’ native language as a resource during instruction until students’ proficiency in the second language has developed to an adequate level (see Hakuta, 1999). The link between language and mathematics is especially evident in the case of word problems. According to a prominent model of arithmetic word problem solving, the first step in learning to solve such problems is for children to form a situational model of a given problem by structuring its relevant features. In a next step, they need to extract a mathematical problem model by reframing the linguistically coded relations of the situational model as mathematical relations. Finally, the children need to do the mathematical calculations, then interpret and validate the results, and, eventually map them onto the actual problem (c.f. Reusser, 1997; Verschaffel, Greer, & De Corte, 2002). Given this model of arithmetic word problem solving, the influence of language on achievement could be threefold: First, instructional processes in the classroom are highly language based, with co-construction processes such as the negotiation of concepts and norms as well as definitions, explanations, and argumentations taking place during mathematics instruction (Elbers & de Haan, 2005). Thus, low proficiency in the instructional language may hamper students’ initial comprehension of the steps needed to carry out the mathematical modeling process involved in solving a word problem. Second, there are certain linguistic structures which imply mathematical structures and operations, thus providing essential information on the mathematical model within the task statement. Children with a low proficiency in the language in

COGNITIVE BENFITS AND COSTS OF BILINGUALISM

which the word problem is presented may not recognize and interpret these verbal structures adequately. For example, the appropriate understanding of relational terms like “more than” or “less than” is critical for extracting a mathematical model (Stern & Lehrndorfer, 1992). In fact, simplifying the linguistic structure of word problems presented in English has been shown to result in an increase in the performance of learners of English as a second language while it had no effect on native speakers (Abedi & Lord, 2001). Finally, problems in comprehending the task on a surface level (i.e., a noun or a verb may be unknown) or the need to translate certain aspects of a problem may result in a greater load of working memory resources which, as a result, are not available for carrying out arithmetic calculations (e.g., Shaftel, BeltonKocher, Glasnapp, & Poggio, 2006). While the latter two aspects refer to effects of language on testing in a narrow sense (comprehension of the problems to be solved), the first aspect involves the social co-construction of meaning during instruction that exerts a cumulative influence on individuals’ academic achievement.

Cognitive Consequences of Bilingualism The so-called threshold hypothesis proposed by Cummins (1979) particularly emphasized cognitive disadvantages due to low language proficiency. Here, negative effects of bilingualism may be particularly prevalent in bilingual speakers who have deficits in both languages (semilinguals or weak bilinguals). Given the findings reported earlier, two types of cognitive costs associated with bilingualism in the context of academic learning are possible: One type of cost results from low proficiency in the instructional language, the other one from the need to switch between languages of instruction and retrieval. In both cases, cognitive costs refer to the additional mental effort that leads to disadvantages in academic learning or performance. In the case of language switching, as suggested previously, additional information processing is required in the language of retrieval. In the case of low language proficiency, more cognitive resources are taken up by, for example, language comprehension or repeated modeling processes due to incorrect mappings. Thus, fewer cognitive resources will be available for the actual learning process. Beyond postulating negative consequences for students with low language proficiency, Cummins also included in the threshold hypothesis the assumption that there are positive cognitive consequences of speaking two (or more) languages, which are expected to occur especially in bilingual speakers who are highly proficient (i.e., native-like) in both of their languages (so-called balanced bilinguals). According to Cummins, at a lower level of bilingual proficiency with only one of the two languages at the level of native speakers, the negative cognitive consequences of bilingualism may be avoided (in so-called dominant bilinguals). Several experimental studies in which both linguistic and nonlinguistic tasks were used have indeed provided evidence that balanced bilinguals will show specific cognitive benefits. In particular, these bilinguals are assumed to have developed an improved ability of executive control due to their daily practice of keeping their two language systems apart in everyday situations (Bialystok, 1988; 1999; 2009; Bialystok et al., 2006, 2008; Carlson & Meltzoff, 2008; Cromdal, 1999; Martin-Rhee & Bialystok, 2008; Rodriguez-Fornells et al., 2006). According to Bialystok, this practice helps them to “selectively attend to specific aspects of

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representation, particularly in misleading situations” (Bialystok, 2001, p. 131). One commonality is underlying all the tasks having revealed a bilingual advantage: They require subjects to respond to cues that are embedded in a misleading context and thus create a cognitive conflict. A cognitive conflict can arise because two competing cues have to be monitored (e. g., ignoring the position of the cue on a screen when responding to the color of the cue) or because the habitual response has to be suppressed in favor of a less familiar one (e. g., labeling pictures of the sun as “night”). In a series of experiments, Martin-Rhee and Bialystok (2008) found that bilinguals’ advantage is due to their improved control of attention rather than to the inhibition of a habitual response (see also Carlson & Meltzoff, 2008; Colzato et al., 2008). They argued, “For bilinguals, their two linguistic systems function as bivalent representations, offering different, potentially competing response options to the same intention or goal. To manage this conflict, bilinguals must attend to the relevant language system and ignore the unwanted system to assure fluency in speech production” (Martin-Rhee & Bialystok, 2008, p. 85). Given the findings on balanced bilinguals’ improved ability to monitor competing cues, it is extremely interesting to ask whether this cognitive advantage of bilingualism also applies to learners with high proficiency in one language only (dominant bilinguals). According to Bialystok and colleagues, the routine of switching between two (or more) symbol systems and the ability to do so in a controlled and adequate manner (in terms of keeping the symbol systems apart) may be the mechanism that underlies the gain in attentional control skills in the bilingual brain. Therefore, we posit that as soon as a basic communicative level is established in both languages, switching becomes a routine, and attentional control skills are trained in the process. Turkish–German bilingual students, for example, typically switch languages between home and school (Stanat, 2008). We may thus expect the cognitive benefits of bilingualism for executive control to show not only in highly proficient bilinguals but also in bilinguals with different degrees of proficiency in their languages. To our knowledge, there have been no published studies so far focusing on whether dominant bilingual children in general and students with an immigrant background in particular (i.e., a group that is usually associated with academic difficulties) show any beneficial cognitive consequences due to their bilingual experience.

Background of the Present Study In the present research, we attempted to disentangle the possible costs and benefits of bilingualism with respect to the mathematical achievement of students with an immigrant background. We were particularly interested in the nature of the cognitive consequences of bilingualism for immigrant children with a low socioeconomic status, as recent discussions in educational science and policy have typically focused on their academic problems. We believe that detecting cognitive advantages in this bilingual group would provide an additional perspective on their academic situation as well as possible means of support. We used a series of mathematical word problems as a measure of mathematics achievement. Children’s performance on arithmetic word problems is considered a reliable predictor of their later mathematical competence. In fact, the Munich Longitudinal Study on the Genesis of Individual Competencies (LOGIK) revealed that

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individual differences in math achievement during primary school, as measured with word problems, predict differences in math achievement at the end of secondary school significantly better than differences in IQ do (correlation coefficients: .64 vs. .04 at age 8 years; Stern, 2009). Furthermore, solving mathematical word problems may not only be an ecologically valid but also a sensitive measure of the cognitive benefits of bilingualism for attentional control processes (Lee, Ng, & Ng, 2009; Pasolunghi, Cornoldi, & De Liberto, 1999). As illustrated earlier, problem solving in mathematics is characterized by a process of cognitive modeling. In order to construct an adequate situational and mathematical model, students need to identify and extract the relevant pieces of information from a given problem context and, at the same time, suppress any misleading or irrelevant linguistic or numerical information. Research on word problems has shown that solution rates of different problems with a mathematically isomorphic structure differed strongly according to the level of the ambiguity of the context. Staub and Reusser (1995), for example, found that first and third graders’ solution rates decreased dramatically when the problem “Joe had 3 marbles. Then Tom gave him 5 more marbles. How many marbles does Joe have now?” was reworded into “Today Dane got 11 marbles from Susan. Yesterday Dane found 5 marbles. How many marbles does Dane have now?” In the present research, we thus used two types of mathematical word problems: common problems and word problems with distractors. The former type was derived from a set of empirically tested word problems with established solution rates for primary school students of various ages (Stern, 1998). For the latter type, we artificially enriched common problems by including numerical information that was not needed for solving the problem (distractors): for example, “Maria had 3 marbles. Altogether, the marbles cost 90 cents. Then Hans gave her 5 marbles. They cost 1.50 euros. How many marbles does Maria have now?” In this case, attentional control processes are particularly called for as children need to suppress the misleading numerical information in order to extract the relevant data. To assess language skills, we used a direct measure of oral proficiency instead of the reading comprehension measure that has typically been used in previous studies. As reported previously, findings from international comparative studies on school achievement showed a close relation between bilingual students’ skills in the instructional language and their achievement in different domains including mathematics. In these studies, however, language proficiency was indicated only by reading comprehension rather than being assessed with language tests. Reading comprehension is a global indicator of language proficiency as its acquisition also depends on cognitive skills unrelated to language but critical to school-related learning. In fact, it has been shown that performance of poor readers on math tests increased significantly when the test items were presented orally (e. g., Helwig, Rozek-Tedesco, Tindal, Heath, & Almond, 1999). Furthermore, language proficiency was usually only measured for the language of instruction but not for the native language. Therefore, it remains unclear what types of bilinguals in terms of Cummins’ framework (i.e., balanced, dominant, or weak) were tested and how this might relate to students’ performance. In the present study, we thus used oral proficiency measures in both the instructional and the native language, and we presented the mathematical word problems orally.

Research Questions and Hypotheses Drawing on the theoretical insights and empirical evidence outlined, we examined whether in a group of bilingual students the level of proficiency in the instructional language predicted achievement on mathematics word problems in the same language. We expected language proficiency to be closely related to the solution rate of word problems even when the influence of arithmetic skills, general cognitive ability, and socioeconomic status was controlled. We also expected that our measure of (oral) proficiency in the language of instruction would be a stronger predictor of mathematics problem solving in that language than reading comprehension. For word problems in students’ native language, we expected an effect of proficiency in the language of instruction beyond the impact of students’ proficiency in their native language. Furthermore, we tested whether bilingual children with an immigrant background would show a cognitive benefit, as compared with monolinguals, concerning attentional control processes. This effect of bilingualism was expected to be evident only in those tasks which especially require attentional control skills, after the effects of general cognitive ability, socioeconomic status, and arithmetic skills were controlled.

Method Participants Participants were 78 third graders from five primary schools in urban districts of Berlin and Frankfurt with a high percentage of Turkish immigrants. Of the participants, 44 were bilingual students with a Turkish–German background and 34 were German monolinguals. The bilingual sample consisted of 20 female and 24 male students with a mean age of 8.5 years. In the group of monolingual students, there were 14 female and 20 male students with a mean age of 8.6 years. The Turkish–German bilingual students and half of the German monolingual students were recruited from the same classes within three schools in Berlin. The other half of the monolingual sample was recruited from two schools in Frankfurt. The sample consisted of three subsamples. Two samples were tested in two consecutive school years (2007 and 2008, with the testing done between January and March in both years) in the same schools (Berlin). The third sample was tested in January 2011 (Frankfurt). Subsample 1 consisted of 43 students (32 bilingual, 11 monolingual); Subsample 2, tested in the consecutive year, consisted of 16 additional students (12 bilingual, four monolingual). Subsample 3 contained only monolinguals (19 students). Comparisons with regard to the control measures used (socioeconomic status, cognitive ability, mathematics achievement; see the instruments listed in the Materials section) revealed no statistically detectable differences between the monolingual and bilingual students of Subsamples 1 and 2 as well as between the monolingual students of Subsamples 1, 2, and 3. Therefore, we will refer to the entire sample in the following sections. As we used the sample of monolingual students as a comparison group when estimating the effects of bilingualisms in immigrant students, it was pivotal to establish the comparability of the monolingual and bilingual samples on the various control measures. In the three schools in Berlin selected for this study, the language of instruction was German; additionally, beginning in first


grade, Turkish language classes were provided for students with Turkish background. Although this was not a bilingual school setting, we assumed that the additional instruction in the Turkish language would contribute to the development of a reasonable level of Turkish proficiency in the participating bilingual students.

Materials The instruments employed can be divided into language measures (language proficiency, reading comprehension), measures of the dependent variable (mathematical word problems), and control measures (cognitive ability, arithmetic skills, socioeconomical status). Language measures. Test of language proficiency. To assess participants’ language proficiency we applied the Bilingual Verbal Ability Test (BVAT; Munõz-Sandoval, Cummins, Alvarado, & Ruef, 1998) which is a standardized test originally developed for measuring bilinguals’ oral language proficiency. Its three subtests are derived from the Woodcock Language Proficiency Battery–Revised (Woodcock, 1991). It is suited for persons from age five to adulthood. The three subtests of picture vocabulary, oral vocabulary–synonyms/antonyms, and verbal analogies were administered to the bilingual group in both the Turkish and the German versions, and to the monolingual group in the German version only. The German version and the Turkish version are comparable with regard to the range of vocabulary tested; yet, specific vocabulary items may differ in difficulty according to their frequency of use. In the first subtest, picture vocabulary, the children had to point at one of four pictures that correctly showed the word the experimenter had presented orally. In the second subtest, oral vocabulary, the children had to name synonyms to (orally) given words or antonyms to (orally) given words. In the subtest of verbal analogies, the children had to finish sentences on the basis of analogies such as “hungry is to eat like tired to . . . (sleep).” As required in the standardized testing procedure, individual testing in each of the subtests continued until a child made four consecutive errors. The German and the Turkish versions of the BVAT used in this study were published translations of the original English version, based on the so-called consensus translation and standardization process (see Munõz-Sandoval et al., 1998). Reliabilities for each subtest are only reported for the English standardized version: rtt ⫽ .85 (picture vocabulary), rtt ⫽ .87 (oral vocabulary), and rtt ⫽ .86 (verbal analogies). Test of reading comprehension. The Ein Leseversta¨ndnistest fu¨r Erst- bis Sechstkla¨ssler (ELFE 1– 6; Lenhard & Schneider, 2006) is a German reading comprehension test for children in Grades 1– 6. It consists of three subtests: (a) word comprehension, (b) sentence comprehension, and (c) text comprehension. For the following analyses, the subscale text comprehension was used as an indicator of students’ reading comprehension in the context of word problems (reported reliability of .86; Lenhard & Schneider, 2006). In this test, students have to read a short paragraph on a topic of everyday life and then have to answer several comprehension questions on the text in multiple-choice format. This test was administered in German only. Measures of the dependent variable: Mathematical word problems. To assess mathematical problem solving abilities, we used two types of mathematical word problems: problems without distractors and problems with distractors.

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Word problems without distractors. A set of nine word problems, consisting of three exchange problems, three comparison problems, two complex comparison problems, and one complex combination problem, was employed. All problems were derived from a set of empirically tested word problems with established solution rates for primary school students of various ages (Stern, 1998). The categorization of word problems (addition and subtraction) with respect to the situational model (exchange, comparison, and combination) was originally provided by Riley, Greeno, and Heller (1983). Solution rates for the different kinds of problems were obtained in several large-scale studies with children from kindergarten to third grade (e.g., Riley & Greeno 1988; Stern, 1998). For the present study, it was thus possible to choose a set of word problems which was well-balanced with respect to the expected solution rates of the sample of third-graders: According to previous studies, the exchange problems ranged in item difficulty from .8 to 1.0; the comparison problems ranged from .75 to 1.0, and the complex combination problem was reported with an item difficulty of .58. All problems had been presented in written format in previous studies. Combination problems require students to find the combined amount of two quantities or the partial amount of a sum. The complex version requires an additional step of calculation as more than two quantities need to be considered: for example, “Paul and Tina have 6 euros altogether. Tina has 4 euros. Kai and Nina have 9 euros altogether. Nina has 3 euros. How many euros do Paul and Kai have altogether?” Exchange problems are characterized by the fact that a certain amount of objects changes hands: for example, “Sabine had 3 euros. Then Philip gave her 5 euros. How many euros does Sabine have now?” There are three versions of exchange problems—final quantity unknown, exchange quantity unknown, and initial quantity unknown—all of which were used in this study. Finally, comparison problems require students to consider the difference between two quantities. These problems are the most difficult ones for students to solve because besides a concept of cardinal numbers, the problems additionally require an understanding of relational numbers: for example, “Beate has 5 stickers. Max has 8 stickers. How many more stickers does Max have than Beate?” There are three kinds of comparison problems— quantity of difference unknown, quantity of comparison unknown, and quantity of reference unknown—all of which were presented in this study. The complex versions of the comparison problems again require an additional step of calculation (see the Appendix for an overview). Word problems with distractors. In addition to the word problems without distractors, we constructed a parallel set of nine word problems that were artificially enriched by numerical information that is not necessary for solving the problem (distractors). As discussed earlier, these problems were specifically designed to allow us to investigate the hypothesized differences in executive control between monolingual and bilingual children; for example, “Maria had 3 marbles. The marbles cost 90 cents. Then Hans gave her 5 marbles. They cost 1.50 euros. How many marbles does Maria have now?” The two sets of nine problems each were then translated into Turkish by a fluent speaker of Turkish and German. The resulting set of nine word problems without distractors and nine word problems with distractors showed a satisfactory internal consistency (e.g., Field, 2009), with Cronbach’s ␣ ⫽ .83 for the German

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versions and Cronbach’s ␣ ⫽ .76 for the Turkish versions. All word problems were administered orally in order to avoid any direct influence of students’ reading comprehension skills (see Helwig et al., 1999). For each word problem, a score of 2 was assigned if a child provided a correct solution after the first presentation of the word problem. If the problem had to be presented a second time (due to nonresponse, incorrect response, or misunderstanding after the first presentation), a score of 1 was assigned for a correct solution. After two incorrect answers or nonresponses, a score of 0 was assigned. Each child could thus attain a maximum of 18 points per set of word problems (with or without distractors). Control measures. Test of cognitive ability. To control for students’ cognitive ability, we administered a subtest of a widely used measure of intelligence (Grundintelligenz Skala 2/Culture Fair Intelligence Test 20 (CFT20); Wei␤, 1998) which is standardized for children from Grades 3–10. The CFT20 is a nonverbal test and is considered to be culture-fair. The four employed subscales of (a) series continuation, (b) classification, (c) matrices, and (d) topologic reasoning were presented in a graphical format. Its reported reliability for children in Grade 3 is .90 (Weiß, 1998). Test of arithmetic skills. We administered the German Mathematics Test for Second Grade (DEMAT 2⫹; Krajewski, Liehm, & Schneider, 2004) as an indicator of students’ ability to perform arithmetic operations. Only subtests that test the basic arithmetic operations of addition and subtraction were used. The children were required to work on eight arithmetic problems that dealt with two-digit numbers up to 99. The reported reliability (internal consistency) for the chosen subtest is .69 (Krajewski et al., 2004). Indicator of socioeconomic-status (SES). Finally, to control for children’s SES, we used the “books at home” index, where children are asked to estimate the number of books their family owns. This measure is considered to be a quite valid estimation of family SES (Lehmann & Nikolova, 2005; Verhoeven & Aarts, 1998). Children in Grade 3 can answer this question quite reliably (Schnepf, 2004). In the item that was presented, the children could choose between five pictures showing book shelves with about 10 books, 25 books, 100 books, 200 books, and more than 200 books.

Results In order to estimate the differences between the groups of monolingual students (N ⫽ 34) and bilingual students (N ⫽ 44) in the language measures and the control measures, we conducted two separate multivariate analyses of variance (MANOVAs), with univariate post hoc tests for the language measures and the control measures. There was no significant difference between the monolingual and the bilingual groups with respect to the control measures of books at home, cognitive ability, and arithmetical skills, ⌳ ⫽ .903, F(3, 74) ⫽ 2.66, p ⬎ .05. As expected, we found a significant difference in the MANOVA on language proficiency (reading comprehension and oral proficiency) between the two groups, ⌳ ⫽ .688, F(2, 75) ⫽ 16.98, p ⬍ .001; ␩2p ⫽ .31. The univariate follow-up ANOVAs revealed significant differences in oral proficiency—BVAT in German, F(1, 76) ⫽ 34.10, p ⬍ .001, ␩2p ⫽ .31; 95% CI [11.01, 22.42]—as well as in reading comprehension—ELFE 1– 6, F(1, 76) ⫽ 8.63, p ⬍ .01, ␩2p ⫽ .10; 95% CI [.85, 4.44]— both in favor of the monolinguals. For the group of bilingual students, the difference between scores of the German and Turkish BVAT versions suggests that these students were more proficient in German than in Turkish. Table 1 provides a summary of the means and standard deviations for the two groups. Tables 2 and 3 show the intercorrelations between the various measures for bilinguals and monolinguals separately.

Research Question 1: Does the Level of Proficiency in the Instructional Language Predict Achievement on Mathematical Word Problems Presented in That Language? In order to estimate the contribution of students’ German language proficiency to their solution rate of ordinary mathematical word problems (i.e., problems without distractors), we conducted a series of multiple regression analyses with the sample of bilingual students Table 1 Means and Standard Deviations of the Groups of Monolingual and Bilingual Students Monolinguals Variable

Procedure All tests were administered in two individual testing sessions of approximately 55 min each for bilingual children and two testing sessions of approximately 40 min each for monolingual children. Between Session 1 and Session 2, there was at least 1 week. The sessions took place in a separate testing room within the school setting. In Session 1, the tests were administered in the following order: collection of background data (name, age, gender, SES), word problems in German, language proficiency in German, and reading comprehension in German. In Session 2, we administered a test of cognitive ability, arithmetic skills, and, for the bilingual students, word problems in Turkish and language proficiency in Turkish. The sequence of Turkish and German sessions was alternated randomly for bilingual students.

Socioeconomic status (no. of books at home) Cognitive ability (CFT20) Arithmetic ability (DEMAT 2⫹) Reading comprehension (ELFE 1–6) German proficiency (BVAT German) Turkish proficiency (BVAT Turkish) Word problems without distractors (German) Word problems with distractors (German) Word problems without distractors (Turkish)

Bilinguals

M

SD

M

SD

3.03 25.00 4.82 9.85 71.76 —

1.00 5.03 2.51 4.03 13.16 —

2.57 23.68 3.80 7.20 55.05 40.91

0.97 6.02 2.00 3.87 12.04 9.92

11.82

2.51

9.34

3.68

7.24

2.34

6.48

3.49

—

—

9.39

3.27

Note. CFT20 ⫽ Grundintelligenz Skala 2/Culture Fair Intelligence Test 20; DEMAT 2⫹ ⫽ German Test of Mathematics for Grade 2; ELFE 1– 6 ⫽ Ein Leseversta¨ndnistest fu¨r Erst- bis Sechstkla¨ssler [A reading comprehension test for first to sixth graders]; BVAT ⫽ Bilingual Verbal Ability Test.


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Table 2 Intercorrelations Among Measures for the Group of Monolinguals Variable 1. 2. 3. 4. 5. 6. 7. ⴱ

Books at home Cognitive ability Arithmetic ability Oral proficiency (German) Reading comprehension (German) Word problems without distractors Word problems with distractors

p ⬍ .05.

ⴱⴱ

1

2

3

4

5

6

7

— .289 ⫺.118 .286 ⫺.036 .050 .113

— .297 .307 .326 .481ⴱⴱ .179

— .346ⴱ .561ⴱⴱ .608ⴱⴱ .310

— .382ⴱ .258 .179

— .483ⴱⴱ .218

— .376ⴱ

—

p ⬍ .01.

only. In the main analyses, we employed only problems without distractors as the dependent variable because these are the ones that have been empirically validated in other studies. However, we expected language proficiency to be the crucial factor for both types of problems. In Model 1, we included the control variables of SES, cognitive ability, and arithmetic skills as they have been established to be influences on mathematical reasoning processes in previous studies (e.g., OECD, 2006; Stern, 1998). In Model 2, we additionally included the indicator of students’ reading comprehension scores in German because reading comprehension has typically been used as an estimate for language proficiency (e.g., OECD, 2006). In Model 3, we additionally included students’ scores of German language proficiency in order to estimate the additional contribution of students’ German language proficiency. We used “forced entry” as the method of entering predictors into the regression models. Table 4 lists the results of the regression analyses for the three models. Altogether, there was a marginally significant change of R2 in Model 2 compared with in Model 1 (p ⫽ .058), and a significant change of R2 in Model 3 compared with in Model 2 (p ⫽ .002). Thus, language proficiency explained a significantly larger amount of variance in the solution rates of word problems than did reading comprehension alone. In Model 3, arithmetic skills and German language proficiency were the strongest predictors for students’ achievement in mathematical word problems, with ␤ ⫽ .384 and ␤ ⫽ .430, respectively. All other predictors, including reading comprehension, remained insignificant in Model 3. The results of this regression analysis thus indicate that the level of proficiency in the instructional language, together with students’ arithmetic skills, is strongly predictive of students’ ability to solve mathematical word problems. Additional analyses in which

problems with distractors were used as the dependent variable revealed the same pattern as the analyses of problems without distractors: In Model 3, arithmetic ability and German language proficiency remained the strongest predictors (␤ ⫽ .378 and ␤ ⫽ .403, respectively; both p ⬍ .01; total R2 ⫽ .437). In a second step, we examined whether the effect of the instructional language also predicted problem solving in students’ native language (Turkish). To do this, we tested in a series of regression analyses (method: forced entry) whether there was an effect of the instructional language of German beyond the effect of bilingual students’ Turkish language proficiency on their solving of word problems presented in Turkish. In addition to Model 1 (which was the same as in the first regression analysis), we included language proficiency in the language of testing (Turkish) in Model 2 and, as a further predictor, language proficiency in German in Model 3. Language proficiency in German was entered last in order to allow us to investigate the additional impact of instructional language proficiency. Table 5 lists the results of Models 1–3. There was a significant change in R2 from Model 1 to Model 2 (p ⫽ .004), indicating the impact of the language of testing. The inclusion of German language proficiency in Model 3, however, did not significantly change R2, although it represents the second best predictor after Turkish language proficiency in Model 3 (Turkish: ␤ ⫽ .381; German: ␤ ⫽ .214). Altogether, the total R2 of Model 3 was moderate, with a total amount of 32% of variance explained. In addition, in comparison to the regression analyses with German word problems, the contribution of arithmetic skills seemed to be diminished, suggesting that factors not represented in the model contributed to problem solving in Turkish to a greater degree than

Table 3 Intercorrelations Among Measures for the Group of Bilinguals Variable 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. ⴱ

Books at home Cognitive ability Arithmetic ability Oral proficiency (German) Reading comprehension (German) Oral proficiency (Turkish) Word problems without distractors (German) Word problems with distractors (German) Word problems without distractors (Turkish) Word problems with distractors (Turkish)

p ⬍ .05.

ⴱⴱ

p ⬍ .01.

1

2

3

4

5

6

7

8

9

10

— ⫺.080 .133 .242 ⫺.007 .095 .178 .048 ⫺.034 .070

— .477ⴱⴱ .122 .183 ⫺.030 .241 .012 .192 .215

— .341ⴱ .472ⴱⴱ .155 .579ⴱⴱ .445ⴱⴱ .315ⴱ .316ⴱ

— .449ⴱⴱ .281 .613ⴱⴱ .535ⴱⴱ .354ⴱ .290

— .321ⴱ .479ⴱⴱ .451ⴱⴱ .364ⴱ .169

— .225 .147 .450ⴱⴱ .427ⴱⴱ

— .760ⴱⴱ .348ⴱ .282

— .468ⴱ .339ⴱ

— .779ⴱⴱ

—


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Table 4 Regression Analyses (Models 1–3) With Bilingual Sample on Predictors of Word Problems Without Distractors (German) Variable Model 1 SES Cognitive ability Arithmetic skills Model 2 SES Cognitive ability Arithmetic skills Reading comprehension Model 3 SES Cognitive ability Arithmetic skills Reading comprehension Oral proficiency (German)

␤

R2

R2 change

p

.346

.346

.001

.099 ⫺.027 .579ⴱⴱⴱ .404

.058

.134

VIF

.956 .752 .743

1.046 1.330 1.345

.948 .748 .791 .769

1.055 1.336 1.691 1.301

.887 .748 .584 .662 .727

1.127 1.336 1.711 1.511 1.376

.058

.121 ⫺.008 .437ⴱⴱ .275 .539

Tolerance

.002

.023 ⫺.013 .385ⴱ .107 .430ⴱⴱ

Note. SES ⫽ socioeconomic status; VIF ⫽ variance inflation factor. ⴱ p ⬍ .05. ⴱⴱ p ⬍ .01. ⴱⴱⴱ p ⬍ .001.

in German. Similar to the first regression, an additional analysis with the problems with distractors as the dependent variable resulted in a comparable pattern, presenting Turkish proficiency as the strongest predictor (␤ ⫽ .380, p ⬍ .05, total R2 ⫽ .272) while the contribution of German language proficiency decreased further (␤ ⫽ .116, ns). Thus, the findings of Model 3 suggest that the importance of the instructional language cannot be simply generalized to problem solving in another language (i.e., the students’ native language). Rather, it was proficiency in the language of testing that exerted the strongest influence on students’ achievement in both German and Turkish.

Research Question 2: Are There Any Cognitive Benefits of Bilingual Students as Compared With Monolinguals? To investigate whether benefits of bilingualism with regard to mathematics problem solving could be found in our sample, we

compared the groups of bilingual and monolingual students on word problems with and without distractors by performing a repeated measures ANOVA with Type of Word Problem as within-subjects factor (problems without distractors vs. problems with distractors) and Language Group as between-subjects factor (bilinguals vs. monolinguals). As covariates we entered SES, cognitive ability, and arithmetic skills. As expected, the results showed a significant interaction of Type of Word Problem ⫻ Language Group, ⌳ ⫽ .939, F(1, 73) ⫽ 4.76, p ⬍ .05, ␩2p ⫽ .06, while the main effects of Type of Problem, ⌳ ⫽ .999, F(1, 73) ⫽ 0.06, p ⬎ .05, and Language Group, F(1, 73) ⫽ 1.79, p ⬎ .05, were not significant. Considering the influence of the control variables, there was an interaction of Type of Problem ⫻ Cognitive Ability, ⌳ ⫽ .944, F(1, 73) ⫽ 4.32, p ⬍ .05, ␩2p ⫽ .05, which can be traced back to the close relation between solutions of problems without distractors and cognitive ability (see Tables 2 and 3). Looking at the mean scores for both groups on word problems with and without distractors (see Table 1 and Figure 1),

Table 5 Regression Analyses (Models 1–3) With Bilingual Sample on Predictors of Word Problems Without Distractors (Turkish) Variable Model 1 SES Cognitive ability Arithmetic skills Model 2 SES Cognitive ability Arithmetic skills Oral proficiency (Turkish) Model 3 SES Cognitive ability Arithmetic skills Oral proficiency (Turkish) Oral proficiency (German)

␤ ⫺.071 .041 .305† ⫺.096 .094 .217 .429ⴱⴱ ⫺.135 .091 .157 .381ⴱⴱ .214

R2

R2 change

p

.106

.106

.207

.284

.319

Note. SES ⫽ socioeconomic status; VIF ⫽ variance inflation factor. p ⬍ .10. ⴱ p ⬍ .05. ⴱⴱ p ⬍ .01.

†

.176

.037

Tolerance

VIF

.956 .752 .743

1.046 1.330 1.345

.953 .743 .719 .959

1.050 1.346 1.390 1.043

.915 .743 .673 .905 .797

1.092 1.346 1.486 1.104 1.254

.004

.161

COGNITIVE BENFITS AND COSTS OF BILINGUALISM Monolingulas Bilinguals 16

Scores on Word Problems

14 12 10 8 6 4 2 0 Without Distractors

With Distractors

Figure 1. Means and standard deviations on German word problems with distractors and without distractors for monolinguals and bilinguals. Error bars indicate standard deviation.

the interaction Type of Problem ⫻ Language Group can be interpreted in terms of a relative benefit for the bilingual students with respect to word problems with distractors. In other words, the substantial disadvantage of bilinguals for problems without distractors relative to monolinguals diminished for problems with distractors. Two subsequent post hoc ANOVAs (same covariates as in the former analyses) confirmed that the monolingual students outperformed the bilingual students on word problems without distractors, F(1, 73) ⫽ 5.07, p ⬍ .05, ␩2p ⫽ .06, 95% CI [0.16, 2.72], while no significant group differences were obtained for word problems with distractors, F(1, 73) ⫽ 0.04, p ⬎ .05, ␩2p ⫽ .00, 95% CI [⫺1.22, 1.52]. The stronger language proficiency in the language of testing apparently contributed to the monolinguals’ advantage on problems without distractors. Yet for problems with distractors, the bilingual students seem to partly compensate for their language deficits by displaying an enhanced ability in attentional control. The bivariate correlations of German language proficiency with performance on problems with distractors varied marginally between the groups of monolingual and bilingual students: rmonolingual ⫽ .179; rbilingual ⫽ .535, p ⬍ .10. This finding points to the importance of language proficiency mediating the effect of achievement on problems with distractors, suggesting that the cognitive benefit of bilingualism is greater in bilinguals with high language proficiency than in those with low proficiency.

Exploratory Analysis: Are There Cognitive Costs When the Language of Instruction Differs from the Language of Testing? Finally, we conducted an exploratory data analysis to test whether bilingual students face specific cognitive costs (switching costs) when the language of retrieval differs from the language of instruction. This analysis picks up on Research Question 1 (second regression analysis) regarding the influence of proficiency in the instructional language for mathematical achievement in students’

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native language. In the regression analysis, the addition of the variable of instructional language in Model 3 did not contribute to a significant change of R2 in Model 2. This seems to suggest that there are no switching costs in mathematical word problem solving in bilingual students of this age since achievement in word problems is apparently influenced solely by language proficiency in the language of testing. However, in previous research on this issue (e.g. Marian & Fausey, 2006; Saalbach et al., 2010; Spelke & Tsivkin, 2001), highly proficient bilinguals have been used to reveal switching costs. Our sample of bilingual students, in contrast, comprised students of varying proficiencies in native language and instructional language. Thus, having a group of rather proficient bilinguals performing on a level comparable to that of monolinguals may be needed to test the existence of switching costs. In contrast, students with a weak command in the instructional language may actually use their native language in order to compensate for deficits in the instructional language when solving word problems in the instructional language. In order to test these hypotheses, we divided our bilingual sample into three equal parts (N ⫽ 14) by rank ordering their scores on German BVAT. The upper third was labeled dominant bilinguals, and the lowest third was labeled weak bilinguals. Leaving the middle third aside, we now had two groups that we analyzed in detail. While the two groups varied in their Turkish proficiency, we did not use this variable as a criterion for group assignment as the distribution was restricted in variance in the entire sample (see Table 6 for descriptives of the two resulting groups). A multivariate ANOVA revealed no significant differences between the groups regarding the measures of SES, cognitive ability, and arithmetic skills. A repeated measures ANOVA with Language of Testing (problems without distractors in German vs. problems without distractors in Turkish) as a within-subjects factor, Bilingual Proficiency (dominant vs. weak) as a between-subjects factor, and Turkish proficiency as a covariate revealed a multivariate main effect for Language of Testing, ⌳ ⫽ .779, F(1, 25) ⫽ 7.09, p ⬍ .05, ␩2p ⫽ .22, a significant main effect for Bilingual Proficiency F(1, 25) ⫽ 21.47, p ⬍ .001, ␩2p ⫽ .46, and a significant interaction of Language of Testing ⫻ Bilingual Proficiency, ⌳ ⫽ .577, F(1, 25) ⫽ 18.35, p ⬍ .001, ␩2p ⫽ .42. In addition, there was a significant

Table 6 Means and Standard Deviations of Bilingual Subgroups Dominant (n ⫽ 14)

Weak (n ⫽ 14)

Variable

M

M

German language proficiency Turkish language proficiency SES Cognitive ability Arithmetic skills Word problems without distractors (German) Word problems without distractors (Turkish) Word problems with distractors (German) Word problems with distractors (Turkish)

68.07 43.00 2.86 23.79 4.71 12.43 10.36 8.64 6.71

SD

SD

5.82 40.03 5.45 9.80 36.00 8.05 1.02 2.29 1.07 6.81 22.64 6.91 1.89 3.14 1.87 2.31 6.57 1.78 2.80 7.93 2.76 3.45 4.00 2.11 3.51 4.86 3.18

Note. Dominant bilinguals ⫽ bilingual speakers who have high proficiency in one language only; weak bilinguals ⫽ bilingual speakers who have deficits in both languages; SES ⫽ socioeconomic status.


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interaction between Language of Testing and the covariate of Turkish proficiency, ⌳ ⫽ .792, F(1, 25) ⫽ 6.56, p ⬍ .05, ␩2p ⫽ .20. Univariate follow-up tests revealed that the dominant bilinguals outperformed the weak bilinguals on German word problems, F(1, 25) ⫽ 49.24, p ⬍ .001, ␩2p ⫽ .66, 95% CI [⫺7.76, ⫺4.24], which is consistent with the regression analyses underlining the importance of language proficiency for problem solving in the respective language. Moreover, there were no significant differences for the two subgroups on Turkish word problems, F(1, 25) ⫽ 15.04, p ⬎ .05, 95% CI [⫺3.78, 0.61], which again suggests that the role of the instructional language does not generally extend to testing in the native language. More important for the explorative question, however, is the interaction of Language of Testing ⫻ Bilingual Proficiency, which is depicted in Figure 2. Post hoc tests for the differences between achievement scores in Turkish and German for each group revealed that the group of dominant bilinguals scored significantly higher when tested in German than when tested in Turkish, t(13) ⫽ 3.31, p ⬍ .01, d ⫽ 0.80, 95% CI [0.72, 3.42], while there were no statistically significant differences for the group of weak bilinguals, d ⫽ 0.58; 95% CI [⫺3.19, 0.47]. However, the latter group showed a reversed trend of switching costs, with higher means when tested in Turkish than when tested in German. For problems with distractors, we found the same pattern of results as for problems without distractors: Language of Testing, ⌳ ⫽ .781, F(1, 25) ⫽ 7.00, p ⬍ .05, ␩2p ⫽ .22; Language of Testing ⫻ Bilingual Proficiency, ⌳ ⫽ .749, F(1, 25) ⫽ 8.39, p ⬍ .01, ␩2p ⫽ .25; Bilingual Proficiency, F(1, 25) ⫽ 8.52, p ⬍ .01, ␩2p ⫽ .25. Again, dominant bilinguals scored higher when tested in German than when tested in Turkish, while the reverse trend was observable with weak bilinguals (see Table 6 for means and standard deviations). German Word problems Turkish Word problems

16

Scores on Word Problems

14 12 10 8 6 4 2 0 Dominant Bilinguals

Weak Bilinguals

Figure 2. Interaction of Language of Testing ⫻ Bilingual Proficiency for word problems without distractors. Dominant bilinguals ⫽ bilingual speakers who have high proficiency in only one language; weak bilinguals ⫽ bilingual speakers who have deficits in both languages. Error bars indicate standard deviation.

This pattern of results is partly consistent with the findings of previous research (e.g. Marian & Fausey, 2006; Saalbach et al., 2010; Spelke & Tsivkin, 2001). Only students from the dominant bilingual group performed better when the instructional language and the testing language matched. It is interesting that students in the low-proficiency group tended to show higher scores when the problems were presented in Turkish rather than German. In other words, our results suggest that the effect of instructional language on bilinguals’ performance in mathematical problem solving is mediated by the proficiency in that language: Bilinguals with a high command of German did significantly better on problems presented in German than on problems presented in Turkish, while students with a weak command of German tended to be better in Turkish. The latter result might reflect bilingual students’ tendency to use their native language in order to compensate for deficits in the instructional language when solving word problems.

Discussion In the present research, we examined the relation between bilingualism and school-related learning. In particular, we aimed to disentangle the possible costs and benefits of bilingualism with respect to learning mathematics for students with an immigrant background. To this end, we compared their achievement on mathematical word problems with that of their monolingual peers. First of all, our study replicated previous findings on the important role of language proficiency for children’s learning. In contrast to previous studies, however, where language assessment was mostly realized by self-reports or via the assessment of reading comprehension, we used a measure that directly tapped (oral) language proficiency. Regression analyses with the sample of bilingual students revealed a close relation between proficiency in German and performance on mathematical word problems in that language. In fact, language proficiency explained as much variance as arithmetic skills did and far more variance than was explained by cognitive ability, SES, and reading comprehension. The contribution of reading comprehension was rendered insignificant when our measure of language proficiency was entered in Model 3 of the regression analysis. In sum, students who were rather proficient in German were much more likely to solve mathematical word problems in German correctly than students with poor German language skills. A similar pattern appeared for Turkish word problems as a criterion. Again, proficiency in this language (students’ native language) was the crucial predictor. An additional effect of the instructional language (German) was not found for mathematics problem solving in students’ native language. Accordingly, we cannot presume a generalized effect of learning processes in German to solution processes in Turkish. Rather, as we will describe, a differentiated picture of cognitive costs when switching between the language of instruction and the language of retrieval emerged in our analyses. Second, our data indirectly support the assumption that bilingualism has beneficial effects even for students who are not highly proficient in both languages and who are from a lower socioeconomic background (as indicated by the district where their schools were located and our measure of SES). Although bilinguals showed substantially lower German language proficiencies than monolinguals, they performed as well as their monolingual peers on problems with distractors. In contrast, as expected from the


regression analyses, bilinguals’ achievement on ordinary word problems was significantly lower than that of monolinguals. This suggests that the costs associated with word problems that required executive control skills (problems with distractors) were less prominent for bilinguals than for monolinguals.

The Role of Language Proficiency in Learning Mathematics Our data confirm the important role of language proficiency in school-related learning established by prior research in general and in solving mathematical word problems, in particular. As described in the introduction, the influence of language proficiency on problem solving in mathematics may be threefold: languagebased knowledge transfer for mathematical procedures acquired in the classroom, knowledge of relational terms and logical operators during the modeling process, and linguistic comprehension of the information in the tasks on a surface level. In the present study, the role of language cannot be traced back to comprehension difficulties on a surface level as we used only very frequently used words and simple grammatical structure. Rather, our findings suggest that the difficulties shown by children with poor skills in the instructional language may be due to their difficulties in understanding the content of prior classroom instruction and in constructing a mathematical model of the problem at hand. Previous research has usually focused on bilingual students’ difficulties following classroom instructions (e.g., Elbers & de Haan, 2005). Our data confirm the importance of the mastery of instructional language for mathematical problem solving. However, our results also suggest that this importance may be, at least partly, attributed to the role of language during the modeling of the problem at hand reflected by a significant influence of the language of testing. In other words, language proficiency affects mathematical achievement on both the level of instruction and the level of testing. In the future, researchers should therefore be aware that testing low-proficiency bilingual students in their instructional language may lead to underestimation of their competencies and thus testing in their native language may be more appropriate. This is also supported by our finding that students with a weak command of the instructional language have the tendency to show better performance when tested in their native language. Overall, our findings make a strong case for improving bilingual children’s language skills in the language of instruction. Thus, efforts to make language assessment obligatory before schooling seem to be a step in the right direction, provided these assessments are followed by appropriate training programs (see Fried, 2008).

Cognitive Costs of Bilinguals An exploratory analysis revealed that bilinguals with a high proficiency in one of the languages (i.e. dominant bilinguals) scored higher when the language of instruction and the language of testing were the same (i.e., German) than when they differed (i.e., Turkish word problems), which partly confirms previous findings on the cognitive costs that accompany the switching between languages of instruction and language of retrieval (e.g., Spelke & Tsivkin, 2001). However, bilingual students with a weak command of German (and Turkish) tended to show better performance when tested in Turkish than when tested in German. Our results thus

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indicate that, given a low proficiency in German, these students may find that the Turkish language functions as a resource in the mathematical modeling process, where the students’ native language may support more adequate mathematical modeling. We should mention that in the group of weak bilinguals, students’ language proficiency in both the native and the instructional language may be considered deficient. Since these students nevertheless performed better when tested in Turkish, we can presume that further factors such as motivation or the frequency of academic language use in Turkish may have influenced their performance on Turkish word problems. This interpretation would be in line with results of our regression analyses on Turkish word problem in Research Question 1. Overall, the effects of language switching in general and the role of the native language for students of low language proficiency in particular should be investigated in more detail.

Cognitive Benefits of Bilinguals Bilingualism in immigrant children does not come with costs only. In our study, we indirectly replicated findings that have shown the cognitive benefits of bilingualism in terms of bilinguals’ executive control skills (see Adesope et al., 2010; Bialystok, 2009; Bialystok et al., 2009), even for a sample of rather weak bilinguals. As analyses of students’ socioeconomic background, their cognitive ability, and their arithmetic skills showed, the group of bilinguals was comparable to the group of monolingual children on these measures while their language proficiency was far below (⬎1 SD). Our results showed that immigrant children with a substantially lower proficiency in German than that shown by their monolingual peers performed equally well as their monolingual peers on tasks designed to test executive control while attaining significantly lower scores than their monolingual peers on ordinary word problems. These findings are quite in line with the conjectures put forward by Bialystok (1988; 1999). Thus, even nonbalanced forms of bilingualism have the potential to promote cognitive benefits. As has been assumed in previous research, it is not the level of language proficiency as such but most likely the routine of keeping apart two language systems by frequent usage and exposure that promote improved executive control skills. This is supported by a recent finding of cognitive gains in 7-month-old infants who were exposed to a bilingual home environment (Kovać & Mehler, 2009). It would thus be extremely interesting to study whether an index of frequency of language use would be a better predictor for cognitive benefits of bilingualism in children than a proficiency measure. Unfortunately, this type of data was not available in the present study. Furthermore, the strong relation between bilinguals’ language proficiency and performance on problems with distractors suggests that cognitive benefits of bilingualism are greater in bilinguals with high language proficiency than in those with low proficiency. It is a question for further investigations whether and to what extent proficient bilinguals from this student population, comparable to the balanced bilinguals commonly investigated in studies on cognitive benefits, can even outperform their monolingual peers on complex mathematical problem with distracting information. Taken together, the results of this study are promising. Our findings emphasize the importance of language proficiency for academic success. We are aware that due to the small sample size


558

of the present study, broad generalizations need to be drawn with caution. However, since the results are generally in line with previous findings, we are certain that our findings are noteworthy with respect to the understanding of the relation among bilingualism, cognitive development, and academic performance. Further, they may offer some practical implications. For example, our results can be seen as a support for many Western European countries’ current efforts to promote language training programs for children with an immigrant background prior to school entry or in their first years at school. Furthermore, our study confirmed the intricate relationship between language and mathematics. Those who design programs to promote language skills in students with an immigrant background should therefore consider including explicit training in solving mathematical word problems, that is, in extracting a mathematical model from a situational context (Kroesbergen, Leseman, Scheele, & Mayo, 2009). Finally, our findings suggest that bilingualism in children with an immigrant background should be considered a resource rather than a burden. The cognitive advantage of speaking two languages may compensate for some of the disadvantages that occur with lower skills in the instructional language as well as with low SES. Therefore, promoting native language skills in addition to instructional language proficiency is a reasonable step to take not only because bilingualism can play important role in the development of a binational identity (see also Cummins, 2004) but also because it offers the advantage of cognitive benefits.

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Appendix Mathematical Word Problems With and Without Distractors (English Translations) Type of word problem

Without distractors

With distractors

Exchange problems Final quantity unknown

Sabine had 3 euros. Then Philipp gave her 5 euros. How many euros does Sabine have now?

Exchange quantity unknown

Julia had 2 crayons. Then Tobi gave her some crayons. Now Julia has 9 crayons. How many crayons did Tobi give her? At first Maria had some marbles. Then she gave 2 marbles to Hans. Now Maria has 6 marbles. How many marbles did Maria have at first?

Initial quantity unknown

Maria had 3 marbles. Together these marbles cost 90 cents. Then Hans gave here 5 marbles. These cost 1.50 euros. How many marbles does Maria have now? Lisa had 2 euros. Then Kai gave her some euros. Lisa wants to buy herself 8 crayons. Now Lisa has 9 euros. The crayons cost 10 euros altogether. How many euros did Kai give her? Tina had some pieces of chewing gum. Then she gave 2 pieces of chewing gum to Karl. One pack of chewing gums costs 80 cents. Now Tina has 6 pieces of chewing gum. She wants to buy herself 4 new packs. How many pieces of chewing gum did she have at first?

Comparison problems Quantity of difference unknown

Beate has 5 stickers. Max has 8 stickers. How many stickers does Max have more than Beate?

Quantity of comparison unknown

Maria has 3 crayons. Hans has 4 crayons more than Maria. How many crayons does Hans have? Lisa has 9 balloons. She has 4 balloons more than Lennart. How many balloons does Lennart have?

Quantity of reference unknown

Meike has 5 marbles. She is 11 years old. Tom has 8 marbles. He is 4 years older than Meike. How many marbles does Tom have more than Meike? Julia is 5 years old. She weights 25 kilo. Tobi is 4 years older than Julia. He weights 45 kilo. How old is Tobi? Maria has 9 marbles. She gets 12 euros pocket money. Maria had 4 marbles more than Hans. Hans gets 10 euros pocket money. How many marbles does Hans have?

Complex comparison problem Meike has 2 euros. Tim has 5 euros more than Meike. How many euros do Meike and Tim have together? Three-person comparison

Claudia has 5 playing balls. She has 5 playing balls less than Thomas. Thomas has 2 playing balls more than Oliver. How many playing balls does Oliver have?

(Appendix continues)

Maria has 2 playing balls. They cost 10 euros altogether. Hans has 5 playing balls more than Maria. They cost 15 euros altogether. How many playing balls do Hans and Maria have together? Lena has 3 euros. She is 8 years old. Karl is 2 years older than Lena. Lena has 4 euros less than Karl. Karl has 3 euros more than Timo. How many euros does Timo have?


561

Appendix (continued) Type of word problem

Without distractors

With distractors

Complex combination problem Total quantity unknown and subset unknown

Paul and Tina have 6 euros together. Tina has 4 euros. Kai and Nina have 9 euros altogether. Nina has 3 euros. How many euros do Paul and Kai have together?

Kai and Lena have 6 drops together. Lena has 4 drops. Both children are 12 years old. Karl and Sabine have 9 drops together. Sabine has 3 drops. Karl and Sabine are 2 years younger than the others. How many drops have Kai and Karl together?

Received April 8, 2010 Revision received March 8, 2011 Accepted March 18, 2011 䡲

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