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The relationship between linguistic skills and arithmetic knowledge
Learning and Individual Differences 23 (2013) 87–91
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Learning and Individual Differences journal homepage: www.elsevier.com/locate/lindif
The relationship between linguistic skills and arithmetic knowledge Rose K. Vukovic a,⁎, Nonie K. Lesaux b a b
Department of Teaching & Learning, New York University, 239 Greene St 2nd Floor, New York, NY 10003, United States Harvard Graduate School of Education, 14 Appian Way, Cambridge, MA 02138, United States
a r t i c l e
i n f o
Article history: Received 5 May 2011 Received in revised form 13 July 2012 Accepted 10 October 2012 Keywords: Verbal ability Phonological decoding Arithmetic Language
a b s t r a c t Although language is implicated in children's mathematical development, few studies have focused specifically on how different linguistic skills relate to children's mathematical performance. Building on the model proposed by LeFevre et al. (2010), this study examined how general verbal ability and phonological skills were differentially related to children's arithmetic knowledge. Third grade children (N=287) were assessed on verbal analogies, phonological decoding, symbolic number skill, procedural arithmetic, and arithmetic word problems. Using mediation analyses, the results indicated that verbal analogies were indirectly related to arithmetic knowledge through symbolic number skill, whereas phonological decoding had a direct relationship with arithmetic performance. These results suggest that general verbal ability influences how children understand and reason with numbers, whereas phonological skills are involved in executing conventional arithmetic problems. © 2012 Elsevier Inc. All rights reserved.
1. Introduction Much like reading, learning and doing mathematics are steeped in oral and written language (Adams, 2003; Adams & Lowery, 2007; Schleppegrell, 2007). For instance, mathematics instruction in classrooms depends primarily on oral explanations and interactions, and the delivery of mathematics curriculum often occurs via written text (Bielenberg & Wong Fillmore, 2004/2005; Schleppegrell, 2007). Furthermore, mathematics consists of a specialized vocabulary: words such as volume, ruler, plot, and product, have different meanings in mathematics than when used in everyday language (Adams, 2003). In turn, to be mathematically proficient, children must develop a language that allows them to participate not only during mathematics instruction, but also to engage quantitatively with the world outside the classroom. Other than the obvious example of word problems, however, the significant language demands of mathematics have until recently been overlooked by researchers and practitioners. A growing body of studies indicate that the same underlying processes that are important for reading, particularly phonological processing, are important for mathematics (e.g., Hecht, Torgesen, Wagner, & Rashotte, 2001; Jordan, Kaplan, & Hanich, 2002; Simmons & Singleton, 2008). Indeed, it has even been suggested that children's mathematical difficulties might actually reflect deficient linguistic processes as opposed to deficits in quantitative processes (LeFevre et al., 2010; Vukovic, 2012). This supposition is in part supported by neuropsychological evidence showing that in addition to a specialized quantitative circuit ⁎ Corresponding author. Tel.: +1 212 998 5205; fax: +1 212 995 3636. E-mail addresses: [email protected] (R.K. Vukovic), [email protected] (N.K. Lesaux). 1041-6080/$ – see front matter © 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.lindif.2012.10.007
in the parietal lobe, a linguistic circuit in the left angular gyrus supports the manipulation of numbers in verbal form (Dehaene, Piazza, Pinel, & Cohen, 2003), suggesting that there is a linguistic basis for some aspects of mathematics. Furthermore, children tend to have more difficulty with language-based mathematical tasks (e.g., number facts, word problems) than with tasks that have fewer language demands (e.g., nonverbal calculation, number sense) (Jordan & Levine, 2009; Jordan, Mulhern, & Wylie, 2009; Locuniak & Jordan, 2008). Taken together, these findings suggest an inextricable link between linguistic skills and mathematical performance. The nature of this relationship, however, remains underspecified. More specifically, although linguistic skills are implicated in mathematical cognition, few studies have systematically examined how different linguistic skills are related to children's mathematical performance. A deeper understanding of the relationship between linguistic skills and children's mathematical performance is necessary to provide theoretical insight and practical guidance into how to best support children's mathematical development. To begin to increase the specificity of our understanding of the role of linguistic skills in mathematical cognition, this study examined how general verbal ability and phonological skills were differentially related to children's arithmetic knowledge. We focused specifically on arithmetic because of its foundational role for the more complex mathematics children encounter in middle and high school. 1.1. Linguistic skills and arithmetic knowledge The bulk of studies investigating the relationship between linguistic skills and children's arithmetic knowledge have focused on phonological processes, because completing simple arithmetic problems requires the retrieval of phonological codes, as well as encoding and
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maintaining phonological representations in immediate awareness (Geary, 1993; Simmons & Singleton, 2008). Weaknesses in phonological processing are therefore thought to hinder the development of tasks that rely on the manipulation and storage of verbal codes, such as counting and solving simple arithmetic problems. Studies show that phonological processes are indeed important for children's arithmetic development (e.g., Fuchs et al., 2005; Hecht et al., 2001; Simmons & Singleton, 2008). This phonological representation hypothesis also helps explain the finding that many children with reading difficulties also have difficulty with arithmetic (e.g., Dirks, Spyer, van Lieshout, & de Sonneville, 2008; Rubinsten, 2009; Simmons & Singleton, 2008). Although an important starting point, the phonological representation hypothesis does not fully account for the link between linguistic skills and arithmetic performance. For instance, there exist children with arithmetic difficulties who are nonetheless good word readers (and therefore tend to have age-appropriate phonological skills) and vice versa (e.g., Fuchs, Fuchs, & Prentice, 2004; Landerl, Fussenegger, Moll, & Willburger, 2009), suggesting that phonological skills are not the sole influential factor. Moreover, Jordan and colleagues have found that good readers with arithmetic difficulties use their strengths in verbal reasoning to complete arithmetic problems, suggesting that linguistic skills beyond phonological processing are involved in arithmetic performance (e.g., Jordan & Hanich, 2003). Indeed, there is evidence that more than just an issue of vocabulary, the language used in arithmetic word problems influences how children represent and solve these problems (e.g., Abedi & Lord, 2001; Brissiaud & Sander, 2010). Together, these findings suggest that general verbal ability plays a different role in children's arithmetic performance than phonological skills. Few studies, however, have specifically contrasted the unique effects of general verbal ability and phonological skills on children's arithmetic performance. In a relevant study, LeFevre et al. (2010) proposed a theoretical model whereby linguistic skills influence mathematical cognition indirectly through symbolic number skill— the number and quantitative skills that depend on the formal number system but do not require knowledge of formal mathematics (see also Jordan, Glutting, & Ramineni, 2010). Examples of symbolic number skill include number identification and numerical reasoning. In their analyses, LeFevre et al. (2010) examined how a linguistic composite that tapped vocabulary, phonological skills, and symbolic number skill (i.e., number identification) directly predicted mathematical outcomes. The authors found that the linguistic composite measured in children at 4.5-years old explained unique variance in procedural arithmetic, numeration, geometry, and measurement at 7.5-years old. That the linguistic composite included general verbal ability, phonological skills, and symbolic number skills, however, makes it difficult to disentangle how different linguistic skills relate to symbolic number skill, as well as the direct and indirect effects of general verbal ability and phonological skills on mathematical cognition. As such, the model proposed by LeFevre et al. provides the basis for investigating the unique role of general verbal ability and phonological skills in children's arithmetic knowledge. 1.2. Present study Building on previous research, we tested the mediation model proposed by LeFevre et al. (2010; see Fig. 1), which makes three predictions. First, children's linguistic skills should be related to children's symbolic number skill (path a). Second, symbolic number skill should be related to arithmetic knowledge (path b). Finally, linguistic skills should have an indirect effect on arithmetic knowledge through symbolic number skill (path ab). We expected this model to hold for general verbal ability, consistent with the predictions of LeFevre et al. By contrast, we hypothesized that phonological skills would maintain a direct effect on arithmetic knowledge (path c 1), given that phonological processes are involved in the storage and
Symbolic Number Skill path b
path a path c¹ Linguistic Skills
Arithmetic Knowledge
Fig. 1. Predicted relationships among linguistic skills, symbolic number skill, and arithmetic knowledge.
retrieval of numbers from long-term memory (e.g., Geary, 1993; Simmons & Singleton, 2008). 2. Method 2.1. Participants The participants were 287 (134 girls) third graders (mean age = 8.60 years, SD = 4 months). The children attended five elementary schools located primarily in working class neighborhoods in an urban Canadian city. The mathematics curriculum was the same across the five schools and emphasized a balanced approach between conceptual understanding and procedural skills. The children were 53.0% majority culture (n = 152), 14.3% Canadian indigenous persons (n = 41), 13.9% Middle Eastern (n = 40), 7.7% Asian (n = 22), and 11.1% other (n = 32). There were no differences by demographic group on the study measures. These participants have been reported on previously (Author). 2.2. Materials 2.2.1. Control variables In each set of analyses, we included control variables to provide for a more stringent test of our models. LeFevre et al. (2010) hypothesized visual–spatial working memory as a distinct pathway to mathematics separate from a linguistic pathway. Although we did not have a measure of visual–spatial working memory, we controlled for working memory using backwards digit span (Wechsler, 1991) and we controlled for visual–spatial thinking with the Block Rotation test of the Woodcock–Johnson Third Edition (WJ-III; Woodcock, McGrew, & Mather, 2001). 2.2.2. Linguistic skill LeFevre et al. (2010) hypothesized that the linguistic basis of mathematics stems from a general language system as opposed to a mathematics-specific language system. As such, we also used domaingeneral linguistic measures. 2.2.2.1. General verbal ability. Consistent with LeFevre et al. (2010), we hypothesized that general verbal ability also reflects children's ability to acquire language-based number skills. Verbal analogies reflect acquired knowledge and skill associated with language and its everyday use and is thus considered a good proxy for overall verbal ability (e.g., Flanagan, Ortiz, Alfonso, & Mascolo, 2002; Primrose, Fuller, & Littledyke, 2001). We used the Analogy subtest of the WJ-III Reading Vocabulary test (Woodcock et al., 2001) as a measure of general verbal ability. With this task, children solve an analogy read aloud by an examiner (e.g., pencil is to lead as pen is to …). The publisher reports reliability between .88 and .90. 2.2.2.2. Phonological skills. Consistent with others (Geary, 1993; LeFevre et al., 2010; Simmons & Singleton, 2008), we also hypothesized that encoding and manipulating numerical symbols are similar to the processes used when encoding and manipulating lexical symbols. We used the Word Attack test of the WJ-III (Woodcock et al., 2001) to assess phonological processing—phonological decoding specifically.
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With this test, children pronounce pseudowords that conform to English spelling rules (e.g., flib, bungic). The publisher reports reliability between .89 and .94. 2.2.3. Symbolic number skill LeFevre et al. (2010) hypothesized that children's symbolic number skill revolves around knowledge of number and they used a measure of number identification. We elected to use a more ageappropriate measure that also tapped knowledge of the number system but was not a test of conventional mathematics. With the Number Series subtest of the WJ-III Quantitative Concepts test (Woodcock et al., 2001), children use inductive and deductive reasoning to determine the next number in a sequence (e.g., 2, 5, 8, __). The publisher reports reliability between .88 and .89. 2.2.4. Arithmetic knowledge We assessed two forms of arithmetic typically encountered in third grade classrooms: procedural arithmetic and arithmetic word problems. 2.2.4.1. Procedural arithmetic. The Calculation test of the WJ-III: Research Edition (Woodcock, McGrew, & Mather, 1999) measures the degree to which students have mastered addition and subtraction facts and are able to use procedural skills to solve whole number arithmetic problems. With this task, children complete a series of written calculation questions involving single- or multi-digit arithmetic problems presented in horizontal or vertical format. The publisher reports reliability between .80 and .87.
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Table 1 Correlations among variables (N = 287). Variable
1. 2. 3. 4. 5. 6. 7.
Verbal ability Phonological skills Symbolic number skill Procedural Arithmetic Arithmetic word problems Working memory Visual–spatial ability
Correlations 1
2
3
4
5
6
– .41⁎⁎⁎ .38⁎⁎⁎ .24⁎⁎⁎ .36⁎⁎⁎ .25⁎⁎⁎ .18⁎⁎
– .39⁎⁎⁎ .38⁎⁎⁎ .45⁎⁎⁎ .37⁎⁎⁎ .10⁎
– .51⁎⁎⁎ .60⁎⁎⁎ .36⁎⁎⁎ .22⁎⁎⁎
– .51⁎⁎⁎ .27⁎⁎⁎ .20⁎⁎
– .35⁎⁎⁎ .20⁎⁎
– .23⁎⁎⁎
⁎ p b .05. ⁎⁎ p b .01. ⁎⁎⁎ p b .001.
3.1. Procedural arithmetic The path from linguistic skills to symbolic number skill—path a—was statistically significant only for verbal analogies, indicating that phonological decoding did not meet the criteria for mediation. The path from symbolic number skill to procedural arithmetic—path b—was statistically significant. Holding symbolic number skill and controls constant, only the indirect effect (path ab) of verbal analogies on procedural arithmetic was statistically significant. By contrast, phonological decoding had a direct relationship with procedural arithmetic—path c1—even with control variables in the model. 3.2. Arithmetic word problems
2.2.4.2. Arithmetic word problems. With the Applied Problems test of the WJ-III: Research Edition (Woodcock et al., 1999), children solve practical problems in mathematics read aloud by the experimenter (e.g., If Diana saved a nickel each day for one week, how much money would she have at the end of that week?). Although several of the early items on this test include questions that do not assess word problem solving (e.g., telling time or temperature, counting money), for third graders, a large majority of the questions are word problems involving whole number arithmetic. The publisher reports reliability between .91 and .93.
The path from linguistic skills to symbolic number skills—path a—was statistically significant for both verbal analogies and phonological decoding. The path from symbolic number skill to procedural arithmetic—path b—was also statistically significant. Holding constant the control variables, the indirect effect of both explanatory variables were statistically significant (path ab). Phonological decoding also maintained a statistically significant direct relationship with word problems (path c1).
2.3. Procedure
Although linguistic skills are implicated in mathematical cognition, few studies have systematically examined how different linguistic skills are related to children's mathematical performance. The bulk of extant evidence suggests that phonological processing in particular influences children's arithmetic knowledge (see Simmons & Singleton, 2008). Little research, however, has examined whether other aspects of language are important for arithmetic even though language more broadly is implicitly involved in learning mathematics. Building on the model proposed by LeFevre et al. (2010), we examined how general verbal ability and phonological skills differentially related to children's arithmetic knowledge. LeFevre et al. (2010) hypothesized that linguistic skills—from phonological processes to vocabulary to verbal reasoning to listening comprehension—influence mathematical cognition through symbolic number skill. By contrast, our results suggest that phonological skills directly influence arithmetic skills whereas language more broadly impacts arithmetic skills through symbolic number skill. Thus, general verbal ability and phonological skills may not necessarily influence children's mathematical cognition—arithmetic in this case—in the same way. Specifically, whereas verbal analogies shared an indirect relationship with arithmetic knowledge, phonological decoding was related to arithmetic skills primarily directly, presumably through its role in storing and retrieving numbers from long-term memory (Fuchs et al., 2006; Koponen, Aunola, Ahonen, & Nurmi, 2007; Simmons & Singleton, 2008). The current findings therefore advance the model proposed by LeFevre et al. (2010) to indicate that general verbal ability appears to impact children's performance by influencing
Children were assessed in the winter of their third grade year. Children were individually assessed on all measures except for procedural arithmetic, which was a group administered task. 3. Results As shown in Table 1, the control variables (i.e., working memory, visual–spatial thinking) were significantly correlated with the response variables (i.e., procedural arithmetic, arithmetic word problems), thereby confirming their inclusion as controls. Both explanatory variables (i.e., verbal analogies and phonological decoding) were significantly correlated with symbolic number skill and with the response variables, and the response variables were correlated with each other. As such, when examining the effects of an explanatory variable (e.g., verbal analogies) on a response variable (e.g., procedural arithmetic), we included as controls the other explanatory and response variable. This provided more stringent analyses in light of the cross-sectional design of our study. The primary analyses examined the direct and indirect effects of linguistic skills on arithmetic knowledge, controlling for working memory and visual–spatial thinking (as well as the reciprocal explanatory and response variable). We followed Baron and Kenny (1986) and Preacher and Hayes (2008) to obtain estimates for direct and indirect effects and used bootstrapping to construct 95% confidence intervals for indirect effects. Table 2 summarizes the results.
4. Discussion
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Table 2 Direct and indirect effect of verbal skills on arithmetic knowledge (N = 287). Response variablea Procedural arithmetic EV = Verbal analogiesb EV = Phonological decodingc Word problems EV = verbal analogiesd EV = phonological decodinge
Effect of EV on M (path a)
Effect of M on RV (path b)
.16 (.06)⁎⁎ .03 (.02)
.39 (.09)⁎⁎⁎ .39 (.09)⁎⁎⁎
.22 (.06)⁎⁎⁎ .04 (.02)⁎
1.04 (.15)⁎⁎⁎ 1.04 (.15)⁎⁎⁎
Direct effect (path c1)
Indirect effect (path ab)
Bootstrapped 95% CI
−.06 (.09) .07 (.03)⁎
.0594 .0099
.0074 to .1370 −.0043 to .0272
.2403 .0389
.0688 to .3757 .0041 to .0780
.26 (.16) .15 (.05)⁎⁎
Note. Values within parentheses indicate standard error. A 95% confidence interval that does not include zero is a statistically significant indirect effect. EV = explanatory variable; M = mediator, which for these analyses was symbolic number skill; RV = response variable. a In all analyses, working memory and visual–spatial thinking ability were controlled. b F(6, 280) = 24.29, p b .001, R2 = .3423. Arithmetic word problems and phonological decoding were also controlled. c F(6, 280) = 24.29, p b .001, R2 = .3423. Arithmetic word problems and verbal analogies were also controlled. d F(6, 280) = 40.46, p b .001, R2 = .4644. Procedural arithmetic and phonological decoding were also controlled. e F(6, 280) = 40.46, p b .001, R2 = .4644. Procedural arithmetic and verbal analogies were also controlled. ⁎ p b .05. ⁎⁎ p b .01. ⁎⁎⁎ p b .001.
the mathematical thinking that involves the symbolic number system, and phonological skills appear necessary for executing conventional arithmetic tasks. Whether this is the case for other domains of mathematical cognition remains to be determined. That verbal analogies were related to mathematical thinking specifically raises the interesting idea that mathematical cognition might actually reflect language-related skills and processes, a notion also suggested by LeFevre et al. (2010). By contrast, others have argued that language facilitates but does not dictate mathematical thinking (Gelman & Butterworth, 2005; Jordan, Levine, & Huttenlocher, 1995). Indeed, there is a growing body of research demonstrating that an innate numerical processing ability exists independent of language and that this number sense might be foundational for conventional mathematics (e.g., Dehaene, 1997; Halberda, Mazzocco, & Feigenson, 2008). In fact, Mazzocco, Feigenson, and Halberda (2011) hypothesized that the ability to verbalize the numerical information elicited by the nonverbal number sense system mediates the relationship between the nonverbal and verbal number systems. Thus, it may be that while number sense provides the basis from which children can think mathematically, general verbal ability shapes the way children represent the symbolic numerical and quantitative skills necessary for conventional mathematics. Of course, we did not have a measure of number sense in this study so more research is needed to understand how linguistic skills interact with number sense to influence children's mathematical development. 4.1. Limitations and future research The current study has limitations that necessarily raise questions for future research. First and foremost, we used only a single indicator of both general verbal ability and phonological skills. Although our results partially supported the predictions of LeFevre et al. (2010), more research is needed with multiple measures of linguistic skills to determine the full extent of the relationship between linguistic skills and mathematical cognition. Similarly, this study did not include a measure of visual–spatial working memory or number sense, each of which LeFevre et al. proposed represent a distinct pathway to mathematics. Although we partially accounted for this by controlling for visual–spatial thinking, verbal working memory, and concurrent arithmetic skill, more research is needed to understand how linguistic skills interact with other pathways to explain children's mathematical cognition. Given that the third grade children in this study had been exposed to multiple years of formal mathematics instruction, future research is also needed to understand the role of instruction in facilitating the coordination between nonverbal and verbal number systems. Finally, the cross-sectional design of this study precludes causal claims. This is especially important to acknowledge because mediation models assume both that explanatory variables (i.e., linguistic
skills) cause the mediator (i.e., symbolic number skill) and that the mediator causes the response variable (i.e., conventional mathematics). Although our study design provided a stringent test of the model proposed by LeFevre et al., longitudinal and experimental studies are needed to replicate our findings to confirm the causal ordering. 4.2. Conclusion The findings from this study suggest that general verbal ability is involved in how children reason numerically whereas phonological skills are involved in executing arithmetic problems. This relationship should continue to be examined across developmental stages, especially as the language demands during mathematics instruction and in mathematics textbooks intensify with increasing years of schooling (Bielenberg & Wong Fillmore, 2004/2005; Lager, 2006; RAND Mathematics Study Panel, 2003). Although mathematical thinking can exist independent of language, children most often need language to express, understand, and learn mathematics, which begs a nuanced understanding of its role in children's ability to express, understand, and learn mathematics. Such knowledge is necessary in order to inform targeted instruction and interventions that address the needs of learners struggling with mathematics. References Abedi, J., & Lord, C. (2001). The language factor in mathematics tests. Applied Measurement in Education, 14, 219–234, http://dx.doi.org/10.1207/S15324818AME1403_2. Adams, T. L. (2003). Reading mathematics: More than words can say. The Reading Teacher, 56(8), 786–795. Adams, T. L., & Lowery, R. M. (2007). An analysis of children's strategies for reading mathematics. Reading & Writing Quarterly, 23, 161–177, http://dx.doi.org/10.1080/ 10573560601158479. Baron, R. M., & Kenny, D. A. (1986). The moderator–mediator variable distinction in social psychological research: Conceptual, strategic, and statistical considerations. Journal of Personality and Social Psychology, 51, 1173–1182, http://dx.doi.org/10.1037/00223514.51.6.1173. Bielenberg, B., & Wong Fillmore, L. (2004/2005). The English they need for the test. Educational Leadership, 62, 45–49. Brissiaud, R., & Sander, E. (2010). Arithmetic word problem solving: A situation strategy framework. Developmental Science, 13, 92–107, http://dx.doi.org/10.1111/j.14677687.2009.00866.x. Dehaene, S. (1997). The number sense: How the mind creates mathematics. New York: Oxford University Press. Dehaene, S., Piazza, M., Pinel, P., & Cohen, L. (2003). Three parietal circuits for number processing. Cognitive Neuropsychology, 20, 487–506, http://dx.doi.org/ 10.1080/02643290244000239. Dirks, E., Spyer, G., van Lieshout, E. C., & de Sonneville, L. (2008). Prevalence of combined reading and arithmetic disabilities. Journal of Learning Disabilities, 41, 460–473, http://dx.doi.org/10.1177/0022219408321128. Flanagan, D. P., Ortiz, S. O., Alfonso, V. C., & Mascolo, J. T. (2002). The achievement test desk reference (ATDR): Comprehensive assessment and learning disabilities. Boston: Allyn & Bacon. Fuchs, L. S., Compton, D. L., Fuchs, D., Paulsen, K., Bryant, J. D., & Hamlett, C. L. (2005). The prevention, identification, and cognitive determinants of math difficulty.
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Contents lists available at SciVerse ScienceDirect
Learning and Individual Differences journal homepage: www.elsevier.com/locate/lindif
The relationship between linguistic skills and arithmetic knowledge Rose K. Vukovic a,⁎, Nonie K. Lesaux b a b
Department of Teaching & Learning, New York University, 239 Greene St 2nd Floor, New York, NY 10003, United States Harvard Graduate School of Education, 14 Appian Way, Cambridge, MA 02138, United States
a r t i c l e
i n f o
Article history: Received 5 May 2011 Received in revised form 13 July 2012 Accepted 10 October 2012 Keywords: Verbal ability Phonological decoding Arithmetic Language
a b s t r a c t Although language is implicated in children's mathematical development, few studies have focused specifically on how different linguistic skills relate to children's mathematical performance. Building on the model proposed by LeFevre et al. (2010), this study examined how general verbal ability and phonological skills were differentially related to children's arithmetic knowledge. Third grade children (N=287) were assessed on verbal analogies, phonological decoding, symbolic number skill, procedural arithmetic, and arithmetic word problems. Using mediation analyses, the results indicated that verbal analogies were indirectly related to arithmetic knowledge through symbolic number skill, whereas phonological decoding had a direct relationship with arithmetic performance. These results suggest that general verbal ability influences how children understand and reason with numbers, whereas phonological skills are involved in executing conventional arithmetic problems. © 2012 Elsevier Inc. All rights reserved.
1. Introduction Much like reading, learning and doing mathematics are steeped in oral and written language (Adams, 2003; Adams & Lowery, 2007; Schleppegrell, 2007). For instance, mathematics instruction in classrooms depends primarily on oral explanations and interactions, and the delivery of mathematics curriculum often occurs via written text (Bielenberg & Wong Fillmore, 2004/2005; Schleppegrell, 2007). Furthermore, mathematics consists of a specialized vocabulary: words such as volume, ruler, plot, and product, have different meanings in mathematics than when used in everyday language (Adams, 2003). In turn, to be mathematically proficient, children must develop a language that allows them to participate not only during mathematics instruction, but also to engage quantitatively with the world outside the classroom. Other than the obvious example of word problems, however, the significant language demands of mathematics have until recently been overlooked by researchers and practitioners. A growing body of studies indicate that the same underlying processes that are important for reading, particularly phonological processing, are important for mathematics (e.g., Hecht, Torgesen, Wagner, & Rashotte, 2001; Jordan, Kaplan, & Hanich, 2002; Simmons & Singleton, 2008). Indeed, it has even been suggested that children's mathematical difficulties might actually reflect deficient linguistic processes as opposed to deficits in quantitative processes (LeFevre et al., 2010; Vukovic, 2012). This supposition is in part supported by neuropsychological evidence showing that in addition to a specialized quantitative circuit ⁎ Corresponding author. Tel.: +1 212 998 5205; fax: +1 212 995 3636. E-mail addresses: [email protected] (R.K. Vukovic), [email protected] (N.K. Lesaux). 1041-6080/$ – see front matter © 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.lindif.2012.10.007
in the parietal lobe, a linguistic circuit in the left angular gyrus supports the manipulation of numbers in verbal form (Dehaene, Piazza, Pinel, & Cohen, 2003), suggesting that there is a linguistic basis for some aspects of mathematics. Furthermore, children tend to have more difficulty with language-based mathematical tasks (e.g., number facts, word problems) than with tasks that have fewer language demands (e.g., nonverbal calculation, number sense) (Jordan & Levine, 2009; Jordan, Mulhern, & Wylie, 2009; Locuniak & Jordan, 2008). Taken together, these findings suggest an inextricable link between linguistic skills and mathematical performance. The nature of this relationship, however, remains underspecified. More specifically, although linguistic skills are implicated in mathematical cognition, few studies have systematically examined how different linguistic skills are related to children's mathematical performance. A deeper understanding of the relationship between linguistic skills and children's mathematical performance is necessary to provide theoretical insight and practical guidance into how to best support children's mathematical development. To begin to increase the specificity of our understanding of the role of linguistic skills in mathematical cognition, this study examined how general verbal ability and phonological skills were differentially related to children's arithmetic knowledge. We focused specifically on arithmetic because of its foundational role for the more complex mathematics children encounter in middle and high school. 1.1. Linguistic skills and arithmetic knowledge The bulk of studies investigating the relationship between linguistic skills and children's arithmetic knowledge have focused on phonological processes, because completing simple arithmetic problems requires the retrieval of phonological codes, as well as encoding and
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maintaining phonological representations in immediate awareness (Geary, 1993; Simmons & Singleton, 2008). Weaknesses in phonological processing are therefore thought to hinder the development of tasks that rely on the manipulation and storage of verbal codes, such as counting and solving simple arithmetic problems. Studies show that phonological processes are indeed important for children's arithmetic development (e.g., Fuchs et al., 2005; Hecht et al., 2001; Simmons & Singleton, 2008). This phonological representation hypothesis also helps explain the finding that many children with reading difficulties also have difficulty with arithmetic (e.g., Dirks, Spyer, van Lieshout, & de Sonneville, 2008; Rubinsten, 2009; Simmons & Singleton, 2008). Although an important starting point, the phonological representation hypothesis does not fully account for the link between linguistic skills and arithmetic performance. For instance, there exist children with arithmetic difficulties who are nonetheless good word readers (and therefore tend to have age-appropriate phonological skills) and vice versa (e.g., Fuchs, Fuchs, & Prentice, 2004; Landerl, Fussenegger, Moll, & Willburger, 2009), suggesting that phonological skills are not the sole influential factor. Moreover, Jordan and colleagues have found that good readers with arithmetic difficulties use their strengths in verbal reasoning to complete arithmetic problems, suggesting that linguistic skills beyond phonological processing are involved in arithmetic performance (e.g., Jordan & Hanich, 2003). Indeed, there is evidence that more than just an issue of vocabulary, the language used in arithmetic word problems influences how children represent and solve these problems (e.g., Abedi & Lord, 2001; Brissiaud & Sander, 2010). Together, these findings suggest that general verbal ability plays a different role in children's arithmetic performance than phonological skills. Few studies, however, have specifically contrasted the unique effects of general verbal ability and phonological skills on children's arithmetic performance. In a relevant study, LeFevre et al. (2010) proposed a theoretical model whereby linguistic skills influence mathematical cognition indirectly through symbolic number skill— the number and quantitative skills that depend on the formal number system but do not require knowledge of formal mathematics (see also Jordan, Glutting, & Ramineni, 2010). Examples of symbolic number skill include number identification and numerical reasoning. In their analyses, LeFevre et al. (2010) examined how a linguistic composite that tapped vocabulary, phonological skills, and symbolic number skill (i.e., number identification) directly predicted mathematical outcomes. The authors found that the linguistic composite measured in children at 4.5-years old explained unique variance in procedural arithmetic, numeration, geometry, and measurement at 7.5-years old. That the linguistic composite included general verbal ability, phonological skills, and symbolic number skills, however, makes it difficult to disentangle how different linguistic skills relate to symbolic number skill, as well as the direct and indirect effects of general verbal ability and phonological skills on mathematical cognition. As such, the model proposed by LeFevre et al. provides the basis for investigating the unique role of general verbal ability and phonological skills in children's arithmetic knowledge. 1.2. Present study Building on previous research, we tested the mediation model proposed by LeFevre et al. (2010; see Fig. 1), which makes three predictions. First, children's linguistic skills should be related to children's symbolic number skill (path a). Second, symbolic number skill should be related to arithmetic knowledge (path b). Finally, linguistic skills should have an indirect effect on arithmetic knowledge through symbolic number skill (path ab). We expected this model to hold for general verbal ability, consistent with the predictions of LeFevre et al. By contrast, we hypothesized that phonological skills would maintain a direct effect on arithmetic knowledge (path c 1), given that phonological processes are involved in the storage and
Symbolic Number Skill path b
path a path c¹ Linguistic Skills
Arithmetic Knowledge
Fig. 1. Predicted relationships among linguistic skills, symbolic number skill, and arithmetic knowledge.
retrieval of numbers from long-term memory (e.g., Geary, 1993; Simmons & Singleton, 2008). 2. Method 2.1. Participants The participants were 287 (134 girls) third graders (mean age = 8.60 years, SD = 4 months). The children attended five elementary schools located primarily in working class neighborhoods in an urban Canadian city. The mathematics curriculum was the same across the five schools and emphasized a balanced approach between conceptual understanding and procedural skills. The children were 53.0% majority culture (n = 152), 14.3% Canadian indigenous persons (n = 41), 13.9% Middle Eastern (n = 40), 7.7% Asian (n = 22), and 11.1% other (n = 32). There were no differences by demographic group on the study measures. These participants have been reported on previously (Author). 2.2. Materials 2.2.1. Control variables In each set of analyses, we included control variables to provide for a more stringent test of our models. LeFevre et al. (2010) hypothesized visual–spatial working memory as a distinct pathway to mathematics separate from a linguistic pathway. Although we did not have a measure of visual–spatial working memory, we controlled for working memory using backwards digit span (Wechsler, 1991) and we controlled for visual–spatial thinking with the Block Rotation test of the Woodcock–Johnson Third Edition (WJ-III; Woodcock, McGrew, & Mather, 2001). 2.2.2. Linguistic skill LeFevre et al. (2010) hypothesized that the linguistic basis of mathematics stems from a general language system as opposed to a mathematics-specific language system. As such, we also used domaingeneral linguistic measures. 2.2.2.1. General verbal ability. Consistent with LeFevre et al. (2010), we hypothesized that general verbal ability also reflects children's ability to acquire language-based number skills. Verbal analogies reflect acquired knowledge and skill associated with language and its everyday use and is thus considered a good proxy for overall verbal ability (e.g., Flanagan, Ortiz, Alfonso, & Mascolo, 2002; Primrose, Fuller, & Littledyke, 2001). We used the Analogy subtest of the WJ-III Reading Vocabulary test (Woodcock et al., 2001) as a measure of general verbal ability. With this task, children solve an analogy read aloud by an examiner (e.g., pencil is to lead as pen is to …). The publisher reports reliability between .88 and .90. 2.2.2.2. Phonological skills. Consistent with others (Geary, 1993; LeFevre et al., 2010; Simmons & Singleton, 2008), we also hypothesized that encoding and manipulating numerical symbols are similar to the processes used when encoding and manipulating lexical symbols. We used the Word Attack test of the WJ-III (Woodcock et al., 2001) to assess phonological processing—phonological decoding specifically.
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With this test, children pronounce pseudowords that conform to English spelling rules (e.g., flib, bungic). The publisher reports reliability between .89 and .94. 2.2.3. Symbolic number skill LeFevre et al. (2010) hypothesized that children's symbolic number skill revolves around knowledge of number and they used a measure of number identification. We elected to use a more ageappropriate measure that also tapped knowledge of the number system but was not a test of conventional mathematics. With the Number Series subtest of the WJ-III Quantitative Concepts test (Woodcock et al., 2001), children use inductive and deductive reasoning to determine the next number in a sequence (e.g., 2, 5, 8, __). The publisher reports reliability between .88 and .89. 2.2.4. Arithmetic knowledge We assessed two forms of arithmetic typically encountered in third grade classrooms: procedural arithmetic and arithmetic word problems. 2.2.4.1. Procedural arithmetic. The Calculation test of the WJ-III: Research Edition (Woodcock, McGrew, & Mather, 1999) measures the degree to which students have mastered addition and subtraction facts and are able to use procedural skills to solve whole number arithmetic problems. With this task, children complete a series of written calculation questions involving single- or multi-digit arithmetic problems presented in horizontal or vertical format. The publisher reports reliability between .80 and .87.
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Table 1 Correlations among variables (N = 287). Variable
1. 2. 3. 4. 5. 6. 7.
Verbal ability Phonological skills Symbolic number skill Procedural Arithmetic Arithmetic word problems Working memory Visual–spatial ability
Correlations 1
2
3
4
5
6
– .41⁎⁎⁎ .38⁎⁎⁎ .24⁎⁎⁎ .36⁎⁎⁎ .25⁎⁎⁎ .18⁎⁎
– .39⁎⁎⁎ .38⁎⁎⁎ .45⁎⁎⁎ .37⁎⁎⁎ .10⁎
– .51⁎⁎⁎ .60⁎⁎⁎ .36⁎⁎⁎ .22⁎⁎⁎
– .51⁎⁎⁎ .27⁎⁎⁎ .20⁎⁎
– .35⁎⁎⁎ .20⁎⁎
– .23⁎⁎⁎
⁎ p b .05. ⁎⁎ p b .01. ⁎⁎⁎ p b .001.
3.1. Procedural arithmetic The path from linguistic skills to symbolic number skill—path a—was statistically significant only for verbal analogies, indicating that phonological decoding did not meet the criteria for mediation. The path from symbolic number skill to procedural arithmetic—path b—was statistically significant. Holding symbolic number skill and controls constant, only the indirect effect (path ab) of verbal analogies on procedural arithmetic was statistically significant. By contrast, phonological decoding had a direct relationship with procedural arithmetic—path c1—even with control variables in the model. 3.2. Arithmetic word problems
2.2.4.2. Arithmetic word problems. With the Applied Problems test of the WJ-III: Research Edition (Woodcock et al., 1999), children solve practical problems in mathematics read aloud by the experimenter (e.g., If Diana saved a nickel each day for one week, how much money would she have at the end of that week?). Although several of the early items on this test include questions that do not assess word problem solving (e.g., telling time or temperature, counting money), for third graders, a large majority of the questions are word problems involving whole number arithmetic. The publisher reports reliability between .91 and .93.
The path from linguistic skills to symbolic number skills—path a—was statistically significant for both verbal analogies and phonological decoding. The path from symbolic number skill to procedural arithmetic—path b—was also statistically significant. Holding constant the control variables, the indirect effect of both explanatory variables were statistically significant (path ab). Phonological decoding also maintained a statistically significant direct relationship with word problems (path c1).
2.3. Procedure
Although linguistic skills are implicated in mathematical cognition, few studies have systematically examined how different linguistic skills are related to children's mathematical performance. The bulk of extant evidence suggests that phonological processing in particular influences children's arithmetic knowledge (see Simmons & Singleton, 2008). Little research, however, has examined whether other aspects of language are important for arithmetic even though language more broadly is implicitly involved in learning mathematics. Building on the model proposed by LeFevre et al. (2010), we examined how general verbal ability and phonological skills differentially related to children's arithmetic knowledge. LeFevre et al. (2010) hypothesized that linguistic skills—from phonological processes to vocabulary to verbal reasoning to listening comprehension—influence mathematical cognition through symbolic number skill. By contrast, our results suggest that phonological skills directly influence arithmetic skills whereas language more broadly impacts arithmetic skills through symbolic number skill. Thus, general verbal ability and phonological skills may not necessarily influence children's mathematical cognition—arithmetic in this case—in the same way. Specifically, whereas verbal analogies shared an indirect relationship with arithmetic knowledge, phonological decoding was related to arithmetic skills primarily directly, presumably through its role in storing and retrieving numbers from long-term memory (Fuchs et al., 2006; Koponen, Aunola, Ahonen, & Nurmi, 2007; Simmons & Singleton, 2008). The current findings therefore advance the model proposed by LeFevre et al. (2010) to indicate that general verbal ability appears to impact children's performance by influencing
Children were assessed in the winter of their third grade year. Children were individually assessed on all measures except for procedural arithmetic, which was a group administered task. 3. Results As shown in Table 1, the control variables (i.e., working memory, visual–spatial thinking) were significantly correlated with the response variables (i.e., procedural arithmetic, arithmetic word problems), thereby confirming their inclusion as controls. Both explanatory variables (i.e., verbal analogies and phonological decoding) were significantly correlated with symbolic number skill and with the response variables, and the response variables were correlated with each other. As such, when examining the effects of an explanatory variable (e.g., verbal analogies) on a response variable (e.g., procedural arithmetic), we included as controls the other explanatory and response variable. This provided more stringent analyses in light of the cross-sectional design of our study. The primary analyses examined the direct and indirect effects of linguistic skills on arithmetic knowledge, controlling for working memory and visual–spatial thinking (as well as the reciprocal explanatory and response variable). We followed Baron and Kenny (1986) and Preacher and Hayes (2008) to obtain estimates for direct and indirect effects and used bootstrapping to construct 95% confidence intervals for indirect effects. Table 2 summarizes the results.
4. Discussion
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Table 2 Direct and indirect effect of verbal skills on arithmetic knowledge (N = 287). Response variablea Procedural arithmetic EV = Verbal analogiesb EV = Phonological decodingc Word problems EV = verbal analogiesd EV = phonological decodinge
Effect of EV on M (path a)
Effect of M on RV (path b)
.16 (.06)⁎⁎ .03 (.02)
.39 (.09)⁎⁎⁎ .39 (.09)⁎⁎⁎
.22 (.06)⁎⁎⁎ .04 (.02)⁎
1.04 (.15)⁎⁎⁎ 1.04 (.15)⁎⁎⁎
Direct effect (path c1)
Indirect effect (path ab)
Bootstrapped 95% CI
−.06 (.09) .07 (.03)⁎
.0594 .0099
.0074 to .1370 −.0043 to .0272
.2403 .0389
.0688 to .3757 .0041 to .0780
.26 (.16) .15 (.05)⁎⁎
Note. Values within parentheses indicate standard error. A 95% confidence interval that does not include zero is a statistically significant indirect effect. EV = explanatory variable; M = mediator, which for these analyses was symbolic number skill; RV = response variable. a In all analyses, working memory and visual–spatial thinking ability were controlled. b F(6, 280) = 24.29, p b .001, R2 = .3423. Arithmetic word problems and phonological decoding were also controlled. c F(6, 280) = 24.29, p b .001, R2 = .3423. Arithmetic word problems and verbal analogies were also controlled. d F(6, 280) = 40.46, p b .001, R2 = .4644. Procedural arithmetic and phonological decoding were also controlled. e F(6, 280) = 40.46, p b .001, R2 = .4644. Procedural arithmetic and verbal analogies were also controlled. ⁎ p b .05. ⁎⁎ p b .01. ⁎⁎⁎ p b .001.
the mathematical thinking that involves the symbolic number system, and phonological skills appear necessary for executing conventional arithmetic tasks. Whether this is the case for other domains of mathematical cognition remains to be determined. That verbal analogies were related to mathematical thinking specifically raises the interesting idea that mathematical cognition might actually reflect language-related skills and processes, a notion also suggested by LeFevre et al. (2010). By contrast, others have argued that language facilitates but does not dictate mathematical thinking (Gelman & Butterworth, 2005; Jordan, Levine, & Huttenlocher, 1995). Indeed, there is a growing body of research demonstrating that an innate numerical processing ability exists independent of language and that this number sense might be foundational for conventional mathematics (e.g., Dehaene, 1997; Halberda, Mazzocco, & Feigenson, 2008). In fact, Mazzocco, Feigenson, and Halberda (2011) hypothesized that the ability to verbalize the numerical information elicited by the nonverbal number sense system mediates the relationship between the nonverbal and verbal number systems. Thus, it may be that while number sense provides the basis from which children can think mathematically, general verbal ability shapes the way children represent the symbolic numerical and quantitative skills necessary for conventional mathematics. Of course, we did not have a measure of number sense in this study so more research is needed to understand how linguistic skills interact with number sense to influence children's mathematical development. 4.1. Limitations and future research The current study has limitations that necessarily raise questions for future research. First and foremost, we used only a single indicator of both general verbal ability and phonological skills. Although our results partially supported the predictions of LeFevre et al. (2010), more research is needed with multiple measures of linguistic skills to determine the full extent of the relationship between linguistic skills and mathematical cognition. Similarly, this study did not include a measure of visual–spatial working memory or number sense, each of which LeFevre et al. proposed represent a distinct pathway to mathematics. Although we partially accounted for this by controlling for visual–spatial thinking, verbal working memory, and concurrent arithmetic skill, more research is needed to understand how linguistic skills interact with other pathways to explain children's mathematical cognition. Given that the third grade children in this study had been exposed to multiple years of formal mathematics instruction, future research is also needed to understand the role of instruction in facilitating the coordination between nonverbal and verbal number systems. Finally, the cross-sectional design of this study precludes causal claims. This is especially important to acknowledge because mediation models assume both that explanatory variables (i.e., linguistic
skills) cause the mediator (i.e., symbolic number skill) and that the mediator causes the response variable (i.e., conventional mathematics). Although our study design provided a stringent test of the model proposed by LeFevre et al., longitudinal and experimental studies are needed to replicate our findings to confirm the causal ordering. 4.2. Conclusion The findings from this study suggest that general verbal ability is involved in how children reason numerically whereas phonological skills are involved in executing arithmetic problems. This relationship should continue to be examined across developmental stages, especially as the language demands during mathematics instruction and in mathematics textbooks intensify with increasing years of schooling (Bielenberg & Wong Fillmore, 2004/2005; Lager, 2006; RAND Mathematics Study Panel, 2003). Although mathematical thinking can exist independent of language, children most often need language to express, understand, and learn mathematics, which begs a nuanced understanding of its role in children's ability to express, understand, and learn mathematics. Such knowledge is necessary in order to inform targeted instruction and interventions that address the needs of learners struggling with mathematics. References Abedi, J., & Lord, C. (2001). The language factor in mathematics tests. Applied Measurement in Education, 14, 219–234, http://dx.doi.org/10.1207/S15324818AME1403_2. Adams, T. L. (2003). Reading mathematics: More than words can say. The Reading Teacher, 56(8), 786–795. Adams, T. L., & Lowery, R. M. (2007). An analysis of children's strategies for reading mathematics. Reading & Writing Quarterly, 23, 161–177, http://dx.doi.org/10.1080/ 10573560601158479. Baron, R. M., & Kenny, D. A. (1986). The moderator–mediator variable distinction in social psychological research: Conceptual, strategic, and statistical considerations. Journal of Personality and Social Psychology, 51, 1173–1182, http://dx.doi.org/10.1037/00223514.51.6.1173. Bielenberg, B., & Wong Fillmore, L. (2004/2005). The English they need for the test. Educational Leadership, 62, 45–49. Brissiaud, R., & Sander, E. (2010). Arithmetic word problem solving: A situation strategy framework. Developmental Science, 13, 92–107, http://dx.doi.org/10.1111/j.14677687.2009.00866.x. Dehaene, S. (1997). The number sense: How the mind creates mathematics. New York: Oxford University Press. Dehaene, S., Piazza, M., Pinel, P., & Cohen, L. (2003). Three parietal circuits for number processing. Cognitive Neuropsychology, 20, 487–506, http://dx.doi.org/ 10.1080/02643290244000239. Dirks, E., Spyer, G., van Lieshout, E. C., & de Sonneville, L. (2008). Prevalence of combined reading and arithmetic disabilities. Journal of Learning Disabilities, 41, 460–473, http://dx.doi.org/10.1177/0022219408321128. Flanagan, D. P., Ortiz, S. O., Alfonso, V. C., & Mascolo, J. T. (2002). The achievement test desk reference (ATDR): Comprehensive assessment and learning disabilities. Boston: Allyn & Bacon. Fuchs, L. S., Compton, D. L., Fuchs, D., Paulsen, K., Bryant, J. D., & Hamlett, C. L. (2005). The prevention, identification, and cognitive determinants of math difficulty.
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