The Contributions of Domain-General and Numerical Factors to Third-Grade Arithmetic Skills and Mathematical Learning Disability.

Reasoning

Relationships between mathematics and reasoning skill have long been proposed, and mathematical reasoning accounts for differences in mathematics achievement independently of computational skill ( Nunes et al., 2007 ). Raven’s Colored Progressive Matrices (CPM, Raven, 2008 ) is a domain-general reasoning test that was originally developed to measure a component of g , the general factor common to different mental tests ( Spearman, 1927 ). The relation between CPM and mathematical achievement was shown in an epidemiological study of 9- and 10-year-olds ( Lewis, Hitch, & Walker, 1994 ): Most (66%) children with poor achievement in arithmetic had CPM scores in the lowest quartile.

Processing Speed

Processing speed, the rapidity of execution of mental operations, has been proposed as a general factor underlying individual differences in cognition since Galton. Increases in processing speed have been suggested to underlie age-related cognitive development, including working memory functioning ( Fry & Hale, 1996 ; Kail, 1991 ). Processing speed accounts for variation in mathematical skills independently of working memory in some studies (e.g., Andersson, 2010 ; Bull & Johnston, 1997 ; Chan & Ho, 2010 ; Fuchs et al., 2006 ; Hecht, Torgesen, Wagner, & Rashotte, 2001 ; but note Geary et al., 2008 ). Processing speed is more related to basic calculation than written arithmetic or story problems ( Chan & Ho, 2010 ; Fuchs et al., 2006 ).

Oral Language

Children’s first encounter with numbers is through learning to count and mastering the number word sequence. Oral language is the principal medium of instruction in elementary school. Both suggest that oral language ability is likely to affect the development of mathematical skills and knowledge. Consistent with this, oral language skills independently account for variation in mathematical skills ( Cowan, Donlan, Newton, & Lloyd, 2005 ; Fuchs et al., 2006 ).

Summary

All domain-general factors are associated with variation in mathematical skills. Although much variance is shared by domain-general factors, some studies indicate particular factors make unique contributions (e.g., Bull & Johnston, 1997 ). The contributions depend on the arithmetical skill (e.g., Fuchs et al., 2006 ). Working memory functioning is often highlighted in relation to arithmetical skills, but this might be because it has been studied more than other domain-general factors.

Relationships between domain-general factors and arithmetical skills might be direct or indirect. Indirect relationships would reflect associations with either numerical factors or other domain-general factors. These possibilities are not exclusive: a zero-order correlation between a domain-general factor and an arithmetical skill might reflect a combination of direct and indirect relationships. By including measures of each domain-general factor and numerical factors in models of variation, this study attempts to assess the nature of the relationships.

Multidigit Skills: Number System Knowledge

The meaning of numbers is determined by their relations to other numbers in the number system. Children typically start school with some knowledge of the number word sequence and the names of numerals ( Siegler & Robinson, 1982 ). During elementary school, they master the system for combining number words and the Hindu-Arabic system for representing numbers with numerals. This enables them to generate accurate counting and numeral sequences from numbers they have not experienced ( Skwarchuk & Anglin, 2002 ).

Number system knowledge has been assessed by the Number Knowledge test ( Griffin, 1997 , 2005 ) and count sequence tasks requiring children to count up and down from specified points (e.g., Cowan et al., 2005 ). Deficient number system knowledge in kindergarten is the best predictor of subsequent mathematics difficulty ( Gersten et al., 2005 ) and, in combination with other measures, number system knowledge predicts later growth in mathematics up to third grade ( Jordan et al., 2007 ; Jordan, Kaplan, Ramineni, & Locuniak, 2009 ).

Ignorance of place value has been found to discriminate children with mathematics difficulties from their peers ( Chan & Ho, 2010 ), and variation in second grade children’s count sequence knowledge, which principally concerns numbers above 100, substantially correlates with single digit calculation ( Cowan et al., 2005 ).

Multidigit Skills: Estimation

Estimation is involved in a variety of approximation tasks including judging measurements in standard units, generating ball park answers to computations, and assigning numbers to quantities without counting. Number line estimation is an approximate number task in which a line with numerals at the endpoints is presented and children either estimate the position of target numbers or estimate the number corresponding to target marks.

Although the cognitive mechanisms underlying numeral placements are debated (e.g., Barth & Paladino, 2011 ; Opfer, Siegler, & Young, 2011 ), there is no dispute that accuracy of number line estimates correlates substantially with other forms of pure numerical estimation ( Booth & Siegler, 2006 ) and general math achievement ( Ashcraft & Moore, 2012 ; Booth & Siegler, 2008 ; LeFevre et al., 2010 ; Schneider, Grabner, & Paetsch, 2009 ; Siegler, Thompson, & Schneider, 2011 ).

The development of a mental number line that faithfully represents the relations between numbers has also been suggested to underlie the development of number system knowledge ( Case et al., 1996 ), and the need to understand the number system is recognized for successful estimation ( Siegler & Opfer, 2003 ). Therefore, the extent to which number system knowledge and estimation accuracy independently contribute to mathematics achievement is uncertain. Number estimation has been found to be impaired in children with number difficulties ( Geary et al., 2008 ).

Summary

General arithmetic skills are related to all the numerical factors. Previous research on single digit processing more often finds relations with numeral magnitude comparison than with quantity enumeration. This may reflect how the measures are derived from performance. In the present study, we combine accuracy with speed to yield efficiency measures. To reduce covariation of single digit processing with domain-general factors, we do not include Stroop trials in the numeral comparison task.

Domain-general factors are hypothesized to affect the development of both number system knowledge and estimation. This yields the prediction of relations between individual variation in domain-general functioning and in these multidigit skills. What is uncertain is the extent to which the multidigit skills account for variation in arithmetical skills independently of domain-general factors and each other. As with the domain-general factors, relationships between numerical factors and arithmetical skills might be direct or indirect, and these are not mutually exclusive. Direct relations would be evidenced by unique contributions to variance, independently of the other factors. Indirect associations might be due to relationships with other numerical or domain-general factors. This study uses models of variation that include the other factors to assess these relationships.

Arithmetical Skills

This study compares the contributions of general and specific factors to explaining variation in three different measures of arithmetic skill: basic calculation fluency, written multidigit computation, and arithmetic word problems. The rationale for the selection of these is as follows. Basic calculation fluency is chosen as it consistently correlates highly with more general measures of math achievement (e.g., Durand et al., 2005 ), is frequently impaired in children with math difficulty (e.g., Russell & Ginsburg, 1984 ), and lastly, is a timed measure as children are only credited for correct answers given in less than 3 s. Basic calculation fluency has always been emphasized in elementary education as it is believed to be crucial for competence in both mental and written arithmetic. As a timed measure, it is likely to be associated with differences in processing speed either at the domain-general level or at the domain specific level of simple number processing.

Written arithmetic involves multidigit computation. Developing competence in written arithmetic remains a key aspiration of the early elementary curriculum and written arithmetic items feature in both curriculum tests and standardized measures of mathematics achievement such as the Wechsler Individual Achievement Test–Second UK Edition (WIAT-II UK; Wechsler, 2005 ). Written arithmetic is also chosen because individuals can show marked discrepancies between their skill in mental and written arithmetic ( Dowker, 2005 ). This may be because written arithmetic is susceptible to procedural bugs and visuo-spatial deficits ( Raghubar et al., 2009 ). Visuo-spatial deficits were predicted on the basis of early research on number difficulties ( Geary, 1993 ), but subsequent research has yielded less support for them than fact retrieval and procedural deficits ( Geary, 2010 ).

Arithmetic word problems require children to understand a set of verbally expressed propositions, identify the relevant computational problem, and execute it. They are included because competence with word problems has long been perceived by math educators as evidence of the ability to apply arithmetic ( Verschaffel, Greer, & De Corte, 2000 ). Also, they have been the focus of study in previous investigations of cognitive correlates (e.g., Fuchs et al., 2006 ; Swanson & Beebe-Frankenberger, 2004 ). Word problems accuracy is more related than basic calculation proficiency to oral language skill and general ability, and this is consistent with the difference in cognitive demands made by these skills ( Fuchs et al., 2006 ).

On the basis of previous work, we anticipate that across the three outcomes the domain-general factors will vary in their contributions. In contrast, we hypothesize that both number system knowledge and estimation will contribute to all three arithmetical outcomes. The theory behind number system knowledge ( Siegler, 1996 ) posits a bidirectional relationship between conceptual knowledge about the number system and even simple computational skill. For example, insight into the number sequence yields knowledge of arithmetical facts such as n + 1, n − 1, and n − ( n − 1). Place value understanding is important for understanding written arithmetic procedures. As estimation skill supports computational estimation then it should support successful monitoring of the execution of procedures in written arithmetic.