DISABILITY

Math

Michèle M. M. Mazzocco, Jenny Yun-Chen Chan

& Emily O. Prager

What is Specific Learning Disabilities in mathematics?

There is much debate surrounding the definition of Specific Learning Disabilities in mathematics (SLD-math). This debate centers on the inconsistent terminology used to describe mathematics-related disabilities and difficulties (as reviewed by Lewis & Fisher, 2016; and Mazzocco, 2007), the many cognitive abilities that synergistically support mathematics learning and performance (Berch & Mazzocco, 2007), the complexity of mathematics as a discipline, and the multi-dimensionality of mathematics skills and processes (e.g., Petrill et al., 2012). The potential under - pinnings and manifestations of mathematics disabilities are reflected in the broad, skill-based diagnostic criteria for SLD-math reported in the DSM-5 (American Psychiatric Association, 2013). Similar to other Specific Learning Disorders addressed in this volume (i.e., SLD-reading or writing; see Chapters 6 and 7), the DSM-5 defines SLD-math as:

a developmental disorder that begins by school-age, . . . involves ongoing problems learning key academic skills (like) . . . math calculation and math problem solving, (and) is not simply a result of lack of instruction or poor instruction.

Similar criteria are reflected in federal and state level definitions of specific learning disability. These definitions typically refer to difficulties in mathematics calculation or problem solving as indicative of SLD, but onlyif those difficulties cannot be attributed to language, emotional, or intellectual impairments. These specific criteria are not adopted by all researchers of SLD-math¹, although many researchers share the belief that a classification of “mathematics learning disability” (MLD) implicates persistent, biologically-influenced and brain-based developmental

difficulties in mathematics (Butterworth, 2005). Of these criteria, persistentdifficulty with mathematics is a notable clinical marker; that is, the difficulties must not simply surface when learning novel mathematics skills or topics, must not be evident at only a single assessment, and must not be limited to only self-reported “difficulty”

on mathematics tasks that reflect developmentally appropriate mental exertion required to master new mathematics principles or execute procedures. Thus, mathematics difficulties associated with SLD-math are typically persistent and severe, often manifested as standardized performance at or below the bottom fifteenth percentile.

Defining characteristics

The difficulties that characterize SLD-math persist despite adequate learning and practice opportunities and, often, ample effort. These mathematics difficulties may manifest as continued reliance on immature strategies (e.g., solving simple addition by counting fingers) that age-mates abandon for more efficient and over-learned alternatives (e.g., retrieving addition facts from memory; Geary, 2011), unsuccessful application of otherwise appropriate procedures (e.g., failure of execution), or inappropriate procedures supported by misconceptions (e.g., Mazzocco, Myers, Lewis, Hanich, & Murphy, 2013). Importantly, persistent difficulty may be masked by apparent success based on rote memory of newly introduced mathematics facts or procedures. That is, some children with SLD-math (and superior verbal memory) may overlearn number combinations when repeated practice is an introductory instructional focus, but these overlearned facts may extinguish once repeated focus is no longer instructionally supported (Murphy & Mazzocco, 2008).

Despite agreement that SLD-math behavioral indicators are severe and persistent, there is some disagreement in whether the etiology of SLD-math is domain specific (e.g., linked to numerical processing deficits; e.g., Butterworth, 2005) or domain general (e.g., a persistent but indirect consequence of linguistic, spatial, or Executive Function skills, e.g., Geary, 1993, 2011) and whether SLD-math can be differentiated as primary vs secondary disorders (e.g., Kaufmann et al., 2013).

But these disagreements may rest on a false dichotomy. There is growing acceptance that domain-specific andgeneral skills play essential roles in mathematics abilities and disabilities (as reviewed by Hohol, Cipora, Willmes, & Nuerk, 2017), that their relative contribution to mathematics difficulties is subject to developmental andindividual differences, and that these variations may underlie distinct subtypes (e.g., Geary, 1993, 2011). Thus, to understand SLD-math, it is essential to recognize that its heterogeneity is a defining feature (as reviewed by Kaufmann et al., 2013).

Given this heterogeneity, it is not surprising that terms like mathematics learning disabilities, mathematics difficulties (MD), and arithmetic disorderhave all been used in basic and clinical studies of SLD-math since the late 1980s, sometimes with different referents or foci (reviewed by Lewis & Fisher, 2016; Murphy, Mazzocco, Hanich, & Early, 2007). For example, some researchers view MLD as synonymous with dyscalculia, a term specifying impaired numerical processing (e.g., Butterworth,

2005; Mazzocco, Feigenson, & Halberda, 2011); and differentiate it from indirect, domain-general etiologies of mathematics difficulties evident in children with per - sistently low (but not deficient) achievement in mathematics (e.g., Geary, Hoard, Byrd-Craven, Nugent, & Numtee, 2007; Murphy et al., 2007). Other researchers interpret both terms (i.e., those focused on mathematics disabilities or difficulties) as domain-general mathematics difficulties that contrast with the domain-specific dyscalculia (Rubinsten & Henik, 2009). Some researchers explicitly focus on all students with mathematics difficulties rather than differentiating between the terms disabilityversus difficulty, in view of the need for instructional support or intervention shared by all such students (e.g., Jordan, Hanich, & Kaplan, 2003). Each of these approaches has empirical and practical merits.

It is useful to be aware of this variation in terminology when evaluating perceived inconsistencies in the SLD-math literature, and that inconsistent terminology does not implicate disagreements about essential constructs. The distinctions researchers draw regarding SLD-math-related constructsof a disability vs. other difficulties mask a general agreement that SLD-mathematics is only one of many factors that account for the challenges many children face with mathematics. Additional obstacles to learning mathematics may include inadequate learning opportunities, low mathematics motivation, or low mathematics self-efficacy (e.g., Tosto, Asbury, Mazzocco, Petrill, & Kovas, 2016). Combined, these sources account for high incidence rates for mathematics underachievement based on nationally representative cohorts (e.g., up to 50%, NAEP, 2015). In contrast, the reported prevalence of SLD-math is approximately 6 to 13% of all school age children (e.g., Barbaresi, Katusic, Colligan, Weaver, & Jacobsen, 2005; Shalev, 2007).

Collectively, reported manifestations of SLD-math include slowed, deficient, and/or alternative numerical, spatial, or logical-reasoning processing skills, relative to peers without SLD-math. For example, individuals with SLD-math are more likely than their peers to have slow or inaccurate processing of non-symbolic number (but see the Current debates section of this chapter), seen in less accurate discrimination between large quantities (e.g., choosing the more numerous of two sets). Since intact non-symbolic number sense skills typically emerge in early childhood and are, by definition, independent of symbols, these intuitive rather than learned abilities may be among the earliest indicators of SLD-math. Although non- symbolic number skills may be refined over time (Halberda, Ly, Wilmer, Naiman,

& Germine, 2012), the imprecision of these skills may persist through adolescence and adulthood among persons with dyscalculia (Piazza et al., 2010). Another early indicator of SLD-math is slower or less accurate processing of symbolicnumbers (such as number words such as “five” and their corresponding Arabic numerals, such as

“5”), manifested as delayed development in forming immediate and reliable associations between these symbols with their corresponding quantity, or in access to these associations (De Smedt & Gilmore, 2011), and resulting in life-long diminished automaticity in comparing numbers of different magnitudes or simple arithmetic (e.g., Mazzocco, Devlin, & McKenney, 2008; Price, Mazzocco, & Ansari,

2013). Individuals with SLD-math are also more likely to have difficulty processing the relation between numbers, such as seen in delayed or deficient performance on ordinality (Rubinsten & Sury, 2011) or number line estimation tasks that require determining approximately where on a physical number line a specific digit should be placed (e.g., Geary, Hoard, Nugent, & Byrd-Craven, 2008). Difficulties with more complex numerical properties such as ratios and proportions suggest misconceptions and a delay in the typical progression in proportional reasoning, at least for some children with SLD-math (Mazzocco et al., 2013). Considered together, these numerical processing differences may contribute to delayed or life- long challenges in arithmetic fluency (e.g., Geary, Hoard, Nugent, & Bailey, 2012;

Mazzocco et al., 2008; Tolar, Fuchs, Fletcher, Fuchs, & Hamlett, 2016) manifested as relying on less efficient strategies and thus requiring significantly more time to successfully complete even simple arithmetic calculations.

Beyond basic numerical skills, individuals with SLD-math show less flexibility in word problem solving (Fuchs et al., 2006) and in logical–reasoning abilities in general (Morsanyi, Devine, Nobes, & Szücs, 2013). As discussed later in this chapter, these reasoning abilities may be viewed as either domain-specific or domain-general. Finally, children and adults with SLD-math are also more likely than their peers to report experiencing mathematics anxiety, although mathematics anxiety is also reported in typically achieving and mathematically precocious youths (e.g., Tsui & Mazzocco, 2007) and is likely to have bidirectional relations with mathematics achieve ment (Carey, Hill, Devine, & Szücs, 2016) that interact with mathematics motivation (Wang et al., 2015). In this chapter, we focus on the relation between these cognitive manifestations of SLD-math and the roles of Working Memory and other components of Executive Function skills.

Working Memory (WM) and related Executive Function (EF) deficits

Working Memory involves the active mental storage and manipulation of infor - mation subject to capacity constraints. It is, therefore, highly relevant to mathematics calculations and problem solving; as a component of information processing and long-term memory development, it also supports learning mathematics. It is not surprising, therefore, that poor Working Memory is a primary cognitive signature of SLD-math (e.g., Bull, Johnston, & Roy, 1999). A diagnosis of SLD-math is not, however, definitive evidence of an individual’s Working Memory function or capacity. This is because Working Memory and mathematics are independent constructs with bidirectional relations subject to developmental and individual differences that vary further across mathematical skills or tasks (e.g., Alloway &

Passolunghi, 2011).

Here we briefly consider the role of Working Memory in the development of mathematical thinking, and then in SLD-math. Since much of the research on SLD-math is based on Baddeley & Hitch’s model (1974; but see Berch, 2008), we organize our discussion on the three following primary components of this model:

the phonological loop, visuospatial sketchpad, and central executive. Briefly, the phonological loop maintains verbal or auditory information in Working Memory, the visuospatial sketchpad maintains visual or spatial information in Working Memory, and the central executive coordinates both components by focusing attention on relevant aspects of a task and interfacing with long-term memory (Baddeley, 2012). The functions of the central executive include inhibitory control (hereafter, inhibition) and cognitive flexibility, skills that some researchers conceptualize as Executive Function skills (Diamond, 2013). Although we refer to all three components separately, we acknowledge the challenge inherent in differentiating between them, especially in early childhood.

Working Memory and mathematics

Geary (2011) and others have found that each of these components plays a unique role in mathematical thinking in typically achieving individuals. For instance, the phonological loop (verbal Working Memory) supports basic processes that involve articulating numbers. In early childhood, verbal Working Memory supports learning the verbal number sequence and, later, it supports fact retrieval. These skills are not simply the development of vocabulary, because they are foundational for number concepts. Non-numerical verbal Working Memory tasks predict number skills performance and growth. For example, preschoolers’ listening recall is correlated with performance on quantity comparison and combination (Purpura, Schmitt, & Ganley, 2017). Verbal Working Memory has also been correlated with early math fluency and word problem solving in children ages 5 to 7 years (Martin, Cirino, Sharp, & Barnes, 2014), and with exact arithmetic problem solving in adults (Kalaman & Lefevre, 2007).

This relation between mathematics and verbal Working Memory is not exclusive. The visuospatial sketchpad(or visuospatial Working Memory) also supports mathematics skills, as suggested by the correlations with growth on early numeracy from kindergarten to first grade (Toll, Kroesbergen, & Van Luit, 2016), the development of number line concepts, translating word problems into mathematical equations, approximate number comparison, and mapping numbers onto space. The central executivesupports inhibiting irrelevant associations from entering Working Memory (such as during fact retrieval, switching operations during long division, or solving word problems), and has been correlated with more sophisticated strategies for solving addition problems (e.g., min counting, decomposition; Geary, Hoard, & Nugent, 2012) and general mathematics ability (e.g., Mazzocco & Kover, 2007). Adults’ ability to update information (verbal or spatial) in Working Mem- ory is important for solving mental addition problems (as shown using dual- task experimental methods; Hubber, Gilmore, & Cragg, 2014). This ability predicts adults’ concurrent performance on multi-digit mental multiplication (Han, Yang, Lin, & Yen, 2016), suggesting the lifelong relevance of central executive in solving complex arithmetic, especially when it involves regrouping (referred to as “carrying”

in earlier generations of mathematics instruction).

Working Memory and SLD-math

Given empirically-supported associations between WM and mathematics per se, it is not surprising that individuals with SLD-math manifest WM difficulties. But does that association implicate causal pathway(s) to SLD-math? The answer to this question is unclear. Collectively, persons with SLD-math show deficits in all three components of Working Memory (e.g., Geary et al., 2007), at early and later grades.

Their deficits may be severe, on average at the 16th percentile (Geary et al., 2007;

Swanson, 1993). Using the Working Memory Test Battery for Children (WMTB- C; Pickering & Gathercole, 2001), Geary, Hoard, Nugent, & Bailey (2012) showed that while the Working Memory span of all children in their study increased from Grade 1 to 5, it remained significantly lower among children with SLD-math (relative to typically achieving or even low achieving students) at both grade levels, and for all three components of Working Memory (except a non-significant difference in the visuospatial sketchpad at Grade 5). In that study, the central executive mediated growthin mathematics achievement over time, supporting the notion that WM supports both doingand learningmathematics.

Reports of poor verbal Working Memory in children and adults with SLD- math are often based on tasks such as forward recall of words or sentences, or backward recall of digits. (In some studies, backward recall tasks are used as measures of the central executive.) This poses some challenges for inferences concerning causal pathways, specifically, in determining if underlying difficulties are based in WM or numerical processing. As aforementioned, verbal Working Memory supports counting (i.e., the verbal number sequence) and, later, fact retrieval; so, its impairment should increase early counting errors. Kindergarteners at risk for SLD-math do have counting difficulties, manifested as being less likely than their peers to detect an examiner’s deliberate counting errors, especially when errors occur early versus later during the verbal counting sequence (Geary et al., 2007) (which poses greater verbal Working Memory demands). Individual differences in verbal Working Memory in children with SLD-math (as measured by word recall backward) predict growth in early numeracy during kindergarten (as measured by the Early Numeracy Test—Revised, which includes items such as number comparison, counting, seriation, and estimation) (Toll & Van Luit, 2014) such that better verbal Working Memory is associated with more growth in early numeracy. Evidence in a study of fourth graders with SLD-math impli- cates smaller digit spans than typically achieving peers (Passolunghi & Siegel, 2001;

2004), which may impair arithmetic performance. Likewise, among adults with SLD-math, verbal Working Memory accounts for more variance on arithmetic performance than does visuospatial Working Memory (Wilson & Swanson, 2001).

(This is in contrast with stronger visuospatial versus verbal Working Memory associations with arithmetic among young typically achieving students (e.g., Alloway & Passolunghi, 2011).)

Studies of the association between visuospatial Working Memoryand mathematics typically rely on measures such as maze memory and location recall (e.g., matrix

recall or the Corsi block task), and numerous studies show deficits on such tasks among children with vs. without SLD-math (Geary et al., 2007; Kyttälä, Aunio, and Hautamäki, 2010). For instance, performance on the Odd One Out task (which requires pointing to the non-matching item within a set of shapes and then recalling its location) measured at the end of kindergarten retrospectively predicted early kindergarten performance on the Early Numeracy Test—Revised an average of 15 months earlier (Toll & Van Luit, 2014), and performance on location recall (on the Corsi block task) and maze memory (when duplicating a previously shown maze solution) was associated with addition and number line estimation perform - ance during first grade (Geary et al., 2007).

Different measures are used to assess skills of the central executive. These tasks typically require that participants selectively attend to and remember relevant details while inhibiting pre-potent responses (e.g., identify the direction of an arrow amidst other arrows of varying directions), transform or “update” information based on the task instructions (such as backward digit/letter/block recall), or flexibly switch response sets (such as judging the veracity of sentences, then recalling the last word of each of the previous two or three sentences). Whereas verbal and visuospatial Working Memory each correlate with and predict specific mathematics domains, performance on the central executive—which is also significantly less efficient among children with versus without SLD-math (Geary et al., 2007)—is correlated with overall mathematics achievement, at most grade levels (e.g., Geary et al., 2007;

Mazzocco & Kover, 2007). Among children with SLD-math, performance on central executive tasks predicts performance on counting error detection and addition fact retrieval errors in kindergarten (Geary et al., 2007), number line estimation in second grade (Geary et al., 2008), growth in problem solving in third grade (Swanson, Jerman, & Zheng, 2008), and calculation fluency in third to sixth grades (Mabbott & Bisanz, 2008), suggesting the involvement of the central executive in overall as opposed to specific mathematics domains.

Three models of SLD-math outcomes

Although mathematics and Working Memory skills each function along their respective continua, here we illustrate how these skills broadly conform to one of four alternative combinations (Figure 8.1), three of which may manifest as SLD-math. Generally speaking, intact Working Memory paired with intact mathe - matical abilities may synergistically support age-appropriate if not optimal or even precocious mathematical abilities, problem solving processes, and achievement (Figure 8.1d; e.g., Hoard, Geary, Byrd-Craven, & Nugent, 2008). When paired with weak or aberrant mathematics abilities, intact Working Memory may support compensatory routes to successful problem solving or mathematics achievement, in which case SLD-math may be masked despite deficient numerical processing (e.g., Murphy & Mazzocco, 2008), at least for some children (Figure 8.1b). An alternative outcome of this latter combination (also corresponding to Figure 8.1b) is, however, that weak mathematics skills (such as symbolic number automaticity) may increase

the Working Memory demands of mathematics tasks to the extent that otherwise intact Working Memory skills are insufficient. Likewise, Working Memory limitations may interfere with otherwise intact mathematical abilities when tasks inherently require greater Working Memory demands (Figure 8.1c), such as exact vs. approximate arithmetic (Kalaman & Lefevre, 2007)—consistent with notions of domain-general SLD-math. It is possible that the manifestation of persons whose profiles conform to either of the last two scenarios may appear quite similar (such as marked by effortful, inefficient, and error prone calculations), despite the poss - ibility that poor Working Memory orimpaired basic mathematics abilities drive the SLD-math profiles across the two situations. Finally, if Working Memory limitations accompany poor mathematics skills, this combination may further impede the mathematics weakness (Figure 8.1a), potentially leading to severe SLD-math.

These different etiologies may result in shared behavioral SLD-math profiles (Kroesbergen & van Dijk, 2015; Rubinsten & Henik, 2009) or, collectively, SLD- math subtypes. Across all four combinations of mathematics and Working Memory ability depicted in the figure, variations occur at the level of precise mathematics or Working Memory skills. Generally, the relations between mathematics and Working Memory may reflect direct or mediating roles of verbal Working Memory, visual–spatial Working Memory, inhibitory control, or cognitive flexi - bility. Importantly, the mathematics achievement outcomes themselves contribute to the efficiency of mathematics learning or problem solving (as indicated by the bidirectional arrows in Figure 8.1), and to the well-established finding of widening achievement gaps between students with versus without learning disability.

As discussed in this section, there is evidence that each of these WM components is implicated in SLD-math. Moreover, the relations are dynamically influenced by additional factors such as math anxiety, which overtaxes Working Memory, or language of instruction for dual language learners. Theoretically, the profiles for children with SLD-math are more likely to fall within panels a, b, or c, depicted in Figure 8.1, because difficulties in mathematical skills (such as number processing) or Working Memory may manifest as SLD-math (e.g., Kroesbergen & van Dijk, 2015). Variation within and across each of these panels is likely, based on SLD- math heterogeneity. As one component of this variation, only some children whose profile conforms to either panel b or c will manifest SLD-math, but it is likely that most children whose profile aligns with panel a will manifest SLD-math. Children whose profile aligns with panel d are less likely to manifest SLD-math, but may experience math difficulties associated with the many other influences on mathematics achievement such as specific language impairment, low mathematics motivation, or inadequate instruction, but these would not be considered a specific learning disability in mathematics.

SLD-math, Working Memory, and development

As aforementioned, other factors may explain some of the variation in findings on Working Memory and SLD-math. Here we posit that another critical factor to