Working memory in children’s math learning and its disruption in dyscalculia

Abstract

Working memory (WM) plays an essential role in children’s mathematical learning. WM influences both the early foundational phases of number knowledge acquisition and subsequent maturation of problem solving skills. The role of individual WM components in mathematical cognition depends not only on problem complexity but also on individual differences in mathematical abilities. Furthermore, the contributions of individual WM components change dynamically over development with visuospatial processes playing an increasingly important role in learning and enhancing mathematical proficiency. Convergent findings from neuroimaging studies are now providing fundamental insights into the link between WM and mathematical cognition, and the mechanisms by which poor WM contributes to learning disabilities. Evidence to date suggests that visuospatial WM is a specific source of vulnerability in children with mathematical learning disabilities and needs to be considered as a key component in cognitive, neurobiological, and developmental models of typical and atypical mathematical skill acquisition.

Introduction

Many aspects of children’s academic skill acquisition require access to working memory (WM) resources [1–3]. In no academic domain is this truer than in mathematical cognition where problem solving abilities depend on the capacity to efficiently manipulate quantity representations in WM [4^••,5]. Over three decades of behavioral research have established that numerical problem solving tasks place strong demands on the active maintenance and manipulation of task-relevant information in WM [5,6]. Cross-sectional and longitudinal studies are providing new insights into the role of individual WM components at different stages of mathematical skill acquisition. Deficits in WM in children with dyscalculia contribute to weaknesses in the representation of quantity information, as well as the ability to manipulate this information during numerical problem solving [7^••]. Convergent findings from neuroimaging studies provide fundamental insights into the link between WM and mathematical cognition, and the mechanisms by which poor WM contributes to dyscalculia. A common neural locus of deficits in visuospatial quantity representations and visuospatial WM likely contributes to both numerical magnitude judgment and arithmetic problem solving deficits in children with dyscalculia.

Working memory in children’s mathematical cognition and learning

The particular emphasis on WM in developmental studies has its origins in children’s immature problem solving abilities, which require them to break down numerical problems into more basic components. The use of such strategies necessitates greater reliance on WM systems for problem solving in children. For example, children rely more on counting strategies during simple arithmetic problem solving and need to access multiple WM components including short-term storage and rule-based manipulation and updating of the contents of stored information [8]. With increased proficiency and a switch to fact retrieval strategies there is less demand and need for WM resources [9,10]. The link between WM and children’s mathematical cognition and learning has largely been based on Baddeley and Hitch’s influential multicomponent model [11,12]. Briefly, this model includes a central executive component, responsible for high-level control, monitoring, and task switching, along with two subordinate, modality-dependent components, important for short-term storage of verbal and visuospatial information, respectively [11]. Crucially, all three components of WM can be distinguished from an early age [13].

Developmental studies using the Baddeley and Hitch model have predominantly reported a strong link between the central executive and visuospatial WM components and math abilities [9,14–17] (Simmons et al., 2012). The effects of phonological WM have generally been much weaker, and are typically more evident during very early stages (ages 4–5), when phonological representations for numbers are still weak and word-based problem solving places greater demands on reading comprehension. In a detailed cognitive analysis of the factors that contribute to mathematical abilities, Szucs and colleagues found strong relations between visuospatial WM measures, but not phonological WM measures, and mathematical abilities in a large well-characterized group of 9 year-old children [18].

Longitudinal studies have expanded on these findings and shown that the central executive component predicts performance on single-digit addition tasks in grades 1 to 3 as well as faster transitions from simple (e.g., counting) to sophisticated (e.g., decomposition) solution strategies [16]. Similarly, in a large sample of 673 children, Lee and Bull found that WM updating capacity in kindergarten predicted growth rate of math abilities (numerical operations) in subsequent grades [19].

It is important to note that the role of individual WM components depends not only on task complexity but also on children’s developmental stage. The changing role of WM components can be detected even in a 1-year time-window between ages 8 and 9. Meyer and colleagues found that while the central executive and phonological components of WM predicted mathematical abilities in second graders, it was the visuospatial component that predicted abilities in third graders [17]. Similarly, Li and Geary reported individual differences in the growth rate of visuospatial WM during childhood. Notably, they found that these differences became increasingly important for learning over time [20]. Nuerk and colleagues examined longitudinal changes associated with multiplication fact retrieval [21^• ]. They found that multiplication task performance was correlated with verbal WM in third graders but with visuospatial WM in grade four. Taken together, these patterns of relationships suggest that the contributions of individual WM processes change dynamically over development with visuospatial WM processes playing an increasingly important role in enhancing mathematical proficiency.

Working memory and fronto-parietal systems associated with children’s mathematical cognition

Functional neuroimaging research has revealed significant overlap in multiple parietal and prefrontal cortex regions involved in WM and numerical problem solving [22–24]. Overlapping patterns of activation have most prominently been detected in the supramarginal gyrus and intraparietal sulcus in the posterior parietal cortex, the premotor cortex, and the ventral and dorsal aspects of the lateral prefrontal cortex (Figure 1). It is important to note, however, that the common patterns of fronto-parietal cortex engagement during WM and numerical problem solving cannot be conflated with shared neural mechanisms, and research on this topic has used both correlational and causative analyses to gain a deeper understanding of the shared neural mechanisms [25].

Figure 1.

Common patterns of fronto-parietal network activations elicited by numerical, arithmetic, working memory and visuospatial processing tasks. Results from meta-analysis conducted using Neurosynth (www.neurosynth.org) with the corresponding search terms.

Neuroimaging studies in typical and atypical development are helping to provide a more mechanistic understanding of the link between individual WM components and brain responses associated with mathematical problem solving. The involvement of WM in mathematical cognition had initially been surmised based on overlapping responses in posterior parietal cortex and prefrontal cortex in the two domains [26–29]. Studies of typical development provided initial evidence for the changing role of WM with age. For example, Rivera and colleagues found that relative to adolescents and young adults, children engage the posterior parietal cortex less, and the prefrontal cortex more, when solving arithmetic problems [29], likely reflecting the increased role of visuospatial WM processes, and concurrent decrease in demands on cognitive control with age. Other studies have more directly addressed the link between WM abilities and numerical problem solving skills. Dumontheil and Klingberg [30] found that activity in the intraparietal sulcus during a visuospatial WM task predicted arithmetic performance two years later in a sample of 6- to 16-year-old children and adolescents. This finding further reinforces the link between visuospatial WM and numerical problem solving and suggests a common underlying process in the intraparietal sulcus subdivision of the posterior parietal cortex.

More detailed analyses of the neural correlates of individual components of WM have provided evidence for the fractionation of neurofunctional systems associated with distinct WM components during numerical problem solving [23^••]. Analysis of the relation between the central executive, phonological and visuospatial components of WM and brain activation during an arithmetic verification task in a large (N = 74) group of 7 to 9-year-old children revealed that visuospatial WM is the strongest predictor of mathematical ability in children in this age group and is associated with increased arithmetic complexity-related responses in left dorsolateral and right ventrolateral prefrontal cortices as well as in the bilateral intra-parietal sulcus and supramarginal gyrus in posterior parietal cortex (Figure 2). This neurobiological finding confirms a pivotal role of visuospatial WM during arithmetic problem-solving in primary-school children.

Figure 2.

Functional dissociations and overlap between brain areas associated with each of the three components of working memory in 7 to 9-year-old children (N = 74). The neural correlates of the central executive (CE), phonological (PL) and visuo-spatial (VS) components of working memory were examined by contrasting brain responses to two different types of addition problems that differed in complexity. Overlap between the CE and VS components was observed only in left supramarginal gyrus (SMG); overlap between CE and PL components was observed only in the left intra-parietal sulcus (IPS); no overlap was observed between VS and PL components. Negative correlation between activity and PL ability is not depicted. No overlap for VS and PL (magenta) was observed. Bottom panel: coronal slices depict regions of interest selected as overlap in correlations of activity and individual working memory components. Scatter plots are based on functional clusters identified using whole-brain regression analysis, and are provided for the purpose of visualization. L, left.

Source: [23^••].

Metcalfe and colleagues also found that visuospatial WM and the central executive component were associated with largely distinct patterns of brain responses during arithmetic problem-solving, and overlap was only observed in the ventral aspects of the left supramarginal gyrus in the posterior parietal cortex, suggesting that this region is an important locus for the integration of information in WM during numerical problem solving [29,31–35].

Finally, there is also evidence that immature prefrontal control systems associated with central executive functions may contribute to weaker math skills in children. Supekar and colleagues used dynamic causal analysis to probe interactions between the prefrontal and parietal cortices in children and adults [36]. They found that despite higher levels of activation, the strength of causal regulatory influences from the fronto-insular control network to the posterior parietal cortex was significantly weaker in children and weak signaling mechanisms contributed to lower levels of performance in children, compared to adults. More broadly, immature prefrontal control systems may contribute to weaknesses in the ability to inhibit irrelevant information such as arithmetic facts or operations during numerical problem solving [4^••,37,38,39^•].

Working memory disruption in children with dyscalculia

Studies of children with dyscalculia provide a unique window into the role of WM in numerical cognition. Dyscalculia is a specific deficit in arithmetic ability in the presence of preserved intellectual and verbal abilities [40–43]. Children with dyscalculia show poor performance on a broad range of numerical tasks, including magnitude judgment [44–47] and enumeration [4^••,48,49]. They also lag behind their typically developing peers in basic arithmetic problem solving skills [4^••,50]. There is growing evidence that deficits in WM can contribute to multiple aspects of dyscalculia, encompassing not only complex arithmetic problem solving but also basic quantity representation [4^••,51^••].

Multiple experimental paradigms across extended periods of early skill acquisition in the domains of number sense and arithmetic have highlighted the involvement of visuospatial WM in developmental models of dyscalculia. At a more fundamental level, deficits in visuospatial WM can influence the ability to engage and manipulate representations of magnitude on a mental number line and estimate non-symbolic quantity. Furthermore, other areas of difficulty that define the profile of children with dyscalculia, such as counting and subitizing, may have their roots in visuospatial WM deficits [4^••,49]. Convergent with these observations, several lines of evidence point to disruptions in visuospatial WM in children with dyscalculia. Even when they are matched with typically developing children on general intelligence, reading and other cognitive measures, children with dyscalculia demonstrate lower visuospatial WM despite preserved phonological and central executive WM abilities [52^•]. Furthermore, Swanson et al. [53] found deficits in visuospatial, but not in other WM components, differentiating children with dyscalculia from children with reading difficulties. Consistent with these findings, Rotzer et al. [54] found that children with dyscalculia had lower scores than typically developing children on a Corsi Block-Tapping Test. Thus, visuospatial WM deficits appear to be a specific source of mathematical difficulty in dyscalculia.

Visuospatial working memory and fronto-parietal impairments in children with dyscalculia

The importance of visuospatial WM and associated fronto-parietal processing during arithmetic problem-solving is further highlighted by neuroimaging studies in children with dyscalculia. Rotzer et al. [54] found that compared to typically developing children, children with low math abilities had lower visuospatial abilities and lower activity levels in the right anterior intraparietal sulcus, inferior frontal gyrus, and insular cortex during a visuospatial WM task. Ashkenazi and colleagues [52^•] identified impaired WM components in children with dyscalculia and then examined their role in modulating brain responses to numerical problem solving (Figure 3). Children with dyscalculia had specific deficits in visuospatial WM in addition to deficits in arithmetic task performance. Crucially, activations in intraparietal sulcus, and dorsolateral and ventrolateral prefrontal cortices were positively correlated with visuospatial WM ability in typically developing children, but no such relation was seen in children with dyscalculia. This result suggests that children with dyscalculia fail to appropriately exploit visuospatial WM resources during problem solving. While still preliminary, extant findings point to the intraparietal sulcus as a common locus of visuospatial WM deficits and arithmetic problem solving deficits in children with dyscalculia. On the basis of these and other related findings, we have suggested that parietal cortex mechanisms for storing and manipulating quantity representations are impaired in dyscalculia [23^••,55^••,56^••].

Figure 3.

Children with dyscalculia do not use visuospatial working memory resources appropriately during arithmetic problem solving. (A) Brain areas in which activity during arithmetic problem solving was significantly correlated with visuo-spatial working memory abilities in the typically developing (TD) and developmental dyscalculia (DD) groups. (a) In the TD group, Block Recall, a measure of visuo-spatial working memory, was correlated with activity in bilateral middle frontal gyrus (MFG), left inferior frontal gyrus (IFG), right anterior insula (AIC), anterior, middle and posterior cingulate cortex and precuneus, bilateral intraparietal sulcus (IPS), right fusiform gyrus, left temporal pole and the cerebellum. No negative correlations were observed in the TD group. (b) In the DD group, Block Recall was negatively correlated with activity in left postcentral gyrus. No positive correlations were observed in the DD group. (B) Fronto-parietal cortical areas where the relation between activity during arithmetic problem solving and visuo-spatial working memory abilities differed significantly between the TD and DD groups. (a) Prefrontal cortex. In TD children, left inferior frontal gyrus (IFG) and right middle frontal gyrus (MFG) showed significant positive correlation between activation during Complex addition problems and Block Recall, a measure of visuo-spatial working memory. In contrast, correlations were nonsignificant in children with DD. (b) Parietal cortex. In TD children, the left intra-parietal sulcus (IPS), and right supramarginal gyrus (SMG) showed significant positive correlation between activation during arithmetic problem solving and Block Recall. In the DD group there were no significant correlations (*P < .05, **P < .01).

Source: [52^•].

Conclusion

WM plays an integral role in children’s math learning and development of problem solving abilities. The role of individual WM components in mathematical cognition is learning-stage dependent, both in terms of proficiency and age. Behavioral and neuroimaging studies are converging on the idea that the contributions of individual WM processes and their neural substrates change dynamically over development, with visuospatial WM processes playing an increasingly important role in learning and enhancing mathematical proficiency. Although the role of the visuospatial component of WM has often been considered secondary to that of the central executive component in typical arithmetic skill acquisition, and has generally been neglected in prior accounts of dyscalculia and math learning disabilities, recent studies suggest that visuospatial WM is a critical component for successfully building quantity representations and efficiently manipulating them during problem solving. These processes are important at all stages of learning and skill acquisition, and are significantly disrupted in children with dyscalculia.

Phonological WM appears most prominently in the earliest stages of learning the verbal mapping of quantity representations and later gives way to visuospatial WM processes important for the representation and manipulation of quantity information in short-term memory. The central executive system helps scaffold the early stages of learning by providing support for building new semantic representations. The central executive component is also required at subsequent stages for more complex problem solving procedures, including the active maintenance of intermediate results and rule-based problem solving.

Within the neurocognitive framework highlighted in this review, the engagement of the intraparietal sulcus and supramarginal gyrus in the posterior parietal cortex, and the ventral and dorsal aspects of the lateral prefrontal cortex changes dynamically with problem complexity and developmental stage. Findings to date suggest that the intraparietal sulcus plays an essential role not only in quantity representations but also in maintaining quantity-related information in short-term WM. Rule-based manipulation of these representations in WM is in turn supported by multiple prefrontal cortical areas, with the supramarginal gyrus as a key locus for integrating frontal control systems with quantity representations supported by the intraparietal sulcus. Together, they provide multiple functional circuits that support essential WM processes in children’s mathematical cognition.

A challenging question for future research is to understand how WM processes are used dynamically to support different types of mathematical learning and how they change with different stages of development. Addressing this question will require developing appropriate computational models of dynamic causal interactions between brain regions, analyzing different stages of information processing, and utilizing more appropriate experimental designs that involve the controlled manipulation of quantity representations in WM [57]. Finally, training studies also have the potential to inform causal links between WM processing and mathematical learning [55^••].