Cognitive subtypes of mathematics learning difficulties in primary education

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Highlights

  • MLD is a heterogeneous disorder.

  • The seven number-specific processing skills are reducible to four factors.

  • Six clusters were found with unique patterns of cognitive strength and weaknesses.

  • The access deficit subtype is characterized by the weakest arithmetic performance.

  • None of the subtypes was particular to a specific age group or gender.

Abstract

It has been asserted that children with mathematics learning difficulties (MLD) constitute a heterogeneous group. To date, most researchers have investigated differences between predefined MLD subtypes. Specifically MLD children are frequently categorized a priori into groups based on the presence or absence of an additional disorder, such as a reading disorder, to examine cognitive differences between MLD subtypes. In the current study 226 third to six grade children (M age = 131 months) with MLD completed a selection of number specific and general cognitive measures. The data driven approach was used to identify the extent to which performance of the MLD children on these measures could be clustered into distinct groups. In particular, after conducting a factor analysis, a 200 times repeated K-means clustering approach was used to classify the children's performance. Results revealed six distinguishable clusters of MLD children, specifically (a) a weak mental number line group, (b) weak ANS group, (c) spatial difficulties group, (d) access deficit group, (e) no numerical cognitive deficit group and (f) a garden-variety group. These findings imply that different cognitive subtypes of MLD exist and that these can be derived from data-driven approaches to classification. These findings strengthen the notion that MLD is a heterogeneous disorder, which has implications for the way in which intervention may be tailored for individuals within the different subtypes.

Introduction

Today, children are required to make decisions based on simple number and quantity information every day (Dowker, 2005). Yet approximately 6% of the school-aged children do not have sufficient mathematics skills, despite being of normal intelligence (Desoete et al., 2004, Gross-Tsur et al., 1996). Still, higher prevalence rates have even been reported when using different methods or more lenient criteria (Barbaresi et al., 2005, Mazzocco and Myers, 2003).

The operationalization and cut-off scores used to define mathematics learning difficulties (MLD) have varied substantially (Moeller, Fischer, Cress, & Nuerk, 2012). Note that as Mazzocco, Feigenson, and Halberda (2011) did, we consider MLD and dyscalculia to be synonymous in this article. We prefer to use MLD in the paper, given that we did not measure mathematics performance multiple times and therefore cannot speak to the stability of the mathematics deficit. At present, most researchers agree that children with MLD experience severe difficulties in encoding arithmetic facts into long-term memory (e.g., Geary, 1993, Rousselle and Noël, 2007). Specifically, while typically developing children shift from the use of effortful procedures to solve arithmetic problems, such as finger counting or breaking problems down into multiple steps, to the fast retrieval of facts from long-term memory, children with MLD persist in the use of non-retrieval strategies to solve arithmetic problems.

Arithmetic is a complex ability composed of a variety of skills which seem to rely on different cognitive processes (Dowker, 2005). Accordingly it has been proposed that MLD is likely to be a heterogeneous disorder (Geary, 2010, Kaufmann and Nuerk, 2005, Rubinsten and Henik, 2009). A data-driven study by Von Aster (2000) supports this proposition. Specifically, Von Aster assessed the basic number processing and calculation skills of 93 primary school children who performed poorly in mathematics. Employing a clustering approach, Von Aster (2000) differentiated a poor performance cluster and three different dyscalculia clusters. The latter clusters consisted of children who scored more than one standard deviation below the mean test score of the normal population on at least one subtest. Children in the Arabic subtype exhibited deficits on a number transcoding task and a number comparison task. The cognitive profile of the verbal subtype was characterized by severe problems on a counting task and weak subtraction skills. The children in the pervasive subtype displayed impairments on almost all measures.

In most other MLD classification studies (Jordan et al., 2003, Rourke, 1993, Shalev et al., 1997), researchers have applied a top-down, a priori approach. They examined the cognitive profiles of MLD subtypes which were specified beforehand based on a priori assumptions derived from prior studies and theories. Consequently they limit the number of subtypes in advance, which could have led on the one hand to a failure to identify all subtypes and on the other hand to the aggregation of two MLD categories with distinct underlying features into one predefined subtype. Moreover, only few studies focused on number-specific cognitive processes (e.g., counting), despite that empirical research has underlined the importance of including these processes in MLD studies (Price & Ansari, 2012). Therefore the current study implemented a data-driven approach, administering a variety of basic number-specific and general cognitive processing tasks to distinguish cognitive subtypes of MLD in primary education.

Knowledge of distinguishable subtypes is crucial to the development of custom-built interventions and the refinement of MLD definitions (Mazzocco and Myers, 2003, Wilson and Dehaene, 2007). Better understanding the nature of MLD is a prerequisite for the formulation of definitions detailing the specific cognitive mechanisms which are a positive indicator of MLD, instead of stating what a disorder is not (e.g. the IQ-discrepancy criteria) (Kavale and Forness, 2000, Stuebing et al., 2002).

Besides being based on a data-driven classification approach, definitions of learning difficulties subtypes should describe the cognitive processes impaired (King et al., 2007, Skinner, 1981). Currently several cognitive processes have been frequently associated with MLD, but no comprehensive picture has emerged. The seemingly incompatible findings have led to the formulation of diverging theories (Andersson and Östergren, 2012, Szücs et al., 2013a).

Initially, researchers focused on the relationship between MLD and general cognitive processes, e.g., working memory, but most of the examined general cognitive processes were not found to be related to MLD (Price & Ansari, 2012). Exceptions were children's working memory and intelligence (IQ), which were frequently, though not consistently, reported to be associated with MLD (e.g., Andersson and Lyxell, 2007, D’Amico and Guarnera, 2005).

To automatize mental calculations, humans are required to keep the problem in verbal working memory while they compute the answer in order to build long-term associations (Geary, 1993). Furthermore, researchers hypothesize that numbers are spatially coded (e.g., Dehaene, 1992) and therefore the processing of numbers is thought to be supported by visuo-spatial working memory skills. Numerous empirical studies comparing the working memory capacities of children with and without MLD reported deficits in visuo-spatial working memory, but not verbal working memory among children with MLD (D’Amico and Guarnera, 2005, McLean and Hitch, 1999, Passolunghi and Mammarella, 2010, Szücs et al., 2013a). Nonetheless, other studies did find verbal working memory deficits in MLD children (Andersson and Lyxell, 2007, Geary et al., 2007, Kyttälä et al., 2010, Rosselli et al., 2010). To date, there is no satisfactory explanation which can account for these inconsistent findings. It is possible that deficient working memory capacities do not underlie severe calculation problems in all MLD children (Rousselle & Noël, 2007) and hence, the conflicting findings are attributable to the use of divergent MLD samples across studies. Data-driven classification studies could shed light on these inconsistent patterns of data by examining whether cognitive processing profiles with and without working memory weaknesses can be delineated in a sample of MLD children.

As noted by Geary (2011), children's IQ level should not be ignored when trying to explain MLD. It has been often associated with inter-individual differences in mathematics achievement and growth (e.g., Primi, Ferrão, & Almeida, 2010). Yet, contrasting the mathematics achievement of typically achieving children and children with MLD, Geary (2011) found that after controlling for IQ, the achievement gap disappeared for children having a low IQ, but remained for children having an average IQ. This has two implications, namely that a low IQ might be a subtype specific characteristic and that in some children with MLD, achievement is related to cognitive factors other than IQ.

As a response to the inconsistent findings discussed above, researchers’ attention has increasingly shifted toward identifying basic number-specific cognitive processes that are insufficiently developed in MLD children. Number-specific cognitive processes, as opposed to general cognitive processes, are postulated to support specifically mathematics achievement (Butterworth, 2010). Many different tasks have been used to measure these processing skills. Some of them repeatedly proved to be significantly associated with inter-individual differences in children's mathematics achievement and MLD.

Magnitude comparison tasks have been used to assess children's understanding of non-symbolic (e.g. dots) and symbolic (e.g. 5) magnitudes (Mundy & Gilmore, 2009). A trademark of the non-symbolic and symbolic magnitude comparison task is the numerical distance effect (NDE) (Moyer and Bayer, 1976, Mundy and Gilmore, 2009). This effect refers to the finding that the comparison of magnitudes that are close together (e.g. 6 and 7) is more error prone and slower than the comparison of magnitudes that are relatively far apart (e.g. 5 and 9). The NDE has been hypothesized to index the representational precision of an inborn approximate number system (ANS), which provides a fuzzy representation of numerosities (“the numeric properties of a set of items in the real world”; Fayol & Seron, 2005, p. 3) (Dehaene, 2011). In this representational system, numerical magnitudes are thought to overlap with one another giving rise to the numerical distance effect and an approximate rather than exact representation. Since the signatures of the ANS, such as the distance effect, can be found in both, non-symbolic and symbolic magnitude comparison processing tasks, it has been contended that symbolic and nonsymbolic processing are both underpinned by the inborn ANS, which has been hypothesized to provide the foundations for the development of mathematical skill (Dehaene, 2011, Piazza, 2010). In view of this, researchers hypothesized that MLD in children are attributable to ANS deficits, which lead to non-symbolic and symbolic processing difficulties (Mazzocco et al., 2011, Piazza, 2010). In line with this, Landerl, Bevan, and Butterworth (2004) observed that in a group of 8–9 year olds, MLD children performed significantly weaker than average achievers on multiple number processing tasks, including a symbolic magnitude comparison paradigm and a dot enumeration paradigm. Numerous other studies reporting that MLD children have significantly weaker non-symbolic and symbolic comparison skills than non-impaired children provide additional support for the hypothesis that MLD are the result of an ANS deficit (Landerl et al., 2009, Mussolin et al., 2010). Contrary to such findings other researchers found MLD children to have impaired symbolic comparison skills, but intact non-symbolic comparison abilities (De Smedt and Gilmore, 2011, Iuculano et al., 2008, Rousselle and Noël, 2007). These findings are more in line with the access deficit hypothesis according to which weak arithmetic skills are due to insufficiently developed symbolic magnitude processing skills. Specifically children with MLD exhibit difficulties in connecting symbolic representations of numerical magnitude, such as Arabic numerals, with the non-symbolic quantities that they represent (Rousselle & Noël, 2007). Given the inconsistent results, Kramer and Landerl (2010) advance the notion that both hypotheses might actually explain MLD, but only of a distinct sub-group.

Comparable to the symbolic comparison task, numerical estimation tasks index symbolic magnitude processing skills, capturing children's ability to translate a symbol into a non-symbolic magnitude and vice versa (Siegler & Booth, 2004). A large variety of estimation tasks have been developed (Ebersbach, Luwel, & Verschaffel, 2013), including numerical estimation tasks (Huntley-Fenner, 2001) and number line measures (Siegler & Booth, 2005). The former asks children to repeatedly estimate the numerical symbol (e.g. 7) which best represents a non-symbolic array (e.g. dots) (Siegler & Booth, 2005). Two previous studies administering the numerical estimation tasks in fourth and ninth grade children found the variability in the estimates given by children for a specific non-symbolic target, to be significantly larger for children with MLD than without MLD (Mazzocco et al., 2011, Mejias et al., 2012). Number line tasks require children to determine the spatial position of a number on a presented line (Siegler & Booth, 2005). Geary, Hoard, and Bailey (2012) and Landerl (2013) examined the developmental trajectories of children's number line estimation skills from second through fourth grade. Both studies observed that children with MLD consistently exhibited less accurate estimations than their typically achieving peers.

A skill imperative to the development of adequate symbolic magnitude processing skills is counting, Landerl et al. (2004) found MLD children to have a steeper reaction time slope when enumerating sets consisting of four dots or more (counting range), but not sets of three dots or less (subitizing range). More recent studies could not replicate these results. Instead, these studies observed MLD children's ability and fluency to enumerate small sets (subitizing range) and not large sets (counting range) of dots to differ significantly from typically developing children (Andersson and Östergren, 2012, Landerl, 2013, Schleifer and Landerl, 2011).

To summarize, in the last decade an increasing amount of researchers attempted to identify which number-specific cognitive processing deficits explain MLD. In this context, diverging hypotheses concerning the particular number-specific cognitive processes impaired in MLD children have been formulated (Andersson & Östergren, 2012). However, not one skill has been found to be consistently impaired in MLD children, speaking against the hypothesis that MLD is a result of a single core deficit (Fias, Menon, & Szücs, 2013). The vast amount of discrepant findings in the literature, combined with the fact that most previous studies have not systematically investigated several MLD theories in one large MLD sample, have made it difficult to understand how a range of general and number-specific cognitive processing measures are associated with MLD. This points to the need of a comprehensive and data-driven analysis that considers the possibility of subtypes. Such an approach allows for the bottom-up inquiry into which distinct cognitive processing profiles characterize MLD children and how separate general and number-specific number processing are not only associated with MLD but also with each other (Mazzocco and Räsänen, 2013, Price and Ansari, 2013).

To analyze children's performance on the discussed number-specific cognitive processing tasks, researchers have used a variety of outcome measures capturing different features of the child's performance on a given task (Price, Palmer, Battista, & Ansari, 2012). Roughly, these measures can be categorized as task-specific effect measures or efficiency measures. Task-specific effect measures, such as the NDE, are hypothesized to tap number-specific representational systems (Maloney, Risko, Preston, Ansari, & Fugelsang, 2010). Efficiency measures, such as mean accuracy and mean reaction time (RT), index a broad range of cognitive processes involved in the processing of number magnitudes.

Yet, researchers have questioned the validity and reliability of task-specific effect measures Inglis and Gilmore, 2014, Maloney et al., 2010, Szücs et al., 2013b. In addition, Landerl (2013) found task-specific effect measures to be less stable than efficiency measures across five assessment points from grade 2 through grade 4. Also, Landerl (2013) observed that contrary to efficiency measures, task-specific effects remained constant across assessment periods indicating that task-specific effect measures reach ceiling level early in development. Lastly, number-specific processing tasks generate generally the same efficiency measures, but different task-specific effects. Consequently, the comparability of task-specific effects measures across tasks is lower than that of efficiency measures. Therefore, in the present study efficiency measures were used as inputs to the analysis.

Section snippets

Participants

The current MLD sample consisted of 226 grade 3–6 children (138 girls; grade 3, n = 41; grade 4, n = 54; grade 5, n = 64; grade 6, n = 67) obtained from a clinical and non-clinical sample. The clinical sample consisted of 76 children who were diagnosed and/or treated at one of six learning disability institutes in the Netherlands. They had been referred to one of the institutes because of enduring weak arithmetic performance. The non-clinical sample consisted of 977 children, who were enrolled in one

Data preparation

Before computing the outcome measures, items of timed tasks with responses faster than 200 ms were excluded. In the estimation task extreme outliers, specifically verbal responses higher than 100, were removed. Next the raw scores of the number-specific cognitive measures1 were transformed into standardized t-scores. The standardized

Discussion

The purpose of the present study was to distinguish subtypes of MLD children based on differences in general and number-specific cognitive processing deficits observed in an MLD sample. Several cognitive processing skills were assessed, which have been frequently associated with MLD in previous empirical studies. Entering these cognitive ‘vulnerability markers’ as input variables into a cluster analysis, revealed that MLD is, as hypothesized, a heterogeneous disorder.

Conclusion

The present results support the view that MLD is a heterogeneous disorder and trying to reduce atypical arithmetic development to one underlying core-deficit is too simplistic. Furthermore, the cluster solution provides little support for the presence of MLD subpopulations which primarily suffer from general cognitive processing deficits and so speak against the notion that MLD is strongly underpinned by domain-general factors, such as working memory. Also, little evidence is found for the

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