Cognitive subtypes of mathematics learning difficulties in primary education

Research in Developmental Disabilities 35 (2014) 657–670

Contents lists available at ScienceDirect

Research in Developmental Disabilities

Cognitive subtypes of mathematics learning difficulties in primary education Dimona Bartelet a,*, Daniel Ansari b, Anniek Vaessen c, Leo Blomert c a

Top Institute for Evidence Based Education Research, Maastricht University, P.O. Box 616, 6200 MD Maastricht, The Netherlands Numerical Cognition Laboratory, Department of Psychology, University of Western Ontario, Westminster Hall, London, ON, Canada N6A 3K7 c Department of Cognitive Neuroscience, Faculty of Psychology and Neuroscience, Maastricht University, and Maastricht Brain Imaging Centre (M-BIC), 6200 MD Maastricht, The Netherlands b

A R T I C L E I N F O

A B S T R A C T

Article history: Received 8 October 2013 Received in revised form 19 December 2013 Accepted 26 December 2013 Available online 22 January 2014

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. ß 2014 Elsevier Ltd. All rights reserved.

Keywords: Basic cognitive processing skills Mathematics learning difficulties Cognitive subtypes Primary school children

1. 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, Roeyers, & DeClercq, 2004; Gross-Tsur, Manor, & Shalev, 1996). Still, higher prevalence rates have even been reported when using different methods or more lenient criteria (Barbaresi, Katusic, Colligan, Weaver, & Jacobsen, 2005; Mazzocco & 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

* Corresponding author. Tel.: +31433884698. E-mail addresses: [email protected] (D. Bartelet), [email protected] (D. Ansari). 0891-4222/$ – see front matter ß 2014 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ridd.2013.12.010

658

D. Bartelet et al. / Research in Developmental Disabilities 35 (2014) 657–670

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 longterm memory (e.g., Geary, 1993; Rousselle & Noe¨l, 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 & Nuerk, 2005; Rubinsten & 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, Hanich, & Kaplan, 2003; Rourke, 1993; Shalev, Manor, & Gross-Tsur, 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 & Myers, 2003; Wilson & 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 & Forness, 2000; Stuebing et al., 2002). 1.1. Cognitive markers of MLD Besides being based on a data-driven classification approach, definitions of learning difficulties subtypes should describe the cognitive processes impaired (King, Giess, & Lombardino, 2007; Skinner, 1981). Currently several cognitive processes have been frequently associated with MLD, but no comprehensive picture has emerged. The seemingly incompatible ¨ stergren, 2012; Szu¨cs, Devine, Soltesz, Nobes, & findings have led to the formulation of diverging theories (Andersson & O Gabriel, 2013). 1.1.1. General cognitive processes 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 & Lyxell, 2007; D’Amico & 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 & Guarnera, 2005; McLean & Hitch, 1999; Passolunghi & Mammarella, 2010; Szu¨cs, Devine, et al., 2013). Nonetheless, other studies did find verbal working memory deficits in MLD children (Andersson & Lyxell, 2007; Geary, Hoard, Byrd-Craven, Nugent, & Numtee, 2007; Kytta¨la¨, Aunio, & Hautama¨ki, 2010; Rosselli, Matute, Pinto, & Ardila, 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 & Noe¨l, 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, Ferra˜o, & 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

D. Bartelet et al. / Research in Developmental Disabilities 35 (2014) 657–670

659

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. 1.1.2. Number-specific cognitive processes 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 & Bayer, 1976; Mundy & 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, Fussenegger, Moll, & Willburger, 2009; Mussolin, Meijas, & Noe¨l, 2010). Contrary to such findings other researchers found MLD children to have impaired symbolic comparison skills, but intact non-symbolic comparison abilities (De Smedt & Gilmore, 2011; Iuculano, Tang, Hall, & Butterworth, 2008; Rousselle & Noe¨l, 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 & Noe¨l, 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, Mussolin, Rousselle, Gre´goire, & Noe¨l, 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 ¨ stergren, 2012; Landerl, (counting range) of dots to differ significantly from typically developing children (Andersson & O 2013; Schleifer & 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 ¨ stergren, 2012). However, not one skill cognitive processes impaired in MLD children have been formulated (Andersson & O 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, & Szu¨cs, 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

660

D. Bartelet et al. / Research in Developmental Disabilities 35 (2014) 657–670

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 & Ra¨sa¨nen, 2013; Price & Ansari, 2013). 1.2. Efficiency vs. task-specific effect measures 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. Taskspecific 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 & Gilmore, 2014; Maloney et al., 2010; Szu¨cs, Nobes, Devine, Gabriel, & Gebuis, 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. 2. Method 2.1. 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 of six elementary schools spread across the Netherlands. Children were labeled as MLD if they performed one standard deviation below the mean norm score (percentile score < 16) on an arithmetic fluency test. Based on the selection criterion 61 (80%) children of the clinical sample were classified as MLD. The 15 children who did not meet the selection criterion were excluded from the study. Applying the selection criterion to the non-clinical sample resulted in the identification of 221 (23%) MLD children. Thus, the initial MLD sample included 282 children. However, 56 (20%) of the children were excluded in subsequent analyses because their verbal and spatial working memory scores were unknown. 2.2. Procedure After schools agreed to participate in the study, they disseminated an information letter to children’s parents, who could indicate their non-approval by returning the information slip. All assessments were conducted by trained research assistants. The arithmetic fluency test was administered centrally in each classroom, while all other tasks were administered individually in a quiet room at school. Except for the nonverbal IQ measure, all cognitive tests were assessed with the aid of a specialized response box containing four buttons, enabling us to record children’s response time in milliseconds for the timed tasks (Dyscalculia Differential Diagnosis [3DM], Blomert, Vaessen, & Ansari, 2013). The reaction time measurement always started with the initiation of a test item. Generally the individual test administration was completed within one session lasting about 60 min. In addition we contacted six specialized learning disability centers. For the clinical sample, the administration procedure was identical to the one maintained in the school sample, except that children were examined at home. 2.3. Cognitive processing tasks All the cognitive processing tasks had good reliabilities, ranging from .88 to .97. For the timed tasks acc-RT scores (# of items correct/s) were calculated to control for speed-accuracy tradeoffs (Salthouse & Hedden, 2002). The total number of correct items was divided by the total response time in seconds. The outcome for the short-term working memory tasks, the estimation task and the number line 0–1000 tasks (see detailed descriptions below) were accuracy (% correct) scores, mean deviation scores and absolute error, respectively, because no reaction time data was available for these tasks. To calculate mean deviation scores the numerosity of an item was subtracted from the participant’s verbal estimate of the numerosity, followed by the division of the numerosity of that same item (Actual Numerosity–Estimated Numerosity/Actual Numerosity). 2.3.1. Dot comparison In this task, consisting of three practice and 64 test items, two arrays of dots were simultaneously displayed on a computer screen in random order. Children were instructed to push the button corresponding to the larger numerosity as

D. Bartelet et al. / Research in Developmental Disabilities 35 (2014) 657–670

661

fast as possible. Numerosities ranged from 1 to 9 (level 1) and 10–39 (level 2) with a distance ratio of .25–.33, .50, .66 or .75 (smaller/larger number). Each ratio was presented 16 times. Displays were controlled for total area filled and total perimeter. For half of the stimuli the total area of the dot arrays is the same, which causes the total perimeter to be greater on the side with more dots. For the other half of the stimuli, the total perimeter of the dots arrays is the same, which causes the total area to be greater on the side with less dots. The side of the larger array of dots was counterbalanced to ensure that for each ratio the larger numerosity was equally often on the right and left hand side. 2.3.2. Arabic numeral comparison Children simultaneously saw two Arabic numerals differing by one of four distance ratios (.25–.33; .50; .66; and .75). After four practice items children responded to 64 test trials containing either one digit numbers (level 1) or two digit numbers (level 2). Also in this task each ratio was presented 16 times and the side of the larger number was counterbalanced for each ratio. Children were required to push the button corresponding to the larger number. 2.3.3. Verbal–Arabic matching During the verbal–Arabic matching task 4 practice items were administered, followed by two levels of 32 items. Depending on the level children heard a one or two digit number word through a headphone, immediately followed by a visual display of an Arabic number on a computer screen. If the numbers matched, children had to push the green button. Otherwise they were supposed to push the red button. Positive and negative responses were equally distributed and randomly ordered across levels. 2.3.4. Dot enumeration Counting was operationalized as a dot enumeration task composed of 6 practice and 45 test items. Each stimulus comprised a black square filled with 1–9 white dots of varying sizes in a non-linear order. Though presented in a random order, no displays containing the same numerosity were presented in immediate succession. Half of the trials were areamatched and half of the trials were perimeter-matched to control for the influence of perimeter and cumulative surface area. One-third of the stimuli displayed quantities falling within the subitizing range (1, 2, or 3) and two-third of the items presented magnitudes falling within the counting range (4, 5, 6, 7, 8, or 9). Children were asked to state their answer aloud, while simultaneously pushing the green button. 2.3.5. Matching objects In the matching objects task children were required to decide which of two quadrilateral planes shown in the bottom of a display contained the same amount of objects as a quadrilateral plane presented in the top. Object quantity ranged from 1 to 6. Children indicated their choice by pushing the corresponding button. After 6 practice items three levels of 15 trials were administered. All planes were controlled for luminance and object size was varied to minimize perceptual cues. Degree of difficulty was increased across levels by enhancing the homogeneity of objects within and between planes. For this purpose different objects were used, specifically four animals (bunny, horse, dog and cat) and four fruits (banana, apple, pear and pineapple). In level one all objects incorporated within an item belonged to the same category. They varied between planes in level two. Finally, in level three objects varied within and between planes. 2.3.6. Estimation Children quickly saw an array of white dots followed by a mask after 750 ms to minimize enumeration or arithmeticbased strategies. They were asked to estimate how many dots were displayed in six practice and 67 test items. If children were hesitant to do so they were encouraged to guess and reminded by the test administrator that counting was not allowed. Displays contained 1, 2, 3, 5, 7, 11 or 16 dots arranged in a random spatial arrangement to avoid processing biases due to the recognition of familiar patterns. As in the dot comparison task, all items were controlled for total area filled and total perimeter. Each numerosity occurred multiple times. Three short breaks were included at fixed time points to avoid fatigue. 2.3.7. Number line 0–1000 In the number line task children saw a number line with a starting point marked by the digit zero and an endpoint labeled by the digit 1000. The target numbers were written above the line in the center of the screen. Before proceeding to the 26 test trials, understanding of the task was checked through the administration of 3 practice items. However, children received no feedback regarding the accuracy of placement to avoid potential calibration. 2.3.8. Baseline response time This measure was composed of 20 test trials in which children saw a row of four empty squares. When an animation figure appeared they had to push the button corresponding to its location as fast as possible. 2.3.9. Verbal short-term working memory Children heard a number of pseudowords through their headphones, while facing away from the computer screen. At the end of each item children heard a specific tone indicating that the pseudoword string ended. Subsequently they were asked

662

D. Bartelet et al. / Research in Developmental Disabilities 35 (2014) 657–670

to repeat the pseudowords of a given item in the same order. After two practice items, the strings length increased across the 13 test trials from 2 to 6 pseudowords. 2.3.10. Spatial short-term working memory In the spatial short-term working memory task a row of four empty squares was displayed on the computer screen. In alternating order a square lighted up red. Children were asked to memorize in which sequence squares turned red. Each item was followed by a mask, which disappeared after children indicated the sequence they memorized by pushing the corresponding buttons. A total of four practice and 13 test items were administered. The number of items in the sequence increased across trials. 2.3.11. Nonverbal IQ The Colored Progressive Matrices is a normed untimed visuo-spatial reasoning test for children in the age range of 5–11 (Raven, Court, & Raven, 1995). Children saw a colored pattern and were asked to select the missing piece out of 6 choices. 2.4. Arithmetic fluency tasks 2.4.1. MLD selection task The selection criterion was based on the total score of the Dutch standardized TempoTest Automatiseren (TTA) (De Vos, 2010). This paper-and-pencil measure is a timed arithmetic fluency task standardized for grade 1 to grade 6. It consists of four subtests: addition, subtraction, multiplication and division. Per subtest children had 2 min to mentally compute as many operations as possible. If children had a composite score below the 16th percentile, they were classified as MLD. 2.4.2. Addition fluency task Children viewed a typed sum along the top of a computer screen. Simultaneously two answers were displayed below the operation. The instruction was to select the correct answer by pushing the corresponding button. Next to three practice stimuli, two levels with 20 items each were included. In the first level the maximum sum was 10, whereas the sum of the operations in level two exceeded 10. 2.4.3. Subtraction fluency task In this task a subtraction operation was presented to children. At the bottom of the computer screen two answers were displayed. Children were asked to push the button corresponding to the correct answer. Also this task was composed of three practice items and two levels of 20 trials each. The minuend was 10 or lower in level one and higher than 10 in level two. 2.4.4. Multiplication fluency task At the top of a computer screen a multiplication equation is displayed. Below this operation two answers are presented and children were required to push the button corresponding to the correct answer. The measure consisted of three practice stimuli and 40 test trials. 2.5. Descriptive statistics of the MLD and normal sample In Table 1 mean age, mean standard score on a nonverbal IQ task, mean percentile score on a mathematics achievement, verbal and spatial short-term working memory test and raw scores on the remaining cognitive measures are reported separately for the MLD and normal sample. MLD children were significantly older than normal achieving children. On all cognitive tasks the typical achieving group significantly outperformed children with MLD.

3. Results 3.1. 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 scores were calculated as a function of raw efficiency score, months of formal schooling, and the interaction of these two variables. The outcome scores of the short-term working memory tasks and the MLD selection task were transposed into percentile scores. Both, standardized t-scores and percentile scores are normalized, reducing the effect of outliers or non-normally distributed data and control for didactic age (months of education) (Vaessen et al., 2010). Next, number-specific tasks requiring a timed motor response were adjusted for general processing speed to eliminate the influence of variation in basic response speed on

1 The number-specific cognitive measures administered were dot comparison, arabic comparison, verbal–arabic matching, matching objects, estimation, number line 0–1000.

D. Bartelet et al. / Research in Developmental Disabilities 35 (2014) 657–670

663

Table 1 Age, arithmetic achievement and performance on number-specific and general cognitive processing tasks of children with MLD and normal achievers.

Age and general cognitive processing tasks Age (months) Nonverbal IQ (ssa 0–10) Verbal short-term working memory (psb 0–100)c Spatial short-term working memory (psb 0–100)c Baseline rt (items/s) Arithmetic performance Arithmetic achievement TTA (ps 0–100) Addition fluency 3DM (items/s) Subtraction fluency 3DM (items/s) Multiplication fluency 3DM (items/s) Number-specific cognitive processing tasks Dot comparison (items/s) Estimation (mean deviation) Counting (items/s) Matching objects (items/s) Verbal-arabic matching (items/s) Arabic comparison (items/s) Number line (% absolute error) a b c

MLD (n = 226)

Typical achieving (n = 767)

M (SD)

M (SD)

131.28 6.13 11.47 35.10 1.58

(14.39) (1.76) (19.10) (26.81) (.35)

T-test

Sig.

126.02 (13.73) 6.68 (1.62)

5.01 4.36

(p < .05) (p < .05)

1.65 (.34)

2.64

(p < .05)

(24.40) (.30) (.23) (.37)

47.59 16.58 16.88 18.37

(p < .05) (p < .05) (p < .05) (p < .05)

(.26) (.04) (.11) (.11) (.28) (.25) (4.80)

3.32 6.22 10.72 9.57 6.99 9.78 6.74

(p < .05) (p < .05) (p < .05) (p < .05) (p < .05) (p < .05) (p < .05)

9.24 .37 .29 .37

(8.62) (.16) (.15) (.21)

59.26 .61 .51 .72

.87 .11 .42 .35 .98 .99 10.37

(.23) (.04) (.09) (.09) (.26) (.23) (6.49)

.93 .09 .49 .42 1.12 1.16 7.23

ss, standard score. ps, percentile score. Task only administered to children with MLD.

Table 2 Pearson correlation between working memory tasks (percentile scores) and number-specific cognitive processing tasks (t-scores). 1 1. 2. 3. 4. 5. 6. 7. 8. 9.

Dot comparison Estimation Counting Matching objects Verbal–Arabic matching Arabic comparison Number line Verbal short-term working memory Spatial short-term working memory

2

3

4

5

6

7

0.00

0.25** 0.24**

0.39** 0.11 0.52**

0.20** 0.15* 0.32** 0.30**

0.35** 0.08 0.46** 0.41** 0.40**

0.15* 0.21** 0.16* 0.16* 0.02 0.15*

8

9 0.02 0.20** 0.04 0.03 0.04 0.08 0.07

0.08 0.22** 0.10 0.12 0.04 0.09 0.20** 0.02

* p < .05. ** p < .01.

task performance. Particularly the baseline response time scores were regressed on the individual number-specific tasks outcome scores. Subsequently, for the MLD sample, the association between number-specific and short-term working memory tasks was determined by means of Pearson correlation analysis (see Table 2). Numerous significant correlations were observed between the number-specific cognitive processing tasks indicating that they might reflect the same underlying construct. Moreover verbal short-term working memory was significantly related to estimation, while spatial short-term working memory was significantly associated with estimation and the number line task.In order to distinguish theoretically meaningful constructs all nine variables (see Table 2) were entered into an explorative factor analysis (principal component with oblimin rotation). The best factor solution for the current MLD sample was determined based on three criteria. Each factor had to have an eigenvalue of at least .70 (Jolliffe, 1986), variables had to have a communality estimate of at least .70 (MacCullum, Widaman, Shang, & Hong, 1999) and the factors were theoretically meaningful. This resulted in a six-factor solution, namely a counting factor (counting task and matching object task), an approximate numerical knowledge factor (dot comparison task and estimation task), a number line factor, a spatial short-term working memory factor, a verbal shortterm working memory factor and an Arabic knowledge factor (Arabic comparison task and verbal–Arabic matching task). The eigenvalues were 2.58, 1.31, 1.05, .95, .80 and .72 respectively. Thus, the seven included number-specific cognitive measures loaded onto four factors. 3.2. Identifying subtypes of MLD Besides children’s nonverbal IQ, the six factors identified and not children’s scores on the individual cognitive processing tasks were entered as variables in the cluster analyses. All the latent variable scores were transformed into standardized

664

D. Bartelet et al. / Research in Developmental Disabilities 35 (2014) 657–670

Fig. 1. Dendogram plot of the agglomerative hierarchical clustering approach (Ward’s method). Note: the dendrogram is collapsed over lower branches to increase visibility of the plot.

Fig. 2. Total-sum-of-squared within cluster distances (mean SUMD) as a function of K-number of clusters.

z-scores to ensure that differences in measurement scale did not influence the results. First, we conducted an agglomerative hierarchical clustering approach (Ward’s method) to determine the number of optimal clusters. This clustering method starts with all individuals representing one case and consecutively combines cases attempting to form clusters which are characterized by the lowest increase in error sum of squares until only one cluster remains (Mooi & Sarstedt, 2011). The result can be described with a dendrogram (Fig. 1) and a plot of the mean sum-of-squares (mean SUMD) as a function of K number of clusters (Fig. 2). Interpreting the height of the different nodes in the dendrogram led to the conclusion that a six cluster solution is the most optimal one. This is further confirmed by the observed ‘elbow’ at K = 6 in the mean SUMD plot. Then, to determine the cognitive profiles of the six different MLD clusters a 200 times repeated K-means clustering approach (Euclidean distance) was conducted using MATLAB (MathWorks, 1999). In this statistical program the K-means algorithm can be executed multiple times from different, randomly chosen starting points in a data set. This has the advantage that the problem of local optima can be circumvented (Steinley, 2003). The cluster solution eventually contained by MATLAB is the global optima, reducing the Euclidean distance more than any of the other cluster options (Steinley, 2003). Current analysis revealed the following six clusters presented in Fig. 3. Cluster 1 (n = 49) consisted of MLD children characterized by average to above average performance on all tasks (z-scores between 0.07 and 0.99) except for number line performance. On this measure their average achievement fell within the low average range (z-score equals 0.40). Therefore it was labeled the weak mental number line subtype. Cluster 2 (n = 33) consisted of MLD children who had strong spatial short-term working memory skills (z-score equals 1.37), next to high-average nonverbal IQ and counting abilities (z-scores of 0.85 and 0.62 respectively). Yet performance on the approximate numerical knowledge and number line tasks fell within the low-average range (z-scores of 0.58 and 0.49 respectively). Given that the profile is characterized by low performance on the tasks relying on the ANS, this cluster was labeled the weak ANS subtype. Cluster 3 (n = 49) comprised MLD children with weak approximate numerical knowledge and spatial short-term working memory proficiency (z-scores of 0.78 and 0.79 respectively), next to average-to-low average performance on the other cognitive measures (z-scores between 0.45 and 0.18). Hence it was assigned the label spatial difficulties subtype. Cluster 4 (n = 36) included MLD children with disabled Arabic knowledge and counting skills (z-scores of 1.03 and 1.28 respectively), despite having relatively strong general cognitive capabilities, as measured by the nonverbal IQ task (z-score equals 0.70). Their performance on the remaining cognitive measures fell within the average range (z-scores of 0.08

D. Bartelet et al. / Research in Developmental Disabilities 35 (2014) 657–670

665

Fig. 3. Mean z-scores on the general and number-specific cognitive processing measures per MLD subtype. Note: ANK = approximate numerical knowledge, C = counting, AK = Arabic numeral knowledge, NL = number line, SM = spatial short-term working memory, VM = verbal short-term working memory, IQ = nonverbal IQ.

666

D. Bartelet et al. / Research in Developmental Disabilities 35 (2014) 657–670

and 0.23). Scoring specifically low on the factors requiring number to numerosity mapping, this group was assigned the label access deficit subtype. Cluster 5 (n = 19) held the fewest MLD children. No number-specific cognitive processing strength or weaknesses could be distinguished as pointed out by the average z-scores (between 0.13 and 0.19) on most tasks. However, they did exhibit very strong verbal short-term working memory skills (z-score of 2.61). Therefore it was labeled the no numerical cognitive deficit subtype. Cluster 6 (n = 40) consisted of MLD children typified by a weak nonverbal IQ (z-score equals 0.80), but strong performance on the number line task (z-score equals 1.48). On the remaining cognitive processing measures children’s achievements fell within the average-to-low average range (z-scores between 0.08 and 0.38). Given that the profile is primarily characterized by a weak IQ, it was labeled the garden-variety subtype. 3.3. External validity of the MLD subtypes To validate the distinguished profiles, a number of external validation variables were selected which were independent of the measure used in the selection criteria. These were (1) three arithmetic and a reading fluency task, (2) the child-specific characteristics age and gender and (3) clinical sample distribution. Arithmetic fluency: ANOVA analyses with post hoc comparisons (bonferroni correction) were conducted for each computerized arithmetic fluency task, controlling for children’s general response speed. A significant difference in performance between the six clusters on addition fluency (3DM) (F(5,220) = 6.74, p < .01), subtraction fluency (3DM) (F(5,220) = 6.04, p < .01) and multiplication fluency (3DM) (F(5,220) = 9.41, p < .01 was found. Closer analyses demonstrated that the access deficit subtype had significantly lower addition and subtraction fluency skills than all other subtypes, except for the no numerical cognitive deficit subtype. Regarding multiplication skills, the access deficit subtype had significantly more difficulties than all other clusters. Reading fluency: ANOVA analyses with post hoc comparisons (bonferroni correction) revealed a significant difference in reading fluency skills between the six clusters (F(5,220) = 3.87, p < .01). Particularly children with the access deficit subtype and the weak ANS subtype had significantly weaker reading fluency skills than the weak mental number line subtype children. Age and gender: Both, age (F(5,220) = 0.60, p > .05) and gender (x2 (5,226) = 6.44, p > .05) were not significantly different between the six MLD types. Note that the distribution of gender in the six distinct clusters was compared using Chi-square analysis. The proportion of boys was 40.8%, 33.3%, 48.5%, 46.9%, 36.8% and 25.0% for ‘‘Cluster one,. . ., and Cluster six’’ respectively. Clinical sample distribution; Finally the distribution of the clusters across the clinical sample, which included the children who were diagnosed and/or treated at one of six specialized learning disability institutes, was investigated. The proportion of clinical children falling within a cluster differed significantly (x2(5,226) = 35.48, p < .01). Specifically 31.1% of the clinical children had an access deficit profile, while another 31.1% displayed a garden-variety profile. The remaining distributions were as follows: 14.8%, 3.3%, 9.8% and 9.8% were classified within the weak mental number line subtype, the weak ANS subtype, the spatial difficulties subtype and the no-deficit subtype respectively. 4. 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. 4.1. The cognitive factors The seven number-specific processing skills assessed, could be reduced to four distinct factors, particularly a counting factor, an Arabic knowledge factor, an approximate numerical knowledge factor and a number line factor. The dot enumeration and matching objects tasks, loading on the counting factor, require children to understand that each exact number word corresponds to a specific numerosity set (Cordes & Gelman, 2005). Although children’s non-symbolic and symbolic comparison tasks are hypothesized to rely both on the ANS (Dehaene, 2011), the fact that they loaded on different factors indicates that these tasks are not measuring the same underlying dimension. The Arabic comparison task loaded together with the verbal–Arabic matching measure on the Arabic knowledge factor. Both measures require children to possess magnitude knowledge of Arabic symbols but also ask children to discriminate two symbolic magnitudes. The dot comparison measure loaded together with the estimation task on the approximate numerical knowledge factor. On the one hand, this is unexpected given that the estimation task is not a pure non-symbolic ANS task since children have to use their symbolic magnitude knowledge to estimate the non-symbolic magnitude. On the other hand, like the dot comparison measure, the estimation task uses non-symbolic stimuli, which might explain why these tasks load on one dimension. However, this does lead to the question why the number line task, requiring children to use their symbolic representations to estimate a non-symbolic magnitude loaded on a different factor. A possible reason could be the inclusion of a spatial positioning requirement in the number line task (Siegler & Booth, 2005). Also note that the target stimulus magnitude is

D. Bartelet et al. / Research in Developmental Disabilities 35 (2014) 657–670

667

continuous and not discrete (Ebersbach et al., 2013). Furthermore, unlike the dot estimation task, the number line task may require a proportional judgment, eliciting children to relate the target number to one or more reference points (e.g. the endpoint 1000) before estimating the correct position of the target number on the number line (Slusser, Santiago, & Barth, 2013). 4.2. The subtypes of MLD The identified factors and children’s IQ clustered together in different patterns of strength and weaknesses, leading to the distinction of six cognitive MLD subtypes. The age and gender distribution across these subtypes was not significantly different. However, of the six identified subtypes, the most specific one was the access deficit group. This group is more severely impaired on arithmetic than any of the other subtypes. In addition, a large proportion of the clinical sample fell within this group. Lastly, the access deficit subtype profile exhibited the strongest deficit, scoring more than one standard deviation below average on the cognitive factors counting and Arabic knowledge. Other subtypes identified were a weak mental number line subtype, a weak ANS subtype, the spatial difficulties subtype, a no numerical cognitive deficit subtype and a garden variety subtype. Thus, our results support the proposition that the cognitive deficits underlying MLD in children is heterogeneous. A popular hypothesis is that MLD originates from a core-deficit in the innate ANS (Mazzocco et al., 2011; Piazza, 2010). An insufficiently matured ANS is believed to be indexed by consistent weak performance on non-symbolic and symbolic ¨ stergren, 2012). However, none of the clusters distinguished was characterized by number processing tasks (Andersson & O weaknesses on all number-specific processing tasks. Two different subtypes, the weak ANS subtype and the spatial difficulties subtype, included children who experienced substantial difficulties (0.6–0.8 SD below mean) on the approximate numerical knowledge factor and the number line measure. Yet, they exhibited average to above average counting and Arabic knowledge abilities. This pattern of strength and weaknesses speaks against the assumption that an impaired ANS drives symbolic number processing deficits. These subtype profiles indicate that non-symbolic and symbolic processing skills might be more distinct than suggested by the ANS hypothesis. Nevertheless, they also provide evidence that an ANS deficit is a cognitive correlate of MLD in some children, although it should be mentioned at the same time that only relatively few (3.3%) of the clinical sample children fell into the ANS subtype. A competing hypothesis assumes that MLD arises from a failure to adequately link the symbolic number representations to its corresponding non-symbolic representation and not from underlying numerosity processing difficulties themselves (Rousselle & Noe¨l, 2007). In line with this view is the cognitive profile of the access deficit subtype. These children exhibit severe difficulties on the Arabic knowledge and counting factors, while the other cognitive abilities were not noticeably weak or strong. It should be mentioned here that we assume that in order to succeed on the tasks constituting the counting and Arabic knowledge factors, children are required to map symbolic to non-symbolic representations. Yet, it has also been postulated that these tasks might not truly measure mapping, but merely symbolic number knowledge (Lyons, Ansari, & Beilock, 2012). The fact that children’s number line performance in the access deficit subtype was, on average, not considerably impaired seems counter-intuitive to the argument that a defective link between a symbolic number and a non-symbolic representation underlies MLD in children. However, the number line measure requires additional cognitive skills beyond the accessing of a numerical quantity represented by an Arabic number. Children have to map symbolic magnitude representations onto a non-numerical quantity, namely a line in space (Siegler & Booth, 2005). Thus children can have weak Arabic knowledge and counting abilities, while possessing intact number line skills and vice-a-versa (Von Aster & Shalev, 2007). In agreement with this, we also singled out a group of children who had low average performance on the number line task, but average to above average skills on the other cognitive processing measures included. Lastly, numerous researchers have advanced a general cognitive deficit hypothesis to explain MLD. According to this hypothesis impaired general cognitive processing skills (e.g. working memory) are the primary source of MLD (Geary, 2004; Szu¨cs, Devine, et al., 2013). Of the subtypes distinguished, only the spatial difficulties subtype experienced considerable general cognitive processing weaknesses. These children performed below average (approaching 1 SD below mean) on the spatial short-term working memory task. No severe subtype-specific impairment of verbal short-term working memory was found. Hence, we conclude that general cognitive processing deficits, indexed by short-term working memory skills, are only associated with MLD in a limited number of children. Moreover the clustering patterns observed for the distinct subtypes indicate that short-term working memory and number-specific processing deficits are independent correlates of MLD. Although the spatial short-term working memory deficit characterizing the spatial difficulties subtype clustered together with deficient approximate numerical knowledge, the number-specific processing deficits did not cluster together with short-term working memory deficits in any of the other subtypes. The profile of two identified subtypes, namely the garden-variety subtype and the no numerical cognitive deficit subtype, do not fit well, even not partially, with any of the previously discussed hypothesis. The garden-variety subtype included children who were characterized by low nonverbal IQ (approaching 1 SD below mean). Children’s intellectual ability has been linked to mathematics achievement (e.g., Primi, Ferra˜o, & Almeida, 2010). Hence, it seems sensible that we find an MLD subtype consisting of children who have weak intelligence. The profile of the no numerical cognitive deficit subtype was mostly flat. Children in this group exhibited, when compared to other children with MLD, approximately average performance on all tasks except for the verbal short-term working memory measure on which they scored very high (2.5 SD

668

D. Bartelet et al. / Research in Developmental Disabilities 35 (2014) 657–670

above mean). This finding is in line with the notion that some children who meet the cut-off criteria for MLD have typically developed general and number-specific cognitive processing skills (Mazzocco, 2001) and their MLD are likely to be associated with other factors, such as education, SES and motivation. Some of the subtypes are similar to subtypes distinguished in a previous data-driven classification study by Von Aster (2000). He identified three MLD and one low performance subtype. The latter group had similar to the no numerical cognitive deficit subtype differentiated in the current study impaired arithmetic skills, but no considerable basic cognitive processing deficits. Von Aster (2000) also found a verbal and Arabic subtype. The former was characterized by counting deficits, while the Arabic group displayed deficits on the tasks requiring children to compare Arabic numbers or transcode verbal number words to digits and vice versa. In the present study, deficits on these skills clustered together into an access deficit subtype. A last subtype identified by Von Aster (2000) was the pervasive group, but none of the six subtypes in the current study displayed noticeable deficits on almost all number-specific cognitive measures administered. The number and profiles of the clusters in Von Aster’s and our study are different, but this could be attributable to variations in the study designs. Von Aster (2000) used a smaller sample size, did not include general cognitive processing variables and he controlled for IQ differences in advance. 5. 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 defective ANS hypothesis. Most support was found for the access deficit hypothesis. One of the subtypes, namely the access deficit subtype, did display relatively severe deficits on the symbolic processing tasks. Moreover, this subtype was the most impaired on arithmetic skills. This indicates that the access deficit subtype is the most specific and also implies that access deficit difficulties are the best cognitive ‘vulnerability markers’ of severe MLD. This study is only the 2nd study to take a data-driven clustering approach in order to better understand the heterogeneity of MLD. In addition, to the best of our knowledge, it is the only classification study examining general and number-specific cognitive processing measures simultaneously to identify MLD subtypes. However, it should be noted that the current subtypes were based on cross-sectional data. Future studies should investigate whether the MLD subtypes are stable over time, since about one-third of the children with MLD do not consistently meet MLD criteria over several years (Mazzocco & Ra¨sa¨nen, 2013). We postulate that the less specific subtypes, such as the garden-variety group and the no numerical cognitive deficit group, include children which would not be defined as MLD if tested multiple times. Further research is also needed to see whether the subtypes distinguished respond differently to tailored interventions. This can shed light on the distinctiveness of the individual subtypes and the causal nature of the relationship between the basic cognitive processing measures and MLD. References Andersson, U., & Lyxell, B. (2007). Working memory deficit in children with mathematical difficulties: A general or specific deficit? Journal of Experimental Child Psychology, 96, 197–228. ¨ stergren, R. (2012). Number magnitude processing and basic cognitive functions in children with mathematical learning disabilities. Learning Andersson, U., & O and Individual Differences, 22, 701–714. Barbaresi, W. J., Katusic, S. K., Colligan, R. C., Weaver, A. L., & Jacobsen, S. J. (2005). Math learning disorder: Incidence in a population-based birth cohort, 1976–82, Rochester, Minn. Ambulatory Pediatrics, 5, 281–289. Blomert, L., Vaessen, A., & Ansari, D. (2013). Differentiaal diagnostiek van dyscalculie: Cognitieve analyse van rekenen. [Dyscalculia differential diagnosis: Cognitive analysis of math]]. Amsterdam, Netherlands: Boom Test Publishers. Butterworth, B. (2010). Foundational numerical capacities and the origins of dyscalculia. Trends in Cognitive Sciences, 14, 534–541. Cordes, S., & Gelman, R. (2005). The young numerical mind: When does it count? In J. I. D. Campbell (Ed.), Handbook of mathematical cognition (pp. 127–142). New York, NY: Psychology Press. D’Amico, A., & Guarnera, M. (2005). Exploring working memory in children with low arithmetical achievement. Learning and Individual Differences, 15, 189–202. De Smedt, B., & Gilmore, C. (2011). Defective number module or impaired access? Numerical magnitude processing in first graders with mathematical difficulties. Journal of Experimental Child Psychology, 108, 278–292. De Vos, T. (2010). Tempo test Automatiseren. Amsterdam, Netherlands: Boom Test Publishers. Dehaene, S. (1992). Varieties of numerical abilities. Cognition, 44, 1–42. Dehaene, S. (2011). The number sense: How the mind creates mathematics (second ed.). New York, NY: Oxford University Press. Desoete, A., Roeyers, H., & De Clercq, A. (2004). Children with mathematics learning disabilities in Belgium. Journal of Learning Disabilities, 37, 50–61. Dowker, A. (2005). Individual differences in arithmetic: Implications for psychology, neuroscience and education. New York, USA: Psychology Press. Ebersbach, M., Luwel, K., & Verschaffel, L. (2013). Comparing apples and pears in studies on magnitude estimations. Frontiers in Psychology, 4(332), 1–6. Fayol, M., & Seron, X. (2005). About numerical representations: Insights from neuropsychological, experimental, and developmental studies. In J. I. D. Campbell (Ed.), Handbook of mathematical cognition (pp. 3–22). New York, NY: Psychology Press. Fias, W., Menon, V., & Szu¨cs, D. (2013). Multiple components of developmental dyscalculia. Trends in Neuroscience and Education, 2, 43–47. Geary, D. C. (1993). Mathematical disabilities: Cognitive, neuropsychological, and genetic components. Psychological Bulletin, 114, 345–362. Geary, D. C. (2004). Mathematics and learning disabilities. Journal of Learning Disabilities, 37, 4–15. Geary, D. C., Hoard, M. K., Byrd-Craven, J., Nugent, L., & Numtee, C. (2007). Cognitive mechanisms underlying achievement deficits in children with mathematical learning disability. Child Development, 78, 1343–1359. Geary, D. C. (2010). Mathematical disabilities: Reflections on cognitive, neuropsychological, and genetic components. Learning and Individual Differences, 20, 130–133.

D. Bartelet et al. / Research in Developmental Disabilities 35 (2014) 657–670

669

Geary, D. C. (2011). Consequences, characteristics, and causes of mathematical learning disabilities and persistent low achievement in mathematics. Journal of Developmental and Behavioral Pediatrics, 32, 250–263. Geary, D. C., Hoard, M. K., & Bailey, D. H. (2012). Fact retrieval deficits in low achieving children and children with mathematical learning disability. Journal of Learning Disabilities, 45, 291–307. Gross-Tsur, V., Manor, O., & Shalev, R. S. (1996). Developmental dyscalculia: Prevalence and demographic features. Developmental Medicine & Child Neurology, 38, 25–33. Huntley-Fenner, G. (2001). Children’s understanding of number is similar to adults’ and rats’: Numerical estimation by 5–7-year-olds. Cognition, 78, B27–B40. Inglis, M., & Gilmore, C. (2014). Indexing the approximate number system. Acta Psychologica, 145, 147–155. Iuculano, T., Tang, J., Hall, C. W. B., & Butterworth, B. (2008). Core information processing deficits in developmental dyscalculia and low numeracy. Developmental Science, 11, 669–680. Jolliffe, I. T. (1986). Principal Component Analysis. New York, NY: Springer. Jordan, N. C., Hanich, L. B., & Kaplan, D. (2003). A longitudinal study of mathematical competencies in children with specific mathematics difficulties versus children with comorbid mathematics and reading difficulties. Child Development, 74, 834–850. Kaufmann, L., & Nuerk, H.-C. (2005). Numerical development: Current issues and future perspectives. Psychology Science, 47, 142–170,. Retrieved from http:// www.questia.com/library/p2811/psychology-science. Kavale, K. A., & Forness, S. R. (2000). What definitions of learning disability say and don’t say: A critical analysis. Journal of Learning Disabilities, 33, 239–256. King, W., Lombardino, L., & Giess, S. (2007). Subtyping of persons with developmental dyslexia via bootstrap aggregated clustering and the gap statistic: Comparison with the double-deficit hypothesis. International Journal of Language & Communication Disorders, 42, 77–95,. Retrieved from http://onlinelibrary.wiley.com/journal/10.1111/(ISSN)1460-6984. Kramer, A., & Landerl, K. (2010, June). Defective number module or access deficit – two different subtypes of dyscalculia? Paper presented at the EARLI SIG22 – Neuroscience and Education, Zu¨rich, Switzerland. Abstract retrieved from http://www.frontiersin.org/10.3389/conf.fnins.2010.11.00048/event_Abstract. Kytta¨la¨, M., Aunio, P., & Hautama¨ki, J. (2010). Working memory resources in young children with mathematical difficulties. Scandinavian Journal of Psychology, 51, 1–15. Landerl, K., Bevan, A., & Butterworth, B. (2004). Developmental dyscalculia and basic numerical capacities: A study of 8- to 9-year-old students. Cognition, 93, 99– 125. Landerl, K., Fussenegger, B., Moll, K., & Willburger, E. (2009). Dyslexia and dyscalculia: Two learning disorders with different cognitive profiles. Journal of Experimental Child Psychology, 103, 309–324. Landerl, K. (2013). Development of numerical processing in children with typical and dyscalculic arithmetic skills – A longitudinal study. Frontiers in Psychology, 4(459), 1–14. Lyons, I. M., Ansari, D., & Beilock, S. L. (2012). Symbolic estrangement: Evidence against a strong association between numerical symbols and the quantities they represent. Journal of Experimental Psychology: General, 141, 635–641. MacCallum, R. C., Widaman, K. F., Zhang, S., & Hong, S. (1999). Sample size in factor analysis. Psychological Methods, 4, 84–99. Maloney, E. A., Risko, E. F., Preston, F., Ansari, D., & Fugelsang, J. (2010). Challenging the reliability and validity of cognitive measures: The case of the numerical distance effect. Acta Psychologica, 134, 154–161. MathWorks. (1999). MATLAB [computer software]. Natick, MA: MathWorks. Mazzocco, M. M. M. (2001). Math learning disability and math LD subtypes: Evidence from studies of turner syndrome, fragile x syndrome, and neurofibromatosis type 1. Journal of Learning Disabilities, 34, 520–533. Mazzocco, M. M. M., & Myers, G. F. (2003). Complexities in identifying and defining mathematics learning disability in the primary school-age years. Annals of Dyslexia, 53, 218–253. Mazzocco, M. M. M., Feigenson, L., & Halberda, J. (2011). Impaired acuity of the approximate number system underlies mathematical learning disability (dyscalculia). Child Development, 82, 1224–1237. Mazzocco, M. M. M., & Ra¨sa¨nen, P. (2013). Contributions of longitudinal studies to evolving definitions and knowledge of developmental dyscalculia. Trends in Neuroscience and Education, 2, 65–73. McLean, J. F., & Hitch, G. J. (1999). Working memory impairments in children with specific arithmetic learning difficulties. Journal of Experimental Child Psychology, 74, 240–260. Mejias, S., Mussolin, C., Rousselle, L., Gre´goire, J., & Noe¨l, M.-P. (2012). Numerical and nonnumerical estimation in children with and without mathematical learning disabilities. Child Neuropsychology: A Journal of Normal and Abnormal Development in Childhood and Adolescence, 18, 550–575. Moeller, K., Fischer, U., Cress, U., & Nuerk, H.-C. (2012). Diagnostics and Intervention in developmental dyscalculia: Current issues and novel perspectives. In Z. Breznitz, O. Rubinsten, V. J. Molfese, & D. L. Molfese (Eds.), Reading, writing, mathematics and the developing brain: Listening to many voices (pp. 233–277). Dordrecht, Netherlands: Springer Science+Business Media B. Mooi, E., & Sarstedt, (2011). A concise guide to market research: The process, data, and methods using IBM SPSS Statistics. London, UK: Springer. Moyer, R. S., & Bayer, R. H. (1976). Mental comparison and the symbolic distance effect. Cognitive Psychology, 8, 228–246. Mundy, E., & Gilmore, C. K. (2009). Children’s mapping between symbolic and nonsymbolic representations of number. Journal of Experimental Child Psychology, 103, 490–502. Mussolin, C., Mejias, S., & Noe¨l, M.-P. (2010). Symbolic and nonsymbolic number comparison in children with and without dyscalculia. Cognition, 115, 10–25. Passolunghi, M. C., & Mammarella, I. C. (2010). Spatial and visual working memory ability in children with difficulties in arithmetic word problem solving. European Journal of Cognitive Psychology, 22, 944–963. Piazza, M. (2010). Neurocognitive start-up tools for symbolic number representations. Trends in Cognitive Sciences, 14, 542–551. Price, G., & Ansari, D. (2012). Developmental dyscalculia: A case for neuroscience in education. BJEP monograph series II, number 8-Educational Neuroscience, 1 (pp. 45–62). Retrieved from http://www.bps.org.uk/publications-new/journals/bjep-monographs/bjep-monographs. Price, G. R., Palmer, D., Battista, C., & Ansari, D. (2012). Nonsymbolic numerical magnitude comparison: Reliability and validity of different task variants and outcome measures, and their relationship to arithmetic achievement in adults. Acta Psychologica, 140, 50–57. Price, G. R., & Ansari, D. (2013). Dyscalculia: Characteristics, causes, and treatments. Numeracy: Advancing Education in Quantitative Literacy, 6(1).. Article 2. Primi, R., Ferra˜o, M. E., & Almeida, L. S. (2010). Fluid intelligence as a predictor of learning: A longitudinal multilevel approach applied to math. Learning and Individual Differences, 20, 446–451. Raven, J., Court, J. H., & Raven, J. C. (1995). Coloured progressive matrices. Oxford, England: Oxford Psychologists Press. Rosselli, M., Matute, E., Pinto, N., & Ardila, A. (2010). Memory abilities in children with subtypes of dyscalculia. Developmental Neuropsychology, 30, 801–818. Rourke, B. P. (1993). Arithmetic disabilities, specific and otherwise: A neuropsychological perspective. Journal of Learning Disabilities, 26, 214–226. Rousselle, L., & Noe¨l, M.-P. (2007). Basic numerical skills in children with mathematics learning disabilities: A comparison of symbolic vs non-symbolic number magnitude processing. Cognition, 102, 361–395. http://dx.doi.org/10.1016/j.cognition.2006.01.005 Rubinsten, O., & Henik, A. (2009). Developmental dyscalculia: Heterogeneity might not mean different mechanisms. Trends in Cognitive Sciences, 13, 92–99. Salthouse, T. A., & Hedden, T. (2002). Interpreting reaction time measures in between-group comparisons. Journal of Clinical and Experimental Neuropsychology, 24, 858–872. Schleifer, P., & Landerl, K. (2011). Subitizing and counting in typical and atypical development. Developmental Science, 14, 280–291. Shalev, R. S., Manor, O., & Gross-Tsur, V. (1997). Neuropsychological aspects of developmental dyscalculia. Mathematical Cognition, 3, 105–120. Siegler, R. S., & Booth, J. L. (2004). Development of numerical estimation in young children. Child Development, 75, 428–444. Siegler, R. S., & Booth, J. L. (2005). Development of numerical estimation: A review. In J. I. D. Campbell (Ed.), Handbook of mathematical cognition (pp. 197–212). New York, NY: Psychology Press. Skinner, H. A. (1981). Toward the integration of classification theory and methods. Journal of Abnormal Psychology, 90, 68–87. Slusser, E. B., Santiago, R. T., & Barth, H. C. (2013). Developmental change in numerical estimation. Journal of Experimental Psychology: General, 142, 193–208.

670

D. Bartelet et al. / Research in Developmental Disabilities 35 (2014) 657–670

Steinley, D. (2003). Local optima in K-means clustering: What you don’t know may hurt you. Psychological Methods, 8, 294–304. Stuebing, K. K., Fletcher, J. M., LeDoux, J. M., Lyon, G. R., Shaywitz, S. E., & Shaywitz, B. A. (2002). Validity of IQ-discrepancy classifications of reading disabilities: A meta-analysis. American Educational Research Journal, 39, 469–518. Szu¨cs, D., Devine, A., Soltesz, F., Nobes, A., & Gabriel, F. (2013). Developmental dyscalculia is related to visuo-spatial memory and inhibition impairment. Cortex. (Advance online publication). Szu¨cs, D., Nobes, A., Devine, A., Gabriel, F. C., & Gebuis, T. (2013). Visual stimulus parameters seriously compromise the measurement of approximate number system acuity and comparative effects between adults and children. Frontiers in Psychology, 4(444), 1–12. Vaessen, A., Bertrand, D., To´th, D., Cse´pe, V., Faı´sca, L., Reis, A., et al. (2010). Cognitive development of fluent word reading does not qualitatively differ between transparent and opaque orthographies. Journal of Educational Psychology, 102, 827–842. Von Aster, M. (2000). Developmental cognitive neuropsychology of number processing and calculation: Varieties of developmental dyscalculia. European Child & Adolescent Psychiatry, 9, S41–S57. Von Aster, M., & Shalev, R. S. (2007). Number development and developmental dyscalculia. Developmental Medicine & Child Neurology, 49, 868–873. Wilson, A. J., & Dehaene, S. (2007). Number sense and developmental dyscalculia. In D. Coch, G. Dawson, & K. W. Fischer (Eds.), Human behavior, learning, and the developing brain: Atypical development (pp. 212–238). New York, USA: Guilford Press.