Research in Developmental Disabilities 35 (2014) 3001–3013

Contents lists available at ScienceDirect

Research in Developmental Disabilities

Arithmetic strategy development and its domain-specific and

domain-general cognitive correlates: A longitudinal study inchildren with persistent mathematical learning difficulties

Kiran Vanbinst *, Pol Ghesquiere, Bert De Smedt

Parenting and Special Education Research Group, Katholieke Universiteit Leuven, Leopold Vanderkelenstraat 32, Box 3765, B-3000 Leuven,

Belgium

A R T I C L E I N F O

Article history:

Received 28 February 2014

Received in revised form 18 June 2014

Accepted 25 June 2014

Available online

Keywords:

Mathematical learning difficulties

Arithmetic strategy development

Numerical magnitude processing

Working memory

Phonological processing

A B S T R A C T

Deficits in arithmetic fact retrieval constitute the hallmark of children with mathematical

learning difficulties (MLD). It remains, however, unclear which cognitive deficits underpin

these difficulties in arithmetic fact retrieval. Many prior studies defined MLD by

considering low achievement criteria and not by additionally taking the persistence of the

MLD into account. Therefore, the present longitudinal study contrasted children with

persistent MLD (MLD-p; mean age: 9 years 2 months) and typically developing (TD)

children (mean age: 9 years 6 months) at three time points, to explore whether differences

in arithmetic strategy development were associated with differences in numerical

magnitude processing, working memory and phonological processing. Our longitudinal

data revealed that children with MLD-p had persistent arithmetic fact retrieval deficits at

each time point. Children with MLD-p showed persistent impairments in symbolic, but not

in nonsymbolic, magnitude processing at each time point. The two groups differed in

phonological processing, but not in working memory. Our data indicate that both domain-

specific and domain-general cognitive abilities contribute to individual differences in

children’s arithmetic strategy development, and that the symbolic processing of numerical

magnitudes might be a particular risk factor for children with MLD-p.

� 2014 Elsevier Ltd. All rights reserved.

1. Introduction

Children with mathematical learning difficulties (MLD) experience difficulties acquiring basic arithmetical skills(Butterworth, Varma, & Laurillard, 2011; Geary, 2004; Mazzocco, 2007). Given that arithmetic comprises a building block forsubsequent growth in mathematics (e.g., Kilpatrick, Swafford, & Findell, 2001) and that difficulties in storing and recallingarithmetic facts (e.g., Berch & Mazzocco, 2007; Jordan, Hanich, & Kaplan, 2003) constitute the hallmark of children with MLD(e.g., Geary, Hoard, & Bailey, 2012), it is crucial to reveal the origin of difficulties in arithmetic strategy development. In thepast decade, most studies on MLD have focused on the cognitive origins of mathematical difficulties in general, i.e. bystudying how numerical magnitude processing (De Smedt, Noel, Gilmore, & Ansari, 2013, for a review), working memory(see Friso-van den Bos, van der Ven, Kroesbergen, & van Luit, 2013, for a meta-analysis) and phonological processing

* Corresponding author. Tel.: +32 16325705.

E-mail address: kiran.vanbinst@ppw.kuleuven.be (K. Vanbinst).

http://dx.doi.org/10.1016/j.ridd.2014.06.023

0891-4222/� 2014 Elsevier Ltd. All rights reserved.

K. Vanbinst et al. / Research in Developmental Disabilities 35 (2014) 3001–30133002

(e.g., Vukovic & Siegel, 2010) are related to children’s performance on general mathematics achievement tests. Yet only a fewstudies investigated the association of these cognitive abilities with specific types of mathematical skill, such as arithmeticstrategy development. This more narrow focus is, however, needed, because it will allow us to explore how numericalmagnitude processing, working memory and phonological processing might contribute to difficulties in children’sacquisition of adequate strategies for solving single-digit arithmetic. This is precisely the aim of the current study. Further, inmany prior studies children with MLD were defined by considering low achievement criteria and not by additionally takinginto account the persistence of the mathematical difficulties. Against this background, we conducted a longitudinal study inwhich arithmetic strategy development, numerical magnitude processing, working memory and phonological processingwere investigated in children with persistent MLD (MLD-p) and typically developing (TD) children.

In the remainder of this introduction, we first describe the arithmetic strategy development in TD children and childrenwith MLD. Next, we discuss how numerical magnitude processing, working memory and phonological processing maycontribute to difficulties in acquiring adequate arithmetic solving strategies. Finally, we present the specific goals of thepresent study.

1.1. Arithmetic strategy development

Many studies have examined arithmetic strategy use during the solution of single-digit arithmetic in TD children (e.g.,Bailey, Littlefield, & Geary, 2012; Barrouillet, Mignon, & Thevenot, 2008). At the earliest stages of arithmetic strategydevelopment, children count all the numbers in a problem (e.g., 1, 2, 3, 4, 5, 6, 7 to solve 3 + 4) and gradually move on to moreadvanced counting procedures, such as counting on from the larger number in the problem (e.g., 4, 5, 6, 7 to solve 3 + 4)(Geary, Bow-Thomas, & Yao, 1992). Through repeated use of counting strategies, children develop representations of basicarithmetic facts, which are stored in their long-term memory (Siegler & Shrager, 1984). Children (and also adults) rely onthese stored fact representations when they evolve to more advanced strategies for solving single-digit additions andsubtractions. These problems are then typically solved either by using an advanced procedural strategy, such asdecomposition (e.g., 7 + 8 = 7 + 3 = 10 + 5 = 15), or by directly retrieving the correct answer from long-term memory (Siegler,1996).

Studies comparing children with and without MLD observed impaired procedural strategy skills in children with MLD(e.g., Geary, 2004, for a review). This impairment is marked by a developmental delay in the shift from immature countingstrategies, such as finger counting, to more advanced procedural strategies, such as decomposing the problem into smallerfacts. Growth modeling data have shown that children with MLD catch up with their typically developing peers over time,narrowing the initial learning gap in procedural strategy skills (Chong & Siegel, 2008).

One of the most robust findings is that children with MLD have difficulties in storing and recalling simple arithmetic facts(e.g., Geary, Hoard, et al., 2012), even despite intensive instructional interventions (e.g., Howell, Sidorenko, & Jurica, 1987).These arithmetic fact retrieval difficulties do not ameliorate with time and constitute persistent deficits in children with MLD(e.g., Geary, Hoard, Nugent, & Bailey, 2012; Jordan et al., 2003; but see Torbeyns, Verschaffel, & Ghesquiere, 2004). Although,there is a lack of universally accepted screening tools for defining MLD, knowledge on classification criteria for MLD isgrowing and it has been suggested that arithmetic fact retrieval deficits may be a useful indicator to include in a diagnosticdefinition of MLD (Geary, 2011).

Questions have been raised about the origin of children’s difficulties in developing adequate arithmetic solutionstrategies. Therefore the aim of this study was to investigate how numerical magnitude processing, working memory andphonological processing contribute to difficulties in arithmetic strategy development.

1.2. Numerical magnitude processing

It is amply evidenced that the ability to represent numerical magnitudes contributes to individual differences inchildren’s mathematical development (see De Smedt et al., 2013; for a review). This ability has typically been investigatedby means of Arabic digit and dot comparison tasks, in which children have to identify the larger of two numerosities.Several researchers have proposed that MLD arise from a fundamental impairment in this ability to represent numericalmagnitudes (Andersson & Ostergren, 2012; Iuculano, Tang, Hall, & Butterworth, 2008; Landerl & Kolle, 2009; Mazzocco,Feigenson, & Halberda, 2011; Piazza et al., 2010). Two hypotheses have been put forward to explain how deficientrepresentations of numerical magnitude are related to MLD. According to the defective number module hypothesis(Butterworth, 2005), MLD originate from a specific deficit in the innate ability to understand and represent numericalmagnitudes. By contrast, the access deficit hypothesis (Rousselle & Noel, 2007) argues that MLD are due to impairments inaccessing semantic numerical representations from Arabic symbols, rather than from difficulties in processing numericalmagnitudes per se. To date, findings remain inconclusive whether the defective number module or the access deficithypothesis explains MLD (see De Smedt et al., 2013; Noel & Rousselle, 2011, for a review). This may be so because theexisting research investigated these hypotheses in the context of children’s general mathematics achievement, yet it islikely that various forms of numerical magnitude processing impact more on some specific aspects of mathematical skillmore than others.

Interestingly, Vanbinst, Ghesquiere, and De Smedt (2012) have recently addressed this issue in typically developing thirdgraders. More specifically, these authors showed that symbolic, but not nonsymbolic, magnitude processing was related to

K. Vanbinst et al. / Research in Developmental Disabilities 35 (2014) 3001–3013 3003

arithmetic strategy use: children with better access to magnitude representations from Arabic symbols retrieved more factsfrom their memory and were faster in executing fact retrieval as well as procedural strategies. It remains to be determinedwhether this can be extended to children with MLD, a core aim of the present study.

1.3. Working memory

Various studies have reported working memory deficits in children with MLD (e.g., Bull, Johnston, & Roy, 1999; Geary,Hoard, Byrd-Craven, Nugent, & Numtee, 2007; McLean & Hitch, 1999; Passolunghi & Siegel, 2004; Szucs, Devine, Soltesz,Nobes, & Gabriel, 2013). These studies have used Baddeley’s three component model of working memory (Baddeley, 1986) astheir framework. At the core of this model is the central executive, which is responsible for the attention-driven control,regulation and monitoring of complex cognitive processes. The model additionally encompasses two slave systems oflimited capacity, which are used for the temporary storage of phonological information (phonological loop) as well as visualand spatial information (visuo-spatial sketchpad).

There has been little agreement on which working memory components are impaired in children with MLD (see Swanson& Jerman, 2006, for a meta-analysis). A possible explanation for these conflicting findings may be that the three workingmemory components affect different aspects of mathematical development to different degrees (Geary et al., 2007). In viewof this, the present longitudinal study narrows down the focus to how working memory is related to arithmetic strategydevelopment.

Difficulties in central executive in children with MLD have been related to the reliance on finger counting and thecommission of counting errors (Geary, Hoard, Byrd-Craven, & DeSoto, 2004). Evidence from dual-task studies haverevealed that the central executive is crucial for keeping track of different steps during arithmetic problem solving(Imbo & Vandierendonck, 2007). In order to compensate for their poor central executive capacities, children withMLD used their fingers to represent addends and to keep track of the counting sequence. Barrouillet and Lepine(2005) further demonstrated that children with low central executive capacities relied less frequently on fact retrievalto solve addition than children with high central executive capacities. This is supported by Wu and colleagues (2008)who found a significant association between the central executive and arithmetic fact retrieval in children withMLD.

The impact of phonological loop and visuo-spatial sketchpad on arithmetic strategy development is less clear (e.g., Friso-van den Bos et al., 2013, for a meta-analysis). Wu et al. (2008) observed no associations between strategy use andphonological loop as well as visuo-spatial sketchpad. By contrast, dual-task studies have shown that an impairedphonological working memory may contribute to difficulties in keeping track of the operands while solving arithmetic withcounting strategies (e.g., Noel, Seron, & Trovarelli, 2004). Correlational data have indicated that the phonological loop mayplay a crucial role in the process of accurately memorizing basic arithmetic facts (Bull & Johnston, 1997; De Smedt et al.,2009).

Turning to the visuo-spatial sketchpad, it has been suggested that poor spatial working memory processes may hinder thestorage and retrieval of arithmetical facts (Rotzer et al., 2009). It is currently unclear how the visuo-spatial sketchpad relatesto children’s arithmetic strategy development.

1.4. Phonological processing

It has been suggested that difficulties in phonological processing may explain MLD (Hecht, Torgesen, Wagner, & Rashotte,2001). This is not unexpected given the comorbidity of math and reading difficulties, and the high correlations oftenobserved between reading and mathematics performance (Shalev, 2007; Simmons & Singleton, 2008). Typically, threedomains of phonological processing are distinguished (Hecht et al., 2001; Wagner & Torgesen, 1987): (1) phonologicalawareness which refers to the conscious sensitivity to the phonological structure of oral language; (2) phonological workingmemory which refers to the short-term storage of phonological representations, analogous to the phonological loop ofBaddeley’s working memory model; (3) rate of access or rapid automatized naming (RAN) which includes children’s capacityto recall phonological representations from long-term memory. It has been demonstrated that these three phonologicalprocessing skills are associated with individual differences in mathematics achievement (Hecht et al., 2001) and that TDchildren outperform children with MLD on each of these domains of phonological processing (Murphy, Mazzocco, Hanich, &Early, 2007; Vukovic & Siegel, 2010).

Cognitive neuroimaging data (Dehaene, Piazza, Pinel, & Cohen, 2003) as well as findings from developmental behavioralstudies have indicated that arithmetic fact retrieval deficits may be the result of deficient phonological processing skills. Ithas been suggested that arithmetic facts might be stored in long-term memory as phonological codes and as a consequence,phonological processing deficits may lead to difficulties in storing and remembering these arithmetic facts from long-termmemory (e.g. Simmons & Singleton, 2008; Vukovic & Lesaux, 2013). For example, De Smedt and Boets (2010) showed inadults with dyslexia without mathematical difficulties that impaired phonological processing skills coincide with fewerarithmetic fact retrieval from long-term memory. Interestingly, De Smedt, Taylor, Archibald, and Ansari (2010) haveobserved a specific association between phonological awareness and arithmetic fact retrieval, indicating that the quality ofchildren’s long-term phonological representations might be related to the efficiency of using fact retrieval when solvingarithmetic problems.

K. Vanbinst et al. / Research in Developmental Disabilities 35 (2014) 3001–30133004

1.5. The present study

Previous studies have shown that children with MLD do not necessarily maintain this low achievement in mathematicsover time (e.g., Mazzocco & Myers, 2003; Mazzocco & Rasanen, 2013) and that children with such transient forms of MLD aremore similar to typically developing children (e.g., Mazzocco & Myers, 2003; Mazzocco & Rasanen, 2013). Therefore, thecurrent longitudinal study combined low achievement and persistency criteria to define children with persistent MLD (MLD-p). Children were included in the MLD-p group, if they performed below the 25th percentile (see also Swanson & Jerman,2006) on a general standardized mathematics achievement test at three time points. A matched control group of TD childrenwas selected on the basis of performance on the standardized mathematics achievement test above the 35th percentile ateach time point. The goal of this longitudinal study was to narrow down the focus from general mathematics achievement toarithmetic strategy development, in order to explore how difficulties in the arithmetic strategy development of childrenwith MLD-p relates to numerical magnitude processing, working memory and phonological processing. Such knowledgecould lead to important breakthroughs in the early identification and treatment of MLD.

We administered a single-digit addition and subtraction task and we simultaneously assessed strategy use with verbalprotocols, to test whether children with MLD-p had difficulties in arithmetic strategy use. Numerical magnitude processingskills were assessed by using numerical magnitude comparison tasks presented in symbolic (digits) and nonsymbolic (dots)formats. Domain-general cognitive skills were assessed by using non-numerical tasks, in order to prevent domain-generaltask performance to be influenced by domain-specific numerical processing. The three components of Baddeley’s workingmemory model were evaluated by a listening recall task (central executive), a corsi block task (visuo-spatial sketchpad) and anonword repetition task (phonological loop). Phonological processing involved the evaluation of phonological awareness(phoneme deletion and spoonerism task), phonological working memory (nonword repetition task) and rate of access (RANtasks).

The comparison of the two groups under study will allow us to investigate the arithmetic strategy development as well asthe domain-specific and domain-general cognitive skills of children with MLD-p. To rule out alternative explanations for theobserved differences between children with MLD-p and TD children, we matched both groups in terms of intellectual ability,age, reading ability, sex and socioeconomic status.

2. Method

2.1. Participants

Participants were selected from a larger ongoing longitudinal study in Flanders (Belgium) and came predominantly frommiddle- to upper middle-class families. The children had no history of developmental disorders or mental retardation, andtheir native language was Dutch. None of the children had repeated a grade. For all participants, written informed parentalconsent was obtained.

All 154 primary school children from the initial sample completed a curriculum-based general standardized mathematicsachievement test (Dudal, 2000), as a screening measure, at three time points: time 1 (September–October 2011), time 2(February–March 2012) and time 3 (September–October 2012). By combining low achievement and persistency criteria, agroup of 14 children with MLD-p (1 boy, 13 girls) was selected from this initial sample and was matched with 14 TD children(4 boys, 10 girls). Children were included in the MLD-p group, if they performed below the 25th percentile (Swanson &Jerman, 2006) on the general standardized mathematics achievement test at the three time points. TD children, had toperform above the 35th percentile on the general achievement test at each time point. Table 1 presents the detaileddescriptive statistics of the sample. Children with MLD-p (8 third graders, 6 fifth graders) and TD children (6 third graders, 8fifth graders) differed in their performance on the general standardized mathematics achievement test (F(1, 26) = 177.23,

Table 1

Descriptive statistics of the sample.

MLD-p group TD group

M SD Range M SD Range

General matha

Time 1 79.60 6.71 65.29–87.39 112.04 6.63 98.66–120.39

Time 2 82.02 5.24 70.15–90.20 110.61 8.10 97.76–121.08

Time 3 81.41 8.35 59.30–89.67 114.46 9.17 96.63–128.66

IQb 95.21 10.49 79–119 100.93 7.86 88–115

Agec 9.19 1.15 7.78–10.57 9.51 1.17 7.80–10.72

Reading abilityd 9.71 3.02 5–14 10.71 2.40 6–15

Note. MLD-p, persistent mathematical learning difficulties; TD, typically developing.a Standardized scores (M = 100, SD = 15).b IQ-score on Raven’s Standard Progressive Matrices.c Mean age in years at Time 1.d Mean standardized reading scores (M = 10, SD = 3) at Time 1.

K. Vanbinst et al. / Research in Developmental Disabilities 35 (2014) 3001–3013 3005

p< .01, h2p ¼ :87). Groups did not differ in terms of intellectual ability (t(26) =�1.63, p = .12), age (t(26) =�.72, p = .48),

reading ability (t(26) =�.97, p = .34), sex (x2 (1, 28) = 2.19, p = .14) and socioeconomic status (x2 (1, 28) = .19, p = .66). Bothgroups of children came from the same elementary schools and were attending the same classes (i.e., the same teachers andmethods). As a result, they were matched in terms of their educational history and environment.

2.2. Materials

Materials consisted of standardized tests, paper-and-pencil-tasks and computer tasks. All computer tasks were designedwith the E-prime 1.0 software (Schneider, Eschmann, & Zuccolotto, 2002).

2.2.1. General mathematics achievement

Children’s general mathematics competence was assessed with a standardized mathematics achievement test from theFlemish Student Monitoring System (Dudal, 2000). This validated series of tests was developed to assess children’s generalmathematics competence by providing untimed achievement tests corresponding to the mathematics curriculum of everygrade of primary school. The tests included various aspects of mathematics, such as number knowledge, understanding ofoperations, simple arithmetic, multidigit calculation, word problem solving, measurement and geometry. For each child, astandardized score (M = 100, SD = 15) was calculated. Reliability coefficients of these tests were: .91 at the start of third grade,.91 in the middle of third grade, .88 at the start of fourth grade, .91 at the start of fifth grade, .90 in the middle of fifth gradeand .91 at the start of sixth grade (Dudal, 2000).

2.2.2. Single-digit arithmetic and strategy use

Children’s arithmetic strategy use was assessed by means of a single-digit addition and subtraction task. Stimuliwere selected from the so-called standard set of single-digit arithmetic problems (Lefevre, Sadesky, & Bisanz, 1996),which excludes tie problems (e.g., 6 + 6) and problems containing 0 or 1 as operand or answer. Only one of each pair ofcommutative problems was selected, resulting in a set of 28 problems per operation. The position of the largest operandwas counterbalanced. Children were asked to perform both accurately and fast. Responses were verbal. A voice keyregistered the child’s reaction time, after which the experimenter recorded the child’s answer. Children could usewhatever strategy they wanted to. On a trial-by-trial basis, the experimenter asked the children to verbally report theirstrategy to solve the arithmetic problem. Similar to other studies in arithmetic (e.g., Torbeyns et al., 2004), strategieswere classified into retrieval (i.e., if the child immediately knew the answer and there was no evidence of overtcalculations), procedure (i.e., if the child indicated that he or she used counting or decomposed the problem into smallersub-problems to arrive at the solution), or other (i.e., if the child did not know how he or she solved the problem). Thisclassification method is a valid and reliable way of assessing children’s arithmetic strategy use (Siegler & Stern, 1998).Two practice trials were presented to familiarize children with task administration. Split-half reliability of the additiontask was .56 for accuracy and .87 for reaction time and split-half reliability of the subtraction task was .73 for accuracyand .95 for reaction time.

2.2.3. Cognitive correlates

2.2.3.1. Numerical magnitude comparison. Children’s numerical magnitude processing skills were measured by means ofsymbolic and nonsymbolic magnitude comparison tasks, consisting of Arabic digits and dot arrays, respectively. In thesetasks children had to compare two simultaneously presented numerical magnitudes, one displayed on the left side of thecomputer screen, and one displayed on the right. Children had to indicate the larger of those two numerical magnitudes bypressing a key on the side of the larger one. The left response key was d; the right response key was k. Children wereinstructed to perform both accurately and fast. Stimuli comprised all combinations of numerosities from 1 to 9, yielding 72trials for each task. The position of the largest numerosity was counterbalanced. The nonsymbolic stimuli were generatedwith the MATLAB script provided by Piazza, Izard, Pinel, Le Bihan, and Dehaene (2004) and were controlled for non-numerical parameters, such as dot size, total occupied area, and density. On one half of the trials, dot size, array size, anddensity were positively correlated with number, and on the other half of the trials, dot size, array size, and density werenegatively correlated. This was done to prevent that decisions were dependent on non-numerical cues or perceptualfeatures. A trial started with a 200 ms fixation in the center of the screen. After 1000 ms, stimuli appeared and remainedvisible until the child responded, except for the nonsymbolic magnitude comparison task, where the stimuli disappearedafter 840 ms, to avoid counting of number of dots. Each trial was initiated by the experimenter with a control key. Reactiontimes and answers were registered by the computer. To familiarize children with the key assignments, three practice trialswere included per task. Split-half reliability of the symbolic task was .76 for accuracy and .94 for reaction time; split-halfreliability of the nonsymbolic task was .67 for accuracy and .72 for reaction time.

2.2.3.2. Working memory. Three tasks from the working memory test battery for children (Pickering & Gathercole, 2001) thathave been widely used in working memory research were administered to tap the components of Baddeley’s workingmemory model. These tasks were the same as in De Smedt et al., 2009). We only selected non-numerical tasks to prevent taskperformance to be influenced by numerical processing.

K. Vanbinst et al. / Research in Developmental Disabilities 35 (2014) 3001–30133006

Central executive. Listening recall (Pickering & Gathercole, 2001), adapted to Dutch, was used to assess the centralexecutive component. In this task, children were asked to judge the correctness of a series of recorded sentences (correct vs.false). They were also instructed to memorize the last word in every sentence, and to recall those words in the presentedorder at the end of each trial. Reliability of this test was .64 (De Smedt et al., 2009).

Visuo-spatial sketchpad. The block recall task (Pickering & Gathercole, 2001) was used to assess the visuo-spatialsketchpad. For each trial, the experimenter tapped out a sequence, at a rate of one block per second, on a board with nineblocks. The child was instructed to reproduce the sequence in the correct order. Reliability of this test was .77 (De Smedtet al., 2009).

Phonological loop. This component was measured by a Dutch adaptation (Scheltinga, 2003) of the nonword repetition testdeveloped by Gathercole, Willis, Baddeley, and Emslie (1994). Each recorded nonword was presented once, and the child wasasked to repeat it immediately after its presentation. Reliability of this test was .81 (De Smedt et al., 2009).

2.2.3.3. Phonological processing. We selected different tasks to measure the three traditional domains of phonologicalprocessing (Hecht et al., 2001), which all have been widely used in research on reading and arithmetic (e.g., Boets et al., 2010for more elaborated task details).

Phonological awareness. Different tasks were used for third- and fifth-graders in order to avoid ceiling effects. In thephoneme deletion task, children were presented with nonwords and they were asked to delete a particular phoneme of thenonword, which resulted in the disclosure of an existing (e.g., DROOS without /d/) or meaningless (e.g., WAPT without /t/)word. Reliability of this task was .92 in third grade (Dandache, Wouters, & Ghesquiere, in preparation) and .79 in fifth grade(Boets et al., 2010). In the spoonerism task, children were required to replace the onset of a word with another consonant inorder to create a new word or nonword (e.g., LIP with /k/ becomes KIP) or they had to swap the consonant onset of two wordsin order to reveal two new words (e.g., SAAI-HOK becomes HAAI-SOK) or nonwords (e.g., GROEN-PLANT becomes PROEN-GLANT). Reliability of this task was .92 in third grade (Dandache et al., in preparation) and .90 in fifth grade (Boets et al.,2010).

Phonological working memory. We used the nonword repetition task described above as a measure of phonologicalworking memory.

Rate of access. Rate of access to phonological information from long-term memory was evaluated by two RAN tasks (vanden Bos, Zijlstra, & Van den Broeck, 2003), which involved the naming of 50 color (black, blue, red, yellow, and green) and 50letter (d, o, a, s and p) stimuli. The time to complete the card was recorded for each task.

2.2.4. Control measures

2.2.4.1. Intellectual ability. Raven’s Standard Progressive Matrices (Raven, Court, & Raven, 1992) was administered as ameasure of intellectual ability. For each child, a standardized score (M = 100, SD = 15) was calculated. Reliability for this testis .88 (Raven et al., 1992).

2.2.4.2. Reading ability. Reading ability was assessed by the standardized Dutch One-Minute-Test version B (Brus & Voeten,1995). In this test, children had to read a list of 116 single words of increasing difficulty as correctly and quickly as possible.The total score was the number of words read correctly within one minute, which was converted in a standardized score(M = 10, SD = 3). Reliability for this test is larger than .90 (Moelands & Rymenans, 2003).

2.3. Procedure

All participants completed the paper-and-pencil-tasks and the computerized tasks individually in a quiet room attheir own school. The order of task administration was fixed for all participants in order not to confound individualdifferences with task order effects. The general mathematics achievement test and Raven’s matrices were group-based. The general mathematics achievement test, the single-digit arithmetic task and the numerical magnitudecomparison tasks were administered at each time point (Time 1: September–October 2011; Time 2: February–March 2012; Time 3: September–October 2012). The remaining measures were only collected at the first timepoint.

3. Results

Trials with incorrect voice-key registration were excluded from the reaction time analyses (<3% of all trials).Reaction time data and strategy frequency data were calculated for the correct responses only. For single-digitarithmetic, trials deviating more than 3 SDs from a participant reaction time on each task were excluded as well as trialswith a reaction time below 500 ms (<.5% of all trials). For the numerical magnitude comparison tasks, trials withreaction times below 300 ms and higher than 5000 ms were additionally discarded (<.5% of all trials). The arithmeticperformance and strategy use of both groups are discussed first, followed by a description of the group differences onthe cognitive correlates.

Table 2

Mean accuracy and reaction time data on the single-digit addition and subtraction task as a function of group.

Task Time 1 Time 2 Time 3

MLD-p TD MLD-p TD MLD-p TD

Addition

Accuracya 93.21 (8.78) 96.14 (5.14) 96.31 (3.59) 99.43 (1.45) 96.68 (4.08) 98.98 (1.67)

Reaction timeb 4072.51 (1408.52) 2259.17 (810.79) 2893.75 (1101.51) 1754.11 (431.59) 2478.29 (796.61) 1490.56 (344.88)

Subtraction

Accuracya 88.93 (12.90) 94.29 (5.21) 95.29 (5.36) 97.86 (3.41) 95.15 (3.90) 98.72 (3.01)

Reaction timeb 5366.81 (2812.27) 2496.78 (809.30) 3511.61 (1453.07) 2056.69 (422.23) 3306.48 (984.75) 1945.35 (458.71)

Note. Standard deviations are presented between parentheses. MLD-p, persistent mathematical learning difficulties; TD, typically developing.a % correct.b Expressed in ms.

K. Vanbinst et al. / Research in Developmental Disabilities 35 (2014) 3001–3013 3007

3.1. Single-digit arithmetic and strategy use

Table 2 displays the mean reaction time and accuracy data in single-digit arithmetic per group across different timepoints. A 2� 3� 2 repeated measures ANOVA with operation (addition vs. subtraction) and time (time 1 vs. time 2 vs. time 3)as within-subject factors and group (MLD-p vs. TD) as between-subjects factor was conducted on accuracy and reaction time.There was a main effect of operation, indicating that additions were solved faster (F(1, 25) = 21.15, p< .01, h2

p ¼ :46) andmore accurately (F(1, 25) = 10.45, p< .01, h2

p ¼ :30) than subtractions. There was a main effect of time indicating that bothaccuracy (F(1, 25) = 7.05, p< .01, h2

p ¼ :22) and reaction time (F(1, 25) = 34.67, p< .01, h2p ¼ :58) improved over

developmental time. As expected, there was a main effect of group: children with MLD-p were systematically slower(F(1, 25) = 21.57, p< .01, h2

p ¼ :46) and less accurate (F(1, 25) = 7.24, p< .01, h2p ¼ :22) than TD children. The accuracy

analyses showed no interaction effects (ps> .15). With regard to reaction time, operation interacted with group membership(F(1, 25) = 5.09, p< .05, h2

p ¼ :17). Post hoc t-tests revealed that group differences were significant for both operations, butthey were slightly larger for subtraction (t(26) = 4.35, p< .01, d = 1.74) than for addition (t(26) = 4.30, p< .01, d = 1.69). Timealso interacted with group (F(1, 25) = 8.31, p< .01, h2

p ¼ :25). Post hoc t-tests indicated that group differences becamesmaller over time, yet children with MLD-p continued to perform more poorly than TD children (all ps< .01). No otherinteractions were observed.

Children’s arithmetic strategy development was examined by investigating the strategy distribution and strategyefficiency. It is important to point out that the same pattern of findings was observed for addition and subtractionseparately. In order to improve clarity, we therefore averaged the data across operations. Strategy distribution wasdetermined by calculating the frequency with which a strategy was used to solve arithmetical problems correctly.Because the frequency of trials belonging to the ‘‘other’’ category was very low (.5%), these trials were excluded fromfurther analyses. The mean frequencies for correct fact retrieval at time 1, time 2 and time 3 were M = 25.79 (SD = 7.11),M = 28.00 (SD = 6.52) and M = 28.50 (SD = 6.71) for children with MLD-p and M = 33.93 (SD = 11.98), M = 34.14(SD = 11.07) and M = 32.07 (SD = 9.75) for TD children. The mean frequencies of procedural strategy use at the differenttime points were M = 24.64 (SD = 9.79), M = 25.23 (SD = 6.72) and M = 24.29 (SD = 7.92) for children with MLD-p andM = 20.38 (SD = 10.28), M = 22.00 (SD = 10.07) and M = 22.71 (SD = 9.68) for TD children, respectively. A 3� 2 repeatedmeasures analysis on the frequencies of correct fact retrieval with time (time 1 vs. time 2 vs. time 3) as within-subjectfactor and group (MLD-p vs. TD) as between-subjects factor revealed a significant main effect of group (F(1, 25) = 4.41,p< .05, h2

p ¼ :15), indicating that TD children used fact retrieval more frequently than children with MLD-p. There wasno main effect of time (F(1, 25) = .34, p = .56) and no Group� Time interaction (F(1, 25) = 2.82, p = .11). This suggeststhat children with MLD-p continued to use less fact retrieval than their TD peers. A repeated measures analysis on thefrequencies of procedural strategy use was not reported, given that the percentages of fact retrieval and proceduralstrategy use are dependent on each other (e.g., more frequent fact retrieval means less frequent procedural strategyuse).

Strategy efficiency was determined by calculating the reaction time and accuracy with which fact retrieval andprocedural strategies were executed. Fig. 1 presents the mean reaction time and accuracy for each strategy per group pertime point. A 2� 3� 2 repeated measures ANOVA with strategy (retrieval vs. procedure) and time (time 1 vs. time 2 vs. time3) as within-subject factors and group (MLD-p vs. TD) as a between-subjects factor was calculated on reaction time andaccuracy. The analysis of the reaction time revealed main effects of strategy (F(1, 23) = 105.67, p< .01, h2

p ¼ :82) and group(F(1, 23) = 15.97, p< .01, h2

p ¼ :41): fact retrieval was executed faster than procedural strategies, and TD children executedall strategies faster than children with MLD-p. There was a main effect of time (F(1, 23) = 37.19, p< .01, h2

p ¼ :62). Post hoc t-tests demonstrated that children’s reaction time of executing all strategies decreased significantly from the first to thesecond time point (p< .01), but not from the second to the third time point (p = .21). Group membership interacted withstrategy (F(1, 23) = 18.78, p< .01, h2

p ¼ :45) and time (F(1, 23) = 8.61, p< .01, h2p ¼ :27). There was a Strategy� Time

interaction (F(1, 23) = 27.58, p< .01, h2p ¼ :55) as well as a Strategy� Time�Group interaction (F(1, 23) = 8.53, p< .01,

h2p ¼ :27). Post hoc t-tests demonstrated that group differences were larger for procedural strategies (t(23) = 4.21, p< .01,

[(Fig._1)TD$FIG]

Fig. 1. Mean error rate and reaction time by strategy per group. Lines represent reaction time on the left y-axis and bars depict error rates on the right y-axis.

K. Vanbinst et al. / Research in Developmental Disabilities 35 (2014) 3001–30133008

d = 1.76) than for fact retrieval (t(25) = 3.13, p< .01, d = 1.25). Group differences became smaller over time, particularly forprocedural strategies, but remained significant at each time point (all ps< .01).

With regard to accuracy, there was a main effect of strategy (F(1, 23) = 7.45, p< .05, h2p ¼ :25) indicating that fact retrieval

was executed more accurately than procedural strategies. There was only a trend toward a main effect of time (F(1,23) = 3.07, p = .09). There was a main effect of group (F(1, 23) = 8.83, p< .01, h2

p ¼ :28), showing that children with MLD-pmade more errors than TD children. Strategy interacted with group (F(1, 23) = 4.93, p< .05, h2

p ¼ :18), suggesting that groupdifferences were larger for procedural strategies (t(23) =�2.77, p< .01, d =�1.16) than for fact retrieval (t(25) =�2.19,p< .05, d =�.88). There was no Time�Group interaction (F(1, 23) = .60, p = .45), suggesting that group differences remainedstable over time. No further interactions were observed (ps> .18).

3.2. Cognitive correlates

3.2.1. Numerical magnitude comparison

The mean reaction time and accuracy on the numerical magnitude comparison tasks are displayed in Fig. 2 for each groupper time point. We calculated a 3� 2 repeated measures ANOVA with time (time 1 vs. time 2 vs. time 3) as within-subjectfactor and group (MLD-p vs. TD) as between-subjects factor on the reaction time and accuracy for each comparison taskseparately.

3.2.1.1. Symbolic magnitude comparison. The analysis of the symbolic reaction time revealed a main effect of time (F(1,25) = 57.13, p< .01, h2

p ¼ :70) indicating that both groups became faster over time (all ps< .01). There was a main effect ofgroup (F(1, 25) = 5.59, p< .05, h2

p ¼ :18) and a significant Time�Group interaction (F(1, 25) = 5.91, p< .05, h2p ¼ :19).

Children with MLD-p were significantly slower than TD children, yet the group differences became smaller over time (Time1: t(26) = 2.49, p< .05, d = .98; Time 2: t(25) = 2.45, p< .05, d = .98; Time 3: t(26) = 1.69, p = .10). With regard to the accuracy,there was a significant main effect of time (F(1, 25) = 5.49, p< .05, h2

p ¼ :18). There was no main effect of group (F(1, 25) = .09,p = .76) and time did not interact with group membership (F(1, 25) = 2.29, p = .14).[(Fig._2)TD$FIG]

Fig. 2. Mean error rate and reaction time by group per numerical comparison task. Lines represent reaction time on the left y-axis and bars depict error rates

on the right y-axis. Small dashes represent Time 1, large dashes represent Time 2 and solid lines represent Time 3.

Table 3

Descriptive statistics on the administered working memory tasks.

Task MLD-p TD Maximum possible t d

M (SD) M (SD)

Central executive

Listening span 6.57 (1.51) 7.14 (1.51) 15 �1.00 �.39

Visuo-spatial sketchpad

Block recall 10.29 (2.27) 10.64 (2.37) 27 �.41 �.16

Phonological loop

Nonword repetition 30.93 (7.44) 36.36 (5.05) 48 �2.26* �.89

Note. MLD-p, persistent mathematical learning difficulties; TD, typically developing.

* p< .05.

K. Vanbinst et al. / Research in Developmental Disabilities 35 (2014) 3001–3013 3009

3.2.1.2. Nonsymbolic magnitude comparison. With regard to reaction time, there was a main effect of time (F(1, 26) = 28.57,p< .01, h2

p ¼ :52), both groups became faster over time (all ps< .01). There was no main effect of group (F(1, 26) = .93, p = .34)and time did not interact with group membership (F(1, 26) = .39, p = .54). The analysis of accuracy on the nonsymbolicmagnitude comparison task showed no main effects of time (F(1, 26) = 1.44, p = .24) or group (F(1, 26) = 1.25, p = .27), and noTime�Group interaction (F(1, 26) = .35, p = .56).

3.2.2. Working memory

The descriptive statistics of the administered working memory tasks are displayed in Table 3 and indicate that only groupdifferences emerged on the nonword repetition task.

3.2.3. Phonological processing

Table 4 presents the descriptive statistics on the administered phonological processing tasks. Children with MLD-p scoredsignificantly lower than TD children on the spoonerism task, nonword repetition task and the naming of letters but not ofcolors.

4. Discussion

Deficits in arithmetic fact retrieval constitute the hallmark of children with MLD. It remains, however, unclear whichcognitive deficits may underpin difficulties in arithmetic strategy development. Many studies defined children with MLD byonly considering low achievement criteria and not by additionally taking the persistence of mathematical difficulties intoaccount. This latter criterion is, however, crucial because many children with low achievement in mathematics do notnecessarily maintain this low achievement over time (e.g., Mazzocco & Myers, 2003; Mazzocco & Rasanen, 2013). Narrowingdown the focus to arithmetic strategy development, the present longitudinal study investigated numerical magnitudeprocessing, working memory and phonological processing in a group of children with MLD-p and a TD group, in order toexplore the associations between these cognitive correlates and arithmetic strategy development. Both groups did not differin terms of intellectual ability, age, reading ability, sex and socioeconomic status. Our longitudinal data showed weakerarithmetic performance in children with MLD-p than in TD children at each time point: children with MLD-p retrieved fewerfacts from their memory and were slower and less accurate in executing fact retrieval as well as procedural strategies. Both

Table 4

Descriptive statistics on the administered phonological processing tasks.

Task MLD-p TD t d

M (SD) M (SD)

Phonological awareness

Phoneme deletiona �.25 (1.30) .38 (1.07) �1.25 �.55

Spoonerisma �.30 (.93) .84 (1.06) �2.73* �1.20

Phonological working memory

Nonword repetitionb 30.93 (7.44) 36.36 (5.05) �2.26* �.89

Rate of access

Color namingc 46.50 (9.50) 43.71 (12.84) .65 .26

Letter namingc 36.43 (9.68) 28.00 (8.04) 2.51* .98

Note. MLD-p, persistent mathematical learning difficulties; TD, typically developing.

* p< .05.a Expressed as z scores.b Maximum possible = 48.c Expressed in seconds.

K. Vanbinst et al. / Research in Developmental Disabilities 35 (2014) 3001–30133010

groups differed in symbolic magnitude processing and phonological processing, but not in nonsymbolic magnitudeprocessing and working memory.

4.1. Arithmetic strategy development

Our group of children with MLD-p solved single-digit addition and subtraction less frequently with fact retrieval andthis fact retrieval deficit persisted over developmental time. The results of this study also converge with prior studies,showing that children with MLD executed strategies less efficiently than TD children (e.g., Geary, Hoard, et al., 2012; Jordanet al., 2003). At all three time points, children with MLD-p executed fact retrieval and procedural strategies slower and withmore errors than TD children. With progressing time, children with MLD-p improved in terms of speed of executingstrategies. Although these improvements narrowed the initial learning gap, group differences remained significant at eachtime point.

4.2. Numerical magnitude processing

The current longitudinal data demonstrate that children with MLD-p show persistent deficits in their ability to compareArabic symbols, but not nonsymbolic dot arrays. These findings are consistent with previous data that observed animpairment in symbolic, but not nonsymbolic, magnitude processing in children with MLD (e.g., De Smedt & Gilmore, 2011;Iuculano et al., 2008; Landerl & Kolle, 2009; Rousselle & Noel, 2007). The current longitudinal data go beyond the previousones by showing that this deficit in symbolic magnitude processing remains stable over time. In all, these data providefurther support for the access deficit hypothesis (Rousselle & Noel, 2007) and suggest that particularly the access tonumerical meaning from Arabic symbols is impaired in children with MLD-p. On the other hand, the present findingscontrast with previous research that has shown impairments in nonsymbolic magnitude processing in children with MLD(e.g., Mazzocco et al., 2011; Mussolin, Mejias, & Noel, 2010; Piazza et al., 2010). It is important to point out that there areconflicting findings on the association between nonsymbolic magnitude processing and mathematics achievement, an issuethat might be explained by recourse to methodological issues, such as the design of stimuli in nonsymbolic tasks (see DeSmedt et al., 2013; for a discussion).

The current findings are in agreement with the findings of Vanbinst et al. (2012), which demonstrated that children’ssymbolic, but not nonsymbolic, magnitude processing skills were associated with individual differences in arithmeticstrategy use, i.e. the frequency of fact retrieval and the speed of executing retrieval and procedural strategies. This allconverges to the hypothesis that particularly the access to numerical meaning from Arabic symbols is key for children’sarithmetic strategy development and suggests that screening symbolic magnitude processing skills might be useful fordetecting children at risk for developing arithmetic fact retrieval deficits (e.g., Nosworthy, Bugden, Archibald, Evans, &Ansari, 2013).

How may this persistent impairment in accessing magnitude representations from Arabic symbols be associated witharithmetic fact retrieval deficits? First, it may be that impairments in symbolic magnitude processing induce a delay in theacquisition of the counting-on-larger strategy (e.g., 4, 5, 6, 7 to solve 3 + 4), as this strategy requires a decision on the largernumber for which access to numerical meaning of the presented Arabic symbols is needed. This delay may lead to difficultiesdeveloping problem–answer associations in long-term memory, which, in turn, hinders children’s transition to advancedsolution strategies. Second, it has been hypothesized that arithmetic facts may be stored in long-term memory in ameaningful way, i.e., according to their magnitude (e.g., Butterworth, Zorzi, Girelli, & Jonckheere, 2001) and it might be thatmeaningful facts are easier to store and to recall from memory (see Robinson, Menchetti, & Torgesen, 2002, for a similarsuggestion). Future empirical research is needed to test these hypotheses more carefully.

4.3. Working memory

We did not observe significant group differences on measures of the central executive and the visuo-spatial sketchpad.While these findings are in accordance with some prior studies (e.g., Geary, Hamson, & Hoard, 2000; Vukovic & Siegel, 2010),they are different from others (e.g., Geary, Hoard, & Nugent, 2012; Szucs et al., 2013). Three explanations might account forthese inconsistencies.

First, studies have used different types of stimuli to measure children’s central executive capacities. Some of these studieshave used numerical tasks (e.g., Chong & Siegel, 2008; Wu et al., 2008), such as a counting span or a backward digit recalltask. Such numerical tasks are problematic in the context of research on MLD, because poor performance on these tasksmight reflect poor numerical processing, poor working memory or both. To overcome this issue, we used a non-numericaltask.

Second, studies vary in the age of the participants under study and it has been reported that the association betweenworking memory and arithmetic skills depends on age (see Friso-van den Bos et al., 2013, for a meta-analysis). The centralexecutive and the visuo-spatial sketchpad may be particularly important during the early stages of learning single-digitarithmetic, and a more diminished reliance on these working memory resources might occur during more automatizedstages of arithmetic strategy development in older children, as in the current sample (De Smedt et al., 2009; Murphy et al.,2007; Raghabur, Barnes, & Hecht, 2010).

K. Vanbinst et al. / Research in Developmental Disabilities 35 (2014) 3001–3013 3011

Third, the existing body of evidence on working memory and mathematics included various types of mathematical skill toexamine their association with working memory. In this study, we focused on one particular skill, i.e. arithmetic strategydevelopment. Because working memory components might affect specific mathematical skills to different degrees (seeGeary et al., 2007; Friso-van den Bos et al., 2013), this might explain the current pattern of findings.

4.4. Phonological processing

Group differences were observed in all three domains of phonological processing: TD children outperformed childrenwith MLD-p on a spoonerism task, a nonword repetition task and a rapid letter naming task. These data suggest that childrenwith MLD-p have deficient phonological processing skills, which corresponds to previous research findings (e.g., Chong &Siegel, 2013; Fuchs et al., 2005; Hecht et al., 2001; Vukovic & Siegel, 2010). Our data also support prior studies which havelinked poor phonological processing skills and arithmetic fact retrieval deficits (e.g., Simmons & Singleton, 2008; Swanson &Sachse-Lee, 2001; Vukovic & Lesaux, 2013).

It is important to emphasize that group differences in phonological processing cannot be explained by poor reading skills,because the group of children with MLD-p did not differ in reading ability from the group of TD children and all scored closeto the population average. This all suggests that the observed group differences do not implicate that children with MLD-pare deficient in phonological processing in the same way as children with reading difficulties or comorbid reading and MLD(also see, Murphy et al., 2007).

It can therefore be suggested that the ability to make fine-grained phonological differentiations may support children’stransition from (time-consuming) counting strategies to direct arithmetic fact retrieval.

4.5. Limitations and future research

When evaluating the current findings, it is important to note that they were based on a small sample size, which is due tothe fact that we adopted a strict criterion for determining MLD-p, i.e. poor performance at three time points. In addition, thegroup of children with MLD-p turned out to be constituted by more girls than boys. While such unequal distribution ofgender might affect the generalization of our findings, data from epidemiological studies (e.g., Else-Quest, Hyde, & Linn,2010; Koumoula et al., 2004) as well as recent evidence (Devine, Soltesz, Nobes, Goswami, & Szucs, 2013) indicate that theprevalence of MLD is similar for girls and boys. Moreover, Devine et al. (2013) also observed no gender differences inmathematics performance. As a result, the current findings can be reliably applied to the broader population of MLD. It is alsoimportant to note that the current conclusions are limited to the specific developmental period under investigation, i.e. thirdto fifth grade. It therefore remains to be determined how numerical magnitude processing, working memory andphonological processing are impaired in MLD during early, less automatized, stages of arithmetic strategy development.

Future research should also consider other cognitive skills that may contribute to difficulties in arithmetic strategydevelopment. For example, it has been suggested that the inhibition of incorrect answers from long-term memory might beparticularly important for successful arithmetic fact development (e.g., Geary, 2004; Fias, Menon, and Szucs, 2013). Recentevidence by Szucs et al. (2013) suggests indeed that children with MLD differ from TD children in inhibition skills. Futurestudies should further unravel how deficient inhibition skills might be related to difficulties in fact retrieval. On the otherhand, non-cognitive correlates, such as math anxiety, should be considered, as they also contribute to poor mathematicsachievement (Ashcraft & Ridley, 2005).

Current data do not allow us to establish causal associations between arithmetic strategy development and cognitivedeficits. It would therefore be interesting to conduct targeted intervention studies (e.g., board games, computer games) inorder to investigate whether training of domain-specific (i.e., symbolic number processing) and/or domain-general (i.e.,phonological processing) cognitive skills leads to more efficient strategy use and to an increasing reliance on fact retrieval.

Acknowledgements

This research was supported by grant G.0359.10 of the Research Foundation Flanders (FWO), Belgium. We would like tothank all participating children, parents and teachers.

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