Identification of children with mathematics learning disabilities (MLDs) using latent class growth analysis Terry T.-Y. Wong a,b, *, Connie S.-H. Ho b , Joey Tang c a Department of Psychological Studies, The Hong Kong Institute of Education, Hong Kong b Department of Psychology, The University of Hong Kong, Hong Kong c Society for the Promotion of Hospice Care, Hong Kong 1. Introduction 1.1. What is mathematics learning disability (MLD)? Mathematics learning disability (MLD), or developmental dyscalculia (DD), has been traditionally described as a disorder of arithmetic skills that reflects a discrepancy between one’s low arithmetical abilities compared with his/her overall intelligence level and chronological age (Mazzocco & Ra ¨sa ¨ nen, 2013). Approximately 7% of the population suffer from this disability (Shalev, 2007). Although numeracy skills are found to influence a person’s educational, financial, and even health status to a similar extent as reading skills (Parsons & Bynner, 2005), there are far fewer research studies on MLD than on reading disabilities (Bishop, 2010; Gersten, Clarke, & Mazzocco, 2007). Because of the limited studies carried out on MLD, there is little consensus on the definition (Kaufmann et al., 2013) and the cognitive profiles of children with MLD (Mazzocco, 2007). Research in Developmental Disabilities 35 (2014) 2906–2920 ARTICLE INFO Article history: Received 9 April 2014 Received in revised form 3 July 2014 Accepted 7 July 2014 Available online Keywords: Numerical cognition Mathematics learning disability Latent class growth analysis Approximate number system ABSTRACT The traditional way of identifying children with mathematics learning disabilities (MLDs) using the low-achievement method with one-off assessment suffers from several limitations (e.g., arbitrary cutoff, measurement error, lacking consideration of growth). The present study attempted to identify children with MLD using the latent growth modelling approach, which minimizes the above potential problems. Two hundred and ten Chinese-speaking children were classified into five classes based on their arithmetic performance over 3 years. Their performance on various number-related cognitive measures was also assessed. A potential MLD class was identified, which demonstrated poor achievement over the 3 years and showed smaller improvement over time compared with the average-achieving class. This class had deficits in all number-related cognitive skills, hence supporting the number sense deficit hypothesis. On the other hand, another low-achieving class, which showed little improvement in arithmetic skills over time, was also identified. This class had an average cognitive profile but a low SES. Interventions should be provided to both low-achieving classes according to their needs. ß 2014 Elsevier Ltd. All rights reserved. * Corresponding author at: Room 19, 1/F, B4, The Hong Kong Institute of Education, 10 Lo Ping Road, Tai Po, New Territories, Hong Kong. Tel.: +852 9878 6870. E-mail address: [email protected](Terry T.-Y. Wong). Contents lists available at ScienceDirect Research in Developmental Disabilities http://dx.doi.org/10.1016/j.ridd.2014.07.015 0891-4222/ß 2014 Elsevier Ltd. All rights reserved.
Transcript
Research in Developmental Disabilities 35 (2014) 2906–2920
Contents lists available at ScienceDirect
Research in Developmental Disabilities
Identification of children with mathematics learning
disabilities (MLDs) using latent class growth analysis
Terry T.-Y. Wong a,b,*, Connie S.-H. Ho b, Joey Tang c
a Department of Psychological Studies, The Hong Kong Institute of Education, Hong Kongb Department of Psychology, The University of Hong Kong, Hong Kongc Society for the Promotion of Hospice Care, Hong Kong
A R T I C L E I N F O
Article history:
Received 9 April 2014
Received in revised form 3 July 2014
Accepted 7 July 2014
Available online
Keywords:
Numerical cognition
Mathematics learning disability
Latent class growth analysis
Approximate number system
A B S T R A C T
The traditional way of identifying children with mathematics learning disabilities (MLDs)
using the low-achievement method with one-off assessment suffers from several
limitations (e.g., arbitrary cutoff, measurement error, lacking consideration of growth).
The present study attempted to identify children with MLD using the latent growth
modelling approach, which minimizes the above potential problems. Two hundred and
ten Chinese-speaking children were classified into five classes based on their arithmetic
performance over 3 years. Their performance on various number-related cognitive
measures was also assessed. A potential MLD class was identified, which demonstrated
poor achievement over the 3 years and showed smaller improvement over time compared
with the average-achieving class. This class had deficits in all number-related cognitive
skills, hence supporting the number sense deficit hypothesis. On the other hand, another
low-achieving class, which showed little improvement in arithmetic skills over time, was
also identified. This class had an average cognitive profile but a low SES. Interventions
should be provided to both low-achieving classes according to their needs.
� 2014 Elsevier Ltd. All rights reserved.
1. Introduction
1.1. What is mathematics learning disability (MLD)?
Mathematics learning disability (MLD), or developmental dyscalculia (DD), has been traditionally described as a disorderof arithmetic skills that reflects a discrepancy between one’s low arithmetical abilities compared with his/her overallintelligence level and chronological age (Mazzocco & Rasanen, 2013). Approximately 7% of the population suffer from thisdisability (Shalev, 2007). Although numeracy skills are found to influence a person’s educational, financial, and even healthstatus to a similar extent as reading skills (Parsons & Bynner, 2005), there are far fewer research studies on MLD than onreading disabilities (Bishop, 2010; Gersten, Clarke, & Mazzocco, 2007). Because of the limited studies carried out on MLD,there is little consensus on the definition (Kaufmann et al., 2013) and the cognitive profiles of children with MLD (Mazzocco,2007).
* Corresponding author at: Room 19, 1/F, B4, The Hong Kong Institute of Education, 10 Lo Ping Road, Tai Po, New Territories, Hong Kong. Tel.: +852 9878 6870.
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1.2. The low-achievement method
Previously, the majority of the studies on MLD employed the low-achievement method of identifying children with MLD(e.g., Mazzocco & Thompson, 2005; Ostad, 1997; Passolunghi & Siegel, 2001). Children are identified as having MLD if theirmathematics achievement scores below a certain percentile while their intellectual functioning falls within the normalrange. However, there is a large variation in the cutoff values used in different MLD studies. A recent study summarized thelatest MLD studies and found that the cutoff values employed in different studies ranged from the 5th to the 45th percentileon standardized mathematics achievement tests (Murphy, Mazzocco, Hanich, & Early, 2007). Clearly, with such a hugediscrepancy in the cutoff values used, these studies usually identify different groups of children. It has been shown that theuse of different cutoff values results in different cognitive profiles of the MLD samples identified (Murphy et al., 2007). Theresulting inconsistency may hinder us from understanding the cognitive profiles of children with MLD.
To minimize the effect of the different cutoff values used, most recent studies used the tripartite method and identifiedthree groups of children according to their mathematics achievement (e.g., Cowan & Powell, 2014; De Smedt & Gilmore,2011; Geary, Bailey, & Hoard, 2009; Mazzocco, Feigenson, & Halberda, 2011b; Murphy et al., 2007). Children who show themost severe deficits in mathematics (e.g., lowest 10th percentile or 1.5 S.D. below the mean) are identified as MLD, whilethose who show milder deficits in mathematics (e.g., between the 11th and 25th percentile or between 1 to 1.5 S.D. belowthe mean) are identified as low-achieving (LA). The cognitive profiles of these two groups are then compared with theirnormally achieving (NA) peers. Studies using the tripartite method usually found significant differences in cognitive profilesbetween the MLD group and the other two groups (e.g., Cowan & Powell, 2014; Mazzocco, Feigenson, & Halberda, 2011a;Mazzocco et al., 2011b; Murphy et al., 2007). With more stringent criteria set for the MLD group, we can be more confidentthat the MLD group identified represents the true MLD population instead of containing some children who only have a milddegree of difficulties. The chance of having false positives is reduced.
Although the low achievement method (with either one or two cutoff values) has been used in most of the studies onMLD, there are several limitations that come along with this method. First, the cutoff values involved in the low-achievementmodel are arbitrary. These cutoff values may be used because they have been commonly used in previous studies instead ofbecause they reflect something truly meaningful. A more important concern is that, as mentioned above, there is a hugerange of cutoff values used in previous studies, and this variation results in different samples of MLD children identified(Murphy et al., 2007). Although the use of the tripartite approach has improved the situation slightly by identifying childrenwith different degrees of difficulties, the issue of arbitrariness is not eliminated. It is possible that children within one groupmay be more different than the children who score around the cutoff boundaries but are classified into different groups. Theuse of an arbitrary cutoff in identifying children with MLD therefore warrants concerns.
Second, when using the low-achievement method, some studies made the classification based on information from asingle time point (e.g., Chan & Ho, 2010; De Smedt & Gilmore, 2011; Landerl, Fussenegger, Moll, & Willburger, 2009). Theuse of information from a single time point is also problematic because of both the measurement error as well as the lackof consideration of children’s learning process. Due to measurement error, children who score around the cutoff valuemay be classified as MLD in one time point while not in the other time point. In Mazzocco and Myers’s (2003) study, forexample, approximately 30% of the children being identified as MLD in one grade were no longer identified as MLD inanother grade. Classifying children as MLD based on a one-off assessment therefore results in substantial classificationerror.
The lack of consideration of the learning process is another issue due to the classification of MLD with data from a singletime point. According to the Response-to-Intervention (RTI) approach of identifying learning disabilities, students who are atrisk for learning disabilities are those who are not able to learn at a reasonable rate and are unresponsive to suitableinterventions (Fletcher, Coulter, Reschly, & Vaughn, 2004). The rate of learning is therefore an important piece of informationin identifying learning disabilities that cannot be measured by any measurement from a single time point. Two previousstudies suggest the importance of incorporating growth in the identification of children with learning disabilities. In Morgan,Farkas, and Wu’s (2009) study, for example, children who had persistent mathematics difficulties in kindergarten showed aslower growth rate in mathematics in elementary school and were more likely to be identified as having MLD in elementaryschool compared with those who showed only transient difficulties or no difficulties. Having persistent difficulties inmathematics in kindergarten implies that these children have slow initial growth in mathematics, thus suggesting theimportance of growth in predicting later outcome. Evidence also comes from the field of reading. For example, in a group ofkindergarteners who participated in a kindergarten phonemic intervention programme, those who turned out to be childrenwith dyslexia later on were found to require a longer time to master the phonemic knowledge taught in the programme. Ontop of that, the number of sessions required for a child to master phonemic skills outperformed the child’s post-interventionphonemic skills in predicting their reading level at later ages (Byrne, Fielding-Barnsley, & Ashley, 2000). The usual practice ofidentifying children with MLD based on the assessment results from a single time point, which ignores the development ofskills with time, therefore warrants cautions.
1.3. The latent class growth analysis
In light of the above limitations of the traditional approach (i.e., arbitrary cutoff, measurement error, lackingconsideration of growth), a new latent growth modelling approach of identifying children with learning disabilities has
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recently been proposed (Boscardin, Muthen, Francis, & Baker, 2008). This approach clusters children into different groupsaccording to their achievement at multiple time points. Latent class growth analysis (LCGA) and growth mixture modelling(GMM) are the two major statistical techniques under this approach (Jung & Wickrama, 2008), with the former assumingthat the variance and covariance estimates for the growth factors within each class are zero, resulting in a moreparsimonious model.
This approach has several advantages over the traditional approach. First, the classification is based on data instead of thearbitrary cutoff points set by researchers. The classification may therefore reflect more genuine differences among theparticipants and may therefore be more meaningful. Second, the situation in which the participant is misclassified as MLDdue to unstable environmental factors (such as being sick during the testing) can be minimized when the results of severalassessments are taken into account. Third, this approach also considers the growth in children’s achievement, which isanother useful indicator of children’s learning disability status.
To demonstrate the reliability of the latent growth modelling approach of classification of learning disabilities,Boscardin et al. (2008) employed GMM to classify children into different classes based on their development ofphonological awareness in kindergarten and their reading development in primary school. It was found that theclassification based on kindergarten data was highly predictive of their later reading status, with a 100% correspondencebetween the learning disability status based on the kindergarten and the elementary school data. This group of childrenshowed worse performance in both phonological awareness and reading throughout the assessment period, with littleimprovement with time, which matched well with our expectation of how typical individuals with dyslexia behaved. Thelatent growth modelling approach of identifying children with learning disabilities therefore provides a possiblealternative for identifying children with MLD. However, to our knowledge, only one study attempted to identify childrenwith MLD using this approach (Geary, Bailey, Littlefield, et al., 2009). The present study therefore attempted to fill this gapby identifying children with MLD based on their development in arithmetic skills over a period of 3 years. The cognitiveprofiles of this group of children were also examined.
1.4. Number-specific cognitive measures
1.4.1. Approximate number system
The approximate number system (ANS) is one of the core systems that allows humans to represent numerositynonsymbolically (Feigenson, Dehaene, & Spelke, 2004). The ANS has been found right at birth (Izard, Sann, Spelke, & Streri,2009) and its precision improves with age (Halberda, Ly, Wilmer, Naiman, & Germine, 2012; Piazza et al., 2010). Neurologicalmodels suggest that the ANS encodes numerosity in an approximate manner along a logarithmically compressed mentalnumber line (Dehaene, Piazza, Pinel, & Cohen, 2003; Piazza, Izard, Pinel, Le Bihan, & Dehaene, 2004). The approximate natureof the ANS means that it allows us to discriminate two numerosities only when the ratio between them is large enough. Thenumber acuity, or the precision of the ANS representation of numerosity, is reflected by the finest ratio between twonumerosities that can be reliably distinguished (i.e., the Weber’s fraction, w).
The discovery of the ANS has stimulated the field of mathematical cognition. Many attempts have been made toinvestigate the relationship between this innate, nonsymbolic ‘number sense’ and humans’ symbolic mathematical skills.The earliest support for the relationship between the two comes from Halberda, Mazzocco, and Feigenson’s (2008) study,which reported a significant relationship between the number acuity of 14-year-old adolescents and their previousmathematics achievement. Later on, more studies verified the above relationship (e.g., Bonny & Lourenco, 2013; Libertus,Feigenson, & Halberda, 2013; Lonnemann, Linkersdorfer, Hasselhorn, & Lindberg, 2013; Wong, Ho, & Tang, under review b;but see Smedt, Noel, Gilmore, & Ansari, 2013 for a summary of studies that did not find a significant relationship between thetwo). The relationship between the ANS and humans’ mathematical skills has also been supported by the studies that foundANS deficits among children with MLD (Landerl et al., 2009; Mazzocco et al., 2011a, 2011b; Wong, Ho, & Tang, under reviewa; but see De Smedt & Gilmore, 2011; Rousselle & Noel, 2007 for the null findings).
1.4.2. Number–numerosity mapping
Number–numerosity mapping has been proposed to be the connecting step between humans’ number acuity and theirunderstanding of the number system (Geary, 2013). It can therefore be perceived as the mechanism of how the ANSinfluences our mathematical skills. Number–numerosity mapping has been measured by counting and estimation tasks inprevious studies (Mejias, Mussolin, Rousselle, Gregoire, & Noel, 2012; Wong et al., under review b). Both types of tasksrequire the participants to associate a numerosity with a number symbol.
Counting can be thought of as the first systematic way for children to learn about number symbols. It allows children tolearn the number symbols by associating these symbols to numerosities. Counting skills also serve as the foundation forchildren to learn arithmetic. Before children are able to directly retrieve the answers to arithmetic problems, they solve basicarithmetic through counting. Even after they are able to directly retrieve answers for arithmetic problems, they sometimesfall back on counting as a back-up strategy. Because of the above reasons, children’s counting skills have been found tocorrelate with their mathematics achievement (e.g., Aunio & Niemivirta, 2010; Reigosa-Crespo et al., 2012). On the otherhand, poor mathematics and poor counting usually go together. Children with MLD have been found to count more slowlythan their peers (Landerl, Bevan, & Butterworth, 2004), and children who count more slowly also perform worse inarithmetic (Reeve, Reynolds, Humberstone, & Butterworth, 2012).
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Estimation is another process in which humans associate a number symbol with a magnitude, such as a numerosity or alength, without the use of real-world knowledge (Booth & Siegler, 2006). Examples of estimation include estimating thenumber of candies in a bottle and the position of 38 on a 0–100 number line. The process of associating a magnitude with anumber symbol is similar to the proposed neurological model in which the magnitude of a number is being represented on amental number line (Dehaene et al., 2003). This resemblance may explain why children’s estimation skills are stronglycorrelated with their mathematics achievement (e.g., Ashcraft & Moore, 2012; Booth & Siegler, 2006; Fuchs et al., 2010) andwhy children (Geary, Hoard, Nugent, & Byrd-Craven, 2008; Mejias, Mussolin, et al., 2012) and adults (Mejias, Gregoire, &Noel, 2012) with MLD tend to perform more poorly in estimation tasks.
1.4.3. Symbolic number processing
The ability to compare the numerical values of numerals has been proposed to be one of the basic numeracy skills(Reigosa-Crespo et al., 2012), which are thought to represent how well the number symbols are being associated to thenumerosity representation (De Smedt & Gilmore, 2011; Rousselle & Noel, 2007). The efficiency of comparing numbers hasbeen found to correlate with one’s mathematics achievement (see De Smedt et al., 2013 for a review), and children with MLDtend to be slower in this task (De Smedt & Gilmore, 2011; Rousselle & Noel, 2007). The performance in the numbercomparison task is one of the defining criteria of developmental dyscalculia in the Dyscalculia Screener developed byButterworth (2003).
1.5. Number sense deficit vs. access deficit hypothesis
By examining the cognitive profiles of children with MLD, we could, at the same time, verify the two major hypothesesconcerning the major deficits among children with MLD. These two hypotheses are the number sense deficit hypothesis(Wilson & Dehaene, 2007) and the access deficit hypothesis (Rousselle & Noel, 2007). While the former suggests that thedeficits among children with MLD originates from a deficit in the number sense (i.e., ANS) that affects all other basicnumeracy skills and hence one’s mathematics achievement, the latter suggests that the deficit causing MLD lies in theinability to access numerosity information from number symbols, leaving the number sense intact. The current literaturereports mixed findings of the two hypotheses (supporting number sense deficit: Landerl et al., 2009; Mazzocco et al., 2011a,2011b; Piazza et al., 2010; supporting access deficit: De Smedt & Gilmore, 2011; Rousselle & Noel, 2007); it is thereforeunclear whether difficulties experienced by those with MLD are originated from a defective number sense or impairedaccess. Both hypotheses predict deficits in tasks that involve number symbols among children with MLD. If a deficit innumber acuity is found among children with MLD, then the number sense deficit is supported. Otherwise, the access deficit issupported.
1.6. The current study
The current study aimed to identify children with MLD using the latent growth modelling approach. LCGA was conductedto classify participants into different groups according to their arithmetic performance over a period of 3 years. The class thatshowed the worst performance over the three-year period would be the potential MLD candidates. The cognitive profiles ofthis class were compared with those of the other classes so that the cognitive deficits of this class of children could beidentified. The number sense deficit hypothesis and the access deficit hypothesis could therefore be compared. A readingmeasure was also included to check if children in different classes also differed in their reading performance.
2. Method
2.1. Participants
Two hundred ten participants (52.4% boys) participated in the present study. They were recruited from 17 differentkindergartens in Hong Kong. Hong Kong children spend 3 years in kindergartens from the age of three. All of the participantswere Chinese-speaking children, with a mean age of 6 years, 1 month (S.D. = 4 months) in the first wave of assessment. Theywere in the third year of kindergarten during the first wave of assessment. Due to attrition, the sample size dropped to 154 inthe fourth wave of assessment (see Table 2 for the sample sizes at each time point).
2.2. Measures
2.2.1. Arithmetic
The arithmetic task, which was designed based on the local mathematics curriculum, consisted of 30 arithmetic items,ranging from single-digit addition to three-digit subtraction to mixture problems (addition and subtraction). With thedevelopment in participants’ arithmetic skills, we included 20 extra items in Time 3 and 4, resulting in a total of 50 items. Theadded items included multiplication, division, and mixture problems of the four operations (see Appendix 1 for the items).The task had adequate internal reliability (Cronbach’s a> 75) and external validity (correlated with the Learning andAchievement Measurement Kit 2.0, a locally normed achievement test, at 0.4 or above).
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2.2.2. Number-related cognitive measures
All of the number-related cognitive measures were conducted on computer.
2.2.2.1. Approximate number system (ANS). The ANS of the participants was assessed using the nonsymbolic comparison taskand the nonsymbolic multiplication task. The nonsymbolic comparison task was adapted from Piazza et al. (2004), in whichthe participants were asked to compare the two arrays of dots presented and to decide on the array containing more dotswithout counting. There were always 16 dots in one of the arrays, while the other array consisted of 10–22 dots. The ratio ofdots involved ranged from 1.0625 to 1.60. Dots were presented in varying dot sizes (diameter ranging from 6 to 58 pixels).The average dot size was directly proportional to the numerosity in half of the trials, while the relationship was reversed inthe other half. There were five practice trials followed by 50 experimental trials. Both the accuracy and the Weber’s fraction(w) were used in the analyses.
The nonsymbolic multiplication task was modified from the nonsymbolic addition task by Gilmore, McCarthy, and Spelke(2010). In each item, participants first saw an array of dots being covered by a shade. The shade then multiplied itself by afactor of 2, 3, or 4, indicating that the array of dots being covered was multiplied by that factor. Finally, another arrayappeared on the other side of the screen, and the participants had to decide which array (the product vs. the comparison)contained more dots. Dots were represented in varying dot sizes (diameter ranging from 6 to 42 pixels). The average dot sizewas directly proportional to the numerosity in half of the trials, while inversely proportional to the numerosity in the otherhalf. A PowerPoint animation was first shown to the participants to familiarize them with the task. There were four practiceitems followed by 24 experimental items.
2.2.2.2. Number–numerosity mapping. The precision of mapping between number symbols and numerosity was assessedusing counting and estimation tasks.
The counting skills of the participants were assessed using the dot–number matching and dot–dot matching tasks. Thedot–number matching task was adapted from Butterworth’s (2003) Dyscalculia Screener. Participants saw an Arabicnumeral on one side of the screen and an array of dots on the other side, and their task was to decide whether the numericalvalue of the numeral matched the number of dots. The dot–dot matching was similar to the dot–number matching, exceptthat the participants saw two arrays of dots this time and they had to decide whether the two arrays were identical innumber. The time limit was 8 s for both tasks. The numerosity involved in both tasks ranged from one to nine. There werefour practice items followed by 36 experimental items in both tasks.
Participants’ estimation skills were assessed using the numerosity production task and the number line estimation task. Thenumerosity production task was adapted from Crollen, Castronovo, and Seron (2011). Participants were first shown an array of100 dots in hardcopy, and they were told about the numerosity. Afterwards, an Arabic numeral appeared on the computerscreen, and the participants’ task was to produce that number of dots by pressing certain keys (e.g., ‘‘J’’ for producing dots and ‘‘F’’for deleting dots). Because this was an estimation task, participants were told to press the keys so that they had to estimate,instead of count, the number of dots produced. The 100-dot hardcopy remained beside the computer so that they could refer toit. Dots were presented in varying sizes (diameter ranging from 6 to 82 pixels). In half of the trials, the average dot size was heldconstant so that the total occupied area was directly related to the target numerosity. In the other half of the trials, the totaloccupied area was held constant so that the average dot size was inversely related to the target numerosity. There were threepractice trials (numerosities of 3, 5, and 7) followed by 20 experimental trials (numerosity ranging from 10 to 98).
The number line estimation task was adapted from Siegler and Booth’s (2004) study. Participants were shown a numberline with 0 and 100 on the two ends. An Arabic numeral was shown above the number line, and the participants were askedto locate that numeral on the correct position on the number line. The time limit was 30 s. There were three practice trials(50, 25, and 75) preceding the 24 experimental trials. Due to the ceiling effect observed in Time 3, the number range in Time 4was changed to 0–1000 instead. For both estimation tasks, linearity, or the fit towards the best-fit linear model, was taken asthe indicator of participants’ estimation skills.
2.2.2.3. Symbolic number processing. The number comparison task from the Dyscalculia Screener (Butterworth, 2003) wasused to assess children’s symbolic number processing skills. Two Arabic numerals were presented on the computer screeneach time, and the participants had to decide which numeral had a greater numerical value. The numerals involved rangedfrom 1 to 9, with a numerical distance between one and four. There were four practice trials and 36 experimental trials. Bothaccuracy and reaction time (considering only the correct trials) were analyzed.
2.2.3. Control measures
2.2.3.1. Nonverbal intelligence. Participants’ nonverbal intelligence was assessed using the short form of Raven’s StandardProgressive Matrices (Raven, 1956). In each item, there was a pattern with a missing piece. Participants had to choose fromamong six to eight pieces a piece that could complete the pattern. A total of 36 items were involved in the short form. The rawscores were then converted to scaled scores based on the local norm.
2.2.3.2. Working memory. The working memory capacity of the participants was assessed by the backward digit span task.Participants were verbally presented with a sequence of digits at a rate of one per second. After listening, they had to recall
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the digit in a backward manner. The recalled sequence had to be completely correct in order to score one mark. There wereseven levels in the task, with two items in each level. Starting with a level of two digits, each successive level had one moredigit to be recalled. The task would be terminated when the participant failed both items in a level. Two practice trials weregiven beforehand to ensure that the participants understood the task.
2.2.4. Reading
The reading achievement of the participants was assessed using the Chinese word reading subtest of the Hong Kong Testof Specific Learning Difficulties in Reading and Writing for Primary School Students – Second Edition, or HKT-P(II) (Ho et al.,2007). The HKT-P(II) is a standardized diagnostic tool for identifying children with dyslexia. The Chinese word readingsubtest consists of 150 two-character Chinese words in ascending difficulty. Participants were asked to read aloud the words.Each correctly read word yielded one mark. The test would be discontinued if the participants failed to get any marks in 15consecutive items. The raw scores were converted into the standardized scores based on the local norm.
2.3. Procedures
A total of 17 kindergartens participated in the present study, with a total sample of 210 recruited initially. Testing wasconducted in the kindergartens in the first wave of assessment, while the following waves of testing were conducted inparticipants’ homes. While the cognitive and arithmetic tasks were conducted in all waves of assessment, the working memory,nonverbal intelligence, and reading measures were conducted in one of the waves only (the first, the second, and the fourthwave, respectively). Waves 2 and 3 were conducted half a year after the previous wave, and wave 4 was one year after wave 3.Each assessment session lasted for approximately 2 h. All of the testing was conducted either by the first author or by trainedexperimenters who were psychology undergraduates. Souvenirs were given to the participants after each testing.
3. Results
3.1. The latent class growth model
Participants were first classified into different classes by the LCGA (Jung & Wickrama, 2008) according to their arithmeticperformance across the four time points using Mplus, Version 6.0 (Muthen & Muthen, 2011). The number of classes in thefinal solution was determined based on both the interpretability of the class structure and the model fit indices. The model fitindices of two- to six-class models are presented in Table 1. While the four-class solution obtained the lowest BayesianInformation Criterion (BIC) value, the p-value of the bootstrapped likelihood ratio test (BLRT) suggested the five-classsolution. Because it has been suggested that the BLRT performs best in selecting models (Nylund, Asparouhov, & Muthen,2007), we followed this criterion and selected the five-class solution. This five-class solution, with adequate sample size inthe smallest group and distinct profiles in different classes, was also interpretable. The entropy of the five-class model was0.736, with the values on the diagonal of the posterior probability matrix (indicating the distinctness of different classes)ranging from 0.725 to 0.964.
The classes in the five-class solution were labelled according to their performance in arithmetic. The top class (n = 14,6.7%) had the highest intercept and obtained the highest scores throughout the four time points. The high class (n = 49, 23.3%)
Table 1
Fit indices of models with different numbers of classes.
No. of classes BIC LMR p-value BLRT p-value Entropy
Note. BIC, Bayesian information criterion; LMR, Lo–Mendell–Rubin; BLRT, bootstrapped likelihood ratio test. The row in boldface indicates the final mode
selected.
Table 2
Sample sizes in the four waves of assessment.
Time Class Total N
Top (1) High (2) Average (3) Low (4) Bottom (5)
1 14 49 109 17 21 210
2 14 43 85 17 20 179
3 12 43 77 17 19 168
4 12 41 69 17 15 154
l
[(Fig._1)TD$FIG]
0
5
10
15
20
25
30
35
40
45
50
4321
Ari
thm
etic
(ra
w s
core
)
Phase
Arithmetic performance of different classes
Top
High
Average
Low
Bottom (MLD)
Fig. 1. Arithmetic performance of different classes. Error bars indicate the standard errors.
T.T.-Y. Wong et al. / Research in Developmental Disabilities 35 (2014) 2906–29202912
had a slightly lower intercept than the top class and always ranked second in terms of arithmetic performance. The averageclass (n = 109, 51.9%) had a relatively low intercept, but they showed the greatest improvement across time, catching up withthe high class at Time 4. The low class (n = 17, 8.1%) had an average intercept, but they showed the slowest growth, with thearithmetic performance at Time 4 being similar to that of the bottom class (n = 21, 10%), whose performance remained theworst across all four time points. Given their consistently poor performance in arithmetic, it was likely that this bottom classrepresented the MLD population (see Fig. 1).
3.2. Class comparisons
After classifying the participants into different classes, the cognitive profiles of the different classes were compared. Theclass comparisons included only those participants who remained in the final sample at Time 4. Outliers (3 or more S.D.sbeyond the class means) were deleted before the analyses. Class identity was the independent variable in all of the analysesbelow. First, the class classification was validated by examining the class difference in arithmetic performance usingrepeated-measures ANOVA. Secondly, performances on the control measures (i.e., nonverbal intelligence and workingmemory) were compared among the classes using ANOVA. Third, the class differences in number-related cognitivecapacities were examined using repeated-measures ANCOVA with the effects of working memory and intelligence beingcontrolled. Finally, an ANCOVA was used to examine whether there were significant class differences in readingachievement, again controlling for the effects of working memory and intelligence.
Given the complexity of the analyses, the focus of the repeated-measures analyses was on the between-subjects effects.For the comparisons with unequal variance in any of the time points involved (i.e., the ratio of the largest group variance tothe smallest group variance> 2), a critical alpha value of 0.01 was used (Tabachnick & Fidell, 2006). Although this happenedin most of our analyses, our conclusions were not affected at all because all class comparisons were highly significant (i.e.,p< 0.01, except for reading). All of the significant between-subjects comparisons were followed up by post hoc analyses withBonferroni adjustment.
3.2.1. Arithmetic
The repeated-measures ANOVA on the class difference in arithmetic performance showed highly significant results,F(4,149) = 190.264, p< 0.001, h2
p ¼ 0:836. The large effect size suggested that more than 80% of the variance in arithmeticperformance was captured by the class membership, thereby validating the latent class classification. The post hoc analysessuggested the best performance in the top class, followed by the high class. Both the average and low classes ranked next,while the two did not differ significantly. Both classes scored significantly higher than the bottom (MLD) class. All of thesignificant comparisons were significant at p< 0.001 (see Table 3).
Another ANOVA was conducted to examine whether the classes also differed in terms of their growth in arithmeticperformance. The class comparison was highly significant, F(4,149) = 309.448, p< 0.001, h2
p ¼ 0:893, and the post hoc
Table 3
Class differences in age, arithmetic, domain-general cognitive measures, and reading achievement.
Measures Analysisa Timeb Performance of the classes Between-subjects effect
Top (1) High (2) Average (3) Low (4) Bottom (5) F h2p Post hoc
a Depending on the variable, different types of ANOVA were conducted (single vs. multiple time points, with vs. without covariates).b The time point in which the measure was conducted.+ p< 0.07.
* p< 0.05.
** p< 0.001.
T.T
.-Y.
Wo
ng
eta
l./Resea
rchin
Dev
elop
men
tal
Disa
bilities
35
(20
14
)2
90
6–
29
20
29
13
T.T.-Y. Wong et al. / Research in Developmental Disabilities 35 (2014) 2906–29202914
analyses suggested that the growth rate was highest for the average class, followed by the bottom (MLD) class (p< 0.001)and then the high class (p = 0.005). Both the top class and the low class showed significantly slower growth than the highclass (ps< 0.001), while the two classes did not differ significantly in terms of growth rate (p = 0.256).
3.2.2. Control measures
The class differences in nonverbal intelligence and working memory were analyzed using ANOVA. Both analyses yieldedsignificant results: F(4,149) = 7.371, p< 0.001, h2
p ¼ 0:165 for nonverbal intelligence; F(4, 149) = 5.609, p< 0.001, h2p ¼ 0:131
for working memory. Post hoc analyses suggested that the top class and the high class did not differ in terms of nonverbalintelligence, with both scoring significantly higher than the average class (ps = 0.038 and 0.044, respectively) and the bottomclass (ps< 0.001). No other class difference in terms of nonverbal intelligence was significant. For working memory, the posthoc analyses suggested significantly higher working memory capacities among the top class, the high class, and the low classcompared with the bottom (MLD) class (ps = 0.010, 0.001, and 0.027, respectively). The high class also had marginallysignificantly higher working memory capacity than the average class (p = 0.060). No other class comparisons were significant(see Table 3).
3.2.3. Number-related cognitive measures
The class differences in the number-related cognitive capacities (i.e., ANS, symbolic–nonsymbolic mapping, and symbolicnumber processing) were analyzed using repeated-measures ANCOVA, with the effects of nonverbal intelligence andworking memory being controlled for. Details of the analyses are shown in Table 4.
3.2.3.1. Approximate number system. The number acuity of the participants was reflected by three indices: accuracy and w innonsymbolic comparison and accuracy in nonsymbolic multiplication. For accuracy in nonsymbolic comparison, theANCOVA suggested a significant between-subjects effect, F(4,147) = 3.728, p = 0.006, h2
p ¼ 0:92. The post hoc analysessuggested that the bottom (MLD) class scored significantly lower than the top and the high classes (p = 0.036 and 0.002,respectively) and marginally significantly lower than the average class (p = 0.057) in this task. No other class comparisonswere significant. The analyses on w also yielded significant findings, F(4,136) = 8.370, p< 0.001, h2
p ¼ 0:198. The post hocanalyses suggested that the bottom (MLD) class had significantly higher w (indicating worse performance) than all otherclasses (p = 0.001 for the comparison with the top class, ps< 0.001 for the comparisons with the other classes), while the w ofthese four classes did not differ significantly from one another. The results seemed to suggest a specific deficit in w among thebottom (MLD) class.
The repeated-measures ANCOVA on the accuracy on the nonsymbolic multiplication task also yielded a significantbetween-subjects effect, F(4,146) = 7.011, p< 0.001, h2
p ¼ 0:161. The post hoc analyses suggested that the performance inthis task divided the classes into two major groups. The first group (consisting of the top and the high classes) scoredsignificantly higher than the second group (consisting of the average, the low, and the bottom classes), with no intra-groupdifferences being significant (ps< 0.04 for all significant comparisons).
3.2.3.2. Number–numerosity mapping. Participants’ performance in number–numerosity mapping was indicated by twomajor skills (counting and estimation), each measured by two indices (accuracy in dot–number and dot–dot matching forcounting, linearity in numerosity production and number line for estimation). For counting, the repeated-measures ANCOVAsuggested a significant between-subjects effect in the accuracy of dot–number matching, F(4,142) = 10.366, p< 0.001,h2
p ¼ 0:226. The post hoc analyses revealed that both the top and the high class counted significantly more accurately thandid the average class (p = 0.012 and p< 0.001, respectively), which also counted significantly more accurately than thebottom (MLD) class (p = 0.019). The low class also counted significantly more accurately than the bottom (MLD) class(p = 0.005), although their performance did not differ from the other classes. The findings from the dot–dot matching taskwere similar, F(4,146) = 11.975, p< 0.001, h2
p ¼ 0:247, with an almost identical pattern in the post hoc analyses (i.e.,Top = High> Average [p = 0.012 and p< 0.001, respectively]> Bottom [p = 0.007], Low> Bottom [p = 0.035]), except that thelow class also counted significantly less accurately than the high class (p = 0.023).
Significant class differences were found in the linearity in both estimation tasks: F(4,146) = 14.277, p< 0.001, h2p ¼ 0:281
for numerosity production; F(4,141) = 10.745, p< 0.001, h2p ¼ 0:234 for number line. The post hoc analyses on the
numerosity production task suggested that the top and the high classes had significantly higher linearity than the averageclass (p = 0.006 and p< 0.001, respectively), which significantly outperformed the bottom (MLD) class (p = 0.002). The highclass also significantly outperformed the low class (p = 0.007), which further outperformed the bottom (MLD) classsignificantly (p = 0.014). Again, the findings were similar for the number line task (Top = High>Average [p = 0.041 and .006,respectively]> Bottom [p< 0.001]), except that the performance of the low class no longer differed significantly from that ofthe high class. Yet, the contrast between the low class’s and the bottom (MLD) class’s performance remained significant(p = 0.002).
3.2.3.3. Symbolic number processing. The number comparison task was used to assess participants’ symbolic numberprocessing. Both accuracy and reaction time were analyzed. The repeated-measures ANCOVA on accuracy suggested asignificant between-subjects effect, F(4,143) = 5.924, p< 0.001, h2
p ¼ 0:142, and the post hoc analyses suggested a specificdeficit among the bottom (MLD) class (ps = 0.001 for the comparisons with the top and high classes, p< 0.001 for the
Table 4
Class differences in domain-specific cognitive abilities examined by repeated-measures ANCOVA.
Measures Timea Performance of the classes Between-subjects effect
Top (1) High (2) Average (3) Low (4) Bottom (5) F h2p Post hoc
a For parental education level, 1 = primary, 2 = junior secondary, 3 = senior secondary, 4 = college, 5 = university or above.b For family income, 1 =<$9000, 2 = $9000–$12,000, 3 = $12,000–$16,000, 4 = $16,000–$20,000, 5 = $20,000–$24,000, 6 = $24,000–$30,000, 7 =>$30,000
All are monthly family income in Hong Kong Dollars.c SES composite refers to the sum of the standardized scores of the three SES measures.+ p< 0.07.
* p< 0.05.
** p< 0.01.
T.T.-Y. Wong et al. / Research in Developmental Disabilities 35 (2014) 2906–29202916
.
comparison with the average class, and p = 0.011 for the comparison with the low class). The analyses on reaction timeyielded a similar but larger between-subjects effect, F(4,141) = 10.721, p< 0.001, h2
p ¼ 0:233, and a similar pattern in thepost hoc analyses (all classes significantly outperformed the bottom [MLD] class: ps< 0.001 for the comparisons with the topand high classes, p = 0.001 for the comparison with the average class, and p = 0.036 for the comparison with the low class),but this time, the high class also significantly outperformed the average class (p = 0.001) and the low class (p = 0.019).
3.2.4. Reading
The reading achievement of the classes was also compared using ANCOVA. The results were significant, but with a smallereffect size, F(4,146) = 3.311, p = 0.013, h2
p ¼ 0:83. The post hoc analyses suggested that the only significant class difference laybetween the high class and the average class, in which the high class did better (p = 0.01). This suggested that neither the lowclass nor the bottom (MLD) class had difficulties in reading. The class classification was therefore arithmetic-specific (seeTable 3).
3.2.5. SES
In all of the above analyses, the low class did not seem to be particularly distinctive in any of the cognitive measures. Thereasons for their slow development in arithmetic therefore remained unclear. An attempt was made to compare the SES of theclasses to see if the SES measures differentiated the low class from the other classes. Three SES measures had been collected,namely father’s and mother’s education level (1 = primary, 2 = junior secondary, 3 = senior secondary, 4 = college, 5 = universityor above) and family income (1 =<$9000, 2 = $9000–$12,000, 3 = $12,000–$16,000, 4 = $16,000–$20,000, 5 = $20,000–$24,000,6 = $24,000–$30,000, 7 =>$30,000, all in Hong Kong Dollars). The results indicated that this class scored the lowest on all threeSES measures, although the differences did not reach statistical significance except for mother’s education, F(4,135) = 4.382,p< 0.01, h2
p ¼ 0:115 (see Table 5). The standard scores of the above SES measures were then summed up to obtain a compositeSES score. The analysis on the composite SES score suggested that this low-achieving class obtained a standard score of�1.713compared with 1.340, 0.582,�0.080, and�0.709 in the top, high, average, and bottom classes, respectively. The between-groupcomparison turned out to be significant, F(4,130) = 3.191, p = 0.015, h2
p ¼ 0:89, with the low class having significantly andmarginally significantly lower SES than the top and high achieving classes, respectively.
4. Discussion
In the present study, a large sample of kindergarteners was assessed on their arithmetic skills four times over a three-yearperiod. An LCGA was conducted to classify the participants according to their arithmetic performance over the four timepoints. A five-class solution emerged, with the lowest performing class being the potential candidates for having MLD.Further analyses of class differences on number-related cognitive measures suggested that this bottom class showed deficitsin most number-related cognitive measures (except for accuracy in nonsymbolic multiplication) compared with theaverage-performing class. Their performances on domain-general cognitive measures and reading, however, werecomparable to those of the average class. On the other hand, the LCGA also identified another class of low arithmeticperformers, who had average initial arithmetic scores but showed the smallest improvement over the 3 years. The cognitiveprofile of this class was not distinctive, but they tended to have lower SES. Theoretical and practical implications arediscussed in the following sections.
4.1. The use of a latent growth modelling approach in identifying children with MLD
The present study was one of the few studies that employed the latent growth modelling approach in identifying childrenwith MLD. Participants were classified into five classes according to their arithmetic performance over 3 years. This approach
T.T.-Y. Wong et al. / Research in Developmental Disabilities 35 (2014) 2906–2920 2917
has several advantages over the traditional approach. First, it reduces the likelihood that children are classified into thewrong group due to an arbitrary cutoff (Boscardin et al., 2008). Secondly, taking multiple measures of achievement intoaccount minimizes the misclassification rate due to measurement error, resulting in higher predictive power (Morgan et al.,2009). Third, the growth of children’s mathematical skills has been taken into account. Previous studies in reading suggestedthat the rate of learning of phonemic skills was more predictive of children’s later reading achievement than the end-of-intervention measure of phonemic skills (Byrne et al., 2000). The same logic may also apply to the case of mathematicslearning. As discussed below, the various aspects of the present findings suggested that the latent growth modellingapproach provides a possible alternative for identifying children with MLD.
First, the present study identified 10% of the participants as having MLD. The prevalence rate is within the range of 3.4% to13.8% obtained in previous studies (see Shalev, 2007 for a summary) that employed an arbitrary cutoff to identify childrenwith MLD. Although a completely different way of identifying MLD was used in the present study, the similarity inprevalence rates obtained across different methods cross-validates each other and suggests that the true prevalence ratemay be within this range.
Secondly, the growth of the MLD class matched well with the theoretical expectations and previous findings. According tothe RTI model, children who are at risk of learning disabilities are those who fail to learn at a reasonable rate and are notresponsive to intervention (Fletcher et al., 2004). Previous studies also suggested that children with MLD showed a shallowergrowth rate compared with their low-achieving and normally achieving peers (Geary, Bailey, Littlefield, et al., 2009; Morganet al., 2009; Murphy et al., 2007). The present study echoed the previous studies and found a shallower growth rate inarithmetic among the MLD class compared with the average class. Although the growth rates of the top and high classes werealso outperformed by the average class, their restricted growth rates could be a result of a ceiling effect, which certainly didnot apply to the MLD class. These converging results suggest that children with MLD not only show a performance lagcompared with their normally achieving peers but also show that this performance lag tends to widen with time.
Third and most importantly, the MLD class identified showed deficits in almost all of the number-related cognitivemeasures in the present study, including the ANS, number–numerosity mapping, and symbolic processing of numbers. Thedeficits in these skills could not be a result of individual differences in general intelligence or working memory capacity,which had been controlled for. Furthermore, the MLD class showed comparable performance with other classes in reading,suggesting that the difficulties of this class are specifically in mathematics. Children with MLD have been shown to havedeficits in the above domains in previous studies (ANS: Landerl et al., 2009; Mazzocco et al., 2011a, 2011b; Piazza et al., 2010;number–numerosity mapping: Geary et al., 2008; Landerl et al., 2004; Mejias, Mussolin et al., 2012; symbolic processing ofnumbers: De Smedt & Gilmore, 2011; Rousselle & Noel, 2007). The current findings therefore converge with those of previousstudies and suggest that the core deficits of MLD may lie in the above-mentioned skills.
In fact, the presence of the above-mentioned deficits might have contributed to the shallower learning curve observed in theMLD class. According to Butterworth, Varma, and Laurillard (2011), dyscalculia, or MLD, can be perceived as a disability arisesfrom the cognitive deficits in numerical processing. Because of these deficits, children with MLD may have difficulties even inbasic processing of numbers, as oppose to the automatic processing of numbers among normally achieving children. Thesedifficulties may result in under-achievement in mathematics either directly or indirectly (e.g., by distracting them from keyelements in the mathematical tasks). With the increasing complexity of mathematical tasks across grades, the situation couldonly get worse. Therefore, unless these basic numerical processing skills have been improved, or compensatory mechanismshave been developed, these deficits are expected to influence children’s learning throughout their lifetime. An increasedperformance gap between children with MLD and their normally achieving peers is therefore expected.
4.2. Number sense deficit vs. access deficit
The present study also provided us with an opportunity to verify whether the core deficit of MLD lies in the number sense,as predicted by the number sense deficit hypothesis (Wilson & Dehaene, 2007), or whether it originates from a deficit inaccessing numerosity information from number symbols, as predicted by the access deficit hypothesis (Rousselle & Noel,2007). The present findings suggest that, on top of the deficits in number–numerosity mapping and symbolic numberprocessing, the MLD class also showed a deficit in their number sense. The number sense deficit hypothesis is thereforesupported. The findings converge with other studies that found significant ANS impairment among children with MLD(Landerl et al., 2009; Mazzocco et al., 2011a, 2011b; Piazza et al., 2010; Wong et al., under review a) and further confirm thelink between the ANS and our symbolic mathematical skills. The impaired number sense may influence the number–numerosity mapping process, which then results in poor symbolic number processing and hence an under-achievement inmathematics (Wong et al., under review b)
4.3. Interventions for children with MLD
Based on the above arguments, a logical way to help children with MLD is to design interventions targeting both numbersense and number–numerosity mapping skills. Some previous attempts have been successful. For example, by trainingadults on some nonsymbolic arithmetic tasks, Park and Brannon (2013) successfully improved participants’ efficiency indoing symbolic arithmetic. A similar training programme for children has also been found to be effective (Hyde, Khanum, &Spelke, 2014). On the other hand, using a number line training game, Kucian et al. (2011) improved the arithmetic
T.T.-Y. Wong et al. / Research in Developmental Disabilities 35 (2014) 2906–29202918
performance of both children with MLD and their normally achieving peers. These successful attempts have providedresearchers, educators, and parents new hope in helping children with MLD.
4.4. The low-achieving class
On top of identifying the MLD class, the present study also identified another low-achieving class. This class of childrenperformed reasonably well in kindergarten, scoring slightly higher than the average class in arithmetic initially. However,they showed the smallest growth rate among the five classes, and their arithmetic scores were caught up to by the MLD classin Time 4. None of the cognitive skills measured in the present study seemed to distinguish this class from the better classes,although it did outperform the MLD class in most domains. However, this class was found to have the lowest SES among thefive classes. The differences in SES between this class and the other classes were substantial, with the composite average SESscore being 1 S.D. below the mean of the second-lowest SES class (i.e., the MLD class). The low SES may explain the slowarithmetic development among this class of children. The results converge with those of other studies that suggest thatchildren from lower SES families suffer from lower mathematics achievement and develop their mathematics skills moreslowly (Jordan & Kaplan, 2006; Morgan et al., 2009). The lack of parental involvement in learning among the low SES familiescould be one of the reasons for the slow development among this low-achieving group. Given that the growth curve of thelow-achieving class was even worse than that of the MLD class, interventions should also target this group of children.However, the intervention they need may differ from that of the MLD children. While children with MLD may needinterventions that target their cognitive skills, simple behavioural interventions such as peer tutoring and increasinghomework supervision may be helpful for the low-achieving class.
It should also be noted that the low-achieving class would not be identifiable using the low-achievement method alone asit was the class’s growth rate, but not its achievement at any time point, that distinguished it from other classes. LCGAtherefore provides a way for educators to identify other groups of children who may fall behind in their learning, whichwould have been ignored using the low-achievement method alone.
4.5. Limitations and future directions
The major limitation of the current study was the relatively small sample size. Given the relatively small initial samplesize, fewer than 20 children with MLD were identified in the final sample, which might limit the generalizability of thefindings. However, the patterns revealed by the class comparisons were clear, and the effect sizes were relatively large (allexcept one comparison in the number-specific cognitive factors yielded medium or larger effect sizes). Furthermore, thefindings from the current study converged well with those obtained in other studies (e.g., Mazzocco et al., 2011a; Piazzaet al., 2010). All of these elements suggested that results of the current study were reliable. By increasing the initial samplesize, future studies may yield a larger MLD sample for further comparisons.
4.6. Conclusions
The present study attempted to identify children into different classes according to their arithmetic performance over 3years. Using LCGA, a five-class structure was obtained, with the lowest performing class being the potential candidates forhaving a MLD. This MLD class, which accounted for 10% of the sample, scored the worst in arithmetic throughout theassessment period, with a shallower growth curve compared with the average class. They also showed deficits in all number-related cognitive processing, including number sense, hence supporting the number sense deficit hypothesis. On the otherhand, another low-achieving class was also identified that had no observable deficits in cognitive processing but had a lowerSES in general. The latent growth modelling approach therefore provides us with a possible alternative for identifyingchildren who had difficulties learning mathematics. Interventions should be provided to these children according to theirneeds.
Appendix 1. Arithmetic task used in the present study
Item no.
Item Item no. Item
1
1 + 2 31 2� 4
2
3 + 4 32 4� 3
3
2 + 7 33 7� 3
4
10 + 4 34 5� 8
5
11 + 6 35 6� 7
6
6 + 13 36 9� 8
7
17 + 6 37 14� 7
8
28 + 37 38 5� 32
9
54 + 38 39 19� 23
10
138 + 129 40 41� 26
11
232 + 173 41 18/3
12
695 + 227 42 15/5
T.T.-Y. Wong et al. / Research in Developmental Disabilities 35 (2014) 2906–2920 2919
Appendix 1 (Continued )
Item no.
Item Item no. Item
13
3–1 43 42/6
14
5–2 44 56/7
15
9–5 45 135/9
16
13–2 46 456/12
17
19–7 47 15 + 6� 8
18
18–12 48 (68–4)/4
19
14–9 49 (28� 7 + 12)/8
20
22–4 50 (207–108)� 6/33
21
40–27
22
102–59
23
227–135
24
741–498
25
5 + 8 + 9
26
24 + 17 + 41
27
20–7–5
28
78–24–19
29
17 + 9–6
30
70–37 + 23
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