Subtypes and comorbidity in mathematical learning disabilities

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Subtypes and comorbidity in mathematical learning disabilities: Multidimensional study of verbal and visual memory processes is key to understanding D. Szu˝cs1 University of Cambridge, Cambridge, United Kingdom Corresponding author: Tel.: +44-01223-767600; Fax: +44-01223-767602, e-mail address: [email protected]

1

Abstract A large body of research suggests that mathematical learning disability (MLD) is related to working memory impairment. Here, I organize part of this literature through a meta-analysis of 36 studies with 665 MLD and 1049 control participants. I demonstrate that one subtype of MLD is associated with reading problems and weak verbal short-term and working memory. Another subtype of MLD does not have associated reading problems and is linked to weak visuospatial short-term and working memory. In order to better understand MLD we need to precisely define potentially modality-specific memory subprocesses and supporting executive functions, relevant for mathematical learning. This can be achieved by taking a multidimensional parametric approach systematically probing an extended network of cognitive functions. Rather than creating arbitrary subgroups and/or focus on a single factor, highly powered studies need to position individuals in a multidimensional parametric space. This will allow us to understand the multidimensional structure of cognitive functions and their relationship to mathematical performance.

Keywords Mathematical development, Mathematical learning disability, Developmental dyscalculia, Working memory development, Mathematical cognition, Mathematics education, Metaanalysis

Progress in Brain Research, ISSN 0079-6123, http://dx.doi.org/10.1016/bs.pbr.2016.04.027 © 2016 Elsevier B.V. All rights reserved.

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Numerous studies have determined that mathematical learning disability (MLD) or developmental dyscalculia (DD) can be connected to weak performance on shortterm memory (STM) and working memory (WM) tasks. Indeed, various aspects of memory function can plausibly be thought to be important for mathematical function and development (hereafter “memory” will refer to both short-term and working memory). Solving even a trivial equation with very small numbers, for example, ((3 + 4) + (2
1))/(2  2), requires a substantial amount of planning, focusing attention on specific parts, retrieving partial results from long-term memory, keeping partial results in mind, and imagining manipulations. All these demands consume immediate memory and executive function (EF) capacity, the later contributing to memory performance. However, the vast array of memory tasks employed in the literature makes it difficult to organize the available body of research. Here, I organize the literature with regard to modality-specific (verbal vs visual) STM and WM subsystems (see a note on the STM/WM separation later). I suggest that this approach uncovers important regularities about comorbidity and MLD subtypes (see, eg, Bartelet et al., 2014, for extended discussion on subtypes) and points to future research directions.

1 MLD AND DD There is no generally agreed upon functional definition of MLD and DD. Some investigators consider these separate disorders, while others consider them as different labels for the same condition. The cause of the terminological confusion is that currently there is no clear generally accepted classification of developmental mathematical weaknesses. In this paper I use the MLD term because most cited papers used this term. I use MLD in a general operational sense, equivalent to the theory-free operational definition of DD, as we defined it previously as “persistently weak mathematical performance of developmental origin, related to the weakness of some kind(s) of cognitive function(s) and/or representation(s); appearing when concurrent motivation to study mathematics and access to appropriate mathematics education is normal” (Szu˝cs and Goswami, 2013). Operationally, children are usually considered to have MLD if they show weaker than average mathematical achievement defined by performing under a criterion level on a mathematics test. Specificity of weakness of mathematical function is often implied in studies, but practically there is variability in whether this specificity is tested. Hence, there is variation in whether comorbidity is allowed in operational definitions. There is no agreement in what control variables (eg, reading and nonverbal IQ), if any, must be in the normal range in order to diagnose MLD. The lack of agreement on criterion validity causes variability in measuring the prevalence of MLD. A recent review of 17 prevalence studies found that estimates ranged between 1.3% and 10.3% (
2 SD to
0.68 SD below the mean in terms of a normal distribution), and their mean was about 5–6% (Devine et al., 2013). The above makes it clear that the often-mentioned heterogeneity of MLD is partly a consequence of the heterogeneous (explicit or implicit) MLD definitions studies use.

ARTICLE IN PRESS 2 WM models in MLD research

As it is unclear what cognitive factors underlie MLD, it is also unclear whether MLD should be conceptualized as representing a quantitative extreme of the cognitive skills associated with mathematical achievement (the extreme tails of distributions), or whether it represents a discontinuous qualitative difference between MLD and typically developing children. A frequent observation is that children with MLD show abnormal structure and function in the intraparietal sulcus (IPS). This abnormality is often explained by assuming that an approximate magnitude representation, frequently called “number sense,” residing in the IPS is impaired in MLD. However, while reviews supporting the number sense theory do not discuss confounds and evidential weaknesses, both behavioral and neuroimaging evidence for this proposal are very weak, or even negative: There are practically no fMRI studies which deliver clear evidence and negative findings seem to outweigh positive ones (see Szu˝cs et al., 2013a,b, for a review). Further, past behavioral data referring the putative impairment of a so-called nonsymbolic number representation are disqualified by major confounds in the stimulus material (Bugden and Ansari, 2015; DeWind et al., 2015; Fuhs and McNeil, 2013; Gilmore et al., 2013; Mix et al., 1997; Szu˝cs et al., 2013a,b). In addition, it is doubtful whether there is a connection between the magnitude representation and school mathematics achievement at all (see Szu˝cs et al., 2014 for analysis). In contrast to the weakly supported number sense theory of MLD, there is a very large volume of behavioral and increasing volume of neuroimaging evidence which supports that MLD and the associated IPS dysfunction are closely related to memory dysfunction. In this paper I organize part of this evidence.

2 WM MODELS IN MLD RESEARCH Most MLD studies rely on the original formulation of the memory model of Baddeley (1986) which assumes that memory function involves domain specific, verbal and visual STM stores, and a domain-general central executive (CE) processing unit. In this model STM is the ability to maintain information in unchanged format for a short while. In contrast, WM refers to the ability to maintain information while carrying out another simultaneous mental process. Put otherwise, STM tasks only require information maintenance, while WM tasks are so-called complex span tasks where a secondary so-called processing task/component, relying on CE function, is also carried out besides maintenance. The original model left CE function largely undefined. A later version assumed that CE performance is determined by a limited capacity attentional control system which coordinates processing of stored items (Baddeley and Logie, 1999). While the above model is near exclusively dominant in MLD research, it is important to see that the field of memory research has many competing models besides Baddeley’s (see, eg, Conway et al., 2008, for a review). In order to organize the material, instead of relying on the original Baddeley memory model here I consider two important new developments. First, recent

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evidence suggests that CE function underlying verbal and visual WM can dissociate and performance on CE tasks may be modality specific (Fougnie and Marois, 2011; Jarvis and Gathercole, 2003; Saults and Cowan, 2007; Shah and Miyake, 1996; Watter et al., 2010). In fact, the dissociations in MLD discussed in this paper also support this view. Second, several studies aimed to define CE function in more detail. An influential study suggests that three EF domains contribute to CE performance: inhibition, attentional focus shifting (shifting), and information updating and monitoring (updating) (Miyake et al., 2000). Various theories seem to fit (parts) of this framework. First, inhibition typically refers to the suppression of unwanted interference from processed items and it has been emphasized by several investigators (Hasher and Zacks, 1988: “access” and “deletion” functions; Cowan, 1995; Kane et al., 2007). Inhibition has been related to “executive attention” skills and to the inhibition of prepotent responses (Engle, 2002; “restraint” in Hasher and Zacks, 1988). Second, updating also relates to the “deletion” function of Hasher and Zacks (1988): Items initially in the focus of attention must be overwritten after becoming irrelevant and a new item should enter the focus of attention. This focus must be maintained precisely. Memory updating performance in N-back tasks reflects the ability to maintain focus. Third, shifting is typically assessed in nonmemory tasks that require volitional control such as the trail-making task (Salthouse, 2011). Most importantly, many researchers question whether STM and WM can be separated clearly. Rather, tasks usually considered to rely on WM can be conceptualized as based on STM and also using an increased amount of attentional and EF function resources (Cowan, 1995; Engle, 2002; Hasher et al., 2007; Kane et al., 2007; Oberauer, 2002). Hence, in general, “classical” WM tasks would be different from STM tasks by increased processing load on some cognitive resources other than mere maintenance. This increased processing may be inherently necessary to carry out some tasks traditionally categorized as “memory tasks.” Here, I take this view but report on STM and WM tasks separately as this is how memory performance was measured in papers. Overall, here I rely on a working model which keeps the phenomenological distinction between the STM and WM “labels,” but I assume that WM measures are distinct from STM measures because of some kind of additional EF processing demands as described earlier. Further, I assume that both STM and WM performance can be modality specific (Fougnie and Marois, 2011; Jarvis and Gathercole, 2003; Saults and Cowan, 2007; Shah and Miyake, 1996; Watter et al., 2010; and see data in this paper). Modality specificity may be the consequence of modality-specific STM processes feeding into WM tasks and/or the consequence of modalityspecific EF. Fig. 1 depicts a potential working model I use here, based on the EF domains identified by Miyake et al. (2000). These EF domains support processing needs in WM tasks by providing resources usually considered to belong to the CE. Here, I consider visual memory as the “whole” domain of visuospatial memory, its potential fractionation will be discussed later.

ARTICLE IN PRESS 3 Verbal and visual memory deficits in MLD

Verbal WM

Phonological decoding verbal IQ

Spatial manipulations

Spatial representations

Symbol manipulation

Symbol grounding

Verbal operations

Verbal representations

Selective attention

Task switching

Magnitude relations

Inhibition Executive functions

Verbal STM

Visual WM Not modalityspecific CE

Visual STM

ModalityModalityspecific CE specific CE

Visual pattern recognition (long-term memory) Spatial skills

Fact retrieval (long-term memory)

FIG. 1 The model of memory function used here and its relation to basic numerical function. Visual and verbal STM enable modality-specific information maintenance. WM is when tasks require EF contribution besides maintenance. Such EF contribution is called “central executive” (CE) function within the framework of memory tasks. WM performance may be modality specific because of the contribution of modality-specific STM processes and/or because some CE processes may be partially or fully modality-specific themselves. Modified from Szu˝cs, D., Devine, A., Soltesz, F., Nobes, A., Gabriel, F., 2014. Cognitive components of a mathematical processing network in 9-year-old children. Dev. Sci. 17, 506–524.

3 VERBAL AND VISUAL MEMORY DEFICITS IN MLD Table 1 summarizes 32 often-cited studies which focused on measuring WM function in MLD children. The table does not represent the outcome of an exhaustive systematic literature search. However, it does provide a wide sample of studies from a number of publication years which are frequently cited in support of various claims about the relation of memory function and MLD and provided clear scores on memory measures. All together these studies tested 665 MLD and 1049 control children. Some studies also tested other groups of children, eg, children with reading problems and MLD and both age-matched and ability-matched control groups. However, here I only focused on the comparison of MLD and control children as these are the categories which consistently occurred in all studies. The studies used a very wide array of tests and used widely varying MLD definition criteria. Some studies used reading and IQ to control for general cognitive abilities, and these control criteria were also very different. Most studies confirmed to Baddeley’s model of WM and tested the CE. However, these CE tests were overwhelmingly tests of verbal memory. Here, I take a modality-specific approach and classify such tests as measures of verbal WM as opposed to tests of visual WM

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Table 1 Major Characteristics of the Studies in the Analysis Number

Authors

Year

MLD Percentile

Control Math Criterion

MLD Reading Criterion

Screening Pool (N)

MLD (N)

Control (N)

Totals (N)

Age of Children in Study

Age Group in Analysis

Reading Matched?

IQ Matched 0

Geary et al.

2007

15%

>50%

—

278

15

46

61

6>7

1

0

Geary et al.

1991

46%

>46%?

—

52

12

26

38

7/8 > 8/9

1

0

0

3

Geary et al.

1999

20%

>59%

?¼52%

114

15

35

50

7

1

1

1

4

Geary et al.

2000

35%

>66%

>40%

114

12

26

38

7/8

1

1

0

5

Geary et al.

2004

30%

>30%?

—

149

58

91

149

7/9/11

2

0

0

6

Geary et al.

2012

?

?

—

231

15

101

116

6 > 10

2

0

0

7

Bull and Johnston

1997

40%

>40%

—

69

32

36

68

7

1

0

0

8

Bull et al.

1999

37%

>40%

—

44

20

24

44

7

1

0

0

9

Hitch and McAuley

1991

18%

>18%

>10%

110

14

14

28

8–9

2

1

1

10

McLean and Hitch

1999

25%

25–75%

25–75%

122

12

12

24

8–9

2

1

0

11

D’Amico and Guarnera

2005

[10%]*

?

>5%

Teachers

14

14

28

9–10

2

1

0

12

Passolunghi et al.

1999

20%

50–80%

—

300

15

18

33

9

2

0

1

13

Passolunghi and Siegel

2001

30%

50–80%

?

280

23

26

49

9

2

1

1

14

Passolunghi and Siegel

2004

30%

50–80%

?

280

22

27

49

10

2

1

1

15

D’Amico and Passolunghi

2009

30%

>30%

>30%

108

12

12

24

10

2

0

1

16

Passolunghi and Mammarella

2010

30%

50–80%

?

293

20

19

39

9

2

1

1

17

Passolunghi et al.

2005

25%?

?

?

?

10

10

20

9

2

0

1

18

Passolunghi and Cornoldi

2008

16%

>84%

—

279

24

30

54

8/10

2

0

1

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1 2

Passolunghi and Mammarella

2012

25%

40–75%

—

320

35

35

70

9–11

2

0

1

20

Schuchardt et al.

2008

16%

>16%

>16%

97

17

30

47

7–10

2

1

0

21

Siegel and Ryan

1989

25%

>30%

>30%

?

36

74

110

9–13

3

0

0

22

2007

SPI*

?

—

138

31

47

78

10

2

0

0

23

Andersson and Lyxell
sz et al. Solte

2007

SPI*

School. perf.*

>40%

Instit.*

7

7

14

17

3

0

1

24

Szu˝cs et al.

2013

15%

16–85%

85–115

1004

12

12

24

9–10

2

1

1

25

Swanson

1993

0.5 SD

?

—

123

19

38

57

10

2

0

0

26

van der Sluis et al.; Exp 2

2005

2 Year delay?

?

?

477

17

18

35

11

3

1

1

27

Swanson and SachseLee

2001

25%

>40%

<25%

73

24

29

53

11

3

0

1

28

Peng et al.

2012

15%

>25%

>25%

805

18

30

48

11

3

1

1

29

Keeler and Swanson; Exp 2

2001

25%

?

?

57

20

18

38

12

3

0

1

30

Andersson and Ostergren

2012

* 7%

* >7%

* >7%

?

20

43

63

12

3

1

0

31

White et al.

1992

30%

40–70%

40–70%

625?

17

50

67

13

3

1

1

32

Wilson and Swanson

2001

25%

>25%

>25%

?

47

51

98

11–52

3

1

0

Notes (*): MLD percentile: In D’Amico and Guarnera (2005) children performed worse than 10% on at least 5 out of 10 maths tests. In Andersson and Ostergren (2012) percentiles on nonstandard tests were used. School perf., defined by school performance. SPI, Receiving Instruction in Special Education Institution. Instit., Diagnosis from special education institution. In the “Age of children in study” column the “>” sign denotes follow-up studies, eg, from 6 to 7 years of age (6 > 7). In the “Reading matched?” and “IQ matched?” columns 1 stands for matched groups and 0 stands for nonmatched groups. Some parameters may be unclear and/or may constitute best guesses marked by “?”

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(eg, Geary et al., 2007, 2012). I considered backward digit span as a measure of verbal WM because it has processing requirements besides mere storage and recall (Geary et al., 1999, 2000). Similarly, backward Corsi span is considered a test of visual WM (eg, Passolunghi and Mammarella, 2010). Some unique papers used a very large battery of memory tasks (Schuchardt et al., 2008). In order to keep parity with other papers in these cases 2–3 of the most frequently used tasks were selected for analysis (typically digit and word span tasks). In some cases counting span (Siegel and Ryan, 1989) and “complex computation span” (Peng et al., 2012) were used as memory tasks. However, tasks requiring computations and counting confound the independent and dependent variables when investigating MLD (Hitch and McAuley, 1991). Hence, I consider these unsuitable for testing memory function in MLD and such tests were not considered when possible. However, in some cases counting span tasks were part of the WMTB-C battery (Pickering and Gathercole, 2001) and only composite scores were reported, so counting span tasks did contribute to some results (Geary et al., 2007, 2012). Related potential effects of these counting span tasks will be discussed later. When both ANCOVA adjusted and nonadjusted means were reported, the nonadjusted ones were used because it is invalid to use ANCOVA “to correct for” group differences (Miller and Chapman, 2001; Porter and Raudenbush, 1987). The studies tested various age groups and sometimes collated results for multiple age groups. I classified studies according to the dominant age groups tested and I grouped them into three age groups (column: age group in analysis; age group 1: about years 6–8; age group 2: about years 9–11; age group 3: above year 11). The meta-analysis reported here computed effects sizes as Hedges’G (Hedges, 1981; see Appendix; see effect sizes in Table S1 (http://dx.doi.org/10.1016/bs.pbr. 2016.04.027)). I combined effect sizes according to a random effects model as this is more suitable for integrating the results of studies with different designs and populations than a fix effects model. The model was implemented as described in Borenstein et al. (2009), pp. 72–74. All calculations were programmed in Matlab 2015b. I compiled the data before any analysis was done and there was absolutely no feedback from analysis into data selection (this is only relevant in a few studies which may have used a large number of custom made or unusual tests, so not all tests were included for the data). I deliberately refrained from potentially misleading statistical significance testing of the data because the overall effect size estimates and considering the actual distribution of the data are more informative.

4 ANALYSIS OF STUDY DATA 4.1 COVERAGE OF MEMORY DOMAINS AND POWER The inset in Fig. 2B shows the number of studies which tested each memory domain. As shown, about half as many (14) studies tested visual STM than verbal STM (22) and less than half as many (11) studies tested visual WM than verbal WM (25). Hence, there is a strong bias for using verbal memory tasks only, especially

The power of studies. (A) The cumulative percent of studies achieving a certain power level (top [red, gray in the print version] axis) and the mean number of MLD (N1) and control participants (N2) vs power (bottom [blue, dark gray in the print version] axis). (B) The power of individual studies in the four memory domains to detect the mean effect sizes in each domain. The long horizontal lines mark power levels of 0.6 and 0.8; the short horizontal dashed lines mark the mean power level in each memory domain. The leftmost markers for each domain depict the mean power in age groups. The other markers denote power in individual studies for the three age groups. The inset shows the number of results for each memory domain.

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FIG. 2

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when testing WM function. This bias is probably the consequence of the dominance of the Baddeley memory model which assumes domain-general WM function. In practice, this means that most studies used verbal tasks to measure supposedly domain-general CE performance. However, especially in light of various recent studies highlighting the specific importance of visual memory for understanding MLD such bias may deprive us from important evidence (Passolunghi and Mammarella, 2010, 2012; Schuchardt et al., 2008; Szu˝cs et al., 2013a,b). Hence, there seems to be a need for more balanced testing of both visual and verbal memory. Power was calculated for independent sample t-tests by the standard Matlab “sampsizepwr” function (Matlab, Statistics Toolbox (C)) based on the reported number of MLD and control participants to detect the mean random effects model effect size (see the following section) computed for each memory domain. The ratio between the participants in the MLD and control groups was taken into account. It is important to consider that published effect sizes are nearly always overestimates of real effect sizes. This is because power is frequently low in psychological studies and a consequence of low power is that only larger effect sizes will reach statistically significant levels (Schmidt, 1996). Because smaller, statistically nonsignificant effect sizes are typically not published, the published literature will then exaggerate effect sizes (that is, low power “paradoxically” leads to larger reported effect sizes than higher power). In consequence, the power values computed here also provide an upper limit on the power of studies. Fig. 2A shows the power distribution (at a ¼ 0.05) of studies in function of the mean of participant numbers in the MLD and control groups. The continuous line shows the percent of studies under a certain power level (cumulative distribution function). For example, about 50% of studies seemed to have power >0.7 with respect to the mean effect sizes computed here. Note that the variation in power for the same mean number of participants in the two groups is due to the often unequal number of participants in MLD and control groups varying across studies. Fig. 2B summarizes the power of studies by the dominant age group tested. Mean power oscillates between 0.55 and 0.85 for the three age groups for the four memory domains. Power is the lowest for verbal STM tasks (note that the mean effect size is the lowest in this task, so higher power is needed to detect this effect size), increases for visual STM tasks, and is the highest for WM tasks. No systematic shift in power is visible for age groups across the board of studies (correlations between power and the age groups studied ranged between
0.07 and 0.26; all n.s.). In summary, the overall power of the MLD memory literature to detect the mean effect sizes computed here (which can be classified as “moderate” to “large”) seems to be similar to the overall level of power to detect similar-sized effects in the psychological literature (Cohen, 1962; Rossi, 1990; Sedlmeier and Gigerenzer, 1989). An additional point is that when judging power here I only considered the number of participants. However, power also depends on the quality of experimental design which is harder to compare across studies. For example, the number of trials used for individual tests significantly affects the amount of error in individual measurements.

ARTICLE IN PRESS 6 Matching reading and IQ in MLD and control groups

5 EFFECT SIZES FROM STUDIES Fig. 3A and B shows the standardized effect sizes (standardized MLD minus control scores) detected by the 32 studies in each memory domain considered. The mean random effects model effect sizes are also marked by horizontal lines (Table 2A shows numerical values and 95% confidence intervals). The colored dotted lines mark mean effect sizes for studies with varying levels of power. The mean effect sizes were surprisingly similar for three domains, except for verbal STM. While these mean effect sizes seem large for psychological studies (Cohen, 1962; Rossi, 1990; Sedlmeier and Gigerenzer, 1989), as noted before, these effect sizes are likely to be overestimates; ie, they provide an upper limit on real effect sizes (Schmidt, 1996).

6 MATCHING READING AND IQ IN MLD AND CONTROL GROUPS As shown in Table 3, important regularities can be noticed when grouping studies according to whether they considered reading and/or IQ when matching MLD and control groups. We can compare results from studies which matched MLD and control groups on reading to studies which did not match these groups on reading by subtracting the absolute effect sizes measured in the later studies from the absolute effect sizes measured in the former ones. Studies which matched reading found
0.48 and
0.35 SD smaller absolute effect sizes in verbal STM and verbal WM tasks than studies which did not match reading. In addition, they also found +0.18 and by +0.30 SD larger effect sizes in visual STM and visual WM tasks than studies which did not match reading (see Fig. 3C). Besides matching reading we can also study how mean effect sizes are modified when matching IQ (Fig. 3D) or matching both reading and IQ (Fig. 3E). Matching IQ on its own seems to have smaller effects: First, the effect size of visual STM discrepancy between MLD and control groups increases by +0.26 SD. Second, the effect size of visual WM discrepancy decreases by
0.16 SD, probably due to the fact of the mixed nature of IQ tests (see later). When matching both IQ and reading, the situation is similar to the one when only reading was matched: The effect size of MLD vs control discrepancy decreases on verbal STM and verbal WM measures by
0.22 and
0.29 SD, respectively, and increases on visual STM and visual WM measures by +0.44 and +0.36 SD. Several conclusions can be drawn from the above observations. First, the verbal STM and verbal WM discrepancy between MLD and control samples decrease substantially if they are matched on reading. Matching on both reading and IQ has some moderate further impact on verbal STM and verbal WM data, but matching IQ on its own has only relatively modest impact. Hence, reading achievement seems the most important control variable to consider when interpreting verbal memory discrepancy between MLD and control groups. This suggests that low readers substantially affect group-level performance on verbal STM and WM tasks when studying MLD samples. Indeed, various studies reported that low readers have verbal STM and verbal

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FIG. 3 Standardized effect sizes in standard deviation (SD) units from the 32 studies in Table 1. (A) Standardized effect sizes in studies sorted according to matching reading and/or IQ (see the rightmost legend). The leftmost markers for each domain denote the power categories (power < 0.5; 0.5  power < 0.8; 0.8  power) for individual studies. The other (smaller) markers denote effect sizes in individual studies. The long horizontal dashed line marks the zero level, the short horizontal black dashed lines mark the mean effect size level in each memory domain, and the short colored (different gray shades in the print version) dotted lines mark the mean effect size level for each power category. (B) The bivariate distribution of effect size and power in individual studies. (C) Mean effect sizes when matching vs not matching reading. (D) Mean effect sizes when matching vs not matching IQ. (E) Mean effect sizes when matching vs not matching reading and IQ. Line styles and colors (different gray shades in the print version) are the same for panels (C–E) as marked in panels (C and D).

ARTICLE IN PRESS 6 Matching reading and IQ in MLD and control groups

Table 2 Mean Effect Sizes from Random Effects Models A. All Data Lower limit Mean Upper limit Tau I2 B. Groups Age group 1 (6–8 years)

Age group 2 (9–11 years)

Age group 3 (>11 years)

Lower Mean Upper Tau I2 Lower Mean Upper Tau I2 Lower Mean Upper Tau I2

Verbal STM

Visual STM

Verbal WM

Visual WM


0.83
0.59
0.36 0.45 65.53


0.93
0.71
0.49 0.27 42.64


0.94
0.75
0.57 0.35 56.09


0.98
0.74
0.49 0.29 52.17

Verbal STM

Visual STM

Verbal WM

Visual WM


1.89
0.88
0.14 0.98 90.47
0.94
0.55
0.16 0.63 78.77
1.10
0.47
0.16 0.63 78.41


1.75
0.71
0.33 0.86 87.40
1.06
0.71
0.36 0.46 68.19 — — — — —


1.77
0.86
0.05 0.87 86.90
1.40
0.89
0.37 0.85 87.16
0.93
0.55
0.17 0.49 75.18

— — — — —
1.39
0.85
0.30 0.61 81.14
1.31
0.65 0.00 0.69 86.33

(A) Mean effect sizes for the whole sample of studies. The lower and upper limits of 95% confidence intervals are also shown. I2 was computed for all studies. If pooled estimates were used for each age group, results differed less than 0.05 (see Borenstein et al., 2009). (B) Mean effect sizes in the three memory domains in the three age groups. I2 was computed independently for each age group. “Lower” and “Upper” stand for the limits of 95% confidence intervals. For age group 1 there were no studies for visual WM. For age group 3 there was only one study for visual STM.

WM deficits (De Beni et al., 1998; Pimperton and Nation, 2010) which is compatible with the analysis here. Second, it also seems that when MLD and control samples are matched on reading, their discrepancy on visual STM and visual WM performance increases. I suggest that the above data depict a double dissociation between two types of MLD supporting some previous conclusions (Schuchardt et al., 2008; Szu˝cs et al., 2013a,b). One type is more associated with reading deficits and is coupled with verbal memory (STM and WM) deficits. This links the frequently reported comorbidity between poor reading and MLD to verbal STM and verbal WM deficit and suggests that the mathematical problems of these children are rooted in their verbal processing skills. The other MLD type seems independent from reading problems and is associated with visual memory (STM and WM) deficits. Both subtypes also seem

289

All

verSTM visSTM verWM visWM

Reading

IQ

Reading + IQ

All Studies

Matched

N.M.

Diff.

Matched

N.M.

Diff.

Matched

N.M.

Diff.

0.59 0.71 0.75 0.74

0.37 0.81 0.59 0.90

0.86 0.63 0.94 0.61


0.48 0.18
0.35 0.30

0.53 0.95 0.74 0.63

0.61 0.70 0.76 0.79


0.08 0.26
0.02
0.16

0.25 0.95 0.54 0.98

0.47 0.51 0.83 0.62


0.22 0.44
0.29 0.36

Studies which matched reading in MLD and control participants reported smaller absolute difference (Diff., matched–not matched groups’ data) between MLD and control groups in verbal STM and WM performance. In contrast, the MLD vs control difference was larger in visual STM and visual WM performance in matched than in nonmatched groups.

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Table 3 The Difference (Diff.) in Absolute Effect Sizes (Random Effects Model) When Comparing Studies Which Matched vs Did Not Match (N.M.) Reading Achievement and/or IQ in MLD and Control Groups

ARTICLE IN PRESS 6 Matching reading and IQ in MLD and control groups

sensitive to matching IQ: The MLD vs control discrepancy further increases when both reading and IQ are matched relative to the case when only reading is matched. In contrast, the MLD vs control discrepancy decreases when both reading and IQ are matched. With regard to the influence of IQ, interpretation can be less clear as the studies used many different IQ tests which of course include a varying proportion of visual and verbal tasks. Hence, matching only on IQ tests may affect both MLD subtypes equally and its impact may be hard to determine clearly from the current data. A related point to note is that because IQ tests may just rely on too many cognitive processes, overall IQ scores may be fairly badly defined measures. For example, in a robust modeling procedure Szu˝cs et al. (2014) found that both Raven’s CPM (Raven, 2008) and the WISC-III Block-design scales (Wechsler, 1991) were less efficient predictors of mathematical achievement scores than a range of better defined, more focused measures (visual STM and WM tests and the trail-making test). Besides the above dissociation there also seems to be an additional level of subtype-independent “baseline” memory impairment. This is because even when reading is matched, verbal WM and verbal STM are still weaker in MLD than in control groups, albeit to a lesser extent than in groups with nonmatched reading. Similarly, when reading is not matched, some visual STM and visual WM impairments remain in MLD groups, albeit to a lesser extent than in groups with matched reading. This baseline memory deficit may be genuine, or may be the consequence of having borderline mixed profile children (eg, poor readers) in groups who then contribute to group-level data. Such children may be close to exclusion criteria but may not reach it. If the baseline deficit is genuine, it may suggest that there is a common memory impairment in both weak and nonweak readers with additional modality-specific impairments in the two MLD subtypes. Notably, the discrepancy between MLD and control groups decreased from 0.59 to 0.37 SD when reading was matched and to 0.25 SD when both reading and IQ were matched (0.25 qualifies as “small” effect size; see Rossi, 1990; Sedlmeier and Gigerenzer, 1989). In contrast, the discrepancy on verbal WM is more preserved, decreasing from 0.75 to 0.59 and 0.54 SD. Therefore, it seems that if reading and IQ are controlled for, effect sizes for verbal STM discrepancy are relatively low. Hence, the verbal STM deficit in MLD may be strongly linked to reading deficits (Schuchardt et al., 2008; Szu˝cs et al., 2013a,b). This has important implications for testing because many studies exclusively test only verbal STM when they study mathematical development in typically developing or MLD children. In light of the current results this is inadequate practice because verbal STM may only appear in one subtype of MLD linked to reading deficit. For example, such a study may aim to “control for” memory impairment by using only a verbal STM test and based on this single test it may conclude that “memory function” is not related to mathematical development or MLD. I suggest that this is a false conclusion because it is highly likely that such selective verbal STM testing actually fails to test other important aspects of memory function for mathematical development and MLD. So, studies should minimally also take visual memory measures.

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6.1 DEVELOPMENTAL PATHWAYS The cognitive architecture may continuously change during development and children are unlikely to have adult cognitive architecture (Karmiloff-Smith, 1995). This is particularly important in mathematics because the content of mathematics itself evolves a lot during childhood. Consequently, even the visual/verbal loading of both study and test material is changing across age groups. Hence, different cognitive functions may be most important for mathematical performance at different ages. So far, some studies testing multiple age groups suggested that verbal STM was more associated with performance in younger children and visual STM was more associated with performance in older children (Alloway and Passolunghi, 2011; Meyer et al., 2010). Others reached opposite conclusions (DeSmedt et al., 2009). Table 2B shows mean effect sizes by age group and Fig. 4 plots individual studies by age group. The boxed markers in Fig. 4 also show average effect sizes for each age group (note that more negative values signify larger [DD minus control] effect sizes). First, effect size changes are unlikely to be related to changes in power (Schmidt, 1996) across age groups as there was no change in power in function of the age groups tested. Second, the data are rather variable, so we cannot really reach clear developmental conclusions. Consequently, we should refrain from selective citing of studies in support of developmental arguments. Rather, in order to clarify developmental pathways, more, highly powered developmental studies are needed. Increasing the power of individual assessment is also crucial because different children in samples may be at different stages of development (Siegler, 1996). It is noteworthy that there are relatively few studies with the youngest age

FIG. 4 Effect sizes in the three memory domains in the three age groups. The long horizontal dashed line marks the zero level; the short horizontal dashed lines mark the mean effect size level in each memory domain. The leftmost boxed (dotted line) markers for each memory domain denote the mean effect sizes for each age group. The other (smaller) markers denote effect sizes in individual studies. The arrow points to an outlier result (see text for details).

ARTICLE IN PRESS 6 Matching reading and IQ in MLD and control groups

group and none of the studies tested visual WM in this age group. Hence, more studies with wider coverage of memory domains are needed, especially with younger children.

6.2 FRACTIONATING SUBTYPES OF VISUAL MEMORY While the above analysis only deals with visual memory as a whole, there is evidence that visual memory can be fractionated further. For example, the visual pattern recognition and spatial components of visual WM seem to develop at different rates (Della Sala et al., 1997; Hamilton et al., 2003; Logie, 1995; Logie and Pearson, 1997; Pickering et al., 2001). Further, static (remembering locations only; eg, visual matrix tasks) and dynamic (remembering locations and the order of presentation; eg, the Corsi Block-tapping task) aspects of visual memory can also be discriminated (Pickering et al., 2001). Some studies reported MLD vs control discrepancy in dynamic STM tasks only (van der Sluis et al., 2005; Exp 2), while others found such difference in both dynamic and spatial tasks but not in visual pattern memory tasks (Passolunghi and Mammarella, 2010; Reukhala, 2001). Some studies with typically developing children suggested that static, dynamic, and visual pattern memory may be associated with different types of mathematical functions (Holmes et al., 2008; Kytt€al€a and Lehto, 2008). Passolunghi and Mammarella (2012) substantially refined findings with static spatial STM, WM, and visual pattern tasks. Both STM/WM tasks had simple and complex versions and the complexity (difficulty) of memory recall was manipulated. The MLD group performed worse than controls on the spatial STM and WM tasks but not on the pattern recognition task. This study also compared a severe MLD group (performing under the 10th percentile in a mathematics test; 12 children) and a mild MLD group (between the 10 and 25th percentiles; 23 children). Severe MLD children performed worse in the more complex spatial recall tasks than in the simpler spatial tasks (see more discussion later).

6.3 FRACTIONATING EFs CE and EF are often used as very wide, badly defined, umbrella terms. Out of the three EF domains potentially contributing to WM performance (Miyake et al., 2000) only inhibition (usually meaning interference suppression) has been studied in detail in MLD. Studies found that perseveration on the Wisconsin Card-Sorting Task correlates with the frequency of memory retrieval errors in arithmetic facts (Bull et al., 1999); MLD children recall more irrelevant and less relevant information than controls (Barrouillet et al., 1997; Passolunghi et al., 1999) and that they show numerous intrusion errors in mental arithmetic (D’Amico and Passolunghi, 2009; Passolunghi and Mammarella, 2012; Passolunghi and Siegel, 2001, 2004; Passolunghi et al., 2005). In addition, Stroop task performance was also found impaired in MLD (Peng et al., 2012; Szu˝cs et al., 2013a,b). Hence, it is likely that weak

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interference suppression contributes to well-known fact retrieval deficits in MLD (Barrouillet et al., 1997; Passolunghi et al., 1999). The above results fit to studies which found that children with poor reading comprehension show deficits in interference suppression in verbal WM tasks (De Beni et al., 1998; Pimperton and Nation, 2010) but not in visuospatial WM tasks (Pimperton and Nation, 2010). Interference suppression deficits in verbal WM tasks were also reported in children with ADHD (Cornoldi et al., 2001; Palladino, 2006; Palladino and Ferrari, 2013). Hence, the question arises whether interference suppression problems appear in a range of developmental disabilities, MLD being one of them. For example, such shared inhibition deficit may explain the baseline level of memory impairment common to both MLD subtypes. It is a further question whether similar inhibition problems manifest in visual and verbal WM tasks in both MLD subtypes. With regard to this it is a major gap in the literature that to date only very few studies tested inhibition in actual visual WM tasks in MLD (Passolunghi and Mammarella, 2012). There is also an obvious need to study other CE functions, shifting and updating, besides inhibition in MLD.

6.4 STUDIES WITH ABILITY-MATCHED YOUNG CONTROLS AND INTERVENTION Very few studies tested math ability-matched control children (Andersson and Lyxell, 2007; Keeler and Swanson, 2001; McLean and Hitch, 1999; Swanson, 1993; Swanson and Sachse-Lee, 2001). One of these studies reported that MLD performed worse than age-matched controls but better than ability-matched controls in verbal/visual WM (Swanson, 1993). Four studies reported that memory performance in ability-matched groups did not differ from MLD in a range of verbal STM/WM and visual WM tasks (Andersson and Lyxell, 2007; Keeler and Swanson, 2001; McLean and Hitch, 1999; Swanson and Sachse-Lee, 2001), except in tasks which required solving addition equations (McLean and Hitch, 1999) or counting (Andersson and Lyxell, 2007). Hence, the majority of studies seem to suggest that MLD children’s memory performance is equivalent to the performance of younger, math ability-matched controls and they are only weaker in tasks requiring computations. This may suggest that MLD children experience a developmental delay in (some kind of) memory performance, and they have underdeveloped computational skills in consequence of their delayed memory development. Developmental delay may be addressed by child friendly interventions, and indeed, there are some intervention studies with promising findings (Holmes et al., 2009; Kroesbergen et al., 2014). Moreover, a play oriented visuospatial task has been shown to correlate with mathematical achievement in 7-year-olds which may also pave the way to child friendly interventions (Nath and Szu˝cs, 2014). An important question is whether specific memory interventions would result in specific mathematical improvement. A critical point is that out of the above studies only one matched reading performance between MLD and controls (McLean and Hitch, 1999) and only two matched IQ

ARTICLE IN PRESS 8 MLD subtypes, network coordination, and individual variability

(Keeler and Swanson, 2001; Swanson and Sachse-Lee, 2001), so conclusions may further be qualified if future studies take these variables into account.

7 PROCESSING NETWORKS AND THE IMPACT OF GENERAL TASK DIFFICULTY I suggest that the MLD profiles identified here are probably related to weak or disrupted processing in various parts of a complex WM/EF network (see also Fias et al., 2013; Szu˝cs et al., 2014). Hence, a single explanatory factor is unlikely to explain MLD. A major goal of research should be to map how parts of the WM/EF network relate to mathematical competence. As argued before (Szu˝cs et al., 2014), specific mathematical weaknesses do not necessarily need specific modular impairment explanations. In order to perform well in mathematical tasks typically a large amount of mental acts have to be carried out in strict succession, each must generate an expected result and then finally one single perfect solution must be generated, typically under time pressure. (Typically, a lot of necessary processing steps are not strictly mathematical, for example, keeping partial information in mind.) Hence, mathematics does not tolerate any errors in the processing system and demands close to perfect performance from an extended processing network crucially relying on well-coordinated memory function. Hence, a minor weakness of memory networks (encoding/maintenance/recall and/or associated EFs) may not be noticeable in other domains but may impair performance in demanding mathematical tasks. Relying on a network view may provide insights into the role of task difficulty in MLD and in developmental disabilities in general. First, it can easily explain the often puzzling heterogeneity of MLD: An extended processing network can be impaired in multiple ways. Because mathematical tasks require the close to perfect function of the whole network, even differential individual minor impairments can lead to “similar” performance decrement in mathematical tasks, but such impairments may be less noticeable in other academic domains. Second, if we define general task difficulty as the general demand on processing network coordination irrespective of the task at hand, then we can assume that because mathematical tasks demand excellent network coordination, they will be particularly vulnerable to any increase in general task difficulty. For example, Passolunghi and Mammarella (2012) found that their severe MLD group was particularly sensitive to increased task complexity.

8 MLD SUBTYPES, NETWORK COORDINATION, AND INDIVIDUAL VARIABILITY The data reviewed here suggest two clear MLD subtypes, one associated with reading and verbal memory problems; the other associated with visual memory problems. Such phenotypes can appear in consequence of various underlying deficits in an extended processing network drawing strongly on memory function (Fig. 5). It seems

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FIG. 5 Potential memory impairments in MLD and the parametric multidimensional study of MLD. Potential MLD impairment patterns (A–F) are explained in the text. Impairments of certain functions (in red, gray in the print version, and bold) are signified by red (gray in the print version) crosses. (G) Schematic representation of all the dimensions to be studied optimally. (H) A more precise representation of the four dimensions to be studied with regard to visual memory performance only.

ARTICLE IN PRESS 8 MLD subtypes, network coordination, and individual variability

that verbal STM impairment (Fig. 5A) is linked with reading deficit, while visual STM impairment (Fig. 5B) is largely independent from reading deficit. First, there may be a baseline EF deficit (Fig. 5C) in both verbal and visual MLD subtypes. For example, such baseline EF impairment may result in poor processing network coordination making MLD children particularly vulnerable to the demands of increasing general task difficulty. It is currently not known whether such EF deficit can equally affect various EFs (eg, inhibition, updating, and shifting), or it primarily affects one EF. Second, EF impairment may cooccur with modality-specific verbal (Fig. 5D) and visual STM (Fig. 5E) impairments defining subtypes and comorbidity with other conditions, like dyslexia. Third, in addition to the above, perhaps EF functions could also show at least partial modality-specific impairments further contributing to MLD profiles (Fig. 5F). Again, it is currently not known whether such modality-specific EF function impairments can be isolated. Currently all the above functional deficits can be imagined to contribute to the data reported here. I suggest that in order to efficiently distinguish between all these potential functional impairments and move our understanding beyond what has been achieved during the past 30 years, it is crucial to take a parametric multidimensional approach: Studies need to test several probable factors in a parametric fashion, also manipulating task difficulty. Fig. 5G provides a schematic representation of some important factors to consider: modality-specific storage (STM) and candidate EFs, potentially also playing a role in general processing network coordination. Both increasing EF demands and set sizes to remember will increase specific taskdifficulty levels. Fig. 5H further exemplifies the parametric multidimensional nature of the testing problem: For example, visual memory set size (capacity to achieve) can be manipulated in tasks and orthogonal to this manipulation the difficulty of inhibitory, updating, and shifting tasks can be manipulated as well. A further complicating factor is that visual memory can also be dissociated into multiple components (pattern memory, spatial memory, static, and dynamic memory) and that general task difficulty (general coordination demand in a processing network) is also a crucial factor. A further complicating factor is individual variability. For example, while there may indeed be some baseline memory deficit in both MLD subtypes indentified, such baseline deficit observed at the group level can also be the consequence of including mixed profile children in groups. For example, as noted, subgroup performance may be influenced by borderline children who are just under/above selection criteria. Hence, it is much more useful to aim to position children in a multidimensional measurement space rather than distribute them into artificially separated subgroups. In fact, currently there is not much evidence that discontinuous subgroups can be defined (see Szu˝cs et al., 2013a,b, 2014, for more analysis). So, it is more likely that children can be positioned in a multidimensional distribution space and the challenge is to understand how position along one dimension affects position along another dimension. Therefore, technically, regression models with larger populations are the most desirable for future studies. Such studies should clearly characterize the robustness of their models, for example, by providing bootstrap

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confidence intervals for parameter estimates rather than just provide fairly noise sensitive point estimates (eg, Szu˝cs et al., 2013a,b, 2014). Another important goal is to increase measurement power at the individual level so that potentially different individual profiles could be discriminated. This is especially important because any interventions designed on the basis of models ultimately must work at the individual level in order to be useful.

8.1 OVERALL CONCLUSIONS Underpowered studies are a serious threat to the validity of data as they overestimate effect sizes, and they have very high false report probability (Ioannidis, 2005; Pollard and Richardson, 1987; Schmidt, 1996). So, in the current well-explored state of the MLD literature underpowered studies are hard to justify. In general, studies should have clear MLD diagnosis criteria, provide standardized mathematics achievement scores, minimally test reading as control variable, and take both verbal and visual memory measures. Relying solely on verbal STM measures is inadequate and can result in falsely rejecting the role of memory processes for MLD. The verbal or nonverbal nature of mathematics and IQ testing instruments should be specified as these are important to consider when interpreting potential modality-specific effects. Data may be difficult to interpret if some of these components are missing. Consequently, the added value of partial studies to the literature is questionable and they may enhance confusion rather than understanding. Using an appropriate number of measures will enable us to take a parametric multidimensional approach to measurement which is key to further our understanding. An important note is that the frequent use of ANCOVA to “correct for” group differences (eg, along IQ and reading scores) should be discontinued because it is an invalid procedure which can preclude clear data interpretation (Miller and Chapman, 2001; Porter and Raudenbush, 1987). With regard to the frequently observed IPS abnormality in MLD (see reviews in Szu˝cs and Goswami, 2013; Szu˝cs et al., 2013a,b), it is important to point out that the IPS is involved in various cognitive functions frequently implicated in numerical tasks, like WM (Coull and Frith, 1998; Culham and Kanwisher, 2001; Dumontheil and Klingberg, 2011; Linden et al., 2003; Rotzer et al., 2009; Todd and Marois, 2004), attention (Coull and Frith, 1998; Davranche et al., 2011; Santangelo and Macaluso, 2011; Vandenberghe et al., 2012), inhibitory function (Cieslik et al., 2011; Mecklinger et al., 2003) and spatial processing (Yang et al., 2013), and the proposed number sense (Dehaene, 1997). Therefore, impairment of any of these functions could plausibly explain IPS abnormalities in DD (Szu˝cs and Goswami, 2013; Szu˝cs et al., 2013a,b). In light of the data reviewed here I suggest that one focus of future neuroimaging research should be evaluating how IPS dysfunction, the frequently observed angular gyrus dysfunction, the recently reported mathematical development— hyppocampus function link (see Quin et al., 2014) and frontal lobe function often related to EF are related to visual and verbal STM and WM as well as EF demands and general task-difficulty effects in MLD. As noted, seemingly specific impairments do not necessitate single factor modular explanations.

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I recommend that all future studies publish power calculations considering the mean effect sizes reported here as a starting point. Because these effect sizes are upper estimates, future MLD memory studies should aim to be able to detect smaller effects.

ACKNOWLEDGMENT This research was supported by a grant from the James S. McDonnel Foundation to D.S.

APPENDIX Effect sizes were computed as defined by Hedges (1981): m1
m1 G¼ SD where m1 stands for the mean performance score of the MLD group, m2 stands for the mean performance score of the age-matched control group, and SD stands for the pooled standard deviation computed as: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðn1
1Þsd21 + ðn2
1Þsd22 SD ¼ n1 + n2
2 where sd1 and sd2 stand for the standard deviations measured in the groups, and n1 and n2 denote the sample sizes in groups.

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