Number Magnitude Processing and Verbal Working Memory in Children with Mild Intellectual Disabilities

ABSTRACT

This study examined if children (M_age = 14.60) with Mild Intellectual Disabilities (MID) display weaknesses in number processing and verbal working memory. An age-matched and mental age-matched (MA, M_age = 6.17) design extended by a group of 9–10-year-olds, and a group of 11–12-year-olds were used. The MID children’s working memory was equal to the MA group but poorer than the other groups. On number tasks, the MID group was faster than the MA group but slower than the other groups. All groups obtained equal Weber fraction scores and distance effects on the number comparison tasks. The MID group performed subitizing and counting faster than the MA group, but slower than the 11–12-year-olds. The results demonstrate that number processing and working memory in children with MID is characterized by a developmental delay, not a deficit. Their main problem is to access the quantitative meaning of Arabic numerals. The development of different types of cognitive abilities is differently affected by educational experience and intellectual ability. The innate number system appears to be unaffected by intellectual capacity or educational experience, while the innate working memory ability is affected by intellectual capacity but not by educational experience. Culturally acquired symbolic number abilities are strongly affected by educational experience.

It is well known that children and adults with intellectual disabilities (ID) struggle to learn even the basics of mathematics such as arithmetic (e.g., Bashash, Outhred, & Bochner, 2003; Hoard, Geary, & Hamson, 1999; Janssen, De Boeck, Viaene, & Vallaeys, 1999). Recently, researchers have begun to study the underlying cognitive processes of these mathematical difficulties more systematically. This research has focused on number processing and working memory as evidence demonstrates that these two cognitive functioning components play important roles in the mathematical development of typical children (Clarke & Shinn, 2004; Fuchs et al., 2010; Geary, 2011; Krajewski & Schneider, 2009; Passolunghi & Lanfranchi, 2012; Träff, Olsson, Skagerlund, & Östergren, 2020). Given this, the purpose of this study was to examine whether children with MID display weaknesses in two key components necessary for learning basic mathematics, that is, number magnitude processing and verbal working memory.

Theoretical models of cognitive functioning problems among children with ID

Similar to prior research focusing on cognitive processing in children with ID, this study sought to test and compare the developmental delay model and the deficit model of intellectual disabilities (Bennett-Gates & Zigler, 1998; Zigler & Balla, 1982). The former model posits that children with ID go through the same developmental trajectory as typical children only slower and they attain a lower asymptotic cognitive level. The deficit model, on the other hand, argues that children with ID experience neurocognitive deficits, and thereby follow an atypical pattern of development and attain a much lower asymptotic cognitive level than typical children (Bennett-Gates & Zigler, 1998; Zigler & Balla, 1982).

In addition to these two models, Kohlberg’s (1968) “Richer General Experience Hypothesis” was examined. This hypothesis states that older children with ID have a richer general experience (e.g., years of schooling) than younger children of the same mental age. The richer general experience of children with ID may enhance certain aspects of their cognitive capacity over and above their mental age.

Number magnitude processing

It is well established that humans are born with a preverbal ability to represent and manipulate quantities (Dehaene, 1997; Gallistel & Gelman, 1992; Gelman & Butterworth, 2005; Wynn, 1992, 1995; Xu & Spelke, 2000). The object tracking system (OTS) and the approximate number system (ANS) constitute two distinct conceptualizations of this innate capacity (Feigenson, Dehaene, & Spelke, 2004). The OTS is a visual-spatial, object-based attention system for keeping precise track of small sets of objects (3–4 objects). It represents objects as distinct individuals in working memory by constructing a one-to-one correspondence between real-world objects and individual mental representations (Feigenson et al., 2004; Piazza, 2010). It allows precise discrimination in the small number range (Piazza, Fumarola, Chinello, & Melcher, 2011) evident by subitizing; the immediate and accurate recognition of numerosities up to four objects (Mussolin, Mejias, & Noël, 2010; Piazza, 2010).

The ANS represents large quantities (>4) on an analog magnitude, in ascending order from left to right. The ANS is approximate and imprecise and imprecision increases with numerical size due to larger spatial representational overlap between quantities (Bugden & Ansari, 2011; Dehaene, 1992; de Hevia, Vallar, & Girelli, 2006).

The OTS and ANS are assumed to provide the foundation of the language-based exact symbolic number system (i.e., number words, Arabic numerals; Butterworth, 1999; Feigenson et al., 2004; Gelman & Butterworth, 2005; Piazza, 2010). Around the age of two, children begin to acquire, by rote, their culture’s number words (one, two …) and successively learn this list of number words in a stable order (i.e., counting sequence). The next important step is to associate each number word with a specific quantity, that is its cardinal value, which enables the child to accurately count sets of objects (Gallistel & Gelman, 1992; Gelman & Gallistel, 1978; Wynn, 1990, 1992).

Non-symbolic number comparison and symbolic number comparison are two standard measures used to assess the ANS, these tasks require the child to select the larger of two magnitudes presented in non-symbolic (i.e., arrays of dots) or symbolic (i.e., Arabic numerals) format (Feigenson et al., 2004; Moyer & Landauer, 1967; Piazza, 2010). A key signature of the performance concerning non-symbolic and symbolic number comparison tasks is the distance effect, that is the selection of the larger of two magnitudes is performed faster and more accurately when the distance is larger (1 vs. 5) compared to smaller (1 vs. 2; Moyer & Landauer, 1967). This effect is assumed to arise as close magnitudes (e.g., 1 vs. 2) show more representational overlap than distant magnitudes (e.g., 1 vs. 5) on the visual-spatial mental number line (i.e., ANS; Dehaene, 1992; de Hevia et al., 2006; Feigenson et al., 2004).

Number magnitude processing in children with intellectual disabilities

To date, three studies have examined number magnitude processing in children with intellectual disabilities of unknown origin using the classical non-symbolic and symbolic number comparison tasks. In two studies, Brankaer, Ghesquière, and De Smedt (2011, 2013) used a chronological-age/ability-level-match design to test and compare the developmental delay model and the deficit model (Bennett-Gates & Zigler, 1998; Zigler & Balla, 1982). Brankaer et al. (2011, 2013) found that 10-year-olds and 8-year-olds with mild intellectual disabilities (MID) were slower than the chronological age-matched comparison group (CA) on the symbolic and non-symbolic comparison tasks (see also Brankaer, Ghesquière, & De Smedt, 2014; Hoard et al., 1999), but performed equally as quickly as the arithmetic ability-matched comparison group (AL). These results provide support for the developmental delay model. However, the MID group in Brankaer et al. (2011) displayed a larger distance effect on the symbolic number task compared to both the CA and AL groups, while the MID group in Brankaer et al. (2013) obtained lower accuracy scores than the CA and AL groups. These two findings are in line with the deficit model. Thus, the evidence with respect to the developmental delay model versus the deficit model regarding number magnitude processing are far from clear-cut but overall provide stronger support for the delay model.

Prior results do not clearly indicate whether children with MID suffer from a delay or a deficit in relation to the core number systems (ANS; OTS) or in their ability to access magnitude information from numerical symbols.

The fact that the 10-year-olds and 8-year-olds with MID in Brankaer et al. (2011, 2013) performed more poorly than the CA group on both the symbolic and non-symbolic magnitude tasks (Brankaer et al., 2011, 2013) suggests that they have problems with the ANS. However, both studies also observed that the MID group’s poorer performance is more pronounced on the symbolic magnitude task than on the non-symbolic magnitude task. Furthermore, Brankaer et al. (2011) found that the MID group remained slower on the symbolic number tasks when controlling for the non-symbolic task, while the MID group’s slower performance on the non-symbolic task disappeared when controlling for the symbolic task. These two findings suggest that the major problem for children with MID is to access the numerical meaning of Arabic numerals rather than the quality of their core number systems (ANS; OTS).

Working memory in children with intellectual disabilities

Working memory, that is the ability to simultaneously process and maintain information in an active state, plays an important role in typical children’s mathematical and numerical learning (Fuchs et al., 2010; Geary, 2011; Hornung, Schiltz, Brunner, & Martin, 2014; Passolunghi & Lanfranchi, 2012; Träff et al., 2020). Hence, researchers studying children with ID have also examined working memory resources in order to account for their mathematical learning problems (Henry & MacLean, 2002; Schuchardt, Gebhardt, & Mäehler, 2010). Most of these studies have employed the three-component model developed by Baddeley (1990, 1997; Baddeley & Hitch, 1974). The main component, the central executive, is a control-attention system responsible for coordinating the performance of simultaneous operations, switching between operations, attending selectively to specific information and inhibiting irrelevant information and retrieving information from long-term memory. The central executive is supported by the visuospatial sketchpad and the phonological loop, which are two storage components for visuospatial and verbal information, respectively (Baddeley, 1990, 1997).

Prior research on children with ID has generated consistent findings showing that they have a lower capacity in all components of working memory than chronological age-matched children (Brankaer et al., 2013; Henry & MacLean, 2002; Schuchardt et al., 2010; Van der Molen, Van Luit, Jongmans, & Van der Molen, 2009). Thus, there is strong evidence that children with ID are marked by a developmental delay in regard to working memory. In contrast, evidence for a working memory deficit in children with ID is scarce as only two studies have provided strong data to support this model. Van der Molen et al. (2009) found that 13-to-17-year-olds with ID were outperformed by mental age-matched children on two central executive working memory tasks (Backward Digit Recall; Listening Recall), one phonological loop task (Nonword Recall), and one visuospatial sketchpad task (Corsi block span). Brankaer et al. (2013) also demonstrated that 8-year-old children with MID have deficits in all three components as they performed more poorly than mental age-matched children on a digit span backward task (i.e., central executive), a digit span forward task (i.e., phonological loop), and a Corsi block span task (i.e., visuospatial sketchpad).

All in all, it appears that the working memory function of children with ID parallels their mental age; thus, they display a developmental delay compared to their peers without ID. It should also be noted that Henry and MacLean (2002) observed important findings showing that 11-to-12-year-olds with ID outperformed younger mental age-matched children on tasks tapping the central executive (listening span) and the visual-spatial sketchpad (pattern span; spatial span). The researchers concluded that these findings are consistent with the Richer General Experience Hypothesis stating that the rich general experience of children with ID may enhance their cognitive capacity over and above their mental age (Kohlberg, 1968).

Evidence suggests that working memory contributes to children’s symbolic number processing skills (e.g., digit comparison; Hornung et al., 2014; Passolunghi & Lanfranchi, 2012; Träff et al., 2020). In view of this, Brankaer et al. (2013) examined if the expected number magnitude processing difficulties in children with MID could be explained by their weak working memory capacity. In order to address this question, Brankaer et al. (2013) computed three ANCOVAs with digit span forward, Corsi block span and digit span backward as covariate, respectively. However, Brankaer et al. (2013) found that when working memory was used as a covariate, the group differences remained unchanged, suggesting that MID children’s problems with number magnitude processing are not explained by their working memory deficits.

The present study

The purpose of this present study was to examine if children with MID display weaknesses in two key components for learning basic mathematics, which are number magnitude processing and verbal working memory.

More specifically, the first aim was to examine if the number magnitude processing and working memory capacity of children with MID are best characterized by a developmental delay or cognitive deficit.

The deficit model predicts that the children with MID will be outperformed by the mental age-matched comparison children on the working memory task and all three aspects of number processing (non-symbolic; symbolic; subitizing and counting).

The developmental delay model predicts that the children with MID will perform equally with the mental age-matched comparison children on the working memory task, as well as all three aspects of number processing. However, they will be outperformed by the chronological age-matched comparison children and perhaps, depending on how far behind they are in their development, by the 5 or 3 years younger comparison groups of average intellectual ability.

The Richer general experience hypothesis predicts that the children with MID will outperform the mental age-matched comparison group on abilities particularly affected by years of schooling, that is symbolic number processing, but not working memory or non-symbolic number processing.

A second aim, derived from Brankaer et al. (2013), was to examine if the assumed weak number processing of children with MID could be explained by their lower working memory capacity.

In order to address the aims and predictions of the study an extended chronological age-matched and mental age-matched design of five groups was employed. In addition to the traditional chronological age-matched and mental age-matched comparison groups, the design included a five-year younger comparison group of typical third graders (9-to-10-year-olds) of average intellectual ability, and a 3-year younger comparison group of typical fifth and sixth graders (11-to-12-year-olds) of average intellectual ability. These two additional groups were included in order to enable a more comprehensive test of the developmental delay model, that is, to examine how far behind the children with MID are in their development.

Method

Participants

A total of 277 children (153 girls) participated in the study. All participating children had Swedish as their native language, normal or corrected-to-normal visual acuity, and no hearing loss. Written consent to participate in the study was acquired from each child’s caregiver prior to testing.

Children with MID

Thirty-six children with MID aged 13–16 (M = 14.60, SD = 0.99) participated in the study. All the children with MID attended special education classes. In order to attend these classes in Sweden, children must be diagnosed with mild to moderate intellectual disability (i.e., an IQ at or below 70). Special education classes in Sweden use a curriculum that is different from the one used by regular schools. An additional criterion for being classified into the MID group was that the child’s standard score on the matrix reasoning sub-test from the Wechsler’s Intelligence Scale for Children (WISC-IV; Wechsler, 2003) was at or below 3.

Mental age-matched comparison group (MA)

This group included 36 typical six-year-olds individually matched to the children with MID on gender and raw scores on the WISC-IV matrix reasoning sub-test (Wechsler, 2003).

Chronological age-matched comparison group (CA)

This group consisted of 98 typical adolescents aged 13–16 (M = 14.63, SD = 0.95). Thirty-three were in their seventh or eighth year of schooling and 32 were in their ninth year of schooling.

Typical 9-to-10-year-olds (T9–10)

This comparison group consisted of 75 typical third graders aged 9–10 years (M = 9.61, SD = 0.29) of average intellectual ability. Each child’s raw score on Raven’s Standard Progressive Matrices was transformed into a percentile score according to the manual (Raven, 1976; Raven, Court, & Raven, 1996). In order to be included in this group the child’s percentile score on Raven’s Standard Progressive Matrices had to be between the 40th ^– 60th percentile (Raven et al., 1996).

Typical 11-to-12-year-olds (T11–12)

This comparison group consisted of 32 typical fifth and sixth graders (M = 11.82, SD = 0.75) of average intellectual ability. Similar to the T9–10 group, the child’s raw score on Raven’s Standard Progressive Matrices was transformed into a percentile score according to the manual (Raven, 1976; Raven et al., 1996). Children with a percentile score between the 16th ^– 84th percentile were included in this group.

Background information and results from the Wechsler’s Intelligence Scale for Children and, the Raven’s Progressive Matrices Test, are displayed for each group in Table 1.

Table 1. Descriptive information for participants by achievement group.

	MID		MA		CA		T9–10		T11–12
	M	SD	M	SD	M	SD	M	SD	M	SD
N (number of boys)	36	(16)	36	(16)	98	(37)	75	(42)	32	(13)
Mean age (in years)	14.60	0.99	6.17	0.26	14.63	0.95	9.61	0.29	11.82	0.75
Matrix reasoning, WISC-IV* (standard scores)	2.22	1.07	9.25	2.03
Matrix reasoning, WISC-IV* (raw scores)	9.25	3.70	10.25	2.54
Ravens Progressive Matrices (percentile)							52.05	6.84	51.34	21.42

MID = Children with Mild Intellectual Disabilities, MA = Mental age-matched control group, CA = chronological age-matched control group, T9–10 = Nine-to-ten-year old typical children, T11–12 = Eleven-to-twelve-year old typical children.

*The Matrix reasoning sub-test from the Wechsler’s Intelligence Scale for Children (WISC-IV; Wechsler, 2003)

Experimental task and procedure

Verbal working memory

In this task, the experiment leaders read sequences of words from a sheet of paper to the child. The child had to decide whether each word was an animal (“Yes”) or not (“No”) before the next word was presented. At the end of the sequence, the child had to recall the words in the sequence in correct serial order. The first span size employed was two increasing to three, four, five, six and seven. All the children were presented up to span size four, but if the child managed to recall span four in correct serial order, testing continued until the child failed both trials of the same span length. Verbal working memory span was measured as the longest sequence of words remembered in correct serial order, plus 0.50 points extra if the child could remember both trials in her/his longest span size. Thus, the maximum score was 7.50.

Non-symbolic number magnitude comparison

This task was administered using the Panamath software (v.1.21; Halberda & Feigenson, 2008). Two arrays, each containing between 5–21 blue and yellow dots, were presented on the screen. The task was to determine which one of the two arrays had more dots, as rapidly as possible without making an error. The child responded by pressing the A key or L key. The child had to press the space bar to present the next trial. Prior to each trial, a fixation cross was displayed in the middle of the screen. Four ratios (1.28; 1.46; 1.75; 2.75) were presented 12 times each, resulting in 48 trials. Surface area varied on half the trials, along with dot size. This was to control for confounding variables and ensure attention to numerosity. The software registered accuracy and response time for each trial and calculated a Weber fraction (w), as an estimate of ANS acuity, which is based on accuracy for each of the four different ratios. The w is calculated by using Weber’s law ((A-B)/B where A and B are the quantities being compared) as it has been found that ANS acuity adheres to psychophysical laws and thus, can be understood in terms of Weber’s law. The w is an index of how much two sets of objects needs to differ so that an individual can reliably notice a difference between them (Halberda & Feigenson, 2008; Libertus, Feigenson, & Halberda, 2011).

Symbolic number magnitude comparison

This task was administered using the SuperLab PRO (v.4.5). Two single-digit numbers were presented simultaneously in the center of the computer screen. The task was to determine as rapidly as possible without making an error, which one of the two numbers was the numerically larger one. The child responded by pressing either A key to indicate the number on the left or the L key for the number on the right. Between each trial, a fixation-cross appeared in the middle of the screen for 1000 ms. Two distances between the numbers were used: 1 (1–2, 2–3, 3–4, 4–5, 5–6, 6–7, 7–8, 8–9) and 4–5 (1–6, 2–7, 3–8, 4–9, 1–5, 2–6, 3–7, 4–8) and each number pair was presented twice (e.g., 8–9 and 9–8) generating a total of 32 trials. The SuperLab software registered accuracy and response time for each trial.

Subitizing and counting

In this task, arrays of randomly arranged black dots were shown on the computer screen. The task was to state the number of dots displayed as quickly as possible without making any errors. The experiment leader pressed the mouse button after the child had stated the number of dots. This test was administered using SuperLab PRO (v.4.5), which recorded response time for each trial. The tester recorded the answers given, and the dots remained on the screen until the participant had provided an answer. The experimenter had to press the space bar to present the next trial which commenced with an empty screen for 1000 ms. The number of dots displayed ranged from 1 to 8, each appearing twice but in different patterns for a total of 16 trials. For each child, response time measures (correct responses only) were created for the subitizing measure (1–4 dots) and counting measure (5–8 dots).

General procedure

All instructions were given orally, read aloud from a printed manuscript to ensure that all participants were given the same instructions. All computer-administered tasks were run on PC laptops. Prior to testing, verbal consent was acquired from each participating child. The child was also informed that she/he could ask for a break in testing, as well as opt-out of the study completely, whenever she/he wanted with no questions asked. All the children were tested in familiar rooms at their respective schools.

Results

Means and standard deviations for the verbal working memory task and the number magnitude processing tasks by group are displayed in Table 2. In order to address the research aims, a number of one-way and two-way ANOVAs were computed to examine group and interaction effects. As the group differences between the MA, CA, T11–12, and T9–10 groups were of minor interest, these differences were only reported but not discussed further. The Šidák test procedure were used as post hoc test and ${\eta }_p^2$ and Cohen’s d were used as effect sizes (Cohen, 1988). An ${\eta }_p^2$ of .25 or larger represents a large effect, whereas an ${\eta }_p^2$ between .10 and .24 is of medium size. An ${\eta }_p^2$ of .09 or less is considered to be small. A Cohen’s d of 0.80 or larger represents a large effect, whereas a Cohen’s ds between 0.30 and 0.79 is of medium size. A Cohen’s d of 0.20 or less is considered to be small.

Table 2. Descriptive statistics for the experimental task by group.

	MID		MA		CA		T9–10		T11–12
	M	SD	M	SD	M	SD	M	SD	M	SD
Verbal working memory	2.91	0.41	2.54	0.90	4.13	0.85	3.84	0.67	3.58	0.69
Non-symbolic Number magnitude Weber fraction	0.21	0.23	0.27	0.23	0.21	0.13	0.20	0.08
RT (ms)	1640	723	2624	1588	681	198	1034	328
Symbolic Number magnitude total	1529	796	2567	1336	605	128	902	303
Distance 1	1599	832	2812	1726	649	141	953	273
Distance 4–5	1417	747	2305	1257	560	107	852	339
Distance effect	182	289	507	1409	90	63	101	148
Total accuracy	29.70	4.59	26.28	5.25	30.19	1.41	30.16	1.87
Subitizing 1–4 objects	1550	364	2176	1034			1354	297	999	237
Counting 5–8 objects	4067	2256	5618	1918			3259	609	2462	517

Due to the large number of F-tests computed (22), a Bonferroni correction was adopted in order to control for experiment-wise alpha inflation and Type I error. The correction resulted in a significant alpha level of p < .0023.

Verbal working memory

The one-way ANOVA performed on this task revealed a large group effect, F(4, 269) = 39.176, p < .001, ${\eta }_p^2$ = .368. Post hoc testing showed that the scores of the MID and MA groups were equal (p = .346), while the MID group was outperformed by the CA group (d = 1.83), the T11–12 group (d = 1.18), and the T9–10 group (d = 1.67). The MA group was also outperformed by the CA group, the T11–12 group, and the T9–10 group. The T11–12 group was outperformed by the CA group, but the scores of the CA and the T9–10 groups were equal (p > .05).

Non-symbolic number magnitude

The ANOVA performed on the weber fraction scores did not reveal any significant group differences, F(3, 235) = 1.479, p = .221, ${\eta }_p^2$ = .019, but a large group effect emerged on the RT measure, F(3, 235) = 73.006, p < .001, ${\eta }_p^2$ = .481. Post hoc testing demonstrated that MID group was faster than the MA group (d = 0.80), but slower than the CA group (d = 1.81) and the T9–10 group (d = 1.08). The MA group was slower than the T9–10 group, who in turn was slower than the CA group.

Symbolic number magnitude

The response times were analyzed using a 4 (groups) by 2 (distance) mixed ANOVA to detect any group differences and whether the four groups displayed the distance effects to the same extent (i.e., group x distance interactions). The ANOVA revealed a large group effect, F(3, 232) = 94.290, p < .001, ${\eta }_p^2$ = .549, a medium-sized distance effect, F(1, 232) = 29.291, p < .001, ${\eta }_p^2$ = .112, and a small group x distance interaction effect, F(3, 232) = 5.277, p = .002, ${\eta }_p^2$ = .064. Post hoc testing displayed that the MID group was faster than the MA group (d = 0.94), but slower than the CA group (d = 1.62) and the T9–10 group (d = 1.04). The MA group was slower than the T9–10 group, who in turn was slower than the CA group.

In order to evaluate the group x distance interaction effect, a distance effect index (see Table 2) was calculated for each child by subtracting the RT on the large numerical distance (4–5) condition from the smaller numerical distance (1) condition. This index was analyzed using a one-way ANOVA and showed a significant group effect, F(3, 232) = 5.2799, p = .002, ${\eta }_p^2$ = .064. The MA group displayed a significantly larger distance effect compared to the CA group, and T9–10 group, but not the MID group (p = .102). No other group differences reached significance (ps > .05).

An additional one-way ANOVA was computed on the accuracy scores which revealed a significant medium-sized group effect, F(3, 232) = 16.903, p < .001, ${\eta }_p^2$ = .179, which was due to the fact that the MA group displayed lower scores compared to all other groups. No other group differences reached significance (ps > .05).

In order to further contrast the difference between non-symbolic and symbolic magnitude processing in children with MID, two ANCOVAs were computed on the symbolic number magnitude task and the non-symbolic number magnitude task with the other task as a covariate. The MID children (adjusted M = 1225) still performed slower on the symbolic number magnitude task when controlling for the non-symbolic number task, F(3, 228) = 24.422, p < .001, ${\eta }_p^2$ = .243, compared to the CA group (adjusted M = 816; d = 0.72), and still performed faster than the MA group (adjusted M = 1976; d = 0.68). However, the mean difference between the MID children and the T9–10 children (adjusted M = 967) was no longer statistically significant (p = .159). The ANCOVA revealed a large contribution from the non-symbolic number task, F(1, 228) = 66.165, p < .001, ${\eta }_p^2$ = .225.

The ANCOVA on the non-symbolic number task displayed a large contribution from the symbolic number magnitude task. F(1, 228) = 66.165, p < .001, ${\eta }_p^2$ = .225, and a small group effect, F(3, 228) = 8.014, p < .001, ${\eta }_p^2$ = .095. Post hoc testing demonstrated that the MID group (adjusted M = 1462) no longer performed faster than the MA group (adjusted M = 1799; p = .285) or slower than the T9–10 group (adjusted M = 1154; p = .168). However, the MID children still performed slower than the CA group (adjusted M = 959; d = 0.95).

Subitizing and counting

The one-way ANOVA performed on the subitizing measurement (1–4 dots) yielded a significant and large group effect, F(3, 165) = 29.101, p < .001, ${\eta }_p^2$ = .346. The MID group performed faster than the MA group (d = 0.81), equal to the T9–10 group (p = .427), but slower than the T11–12 group (d = 1.79). The MA group was significantly slower than all other groups, while the T11–12 group was significantly faster than all other groups.

On the counting measurement (5–8 dots) a large group effect emerged F(3, 165) = 33.389, p < .001, ${\eta }_p^2$ = .378. The MID group performed faster than the MA group (d = 0.74), but slower than the T9–10 group (d = 0.49), and the T11–12 group (d = 0.98). The MA group was significantly slower than all the other groups, while the T11–12 group was significantly faster than all the other groups.

Could weak working memory account for the MID group’s slower number magnitude processing?

In order to examine whether the slower performance of the MID group on the number magnitude tasks could be attributed to their poorer working memory capacity ANCOVAs were computed using the verbal working memory task as a covariate.

The ANCOVA computed on the symbolic number magnitude task revealed a large group effect, F(3, 229) = 47.587, p < .001, ${\eta }_p^2$ = .384, and a small but significant contribution from the working memory task, F(1, 229) = 14.878, p < .001, ${\eta }_p^2$ = .061. Sidak post hoc tests showed that the MID group (adjusted M = 1285) was still faster than the MA group (adjusted M = 2319; d = 0.94), and still slower than the CA group (adjusted M = 703; d = 1.02). However, the group difference between the MID group and the T9–10 group (adjusted M = 941) disappeared (p = .060). The MA group was still slower than the T9–10 group, while the difference between the T9–10 group and the CA group disappeared (p = .057).

The ANCOVA performed on the RT measure for the non-symbolic number magnitude task showed a large group effect, F(3, 234) = 40.441, p < .001, ${\eta }_p^2$ = .341, but no contribution from the working memory task, F(1, 234) = 1.770, p = .185, ${\eta }_p^2$ = .008. Thus, the MID group (adjusted M = 1540) was still faster than the MA group (adjusted M = 2536; d = 0.81) and still slower than the CA group (adjusted M = 720; d = 1.55) and the T9–10 group (adjusted M = 1050; d = 0.87). The MA group was still slower than the T9–10 group, who in turn was slower than the CA group.

The ANCOVA computed on the subitizing measurement demonstrated a large group effect, F(3, 161) = 15.132, p < .001, ${\eta }_p^2$ = .220, and a small contribution from the working memory task, F(1, 161) = 12.337, p = .001, ${\eta }_p^2$ = .071. The MID group (adjusted M = 1454) still performed faster than the MA group (adjusted M = 2001; d = 0.71) and equal to the T9–10 group (adjusted M = 1461, p = 1.00), but were still slower than the T11–12 group (adjusted M = 1050; d = 1.31). The MA group was still significantly slower than all other groups, while the T11–12 group was still faster than the T9–10 group.

On the counting measurement a large group effect emerged F(3, 161) = 16.626, p < .001, ${\eta }_p^2$ = .237, and a small contribution from the working memory task, F(1, 161) = 11.241, p = .001, ${\eta }_p^2$ = .065. The MID group (adjusted M = 3928) still performed faster than the MA group (adjusted M = 5190; d = 0.60), equal to the T9–10 group (adjusted M = 3520; p = .772), but still slower than the T11–12 group (adjusted M = 2587; d = 0.82). The MA group was still significantly slower than all the other groups, while the T11–12 group was still significantly faster than all the other groups.

Discussion

This study addressed two aims regarding number magnitude processing and working memory in children with MID. The first aim was to examine if number magnitude processing and working memory capacity of children with MID are best characterized by a developmental delay or cognitive deficit. The second aim was to examine if the assumed weak number magnitude processing of children with MID could be explained by their weak working memory capacity. The results will now be discussed in relation to the contrasting models and hypotheses.

A developmental delay or a deficit?

The MID children’s slower performance on symbolic and non-symbolic magnitude comparison tasks compared to the CA and the T9–10 comparison groups corroborate and extend prior research (cf. Brankaer et al., 2011, 2013, 2014; Hoard et al., 1999). However, the findings that the MID group performed these two tasks faster than the MA comparison group and obtained higher accuracy scores than the MA comparison group on the symbolic number magnitude task, but similar accuracy scores as the CA and T9–10 comparison group contradicts prior studies by Brankaer et al. (2011, 2013). Moreover, the finding that the MID children’s symbolic numerical distance effect and weber fraction scores of the non-symbolic number magnitude task were equal to the MA, CA, and T9–10 comparison groups are important findings inconsistent with Brankaer et al. (2011).

Important and novel findings were observed on the subitizing and counting task as the MID group’s performance on the subitizing (1–4 dots) and counting (5–8 dots) measurements, were faster than the MA comparison group, but slower than the T11–12 comparison group. Compared to the T9–10 comparison group, the MID group was equally fast on the subitizing measurement but slower on the counting measurement. These results extend prior research by showing that children with a mild intellectual disability also experience difficulties with these aspects of number processing not only with number magnitude comparison.

The combined results of the number processing tasks and the working memory task provide strong support for the developmental delay model as the MID group display poorer performance on almost all measurements not only compared to the CA comparison group but also compared to the T9–10 and T11–12 comparison groups.

The overall findings from the number processing tasks also provide evidence for Kohlberg’s (1968) “Richer General Experience Hypothesis” as the MID children outperformed the MA children on all measurements, except for the weber fraction score and the numerical distance effect measurement. The sample of children with MID used had a mean age of 14.60 years and they were in their seventh, eighth, or ninth year of schooling (i.e., special education classes). The children in the MA comparison group, on the other hand, had a mean age of 6.17 years and they had not yet started their formal schooling. Thus, considering the differences in chronological age and years of schooling it is obvious that the MID children have considerably more general experience than the MA children, which enable them to develop number processing skills and knowledge over and above their mental age.

This study provides some answers to the question; How far behind are children with MID in their development of number processing? The results of the number comparison tasks and the counting measurement (5–8 dots) demonstrate that despite their seven-to-nine-years of schooling their performance level is below typical nine-to-ten-years-old children who are in their third year of schooling. Thus, the children with MID suffer from a considerable developmental lag concerning these aspects of number processing. Their developmental delay regarding subitizing is less extensive as they appear to be at the same level as typical 9-to-10-years-old children but below typical 11-to-12-years-old children who are in their fifth or sixth year of schooling. The MID children performed as equally well as the CA comparison group on the weber fraction and the numerical distance effect measures suggesting that their development of the core number magnitude representation (i.e., the ANS; Dehaene, 1992) is more or less unaffected by their intellectual disability.

Contrary to Van der Molen et al. (2009) and Brankaer et al. (2013), the MID children’s verbal working memory performance was equal to the MA children. However, similar to prior research, the MID group performed more poorly than the CA comparison group (Brankaer et al., 2013; Henry & MacLean, 2002; Schuchardt et al., 2010; Van der Molen et al., 2009). A novel and important finding is that the MID children also were outperformed by the T11–12 and the T9–10 comparison groups on the verbal working memory task. Thus, the MID children’s level of working memory development is below the level of typical nine-to-ten-years-old children of average intellectual ability. These findings are in line with the developmental delay model but contradict the deficit model. Furthermore, contrary to Kohlberg (1968), the MID children’s richer general experience had not enhanced their verbal working memory capacity, suggesting that this type of innate cognitive ability is less malleable by general experience as well as by education (cf. Henry & MacLean, 2002). Instead, the MID children’s verbal working memory capacity appear to parallel their mental age, which is consistent with the view that executive working memory functions are related to intelligence or constitute one aspect of the intelligence construct (Ackerman, Beier, & Boyle, 2005; Conway, Kane, & Engle, 2003; Jaeggi, Buschkuehl, Jonides, & Perrig, 2008).

Problems with the core number systems or accessing magnitude information from numerical symbols?

Consistent with the assumption that children with MID have a weakness in the core number system, the MID group performed symbolic and non-symbolic magnitude comparison more slowly than the CA and the T9–10 comparison groups (Brankaer et al., 2011, 2013). Furthermore, the slower performance of the MID children compared to the CA comparison group on the symbolic and non-symbolic number magnitude tasks remained when including the other magnitude task as a covariate in the ANCOVA (see Brankaer et al., 2011). The results of the two ANCOVAs demonstrate that the MID children’s slower number processing performance is not due to a general slowness.

Importantly and contrary to a weakness in the core number system, the MID children performed equally to the CA comparison group on the weber fraction measure of the non-symbolic magnitude comparison task. Furthermore, the MID children’s numerical distance effect was equal to the distance effect of the CA comparison group. These two findings suggesting that their core number magnitude representation (i.e., the ANS; Dehaene, 1992) is more or less unaffected by their intellectual disability.

The fact that the 6-year-olds (MA), the 14-year-olds MID, CA) and 9-to-10-years-olds (T9–10) displayed equal Weber fraction values disputes the assumptions that the acuity of the ANS sharpens with biological age and/or educational experience (Dehaene, 1997; Halberda & Feigenson, 2008; Izard, Sann, Spelke, & Streri, 2009; Noël & Rousselle, 2011). However, the biological age maturation position is compatible with these results if it is assumed that the sharpening of the ANS primarily takes place up to 6 years of age as demonstrated by Halberda and Feigenson (2008). Noël and Rousselle (2011) position, on the other hand, the assumption that mathematical educational experience is critical for the sharpening of the ANS is difficult to reconcile with these results due to considerable differences in educational experience. The 6-year-olds (MA group) had not yet started formal schooling while the children in the CA comparison group had seven to 9 years of schooling. This lack of association between ANS acuity (Weber fraction scores) and mathematical educational experience indicates that the approximate core number system, the ANS, is not critically involved in children’s development of formal mathematical skills (cf. Butterworth, 2010; Núñez, 2017a, 2017b).

The MID children’s slower performance on the counting measurement (5–8 objects), a task that entails coding numerosities into counting words, further demonstrates that they have problems with accessing magnitude information from numerical symbols.

All in all, the present data provide strong evidence that the primary problem for children with MID is a developmental lag in their ability to access the quantitative meaning of numerical symbols.

The children with MID were outperformed by the T11–12 comparison group on the subitizing task but performed equal to the T9–10 comparison group. Thus, children with MID appear to have developmental problems with their object-tracking system, responsible for the instantaneous and precise apprehension of numerosities of up to four objects.

Can low verbal working memory capacity account for the MID children’s poor number processing?

Akin to Brankaer et al. (2013), when controlling for performance on the working memory task, the MID group was still slower than the CA comparison group on the symbolic number magnitude task and also slower than the T11–12 comparison group on the counting measure (5–8 dots). However, the group differences between the MID group and the T9–10 comparison group disappeared, suggesting that the MID children’s developmental lag regarding these two aspects of symbolic number processing may partially be attributed to their lower verbal working memory capacity.

In contrast, controlling for performance on the verbal working memory task did not eliminate the slower performance of the MID group compared to the CA and T9–10 comparison groups on the non-symbolic number magnitude task. Neither did working memory performance eliminate the group difference between the MID group and the T11–12 comparison group on the subitizing measurement. These two non-significant findings indicate that low levels of working memory capacity do not account for the MID children’s developmental lag in non-symbolic number magnitude processing and subitizing (cf. Brankaer et al., 2013).

Limitation section

Some methodological limitations should be acknowledged. The study did not include measures tapping all components of Baddeley’s working memory model. By including additional measures, it would have been possible to provide a more comprehensive description of the MID children’s working memory capacity. In view of the developmental delay model and the richer general experience hypothesis, it would have been interesting to include more complex number processing tasks such as number line estimation and double-digit number comparison. It is reasonable to assume that complex number processing abilities are more affected by educational experience than basic abilities such as non-symbolic number comparison and subitizing.

Conclusions

The present evidence demonstrates that number magnitude processing in children with MID is characterized by a developmental delay, not a deficit. Their developmental lag in years varies between three-to-five-years depending on the task at hand. It is important to note that the MID children do not appear to have a delay concerning the quality of their core number magnitude representation. Instead, their main problem is to access the quantitative meaning of Arabic numerals (i.e., developmental lag in access).

The MID children’s verbal working memory capacity is marked by a 5-year developmental delay. The developmental lag regarding certain aspects of symbolic number processing (symbolic number magnitude comparison; counting) can partially be attributed to the MID children’s lower verbal working memory capacity.

This study shows that the development of different types of cognitive abilities are differently affected by educational experience and intellectual ability. The innate non-symbolic number processing ability (i.e., ANS acuity) appears to be unaffected by intellectual capacity or educational experience, while the innate verbal working memory ability is affected by intellectual capacity but not by educational experience. Culturally acquired symbolic number processing abilities, on the other hand, are strongly affected by educational experience.

Educational implications

The present results show that mathematical instructions for children with MID exert positive effects on their number magnitude processing abilities as they performed over and above their mental age. However, in order to reduce their rather substantial developmental lag in symbolic number magnitude processing speed interventions should primarily focus on exercises strengthening the connection between number words and Arabic numerals and the numerosities they represent. It is important that these exercises are performed under time pressure in order to enhance MID children’s number processing speed, as their underlying magnitude representations appear to parallel their chronological age. Thus, some type of educational software such as the Number Race (Wilson, Revkin, Cohen, Cohen, & Dehaene, 2006) or Rescue Calcularis (Kucian et al., 2011) could be used for this purpose. These softwares’s have previously been successfully used to enhance symbolic number magnitude processing skills in children with low number skill (e.g., DD; Käser et al., 2013; Kucian et al., 2011; Räsänen et al., 2009; Wilson, Dehaene, Dubois, & Fayol, 2009; Wilson et al., 2006).

Ethics

This research project has been approved by the Regional Ethics Committee in Linköping, Sweden (Protocol Number 2016/318-31)

Disclosure statement

The authors report no conflict of interest.

Data availability statement

The main and corresponding author has full access to all the data in the study and takes responsibility for the integrity of the data and the accuracy of the data analysis.

Additional information

Funding

This work was supported by the Swedish Research Council under Grant 2015-02157, and under Grant 721-2011-2872.