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Research in Developmental Disabilities

Approximate additions and working memory in individuals with Down syndrome Carmen Belacchi a,*, Maria Chiara Passolunghi b, Elena Brentan c, Arianna Dante c, Lara Persi a, Cesare Cornoldi c a b c

Department of Human Sciences, University of Urbino ‘‘Carlo Bo’’, Italy Department of Life Sciences, University of Trieste, Italy Department of General Psychology, University of Padova, Italy

A R T I C L E I N F O

A B S T R A C T

Article history: Received 1 December 2013 Received in revised form 26 January 2014 Accepted 28 January 2014 Available online 3 March 2014

There is some evidence that individuals with Down syndrome (DS) may have a poorer mathematical performance and a poorer working memory (WM) than typically developing (TD) children of the same mental age. In both typical and atypical individuals, different aspects of arithmetic and their relationships with WM have been largely studied, but the speciﬁc contribution of WM to the representation and elaboration of non-symbolic quantities has received little attention. The present study examined whether individuals with DS are as capable as TD children matched for ﬂuid intelligence of estimating numerosity both of single sets and of added sets resulting when two sequentially presented sets are added together, also considering how these tasks related to verbal and visuospatial WM. Results showed that the DS group’s performance was signiﬁcantly worse than the TD group’s in numerosity estimation involving one set, but not when estimating the numerosity resulting from the addition. Success in the addition task was related to success in the working memory tasks, but only for the group with DS; this applied especially to the visuospatial component, which (unlike the verbal component) was not impaired in the group with DS. It is concluded that the two numerosity tasks involve different processes. It is concluded that the arithmetical and working memory difﬁculties of individuals with DS are not general, and they can draw on their WM resources when estimating the numerosity of additions. ß 2014 Elsevier Ltd. All rights reserved.

Keywords: Mathematical skills Down syndrome Working memory

1. Introduction When teaching and supporting cognitively impaired individuals, it is crucial to examine to what extent they can learn not only formal mathematics, an area where they typically have severe difﬁculties (Buckley, 1985; Pieterse & Treloar, 1981), but also intuitive mathematics, which may help them to adapt to the demands of the environment. Intellectually disabled individuals frequently have to make decisions based on estimating the numerosity of sets, and particularly the numerosity resulting from the sum of successively presented sets. For instance, they may have taken

* Corresponding author. Tel.: +390722305813. E-mail address: [email protected] (C. Belacchi). http://dx.doi.org/10.1016/j.ridd.2014.01.036 0891-4222/ß 2014 Elsevier Ltd. All rights reserved.

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some chocolates from a dish a few times and need to estimate how many they have taken altogether without being able to see what they had taken previously. The present study examines to what extent individuals with an intellectual disability due to Down syndrome (DS) are capable of estimating numerosity of single and added sets, and the role of underlying cognitive mechanisms, and working memory in particular, in supporting this estimation. There is robust evidence to show that the estimation of the numerosity of a single set relies on an innate predisposition shared by humans and animals, founded on an early, non-symbolic system of numerical representation (Carey, 2001; Feigenson, Dehaene, & Spelke, 2004). Several studies (e.g. Dehaene, 1997; Gallistel, 1990; Meck & Church, 1983) have also proposed that this approximate, abstract and pre-verbal representation of numerosity is processed by a speciﬁc cognitive component, which has been called the approximate number system (ANS). The precision of this representation is expressed in terms of ‘‘number acuity’’ (Izard & Dehaene, 2008; Piazza, Izard, Pinel, Le Bihan, & Dehaene, 2004). Some authors consider the ANS as the basis of exact arithmetic (Barth et al., 2006; Barth, La Mont, Lipton, & Spelke, 2005; Gilmore, McCarthy, & Spelke, 2007). When children learn number words in counting routines, the ANS should establish connections with the related symbols that begin to acquire meaning (Dehaene, 1992; Dehaene & Cohen, 1995, 1997; Gallistel & Gelman, 1992). The capacity to represent and mentally manipulate approximate numerosity could be the key to learning arithmetic, that gradually develops with the support of language and instruction (e.g. Butterworth, 1999; Gelman & Gallistel, 1978; Lemer, Dehaene, Spelke, & Cohen, 2003) and thus predict success in mathematical learning (see Passolunghi & Lanfranchi, 2012). However exact and approximate representations of numerosity can be distinguished, as proposed by Butterworth (1999) who hypothesized that a speciﬁc impairment of the ‘‘number module’’, which enables the representation of exact numerosity, is at the root of dyscalculia. Consistent with this view, Iuculano, Tang, Hall, and Butterworth (2008) found that children with dyscalculia are impaired in tasks of exact numerosity, but not in approximate number tasks (see also Rousselle & Noe¨l, 2007), but this observation was contrasted by Piazza et al. (2010) who found that also approximate estimation may be poor in developmental dyscalculia. In sum, there is evidence that developmental dyscalculia may be related with a difﬁculty to estimate numerosities and this evidence could be used for exploring the factors underlying the arithmetic difﬁculties experienced by other disabled group, like individuals with Down syndrome (DS). In fact there is evidence of individuals with DS behaving poorly in mathematics (e.g. Abdelahmeed, 2007; Gelman & Cohen, 1988), also revealing a different pattern of behavior from TD children of comparable mental age varying according to the type of task. Marotta, Viezzoli, and Vicari (2006) found, for example, that individuals with DS were differently comparable with TD children of different ages, depending on the mathematical tasks they were presented. In a recent study Sella, Lanfranchi, and Zorzi (2013), by administering two delayed match-to-sample tasks, also found that individuals with DS were more weak in some numeric tasks than in other ones. In particular, individuals with DS were poorer than children matched for mental age in discriminating small numerosities (within the subitizing range) but not in discriminating large numerosities, offering thus support to a previous evidence collected by Camos (2009). The fact that individuals with DS are poorer than TD children in some tasks than in other tasks could also be the effect of a speciﬁc developmental trajectory in the cognitive competences of individuals with DS, coinciding with a different involvement of verbal or visuospatial processes in mathematical tasks at different ages (Paterson, Girelli, Butterworth, & Karmiloff-Smith, 2006; Vicari, Marotta, & Carlesimo, 2004). In fact, among the various components that are related to mathematics but not domain-speciﬁc, verbal and visuospatial WM seem to be particularly important. The relationship between WM and arithmetic has been largely documented in both TD and atypically developing individuals, although the speciﬁc contribution of WM to the representation and elaboration of non-symbolic quantities has attracted relatively little interest (see Geary, Hoard, Nugent, & Byrd-Craven, 2007 vs. Butterworth & Reigosa, 2007). Many studies have used the working memory model proposed by Baddeley (1986, 2007) as a framework in which to study mathematical abilities. In fact, the two modality-speciﬁc systems of WM – the phonological loop and the visual-spatial sketchpad – specialized in processing language-based and visuospatial information, respectively, seem both involved in arithmetic. In particular, in the case of numerosity estimation resulting from the addition of sets maintained in memory, both the verbal component (verbalization and counting processes) and the visuospatial component (to recall the sets) of WM may be involved, similarly to what happens in the case of the mental addition of symbolic quantities. In fact, in the case of approximately adding symbolic numbers, both WM components have been shown to have a role – using a dual task paradigm – by Caviola, Mammarella, Cornoldi, and Lucangeli (2012) in TD children, and by Mammarella, Caviola, Cornoldi, and Lucangeli (2013) in children with developmental dyscalculia. In fact, approximate calculation in children seems to involve WM to a larger extent than exact calculation, unlike the situation seen in adults (Kalaman & LeFevre, 2007). Despite the obvious differences between adding numbers or adding analogical sets, both the two addition tasks could at least partly share a similar WM involvement. Studies on the relationship between WM and addition of analogical sets in individuals with DS could thus take advantage of the knowledge gained in the large body of research on the WM weaknesses of individuals with DS, that are severe but selective, affecting verbal WM to a larger extent than visuospatial WM, and with further differentiations within each of these components (Bargagna, Perelli, Dressler, & Pinsuti, 2004; Carretti, Lanfranchi, & Mammarella, 2013; Jarrold, Baddely, & Hewes, 2000; Kanno & Ikeda, 2002; Lanfranchi, Cornoldi, & Vianello, 2004; Seung & Chapman, 2004; Vicari et al., 2004).

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2. The present study The present study assessed verbal and visuospatial WM and numerosity estimation abilities in individuals with DS and preschoolers matched for ﬂuid intelligence. We tested the hypotheses that individuals with DS may do less well than TD children only in some aspects (verbal WM and estimating the numerosity of a set), but not in others (visuospatial WM and estimating the numerosity of the sum of two successively presented sets), and that the pattern of relationships between WM and approximate addition would differ between the two groups. Previous studies focused mainly on estimating the numerosity of a perceptually available single set. However, approximate addition in everyday life typically requires to approximately estimating the sum of sets presented in succession (and no longer visible), and it is not clear whether in this case results are similar and whether success in approximate additions relies on numerosity estimating ability or is also supported by other cognitive mechanisms, and WM in particular. In fact, estimating the numerosity of a sum obtained by adding successively presented sets may not necessarily demand a high number acuity, but must focus on a plausible approximation criterion. Therefore, it may not involve the same skills as for accurately identifying the numerosity of a visible, single set, but rely instead on other abilities, and especially working memory, verbal (to recall some simple verbalizations of the numerosities) and especially visuospatial (to recall the overall analogical sets). Furthermore, the relationship between WM and estimating additions may be different in young children such as typically developing (TD) preschoolers, and in older intellectually impaired individuals with DS, who have basic deﬁcits but also the advantage of a longer experience with addition at school and in everyday life. Preschoolers have yet to be taught additions and may estimate the numerosity of sums of sets directly, whereas DS adolescents and young adults, despite a similar performance to the preschoolers in terms of ﬂuid intelligence, can rely to a lesser extent on their domainspeciﬁc abilities and to a greater extent on their memory. Thus, by drawing from their longer experience of the day-to-day need to add up natural sets, and taking advantage of their good visuospatial WM, individuals with DS could rely more than TD children matched for mental age on their memory resources and then cope successfully with estimating the numerosity of sums of sets. Conversely, individuals with DS would have more difﬁculty with numerical acuity tasks more directly related with their arithmetic difﬁculties, and less likely to beneﬁt from the mediation of WM processes. In the present study, to examine these issues, a group of individuals with DS and a matched control group of TD children were asked to complete a task that involved estimating the numerosity of a set of dots, adapted from a task proposed by Berteletti, Lucangeli, Piazza, Dehaene, and Zorzi (2010), and another task (approximate addition), adapted from Iuculano et al. (2008), that involved estimating from memory the numerosity resulting from the sum of two successively presented sets of dots. Participants were also administered a number of verbal and visuospatial working memory tasks, adapted from Lanfranchi et al. (2004), to test the abilities that might support their performance in the estimation tasks and in particular in the approximate addition. 3. Method 3.1. Participants The study involved 42 individuals with DS (18 M, 24 F), with a mean chronological age of 18.8 years (range: 15.0– 29.11), compared with 42 TD children (22 M, 20 F) with a mean chronological age of 5.4 years (range: 4.10–7.6) matched for sociocultural level (all children came from the same sociocultural contexts in areas of Northern and Central Italy) and ﬂuid intelligence, as measured with Raven’s Colored Progressive Matrices (CPM, Raven, Court, & Raven, 1992, Italian standardization by Belacchi, Scalisi, Cannoni, & Cornoldi, 2008). The mean CPM scores in the DS and TD groups did not differ signiﬁcantly, being 16.38 (SD = 1.81) and 17.14 (SD = 2.87), respectively [F(1,82) = 2.12, p = .149, h2p ¼ :025, Cohen’s d = .32]. We preferred to base the individuation of mental age of individuals with DS and their matching with TD children on a new administration of an intelligence test (CPM) as diagnosis and assessment reports of individuals with DS were different (and sometimes incomplete) as they came from different local Clinical Services using different or not well speciﬁed procedures. The TD group, based on the most recent Italian norms of CPM (Belacchi et al., 2008) had a mean IQ very close to 100 (mean IQ = 103.93; SD = 6.75) and the two groups had a mean mental age of approximately 5 years and 6 months. The two groups were on the contrary different for schooling as the TD had received little or no instruction on arithmetic, whereas the group with DS had experienced 8 and sometimes more years of special instruction, including some basic elements of arithmetic. In fact, the typical Italian school curriculum for mentally handicapped children includes an introduction to arithmetic operations and some practice with counting and written calculation. The study met the ethical requests deﬁned by the Italian Association for Psychology and was based on the parents’ consent for the administration of the tasks. 3.2. Materials and procedure Both groups were administered tasks to measure their ability to estimate numerosities of single and multiple sets (using tasks adapted from Berteletti et al., 2010 and Iuculano et al., 2008), and their working memory, both verbal (VWM) and

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visuospatial (VSWM) (using the tasks in Lanfranchi et al., 2004, in the versions modiﬁed by Lanfranchi, Jerman, and Vianello (2009) and Carretti, Belacchi, and Cornoldi (2010). 3.2.1. Arithmetic tasks Both practice and experimental trials were based on the manual presentation of A4 sheets of papers in order to make the task more friendly and understandable to children. Instructions stressed that the task was to decide which set included more dots and to give the response by pointing to that set. In the approximate addition task the experimenter insisted on the fact the numerosity of one set corresponded to the result of the addition and gave, if necessary, concrete examples of the fact that the resulting set should include all the elements present in the two to be added sets. All children demonstrated, during the practice trials, to have well understood the tasks. 1. Numerosity estimation: the task involved participants pointing quickly to which of two different conﬁgurations contained more black dots without counting the dots. Each stimulus consisted of two circles, presented for approximately 4 s, one containing a ﬁxed number of dots (16), and the other containing a variable number of dots (8, 10, 12, 14, 18, 20, 22 and 24). Three examples of the stimuli were shown to enable participants to become familiar with the task, which sufﬁced to ensure that they understood what they had to do. The black dots could be distributed all over circles, occupying the whole surface, or in clusters so as to occupy only a part of the area, for a total of 32 comparisons. One point was awarded for each correct answer, 0 for each incorrect answer, and the ﬁnal score was the sum of the points obtained, which ranged from a minimum of 0 to a maximum of 32. 2. Approximate addition (based on estimating the numerosity of a set resulting from the combination of two sets presented in succession). The task involved estimating the numerosity of the blue dots obtained after adding two different sets together by comparison with a reference set having the same numerosity as the reference set used in the previous numerosity estimation task (16 red dots) for a total of 16 experimental trials. Half of the trials involved ‘moving and hiding dots behind a box’ and the other half involved a ‘moving occluder, hiding the two blue sets, one at a time’. As in the numerosity estimation task, participants were asked not to count in order to give their answer, and counting was prevented by the rapid presentation of each set (approximately 2.5 s). Three items were used as examples, and it was assumed that participants had understood the task if they responded correctly. There were more red dots in half of the tests, more blue ones in the other half. The numerosity of the red dots was constant (16 dots), while that of the blue ones varied (being 8 = 2 + 6, 3 + 5; 12 = 3 + 9, 5 + 7; 20 = 5 + 15, 9 + 11; 24 = 8 + 16, 22 + 2)). The dots were evenly distributed inside the circles. One point was awarded for each correct answer, 0 otherwise. The minimum score was 0, the maximum 16. 3.2.2. WM tasks We used two verbal and two visuospatial tasks from the battery adopted at the University of Padova for assessing WM in intellectually disabled individuals, based on the standard span procedure and suspending the task after a participant made two errors at the same difﬁculty level. The score was computed on the basis of the number of perfectly recalled items (for detailed presentations of the tasks and their reliability, see also Carretti et al., 2010; Lanfranchi, Baddeley, Gathercole, & Vianello, 2012; Lanfranchi, Carretti, Spano`, & Cornoldi, 2009; Lanfranchi et al., 2004). Verbal WM1: this task involves immediately repeating orally presented lists of words, in the same order of presentation. The words were highly familiar (e.g. ball, moon) and well known to any child. The word lists were increasingly long, ranging from a minimum of 2 to a maximum of 5 elements to remember, with two different lists for each level of difﬁculty. An example was provided, then it was assumed that the task had been understood. One point was awarded if the sequence of words was repeated in the same order of presentation, 0 if the words were incorrect, incomplete or presented in a different order. The ﬁnal score was obtained from the sum of the single scores obtained for each series, and ranged from a minimum of 0 to a maximum of 8. Verbal WM2: this task involves repeating the ﬁrst word in one or two lists simultaneously presented both orally and visually, written in upper case in a standard font, using colored cards (red for the single list, red and green for the pairs of lists). There were two levels of difﬁculty for the experimental condition with single lists (containing two or three words) and two levels for the condition with pairs of lists (with two or three words in each list). For each level of difﬁculty there were two different sets of word lists. An example was administered, comprising a single list, after which it was assumed that the requirements of the task had been understood. The pairs of lists were subsequently only presented if participants had responded correctly for the single lists. One point was awarded if participants recalled the ﬁrst word(s) correctly, 0 if their answer was incorrect, incomplete or in a different order. The overall score was the sum of the single scores obtained for the different levels and ranged from a minimum of 0 to a maximum of 8. VSWM1: this task involved the participant remembering the positions of two or three green squares shown on a chessboard 2 2, 3 3 and 4 4 squares in size. After 15–20 s of looking at the stimulus, participants were given an empty chessboard on which to mark the previously seen conﬁguration. An example was presented ﬁrst to ensure that participants understood the task. For the 3 3 and 4 4 matrices, there were two different levels of difﬁculty involving two or three green squares. Each level consisted of two different, equally difﬁcult conﬁgurations. One point was awarded if participants were able to remember the sequence presented correctly, 0 if their positioning of the squares was wrong, or incomplete. The

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total score was obtained from the sum of the single scores for the different conﬁgurations, ranging from a minimum of 0 to a maximum of 10. VSWM2: this task involved the participant recalling the ﬁrst stop along the path(s) covered by one or two toy frogs on a 4 4 chessboard. After the frogs had completed their journey, participants were asked to indicate on the chessboard only the ﬁrst box occupied by the ﬁrst frog (and the ﬁrst box occupied by the second frog, if any). There were two levels of difﬁculty for one frog’s path (comprising two or three jumps), and three levels when two frogs were involved (comprising three or four jumps for each frog). There were two different, equally difﬁcult paths for each level. One point was awarded only if participants remembered all the information required. The ﬁnal score was the sum of the points obtained and ranged from a minimum of 0 to a maximum of 10. 4. Results 4.1. Group comparisons For the numerosity estimations, we ﬁrst collected information on the reliability of the two adaptations of the tasks, ﬁnding that Cronbach’s alpha was not particularly high but acceptable for the goals of the study (.71 for the numerosity estimation for a single set, and .53 for the numerosity estimation of the sum of two sets). Table 1 presents the mean scores of the two groups in the two tasks. As it can be seen the numerosity estimation of a single set was easier than the other task for controls, but not for the DS group. In fact a mixed 2 2 ANOVA on the raw scores, with Group (TD vs. DS) as a between subject factor and type of task (NE vs. AA) as a within subject factor, found a signiﬁcant interaction between groups and tasks F(1,82) = 5.83, p = .018, h2p ¼ :066. Separate comparisons between groups for the two tasks showed that the two groups differed signiﬁcantly in the numerosity estimation of a single, but not when they estimated the sum of two sets. As the tasks and their number of items were different, we also run separate analyses for each task between the groups and within each group using the percentage scores and we obtained similar results. In fact we found that children with DS had a signiﬁcantly poorer performance than TD children in the numerosity estimation task: 75.45% (SD = 9.25) vs. 81.40% (SD = 7.17) of correct responses; Student’s t = 3.296, p < .001), whereas the two groups did not signiﬁcantly differ in the case of the percentages of correct responses in the approximate addition: 75.45% (13.76) vs. 77.38% (SD = 10.51), Student’s t = .724, p = .47). When we compared the two tasks separately within each group we found that TD children performed signiﬁcantly better in the numerosity estimation task (t = 2.105, p < .041) than in the mental addition task, whereas no difference was found in the group with DS that had exactly the same percentage of correct responses in the two tasks. Table 2 shows the mean scores (and SDs) for the two groups in the four WM tasks, and in the two scores obtained from the sums of the two verbal and the two visuospatial WM scores, respectively. The comparison between the two groups showed that they differed signiﬁcantly in terms of the sum of their verbal WM scores [F(1,82) = 16.84, p < .001, h2p ¼ :17, Cohen’s d = .90], but not of their visuospatial WM scores [F(1,82) = 0.24, p = .623, h2p ¼ :003, Cohen’s d = .11]. The same pattern emerged when the four tasks were considered separately because the DS group did signiﬁcantly worse in both verbal tasks – where F was 21.42, p < .001 and 7.73, p < .01, respectively, for tasks 1 and 2 – but not in the visuospatial tasks with a slightly poorer performance than controls in the ﬁrst task but even a slightly better performance in the second task (p = .173 and p = .732, for tasks 1 and 2, respectively).

Table 1 Mean scores (SDs in parentheses) of the individuals with Down syndrome (DS) and of the typically developing (TD) controls in the numerosity tasks. Groups

Numerosity (0–32)

Addition (0–16)

TD DS F p

26.05 (2.29) 24.14 (2.96) 10.865 .001 .117

12.38 (1.68) 12.07 (2.20) .524 .471 .006

h2p

Table 2 Mean scores (SDs in parentheses) of the individuals with Down syndrome (DS) and of the typically developing (TD) controls in the four Working Memory tasks and in the two summed scores. Groups TD DS * p < .01. *** p < .001.

Verbal WM1 4.71 (1.02) 3.60 (1.19)

***

Verbal WM2 5.95 (2.63) 4.52 (2.04)

*

VSWM1 5.17 (1.82) 4.48 (2.70)

VSWM2 6.00 (2.71) 6.21 (3.01)

WM-verbal

VSWMS ***

10.67 (3.03) 8.12 (2.65)

11.17 (3.68) 10.69 (5.05)

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Table 3 Correlations between WM (verbal and visuospatial), numerosity estimation and estimation of addition respectively for the DS and the TD groups. WM-V

Task Verbal WM VSWM Addition Numerosity

WM-VS

–

.441 – .022 .111

.560** .069 .037

Addition

**

Numerosity

**

.433 .528**

.082 .272 .223 –

– .058

DS, above diagonal; TD, below diagonal. ** p < .01. Table 4 Results of the regression analysis examining the predictive power of the WM tasks for success in the numerosity estimation of addition.

1

2

3

Model

B

Std. error

(Constant) WM_V WM_VS

10.077 .090 .119

.692 .074 052

(Constant) WM_V WM_VS Groups

10.106 .088 .120 .028

(Constant) WM_V WM_VS Groups WM_V Group WM_VS Group

12.132 .066 .041 3.683 .140 .223

Beta

T

p

.144 .268

14.570 1.227 2.291

.000 .223 .025

.838 .082 .053 .452

140 .269 .007

12.061 1.069 2.255 .063

.000 .288 .027 .950

1.081 .111 .091 1.424 .161 .110

.105 .091 .948 .322 .736

11.224 .592 .443 2.586 .868 2.023

.000 .556 .659 .012 .388 .046

Dependent variable: numerosity estimation of addition.

4.2. Correlations between numerosity and WM tasks Table 3 shows the correlations between WM (verbal and visuospatial), numerosity estimation of single sets, and numerosity estimation of the sum of two sets. To simplify the presentation of the results, only the correlations for the pooled scores are presented, but the correlations showed much the same patterns when the four different tasks were considered separately. A look at the correlations shows how the two numerosity estimation tasks differ substantially, as they are only moderately correlated in both groups (.058 in the TD group, .223 in the DS group) and they are differently related with WM. Concerning this latter aspect, given the clear relationship between WM and the estimation of sums in the DS group, but not in the TD group, we compared the correlation values obtained for the two groups. The difference between the two values was high for both verbal and visuospatial WM, but in the case of verbal WM it only approached signiﬁcance (p = .082), despite the considerable difference between the values (.364). As for the difference between the two correlation values in the case of VSWM, this value was particularly high (.55), and highly signiﬁcant (p = .007). The overlap between the two correlations (with a 95% CI) is small and the difference did not touch the zero line. 4.3. Mediation model To further support the hypothesis that participants’ groupings mediated the relationship between their WM and their ability to estimate the numerosity of multiple sets, we developed a hierarchical regression model (see Table 4). In the ﬁrst step we input the two summed WM scores (for verbal WM and VSWM), then in the second we introduced the groups as a dummy variable (0 TD, 1 DS), and in the third step we added the two interaction effects (verbal WM group and VSWM group). The three models derived from the inclusion of the 1st, 2nd or 3rd step, were all signiﬁcant [Fs > 3.91, p < .012]. In particular, the proportion of variance explained in the model based on the ﬁrst step was 12.8% [R2 = .128, F(2,81) = 5.94, p = .004] and the proportion did not change after the second step [R2change ¼ :000, F(1,80) = 0.004, p = .950]. On the other hand, the model based on all three steps – taking the interactions into account as well – explained a substantial part of the variance, i.e. 21.6% [R2change ¼ :088, F(2,78) = 4.39, p = .016]. In particular, the signiﬁcance of the beta value affecting the interaction between visuospatial WM and the study groups further supports how important visuospatial WM is to the DS group when it comes to estimating the numerosity of multiple sets. 5. Discussion and general conclusions Intellectually impaired individuals such as those with DS may do badly in logical and arithmetical tasks, encountering difﬁculties in all cases where they have to handle quantities. Their difﬁculties may be partially overcome, however, and

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substantial errors can be avoided if they are at least able to estimate the numerosity of analogical sets. The present study focused on whether individuals with DS are able to estimate the numerosity of sets, and in particular the numerosity of the sum of two sequentially presented sets, as well as typically developing children matched for ﬂuid intelligence. Our results showed that the DS group was not advantaged, as it happened in controls, by the single set estimation. In fact the DS group did signiﬁcantly worse than the TD group in estimating the numerosities of single sets, but not the numerosities resulting from additions, reaching the same level of performance in the two tasks. These ﬁndings shed light on a factor that may contribute to the arithmetical difﬁculties often seen in individuals with DS who are typically better in reading than in numerical literacy (Buckley, 1985; Pieterse & Treloar, 1981). Despite the fact that education may enhance the acuity of the nonverbal approximate number system (Piazza, Pica, Izard, Spelke, & Dehaene, 2013) and the participants with DS of the present study had already followed formal instruction, they nonetheless performed less well than preschoolers, suggesting that a difﬁculty in estimating numerosity of single sets is a critical weakness associated with DS that could explain at least a part of their difﬁculties in arithmetic, like it has been seen in developmental dyscalculia. Conversely, we did not ﬁnd a difference between the DS and TD groups when it came to estimating the numerosity of the sum of two sets. A direct comparison between the performances in the two different arithmetic tasks administered in the present study must be very cautious as – in order to follow the methodology adopted in the previous studies and to have tasks comparable in difﬁculty – the two tasks differed on many respects, including the number of trials and the size of the differences between the two compared sets (in particular the minimal difference between two sets was of two dots in the case of numerosity estimation and of four dots in the case of approximate addition). However it is out the discussion the fact that, compared to controls, children with DS were not poorer in the case of approximate addition. This result offers important insight on the potential arithmetic abilities of individuals with DS that can have implications for everyday life. The result could be due to a series of concurrent factors, including school experience introducing the individuals with DS to addition, as also suggested by the moderate correlation we found in the DS group between age and success in the task (r = .23). The reference to school experience, impossible in the case of controls, could also at least in part explain why individuals with DS approached the approximate addition task in a different way from preschoolers, relying to a larger extent on mediating WM processes, a result we comment below. The results of WM tasks replicated previous evidence showing that individuals with DS generally perform poorly in verbal WM tasks, whatever of the type of WM task and the degree of control it requires, but not in VSWM tasks (see e.g. Lanfranchi et al., 2004; Lanfranchi Jerman, Dal Pont, Alberti, & Vianello, 2010). In fact, the performance in the VSWM of the DS group was good and similar to that of the TD children, with minor differences between the two tasks. On this respect it must be noticed that the DS group appeared especially advantaged in the case of the spatial sequential task requiring to remember the ﬁrst position, thus replicating previous observations (Carretti et al., 2013; see also Lanfranchi, Jerman, et al., 2009). The administration of WM tasks together with the arithmetic tasks offered the opportunity of seeing that the DS group’s success in approximate addition related to a mediated process involving the use of WM, especially in its visuospatial component, which thus emerged as a good predictor for this group of the outcome of the addition task. This result can be explained also with reference to the nature of the approximate addition task we required where – in order to reproduce typical everyday tasks – the to be added sets were not perceptually present when deciding about the result of the addition. This result could also explain why also in the estimation of the numerosity of single sets, in relationship with the request of maintaining them in memory, individuals with DS may be as good as children matched for mental age (Sella et al., 2013). However it must be noted that a relationship between WM and approximate addition was clearly evident only for the DS group. In fact, the correlation between the results in the WM tasks and the approximate addition task was much greater for the DS group than for the TD group, and the difference between the correlation values for the two groups was highly signiﬁcant in the case of visuospatial WM, which is a well-known strong point for individuals with DS. From a general point of view, our results provide important suggestions because they show that, in populations where mathematical education has not played a central part, there is a substantial independence between the accurate estimation of the numerosity of a single set and the estimation from memory of the numerosity of the sum of two previously presented sets. This independence is jointly supported in the present study by the low correlation emerging between the two tasks and the facts that the DS group’s performance was only impaired in one case, only one of these tasks was substantially related with WM and only one of these tasks showed a similar pattern of relationships with WM in the two groups considered here. This independence however seems limited to particular subgroups of children who make a more limited use of skills learned at school, as research by Iuculano et al. (2008) revealed a high correlation in 8- to 9-year-old children between numerosity estimations in a single set and those obtained by adding two sets together. Our study has a number of limitations (including the characteristics of the groups, and the choice of tasks and stimuli) and consequently needs to be further replicated and extended, also in order to understand the factors producing partly different results (see Camos, 2009). For example, we noticed that the group with DS was less in difﬁculty in the estimation of a single set numerosity when the dots were concentrated in a part of the sheet than when they were distributed in the sheet, a result that should be more systematically studied. However our ﬁndings provide important information and suggest an optimistic outlook on intellectually disabled individuals’ ability to learn arithmetical competence, an ability that has sometimes been underestimated in intellectually disabled individuals and in particular in Down syndrome. In particular results have clear educational implications as they suggest that individuals with DS can achieve good results if their strengths are recognized and developed.

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