Basic number processing deficits in developmental dyscalculia: Evidence from eye tracking

Cognitive Development 24 (2009) 371–386

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Cognitive Development

Basic number processing deficits in developmental dyscalculia: Evidence from eye tracking K. Moeller a,∗, S. Neuburger a, L. Kaufmann b,c, K. Landerl a, H.-C. Nuerk a a b c

Institute of Psychology, Eberhard Karls University, Friedrichstrasse 21, 72072 Tuebingen, Germany Clinical Department of Pediatrics IV, Division Neuropedriatrics, Innsbruck Medical University, Austria Institute of Psychology, University of Salzburg, Salzburg, Austria

a r t i c l e

i n f o

a b s t r a c t Recent research suggests that developmental dyscalculia is associated with a subitizing deficit (i.e., the inability to quickly enumerate small sets of up to 3 objects). However, the nature of this deficit has not previously been investigated. In the present study the eye-tracking methodology was employed to clarify whether (a) the subitizing deficit of two boys with dyscalculia resulted from a general slowing in the access to magnitude representation, or (b) children with dyscalculia resort to a back-up counting strategy even for small object sets. In a dot-counting task, a standard problem size effect for the number of fixations required to encode the presented numerosity within the subitizing range was observed. Together with the finding that problem size had no impact on the average fixation duration, this result suggested that children with dyscalculia may indeed have to count, while typically developing controls are able to enumerate the number of dots in parallel, i.e., subitize. Implications for the understanding of developmental dyscalculia are considered. © 2009 Elsevier Inc. All rights reserved.

In modern society, academic and professional success is affected by skills in both written language (reading and spelling) and numeracy (Bynner & Parsons, 1997; Parsons & Bynner, 2005). Although it is widely acknowledged that phonological skills may be a building block for the acquisition of reading (Shaywitz & Shaywitz, 2005; Uppstad & Tonnessen, 2007), much less is known about the neuro-cognitive foundations of arithmetical skills. It has been suggested that very basic numerical

∗ Corresponding author. E-mail address: [email protected] (K. Moeller). 0885-2014/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.cogdev.2009.09.007

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skills may underlie the development of numerical cognition (Halberda, Mazzocco, & Feigenson, 2008; Holloway & Ansari, 2009; Kaufmann, Handl, & Thoeny, 2003). Preverbal infants are already capable of making number-based discriminations, which strongly suggests that basic number processing is largely independent of language skills (Brannon, 2005; Gelman & Butterworth, 2005; Wynn, 1992). In this context the term basic number processing denotes the ability to discriminate and estimate numerical quantities (non-symbolic number processing) and may be distinguished from complex calculation skills involving the manipulation of Arabic symbols. In older children and adults, the ability to quickly enumerate small object sets (up to 3 or 4 items) is known as subitizing (Mandler & Shebo, 1982) and may be distinguished from the more effortful enumeration of larger set sizes. Generally, response latencies in naming the quantity of objects in sets up to 3 or 4 do not increase reliably but are rather constant while those for larger object sets increase linearly as a function of set size (most probably reflecting the counting process; Trick & Pylyshyn, 1993). It has been suggested that subitizing reflects very basic numerical knowledge that can be observed in infants and may serve as a cognitive scaffold for the development and acquisition of more complex numerical skills such as counting (Benoit, Lehalle, & Jouen, 2004; Klahr & Wallace, 1976; Starkey & Cooper, 1995) and even arithmetic skills (Hannula, Räsänen, & Lehtinen, 2007; Lipton & Spelke, 2003; Sophian, 1987; Wynn, 1992). With increasing age, children not only learn how to count and calculate, but concomitantly develop more precise number representations (Moeller et al., 2009; Opfer & Siegler, 2007), facilitating quick and accurate access to representations of number magnitude and arithmetic facts. However, children suffering from developmental dyscalculia are slower and more error prone in accessing number magnitudes (Rubinsten & Henik, 2006). Wilson and Dehaene (2007) have proposed that such basic numerical deficits, i.e., impaired access to numerical magnitude representation, may stem from number magnitude representation being too indefinite to provide fast and accurate activation of a single number. Such noisy number representations are assumed to underlie the frequently observed performance deficiencies of children with dyscalculia in number comparison tasks requiring the child to indicate the numerically larger number (Landerl, Bevan, & Butterworth, 2004; Landerl, Fussenegger, Moll, & Willburger, 2009; Rousselle & Noël, 2007; Rubinsten & Henik, 2006; see Kaufmann & Nuerk, 2006a, 2008, for similar deficits in ADHD children). Even in most basic tasks such as dot counting, preliminary evidence suggests that children with dyscalculia may have particular deficits in producing the correct numerosity of objects. Landerl et al. (2004) observed that the increase of response latencies in counting 4–10 dots was steeper for children with dyscalculia compared to typically developing children. More interestingly, even within the subitizing range of 1–3 dots the RT slopes of children with dyscalculia and those with a double deficit (dyslexia and dyscalculia) were descriptively steeper than the slopes for control children. Although this difference was not statistically significant, probably due to power problems, a reliably steeper slope within the subitizing range has recently been observed for children with dyscalculia by Schleifer and Landerl (submitted for publication). Here, increase in the dependent variable (e.g., RT) as a function of the independent variable (e.g., set size), as formalized by the regression slope b of a linear regression (a = bx + c), is used to indicate either subitizing or counting processes. The regression slope was chosen as it represents a direct measure of the kind of relation between an independent and dependent variable. In the context of the subitizing/counting differentiation, subitizing is usually assumed when the slope b within the subitizing range of 1–3 dots is either not significantly different from 0, or, using a somewhat weaker criterion, at least significantly smaller than the slope within the counting range (more than 3 dots). In contrast, counting is generally indicated by a much steeper slope b, significantly larger than 0 or at least reliably steeper than that for the subitizing range, meaning that RT increases reliably and linearly with each dot added to the set. Current empirical findings corroborate the hypothesis that deficient numerosity processing is a key cognitive deficit of individuals diagnosed with developmental dyscalculia (Butterworth, 2005; Landerl et al., 2004; Wilson & Dehaene, 2007). Converging evidence comes from recent brain imaging studies. Relative to controls, activation of number sensitive regions within the parietal lobe were less pronounced in children with developmental dyscalculia when they were asked to indicate the numerically larger of two dot patterns (Price, Holloway, Räsänen, Vesterinen, & Ansari, 2007; but see Kucian

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et al., 2006). Moreover, recent structural MRI studies also index that in children with developmental dyscalculia parietal grey matter, which is supposed to subserve basic numerical processing, is reduced, compared to typically developing controls (Rotzer, Kucian, von Aster, Klaver, & Loenneker, 2008). In summary, children suffering from developmental dyscalculia not only present deficits in complex numerical skills but also in the most basic numerical tasks such as subitizing. However, there is currently no research investigating the nature of this subitizing deficit. The present study addressed this question by employing eye-movement data for the first time in dyscalculia research. To illustrate in what way eye movements may be more useful than simple reaction times and error rates, we briefly outline how eye movement data have already been incorporated to investigate other developmental disorders as well as basic numerical processing in adults. 1. Learning disorders and eye tracking In research on learning disorders, eye tracking is becoming a more common method to gain online information on cognitive processing. Analysing eye movements is of special interest as it provides information on temporal as well as spatial aspects of how individuals approach a certain task (Rayner & Pollatsek, 1989). Especially in the field of developmental dyslexia, eye-tracking studies have lately helped to better understand children’s problems in word recognition. Individuals with dyslexia have repeatedly been demonstrated to make shorter eye movements (saccades) during reading, resulting in a higher number of fixations (De Luca, Borrelli, Judica, Spinelli, & Zoccolotti, 2002; Hutzler & Wimmer, 2004). In addition, a number of studies indicated longer mean fixation durations and a higher number of regressive saccades (De Luca, Di Pace, Judica, Spinelli, & Zoccolotti, 1999; Heller, 1979; see Rayner, 1998, for a review). These findings clearly reflect the processing difficulties that individuals with dyslexia experience during reading, difficulties that could only be identified by evaluating eye-fixation behaviour. Moreover, for ADHD, first studies using eye-tracking paradigms pinpointed impairments in aspects of effortful visual selective attention (Mullane & Klein, 2008), problems in inhibiting irrelevant distractors in visual search tasks (Van der Stigchel, Merten, Meeter, & Theeuwes, 2007) and deficits in visuo-spatial working memory (Rommelse et al., 2008) to be of particular prevalence. Despite these interesting findings, eye tracking has so far not been applied to developmental dyscalculia. However, studies exploring eye-movement behaviour during dot counting in adults and children with typical arithmetic skills have appeared recently. Sophian and Crosby (2008) used eye tracking to show that not only counting numerosities higher than 4, but also subitizing of 3 or less items, requires visual attention, a finding that is inconsistent with the assumption that subitizing is a preattentional mechanism (Trick & Pylyshyn, 1993, 1994). Moreover, Watson, Maylor, and Bruce (2007) were the first to show a tight coupling between saccadic frequencies and response latencies in adults. In line with what is generally reported for response latencies, a distinct increase of saccadic frequency from subitizing to counting range was observed. Recently, Schleifer and Landerl (submitted for publication) extended this finding, observing a similar differentiation between subitizing and counting in the eye-movement patterns of typically developing children of different age groups. These findings indicate that eye-fixation behaviour has been shown capable of distinguishing between subitizing and counting to examine the nature of the subitizing process and – for other learning disorders – of identifying qualitative differences between typical and atypical cognitive development. In the present study we employ the eye-tracking methodology to investigate whether qualitative differences between typical and atypical development can also be observed in the domain of basic numerical cognition, i.e., in a dot-counting task. 2. Objectives of the present study We have introduced evidence that children with dyscalculia may be impaired in subitizing and/or counting (Landerl et al., 2004; Schleifer & Landerl, submitted for publication). As the existing evidence is not entirely conclusive, the first objective of the study was to examine whether individuals with dyscalculia indeed are impaired in subitizing and counting with respect to overall RT. In particular, it was of interest whether the increase in RT within the subitizing range is steeper for children with dyscalculia than for typically developing children, indicating a subitizing impairment.

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The second objective pertains to the nature of this impairment. We undertook to investigate whether children with dyscalculia were qualitatively or only quantitatively different from typically developing controls in subitizing. In our view, two reasons for such basic numerical impairment can be distinguished. The first is that children with dyscalculia suffer from an impairment in accessing the magnitude representation (possibly upon a mental number line) of a number. They experience greater difficulty in activating the exact number representation that corresponds to a non-symbolic magnitude representation. Such difficulties may be more pronounced for larger problem sizes even within the subitizing range because the tuning curves of the magnitude representations are wider for larger representations (Nieder, 2005).1 If the tuning curves for children with dyscalculia were even noisier than those for typically developing children, access should not only be generally harder (resulting in an overall slowing) but it should be increasingly harder as problem size increases because the tuning curves overlap even more. A second hypothesis explaining a steeper slope in the subitizing range for individuals with dyscalculia is that there is a qualitative difference. Children with dyscalculia may not be able to subitize (i.e., encode the cardinality of up to 3 dots in parallel). Because of this deficit they may need to rely on back-up strategies (Fuson, 1988) such as simply counting even 1–3 dots. Clearly, a counting procedure would result in a more pronounced increase of RT with dot number, while subitizing (in typically developing children) would not result in such a steep RT increase. Both accounts would predict a steeper slope for children with dyscalculia within the subitizing range for RT data. However, they may be distinguished by their corresponding eye-fixation pattern. When children with dyscalculia rely on counting as a back-up strategy, they should exhibit a larger number of fixations for larger problem sizes even within the subitizing range, as counting is an attentional process and eye fixations typically mirror attentional engagements (Shepherd, Findlay, & Hockey, 1986; see Duc, Bays, & Husain, 2008 for a review). However, this may not apply to the first account, as it does not postulate any difference in overt attention shifting but rather proposes that more time is needed to extract the numerical meaning from the visual information of dots, i.e., to resolve the noise when accessing exact number representations upon the mental number line. Therefore, children may not fixate one point after the other, as it may not be the problem to encode the dot pattern. Rather, they may fixate longer for a larger number of dots because it takes them longer to retrieve the cardinality of this dot pattern within the subitizing range. As there are no data on eye-fixation behaviour from dyscalculia research, results from dyslexia research may serve as comparative data. Here, longer fixation durations have repeatedly been observed for dyslexics (Heller, 1979) and have been interpreted to reflect impaired access to semantics via the extraction of visual information (Judica, De Luca, Spinelli, & Zoccolotti, 2002). Following this rationale, fixation duration of individuals with dyscalculia may increase with increasing problem size as the representation of number magnitude is noisier for larger problem sizes (Brysbaert, 1995; Dehaene, 2001; Nieder, 2005). These two accounts are investigated here by examining not only RT but also the number of fixations and the average fixation durations for both typically developing children and children with dyscalculia in a dot-counting task. Specifically, we ask whether we can reliably observe: (a) a larger RT slope in children with dyscalculia than in typically developing children; (b) a larger slope for the number of fixations for children with dyscalculia; and/or (c) a larger slope for the average fixation duration for children with dyscalculia? In a secondary analysis we examine these three measures for three dots in particular because three dots represent the largest problem size within the subitizing range and therefore the differences between typically developing children and children with dyscalculia should be most pronounced for 3 dots. Finally, we also examine these three measures for the counting range (4–8 dots), as sometimes differences between normally developing children and children with dyscalculia have been observed in the counting range, as well (Landerl et al., 2004).

1 Note that the latter assumption is not arbitrary; rather number neurons with different tuning curves have been repeatedly reported (Nieder, 2005; Nieder, Diester, & Tudusciuc, 2006; Nieder & Miller, 2003) and a greater width of tuning curves for number neurons corresponding to larger problem sizes has also been successfully modelled to explain human RT data (Verguts & Fias, 2004; Verguts, Fias, & Stevens, 2005).

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Table 1 Performance (in z-scores) in achievement tests (HAWIK-III, SLS, HRT) for the control group and the two participants with dyscalculia, F.K. and S.S., is summarized in Panel A. Panel B gives the results of the visual attention assessment of F.K. and S.S. (KITAP). (A) Achievement tests

Control group

F.K.

S.S.

Pretest HAWIK-III subtests Reading (SLS) Arithmetic (HRT)

M = 1.0 (SD = 0.7) M = 1.1 (SD = 0.5) M = 0.5 (SD = 0.2)

1.2 0.5 −1.2

1.5 0.3 −1.3

At day of testing Addition subtest (HRT) Subtraction subtest (HRT)

M = 0.7 (SD = 0.7) M = 0.6 (SD = 0.6)

−1.3 −1.4

−1.1 −1.5

(B) Attention test battery (KITAP)

F.K.

Alertness Median RT Variance

S.S. (before test)

S.S. (after test)

1.8 1.6

1.8 2.2

– –

Flexibility Median RT False alarms

2.0 −0.9

1.5 0.1

– –

Sustained attention Median RT False alarms Misses

1.9 −0.7 0.2

1.2 −0.7 2.0

1.2 1.8 2.0

3. Method 3.1. Participants Participants were two boys with dyscalculia, F.K. (age 10–10) and S.S. (age 10–7), and an agematched control group of 8 normally developing children (2 girls and 6 boys). Mean age of the control group was 10–5 years with a standard deviation of 3 months. All children were recruited from a large sample of Austrian children tested for another research project examining the prevalence as well as the cognitive basis of comorbidities between developmental dyslexia and dyscalculia. Intelligence scores (subtests Block Design and Similarities of the HAWIK-III, Tewes, Schallberger, & Rossmann, 1999) and reading ability (assessed by the SLS 1–4, Mayringer & Wimmer, 2006) as well as scores on a standardized arithmetic test (subtests addition, subtraction, multiplication, division, magnitude comparison, and fill-in equations of the HRT; Haffner, Baro, Parzer, & Resch, 2005) were available (see Table 1, Panel A). Developmental dyscalculia of F.K. and S.S. was indicated by their below-average performance in the HRT ( > 1 SD below age norm), whereas each child in the control group achieved a test score within the normal range. As seen in Table 1 (Panel A) F.K. and S.S. showed no deficits in the two other cognitive achievement tests, i.e., they presented neither reading problems nor impairments in general cognitive abilities. As the assessment of cognitive abilities in the context of the comorbidity project took place approximately six months before the current investigation, the Addition and Subtraction subtest of the HRT were administered again to verify arithmetic achievement of the control children as well as the arithmetic deficits of F.K. and S.S. (see Table 1, Panel A). As it is important to rule out perceptual and/or attentional deficits accounting for differing fixation patterns in participants with dyscalculia, the visual attention capabilities of the two children with dyscalculia were assessed by three subtests of a standardized test battery for attentional functioning (subtests Alertness, Flexibility, and Sustained Attention of the KITAP, Zimmermann, Gondan, & Fimm, 2002). As can be observed from Table 1 (Panel B) neither S.S. nor F.K. exhibited any perceptual/attentional impairment in this assessment. It is important to note that S.S. was also diagnosed with ADHD. At the time of testing S.S.’s medication (i.e., Concerta) was well-adjusted so that he did not present with any behavioural abnormalities.

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As Rubinsten, Bedard, and Tannock (2008) have observed no effects of methylphenidate (such as Concerta) on basic numerical capabilities involving the processing of quantities, we are confident that S.S.’s numerical performance was not altered by his medication. However, to ensure that his eye-fixation behaviour was not driven by a decline of attentional processing resources, the Sustained Attention subtest of the KITAP (Zimmermann et al., 2002) was administered again after S.S. had completed the dot-counting task. The results obtained in this second evaluation did not differ substantially from those of the first testing (see Table 1, Panel B), corroborating the validity of the eye-fixation data. 3.2. Apparatus An EyeLink 1000 eye-tracking device (SR Research, Mississauga, Ontario, Canada) was used to record participants’ eye movements throughout the dot-counting task. The EyeLink 1000 provides a spatial resolution of less than 0.5◦ of visual angle at a sampling rate of 1000 Hz. Stimuli were displayed on a 21 monitor driven at a refresh rate of 120 Hz with resolution set to 1024 × 768 pixels. Children had to place their heads in a forehead rest throughout the experiment to keep viewing distance (50 cm) and viewing angle constant. Participants had to name the number of black dots (1.25 cm in diameter) within a white square (9.6 cm × 9.6 cm). Numerosities between 1 and 8 dots were presented eight times each amounting to a total of 64 critical trials. To prevent children from associating a certain layout of dots with any given numerosity, eight different layouts of each dot pattern were generated and used in the experiment. 3.3. Procedure All children were tested individually in a quiet room. Before the critical dot-counting task, the Addition and Subtraction subtests of the HRT were administered again. Then participants were seated and the eye-tracker was adjusted using a nine-point calibration cycle to maximize spatial resolution. At this point children were instructed to press the response button as soon as they knew the number of dots in the display and to name the number presented at the same time. Thereby, it was intended to ensure RT to be a measure of the processing time only after subitizing/counting was completed.2 By pressing the response key the dots disappeared and a backward mask was displayed for 500 ms followed by the fixation point of the next trial. Whether the child’s response was correct or not was recorded by the experimenter with no feedback given as to the correctness of the responses. The experiment started with 5 randomly chosen practice trials, followed by the 64 critical trials separated into two blocks of 32 trials each. Dot patterns were presented in randomized order. At the conclusion of the experiment, participants received a small monetary compensation. 4. Results 4.1. Method of analysis The influence of the increasing number of dots in the display was evaluated using response latencies (RT) as the dependent measure of manual performance as well as the number of fixations (NFix) and average fixation duration (Fix Dur) as the dependent measures of eye-fixation behaviour. Separate analyses were conducted covering (a) the subitizing range, i.e., naming 1–3 dots, and (b) the counting range, i.e., naming 4–8 dots. Following the approach by Lorch and Myers (1990), the regression slopes predicting either of the three dependent variables on the basis of the number of to-be-named dots was computed separately for the subitizing and the counting range for each individual participant.

2 The reliability of this proceedure is corroborated by two main arguments. First, in exactly the same task Schleifer and Landerl (submitted for publication) used both this kind of RT measurement as well as a voice key to determine RT and observed (a) identical results in separate analyses for these two RT measurements and (b) that the two measures correlated with r = .9. Moreover, this kind of RT measuring has already been used successfully in dyslexia research (Landerl, Wimmer, & Frith, 1997) for the case of word/nonword reading.

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Fig. 1. Mean RT (Panel A), mean number of fixations (Panel B), and the average fixation duration (Panel C) for each number of dots, separated for the control group as well as the two participants with dyscalculia F.K. and S.S. Error bars indicate 1 SD.

Subsequently, the regression slopes of the two boys with dyscalculia, F.K. and S.S., were compared to that observed for the typically developing control group using a variant of the t-test adapted to the comparison of a single participant’s performance to that of a small control sample in single case studies (Crawford & Howell, 1998; Crawford, Howell, & Garthwaite, 1998). Generally, regression slopes of F.K. and S.S. falling outside the second and third quartile of the control group’s performance were considered conspicuous. 4.2. Descriptive statistics Generally, error rates were very low in the control group, with a mean of 1.6% (range: 0.0–4.7%). As the error rate of F.K. (4.7%) did not exceed that of the controls and S.S. did not commit any error at all in the dot-counting task, evaluation focused on RT, number of fixations, and average fixation duration. Only manual RT as well as eye-fixation data from trials responded to correctly were incorporated in the following analyses. Descriptive statistics for the variables analysed are depicted in Fig. 1: Panel A gives the mean RT for each number of dots separated for the control group and the two participants with dyscalculia (F.K. and S.S.). In a similar manner the mean number of fixations as well as the average fixation duration per number of dots are shown in Panels B and C, respectively. 4.3. Subitizing range 4.3.1. Control sample In the subitizing range a mean regression slope of ˇ = 27.17 (SEM: 19.56; Fig. 2, Panel A) was observed for RTs, which was not reliably different from 0 [t(7) = 1.39, p = .21]. For number of fixations and average fixation duration, mean regression slopes were ˇ = 0.08 (SEM: 0.09, Fig. 2, Panel B) and ˇ = 1.17 (SEM: 12.67, Fig. 2, Panel C), respectively. As observed for the RT slope, neither the mean regression slope of the number of fixations [t(7) = 0.92, p = .39] nor that of the average fixation duration [t(7) = 0.09, p = .93] deviated significantly from 0. Finally, the mean number of fixations required to encode three dots was ˇ = 1.41 with a SEM of 0.15 (Fig. 3). 4.3.2. Participant S.S. The regression slopes for both RT (ˇ = 66.69; t = 0.72; percentile: 76; Fig. 2, Panel A) as well as number of fixations (ˇ = 0.63; t = 2.16; percentile: 97; Fig. 2, Panel B) were higher than the regression slopes

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Fig. 2. Regression slopes of the two participants with dyscalculia, F.K. and S.S. (as indicated by the symbols  and 䊉, respectively) for RT (Panels A and D), the number of fixations (NFix, Panels B and E) and the average fixation duration (Fix Dur, Panels C and F) separated for the subitizing and counting range (upper and lower charts, respectively). Shaded areas represent the second and third quartile of the control group’s performance. Unlike in usually presented box plots, the mean of the regression slopes observed for control children is depicted and not the median to enhance coherence with the text. Additionally, error bars do not indicate the lowest and highest slope obtained but rather reflect 1 SD to provide another criterion for evaluating the degree to which the performance of the two children with dyscalculia differed from that of the control children.

observed for the control group and fell inside the fourth quartile of the control group’s performance. However, this pattern of results was not found for the regression slope of average fixation duration. Instead, a slope of ˇ = −46.83 (t = −1.35; percentile: 11; Fig. 2, Panel C) indicated that the average fixation duration of S.S. decreased as the number of dots within the subitizing range increased, whereas this was not the case in the control group. Evaluating the number of fixations required to encode 3 dots showed that S.S. made more fixations than the control group in these trials (2.13 vs. 1.41, t = 1.70; percentile: 94; Fig. 3). 4.3.3. Participant F.K. As observed for S.S., the regression slopes for RT (111.11; t = 1.53; percentile: 92; Fig. 2, Panel A) and number of fixations (0.26; t = 0.71; percentile: 76; Fig. 2, Panel B) of F.K. were greater than the slopes of the control group, i.e., falling in the fourth quartile of the control group’s performance. Also, in line with the data of S.S., this was not the case for the regression slope of average fixation duration, which fell in the third quartile of the performance of controls (ˇ = 6.27; t = 0.14; percentile: 56; Fig. 2, Panel C). Examining the number of fixations made while encoding 3 dots revealed that the number of

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Fig. 3. The number of fixations required by F.K. and S.S. to encode 3 dots can be observed in relation to the number of fixations of the control sample. Again, the shaded area indicates the second and third quartile around the mean and error bars reflect 1 SD.

fixations of F.K. was also greater in comparison to the control group (2.14 vs. 1.41, t = 1.72; percentile: 94; Fig. 3).3 4.4. Counting range 4.4.1. Control sample For RT, in the counting range a mean regression slope of ˇ = 468.21 (SEM: 38.94, Fig. 2, Panel D) was observed, which differed significantly from 0 [t(7) = 12.02, p < .001]. Furthermore, the mean regression slope for the number of fixations as well as the average fixation duration were ˇ = 1.59 (SEM: 0.07, Fig. 2, Panel E) and ˇ = −18.02 (SEM: 5.97, Fig. 2, Panel F), respectively; both were reliably different from 0 [number of fixations: t(7) = 22.64, p < .001; average fixation duration: t(7) = 3.02, p < .05]. 4.4.2. Participant S.S. The regression slope for RT (449.15; t = −0.17; percentile: 44; Fig. 2, Panel D) fell in the second quartile of the control group’s performance and thus did not differ from the latter. However, the slope for the number of fixations (1.41; t = −0.94; percentile: 19; Fig. 2, Panel E) was smaller than that observed for the controls. This was accompanied by a less pronounced negative slope for the average fixation duration (ˇ = −3.63; t = 0.14; percentile: 80; Fig. 2, Panel F) compared to the controls. This result indicated that S.S.’s number of fixations did not increase as steeply with the increasing number of dots as did the number of fixations of the control group. However, his average fixation durations did not decrease in length as strongly as did the fixation durations of the control group. Taken together, S.S. made not more but longer fixations as the number of dots increased.

3 Note that these results cannot be accounted for by differences in reading ability. In a regression analysis each of the three dependent variables (RT slope, NFix slope, average fixation duration slope, and NFix for three dots) was predicted by the reading score. Thereby, the resulting residuals provide a measure of task performance free from influences of reading ability. Evaluation of these residuals showed that the results did not differ substantially form those obtained for the actual values. Within the subitizing range the residuals for F.K. and S.S. were reliably larger than that of the control group for the number of fixations while encoding three dots (F.K.: t = 1.74; percentile: 94; S.S.: t = 1.70; percentile: 93) and for the NFix slope (F.K.: t = 0.74; percentile: 76; S.S.: t = 2.21; percentile: 97). Moreover, the residuals of the average fixation duration slope differed reliably from the mean of the control group for S.S. but not for F.K. (F.K.: t = 0.14; percentile: 45; S.S.: t = 1.35; percentile: 89). Finally, unlike in the analysis for the actual RT slope, the residual for S.S. fell into the third quartile with the residual for F.K. located in the fourth. However, inspection of the percentile for S.S. showed that it was at the upper end of the third quartile and very close to the percentile observed in the analysis of the actual RT slope (F.K.: t = 1.34; percentile: 90; S.S.: t = 0.61; percentile: 73 vs. 76 in the original analysis).

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4.4.3. Participant F.K. Comparable to the results for S.S., the regression slope for RT (ˇ = 466.03; t = −0.02; percentile: 49; Fig. 2, Panel D) fell inside the second quartile of the control’s performance. Moreover, the slope for the average fixation duration (ˇ = −10.13; t = 0.47; percentile: 67; Fig. 2, Panel F) also did not show any peculiarity. However, the regression slope for the number of fixations (ˇ = 1.71; t = 0.63; percentile: 73; Fig. 2, Panel E) indicated that F.K.’s number of fixations tended to increase more rapidly as the number of dots increased, compared to the average fixation duration of the control children.4 5. Discussion The aim of the current study was to evaluate whether children with dyscalculia are impaired in the very basic ability of discriminating and estimating non-symbolic numerical quantities. To pursue this issue, it was of interest whether the performance of two boys with dyscalculia in a dot enumeration task differed either qualitatively or quantitatively from the performance of an age-matched control sample. As our hypothesis mainly involved performance within the subitizing range, the results regarding the enumeration of 1–3 dots are discussed first followed by a discussion of the results for the counting range. 5.1. A subitizing deficit in developmental dyscalculia? The present results suggest that children with dyscalculia are qualitatively different from typically developing children in basic number processing, i.e., in subitizing. Within the subitizing range of 1–3 dots, the RT slope of the two boys with dyscalculia was reliably larger than that of the control children (the latter not being significantly different from 0). This difference confirmed that children with dyscalculia are indeed impaired in subitizing. Eye movements were used for the first time in children with dyscalculia to examine the nature of this subitizing impairment. In line with our hypothesis, the slope of the number of fixations was larger for children with dyscalculia than for control children. This indicated that for children with dyscalculia dot enumeration within the subitizing range is an attentive process and qualitatively different from that of control children: rather than subitizing (i.e., encoding the cardinality of up to 3 dots in parallel), this increase in the number of fixations indicates a sequential back-up strategy, namely that the two children with dyscalculia were possibly counting at least in some proportion of the trials (see Bruandet, Molko, Cohen, & Dehaene, 2004 for a similar interpretation based on RT). This account is further strengthened by the specific analysis for three dots, the largest problem size in the subitizing range. For three dots, the number of fixations required by children with dyscalculia was more than one SD larger than that of the control group, although only eight trials with 3 dots were presented. This suggests that normally developing children use a much more efficient strategy for 3 dots in particular, that is to say (parallel) subitizing rather than sequential counting. These results also held when controlling for the somewhat lower, although still normal, reading scores of the two dyscalculic boys. This indicates that the conspicuous performance of the two boys with dyscalculia within the subitizing range and in particular their eye-fixation behaviour cannot be accounted for by their relatively lower reading ability. In turn it strengthens our argument that impaired basic (subitizing) processes in developmental dyscalculia cannot be accounted for by concomitant deficits such as deficient processes associated with a somewhat lower reading performance. Finally, we have suggested that access to the corresponding number magnitude representation may be noisier in children with dyscalculia. In dyslexia research, prolonged fixation durations (Hutzler &

4 Note that the results of the analysis of the regression residuals after partialing out reading ability again mirrored that of the analysis of the actual values. For both participants with dyscalculia the residuals of the RT slope did not differ from that of the control group (F.K.: t = 0.02; percentile: 51; S.S.: t = 0.17; percentile: 57). Moreover, the diverging pattern of results for the NFix slope was replicated in the residual analysis (F.K.: t = 0.63; percentile: 26; S.S.: t = 0.92; percentile: 81). Finally, results identical to those for the actual values were also obtained in the residual analysis for the average fixation duration slope. Here, no difference between the residual for F.K. as compared to the control group was present, while for S.S. the residual was reliably larger than that of the control group (F.K.: t = 0.47; percentile: 67; S.S.: t = 0.86; percentile: 80).

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Wimmer, 2004) have been interpreted to reflect impaired access to the representation of a word or its subcomponents (Judica et al., 2002). When a similar logic applies to dyscalculia the noisier representation of numerical magnitude may prolong average fixation durations for larger problem sizes within the subitizing range. However, the data did not support this latter hypothesis since the slopes of the average fixation duration for children with dyscalculia were not steeper than the slopes of the control children. Moreover, even when the largest problem size in the subitizing range was specifically examined, no difference between either boy with dyscalculia and the control sample could be observed. Taken together the results for subitizing are meaningful in two respects. When compared to the control group, both children with dyscalculia exhibited a problem size effect within the subitizing range for RT and number of fixations; however, they did not differ in terms of average fixation duration. 5.2. Performance within the counting range The results for the counting range were much less consistent although some deviations for the children with dyscalculia could be observed. Consistent with the observations of Schleifer and Landerl (submitted for publication) neither of the two children with dyscalculia showed a larger RT slope in the counting range, compared to controls. Moreover, although both children with dyscalculia were deviant in their slopes for the number of fixations, these differences were not similar; while the slope of S.S. was less pronounced, i.e., flatter, than the slope observed for the control group, the slope of F.K. tended to be steeper compared to the slope of the controls for this variable. Finally, S.S. showed a less negative slope for average fixation durations, suggesting that for him the cognitive processes involved with a particular magnitude (e.g., access to the mental number line) became more difficult for larger problem sizes. Hence, both children also exhibited deficits in the counting range as compared to control children. For F.K. these were similar to the difficulties in the subitizing range, the encoding of non-symbolic dot patterns was deficient as indicated by a particularly higher number of fixation for larger problem sizes. It is important to note that in this child with dyscalculia the number of fixations was even larger than the number of dots itself (e.g., 9.5 fixations for 8 dots, Fig. 1, Panel B) indicating frequent corrective eye movements during the counting process. In the other child (S.S.) the slope of the average fixation duration was different from the control group, indicating that for larger problem sizes the average fixation duration was relatively high. This is consistent with the magnitude representation access account, suggesting that for larger problem sizes S.S. needs extra time to encode the cardinality of the dots because access to the exact magnitude representation may be noisier than for the control children. Thus, in both children with dyscalculia dot counting seems to be impaired in the counting range as well, but for different reasons. 5.3. Is the subitizing deficit of children with dyscalculia a perceptual deficit? One could argue that the steeper slope of the children with dyscalculia in subitizing does not indicate a numerical deficit but rather a perceptual deficit because these children need longer to extract information from complex scenes. As the encoding of three dots is more complex than encoding of one dot, they may produce a subitizing slope while children with better perceptual skills are not hindered by the complexity of three dots. This account is rather unlikely in the case of the current study, as the two children with dyscalculia were assessed by a battery of visual attention tasks (e.g., visual alertness, sustained attention) with visual settings more complex than 1, 2 or 3 dots. Both children with dyscalculia did not exhibit any perceptual and/or attentional impairment in those visual attention tasks. As perception of three dots was much easier than any of the attention tasks and because both children were impaired in encoding the number of three dots, we do not see how simple perceptual processes can account for these differences. Rather, we assume that the difficulties of children with dyscalculia result from a deficient comprehension of numerosity, i.e., encoding, representing and/or naming the number of dots.

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5.4. What is the nature of the numerical deficit in subitizing? Landerl et al. (2004) observed that children with dyscalculia are slower in naming symbolic numbers than controls. If such a naming deficit were related to problem size in our children, slower naming, i.e., slower production of the corresponding number words, as already observed for children with dyscalculia (Landerl et al., 2004), might explain the RT pattern of our results. Encoding three dots for children with dyscalculia may be harder due to a problem size effect in number naming (see Brysbaert (1995) and Pynte (1978) for such results in adults), suggesting that the number word three may take longer to produce than the number word one. However, we think that our eye-movement data are inconsistent with such a problem-size account. A naming deficit for larger problem sizes can in our view not account for the steeper slopes for number of fixations in children with dyscalculia. If encoding non-symbolic magnitude and representing this magnitude was similar to that for controls, there is no reason why a worse production of the corresponding number word should lead to a problem size effect in encoding the number of dots. Rather the eye fixation data suggested that the children with dyscalculia in our study exhibited a non-symbolic magnitude encoding deficit. If access to the mental number line (i.e., semantic magnitude representation) was most relevant for the deficits of the two individuals with dyscalculia, this access should take longer because the representation is noisier. This should be reflected in longer fixation durations, in particular for larger problem sizes. However, the average fixation durations did not differ between children with dyscalculia and normal controls. Only the number of fixations was more strongly related to problem size in the subitizing range for the children with dyscalculia. This suggests that the encoding of non-symbolic quantity is somehow impaired in our children with dyscalculia. We suppose that they cannot encode small quantities of dots in parallel but rather rely on a back-up counting strategy. At this point, an approach often applied in reading research may provide further evidence for differentiating qualitative from quantitative discrepancies in children with dyscalculia and normally developing children. Here, it is of interest whether the anomalies observed in the reading habits of children with dyslexia (i.e., slower, erroneous reading, eye-movement characteristics, etc.) reflect either a general slowing in the development of reading skills or an abnormal dysfunctional development. Therefore, the reading characteristics of children with dyslexia are usually not only contrasted to a sample of age-matched controls but also to a sample of younger children matched for their score in a test of reading ability. When the observed reading characteristic of the children with dyslexia mirrors that of the younger reading score matched children, this argues for a delayed but in itself normal development of reading abilities. In contrast, the assumption of abnormal development is corroborated when the children with dyslexia exhibit deficiencies or problems not observed in normally developing children even at an earlier age. Transferred to the case of developmental dyscalculia, this means that in future research it would be particularly interesting to compare the eye-fixation behaviour of children with dyscalculia to the eye-fixation pattern of a control group matched for their numerical abilities, a procedure which has been employed in dyslexia research for years (Ellis, McDougall, & Monk, 1998; Landerl, Wimmer, & Frith, 1997; Snowling, 1981; Ziegler, Perry, Ma-Wyatt, Ladner, & Schulte-Körner, 2004), but – to the best of our knowledge – has not been used in dyscalculia research so far. This would help to distinguish delayed from deviant numerical development. In the present case, the steeper slope of average fixation durations observed for S.S. within the counting range may indicate a delayed development rather than a dysfunction. The average fixation durations of S.S. increased more strongly than that of the control group with increasing set sizes. This can be interpreted as an indication of a noisier magnitude representation in S.S. As it is assumed that the representation of magnitude becomes more accurate with increasing age, the problem of S.S. when it comes to the access to number magnitude may index a developmental delay. However, as we cannot provide eye-fixation data of HRT-matched controls, this proposition remains to be tested by future studies. 5.5. Developmental dyscalculia – a core deficit in magnitude processing? At first the postulate of a non-symbolic deficit in dyscalculia seems to be at odds with the recent study of Rousselle and Noël (2007) who observed deficits in symbolic but not in non-symbolic tasks.

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However, a closer look at this study can resolve this apparent contradiction. In their non-symbolic comparison task Rousselle and Noël (2007) did not use dot patterns from the subitizing range of 1–3 dots. In contrast, we observed the most reliable differences between individuals with dyscalculia and controls, particularly in the subitizing range. A reconciliation of these studies could therefore be achieved along the lines that the most prevalent differences between children with dyscalculia and controls are observed in processing small non-symbolic sets, while approximate processing of larger dot patterns may not be as severely impaired. Such an account would be consistent with the notion of two non-symbolic systems (Feigenson, Dehaene, & Spelke, 2004; Xu, 2003), one dedicated to the exact processing of small quantities (i.e., an object tracking system, cf. Xu, 2003) and the other one preferentially tuned to approximate processing of large quantities (i.e., a number estimation system; Xu, 2003). This account is further corroborated by our own results for the counting range, where the differences between children with dyscalculia and normal controls tended to be smaller and much less consistent. One child with dyscalculia differed in the number of fixation slope, while the other differed in the average fixation duration slope. This suggests that one boy with dyscalculia may have a problem in encoding dot patterns during the counting process as well, whereas the other boy with dyscalculia may experience problems in accessing the representation of larger magnitudes and/or their corresponding number words. Such differences could have two implications: (a) different children with dyscalculia may be impaired in a task as simple as dot enumeration for different reasons and (b) group differences between individuals with dyscalculia and controls may be obscured when data are averaged over different individuals with dyscalculia. These implications bring up the issue of generalizability of the current results. Taking into consideration that the sample of children with dyscalculia assessed in the current study was small (n = 2), future studies are necessary to replicate and validate the current findings. However, while the pattern of impairments was divergent for S.S. and F.K. within the counting range, they exhibited a comparable performance pattern within the subitizing range. From this one might speculate that developmental dyscalculia may be associated with a rather general impairment in the ability to subitize but may become a heterogeneous condition for abilities going beyond an innate object tracking system (Antell & Keating, 1983; Wynn, 1998). Nevertheless, more empirical evidence is required to evaluate these hypotheses. Our conclusions relate to the ongoing debate on the nature and the origin of developmental dyscalculia. Recent reviews emphasize that it may be misleading to adopt a single-deficit view of developmental dyscalculia (Kaufmann & Nuerk, 2006b, 2007; Rubinsten & Henik, 2009; Wilson & Dehaene, 2007; see also Mazzocco & Myers, 2003). Upon conducting a thorough literature review, Rubinsten and Henik (2009) suggest that developmental dyscalculia represents a heterogeneous disorder, both with respect to behavioural and neuro-cognitive phenotypes. Importantly, the authors stress the need to differentiate a “pure” form of developmental dyscalculia (associated with a rather circumscribed difficulty to process numerosity) from (a) a broader form of mathematical learning disability (encompassing numerosity processing deficiencies and non-numerical deficiencies including attention, executive functions and visual-spatial skills), and (b) a dyscalculia accompanied by comorbid learning disorders such as dyslexia or ADHD. Based on the latter differentiation, F.K. presents with a pure form of developmental dyscalculia while S.S. exhibits the co-morbid condition. Interestingly, their eye-fixation behaviour was indicative for the two boys’ representing different diagnostic subtypes by revealing (a) a comparable deficit within the subitizing range (reflecting the diagnostic validity of dyscalculia) and (b) distinct processing impairments within the counting range (suggestive of noisier magnitude representations in the child with co-morbid ADHD; see also Kaufmann & Nuerk, 2008). Taken together, the current data emphasize the need to carefully diagnose, take into account and potentially partial out possible co-morbidities when investigating developmental dyscalculia. Furthermore, our study may prove a good example of how single-case evaluations are informative when examining neuro-cognitive phenotypes of different manifestations of developmental dyscalculia. 6. Conclusions The present data have some important implications for the theoretical understanding of developmental dyscalculia. First, children with dyscalculia diagnosed with symbolic processing tasks such as

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the HRT can indeed have deficiencies in basic non-symbolic task such as subitizing. Second, the present eye-fixation data indicated that this deficit is neither located at the level of number word production nor at the level of accessing number magnitude on a mental number line but rather at a level of quick automatic and parallel encoding of non-symbolic quantities. Third, the current eye-fixation data also indicated that different children with dyscalculia may have different deficiencies, which may lead to problems when averaging over a group of children with dyscalculia. At a more general level, the present study shows that the use of the eye-tracking methodology in dyscalculia research may enhance the understanding of dyscalculia in a similar way as eye-tracking studies have aided the understanding of dyslexia and ADHD. 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