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Psychology Science, Volume 47, 2005 (1), p. 172 - 183
Number sense in children with visuospatial disabilities:
orientation of the mental number line
JAN BACHOT1,WIM GEVERS2,WIM FIAS2,HERBERT ROEYERS3
Abstract
Various sources of information demonstrate a tight link between visuospatial and nu-
merical disabilities. In the past, this link has been attributed to the involvement of visuospa-
tial peripheral support systems like for instance visuospatial working memory in numerical
cognition. However, it is also possible that the association of visuospatial and numerical
abilities has a more basic origin. The basic mental representation of numerical magnitude has
been shown to take the form of an oriented mental number line in normally developing chil-
dren and adults, as evidenced by the SNARC effect (preference for left-hand responses to
small numbers and right-hand responses to large numbers). To investigate the possibility of
abnormal spatial number coding in children with combined visuospatial and numerical dis-
abilities, we measured the SNARC effect in a visuospatial disability group (VSD) during a
number comparison task (smaller or larger than 5) and compared it to a matched control
group. A SNARC effect was obtained in the control group but not in the visuospatial disabil-
ity group. This result is a first indication that the link between visuospatial and numerical
disabilities may be mediated by a basic abnormality in representing numerical magnitudes on
an oriented mental number line.
Key words: SNARC, visuospatial disability
1 J. Bachot, Center of Mental Health Andante, Herculusstraat 17, B-2600 Antwerp, Belgium; Phone: +32 3-
2703773, Fax: +32 3-2703774; E-mail: jan.bachot@andante.be
2 Department of Experimental Psychology, Ghent University, Ghent, Belgium
3 Department of Experimental Clinical and Health Psychology, Ghent, Belgium
Number sense in children with visuospatial disabilities:
orientation of the mental number line
173
The relation between visuospatial problems and mathematical disability has been primar-
ily attributed to dysfunctions at the level of peripheral support systems, like working mem-
ory, visual imagery (Keeler & Swanson, 2001; Reuhkala, 2001) or visuospatial attention,
causing deficiencies in retrieval and/or procedural operations (for review see Geary &
Hoard, 2001). In this paper we do not concentrate on these visuospatial functions supporting
mental arithmetic, but we focus on the central representation of numerical meaning, being
the core component of numerical and mathematical knowledge (Dehaene, 1992; Wynn,
1998), and show that the spatial component of this central representation is affected in chil-
dren with visuospatial disabilities.
Recent studies show that animals (Nieder, Freedman, & Miller, 2002; Sawamura, Shima,
& Tanji, 2002) and human infants (Wynn, 1996) are able to represent numerosity and to
discriminate it from other numerosities. The most important variable determining quantifica-
tion performance is the numerical distance between the numerosities to be discriminated:
discriminability increases with increasing difference between the two sets of elements. Put
more simply: a set of 2 elements can be more easily discriminated from a set of 4 elements
than from a set of 3 elements. Moreover, this distance effect is modulated by numerical size.
For larger numerosities, the numerical distance between the to-be-compared sets needs to be
larger in order to be discriminable (Xu & Spelke, 2000).
With further development the capacity of the number line representation is extended and
its representational accuracy increased (Dehaene, 2001; Verguts & Fias, 2004). Mastery of
symbol systems like language, and particularly the number word list and its integration with
explicit counting procedures (Gallistel & Gelman, 1992) can be expected to be the primary
determinants of the development of the initial representation to a more efficient system.
Importantly, when healthy adults compare quantities, be they presented as collections of
items (Buckley & Gillman, 1974; Fias, Lammertyn, Reynvoet, Dupout, & Orban, 2003) or
as symbols expressed in Arabic digit format (e.g. Moyer & Landauer, 1967) or verbal mo-
dality (Dehaene, 1992), numerical distance remains the most important determinant of per-
formance. Even after extensive training the distance effect does not disappear (Dehaene,
1997). This strongly suggests that the representations underlying adult performance originate
from the initial representational ability available in early childhood.
Yet, increased capacity and accuracy is not the only development of the number line rep-
resentation. Recent research shows that the number line representation becomes spatially
coded: small numbers are associated with left and large numbers with right. This spatial
coding was first demonstrated by Dehaene, Bossini, and Giraux (1993) in normal adults as a
Spatial Numerical Association of Response Codes effect: when doing parity judgment with
key presses as response, subjects exhibited faster left hand than right hand responses to small
numbers, the reverse being true for large numbers. Further research showed that the effect is
not specific to parity judgment and is robustly observed in other tasks (e.g. Fias, Brysbaert,
Geypens, & d’Ydewalle, 1996; Fias, Lauwereyns, & Lammertyn, 2001). In a developmental
study, Berch, Foley, Hill, and Ryan (1999) showed that the SNARC effect in parity judg-
ment emerges from the third grade. However, it should be noted that parity judgment is a
task which requires relatively advanced mathematical knowledge. Therefore, the actual
spatial coding of magnitude representations may occur at a younger age (Fias & Fischer,
2005), although a critical age has not been determined. It has been hypothesized that the
SNARC effect depends on reading habits: Lebanese subjects who are used to read from right
to left tend to exhibit an inverse SNARC effect (large-left and small-right; Zebian, in press).
J. Bachot, W. Gevers, W. Fias, H. Roeyers
174
It is hard to tell, though, whether this reversal is a direct consequence of reading direction or
from a more general culturally-determined habit of ordering information from right to left
(Tversky, Kugelmass, & Winter, 1991).
In sum, the normal core mental representation of numerical magnitude takes the form of
an oriented number line, with two behavioral effects emanating from this kind of representa-
tion: distance and SNARC effect. The distance effect is present in the initial stages of devel-
opment whereas the spatial coding of the number line is acquired later.
In this study, we address the question to what extent the spatial coding of the mental
number line develops normally in children with visuospatial disabilities, with otherwise
normal verbal skills, who showed also problems on arithmetic, next to a much better level on
reading. To exclude differences in automaticity from our results, the SNARC effect was
measured in the context of a magnitude comparison task, which requires access to the mental
number line for correct task performance.
Method
Participants
The sample consists of 32 children in the range from 7 to 12 years. Half of them be-
longed to the group of children with visuospatial disability (VSD group), the other half was a
matched control group. The VSD group consisted of 16 children (mean age = 9,29;
SD=1,21), who were referred to a Centre of revalidation or a Mental Health Centre for chil-
dren and adolescents with complaints of emotional and/or learning disorders. They were
called upon post-hoc and were submitted to a series of neuropsychological tasks. They had
relatively low scores on Performance IQ compared to normal Verbal IQ of the WISC-R and
WISC-III going together with visuospatial disabilities and dyscalculia (see Table 1). They all
had low scores on the block design subtest.
They also had difficulties with visuospatial tasks such as the spatial task of the PMA
(Thurstone & Thurstone, 1962), the Judgement of Line Orientation (Benton, Hamsher,
Varney, & Spreen, 1983), the Developmental Test of Visual Motor Integration (Beery,
1997). They performed the latter in a deficient way without having difficulties on the sup-
plemental developmental tests of Visual Perception and Motor Coordination. Children with
problems in motor coordination were excluded from the experimental group. Moreover, we
administrated a test for visuo-spatial working memory, the VSS (de Sonneville, 2001). They
also did an arithmetic task for complex addition and number concepts (KRT, Cracco, 1993),
a task on simple automatized number facts (TTR, De Vos, 1992), probably based on retrieval
from rote verbal memory (Dehaene, 1992) and a reading test on words (Brus, One-Minute
Test; Brus & Voeten, 1979) and nonsense words (Klepel-pseudo-word test; Van den Bos,
Spelberg, Scheepstra, & De Vries, 1994) at their school level. Comparing mean results of the
Arithmetic test and those of reading (see Table 1), the discrepancy is obvious.
The control group consisted of 16 children matched by gender and age (M=9.24,
SD=1.15). These children were recruited from local schools on the basis of both the teach-
ers’ referrals and the test scores on the WISC-R or WISC-III as a measure of general intelli-
gence (Spreen & Strauss, 1991) and on reading tests. Verbal intelligence was matched to the
Number sense in children with visuospatial disabilities:
orientation of the mental number line
175
Table 1:
Means, SD and F-values of VSD and Control groups4 on the variables.
Test5VSD (n=16)
M(sd)
Control (n=16)
M(sd)
F (1)
WISC
VIQ 108.63 (9.28) 106.19 (5.81) 0.79
PIQ 88.56 (10.56) 107.19 (9.21) 28.28**
BP 7.88 (1.74) 10.06 (1.81) 12.13**
Visuospatial tasks (z-score)
PMA -0.36 (0.97) 0.66 (0.93) 9.20**
JLO -0.60 (0.62) 0.66 (0.84) 23.52**
VMI -0.36 (0.55) 0.68 (0.63) 24.60**
VMIVP 0.41 0.80 1.85
VMIMC 0.33 0.44 0.17
Visuospatial WM (z-score)
VSS -1.90 (1.20) -0.03 (1.04) 22.31**
Arithmetic (z-score)
KRT -1.67 (0.98) -0.04 (0.63) 31.00**
TTR -0.53 (1.45) 0.56(1.68) 3.82
Reading (z-score)
Brus 0.04 (1.49) 0.79 (1.20) 2.46
Klepel 0.06 (0.70) 0.17 (0.53) 0.23
* p<0.05, ** p<0.01
VSD group. (see Table 1). In order to be accepted in the control group, a child’s perform-
ance IQ should not differ from its verbal IQ (maximal difference of 10).
An ANOVA on the standard scores of the two groups revealed a significant difference
between the two groups on all the visuospatial and the complex arithmetic tasks but not on
the reading tasks, being average for both groups (F-values provided in Table 1).
Procedure
In a number comparison task, subjects had to decide whether a presented Arabic number
was larger or smaller than five. Responses were given on an AZERTY keyboard by means of
the letters Q (left) and M (right). Subjects were asked to respond as fast and as accurate as
possible. In the first block, subjects had to respond to small numbers with the left hand and
to large numbers with the right hand. During the second block this response assignment was
reversed.
4 M age : VSD 9.29 (1.21); control 9.24 (1.15); F(1) =0.02, p=0.89
5 BP: Block Design; PMA: Primary Mental Abilities, figure completion; JLO: Judgement of Line Orientation;
VMI: Developmental Test of Visual-Motor Integration; VMIVP: VMI Visual Perception; VMI MC: VMI
Motor Coordination; VSS: Visuospatial sequencing; KRT: Kortrijkse rekentest; TTR: Tempo test rekenen;
Brus: 1 minute reading test; Klepel: reading test with pseudo-verbs.
J. Bachot, W. Gevers, W. Fias, H. Roeyers
176
Arabic numbers ranging from one to nine (except five) were shown at the center of the
screen (height: 1.25°; width: .57°). Each target number was preceded by a fixation cross at
the center of the screen for 500ms. Immediately following offset, the target number appeared
and remained there until response or 7000ms elapsed. After target presentation the screen
remained blank for 500 ms; thereafter a new trial was initiated.
Each target number was presented 10 times, leading to a total of 80 target presentations
per block. Both blocks were preceded by a training list in which each target number was
presented twice. Target numbers were randomized in such manner that the same number
could never be repeated.
Data analysis
The data were analyzed using a regression method proposed by Lorch and Myers (1990,
Method 3). For each subject, for each number and for each side of response, median reaction
time (RT) was calculated. From these median scores, differences in RT (dRT) were calcu-
lated for each subject, subtracting left hand responses from right hand responses. For each
subject, dRTs are then regressed with number magnitude as predictor variable. A t-test de-
cides whether the regression slope differs significantly from zero. Because a clear prediction
is postulated about the direction of the slope (negative correlation between dRT and number
magnitude), all tests are one-sided. The distance effect was analysed using the same method.
Because the distance effect is independent from response side, the regression was performed
on average RT for each number separately. Subsequently, each number was regressed with
the distance towards the reference number five as predictor variable (e.g. distance 4 for tar-
gets 1 and 9, distance 3 for targets 2 and 8, etc…).
Results
Control subjects
Average error rate over subjects was 4.92% with a maximum of 13.75%. There was no
speed-accuracy trade off over the 16 cells of the design (r= .44 n = 16, p<.09), therefore
errors were not analyzed separately. Median RT over targets ranging from one to nine was
809, 792, 814, 844, 897, 833, 777 and 835ms, respectively.
Subsequently, the Lorch and Myers method was applied for revealing possible distance
effects. As a result, the following regression equation was obtained:
Number sense in children with visuospatial disabilities:
orientation of the mental number line
177
RT = 871.27 – 18.47 (Distance)
1A) Distance ef fect, control group
700
750
800
850
900
950
D4 D3 D2 D1 D1 D2 D3 D4
Distance
RT
Obs erved
Fitt ed
with distance contributing significantly [t(15) = -2.03, SD = 36.45, p<.05]. Furthermore, the
distance effect contributed in a reliable manner up to a distance of 3 from the reference num-
ber five [F(1,15) = 5.63; p < . 05]. The analysis to evaluate the presence of a SNARC effect
revealed the following equation:
dRT = 44.76 – 13.97 (Magnitude)
1C) SNA RC ef fect, c ontrol group
-150
-50
50
150
12346789
Numb er
dRT
Obs erved
Fitt ed
with a significant magnitude slope [t(15) = -2.12, SD = 26.35, p < .05].
So, both distance and SNARC effects are present when the control subjects performed
the number comparison task.
VSD group
Average error rate over subjects was 11.33% (maximal: 31%). A positive correlation
over the 16 cells of the design indicated the absence of a speed-accuracy trade-off (r=.37,
n=16, p < .2). Median RT over subjects for each target number ranging from one to nine was
J. Bachot, W. Gevers, W. Fias, H. Roeyers
178
973, 980, 963, 1094, 1116, 997, 992 and 960ms, respectively. The slope value for distance,
obtained from the following regression equation differed significantly from zero, t(15) =
-3.82, SD = 42.87, p < .001.
RT = 1111.66 – 40.97(Distance)
1B) Distance effect, VSD group
900
950
1000
1050
1100
1150
D4 D3 D2 D1 D1 D2 D3 D4
Distance
RT
Observed
Fitted
As can be seen in the Figure, there was only a reliable distance effect between distance
one and two (F(1,15) = 5.19; p < .05). The other distances did not reach significance (all p >
.5).
Finally, the regression analysis was run on the dRTs in order to investigate the SNARC
effect. Although Figure 1d shows a positive slope indicative for a reversed SNARC effect,
the slope did not significantly differ from zero. t(15) =1.35, SD = 58.27, p > .08.
dRT = -90.41 + 19.65 (Magnitude)
1D) SNARC effect, VSD group
-150
-100
-50
0
50
100
150
12346789
Number
dRT
Observed
Fitted
However, in spite of this lack of significance, a significant positive correlation was pre-
sent between the difference in IQ (VIQ minus PIQ) and the magnitude of the slope (r = .52;
p < .05). Hence, the larger the difference between VIQ and PIQ, the more positive the slope
value becomes.
Number sense in children with visuospatial disabilities:
orientation of the mental number line
179
Figure 1:
Upper figures: observed data and regression line representing RT responses as a function of
distance by a) the control group and b) the experimental group. Lower figures: observed data and
regression line representing RT differences between right-handed minus left-handed responses as
a function of magnitude by c) the control group and d) the experimental group.
Direct comparison of visuospatial disability and control group
The VSD group was both slower in RT (t(15) = -2.69; p < .01) and made more errors
(t(15) = -3.43; p < .01) than the control group. In order to find out whether the visuospatial
disability group and the matched control group were different from each other with regard to
the Distance effect and the SNARC effect, dependent sample t-tests on the slopes were per-
formed. For the distance effect, there was a marginal significant difference between both
groups, t(15) = 1.65; SD = 54.48; p <.08. As can be seen in Figure 1a and 1b the slope re-
flecting the distance effect is steeper in the experimental than in the control group. Notice
also that the distance effect is more fine tuned in the control group than in the experimental
group. While the control subjects showed a reliable distance effect up to a distance of three
(F(1,15) = 5.63; MSe = 8793; p < .05), the VSD group only showed a reliable distance effect
when a distance of 1 was compared to a distance of 2 (F(1,15) = 5.19; MSe = 96583; p <
.05).
Furthermore, the SNARC effect was found to reliably differ between the two groups
t(26) = -2.43, p < .05. In order to visualize the individual differences in slope in the two
groups, figure 2 provides a cumulative frequency distribution of the two groups.
J. Bachot, W. Gevers, W. Fias, H. Roeyers
180
Figure 2:
Cumulative frequency distribution representing both the control and the VSD group as a function
of the slope value.
Relative Cumulative Distribution
0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
1
-140
-120
-100
-80
-60
-40
-20
0
20
40
60
80
100
120
140
Negative <- Slope -> Positive
Relative freq uency
VSD
Control
In order to decide whether the values from the VSD group are stochastically more posi-
tive than the values of the control group, a Kolmogorov-Smirnov two-sample one-tailed test
was applied. The largest difference between both groups was .5 [D(16,16) = 16*16*.5 = 128;
p < .05]. From this analysis it can be concluded that the VSD group shows proportionally
more positive values than the control group.
In order to examine the influence of multiple differences and to exclude that the group
differences are due to other variables than those of interest we performed the following mul-
tiple regression analyses. Group membership appeared to contribute significantly to the slope
values for the SNARC effect, t(30) = -2.10; p < .05 but not for the distance effect t(30) =
1.60; p > .1. With regard to the SNARC slopes, entering verbal IQ and age into the regres-
sion did not alter the results. Neither of these factors predicted the slope values whereas
group membership continued to maintain a significant influence t(28) = -1.91; p < .05 (one-
sided).
However, when the scores on each of the visuospatial tasks were taken into considera-
tion, the group difference no longer reached significance, t(26) = -0.83, p > .4, supporting
our conclusion that the missing SNARC slopes in the VSD group rely on deficient spatial
processing. When the arithmetic tasks were entered in the regression, the KRT explained
most of the variance t(28) = -1.84; p < .08 whereas the group differences again disappeared,
t(28) = -0.43; p > .6. Whether deficient visuospatial processing is caused by deficient arith-
metic processing or the other way round can not be determined on the basis of the present
correlational data. Whatever it is, the present data do point to the importance of spatial char-
acteristics in the SNARC effect.
Number sense in children with visuospatial disabilities:
orientation of the mental number line
181
Discussion
In this study, we addressed the question to what extent the spatial coding of the mental
number line develops normally in children with visuospatial disabilities, with otherwise
normal verbal skills, who also showed problems on arithmetic, next to a much better level on
reading. To this end, both the SNARC effect and the distance effect were measured within
this VSD group and compared with a control group matched on age and verbal intelligence.
A first indication for a difficulty with the mapping of the numbers on an internal number
representation was the general finding that performance of the VSD group was both slower
and less accurate compared to the control group.
Additionally, it was shown that the control group and the VSD group differ from each
other with respect to the SNARC effect. In line with Berch et al. (1999), we found that the
control group showed a normal SNARC effect in that small numbers were responded to
faster with the left hand and large numbers with the right hand. In contrast, the VSD group
showed no evidence for this mapping from magnitude to response side. A number of alterna-
tive interpretations regarding the absence of a SNARC effect in the VSD group can be ruled
out.
First, it cannot be attributed to the fact that VSD children would have particular difficul-
ties with a change of response assignment halfway the experiment. Because the congruent
stimulus-response mapping (small left, large right) was administered first followed by the
incongruent mapping, one would then expect an enhancement of the SNARC effect. The fact
that the SNARC effect tends to reverse in the VSD group, can have two reasons. One reason
could be that the number line of the VSD children is misoriented, i.e. from right to left in-
stead of from left to right. The reverse SNARC effect, albeit non-significant, could also be a
reflection of a general training effect because the reversal of the SNARC effect means that
the VSD group was faster in the second half of the experiment than in the first half. If this
would be the case, this suggests that the VSD children are largely insensitive to the specific
S-R mappings. Results from the complementary order of s-r mapping (incongruent followed
by congruent) will allow to distinguish between the two interpretations.
A second possible cause for the absence of a SNARC effect is that the VSD group was
far too heterogeneous to obtain reliable measures. However, this does not seem plausible
because the subjects were carefully selected in order to obtain a homogeneous group. Fur-
thermore, the VSD group was, with the exception of visuospatial properties and arithmetical
abilities, matched with a control group that reliably showed the SNARC effect. Moreover,
two points of evidence suggest that the positive slope value is not attributable to noise. First,
a positive correlation was obtained between the difference in Verbal and Performal IQ (P-V
IQ) and the slope value indicating that the larger the P-V IQ, the more positive the slope
value becomes. Second, a frequency distribution showed that the VSD group yielded propor-
tionally more positive slope values than did the control group.
This link between visuospatial disabilities and an abnormal representation of numerical
magnitudes on an oriented mental number line does not necessarily reflect a direct causal
relationship but may be established via an intervening variable. For instance, the specific
role of peripheral support systems like working memory needs further investigation. In fact
the VSD group performed significantly worse than the control group on a visuospatial work-
ing memory task (VSS). From a developmental point of view it seems not contradictory that
J. Bachot, W. Gevers, W. Fias, H. Roeyers
182
deficits in spatial working memory might contribute to a delay or deficiency in building up
automatized spatial number representations during early school years.
The finding that the VSD group showed no SNARC effect provides us with first evi-
dence that these children exhibit problems in mapping the numbers on a mental number
representation. Converging evidence towards this conclusion were the results obtained for
the distance effect. Whereas the control group showed a distance effect, reliable up to a
distance of three, the VSD group showed only a reliable distance effect for the numbers 4
and 6.
At this point, the results can only be regarded as a first indication of a problem with the
mapping of the numbers on the ‘mental number line’. The reason why there is a tendency for
a reverse orientation of the mental number line (or the mapping towards the mental number
line) remains unclear and cannot be resolved on the basis of the present experiment.
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