Number skills are maintained in healthy ageing

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Number skills are maintained in healthy ageing
Marinella Cappelletti
a,b,
⇑
, Daniele Didino
c,1
, Ivilin Stoianov
d,e,1
, Marco Zorzi
d,f
a
UCL Institute of Cognitive Neuroscience, 17 Queen Square, London WC1N 3AR, UK
b
Goldsmiths College, University of London, New Cross London, SE14 6NW, UK
c
Department of Cognitive and Education Science, University of Trento, via Tartarotti 7, Rovereto, Trento, Italy
d
Department of General Psychology, University of Padova, via Venezia 8, 35131 Padova, Italy
e
Institute of Cognitive Sciences and Technologies, National Research Council, Rome, Italy
f
IRCCS San Camillo Hospital, Venice-Lido, Italy
article info
Article history:
Accepted 29 November 2013
Available online 11 January 2014
Keywords:
Numerical cognition
Ageing
Numerosity perception
Number acuity
Computational modelling
abstract
Numerical skills have been extensively studied in terms of their
development and pathological decline, but whether they change
in healthy ageing is not well known. Longer exposure to numbers
and quantity-related problems may progressively refine numerical
skills, similar to what happens to other cognitive abilities like ver-
bal memory. Alternatively, number skills may be sensitive to age-
ing, reflecting either a decline of number processing itself or of
more auxiliary cognitive abilities that are involved in number
tasks. To distinguish between these possibilities we tested 30 older
and 30 younger participants on an established numerosity discrim-
ination task requiring to judge which of two sets of items is more
numerous, and on arithmetical tasks. Older participants were
remarkably accurate in performing arithmetical tasks although
their numerosity discrimination (also known as ‘number acuity’)
was impaired. Further analyses indicate that this impairment
was limited to numerosity trials requiring inhibiting information
incongruent to numerosity (e.g., fewer but larger items), and that
this also correlated with poor inhibitory processes measured by
standard tests. Therefore, rather than a numerical impairment,
poor numerosity discrimination is likely to reflect elderly’s impov-
erished inhibitory processes. This conclusion is supported by sim-
ulations with a recent neuro-computational model of numerosity
perception, where only the specific degradation of inhibitory
processes produced a pattern that closely resembled older partici-
pants’ performance. Numeracy seems therefore resilient to ageing
but it is influenced by the decline of inhibitory processes
0010-0285/$ - see front matter Ó2013 Elsevier Inc. All rights reserved.
http://dx.doi.org/10.1016/j.cogpsych.2013.11.004
⇑
Corresponding author. Fax: +44 (0)207 916 8517.
E-mail address: m.cappelletti@ucl.ac.uk (M. Cappelletti).
1
These authors contributed equally to this work.
Cognitive Psychology 69 (2014) 25–45
Contents lists available at ScienceDirect
Cognitive Psychology
journal homepage: www.elsevier.com/locate/cogpsych

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supporting number performance, consistent with the ‘Inhibitory
Deficit’ Theory.
Ó2013 Elsevier Inc. All rights reserved.
1. Introduction
Does our ability to use numbers and arithmetical concepts change with ageing? Are these changes
specific to numeracy or do they rather reflect decline of more general cognitive processes such as
attention or inhibitory processes? Numerical skills have been extensively studied in children and
young adults, both in terms of development or impairment following brain lesions (Ansari, 2008;
Cappelletti, 2011). However, little is known about the impact of healthy ageing on numerical skills,
and the few studies that investigated this issue focused mostly on arithmetical abilities, i.e. those
required when solving problems such as 8 9 or 243 + 39. These studies concurred to show that
although older participants can learn new ways to solve arithmetical problems, they show a smaller
repertoire of strategies and are less efficient than younger participants in selecting among them (e.g.
Duverne & Lemaire, 2005;Lemaire & Arnaud, 2008;Geary & Lin, 1998; Salthouse & Kersten, 1993), or
that they do not equally engage the same brain regions as younger participants when performing
arithmetical tasks (El Yagoubi, Lemaire, & Besson, 2005). However, these tasks are typically
multi-componential, requiring several processes such as the retrieval of arithmetic facts, the use of
procedures and the ability to monitor the steps of the problem (Cappelletti & Cipolotti, 2011). It
may therefore be difficult to isolate which specific component may be affected by ageing.
An alternative approach to test the impact of ageing on numeracy skills is to assess other simpler skills
(sometimes referred to as ‘biologically primary skills’, Geary & Lin, 1998) which are thought to be founda-
tional to more complex, education- and language-based numerical and arithmetical abilities. One such
foundational skill isthought to be our capacity to represent approximate number, which is basedon encod-
ing numerosities as analog magnitudes (e.g. Izard, Dehaene-Lambertz, & Dehaene, 2008; Stoianov & Zorzi,
2012), and relies on an ‘approximate number system’ (ANS, Feigenson, Dehaene, & Spelke, 2004). The ANS
is often measured in terms of the ability to discriminate numerosities (e.g. which set has more elements),
also referred to as ‘number acuity’ (Halberda,Mazzocco, & Feigenson, 2008). Number acuity is expressed as
Weber fraction (wf), which reflects the amount of noise in the underlying approximate number represen-
tation (Halberdaet al., 2008; Piazza,Izard, Pinel, Le Bihan, & Dehaene, 2004).The wf is highly variable across
individuals (Halberda, Lya, Wilmerb, Naimana, & Germine, 2012; Halberda et al., 2008; Piazza et al., 2004),
and it refines progressively from infancy to adulthood (Halberda et al., 2008; Halberda & Feigenson, 2008;
Lipton & Spelke, 2003, 2012; Piazza et al., 2010). Whether it continues to improve with age is an open ques-
tion: longer exposure to numbers may refine the approximate number system further, similarly to what
happens to other cognitive abilities like vocabulary and semantic memory (e.g. Hedden & Gabrieli,
2004). Notably, number acuity has been found to correlate with math achievement in children (Halberda
et al., 2008; Mazzocco, Feigenson, & Halberda, 2011a) and to be impaired in children with developmental
dyscalculia (Mazzocco, Feigenson, & Halberda, 2011b; Piazza et al., 2010),
A few previous studies have focused on how healthy ageing participants are able to represent approx-
imate large numerosities (i.e., more than 10 elements), which has sometimes been reported to be well
maintained (e.g. Gandini, Lemaire, & Dufau, 2008; Gandini, Lemaire, & Michel, 2009; Lemaire & Lecac-
heur, 2007; Trick, Enns, & Brodeur, 1996; Watson, Maylor, & Bruce, 2005; Watson, Maylor, & Manson,
2002). Some of these previous results, however, are difficult to interpret. This is because in some cases
the focus was mainly on the strategies used to perform the numerosity task, without reporting finer
quantitative details of older participants’ performance (e.g. Gandini et al., 2008). In other studies, the
long presentation of the stimuli (6 s or even unlimited) may have encouraged processes different from
numerosity estimation, like counting (Gandini et al., 2009; Lemaire & Lecacheur, 2007; Watson, Maylor,
& Bruce, 2007). Likewise, some experimental designs did not control for continuous variables which
inevitably vary when manipulating the numerosity of the display, like the total area covered by the dots,
i.e. cumulative area (for discussion see Piazza et al., 2004). If these continuous variables are not taken into
account, for example if in all trials an increase in numerosity always corresponds to an increase in
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cumulative area (e.g., Gandini et al., 2008), it is unclear whether participants judged changes in numer-
osity or in these continuous variables. Finally, participants’ performance cannot be fully characterised
when only measured as percentage of correct answers rather than in terms of finer psychophysical mea-
sures like the wf (Gandini et al., 2008, 2009; Lemaire & Lecacheur, 2007;Watson et al., 2005).
A recent internet-based mega-study showed a different patter of results, indicating that the ability
to discriminate numerosities (as indexed by the wf) may indeed be sensitive to ageing (Halberda et al.,
2012). This age-related deterioration, which was nevertheless not explained by the authors, may re-
flect either the decline of the approximate number system itself or of more peripheral cognitive pro-
cesses that are involved in discriminating numerosities. For instance, working memory, attention and
inhibitory processes are all critical when discriminating numerosity, for example when retrieving the
quantity of a standard set of items to be compared to a test set or when inhibiting task irrelevant infor-
mation, such as the area or the density of elements correlating with numerosity. Since working mem-
ory, attention and inhibitory processes tend to decline with age (Grady, 2012; Hedden & Gabrieli,
2004; Nyberg, Lövdén, Riklund, Lindenberger, & Bäckman, 2012; Salthouse, Atkinson, & Berish,
2003), they may in turn affect performance in numerosity discrimination tasks. In particular, inhibi-
tory processes have been suggested to decline with age and to underlie age-related impairments in
many cognitive functions according to the ‘Inhibitory Deficit’ Theory (Hasher, Lustig, & Zacks, 2007;
Hasher & Zacks, 1988; Hasher, Zacks, & May, 1999). Among the separate functions of inhibition, the
‘restraint function’ (also called ‘inhibition of dominant responses’, Miyake et al., 2000) is the ability
to control strong responses so that others more appropriate for the task goal can be used (Hasher
et al., 2007). This function, which is sensitive to age decline (Kane & Engle, 2003; May & Hasher,
1998), may be particularly relevant in the numerosity discrimination task. This could be the case of
trials where some continuous variables change orthogonally to numerosity, for instance when fewer
large-size dots have a bigger cumulative area than smaller-size but more numerous dots. In this case,
the task-irrelevant but salient information about cumulative area (Hurewitz, Gelman, & Schnitzer,
2006) has to be controlled in order to correctly discriminate numerosity.
Here we studied young and ageing participants in order to explore the precision of their ANS mea-
sured in terms of number acuity in a numerosity discrimination task. Our study is novel not only be-
cause it systematically measured number acuity in older people but also because it aimed to
determine what underlies the pattern of spared or impaired numerosity processing. We reasoned that
if number acuity does not differ between older and young participants, this may be suggestive of
maintained ANS. In contrast, age-related differences in number acuity may reflect impairments spe-
cific to the number system, or alternatively decline of more general cognitive processes. To distinguish
between these two possibilities, our plan was twofold: first, using established neuropsychological
measures, we planned to investigate arithmetical abilities, which are thought to be linked to the
ANS (Halberda et al., 2008) and are therefore expected to be impaired if the ANS is impaired. Second,
using dedicated and well established tasks, we set to investigate the integrity of older participants’
inhibitory processes and in particular of restraint functions which might contribute to any age-related
difference in numerical abilities. Besides studying numerical abilities in older participants in depth
using behavioural measures based on psychophysics and neuropsychology, a second novel aspect of
our research is to combine these measures with a computational approach. This aimed to establish
the type of condition that may resemble our participants’ performance in the numerosity discrimina-
tion task. Our goal was to better understand whether any impairment in numerosity discrimination
may be due to a global deterioration of the number system or to impaired peripheral processes, spe-
cifically those on which inhibitory functions rely on.
2. Behavioral study
2.1. Methods
2.1.1. Participants
Sixty right-handed neurologically healthy, education-matched participants with normal or
corrected-to-normal vision gave written consent and were paid to participate in our study which
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was approved by the local research Ethics Committee. Participants were selected from the UCL Insti-
tute of Cognitive Neuroscience database because of their age: the 30 young participants had a mean
age of 24.8 years (range 19–36; 13 males); the 30 older participants had a mean age of 65.77 years
(range 60–75; 12 males). Ageing participants were considered neurologically normal on the basis of
self-report and on their performance in the Mini Mental State Examination (MMSE, Folstein, Folstein,
& McHugh, 1975; see Table 1). Information on participants’ education and mathematical education in
particular was collected in order to assess the possible impact of these factors on performance in the
experimental tasks. Data in all participants were collected in one or two testing sessions (in the latter
case, they were 7–10 days apart).
2.2. Numerosity discrimination: experimental tasks and stimuli
Stimulus presentation and data collection were controlled using the Cogent Graphics toolbox
(http://www.vislab.ucl.ac.uk/Cogent) and MATLAB 7.3 software on a Sony S2VP laptop computer with
video mode of 640 480 pixels, and 60 Hz refresh rate.
Following earlier studies that investigated number acuity across the lifespan (Halberda et al.,
2012), we used a version of the numerosity discrimination task that required to judge which of two
intermixed collections of coloured dots was more numerous. Sets of dots were presented in blue
and yellow in each display, with 5–16 dots for each colour. The ratios between the larger and the smal-
ler number of dots were 2:1, 4:3, 6:5 and 8:7, with 40 trials presented for each of the easiest ratios (i.e.
2:1 and 4:3), and 120 for each of the most difficult ratios (6:5 and 8:7) (see Table S1). We used a larger
number of trials in the most difficult ratios in order to increase statistical power (i.e., to obtain more
reliable estimates of individual wf) since we expected lower accuracy in these ratios. A total of 320
trials was presented in 10 blocks, and in each ratio the ‘larger’ set was equally assigned to the two
colours.
For each ratio there was also an equal number of congruent (dot-size controlled) and incongruent
(area controlled) trials presented in random order. Dot-size controlled trials were those in which the
average diameter of the dots in the larger set was equal to the average diameter of the smaller set.
The diameter of a dot ranged approximately between 0.57°and 1.17°of visual angle from a distance
of 57 cm, the average diameter being 0.87°. In these trials the cumulative area of the larger set was
always larger than the cumulative area of the smaller one. In contrast, area controlled trials were those
in which the average diameter of the larger set (which ranged approximately between 0.57°and 1.17°
of visual angle, i.e. 0.87 ± 35%) was smaller than the average diameter of the smaller set (which ranged
approximately between ±35% of the average diameter of the smaller set itself). The average diameter
of the smaller set was selected so that the cumulative area of the two sets was equal.
2.2.1. Procedure
Each trial started with a fixation point for 1500 ms followed by a display of blue and yellow dots for
200 ms after which a question mark appeared to prompt participants to respond (see Fig. 1). Partici-
pants were instructed to make an unspeeded answer indicating which group of dots (blue or yellow)
was more numerous by pressing one of two predefined computer keys (M or N), whose assignment
was randomised between participants. Once an answer was made, the following trial started imme-
diately with the 1500 ms fixation.
2.2.2. Data analysis
For all data, the Shapiro–Wilk test confirmed the normality of the distribution, and the data were
analysed using parametric tests (multiple regressions, Analyses of Variance (ANOVAs) and t-tests). The
data sphericity was tested using the Mauchy Test, and for significant results Greenhouse–Geisser
correction was applied in case of sphericity violation. Significance was set at a pvalue of 0.05.
Following earlier studies, we fitted the response distribution of each participant in the numerosity
discrimination task to obtain individual estimates of number acuity (i.e., the precision of the underly-
ing numerical representation), expressed as internal Weber fraction (wf, e.g. Halberda et al., 2012,
2008; Piazza et al., 2004, 2010. See SI for a detailed description of how the wf was calculated).
Although speed was not stressed in the instructions given to participants, we analysed response times
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Table 1
Older and young participants’ (A) demographic information, and performance in (B) background tasks, (C) number and arithmetic tasks,
(D) executive and inhibitory processes. Percentile,reaction time(RTs), percent corrector Weber Fraction (wf) with sta ndard deviation ( SD).
Task/information Young participants (N= 30) Older participants (N= 30)
A. Demographic information
Age 24.8 years (range 19–36) 65.77 years (range 60–75)
Gender 13 males 12 males
Years of education 18 16
Years of mathematical education 13 12
B. Background
Full IQ (WAIS-R)
a
116.2 (10.8) 118.3 (13.4)
Mini Mental State Examination
b
nt 29.4 (0.6)
Stimulus identification 293.1 ms (33.72) 348.2 ms (34.0)
Vocabulary
a
83%ile (18.5) 94%ile (6.3)
Digit span
a
82%ile (19.6) 86%ile (12.3)
Door recognition
c
87%ile (15.1) 83%ile (14.7)
C. Number and arithmetic
Numerosity discrimination based on
All trials (wf) 0.24 (0.04) 0.30 (0.07)
Congruent trials (wf) 0.238 (0.06) 0.255 (0.08)
Incongruent trials (wf) 0.256 (0.06) 0.367 (0.15)
Number comparison
Accuracy (% correct) 97.1 (2.5) 98.9 (1.6)
RTs small/large distance
d
576 ms (141)/488 ms (76) 698 ms (112)/607 ms (65)
Arithmetic tasks
Arithmetic verification
Accuracy (% correct); RTs 94.2 (4.4); 1127 ms (241) 97.3 (2.9); 1409 ms (387)
Graded Difficulty Arithmetic Test
e
80.6%ile (18.2) 86.3%ile (19.9)
WAIS-R math sub-test
a
83.1%ile (16.4) 86.3%ile (22.8)
Reading and writing numbers 99.5% (1.9) 98.7% (2.1)
D. Executive and inhibitory processes
Number Stroop task
Magnitude comparison
Congruent 99.3%; 478 ms (67) 99.8%; 745 ms (99)
Neutral 97.17%; 514 ms (64) 99.6%; 789 ms (112)
Incongruent 89.1%; 544 ms (66) 96.4%; 866 ms (120)
Physical comparison
Congruent 99.1%; 457 ms (71) 99.8%; 600 ms (76)
Neutral 99.8%; 470 ms (62) 99.8%; 615 ms (79)
Incongruent 93.1%; 505 ms (82) 99%; 687 ms (84)
Word Stroop task
f
Word target
Congruent 98.7%; 462 ms (55) 99.3%; 602 ms (82)
Neutral 99.7%; 472 ms (48) 99.3%; 612 ms (72)
Incongruent 93.3%; 513 ms (68) 98.0%; 696 ms (72)
Colour target
Congruent 98.3%; 467 ms (102) 99.3%; 611 ms (103)
Neutral 95.6%; 473 ms (114) 99.8%; 610 ms (93)
Incongruent 94.8%; 470 ms (80) 98.8%; 725 ms (161)
Attention network test (ANT)
g
Congruent; Incongruent 448 ms (86); 533 ms (59) 658 ms (82); 766 ms (90)
Conflict: incongruent–congruent 85 ms (20) 98 ms (26)
nt = not tested; ms = milliseconds.
a
Wechsler (1995).
b
Folstein et al. (1975); max score: 30.
c
Baddeley A. D., H., & I., 1994.
d
Longer RTs to small number distances indicate normal distance effect [r= 0.4, F(1, 29) = 14.8, p< 0.001], not different from
young participants [t(58) = 8.4, p< 0.001].
e
Jackson and Warrington (1986).
f
Stroop, 1935.
g
Fan et al. (2002).
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to further characterise participants’ performance following other studies using a similar experimental
paradigm and procedure (e.g. Halberda et al., 2012; Piazza et al., 2010).
2.2.3. Results of numerosity discrimination
Accuracy differed across the four numerosity ratios [percent correct, ratio 2:1, Young = 96%
(sd = 0.4), Older = 89% (sd = 0.7); ratio 4:3, Young = 78% (sd = 0.9), Older = 77% (sd = 0.6); ratio 6:5,
Young = 71% (sd = 0.4), Older = 66% (sd = 0.5); ratio 8:7, Young = 65% (sd = 0.6), Older = 62% (sd = 0.4)].
An analysis of the wf in young participants indicated a large variability, such that the wf was on
average 0.24, ranging from 0.16 to 0.33, consistent with previous studies (e.g. Halberda et al., 2008,
2012). Likewise, number acuity in older participants showed large individual differences, being on
average 0.30 and ranging from 0.20 to 0.49.
In a regression analysis with wf as dependent variable and group as independent one, we found
that group was a significant predictor of number acuity [beta = 0.38, SE = 0.16, t= 3.1, p=0.003], as
confirmed by a significant difference between older and young participants [t(58) = 3.2, p= 0.002,
see Fig. 2 left panel]. A more specific analysis looking at the variability of performance in the older
sample showed that within the older group 21 out of 30 participants showed a wf that was one or
two standard deviations below the average of the younger group, indicating defective performance.
This suggests that the group difference we observed was not simply driven by a few participants,
thereby mirroring the results of Halberda et al.’s (2012) internet mega-study.
We also tested whether education in general and mathematical education in particular may predict
number acuity. This is because mathematical expertise (e.g. in accountants and bookkeepers) has pre-
viously been shown to explain differences in memory ability for arbitrary numbers and for grocery-
store prices between expert and non-expert ageing participants (Castel, 2005, 2007). Although our
Fig. 1. Numerosity discrimination task. (A) Participants saw a centrally presented fixation cross for 1500 ms immediately
followed by a brief (200 ms) presentation of a display of yellow and blue dots (for the purpose of the figure displayed in black
and white). Participants were instructed to indicate with a button press whether each trial contained more blue or yellow dots,
with no time constrains. The following trial was displayed immediately after the answer. (B) Example of area-controlled
(incongruent) and dot-size controlled (congruent) trials. Participants were presented with 320 trials; half of these trials
contained yellow and blue dots having on average the same cumulative area and whose diameter was inversely correlated to
numerosity, implying that the least numerous set had the larger dot diameter (incongruent trials, bottom panel of B). The other
half of the trials contained yellow and blue dots of approximately the same average diameter such that the larger display had
larger cumulative area (congruent trials, top panel of B).
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sample did not include mathematical experts, we nevertheless explored whether education and
mathematical education may in part explain the difference in number acuity we observed. However,
we found that neither of these predictors were significant [education: beta = 0.08, SE = 0.004, t= 0.58,
p= 0.6; mathematical education: beta = 0.009, SE = 0.004, t= 0.06, p= 0.9].
The difference in wf might suggest that relative to younger participants, older participants needed a
larger discrepancy between the two stimulus sets to be able to identify the larger, which may be due
to a deteriorated representation of numerosity.
2.3. Arithmetical tasks
Impaired performance in the numerosity discrimination task may reflect a deteriorated represen-
tation of numerosity, which has been suggested to be linked to arithmetical skills, following the pro-
posal that number acuity is foundational to these skills (Halberda et al., 2008). We therefore tested
older and younger participants’ arithmetical abilities, predicting a possible impairment in older par-
ticipants if these skills are linked to ANS.
2.3.1. Stimuli and methods for the arithmetical tasks
Stimuli for the arithmetical tasks were Arabic numbers in the form of single to 3 digits presented
either on the computer, on paper or read aloud by the experimenter. We used the following four tasks:
arithmetic verification, number comparison, multi-digit mental arithmetic, and arithmetical problem
solving.
Arithmetic verification required participants to indicate as fast as possible whether an arithmetic
problem displayed the correct or incorrect answer (using the ‘F’ or ‘J’ keys of the keyboard). Twenty
single-digit problems for each type of operation (addition, subtraction and multiplication) were pre-
sented in separate blocks with no repetition of the same items in immediately successive trials. All
problems had operands below 10; results of the addition and subtraction problems were between 1
and 18, and of multiplication problems were between 6 and 36. Following a 500 ms central fixation
Fig. 2. Behavioural results. Psychometric functions (PF) indicating young (in light gray) and older (in black) participants’
performance in the numerosity discrimination task measured in terms of wf in: (A) all conditions (i.e. average of dot-size
and area-controlled trials), (B) dot-size controlled trials only, and (C) area-controlled trials only. The psychometric
functions plot the probability of choosing ‘more blue’ dots (y-axis) as a function of the ratio between blue and yellow
dots (x-axis).
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cross, each operation was presented for up to 7 s during which participants could answer. When pre-
sented with incorrect results, these were either 1 or 2 units apart from the correct result for addition
and subtraction problems (e.g. 9 + 6 = 13 or 7–2 = 3) or 2 units apart for multiplication problems (e.g.
63 = 16).
Number Comparison required indicating as fast as possible the larger of two Arabic numbers pre-
sented to the left and right of a central fixation point (using the ‘F’ and ‘J’ keys of the keyboard to indi-
cate the larger number appearing on the left or on the right respectively). Thirty-six pairs of single-
digit Arabic numbers (1–9) were individually presented.
2
Stimuli pair were centred along the horizon-
tal line of the computer screen and displayed for 500 ms each to the left or the right of the fixation cross;
they were replaced by a black screen for a maximum of 2500 ms during which participants made an an-
swer. After this, the next trial started immediately. The following numerical distances were used: 1 (e.g.
7 vs 8), 2 (e.g. 3 vs 1), 3 (e.g. 5 vs 2), 4 (e.g. 1 vs 5), 5 (e.g. 4 vs 9). The larger number in each pair was
equally presented to the left and to the right; likewise, correct answers were equally assigned to the left
or the right digit in each pair.
Multi-digit mental arithmetic was tested with the Graded Difficulty Arithmetical Test (GDA, Jack-
son & Warrington, 1986), a standardised test based on two separate blocks of twelve 2 to 3-digit addi-
tion and twelve 2 to 3-digit subtraction problems of progressive difficulty (e.g. from ‘13 + 15’ to
‘243 + 149’). These were orally presented one at a time for an oral answer, which scored 1 point if cor-
rectly produced within 10 s.
Arithmetical problem solving was assessed with the arithmetical subtest of the Wechsler Intelli-
gence Scale (WAIS-R; Wechsler, 1995), consisting of twenty arithmetical problems embedded in a text
and orally presented for an oral answer. Correct answers produced within a maximum time (spanning
from 15 to 60 s depending on the problem) were assigned 1 point.
Performance in these numerical and arithmetical tasks was expressed as accuracy and/or reaction
times (RTs) of correct answers only; reaction times were collected in the number comparison and the
arithmetic verification tasks only.
Since performance in arithmetical skills are often more sensitive to speed than accuracy, and since
speed tends to increase with age (e.g. Salthouse, 1991), we measured older participants’ general speed
of performance in two ways. The first way was based on a simple perceptual task, which required par-
ticipants to respond as rapidly as possible to a dot stimulus appearing on the computer monitor at var-
ious locations. Forty trials were presented, each displaying a dot for 200 ms in random locations on the
left or right of a computer monitor and following an ISI randomly selected between 500 ms and 2 s.
Participants were instructed to make speeded responses whenever the stimulus appeared by pressing
a pre-defined key. Accuracy and response times were recorded.
The second way consisted of measuring older participants’ speed of performance using a two-
choice non-numerical task whose modality of response resembled the type of choice participants
had to make in the number and arithmetical tasks. Specifically, we used responses made in the Stroop
‘neutral’ conditions of the Word Stroop task (averaged across the case where the target colour was
presented on the neutral string ‘XXX’, and where the target word was presented in the neural colour
grey; see also Section 2.4.2.4).
2.3.2. Results of arithmetical tests
Remarkably, we found that young and older participants accuracy did not significantly differ in any
of the number and arithmetical tasks (all P> 0.4, see Table 1). In older participants, proficiency in the
numerosity task correlated with accuracy in the arithmetical tasks [arithmetic verification: r= 0.51,
F(1,29) = 9.7, p< 0.004; multi-digit arithmetic (GDA): r= 0.42, F(1, 29) = 6.1, p< 0.02; arithmetic prob-
lem solving (WAIS): r= 0.36, F(1, 29) = 4.2, p< 0.05], but not in number comparison [r= 0.15,
F(1,29) = 0.6, p= 0.4, ns].
Despite maintained accuracy, older participants were significantly slower than young to perform
the number comparison and the arithmetic verification tasks [t(58) = 13.1, p< 0.001 and t(58) = 3.2,
2
The pairs of Arabic numbers were as follows; distance 1: 1–2, 3–2, 4–3, 5–4, 6–7, 7–8, 8–9; distance 2: 3–1, 2–4, 4–6, 5–3, 7–5,
6–8, 9–7; distance 3: 4–1, 2–5, 3–6, 7–4, 8–5, 6–9, 5–2, 1–4; distance 4: 1–5, 3–7, 4–8, 5–9, 6–2, 7–3, 8–4, 9–5; distance 5: 1–6, 2–
7, 8–3, 9–4.
32 M. Cappelletti et al. / Cognitive Psychology 69 (2014) 25–45

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p< 0.01 respectively). We therefore tested whether this slowness was specific for the number tasks or
whether instead it reflected general slowness. We run a fixed-entry hierarchical regression with the
basic RTs, the choice RTs and group entered as independent variables in different steps, and RTs (sep-
arate for arithmetic verification and number comparison) as the dependent variable. Results showed a
significant effect of group [arithmetical verification: beta = 275.6, SE = 133.5, t= 2.0, p< 0.05;
number comparison: beta = 245.6, SE = 49.2, t= 4.9, p< 0.001], suggesting that older participants
were generally slower than younger. However, a non-significant effect of RTs in the basic perceptual
speed task [arithmetical verification: beta = 1.8, SE = 1.0, t= 1.7, p= 0.09, ns; number comparison:
beta = 0.5, SE = 0.38, t= 1.3, p= 0.18, ns] suggests that basic RTs did not explain participants’ slowness
in the number and arithmetic tasks. Instead, a significant effect of two-choice RTs [arithmetic verifi-
cation: beta = 1.9, SE = 0.39, t= 4.8, p< 0.001; number comparison: beta = 0.32, SE = 0.14, t= 2.2,
p= 0.03] which contributed to 65% and 88% of the variance in the two tasks respectively, accounted
for participants’ slowness in the number and arithmetic tasks.
2.4. The role of other cognitive processes in number acuity
Our results suggest that despite a larger wf, older participants’ numerical and arithmetical skills
were well maintained.This therefore does not support one of our initial hypotheses that a larger wf
may reflect a more general impairment in number and arithmetical processing.
An alternative possibility is that a larger wf may be explained by impairments in inhibiting task-
irrelevant information, an hypothesis that has been previously put forward to account for elderly’s im-
paired performance in other cognitive functions (Hasher & Zacks, 1988; Hasher et al., 1999, 2007). If
this is the case, differences between congruent and incongruent numerosity trials may be expected,
since incongruent trials required inhibiting task-irrelevant information coming from continuous
variables such as the size of the dots that negatively correlated with numerosity (Dakin, Tibber, Green-
wood, Kingdom, & Morgan, 2011; Gebuis & Reynvoet, 2012; Hurewitz et al., 2006). In addition, we also
examined older participants’ inhibitory skills independently from number acuity.
2.4.1. Congruent vs incongruent numerosity trials
To investigate whether older participants’ larger wf may reflect a difficulty to inhibit information
that may be irrelevant to numerosity judgments, separate wf for congruent (dot-size controlled)
and incongruent (area controlled) trials in the numerosity discrimination task were calculated and
compared within and between groups using paired and independent t-tests respectively. Relative to
congruent trials, incongruent numerosity trials resulted in a larger wf in both groups (older: congru-
ent = 0.255 and incongruent = 0.367; young: congruent = 0.238 and incongruent = 0.256), but with a
significant difference between trial types only in the older [t(29) = 3.9, p< 0.001] but not in the young
participants [t(29) = 0.9, p= 0.3, ns], see Fig. 2 middle panel. The wf differed between the two groups
only in the incongruent [t(58) = 3.7, p< 0.001] but not in the congruent trials [t(58) = 0.1, p= 0.4, ns;
see Fig. 2 right panel].
In older participants, age correlated with performance in the incongruent numerosity trials [r= 0.3,
F(1,29) = 2.9, p<0.04] but not with the congruent ones [r= 0.12, F(1, 29) = 0.4, p=0.5, ns].
2.4.2. Numerosity discrimination task with blocked congruent and incongruent trials
Having found that older participants were specifically impaired in the incongruent numerosity
trials as indexed by a significantly larger wf relative to young participants, we aimed to identify which
factors may account for such impairment. One possibility is that older participants’ impaired incon-
gruent numerosity trials might be related to the strategies used to perform these trials. For instance,
in order to identify the more numerous set older participants may have used a strategy based on just
numerosity, and also other strategies, for example based on a salient non-numerical visual feature
(like the area covered by the dots or amount of colour as a proxy for numerosity). These numerosity
and area-based strategies may have been successfully used in the case of incongruent and congruent
trials respectively, but switching between them may have been particularly difficult for ageing partic-
ipants. We reasoned that a difficulty in switching among strategies may be ameliorated by keeping the
congruent and incongruent numerosity trials blocked. We therefore administered participants the
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same numerosity task but with the congruent and incongruent trials in blocked rather than random-
ized order, and with a trial-by-trial feedback to help maintaining correct performance. Since we only
found a large wf in the older participants, we did not test younger participants in this blocked and
feedback-based version of the numerosity task.
3
2.4.2.1. Stimuli and method. We used the same stimuli, design and procedure previously employed (see
Section 2.2), with the following changes. First, the same number (N= 320) and type of trials previously
used were now presented in random order in 5 blocks of 32 congruent trials and 5 blocks of 32 incon-
gruent trials. The 5 blocks of congruent and the 5 blocks of incongruent trials were grouped and the
order was counterbalanced across participants. Second, to encourage correct performance, trial-by-
trial feedback was introduced at the end of each trial, indicating whether participants made a correct
or wrong answer. All the other features of the task were exactly the same as before (see Section 2.2).
Performance was calculated in terms of wf with the same procedure previously used; we also mea-
sured response times, although speed was not stressed in the instructions to participants.
2.4.2.2. Results. The wf and response times obtained in the numerosity task with blocked and with the
random trial presentation were compared in independent t-tests. There was no significant advantage
in presenting trials in a blocked fashion and with feedback relative to the canonical random presen-
tation, indicated by an almost unchanged wf (average random condition: 0.29 vs blocked condition:
0.31; t(11) = 0.3, p= 0.7, ns). This suggests that it is unlikely that higher wf in older participants is sim-
ply due to problems in switching among strategies, for instance from a numerosity-based to an area-
based rule. Further support to this suggestion comes from the analysis of the response times, which
were examined in an ANOVA with presentation type (blocked vs random) and trial type (congruent
vs incongruent) as within-subject factor. This showed no main effect of presentation [F(1,11) = 0.8,
p= 0.3, ns] and of trial type [F(1,11) = 0.7, p= 0.4, ns], and no significant interaction [F(1, 11) = 0.02,
p= 0.8, ns], therefore suggesting no difference between the blocked and the random conditions.
2.4.2.3. Analysis of response times in the numerosity task. The pattern of response times in the numer-
osity discrimination task (data from the main experiment, see Section 2.2) was analysed to assess the
hypothesis that group differences may be due to a speed-accuracy trade-off for the older participants.
We used a multivariate analysis of variance (MANOVA) with response times and accuracy (wf) for the
congruent and incongruent trials entered as dependent variables, and group (young and older) and
type of trial (congruent and incongruent) as fixed factors. Both group and trial type were significant
[respectively: F(2,115) = 16.04, p< 0.001, Wilk’s
K
= 0.78; and F(2,115) = 15.05, p< 0.001, Wilk’s
K
= 0.79], as well as their interaction [F(2,115) = 4.8, p< 0.01, Wilk’s
K
= 0.9]. Specifically, group
had a significant impact on both accuracy [F(1, 116) = 7.3, p< 0.001] and response times
[F(1,116) = 26.3, p< 0.001], and likewise trial type significantly affected accuracy [F(1, 116) = 21.9,
p< 0.00] and response times [F(1, 116) = 9.7, p= 0.002]. Group and trial type also interacted with accu-
racy [F(1,116) = 5.1, p< 0.02] and with response times [F(1,116) = 4.1, p= 0.04]. This is because young
participants were more accurate than older in incongruent (wf: 0.256 vs 0.367; t(58) = 3.7, p<0.001)
but not in congruent trials (wf: 0.238 vs 0.255; t(58) = 0.1, p=0.4, ns), and response times for congru-
ent and incongruent numerosity trials differed in the older [617 ms vs 649 ms, t(29) = 2.3, p=0.02] but
not in the young participants [415 ms vs 427 ms, t(29) = 0.9, p=0.3, ns]. This RTs significant difference
in trial type in the ageing group was maintained even when older participants’ general slowness was
taken into account in a regression analysis. Here response times for the congruent and incongruent
numerosity trials were separately entered as dependent variable in the analysis, with group and
two-choice RTs as independent variables. There was a significant effect of group in the incongruent
trials only [congruent: beta = 0.3, SE = 65.7, t= 1.5, p= 0.13, ns; incongruent: beta = 3.9, SE = 71.8,
t= 2.0, p= 0.048], suggesting that older participants were slower than younger only in the incongruent
trials. In these trials there was no main effect of choice RTs [beta = 0.25, SE = 0.22, t= 1.2, p= 0.2, ns],
indicating that slowness in performing incongruent trials could not be accounted for by general
3
Only 12 older participants could be tested on this additional version of the numerosity task.
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slowness, although this seemed to be case for congruent trials where choice RTs were marginally sig-
nificant [beta = 3.5, SE = 0.2, t= 1.8, p= 0.06]. These findings suggest that older participants did not
trade off their accuracy for response speed. Indeed, both accuracy and RT data revealed a specific dif-
ficulty of older participants in processing incongruent trials.
These results suggest that larger wf in older participants were due to differences in wf in the incon-
gruent but not the congruent trials, possibly because these trials relied upon inhibitory skills. We
therefore aimed to explore these skills in more details in our older participants. We focused on inhi-
bition not only because it is critical for processing the incongruent numerosity trials but also because
it has been suggested to decline with age and to underlie age-related impairments in many other cog-
nitive functions (Hasher & Zacks, 1988; Hasher et al., 1999, 2007).
2.4.2.4. Older participants’ performance in inhibitory tasks. Using standard tests, we specifically focused
on the restraint function among inhibitory processes, i.e. the ability to control strong responses such that
others more appropriate for the task goal can be used (Hasher, Lustig, & Zacks, 2007). We reasoned that
this function may be particularly relevant in the numerosity discrimination task and specifically in the
incongruent trials where information about the area covered by the dots had to be controlled in order to
correctly discriminate numerosity. We used the following standard tasks to assess inhibitory processes
in older participants: Word Stroop, Number Stroop, Attention Network test (ANT).
The word Stroop task requires participants to read as quickly as possible either a word ignoring the
colour of the ink it is printed on (for instance ‘RED’ whether printed in colour red, blue or in grey for a
neutral condition), or to name the colour in which words are printed ignoring their meaning (for in-
stance to name the colour red whether displayed on the word ‘RED’, ‘BLUE’ or on ‘XXX’ for a neutral
condition). There were 60 trials for each task (word or colour). For the word task, stimuli were the
word ‘RED’ or ‘BLUE’, which could appear in red, blue or gray colour; this therefore resulted in a con-
gruent, incongruent or neutral condition (20 trials for each type), depending on whether, for example,
the word ‘RED’ appeared in colour red, or in colour blue or in gray. In the colour task, stimuli were the
word ‘RED’, ‘BLUE’ or a string of ‘XXX’. These could appear in colour red or blue, therefore resulting in a
congruent, incongruent or neutral condition (20 trials for each type), depending on whether, for exam-
ple, the colour red appeared on the word RED or on the word BLUE or on the string XXX.
In each trial, participants saw a centrally presented 500 ms fixation cross, followed by a word stim-
ulus displayed until the participant made an answer or for a maximum of 4000 ms. After this, the fol-
lowing trial started immediately. Participants were asked to decide as quickly as possible whether the
stimulus was the word ‘RED’ or ‘BLUE’ irrespective of the colour (in the word task), or whether it was
displayed in red or blue font irrespective of the meaning of the word (colour task); they were in-
structed to press the left and right arrow keys for blue (word or colour) and red (word or colour)
respectively. The two tasks were presented separately, and the order of the tasks was counterbalanced
across participants.
For each task (word or colour), accuracy and response times were calculated for the three conditions:
neutral (corresponding to the target word printed in grey or the target colour printed on XXX), congru-
ent (target word printed on the corresponding colour or the target colour on the corresponding word),
and incongruent (target word printed on a different colour like ‘RED’ printed in blue, or the target colour
printed on a different word, like colour red on the word blue). These three conditions allowed calculating
an index of ‘congruity’ (i.e. response times of correct answers only in incongruent–congruent trials), a
standard measure of participants’ ability to inhibit task irrelevant information (Stroop, 1935).
The number Stroop task is based on an established paradigm that assesses the automatic process-
ing of numbers as well as inhibitory processes using experimental stimuli that contain congruent and
incongruent information (Henik & Tzelgov, 1982). In two separate tasks, participants viewed a total of
336 pairs of 1–9 Arabic numbers (168 per block) that could vary in magnitude (e.g. 3 vs 2) or physical
size (e.g. 3 vs 2). There were therefore three types of stimuli (each of 36 trials per task): a congruent
stimulus corresponded to a pair of digits in which a given digit was larger in both the relevant and the
irrelevant dimensions; a neutral stimulus was a pair of digits that differed only on the relevant dimen-
sion (magnitude or physical size); an incongruent stimulus consisted of a pair of digits in which one of
the digits was at the same time larger in one dimension (e.g. magnitude) and smaller in the other (e.g.
physical size). Each number stimulus could be paired to itself, therefore consisting of a neutral
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stimulus for the physical size condition (e.g. 2 vs 2), or to another number stimulus which could be
between 1 and 4 units apart. Moreover, the two number stimuli could be of the same physical size,
therefore consisting of the neutral stimulus for the magnitude condition, or they could vary along
two levels of physical size, as stimuli could appear in a vertical visual angle of 0.7°or 0.9°centred
along the horizontal line of the computer screen to the left or the right of the fixation cross.
Participants were required to indicate the larger number in either magnitude or physical size by
pressing either the left or the right arrow if the larger number was presented either to the left or to
the right. Following a 500 ms fixation cross, the number stimuli were presented until the participant
made an answer or for a maximum of 4000 ms. After this, the next fixation cross appeared and the
following trial started immediately. For each task (magnitude or physical size), accuracy and response
times were recorded. Using this experimental design, it is common to find a congruity effect which
consists of facilitation, i.e. faster responding to stimuli in which information about the magnitude
and the physical size of the stimuli is congruent (e.g. 3 vs 2) relative to neutral trials (e.g. 3 vs 3 for
physical comparisons or 3 vs 2 for numerical comparisons), and of interference namely slower re-
sponses to stimuli in which information about the magnitude and the physical size of the stimuli is
incongruent (e.g. 3 vs 2) relative to neutral trials (Henik & Tzelgov, 1982). These effects are thought
to reflect automatic processing of numerical and physical size even when magnitude or size are
task-irrelevant; this is suggested by studies showing that irrespective of which dimension is relevant,
response times for congruent trials are always shorter than response times for incongruent trials, and
for incongruent trials shorter than for neutral trials, although an advantage for congruent relative to
neutral trials has only been observed in the magnitude condition (Henik & Tzelgov, 1982; Tzelgov,
Meyer, & Henik, 1992). This task therefore served the double purpose of measuring participants’ auto-
matic number processing and, critically, also their ability to inhibit task-irrelevant information.
The Attention Network Test (ANT, Fan, McCandliss, Sommer, Raz, & Posner, 2002) examines exec-
utive and inhibitory processes using a variation of the flanker task (Eriksen & Eriksen, 1974) whereby
participants attend to one object while ignoring others (Posner, 1980). In the version used here, a cu-
ing task and a flanker task were combined such that participants responded to cued or un-cued central
targets while ignoring flanking distractors. A total of 288 trials were presented in 3 blocks of 96 trials
each. The stimuli consisted of a target arrow flanked by two arrows on either side, which could appear
in the same direction as the target arrow (congruent condition e.g. ) or in the opposite
direction (incongruent condition e.g. ? ). Following Fan and colleagues’ paradigm (2002),
each arrow was presented at 0.55°of visual angle and separated from the adjacent arrows by 0.06°
of visual angle. The stimuli (the central arrow and the four flankers) consisted of a total 3.08°of visual
angle. Participants were instructed to attend to the middle arrow and to decide whether it was point-
ing to the left or to the right. A trial consisted of the following events: a central fixation cross was first
presented for a random duration between 400 and 1600 ms, and followed by either a 100 ms warning
asterisk cue in the cued trials or by a longer fixation in the un-cued trials, and by a second 400 ms fix-
ation period after which the target and the flankers appeared simultaneously and centrally, at 1.06°of
visual angle either above or below the fixation point. The cue could appear centrally, hence corre-
sponding to a spatially neutral condition or it could precede the target and flankers in the same posi-
tion (the cue was always valid), i.e. at 1.06°of visual angle above or below the fixation point, which
corresponded to a spatially orienting condition. The target and flankers remained on the screen until
the participant responded or for a maximum of 1700 ms. After a response was made, the next trial be-
gan immediately.
Participants had to press as quickly as possible a left-hand key if the central arrow pointed left and a
right-hand key if it pointed right. The task allows measuring three indexes of performance based on
how response times of correct answers are influenced by alerting cues, spatial cues, and flankers: alertness
(cued vs un-cued trials), orienting (central cue vs spatialcue), and conflict(congruent vs incongruent trials
averaged across cued and un-cued, and central vs spatial cue). As conflict is the critical component we
aimed to measure, we limited the analysis of the performance to this index of behaviour.
2.4.2.5. Results of tasks assessing inhibitory processes. Older participants significantly differed from
younger participants in the ability to suppress task-irrelevant information (see Table 1). In both tasks
of the Word Stroop (i.e. where the target was either a word or a colour), older participants were
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significantly slower in incongruent than congruent trials relative to young (word Stroop task:
t(58) = 3.2, p=0.002; colour Stroop task: t(58) = 7.5, p<0.001). Likewise in both the Number Stroop
tasks and in the ANT, older participants showed a larger congruity effect relative to young participants
[Number Stroop tasks: magnitude comparison, t(58) = 3.7, p<0.001; physical size comparison,
t(58) = 2.1, p<0.04; ANT, t(58) = 2.7, p=0.008]. Accuracy was near ceiling in both groups (98.6%
and 98.9% correct in older and young respectively), and did not differ between groups [t(58) = 0.76,
p= 0.54, ns].
Critically, older participants’ poor performance in these tasks correlated with wf in the incongruent
numerosity trials. Hence, the worse the elderly’s performance in these incongruent relative to congruent
trials, the longer weretheir response times in the incongruent trials of the Stroop tasks and of the Attention
Network Test [Word Stroop (averaged over word and colour target conditions): r=0.38,p< 0.04; Number
Stroop (averaged over magnitude and physical conditions): r=0.4,p=0.03;ANT:r=0.37,p=0.04].This
was not the case of performance in the congruent trials which did not correlate with any measure of inhi-
bition p rocesses [Word Stroop : r=0.24,p<0.2,ns;NumberStroop:r=1.5,p=0.4,ns;ANT:r= 0.06, p=0.7,
ns]. The equivalent analyses in young participants showed no link between wf in congruent or incongruent
numerosity trials and response times in any of the above tasks [all P>0.1].
These results suggest a link between wf in the incongruent trials and older participants’ perfor-
mance in tests of inhibitory processes. This in turn may reflect deteriorated inhibitory processes rather
than impoverished number processing, as this would have likely resulted in impaired performance in
other arithmetical tasks. To further support the hypothesis that larger wf may reflect deteriorated
inhibitory processes, we performed a set of computer simulations based on a recent neurocomputa-
tional model of numerosity perception (Stoianov & Zorzi, 2012).
3. Computational modelling study
The connectionist model of numerosity perception developed by Stoianov and Zorzi (2012) is a
deep neural network with two hierarchically organised layers of (hidden) neurons. The first layer re-
ceives input from the image and it consists of uniformly spread center-On local detectors (high fre-
quency spatial filters). The second layer consists of local numerosity detectors that receive input
from the center-On neurons normalised by an inhibitory signal representing the image cumulative
area. The activity of numerosity detectors (layer 2 neurons) is used as input to a simple linear classifier
that is trained to decide whether the input numerosity is greater than a reference number. Perfor-
mance of the model in the numerosity comparison task is evaluated in the same way as human par-
ticipants, that is by computing the model’s number acuity (wf) from the accuracy data.
The main aim of our simulations was to establish what type of impairment would produce the dis-
sociation between intact performance on dot-size controlled (‘‘congruent’’) trials and declined perfor-
mance on area-controlled (‘‘incongruent’’) trials observed in the elderly population. Rather than fitting
the participants’ performance, the simulation aimed to offer an explanation at the level of the mech-
anisms that subtend numerosity discrimination in the model. To provide a closer match to the ANS
task used in the present behavioral study, the model was extended to simulate the comparison of
two variable visual numerosities.
3.1. Computer simulations
3.1.1. Stimuli
Using the method described in Stoianov and Zorzi (2012), we created visual numerosity images of
size 30 30 pixels, each containing up to 32 randomly placed non-overlapping rectangular objects of
variable size and shape
4
(see samples in Fig. 3). Object size varied within each pattern according to a
Gaussian term with mean = 0 and sd = 0.5. We created a database of images containing objects with
4
Note that the range of numerosities (with upper limit of 32 objects) was selected by Stoianov and Zorzi (2012) to allow the
simulation of the typical experimental setting of numerosity comparison studies (e.g., Piazza et al., 2004). This requires a minimum
image size of 30 30 pixels to keep all objects spatially separated by at least one pixel in the largest numerosity condition.
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variable cumulative area that ranged from 32 to 256 pixels at 8 levels, with 200 images for each level of
numerosity (n= 32) and cumulative area (n= 8). We then compiled a training set by drawing 10240 pairs
of images from the image database, ensuring that numerosity and cumulative area were balanced. The
training set was used to train the linear classifier on the numerosity comparison task, while the model’s
performance (i.e., wf) was assessed using an independent set of test images that were not used for train-
ing (as in Stoianov & Zorzi, 2012).
We also created a database of test images that had the same characteristics of the stimuli used in
Experiment 1. In particular, area-controlled test images had constant cumulative area of 100 pixels,
whereas size-controlled test images contained objects with constant size (3 3 pixels). Cumulative
area was not predictive of numerosity in the area-controlled images but it was correlated with the
numerosity in the size-controlled ones. The database contained 200 images for each of the numeros-
ities and conditions (size-controlled vs. area-controlled) of the behavioural study. The test set was cre-
ated by drawing 1100 pairs of test images for each of the two conditions, yielding the same
numerosity pairs (and ratios) used in the behavioural study.
3.1.2. Model details
The simulations extended the simplified version of the model described in the Supplementary
Material of Stoianov and Zorzi (2012) where the numerosity perception network is hard-wired and
training involves only the numerosity discrimination task. The model is depicted in Fig. 3. It is worth
noting that the architecture and the parameters of the model (e.g., size of the receptive fields) were
Fig. 3. Computational model. (A) Hierarchical network for numerosity perception (Stoianov & Zorzi, 2012). The input image
(30 30 pixels) is processed by one layer of center-on detectors with small-size receptive fields. Numerosity detectors at the
top layer have larger receptive fields and compute a local numerosity signal by combining the activity of center-on detectors
with an inhibitory normalization signal conveying the image’s cumulative area (see text for details). Poor numerosity
discrimination in the elderly population can be accounted for by degraded inhibitory normalization. (B) Numerosity comparison
task. Activity of the numerosity detectors obtained for each image of a pair was used as input to a linear classifier, that is a
simple network trained to decide which of the two numerosites was larger.
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identical to those reported in the original study and they were chosen as to faithfully represent the
structure emerged (i.e., self-organised) in the learning model. For the present purposes, a key advan-
tage of the simplified mathematical model is that it clearly identifies the functionality of each model
component, such as the role of inhibitory signals (see below).
The numerosity perception network has two hierarchically organised layers of neurons above the
visual input layer. The first layer (size: 13 13 neurons) receives input from a binary image Iand it
consists of uniformly spread center-On local detectors (i.e. high frequency spatial filters) activated
according to the following equation:
O
ij
¼fðXV
ij
IþbÞ ð1Þ
where V
ij
are 2D-gaussian-shaped receptive fields (
r
= 2) over the image I,bis a bias term (fixed to
1), and f(.) is the logistic function. The second layer (size: 6 6 neurons) consists of uniformly spread
local numerosity detectors activated according to the following equation:
N
kl
¼XW
kl
Ocð2Þ
where W
kl
are 2D-Gaussian-shaped receptive fields (
r
= 6) over the center-ON neurons and cis a
(inhibitory) normalisation term based on the image cumulative area,
c¼log 1 þPI
c
max
 ð3Þ
where c
max
is the maximal cumulative area across the image database. The activity of the numerosity
detectors coarsely represents numerosity. Indeed, Stoianov and Zorzi (2012) showed that the response
profile of the population activity of layer 2 numerosity neurons closely resembled the monotonic cod-
ing of numerosity observed in monkey LIP neurons (Roitman, Brannon, & Platt, 2007) and that it was
invariant to perceptual properties of the stimuli, such as cumulative area.
Stoianov and Zorzi (2012) showed that the activity of numerosity detectors supported human-like
performance in numerosity comparison when used as input to a linear classifier that was trained to
compare numerosities to a fixed reference. More specifically, the activity of the numerosity detectors
elicited by a given input image was fed to a decision layer encoding a ‘‘smaller’’ vs. ‘‘larger’’ judgment.
This simple linear network was trained using the delta rule, which is formally equivalent to the Resc-
orla-Wagner learning rule and has been widely used to account for human learning (Sutton & Barto,
1981). Thus, the model assumes that numerosity comparison is a simple linear task when the input to
the decision is an abstract representation of numerosity. This is line with Dehaene and Changeux
(1993) computational model of the development of early numerical abilities and it fits well with
the observation that the classic numerosity comparison paradigm (with explicit judgment/response)
can be carried out even by 3–4 year old children (Halberda et al., 2008; Piazza et al., 2010). The model
responses, when plotted as a function of numerical ratio, yielded a psychometric function that per-
fectly mirrored those of adult observers, thereby obeying Weber’s law. Moreover, the wf computed
from the model’s response distribution was virtually identical to the mean value reported for adult
observers (e.g., Piazza et al., 2010). The model’s descriptive adequacy was therefore supported by
its match to both behavioral (i.e., human psychophysics) and neural (i.e., single cell recording) data
(see Stoianov & Zorzi, 2012, for further details).
Here we simulated the comparison of two variable numerosities (as in the behavioral paradigm of
Experiment 1 and in the studies of Halberda et al., 2008, 2012) by feeding two internal numerosity
representations (i.e., the activity of the numerosity detectors for each image of a given pair in the
training set) to a linear classifier that was trained to decide which of the two numerosities was larger.
Learning was based on the delta rule and the training database was presented for 100 epochs (when
learning became markedly asymptotic). Model’s performance after training (i.e., model’s wf) was as-
sessed on the image pairs contained in the test set. Because the representation of each numerosity
(activity of numerosity detectors) is intrinsically noisy, we expected greater response variability
(i.e., a higher wf) in this version of the task relative to the original simulations of Stoianov and Zorzi
(2012) where a single numerosity was compared to a fixed reference number.
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3.2. Elderly model
To simulate the declined performance of elderly population, we implemented a degradation to the
network connections by applying a stochastic decay term to the synaptic strengths (i.e., weight val-
ues). This stochastic reduction of synaptic efficacy results in neurons’ decreased responsivity to affer-
ent signals. Decreased responsivity of neurons, which turns into a weaker signal-to-noise ratio (i.e.,
increased neuronal noise) is also critical in the neuromodulation model of ageing of Li and colleagues
(Li, Lindenberger, & Bäckman, 2010; Li, Lindenberger, & Sikström, 2001). In their model, this is ob-
tained by flattening the slope of the neurons’ sigmoid activation function, which is contolled by a sto-
chastic gain parameter. Li and colleagues link the value of the gain parameter to dopaminergic
modulation, which is known to be altered in elderly population.
We implemented two different types of impairment: a global degradation of the numerosity per-
ception network, which involved all network synapses, or a selective degradation of the inhibitory
synapses (in line with the inhibition deficit hypothesis). Inhibition in the model is critical for abstract-
ing numerosity from continuous visual properties, specifically cumulative area (see Eq. (2)). Indeed,
the normalisation signal represents an inhibitory input to the numerosity detectors that is conveyed
through feed-forward inhibitory connections (see Fig. 3). Feed-forward inhibition is known to act as
input normalisation mechanism in real neurons (e.g., Pouille, Marin-Burgin, Adesnik, Atallah, & Scan-
ziani, 2009). In the present model, the excitatory input to numerosity detectors increases both with
numerosity and cumulative area (because large objects will tend to activate more center-on detectors
than small objects) but is normalised by an inhibitory input whose strength also increases with cumu-
lative area (see Stoianov & Zorzi, 2012, for detailed analyses).
Synaptic degradation was obtained by scaling the connection weights by a random value (range [0–
1]) drawn from a Gaussian distribution with mean = mand s.d. = 0.1 (smaller values of mdetermine
stronger impairment). Numerosity comparison in the model was adjusted after the impairment
through a short re-training (5 epochs) on a new dataset created just as the training set and containing
5120 image pairs. The re-training was meant to capture a gradual adjustment of the decision process
to compensate for the aging effect. Indeed, aging is known to modify the decision criteria in two-
choice decision tasks (Ratcliff, Thapar, & McKoon, 2006). The severity of each type of impairment, con-
trolled by the parameter m, was set to yield an average decline in performance (across 30 replications)
that matched the decline observed in human participants. This procedure yielded a global impairment
characterised by m=0.20 and, alternatively, an inhibitory impairment with m=0.25.
We performed 30 replications of each type of impairment (i.e. global vs. inhibitory) and we as-
sessed the model’s performance in numerosity comparison on the test dataset. Individual wfs were
separately computed for the two conditions (size-controlled vs. area-controlled) of the test dataset.
A control population of 30 unimpaired networks was created and assessed after applying the same
readjustment procedure used for the impaired networks (to ensure that the observed differences were
not due differences in training regimen). Test and re-training dataset were independently drawn for
each replication.
4. Results of the computational model
The wf values obtained in the simulation (computed for each replication) were submitted to a
mixed ANOVA with condition (area-controlled vs. dot size-controlled) as within-subject factor and
impairment type (unimpaired, inhibitory, global) as between-subject factor. Both factors, and impor-
tantly, their interaction, affected numerosity discrimination (impairment: F
(2,87)
= 26, p< 0.001; condi-
tion: F
(1,87)
= 212, p< 0.001; interaction: F
(2,87)
= 187, p< 0.001). Planned contrasts revealed that both
types of impairment worsened discrimination performance relative to the unimpaired models, with
overall wf (i.e., across conditions) increasing from 0.249 to 0.313 after global impairment
(t(58) = 9.8, p< 0.001) and to 0.315 after inhibition impairment (t
(58)
= 5.5, p< 0.001), see Fig. 4. Per-
formance of the two types of impaired model was not statistically different (t
(58)
= 0.2, p= 0.85). Crit-
ically, the inhibition impairment specifically worsened discrimination on area-controlled stimuli
(wf = 0.384) relative to dot-size controlled stimuli (wf = 0.246; t
(29)
= 20.6, p< 0.001). Global
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impairment yielded similar discrimination performance in both conditions (area-controlled:
wf = 0.311, dot-size controlled: wf = 0.315; t
(29)
= 0.5, p= 0.59). The unimpaired models showed a
performance comparable to that of our young adult participants in both conditions, with a slight
advantage on the congruent (wf = 0.244) relative to the incongruent (wf = 0.253) stimuli (t
(29)
= 2.1,
p= 0.044). Finally, the inhibition impairment deteriorated performance (relative to that of the
unimpaired models) in the incongruent condition (t
(29)
= 8.6, p< 0.001) but not in the congruent
condition (t
(29)
= 0.2, p= 0.85).
In sum, just as in the elderly human participants, specific impairment of inhibition caused a large
decrease of performance on stimuli in which task-relevant and irrelevant features compete. Con-
versely, an equally strong global impairment caused a decline in performance that was identical across
conditions. Reduced inhibition of irrelevant information is therefore critical to explain the specific pat-
tern of numerosity discrimination performance in elderhood.
5. Discussion
We investigated whether numerical and arithmetical skills may be impaired in healthy ageing. We
first studied whether normally ageing participants had impaired number acuity – which tests a foun-
dational numerical skill measured in terms of Weber fraction, wf – and found that older participants
showed a significantly larger wf relative to young participants. This is in line with the results of Hal-
berda et al.’s internet mega-study (Halberda et al., 2012), and may indicate impaired numerosity pro-
cessing. However, a novel and more detailed analysis showed that this larger wf, i.e. impaired
numerosity processing, could be due to a specific inability to process trials in which numerosity
was incongruent with other measures of continuous quantity (i.e., where the least numerous set con-
tained dots of larger size), whilst congruent trials (i.e., those where the more numerous set had a lar-
ger cumulative area) were performed equally accurately in both populations. This impairment in
incongruent trials did not correspond to difficulties in other numerical or arithmetical problems which
were exceptionally well maintained in older participants.
Whether older participants performed the congruent numerosity trials by processing just numer-
osity or whether they successfully combined information about numerosity with information about
other continuous dimensions like the area covered by the dot stimuli may be difficult to establish. In-
deed, even if numerosity can effectively be extracted in the context of other dimensions such as cumu-
lative area or dot size (e.g., Stoianov & Zorzi, 2012), information from these dimensions can influence
numerosity judgments (Gebuis & Reynvoet, 2012; Hurewitz et al., 2006; Stoianov & Zorzi, submitted
for publication). However, older and younger participants’ wf were very similar in the congruent num-
erosity trials therefore suggesting that either both groups consistently used an area-based strategy to
perform the numerosity task, or alternatively that they both mainly used a numerosity-based strategy.
The first possibility seems unlikely as participants would have had to continuously switch between
Fig. 4. Computational model results. Psychometric functions (PF) indicating the model’s numerosity discrimination perfor-
mance (averaged across networks) for each type of impairment (unimpaired, global and inhibition impaired) and test dataset
(area-controlled vs. size-controlled).
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strategies (e.g. between an area-based and a non area-based strategy), which seems inefficient, and
was not corroborated by differences in response times. Moreover, in older participants this switching
might have resulted in faster response times in the numerosity task with blocked conditions, which
we did not find. The second possibility is that both groups used a numerosity-based strategy which
had to be efficiently combined with inhibitory processes in the numerosity incongruent trials. We sug-
gest that ageing participants may have used a numerosity-based strategy to perform the numerosity
task and that their larger wf can be explained by impoverished inhibitory processes.
This hypothesis was supported by the results of tasks assessing inhibitory processes independent
from numerosity. We found that elderly’s performance in these tasks was significantly different from
younger participants and that it correlated with performance in the incongruent numerosity trials.
Therefore, the more difficult it was to inhibit task-irrelevant information, the worse was the perfor-
mance in the incongruent numerosity trials. We further tested the hypothesis that performance in
incongruent numerosity trials may depend on the integrity of inhibitory processes using a computa-
tional model. This showed that a general degradation of the number system resulted in a larger wf
with no difference between dot-size and area-controlled trials. However, a more specific degradation
of the inhibitory processes resulted in a significantly larger impairment in processing the area-con-
trolled numerosity trials, which resembled our older participants’ performance.
Approximating larger numerosities (i.e. more than 10 elements) or ‘number acuity’ (Halberda et al.,
2008) has been shown to be either well maintained (Gandini et al., 2008; Trick et al., 1996; Watson
et al., 2005) or conversely impaired in older participants (Halberda et al., 2012). However, different
aims or sub-optimal methodological aspects of some of these previous studies make it difficult to
interpret some of the reported results. Similar to these earlier studies, we aimed to investigate the pat-
tern of spared or impaired numeracy skills in elderly participants, but we also sought to provide fur-
ther and more specific information on what exactly lies behind elderly’s number performance. For the
first time, our results of ageing participants’ maintained performance in congruent numerosity trials
but impaired in incongruent ones points at two fundamental processes intrinsic to numerosity dis-
crimination: the abstraction of numerosity in a set and the inhibition of task-irrelevant information.
We suggest that the first of these two processes was maintained in the ageing group as indicated
by an unchanged wf in numerosity congruent trials relative to younger participants and by maintained
arithmetical performance. In contrast, inhibitory processes were partially defective in ageing partici-
pants and we suggest that this is what affected performance in the incongruent numerosity trials.
Such distinction between abstraction and inhibitory processes in numerosity discrimination is likely
to be undetected in younger populations, or even in elderly if the experimental design does not allow
distinguishing between congruent and incongruent trials, or if performance measures do not allow fi-
ner analyses.
Inhibitory processes have so far been suggested to account for elderly’s impaired performance in
several processes, for instance memory and language, based on the ‘Inhibitory Deficit’ Theory (Hasher
et al., 2007; Healey, Campbell, & Hasher, 2008). Here we suggest that these processes may also ac-
count for the impoverished performance we observed in the numerosity trials that required inhibitory
processes in order to be accurately judged. It may also be argued that in ageing participants the
impairment lies at the encoding level (Stoltzfus, Hashe, & Zacks, 1996). That is, older participants
may not have encoded numerosity information properly and this in turn may account for the larger
wf. However, this seems less likely because an encoding problem, which would make information dif-
ficult to retrieve, would have also affected the congruent numerosity trials.
Our novel, combined approach allowed us to avoid the misleading conclusion that ageing affects
numerosity discrimination. Instead, intact number acuity and arithmetical skills in ageing participants
adds to previous observations that despite cognitive decline in some functions, the ageing brain is able
to maintain other skills like vocabulary, syntax and semantic memory (e.g. Hedden & Gabrieli, 2004).
However, differently from these education-related skills whose maintenance may partly be due to
increasing practice with age, undamaged number abilities are likely to reflect the preservation of a
system that is at least partially unrelated to education and language and that is thought to be robustly
embedded in the human brain (Feigenson et al., 2004). This is in line with the evidence that quanti-
fication processes, albeit in a more rudimental form, are already present in children and infants
(Lipton & Spelke, 2003), and are relatively better maintained than other processes following brain
42 M. Cappelletti et al. / Cognitive Psychology 69 (2014) 25–45

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lesions (Cappelletti, Butterworth, & Kopelman, 2012). The idea that numerical and arithmetical skills
are resilient to normal ageing because they rely on a primitive and approximate number system (see
Hasher & Zacks, 1979 for a similar view), is consistent with the idea that primitive systems tend to be
more robust to ageing (Lemaire & Lecacheur, 2007; Trick et al., 1996). This is because primitive
systems tend to be innate or to be acquired earlier in life and this may put them in a stronger position
relative to more complex and recently acquired skills (Trick et al., 1996). It is also possible that the
number system relies on brain areas that are naturally less affected by ageing (Nyberg et al., 2012)
or where decline begins later, a working hypothesis for future studies.
Acknowledgments
This work was supported by a Royal Society Dorothy Hodgkin Fellowship, a Royal Society and a
British Academy Research Grants (M.C), and by Grant No. 210922 from the European Research Council
(M.Z.). We thank Brian Butterworth for his comments on an earlier version of the manuscript.
Appendix A. Supplementary material
Supplementary data associated with this article can be found, in the online version, at http://
dx.doi.org/10.1016/j.cogpsych.2013.11.004.
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