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Approximate Number Sense in Students With Severe Hearing Loss: A Modality-Neutral Cognitive Ability

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The Approximate Number System (ANS) allows humans and non-human animals to estimate large quantities without counting. It is most commonly studied in visual contexts (i.e., with displays containing different numbers of dots), although the ANS may operate on all approximate quantities regardless of modality (e.g., estimating the number of a series of auditory tones). Previous research has shown that there is a link between ANS and mathematics abilities, and that this link is resilient to differences in visual experience ( Kanjlia et al., 2018 ). However, little is known about the function of the ANS and its relationship to mathematics abilities in the absence of other types of sensory input. Here, we investigated the acuity of the ANS and its relationship with mathematics abilities in a group of students from the Sichuan Province in China, half of whom were deaf. We found, consistent with previous research, that ANS acuity improves with age. We found that mathematics ability was predicted by Non-verbal IQ and Inhibitory Control, but not visual working memory capacity or Attention Network efficiencies. Even above and beyond these predictors, ANS ability still accounted for unique variance in mathematics ability. Notably, there was no interaction with hearing, which indicates that the role played by the ANS in explaining mathematics competence is not modulated by hearing capacity. Finally, we found that age, Non-verbal IQ and Visual Working Memory capacity were predictive of ANS performance when controlling for other factors. In fact, although students with hearing loss performed slightly worse than students with normal hearing on the ANS task, hearing was no longer significantly predictive of ANS performance once other factors were taken into account. These results indicate that the ANS is able to develop at a consistent pace with other cognitive abilities in the absence of auditory experience, and that its relationship with mathematics ability is not contingent on sensory input from hearing.
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ORIGINAL RESEARCH
published: 09 June 2021
doi: 10.3389/fnhum.2021.688144
Edited by:
Michele Fornaciai,
International School for Advanced
Studies (SISSA), Italy
Reviewed by:
Ariel Starr,
University of Washington,
United States
Mathieu Guillaume,
Stanford University, United States
*Correspondence:
Justin Halberda
halberda@jhu.edu
These authors have contributed
equally to this work and share first
authorship
Specialty section:
This article was submitted to
Cognitive Neuroscience,
a section of the journal
Frontiers in Human Neuroscience
Received: 30 March 2021
Accepted: 12 May 2021
Published: 09 June 2021
Citation:
Ma H, Bu X, Sanford EM, Zeng T
and Halberda J (2021) Approximate
Number Sense in Students With
Severe Hearing Loss:
A Modality-Neutral Cognitive Ability.
Front. Hum. Neurosci. 15:688144.
doi: 10.3389/fnhum.2021.688144
Approximate Number Sense in
Students With Severe Hearing Loss:
A Modality-Neutral Cognitive Ability
Hailin Ma1,2, Xiaoou Bu2,3, Emily M. Sanford4, Tongao Zeng2and Justin Halberda4*
1College of Education, Shanxi Normal University, Xi’an, China, 2Plateau Brain Science Research Center, Tibet University,
Lhasa, China, 3Faculty of Education, EastChina Normal University, Shanghai, China, 4Department of Psychological
and Brain Sciences, Johns Hopkins University, Baltimore, MD, United States
The Approximate Number System (ANS) allows humans and non-human animals to
estimate large quantities without counting. It is most commonly studied in visual
contexts (i.e., with displays containing different numbers of dots), although the ANS
may operate on all approximate quantities regardless of modality (e.g., estimating the
number of a series of auditory tones). Previous research has shown that there is a link
between ANS and mathematics abilities, and that this link is resilient to differences in
visual experience (Kanjlia et al., 2018). However, little is known about the function of
the ANS and its relationship to mathematics abilities in the absence of other types of
sensory input. Here, we investigated the acuity of the ANS and its relationship with
mathematics abilities in a group of students from the Sichuan Province in China, half
of whom were deaf. We found, consistent with previous research, that ANS acuity
improves with age. We found that mathematics ability was predicted by Non-verbal
IQ and Inhibitory Control, but not visual working memory capacity or Attention Network
efficiencies. Even above and beyond these predictors, ANS ability still accounted for
unique variance in mathematics ability. Notably, there was no interaction with hearing,
which indicates that the role played by the ANS in explaining mathematics competence
is not modulated by hearing capacity. Finally, we found that age, Non-verbal IQ and
Visual Working Memory capacity were predictive of ANS performance when controlling
for other factors. In fact, although students with hearing loss performed slightly worse
than students with normal hearing on the ANS task, hearing was no longer significantly
predictive of ANS performance once other factors were taken into account. These
results indicate that the ANS is able to develop at a consistent pace with other cognitive
abilities in the absence of auditory experience, and that its relationship with mathematics
ability is not contingent on sensory input from hearing.
Keywords: approximate number sense, mathematics, cognition, hearing loss, domain general
INTRODUCTION
Mathematical competence is essential to a wide range of activities in most modern cultures.
Previous studies suggest that math ability predicts a variety of long-term consequences such as
job attainment and success (Rivera-Batiz, 1992), socio-economic status (Ritchie and Bates, 2013),
and health care decisions (Reyna et al., 2009). A wealth of research suggests that individual
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differences in math abilities depend on many factors, including
home learning environment (LeFevre et al., 2010), teacher
characteristics (Klibanoff et al., 2006;Beilock et al., 2010),
and domain-general skills such as IQ, working-memory, and
inhibitory control (Rohde and Thompson, 2007;Gilmore et al.,
2013). Recent evidence suggests that there is also an innate, non-
symbolic sense of quantity that gives rise to our basic numerical
intuitions. A component of this broader number sense emerges
from an evolutionarily and ontogenetically ancient Approximate
Number System (ANS) which is present in human infants in the
first year of life (Xu and Spelke, 2000;Izard et al., 2009;Feigenson
et al., 2013;Starr et al., 2013), as well as exhibited by other
non-verbal populations including monkeys, fish, rats, chicks, and
birds (Feigenson et al., 2004;Agrillo et al., 2012). The ANS is
a mental system of approximate number representations that
is activated during symbolic and non-symbolic number tasks,
which can be modeled by a series of Gaussian curves organized
on a mental number line (Gallistel and Gelman, 1992;Dehaene
et al., 2003;Piazza et al., 2004;Mazzocco et al., 2011a,b). The key
signature of the ANS is that it represents numerical information
in an imprecise way, with the imprecision in its representations
increasing with the numerosity. Indexing this signature, and the
acuity of the ANS, can be captured by a Weber fraction (w)
which varies between individuals, where a smaller Weber fraction
corresponds to higher precision (Pica et al., 2004;Halberda et al.,
2008;Halberda and Feigenson, 2008;Piazza et al., 2010;Libertus
et al., 2011;Odic et al., 2014). In accordance with Weber’s Law,
the difficulty of discriminating two numerosities depends on their
ratio rather than their absolute difference (Piazza et al., 2004). For
example, it is equally difficult to distinguish which of 8 vs. 16 is
larger as it is to distinguish which of 16 vs. 32 is larger. In humans,
ANS acuity increases with age, peaking at around 30 years of age
(Halberda et al., 2012).
Math Achievement and ANS in Students
With Hearing Loss (SHL) and Students
With Normal Hearing (SNH)
Deaf individuals are generally considered to be lagging behind
hearing peers in mathematical tasks across a wide age range
(Ansell and Pagliaro, 2006). They show delays in abstract
counting and scores on standardized tests (e.g., arithmetical
problem solving, logical reasoning, and understanding of
fractional concepts) (e.g., in 2–3.5-year-olds, Pagliaro and
Kritzer, 2010). Mitchell (2008) found that deaf students are
significantly below grade level, exiting high school with about
a 5th–6th grade level of mathematical achievement. Previous
research has shown that because of the impoverished language
environments, their hearing losses and limited access to wide-
ranging numerical experience, many Students with Hearing Loss
(SHL) are deficient in early quantitative concepts (Nunes, 2004;
Kritzer, 2009a;Pagliaro and Kritzer, 2010;Pixner et al., 2014).
Madalena et al. (2015) found that SHL with an early exposure
to a sign language show better performance than those with a
late exposure to the same language. Home environment may
differ between typically hearing families and families with a
SHL. Indeed, typically hearing families increase the probability
of occurrence of informal and natural interactions involving
numerical knowledge unconsciously by questioning, asking for
clarification, or providing additional information in daily life
activities (Kritzer, 2009b;Levine et al., 2010).
The ANS is often assumed to relate to arithmetic performance
throughout childhood, adolescence and the adult years and
current ANS acuity predicts future math ability (Halberda
et al., 2008;Mazzocco et al., 2011b;Libertus et al., 2013).
After controlling for scientific ability, writing ability and
computer proficiency, the correlation between ANS acuity and
mathematical ability of subjects aged 11–85 remained significant
across the lifespan (Halberda et al., 2012). In addition, ANS
acuity contributes to individual differences not only in the general
population, but also in some special groups. Young adults with
William’s Syndrome performed poorly on both symbolic math
and ANS tasks (Libertus et al., 2014), while Wang et al. (2017)
found that ANS acuity was linked to symbolic math performance
in gifted adolescents. Others found that students with specific
math impairment (dyscalculia) performed significantly more
poorly on the ANS task than their typically developing peers;
in other words, less precise ANS representations are related to
difficulty in mathematics broadly (Geary et al., 2008;Piazza et al.,
2010;Bull et al., 2011;Skagerlund and Träff, 2016).
As mentioned above, SHL show a range of mathematical
difficulties but whether the mechanism of this difficulty is the
same as that of students with normal hearing is not known. For
instance, in SHL, it may be that the innate ANS representations
are as precise as their peers, while the mapping between ANS
and more complicated mathematical concepts is delayed due
to reduced access to linguistic and mathematical input. If
the differences found between SHL and SNH in mathematics
performance are due to their differences in experience rather than
a difference in their innate ANS representations, there remains
a question of whether it is due to a general lack of auditory
input, related to delays in access to language or higher-level
math concepts, or due to fundamental differences in information
processing among SHL (Bull, 2008). In the present study, we
aimed to document the potential relationship between hearing
loss (and the many factors that covary with it) and ANS acuity.
ANS and Domain-General Abilities
Because of the potential importance of domain-general abilities
to both formal mathematics success and developing ANS
acuity, we considered multiple examples of such abilities in
the present study.
Inhibition
Inhibition is thought to be important to performance in ANS
tasks. Performance on trials where spatial characteristics can
vary widely, e.g., in stimuli that are congruent or incongruent
with numerical information. For instance, Clayton et al. (2015)
found that people are much more accurate on trials where the
larger set numerically also has the larger convex hull (congruent
trials) than on trials where the opposite is the case (incongruent
trials). Other non-numerical features that can influence responses
on number tasks include surface area, diameter, perimeter, and
density (Dakin et al., 2011;Gebuis and Reynvoet, 2011).
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In order to reduce the extent to which subjects in ANS
experiments can rely on non-numerical cues, one strategy has
been to use two interleaved groups of stimuli: one where the
average dot size is constant across both sets (such that the more
numerous set also has a larger e.g., total dot area), and one where
the total dot area is constant across both sets (such that the more
numerous set has a smaller average dot size; Halberda et al.,
2012). By mixing presentations of trials from these two stimulus
sets together, neither average dot size nor total area is a reliable
predictor of the number of objects throughout the experiment. If
a subject tends to rely on a continuous feature such as dot size,
then they will show different performance on size congruent and
incongruent trials: this feature will help them respond correctly
on congruent trials but will result in worse performance on
incongruent trials, unless they are able to selectively suppress that
signal on incongruent trials. Therefore, in order to consistently
perform well on both congruent and incongruent trials, one must
exert inhibitory control not unlike that required for a Stroop
task. Relative differences in performance between congruent and
incongruent trials may reflect differences in inhibitory control
across subjects. Given that SHL are often reported to have
inhibitory difficulties (Titus, 1995;Traxler, 2000), it may be that
part of the source of SHLs mathematical challenges come from a
relative lack of inhibitory control.
Visual Working Memory
Visual-spatial processing is important for number perception,
possibly because of the important role it plays in the formation
of set representations from visual sets (Paul et al., 2017). In fact,
visual form perception and visual short-term memory have been
found to fully account for the relationship between ANS acuity
and arithmetic performance in some instances (Zhang et al.,
2019). This may be particularly important in young children, who
appear to use visuospatial strategies when performing mental
arithmetic more than older children (McKenzie et al., 2003), and
where visual-spatial short-term memory span increases from 3 to
8 years of age (Pailian et al., 2016), and where visual-spatial short-
term memory span has been found to be selectively predictive of
math success in young children (Bull et al., 2008).
Further support that visual-spatial processing and working
memory are important for number perception comes in the form
of co-occurring challenges with number processing and working
memory. Deficits in visual-spatial working memory have been
found to be associated with numerical magnitude processing
weaknesses in children with mathematical learning disabilities
(Andersson and Östergren, 2012). Children with developmental
dyscalculia have shown math-specific impairments as well as
deficits in visuo-spatial working and short-term memory and
inhibitory control (Szucs et al., 2013). Notably, ANS acuity
differences between typically developing children and children
with developmental dyscalculia have been found to be more
extreme on size-incongruent trials than size-congruent trials.
Because of the role that visual working memory plays in
extracting numerical information from visual scenes, it may
be extremely important to investigate in situations where ANS
acuity varies between populations (Bugden and Ansari, 2015).
Conflicting evidence suggests that visual working memory
capacity cannot fully explain numerical deficits. For instance,
Peng et al. (2017) found that numerical knowledge mediates
the relationship between ANS performance and early arithmetic
abilities, above and beyond that which is explained by visuospatial
processing. Additionally, research on children born extremely
preterm found ANS acuity deficits that were not explainable on
the basis of working memory or attention abilities (Libertus et al.,
2017). Further research is necessary to investigate the extent to
which visual working memory capacity can explain ANS acuity
differences between populations. Given that deaf children have
been shown to have deficits in visual working memory (López-
Crespo et al., 2012), this question is particularly relevant for
the current study.
Attention Network
Numerical processing involves the deployment of attention, more
so for subitizing than for large number processing (Anobile
et al., 2012). In fact, some studies have found that estimation of
large numbers is relatively unaffected by tasks with conflicting
attentional demands (Burr et al., 2010). Nonetheless, spatial
attention has sometimes been found to be involved in ANS task
performance (Anobile et al., 2012).
When studying attention related to other cognitive abilities,
the attentional system is sometimes divided into three separate
components: alerting, orienting, and executive control attention
networks (Fan et al., 2009). The alerting portion refers to
the ability to increase attention at the expected onset of a
new stimulus. The orienting attention network is thought to
explain the ability to select a particular target for attention
among a variety of inputs, whether intentional or through
attention capture. Finally, the executive control network is
thought to detect and resolve conflicts between co-occurring
mental computations.
Attentional network development has been a topic of
particular interest in deaf children (Daza and Phillips-Silver,
2013). The development of the alerting network is thought to
be impaired in the absence of auditory stimulation, while some
components of the orienting attention network are enhanced,
such as moving and engaging. The executive control network
has been found to develop along a similar trajectory to that of
hearing children.
There is known to be a strong relationship between number
processing, math ability, and attention (Anobile et al., 2013).
Like ANS perception, performance on attention tasks has been
found to predict symbolic math achievement in children and was
also predictive of ANS ability (Anobile et al., 2013). Attentional
deficits may be implicated in math-specific disabilities such
as developmental dyscalculia. Therefore, we are interested
in whether similar attentional deficits impact the numerical
processing of SHL, and whether these deficits can be traced to
specific attentional networks.
Summary: Motivations for the Current
Work
Considering the ways in which school mathematics abilities
might be related to the ANS, and vice versa, it is likely
to be a highly interdependent relationship. The Defective
Number Module Hypothesis, perhaps too simply, suggests that
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mathematical deficits may have their roots in innate difficulties
processing non-symbolic number; for instance, an impairment
of the ANS has been proposed as the origin of dyscalculia,
a mathematics-specific learning disability (Butterworth, 2005;
Mazzocco et al., 2011a). Of course, an effect in the opposite
direction might also occur. Such a relationship might be explored
in SHL; not because SHL necessarily have dyscalculia themselves;
but rather, because the reduced exposure of SHL to numerical
concepts in early development may lead to similar problems
(Swanwick et al., 2005). A relatively small number of studies
have investigated the performance of SHL on specific areas of
mathematics (Pagliaro and Kritzer, 2010). The current study
expands this by focusing both on symbolic mathematics and
non-symbolic numerical processing.
In Experiment 1, we tested whether SHLs responses to the
ANS task conform to Weber’s law, and investigated whether their
acuity is affected by size congruency manipulations (e.g., Clayton
et al., 2015). We expected that SHLs ANS responses will follow
Weber’s law, and that they will perform better on size-congruent
than size-incongruent trials – just as SNH.
In Experiment 2, we compared SHL to a population of SNH,
to test whether effects such as size congruency influence ANS
acuity similarly between the two groups. It is possible that
congruency manipulations would be especially detrimental to
SHL, since they may have particular difficulties with inhibition
(Titus, 1995;Traxler, 2000), and inhibition ability is thought
to play an important role in mitigating the influence of size
congruency on number responses (Clayton and Gilmore, 2015;
Norris and Castronovo, 2016). We then explored the extent to
which ANS acuity predicts mathematics ability when taking into
account other factors such as inhibitory control, visual working
memory capacity, and attention network performance. If SHL
perform like other students their age, we would expect to see
Weber fraction uniquely account for mathematics ability, above
and beyond the contributions of these other factors (Halberda
et al., 2008;Chen and Li, 2014;Schneider et al., 2016).
EXPERIMENT 1
Materials and Methods
Participants
One hundred and forty-four students with hearing loss (mean
age = 13.58 years, SD = 2.34, range = 8–18 years; 60 females)
from 6 special education schools participated in the study. All
were enrolled in the third grade to ninth grade. SHL were
prelingually deaf students and exhibited severe (71–90 dB) and
profound hearing loss (>91 dB). All of them were right-handed,
with normal or corrected-to-normal vision and no history of
neurological or psychiatric illness.
ANS Acuity
We administered a version of Panamath (Psychophysical
Assessment of Number-Sense Acuity; www.panamath.org), a
non-symbolic numerical comparison task, to assess the acuity
of children’s ANS. The two spatially intermixed arrays of blue
and yellow dots were presented for 1,200 ms followed by a 200
FIGURE 1 | Accuracy by trial ratio. We found that subjects in Experiment 1
conformed to Weber’s law, where accuracy increases as a function of ratio.
Line represents best-fitting logarithmic relationship between ratio and percent
correct. Gray region represents 95% CI.
ms backward mask, followed by a blank gray screen until the
response was completed. Students were asked to judge whether
more of the dots were blue or yellow. There were between 5 and
21 dots in each array, the ratios were categorized into 4 ratio bins:
1.14, 1.2, 1.33, and 2, with 20 trials in each ratio bin, yielding a
total of 80 trials. To avoid subjects from relying on the cumulative
area of dots, on half of the trials dots were size-confounded, and
on the other half of the trials dots were size-controlled. Notice
that Panamath does not systematically control for all possible
non-numerical cues (e.g., convex hull is only partially controlled
via the total area and dot size manipulations). Our aims were
to test for Weber’s law and (in Experiment 2) to test for the
relationship of ANS acuity to formal math abilities. Our interest
in inhibitory control was test here only by our area manipulation.
Results and Discussion
Overall, subjects in Experiment 1 had relatively high accuracy
on the ANS task (M= 85.8%, SD = 6.2%). We confirmed that
accuracy improved as a logarithmic function of increasing ratio,
as is expected with data conforming to Weber’s law (Dehaene,
2003). We evaluated this by performing a linear regression
predicting subjects’ average accuracy (on both trial types) from
the logarithm of trial ratio. We found that this model significantly
predicted accuracy, β= 0.651, t(574) = 20.55, p<0.001 (see
Figure 1). This result indicates that, among these subjects,
accuracy was dependent upon the trial difficulty as determined
by comparison ratio, consistent with Weber’s law.
Next, we were interested in whether ANS performance
improved with age. We were also interested in whether
performance was better on size-congruent trials than size-
incongruent trials. Both of these effects have been found
repeatedly in previous research on the ANS (e.g., Halberda et al.,
2008, 2012;Clayton et al., 2015;Smets et al., 2015).
We used subjects’ accuracy (for all trials, as well as separately
for size-congruent and size-incongruent trials) to different ratios
(r) to fit their Weber fraction (w) according to the following
psychophysical model, used extensively in previous ANS research
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Ma et al. Approximate Number Sense in SHL
(Pica et al., 2004;Cantlon and Brannon, 2006;Halberda and
Feigenson, 2008;Halberda et al., 2008, 2012;Piazza et al., 2010;
Libertus et al., 2011, 2013, 2014;Odic et al., 2013, 2014;DeWind
et al., 2015;DeWind and Brannon, 2016;Starr et al., 2017;Wang
et al., 2017):
probability correct =11
2erfc r1
w21+r2
The model was fit to each subjects’ data using Maximum
Likelihood Estimation (MLE) in R. Previous research has
indicated that accuracy and response time may index different
abilities (e.g., Halberda et al., 2012), and because we were
interested in the amount of internal noise in our subjects’ number
representations, we focused on using accuracy-based Weber
fractions to test our hypotheses.
In this model, a smaller Weber fraction corresponds to higher
accuracy and therefore better performance. On average, the
subjects in this study had a mean Weber fraction of 0.168, which
is in line with previous research on ANS acuity among 14-
year-olds (the mean age of our participants), who have been
found to have Weber fractions ranging from 0.119 to 0.567
(Halberda et al., 2008).
To evaluate whether performance improved with age, we
performed a linear regression predicting Weber fraction (based
on all trials) from subject age, expecting to see a negative linear
trend (indicating that performance improved with age). Indeed,
that was what we found: increasing age significantly predicted a
decline in Weber fraction, β=0.212, F(1, 142) = 6.69, p= 0.011,
R2= 0.04.
Next, we investigated whether subjects performed differently
on the size-confounded versus size-controlled trials, expecting
that subjects would have higher Weber fractions (i.e., worse
performance) on size-controlled trials than size-confounded
trials. A paired t-test confirmed that subjects had smaller Weber
fractions and therefore performed better on the size-confounded
(M= 0.15, SD = 0.09) than size-controlled (M= 0.19, SD = 0.12)
trials, t(143) = 4.11, p<0.001.
This preliminary study demonstrates our ability to work with
SHL in the relevant schools, and replicates several key findings
from the literature on the ANS.
EXPERIMENT 2
Materials and Methods
Participants
In Experiment 2, we focused on a subgroup of the children
from Experiment 1 and also ran a new group of age-relevant
controls. In order to focus on effects related to symbolic math
development, we relied on the Chinese Rating Scale of Pupil’s
Mathematic Abilities (C-RSPMA; Wu and Li, 2005) which is
normed for children in primary school. For this reason, we
restricted our SHL sample to children in primary school with
complete datasets as well as a new group of control children
with complete data sets. Ninety-seven SHL (Mage = 12.58 years,
SD = 1.95, range = 8–18 years; 38 females) from 6 special
education schools and 97 SNH (Mage = 10.36 years, SD = 1.24,
range = 8–12 years; 47 females) from 1 normal primary school in
Sichuan, China, participated in the study. All were enrolled in the
third grade to sixth grade. The SNH students were approximately
matched to the SHL in grade level (although SHL were on
average older than SNH and had a much wider age range, as
is typical in SHL). SHL were prelingually deaf students and
exhibited severe (71–90 dB) and profound hearing loss (>91
dB). All subjects were right-handed, with normal or corrected-
to-normal vision and no history of neurological or psychiatric
illness. Table 1 shows detailed demographic information on
all participants.
Tasks and Procedure
Chinese rating scale of pupil’s mathematic abilities
The Chinese Rating Scale of Pupil’s Mathematic Abilities (C-
RSPMA; Wu and Li, 2005) based on the Germany Rating
Scale of Pupil’s Mathematic Abilities established by Heidelberg
University was used to assess the primary students’ basic
mathematical competencies.
C-RSPMA is composed of 11 subtests divided into two broad
categories. One category tests mathematics operation such as
addition, subtraction, multiplication, division, blank filling and
comparisons. The other category focuses on skills in spatial
vision and logical thinking, with tasks such as figure writing,
length estimation, block counting, graph counting and figure
connection. For these 11 subtests, students were required to
answer as many items as possible within the stipulated time (1–
3 min, dependent on different subtests). The Cronbachs alpha is
above 0.7, split-half reliability coefficient is 0.83.
Non-verbal IQ
To evaluate children’s non-verbal IQ, we administered the
combined Raven’s Test (CRT-CC3; Wang et al., 2007). This
test contains 72 matrices of increasing difficulty, and a correct
answer yielded one point. Students were required to identify
the missing element that best completes a pattern from six or
eight alternatives.
Inhibition
The Flanker Task was used to measure inhibitory control (Eriksen
and Eriksen, 1974; see Figure 2A). This task measures inhibitory
control by requiring subjects to respond in the direction of a
central arrow while ignoring the sometimes-conflicting direction
of the arrows on either side of it. Each trial started with a
TABLE 1 | Experiment 2 participant demographic information.
SHL SNH
Age (M ±SD) Sex Age(M ±SD) Sex
Group Male Female Male Female
3 grade 11.57 ±1.93 14 9 8.70 ±.56 10 13
4 grade 12.35 ±1.90 15 8 9.84 ±.36 11 11
5 grade 12.14 ±.94 12 10 10.83 ±.39 12 11
6 grade 13.93 ±1.93 18 11 11.72 ±.45 17 12
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FIGURE 2 | The flowchart of the experimental paradigms. (A) The flanker task. (B) Visual delayed match-to-sample task. (C) Attention network task.
fixation cross presented centrally for 500 ms followed by a blank
screen for 500 ms, after which the target and flanking stimuli
appeared. These stimuli were presented for 200 ms followed by a
response window until a response was made up to 1,500 ms later.
A blank screen of 1,500 ms separated each trial. Half of the trials
were congruent (<<<<< or >>>>>), whereas the other half
were incongruent (e.g., <<><< or >><>>). Students were
instructed to respond as accurately and as quickly as possible to
indicate the direction of the centrally presented target arrow by
key press. This task contained a practice block with 12 trials and
two experimental blocks with 60 trials each.
Visual working memory
We used a visual delayed match-to-sample task to measure visual
working memory (Dong et al., 2014;Figure 2B). A fixation
cross was presented for 500 ms followed by a target stimulus,
which was a grid that had some squares highlighted (high load
condition: 4/9 highlighted; low load condition: 2/9 highlighted).
The target stimulus was visible for 1,000 ms, followed by a
blank screen for 2,800 ms (where the participant had to hold
the locations of the target squares in memory). Finally, a probe
stimulus appeared, which consisted of the same grid but with
only one square highlighted. Students indicated by button press
whether or not the probe square appeared in one of the same
locations as was highlighted in the target stimulus. The probe
stimulus was visible for up to 5,000 ms. This task contained
a practice block with 10 trials and one experimental block
with 60 trials.
Attention network
The attention network test is used to measure the efficiency of
the three aspects of attentional networks (i.e., alerting, orienting,
and conflict; Fan et al., 2002;Rueda et al., 2004;Figure 2C).
Each trial began with a fixation presented at the center of the
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Ma et al. Approximate Number Sense in SHL
screen for a random duration between 400 and 1,600 ms, after
which the cue stimulus appeared for 150 ms. Subsequently,
the fixation was again presented for 450 ms followed by a
target stimulus which appeared for a maximum duration of
1,700 ms, followed by feedback for 2,000 ms. Finally, a fixation of
1,000 ms separated each trial. This task consisted of one practice
block of 12 trials and two experimental blocks involving 60
trials each.
The ANT includes four cue conditions (no cue, central
cue, double cue, and spatial cue) and three target conditions
(congruent, incongruent, and neutral). The target stimulus was
a single yellow fish or a horizontal row of five yellow fish which
were presented about 1either above or below fixation. Each
fish subtended 0.58of visual angle and was separated from
neighboring fish by 0.21. The five fish subtended a total of 8.84.
Students were instructed to respond to the direction that a central
fish was facing by button press.
ANS acuity
The procedure to evaluate ANS acuity was the same
as Experiment 1.
Data preparation
The data preparation and Weber fraction modeling for the ANS
results were identical to those used in Experiment 1.
The C-RSPMA was scored following standard protocol to
calculate a Mathematics score for each subject (Wu and Li, 2005).
For the task measuring Non-verbal IQ, the final raw Raven test
scores were converted to a standard IQ score according to the
norm for Chinese children.
For the task measuring Inhibitory Control, we computed a
score based on response time in the Flanker Task. An index
of inhibitory control for each subject was calculated using
the following formula over mean response times in the two
conditions: Score = RTincongruent – RTcongruent. This single value
represents how much longer it took the subject to respond to
incongruent trials than to congruent trials, and therefore a lower
value corresponds to better inhibitory control.
For the visual working memory task, a composite score was
created for working memory performance by combining results
from both accuracy and response time. Across all subjects, we
z-scored average response times on high memory load trials
(correct responses only), average response times on low memory
load trials (correct responses only), average accuracy on high
memory load trials, and average accuracy on low memory load
trials. This resulted in each subject having four values that
indicated how well, relative to other subjects, they performed
on each of these four indices of performance. We then averaged
these four z-scores for each subject to get a single composite score
of performance on the working memory task relative to other
subjects in the sample.
For the Attention Network Task, we calculated a separate
score for the efficiency of the three attentional networks based
on response times to different cue conditions. The efficiency
of three attentional network scores based on the RTs were
calculated using the following formula (see Figure 2C for
cue conditions): Alerting effect = RTnocue RTdoublecue,
FIGURE 3 | Accuracy by trial ratio and group membership. As expected by
Weber’s law, accuracy increased as a function of trial ratio. SNH were, on
average, more accurate than SHL. There was no interaction between these
two variables. Lines represents best-fitting logarithmic relationship between
ratio and percent correct. Shading corresponds to 95% CI.
Orienting effect = RTcentercue RTspatialcue, and Conflict
effect = RTincongruent RTcongruent.
Results
ANS Performance
Once again, our subjects performed fairly well in terms of
accuracy on the ANS task (M= 86.3%, SD = 5.6%). We once
again evaluated whether accuracy was dependent upon trial
ratio to confirm that our results were consistent with Weber’s
law. We used multiple regression predicting accuracy from the
logarithm of trial ratio, group membership (SNH or SHL), and
their interaction. We expected that both groups would show a
significant influence of trial ratio on accuracy, and that there
would be no interaction between the two variables. This was
confirmed: the logarithm of trial ratio significantly predicted
accuracy, β= 0.685, t(772) = 18.25, p<0.001. There was also a
significant effect of group membership, where SNH (M= 87.2%,
SD = 4.6%) had slightly higher accuracy on average than SHL
(M= 85.4%, SD = 6.3%), β= 0.076, t(772) = 2.85, p= 0.004. There
was no interaction between the two, indicating that trial difficulty
impacted both groups the same relative amount, p= 0.579 (see
Figure 3).
Next, we were interested in performance as indexed by model-
fitted Weber fractions. Overall, with one Weber fraction fit to
each subject’s responses to all trials, our subjects had similar
Weber fractions to those we found in Experiment 1 (M= 0.162,
SD = 0.065).
We were interested in whether Weber fraction was affected
by group membership (SHL vs. SNH) and size congruity
(congruent vs. incongruent trials). To test this, we again fit
each subject’s responses with a Weber fraction, separately for
size congruent and incongruent trials. Then we conducted a
two-way ANOVA predicting Weber fraction from hearing and
size congruency, with group membership as a between-subjects
variable and size congruency as a within-subjects variable. Both
main effects were significant. Consistent with the results from
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Ma et al. Approximate Number Sense in SHL
FIGURE 4 | Weber fractions by group membership and size-congruity. Weber
fractions were lower (corresponding to more precise responses) when
subjects were responding to size-congruent trials than when they were
responding to size-incongruent trials. Also, SNH had higher acuity than SHL.
There was no interaction between these two variables. Error bars represent
standard error.
Experiment 1, we found that acuity was better on size-congruent
trials (M= 0.142, SD = 0.080) than on size-incongruent trials
(M= 0.185, SD = 0.100), F(1, 382) = 5.044, p= 0.025, across
the two groups. For group membership, we found that SHL
(M= 0.174, SD = 0.079) had larger Weber fractions than
SNH (M= 0.150, SD = 0.043), F(1, 382) = 7.34, p= 0.007,
indicating that SNH had slightly better acuity. Importantly,
there was no significant interaction between these two factors,
F(1, 382) = 0.29, p= 0.634 (see Figure 4). This indicates
that size-congruency impacted performance equally for subjects
regardless of group membership—which runs counter to the
expectation that difficulty with inhibition would drive especially
worse performance for SHL on size-incongruent trials.
We also verified whether Weber fraction varied with
age in this sample. We investigated this by performing a
linear regression predicting Weber fraction (collapsed across
congruency conditions) from age and group membership.
Consistent with previous research, we found that the model
significantly predicted Weber fractions, F(3, 190) = 7.03,
p<0.001, R2= 0.09 (see Figure 5). Both group membership,
β=0.366, t(190) = 4.24, p<0.001, and age, β=0.293,
t(190) = 2.96, p= 0.003, significantly predicted Weber fractions,
while their interaction was not significant, p= 0.741. Within
both groups, increasing age was linked to decreasing Weber
fractions (meaning older subjects were more precise in their ANS
responses than younger subjects), and the rate of this effect did
not differ between the two groups.
Relationship Between ANS Performance and Other
Tasks
Next, we were interested in the extent to which ANS performance
could be predicted by performance on other related tasks.
We tested each subject in the following domains: Non-verbal
IQ (Raven task), Inhibitory Control (Flanker task), Visual
Working Memory, and Attention Network strength. Using the
data processing techniques described above, this resulted in
FIGURE 5 | Weber fraction by age and group membership. In both groups,
increasing age was linked with decreasing Weber fractions. Line corresponds
to best-fitting linear relationship between Age and Weber fraction for each
group. Shading corresponds to 95% CI.
the creation of the following scores for each subject: Raven
score; Inhibitory Control composite score; Working Memory
composite score; and Conflict, Alerting, and Orienting ANT
scores (for mean scores on each task, see Table 2). We then
predicted Weber fractions from this group of variables, as well as
Hearing group and Age. We included interaction terms between
Hearing group and each other variable to evaluate whether the
pattern of results differed between SHL and SNH.
On the whole, this model explained significant variance in
ANS performance, F(15, 178) = 3.233, p<0.001, R2= 0.148.
We found that Age, β=0.306, p= 0.002, Non-verbal IQ,
β=0.238, p= 0.029, and Visual Working Memory score,
β=0.201, p= 0.046, were each significant predictors when
other variables were taken into account. Score on Conflict
ANT was marginally significant, β=0.150, p= 0.096, and
no other variables were significant, ps>0.116. Increases in
each of these variables corresponded to decreases in Weber
fractions, indicating that students who were older and had
higher Non-verbal IQ, Visual Working Memory capacity, and
scores on the Conflict ANT tended to have better ANS acuity.
Notably, none of the interactions between group membership
and other variables were significant, ps>0.240, indicating that
the relationship between ANS and other task performance was
similar among SHL and SNH.
TABLE 2 | Mean scores on each task by hearing group.
SHL SNH
Mean SD Mean SD
Non-verbal IQ (Raven) 48.60 6.12 53.28 6.04
Inhibitory control 158.05 78.92 165.87 80.29
Visual working memory 0.352 0.79 0.352 0.58
Conflict ANT 131.41 67.63 104.63 55.36
Alerting ANT 29.89 42.77 21.09 39.98
Orienting ANT 1.47 52.90 8.47 47.38
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Interestingly, hearing group membership was no longer
predictive of ANS performance when the other variables
were included, β=0.178, p= 0.116. However, due to
the decreased power associated with the large number of
predictors included in this model, we caution against a strong
interpretation of this result.
Relationship Between ANS and Mathematics
Performance
Finally, we were interested in the degree to which ANS
performance could account for variability in formal mathematics
scores (M= 193.05, SD = 62.06), above and beyond that
which could be accounted for by other related abilities. We
did this by utilizing the suite of predictors tested in the
previous section (Hearing group; Age; Raven score; Inhibitory
Control composite score; Working Memory composite score;
and Conflict, Alerting, and Orienting ANT scores), and used
linear regression to determine whether ANS performance
predicted Mathematics performance once these variables were
taken into account. As in the previous section, the only
interactions included in this model were between hearing group
membership and each other variable, to determine whether
these variables had different explanatory power among SHL
compared to SNH.
For our first model, we regressed Mathematics score over the
suite of these predictor variables, excluding ANS performance.
This model significantly predicted Mathematics ability, F(15,
178) = 27.99, p<0.001, R2= 0.68 (see Table 3 for standardized
coefficients). Of the predictors, only the Attentional Network
scores did not significantly explain some variance in Mathematics
ability; group membership, Age, Non-verbal IQ, Inhibitory
Control, and Visual Working Memory capacity all contributed
to explaining Mathematics performance. SNH (M= 232.88,
SD = 41.88) had significantly higher Mathematics scores on
average than SHL (M= 153.22, SD = 52.76). Increasing
Age, Non-verbal IQ, and Visual Working Memory capacity
corresponded to increases in Mathematics Score. Interestingly,
an increase in Inhibitory Control score corresponded to a
decrease in Mathematics score. No interactions with group
membership were significant, ps>0.191, indicating that the
TABLE 3 | Standardized coefficients from regressions predicting
mathematics score.
Predictor Model 1 βModel 2 β
Hearing group 0.559*** 0.540***
Age 0.149* 0.102.
Non-verbal IQ 0.395*** 0.359***
Inhibitory control 0.125* 0.122*
Visual working memory 0.146* 0.115.
Conflict ANT 0.078 0.101.
Alerting ANT 0.071 0.078
Orienting ANT 0.040 0.035
Weber fraction 0.154**
p<0.1, *p <0.05, **p <0.01, ***p <0.001.
Model 1β, the first model beta-coefficient; Model 2β, the second model beta-
coefficient.
influence of each variable on Mathematics score was similar
for both groups.
We then compared this model to a second model that
included the same predictors and additionally included ANS
performance as indexed by Weber fraction. This model also
explained a significant amount of variance in Mathematics score,
F(17, 176) = 26.28, p<0.001, R2= 0.69. Weber fractions were
significantly predictive of Mathematics score even when other
variables were taken into account, t(176) = 3.03, p= 0.003. Once
Weber fraction was added to the model, Age, Visual Working
Memory capacity and Conflict ANT score became marginally
significant predictors of variance in Mathematics score (likely
due to the shared variance between these predictors and ANS
performance found in the previous section). Hearing group,
Inhibitory Control and Non-verbal IQ remained significant
predictors (see Figure 6 for the individual relationship between
each predictor and Mathematics score). As in the previous model,
there was no interaction between group membership and any of
the other predictors, ps>0.111.
We then checked that the second model explained
significantly more variance than the first model, to confirm
that ANS task performance explained additional variance in our
subjects’ Mathematics scores. An ANOVA comparing these two
models significantly favored the second model, and therefore the
inclusion of ANS task performance, over the first model, F(2,
176) = 4.70, p= 0.010. ANS ability uniquely explained variance
in Mathematics score beyond that which was explained by other
predictors, and did so similarly for both SHL and SNH.
DISCUSSION
To summarize our results, we found that students with hearing
loss (SHL) had lower ANS acuity than control subjects (SNH)—
even though SHL tended to be a bit older. The magnitude of
this effect was decreased when other factors were taken into
account (such as Non-verbal IQ and Visual Working Memory
capacity), indicating that the difference in ANS performance
that we observed may be at least partially due to other factors
that tend to vary between these groups, rather than due solely
to the imprecision of the ANS representations themselves.
All students showed a tendency to perform better on size-
confounded than size-controlled trials, consistent with a role for
inhibitory control. But, a specific role for reduced inhibitory
control to drive especially low ANS acuity in SHL did not
bear out. We found that many factors contributed uniquely to
performance on the Math test, and most importantly, even when
taking these other potential contributing factors into account,
the precision of the ANS (Weber fraction) still accounted for
significant variance in Math score. Therefore, we conclude that
the ANS’s contribution to Math ability in children goes above
and beyond that which can be accounted for by other measures
such as Inhibitory Control, Working Memory capacity, and
Attention Network performance, and, to the extent that we find
unique variance between ANS and symbolic math ability above
and beyond these factors, these abilities may play only a minor
role in modulating the link between the ANS and symbolic
math ability.
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Ma et al. Approximate Number Sense in SHL
FIGURE 6 | Predicting mathematics score. When other variables were taken into account, only group, Non-verbal IQ, Inhibitory Control, and Weber fraction
significantly predicted Mathematics score.
The present study adds further support for the claim that
ANS abilities relate to school math abilities in children, consistent
with previous meta-analyses on the topic (Chen and Li, 2014;
Schneider et al., 2016). Here, we observed this effect in two large
samples of students, controlling for many relevant factors. We
also saw that this effect is important both for typically developing
children and students with hearing loss (SHL). That we saw
accuracy patterns consistent with Weber’s law in our SHL (and
only a small difference in Weber fraction between SHL and SNH
when controlling for other factors) suggests that the ANS is able
to develop somewhat normally in the absence of auditory input.
SHL tended to have lower scores on many of the facilities tested
in the present studies, which raises the possibility that the ANS
deficits we saw are not specific deficits, but rather due to general
developmental challenges that arise for deaf children, such as late-
onset language exposure or reduced access to early mathematics
education (Swanwick et al., 2005;Bull, 2008).
Combining this with the existing result of normal functioning
of the ANS in blind participants (Kanjlia et al., 2018) supports
the suggestion that the ANS is a domain general cognitive
system with representations that abstract away from any
particular modal signal. Although size-congruency influenced
ANS performance in this sample (and convex hull was not
controlled for), given that ANS representations develop in
individuals with vastly different sensory experiences, we argue
that the content of these shared representations must be
something that is preserved across modalities (see also Halberda,
2019). That is, if the ANS is able to develop in both blind
individuals and SHL, and given that links between the ANS and
math ability are observed in both populations, it appears that the
ANS abstracts away from particular modal content. Nonetheless,
the mechanism underlying congruency effects, and whether they
occur at the extraction or response stage, remains a fruitful path
for future study.
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Ma et al. Approximate Number Sense in SHL
As with many previous demonstrations, the present results
suggest a picture of the ANS as a domain general cognitive system
that supports non-symbolic numerical intuitions and relates to
symbolic math abilities.
DATA AVAILABILITY STATEMENT
The raw data supporting the conclusions of this article will be
made available by the authors, without undue reservation.
ETHICS STATEMENT
The studies involving human participants were reviewed and
approved by the Ethics Committee of Tibet University. Written
informed consent to participate in this study was provided by the
participants’ legal guardian/next of kin.
AUTHOR CONTRIBUTIONS
XB and HM: study concept and design. XB, ES, and
TZ: acquisition and analysis or interpretation of data.
XB, ES, JH, and HM: drafting of the manuscript. HM
and JH: obtained funding. ES, XB, JH, and HM: critical
revision of the manuscript for important intellectual content.
All authors contributed to the article and approved the
submitted version.
FUNDING
This work was supported by the Educational Scientific
Research Projects of Tibet (XZJKY19104) and a McDonnell
Scholar Award to JH. This material is also based upon work
supported by the National Science Foundation Graduate
Research Fellowship under grant no. DGE1746891.
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Conflict of Interest: The authors declare that the research was conducted in the
absence of any commercial or financial relationships that could be construed as a
potential conflict of interest.
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