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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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Ma et al. Approximate Number Sense in SHL

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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Ma et al. Approximate Number Sense in SHL

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 SHL’s 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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Ma et al. Approximate Number Sense in SHL

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 SHL’s 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 SHL’s 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 =1−1

2erfc r−1

w√2√1+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 Cronbach’s 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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Ma et al. Approximate Number Sense in SHL

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 1◦either above or below ﬁxation. Each

ﬁsh subtended 0.58◦of 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 = RTno−cue −RTdouble−cue,

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 = RTcenter−cue −RTspatial−cue, 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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Ma et al. Approximate Number Sense in SHL

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

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potential conﬂict of interest.

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