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Developmental Science. 2018;e12645. wileyonlinelibrary.com/journal/desc
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https://doi.org/10.1111/desc.12645
© 2018 John Wiley & Sons Ltd
Received:3March2017
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Accepted:19October2017
DOI:10.1111/desc.12645
PAPER
Young children’s non- numerical ordering ability at the start of
formal education longitudinally predicts their symbolic number
skills and academic achievement in maths
Patrick A. O’Connor | Kinga Morsanyi | Teresa McCormack
School of Psychology, Queen’s University
Belfast, Belfast, UK
Correspondence
PatrickA.O’Connor,Queen’sUniversity
Belfast, School of Psychology, David Keir
Building,MaloneRoad,Belfast,BT95AG,UK
Email: poconnor08@qub.ac.uk
Abstract
Ordinalityisafundamentalfeatureofnumbersandrecentstudieshavehighlighted
the role that number ordering abilities play in mathematical development (e.g., Lyons
et al., 2014), as well as mature mathematical performance (e.g., Lyons & Beilock,
2011). The current study tested the novel hypothesis that non- numerical ordering
ability, as measured by the ordering of familiar sequences of events, also plays an
important role in maths development. Ninety children were tested in their first
school year and 87 were followed up at the end of their second school year, to test
the hypothesis that ordinal processing, including the ordering of non- numerical ma-
terials, would be related to their maths skills both cross- sectionally and longitudi-
nally. The results confirmed this hypothesis. Ordinal processing measures were
significantly related to maths both cross- sectionally and longitudinally, and children’s
non- numerical ordering ability in their first year of school (as measured by order
judgements for everyday events and the parents’ report of their child’s everyday
ordering ability) was the strongest longitudinal predictor of maths one year later,
when compared to several measures that are traditionally considered to be impor-
tantpredictorsofearlymathsdevelopment.Children’severydayorderingability,as
reported by parents, also significantly predicted growth in formal maths ability be-
tween Year 1 and Year 2, although this was not the case for the event ordering task.
The present study provides strong evidence that domain- general ordering abilities
play an important role in the development of children’s maths skills at the beginning
of formal education.
RESEARCH HIGHLIGHTS
• Numerical and non-numerical ordering ability related to formal
maths skills concurrently and longitudinally.
• Non-numerical ordering abilities in the first year of school were the
strongest predictors of maths one year later.
• The study highlights the importance of domain-general ordering
abilities to the early development of formal maths skills.
1 | INTRODUCTION
The relations between order processing abilities and the development
of maths skills have recently attracted the interest of researchers.
Lyons and Beilock (2011) proposed that representing and process-
ing the relative order of numbers is a stepping stone in moving from
approximate representations of number to exact representations.
Separately, other researchers (e.g., Attout, Noël, & Majerus, 2014;
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Attout & Majerus, 2015) have proposed that working memory for
order information is important for early mathematics development.
Ordinalityis a fundamental aspect of the symbolic number sys-
tem, referring to the position in which a numeral is found within the
number sequence. One proposal is that performance in tasks that
tap children’s ability to process symbolic order reflects the extent
to which they have a refined spatialized representation of the num-
ber sequence along a mental number line (Kaufman, Vogel, Starke,
Kremser,&Schocke,2009).However,thissuggestiondoesnotexplain
why performance on non- numerical working memory tasks, which
involve temporarily holding short non- numerical sequences in short-
termmemory,isrelatedtomathsabilities(Attoutetal.,2014;Attout
& Majerus, 2015). Existing findings suggest that the representation of
the ordered number sequence in long- term memory and the ability to
hold and process unfamiliar order information in short- term memory
are both important for maths.
We believe that ordering skills and mathematics might be related
for multiple reasons. Most relevant to young children is the fact that
learningto count involveslearning an orderedsequenceof items.In
addition, even the simplest counting principles (Gelman & Gallistel,
1978),such as the stable order principle (i.e., numerals always have
the same order in a count), and the cardinal principle (i.e., the numeral
applied to the last item in a set represents the number of items in the
set)involvereferencetoordinality.NiederandDehaene(2009)argue
that it is difficult to envisage how children could acquire knowledge
of the symbolic number system, beyond rote learning or other com-
pensatory strategies, if they do not understand the correct order in
which the numbers are arranged. Successful arithmetic performance
is dependent upon both knowledge of the correct order of the num-
bers, and an understanding of the correct order in which mathematical
operations should be carried out. For example, if children are asked
tosolvetheproblem“5−2=?”,toarriveatthecorrectsolutionthey
must understand that they should take 2 away from 5, rather than
vice versa. Thus, calculation itself depends upon temporarily holding
order information in working memory. Processing order information is
alsoessentialforworkingwithmulti-digit numbers.Itcanbe argued,
therefore, that mental representations of order may play a role in the
development of both basic symbolic number knowledge and subse-
quent maths ability, and recent evidence suggests that there is indeed
a relationship between the processing of numerical order relations and
maths achievement in both children and adults.
The most widely used task to assess symbolic ordering ability is the
ordinaljudgementtask(e.g., Goffin&Ansari,2016;Lyons&Beilock,
2011). Participants are shown three numbers on the screen (half of
the pairs or triads are in the correct order, the other half are in the
incorrect order) and they must judge whether the numbers are in the
correctascendingorder,fromlefttoright.Ataskdevelopedtoassess
non- numerical order processing skills is the order working memory
(WM)task(e.g.,Attout&Majerus,2015).Inthistask,participantshear
lists of familiar animal names. The lists range from two to seven ani-
mals in length, and participants must re- create the correct sequence
of animals using cards that represent the animals in the list that they
have just heard. Importantly, the cards given to participants inform
them about both the identity and the number of animals within the list.
Thus, the task makes minimal demands on item memory; participants
mustonlyremembertheorderofitems.Aswillnowbedescribed,sev-
eral studies have indicated that performance on both these types of
order processing tasks is linked to maths ability, suggesting that both
numerical and non- numerical ordering ability may be important for
formal maths skills.
Inalargestudyofchildrenacrossschoolgrades1–6,Lyons,Price,
Vaessen, Blomert, andAnsari (2014) investigated the role of basic
number skills in the development of maths ability. The authors used a
wide range of numerical and non- numerical tasks to investigate what
skills were important for maths at different developmental stages.
They found that the predictive power of numerical ordering ability
(i.e.,the ordinaljudgementtask)increasedacross grades.Attheear-
liest grades, numerical ordering was not a strong predictor of maths,
but by grade 6 (around the age of 12), it was the strongest of all the
predictors.Anotherpaper(Vogel,Remark,&Ansari,2015)reportedno
relationship between distance effects in number ordering and first-
graders’(aroundage6–7)mathematicsperformance.However,Vogel
et al.’s ordering task only contained dyads of numbers, rather than the
triads that are more commonly used in this literature, and it is possible
that the dyad task is less sensitive at detecting the appropriate order
processingskills(althoughseeAttout&Majerus,2015).Overall,these
studies suggest that symbolic ordering ability is important to children’s
maths skills, although the strength of this relationship might change
with development.
Attout etal. (2014) investigatedthe links between verbal WM
abilities (non- numerical item and order WM), numerical magnitude
and order processing abilities and calculation performance at three
different time points: 6 months into the final year of kindergarten
(T1), one year later (T2) and during the second grade of school (T3).
Attoutetal.foundthattheonlyrelationshipbetweenchildren’snu-
merical ordinal judgement and maths was observed cross- sectionally
atT2. Ontheotherhand, children’sperformancein the orderWM
task was cross- sectionally related to maths at each time point, whilst
performance on this task at T1 was longitudinally related to maths at
T2 and T3, suggesting the importance of early non- numerical order
memory to later maths performance. These relationships remained
significant, even after controlling for age, verbal and non- verbal
intelligence.
Arelationshipbetweenorderprocessingandmathshasbeenfound
not only in studies involving typically developing children, but also in
studies involving children with developmental dyscalculia (DD)—a
developmental disorder characterized by difficulties in the retrieval
and storage of arithmetic facts, when no other sensory or intellectual
disabilitiesarepresent (e.g., Butterworth, 2005;vonAster & Shalev,
2007). Attout and Majerus (2015) investigated symbolic and non-
symbolic magnitude and order processing in 8- to 12- year- old children
with DD and a group of typically developing children matched on age,
IQ and reading abilities.The children were given the orderworking
memory task, as well as a calculation task, symbolic and non- symbolic
ordinal judgement tasks (judging whether two sets of lines or numer-
als were in the correct ascending order numerically) and symbolic and
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O’CONNOR et al.
non- symbolic magnitude judgement tasks (judging which of two sets
of lines or numerals was the most numerous). Attout and Majerus
found that the DD group tended to be slower on symbolic magni-
tude and ordering tasks and committed more recall errors in the order
working memory task, suggesting that children with DD may have dif-
ficulties in processing and remembering order information.
Together, the evidence suggests that both numerical and non-
numerical ordering abilities are important to the development of
typical maths skills, and that children with DD have order process-
ing deficits. Whilst the evidence is promising, there are still several
important unresolved issues concerning the link between order pro-
cessingskillsandmaths.Inparticular,we do not know the precise
nature of the order processing skills that are important for maths
development. Two quite distinct types of order processing tasks—
the numerical ordinal judgement task and the order working mem-
ory task—have each shown a link with children’s mathematical skills.
Notably,Attout etal. (2014) found that children’sperformance on
these two types of ordering tasks was not correlated (although see
Attout&Majerus,2015);performanceonthetasksalsoshowedquite
different patterns of cross- sectional and longitudinal relations with
maths skills. This suggests that they draw on different order process-
ingskillsandarerelatedtomathsskillsfordifferentreasons.Indeed,
these tasks differ in two salient respects: (i) in terms of whether they
involve processing of numerical or non- numerical order information
and (ii) in terms of whether they involve retrieving and processing
information from order representations held in long- term memory
versus unfamiliar sequences temporarily held in short- term memory.
Attoutetal.(2014,p.1676)suggestthat“orderWMabilitiespredict
calculation abilities not via access to a common set of (long- term)
ordinal representations but via mechanisms intrinsically associated
withshort-termstoragecapacitiesoforderinformation”.Whatisnot
clear is whether such short- term memory mechanisms are the only
domain- general order processing ones that are important for maths
development, because previous studies with children have not used
tasks involving long- term ordinal representations of non- numerical
information.
Lyons, Vogel, andAnsari (2016), in their review of the literature
examining the links between ordinality and mathematical skills, argue
that there is a paucity of research investigating the relation between
non- numerical ordering abilities and maths. Recent studies with adults
(Morsanyi,O’Mahony,&McCormack,2017; Sasanguie, De Smedt &
Reynvoet,2017;Vos,Sasanguie, Gevers,& Reynvoet,2017) showed
that non- numerical order processing, as measured by month and letter
ordering tasks that required participants to make judgements about
the order of month/letter triads, was very strongly related to adults’
numerical skills, and the distance effects found in these tasks were
also similar to the distance effects found in number ordering tasks.
Thus, the ordering of familiar non- numerical sequences is also related
to maths ability, atleast in adults. In order to investigate this issue
developmentally, in the current study we included tasks that measured
ordering ability involving familiar, non- numerical sequences.
We investigated the ability to process order information regard-
ing familiar non- numerical sequences held in long- term memory by
introducing two measures that have not been used previously. First,
a temporal ordering task, inspired by previous research with young
children (Friedman, 1977, 1990)was employed. The version of the
task that we developed is similar to the number ordering tasks used in
other studies (e.g., Lyons & Beilock, 2011; Lyons et al., 2014), except
that children were shown a pictorial representation of a triad of daily
events rather than numbers. Each test trial was drawn from a set of six
events (waking up, getting dressed, going to school, eating lunch, eat-
ing dinner and going to bed) and children judged whether the order of
the events was correct or not. Second, to assess the role of everyday
non- numerical ordering skills, we developed a new eight- item ques-
tionnaire to assess the extent to which parents agreed or disagreed
that their child could carry out familiar tasks that all included the re-
quirement to follow a set order (such as getting dressed for school).
Our motivationfor using this measure was the existence of clinical
reports of individuals with DD that describe how they often struggle
with everyday tasks that have a strong ordering component (National
CenterforLearningDisabilities,2007).Together,thesetasksprovided
us with a novel way of assessing the relation between domain- general
order processing abilities and emerging maths skills.
In addition to the question of what types of order processing
skills are related to maths at the start of formal education, it is also
of concern that there is a lack of longitudinal research investigating
whether there may be a causal relationship between ordering ability
and the early development of maths skills. This is echoed by Lyons
et al. (2016), who point out that most of the findings concerning the
link between ordering abilities and maths have been based on correla-
tional evidence at a single time point. The only longitudinal study so
farwasconducted byAttoutetal. (2014)who foundseparatecross-
sectional links between both numerical ordering and non- numerical
order working memory and maths, but only a longitudinal link between
order working memory and maths. We employed a longitudinal design
that involved children completing a range of tasks at the very start of
their formal education, and then measuring their formal maths skills
towards the end of their first and second year of school.
We studied children in their earliest years of education to address
a further issue arising from the previous literature concerning the
stage of development at which ordering ability becomes an important
predictorofmathsskills.Studies(e.g.,Attout&Majerus,2015; Lyons
etal., 2014; Morsanyi, Devine, Nobes,& Szűcs, 2013) have consis-
tently shown that order processing is strongly related to maths skills
amongstolderchildren(betweentheagesof8and13). However,as
mentioned above, there are mixed findings regarding whether there
is a strong link between ordering abilities and maths at the start of
formaleducation (Attoutetal.,2014; Lyonsetal.,2014;Vogeletal.,
2015), with Lyons et al.’s (2014) finding that this relation only becomes
pronounced with development. The children in the current study were
between the ages of 4 and 5 when they first participated in the study,
which makes them the youngest sample so far in which the link be-
tween order processing skills and maths ability has been investigated.
It was conductedwith a sample of children from Northern Ireland;
NorthernIrelandhastheyoungestschoolstartingage(4yearsold)of
all the 37 countries participating in Eurydice, the information network
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on education in Europe (Eurydice at NFER, 2012), and one of the
youngest school starting ages in the world.
Finally, it is also important to compare the predictive value of or-
dering tasks with other tasks that are related to mathematical skills
(see e.g., Attoutetal., 2014; Lyons etal., 2014;Vogel etal., 2015).
Giventhe amount ofresearchinterest inwhetherthe ability topro-
cessmagnitudes is related to maths (e.g.,Chen&Li,2014;Gilmore,
McCarthy,& Spelke,2010; Halberda,Mazzocco& Feigenson, 2008;
Holloway&Ansari,2009; Piazzaetal.,2010; Schneideretal., 2017),
the current study included both symbolic and non- symbolic magni-
tude measures.
Insum, theaimofthe currentstudywasto assess therelative
contributions of numerical and non- numerical order processing to
the development of maths skills in children who have just begun for-
malmathsinstruction. Inalongitudinalstudy,childrenweretested
during their first year of primary school and completed a maths as-
sessment at the end of the school year. The same children com-
pleted another maths assessment at the end of their second year
of primary school. The main research question concerned whether
numerical and non- numerical ordering abilities predicted variance in
mathematical skills both cross- sectionally and longitudinally, after
other powerful predictors of early mathematical skills, as well as
children’s verbal and non- verbal intelligence, were taken into ac-
count.Inaddition,thecurrentstudywasthefirsttoinvestigatethe
link between non- numerical ordering tasks including familiar and
everyday sequences and maths performance at the start of formal
education.
2 | METHOD
2.1 | Participants
Ninety children at the start of their first year of primary school educa-
tion were recruited from four schools in the Belfast area (43 females,
Mean age = 4 years 11 months; SD = 3.73 months). Eighty-seven
childrencompletedthemathsassessment(43females,Meanage=6
years 2 months, SD=3.44months)attheendoftheirsecondschool
year.DuetothedemographicsofthepopulationinNorthernIreland,
thevastmajorityofchildrenwereofCaucasianorigin;informationon
their SES is reported below.
2.2 | Materials
2.2.1 | Deprivation measure
Children’s level of socioeconomic deprivation was determined
usingtheNorthernIrelandMultipleDeprivationMeasure(Northern
IrelandStatisticsandResearchAgency,2010).Thismeasureassigns
adeprivationscoretoeachelectoralwardinNorthernIrelandbased
onavarietyofindices.Ahigherscoreindicatesahigherlevelofdep-
rivation for the area. The scores can be interpreted as percentiles
(e.g.,a score of 10 means that thearea is less deprived than 90%
ofallpostcode-basedareaswithin NorthernIreland). Inthe current
sample, deprivation scores ranged from 1.85 to 68.57 (Median =
11.00).Onechilddidnotprovideapostcode,soadeprivationscore
couldnot be calculated. Along with age and both verbal and non-
verbal intelligence, children’s deprivation scores were used as co-
variates in the data analysis.
2.2.2 | IQ
Children’s intelligence was measured using the Vocabulary and
Block Design subtests of the Wechsler Preschool & Primary Scale
of Intelligence – Third UK Edition (WPPSI-III UK; Wechsler, 2003).
Children’sestimatedfull-scaleIQscoreswerecomputedfollowingthe
method outlined in Sattler and Dumont (2004) and were found to be
withinthenormalrange(MeanIQscore=95.92,SD=13.51).
2.2.3 | Order processing measures
Parental Order Processing Questionnaire (OPQ)
Parents were asked to complete an eight- item questionnaire (included
intheAppendix)inwhichtheyindicatedona7-pointLikertscalethe
extent to which they agreed or disagreed with certain statements re-
garding their child’s ability to perform everyday tasks that involved
an order processing element (e.g., “my son/daughter can easily recall
the order in which past events happened”). The items were devel-
oped based on clinical observations regarding the everyday difficul-
ties that individuals with dyscalculia commonly encounter (National
CenterforLearning Disabilities,2007),buttheyweremodifiedto be
appropriate for young children. Five items were scored positively (i.e.,
higher scores indicated better ordering ability), and three items were
scorednegatively.Aprincipalcomponentanalysiswithvarimaxrota-
tion showed that the scale had a two- factor structure, with the posi-
tiveitemsloadingonfactor1(whichexplained 41%ofthevariance),
and the negative items loading on factor 2 (which explained 21%
of the variance). The scale demonstrated good internal consistency
(Cronbach’salpha=.75).Thetotal scorefrom thisscalewasusedas
a measure of children’s ability to carry out everyday tasks requiring a
long- term memory representation of the correct order of sequences.
Five parents did not complete the questionnaire, so no score could be
computed on this measure for their children.
Order working memory (WM) task
This task measured children’s ability to retain serial order infor-
mation. The English version was modelled on a task developed by
Majerusandcolleagues(Attout&Majerus,2015;Attoutetal.,2014;
Majerus,Poncelet, Greffe,&Van der Linden,2006).This taskmeas-
ures children’s ability to retain and manipulate serial order informa-
tion by measuring their ability to re- create the correct sequence of
a list of animal names that were presented to them through a set of
earphones, using cards depicting the animals. The stimuli used were
seven monosyllabic English animal words (bear, bird, cat, dog, fish,
horse, and sheep). The mean lexical frequencies of these words were
established using SUBTLEX- UK word frequencies (SUBTLEX- UK:
Van Heuven, Mandera, Keuleers, & Brysbaert, 2014). SUBTLEX-UK
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presents word frequencies as Zipf values, with values between 1 and
3 representing low frequency words and values between 4 and 7 rep-
resenting high frequency words. The stimuli demonstrated high lexical
frequencyaccording to thesevalues(mean lexicalfrequency=4.94,
range = 4.67–5.19). The stimuli were used to create 24 word lists,
which ranged in length from two to seven words, with four trials per
list length. Each word only appeared once per list and the same 24
lists were presented to all participants. The stimuli were recorded by
a female voice, and an inter- stimulus interval of 650 ms was used.
Meanitemdurationwas565ms(range=407–674ms).Foreachcor-
rectly recalled sequence, children were given a score of 1. Split- half
reliability estimates, using the Spearman- Brown formula, indicated
good reliability (r=.93).
Daily events task
AmodifiedversionofFriedman’s(1990)temporalorderingtaskwas
used to measure children’s ability to judge the correctness of the
order of familiar daily events. Children were first trained on how
to order events using two training sequences (four cards show-
ing a boy playing on a slide, and six cards depicting a sequence in
whichaboypickedupandopenedapresent).Childrenhadtocor-
rectly order both sequences four times before they could proceed
to the next phase of the training, which involved the items of the
experimental sequence. The experimental sequence consisted of six
cards that represented six familiar events that happen during the
day (waking up, getting dressed, going to school, eating lunch, eat-
ing dinner and going to bed). For the training phase, children were
first told what each picture represented and were shown the cor-
rect order by the experimenter. Then the cards were shuffled and
children were asked to recreate the correct order. For the experi-
mental sequence, children learned the names for each of the daily
events and saw the correct order in which these events should go.
Afterthis,childrenweregivenacomputer-basedtaskinwhichthey
were told that they would see any three of the daily events and that
their task was to judge whether the order was correct or not, from
right to left, by pressing a tick or a cross on the touchscreen moni-
tor.Halfofthe24trials(therewere12setsthatwerepresented
twice) showed a triad of events in the correct order, the other half
showedatriadthatwasintheincorrectorder.Childrenweregiven
a score of 1 for each correct answer and a measure of children’s re-
action times, for correct trials only, was also taken. Since each trial
was presented twice, a split- half reliability was calculated using the
Spearman- Brown coefficient, which was found to be adequate (.57).
Due to the relatively high error rate, reliability for RTs for correct
trials was not computed, and the RT measure was not considered
further.
Symbolic number ordering1
This task assessed children’s early knowledge of the order of symbolic
numbers.Childrenwereshownthecorrectsequenceofthenumbers
1–9usingcards. These cards were then shuffled andchildrenwere
asked to re- create the correct forward order (involving two trials). This
procedure was then repeated for the backward sequence of numbers
(twotrials).Intwosubtasks,childrenalsoorderedthenumbersfor-
wards (four trials) and backwards (four trials) from different starting
positions, with a score of 1 given for each correct trial. The proportion
of correct responses was calculated based on performance on all four
oftheorderingtasks.Areliabilityestimateforthetotalscorewashigh
(Cronbach’salpha=.93).
Counting
This task was based on the number sequence elaboration task, as out-
linedinHannulaand Lehtinen(2005).Inthefirst part,childrenwere
asked to count from 1 until the highest number they could think of
(theywerestopped if they reached 50) in two trials.Intwo further
subtasks, children also counted forwards and backwards from differ-
entstartingpoints.Childrencouldcorrectthemselvesonceduringany
trial. The reliability estimate for both forward and backward subtasks
combinedwasgood(Cronbach’salpha=.77).
Giventhestrong correlationbetweencounting until the highest
number and both forward (r(88)=.76,p < .001) and backward count-
ing (r(88) = .65, p < .001), a total counting score was calculated by
adding z- scores for all three counting measures.
2.2.4 | Magnitude processing measures
Non- symbolic addition2
This task measured the ability to represent and manipulate non-
symbolic quantities and was based on the procedure used by
Gilmoreetal. (2010), in whichchildrenviewtwo sets of bluedots
or“marbles”thatacharacterhad,whichappearoneaftertheother
on the left- hand side of the screen, and have to estimate the sum
of the two arrays (sum array) and compare that sum to the quantity
of a third array (comparison array, composed of red dots) that a dif-
ferent character had, which appeared on the right- hand side of the
screen. The numerical ratio of the sum and comparison arrays was
manipulated across the 24 trials (1:2, 3:5, and 2:3), with eight trials
per ratio. The number of dots for both arrays varied from 6 to 45; 6
being the lowest number of dots as this reduced the possibility that
children could subitize the number of dots presented. Perceptual
variables (dot size, density and array size) were also varied, so that
they correlated with numerosity on half the trials (congruent trials)
and were uncorrelated on the other half of the trials (incongruent
trials), reducing the possibility that children may have used percep-
tual information as a cue when judging which array was the most
numerous. Furthermore, the trials were designed in such a way that
it was not possible for the children to perform above chance if they
simply responded on the basis of a comparison between the num-
berofbluedots in the second set and the number of red dots. In
each trial the number of red dots was at least 1.5 times greater than
the number of blue dots in the second set. Nevertheless, the overall
number of blue dots was larger in half of the trials than the overall
number of red dots, whereas in the other half of trials the opposite
wastrue.Inthetask,childrenhadtopressoneoftwobuttonsonthe
touchscreen to indicate which character they thought had the most
marbles. They completed four practice trials, with feedback given
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ontheir performance, followed by24experimentaltrials. Children
were given a score of 1 if they correctly judged which character
had the most marbles. Reliability for this task for accuracy was quite
low, but acceptable (Cronbach’s alpha = .50). One-sample t tests
confirmed that children performed above chance at each ratio [1:2;
t(89)=4.45,p < .001. 3:5; t(89)=3.76,p < .001. 2:3 t(89)=2.93,p
< .001].
Number comparison
Children’sabilitytocomparesymbolicquantitieswasassessedusinga
computer-basedNumberComparisontask(e.g.,Dehaene,Dupoux,&
Mehler,1990)inwhichchildrenwerepresentedwithatargetnumber
(between1and4or6and9)andwereaskedtopressoneoftwobut-
tons to indicate whether they thought that the number on the screen
was bigger or smaller than 5. Each number was presented five times,
in a random order, giving a total of 40 experimental trials. These were
precededbyfourpracticetrials.Childrenwerescored1foreachtrial
in which they correctly judged whether the target number was bigger
or smaller than 5, with reaction time data also obtained. Reliability
estimates for accuracy (Cronbach’s alpha = .88) and reaction times
(Cronbach’salpha=.66)weregood.
2.2.5 | Estimation measure
Number line task
Thenumberline task(Cohen&Blanc-Goldhammer,2011;Laski &
Siegler,2007,Link,Huber,Nuerk,&Moeller,2014;Siegler&Opfer,
2003) was used to assess children’s ability to spatially represent
numbers along a mental number line. This task used the number-
to- position version, in which children used their finger to indicate
the position on the number line where a target number should go.
Thisversion used 1–10 and 1–20 scales, and itwas framed as a
game in which the children had to help Postman Pat to deliver pre-
sentstohouses ondifferentstreets(Aagten-Murphyetal.,2015).
There were six experimental trials, in which the child was asked
to indicate the position of numbers 3, 4, 6, 7, 8 and 9. For the
1–10 number line, the numbers 5 and 10 wereused as the two
practicetrials; forthe1–20numberline, the numbers 10 and20
were used as the two practice trials, whilst the child was asked to
indicate the position of the numbers 4, 6, 8, 13, 15 and 18 in the six
experimental trials, which were presented in a random order. The
numberline was1000pixels long forbothscales Children’serror
for each individual trial was calculated as the distance in pixels be-
tween children’s estimated position and the actual position of the
targetnumber.The averageofchildren’serrorsacross both 1–10
and 1–20 scales was used as the overall measure of estimation
errorforthetask.Areliabilityestimatewascomputed(Cronbach’s
alpha=.70).
2.2.6 | Maths achievement
At the end of their first year of school, children’s maths abil-
ity was assessed by administering a 28- item maths achievement
test, consisting of questions from the calculation subtest of the
Woodcock-JohnsonIIItestsof achievement(Woodcock,McGrew,
&Mather,2001)andfromFormAoftheTestofEarlyMathematics
Ability(TEMA-3;Ginsburg& Baroody, 2003). The questions from
the calculation subtest contained six addition and four subtrac-
tionproblems,whilst thequestionsfromtheTEMA-3includedthe
counting of objects and animals, selecting the next number after a
given number in the counting list, as well as selecting which number
islarger from a choice of two. At the endoftheirsecondyearof
school, children were assessed using the age- appropriate version of
theMathsAssessmentforLearningandTeaching(MALT;Williams,
2005) which consisted of 30 questions, assessing counting and un-
derstanding number (nine questions), knowing and using number
facts (seven questions), calculating (eight questions) and measuring
(sixquestions).Children’srawscoresonbothmathsmeasureswere
used in the analyses. The reliability estimates for the maths measure
attheendofchildren’sfirstyearofschool(Cronbach’salpha=.91)
andfortheMALTattheendofchildren’ssecondyear(Cronbach’s
alpha=.83)werehigh.
2.3 | Procedure
The study received ethical approval from the university department’s
ethics committee. In Session 1, all children completed the Number
Orderingtask, followed by the NumberComparisontask,theAnimal
Racetaskandfinally,theNon-SymbolicAdditiontask.InSession2,chil-
drencompletedtheDailyEvents Ordertask, followedby theWPPSI-
IIIsubtests, then the Baseline Reaction Time task,Countingtaskand
then finally, the Number Line task. The computer- based tasks were
designed using E- Prime Version 2.0. These tasks were presented on
atouchscreen,connectedtoalaptop.Attheendofeachschoolyear
(Time1 =endof year1;Time 2= endofyear 2),childrencompleted
themathsachievement test in small groups of 3–6, in which the ex-
perimenter read out the questions and instructed the children to write
downtheiranswers.Allothertaskswereadministeredindividually.
3 | RESULTS
Descriptive statistics for both accuracy and reaction times are in-
cluded in Table 1. The median number that children were able to
countupto(outof50)was39.Most childrenperformedwellonthe
two numerical ordering tasks (forward and backward counting mean
accuracy = 76%) and on number ordering (82%). Two children per-
formedverypoorlyinthese.Inthenon-numericalorderingtasks,chil-
drendidnotperformquiteaswell.Inthedailyeventstask,children’s
accuracywas65%,whichwasabovechance(t(89)=11.10,p < .001).
In the order working memory task, children on average got 9 trials
correct, meaning that they were able to correctly remember ordered
sequencestoasequence length of 3. Children’s mean score on the
OPQwas44.02outof56,withparentstendingtoratetheirchildren
highly in terms of being able to carry out everyday tasks with a strong
ordering component.
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O’CONNOR et al.
As previously mentioned, children’s accuracy on the non-
symbolic addition task was relatively low, but their performance
on the task was above chance (t(89) = 5.09, p < .001). Children
performedmuchbetteronthenumbercomparisontask.Inthenum-
berlinetask,children’sestimatesonthe1–10scalewereonaverage
about 1.8 numbers away from the target number, whilst their esti-
matesonthe1–20numberlinewereonaverageabout3.4numbers
from the target.
3.1 | Zero- order and partial correlations (after
controlling for age, IQ and socioeconomic status)
between the order and magnitude processing
measures, counting ability and maths achievement
at the end of children’s first year of school
Table 2 shows that vocabulary scores were significantly positively cor-
related with order- processing (order WM, daily events, counting) and
non- symbolic addition and maths scores. Block design scores were
significantly positively correlated with the order- processing measures
(order WM, daily events, number ordering), as well as performance on
the number line task. Finally, higher deprivation scores were signifi-
cantlyrelatedtolowerperformanceonbothIQmeasuresandmaths,
as well as lower performance on the order WM, daily events, number
ordering and number comparison tasks.
Asshown inTable2,thereweresignificant correlationsbetween
general order- processing measures and maths at the end of chil-
dren’s first year of school; children’s maths ability was related to their
scoreson the OPQ,number ordering ability, daily eventstask accu-
racy, countingability and their order working memoryaccuracy. Of
the magnitude measures, only number comparison was found to be
relatedtomaths.Aftercontrollingforage,deprivationscoresandver-
bal and nonverbal intelligence, number comparison performance was
TABLE1 Descriptive statistics for all measures
Measure Minimum Maximum Mean (SD)
Vocabulary (scaled score) 4 17 8.52 (2.10)
Block Design (scaled
score)
4 16 10.12 (3.15)
OrderProcessing
Questionnaire
21 56 44.02 (7.69)
OrderWM 1 16 9.52(4.54)
Daily events accuracy .38 1 .65 (.13)
Symbolic number
ordering
0 1 .82 (.30)
Countingto50 6 50 39(13.15)
Countingforwardand
backward
0 1 .76 (.22)
Non- symbolic addition .30 .88 .56 (.11)
Number comparison
acc.
.40 1 .71 (.19)
Number comparison RT
(ms)
778 6059 2404.04
(1044.16)
Number line task (Mean
scaled error)
64 453 191.52
(74.90)
Baseline RT (ms) 860 2284 1435 (283.71)
Maths (Year 1) 1 28 23.24 (4.88)
Maths (Year 2) 7 29 21.74 (4.71)
TABLE2 Zero- order correlations between all measures
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14)
(1)Age —
(2) Vocabulary .04 —
(3) Block Design .09 .09 —
(4) Deprivation .11 −.41*** −.22* —
(5)OrderProcessing
Q.
.08 .15 .03 −.09 —
(6)OrderWM .17 .22* .30** −.22* .18 —
(7) Daily events −.09 .38*** .29** −.27** −.08 .44*** —
(8) Number ordering .14 .19 .24* −.23* .26* .41*** .24* —
(9)Counting .09 .27** .13 −.10 .15 .54*** .34** .36** —
(10) Non- symbolic
add.
−.23* .24* .12 −.19 −.14 .11 .22* .19 .02 —
(11) Number
comparison
.06 .18 .09 −.22* .20 .28** .34** .29** .29** .15 —
(12) Number line
(Error)
.21* −.02 −.26* .10 .11 −.05 −.15 −.05 −.20 −.14 −.04 —
(13) Maths (Year 1) −.004 .32** .16 −.26* .30** .32** .46*** .40*** .54*** .14 .21* .02 —
(14) Maths (Year 2) .10 .37*** .29** −.29** .28* .23* .41*** .38*** .43** .30** .24* −.17 .69*** —
Note.Taskabbreviation:Add.:addition.Q:Questionnaire.WM:Workingmemory
*p<.05;**p<.01;***p < .001.
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O’CONNOR et al.
no longer significantly related to maths performance (p= .29).OPQ
scores, r(78)=.26,p < .05; number ordering performance, r(78)=.25,
p < .05; daily events accuracy, r(78) = .36, p < .01; counting ability,
r(78)=.43,p < .001; and order WM accuracy, r(78)=.30,p < .01, re-
mained significantly related to maths after controlling for the covariate
measures.
3.2 | Zero- order and partial correlations
between the order and magnitude processing
measures, counting and maths achievement at the
end of children’s second year of school
Table 2 shows that vocabulary, block design and deprivation scores
at T1 weresignificantly related to maths at T2. Children’sT1 OPQ
scores, daily events task accuracy, number ordering ability, order
working memory accuracy, daily events accuracy and counting ability
were related to maths ability at the end of children’s second year of
school. For the magnitude measures, both non- symbolic addition ac-
curacyandnumbercomparisonaccuracywererelatedtomaths.After
controlling for age, deprivation scores and verbal and non- verbal intel-
ligence, the only significant relationships with maths were observed
forOPQscores,r(75)=.24,p < .05; counting ability, r(75)= .24, p <
.05; and number ordering performance, r(75)=.24,p < .05.
3.3 | Bootstrap correlations
Abootstrapprocedure(using10,000samples)wasalsoappliedtoas-
sess the reliability of the relationship between the measures which
had previously been observed as having a significant zero- order and/
or partial correlation with maths, and maths achievement at each time
point. This procedure allowed for a 95% confidence interval to be
computed for the correlations between each measure and children’s
maths ability and if any measure was found to have a significant boot-
strap correlation with maths, then it was considered to be robustly
related to maths achievement. Figure1 shows 95% bootstrap con-
fidence intervals between the measures and maths achievement at
theend ofchildren’sfirst year ofschool,whilst Figure2shows95%
bootstrap confidence intervals between measures and maths achieve-
ment at the end of children’s second year of school.
Figure 1 shows that the measures which had previously shown sig-
nificant zero- order and/or partial correlations with maths at the end
of children’s first year of school also showed significant zero- order
bootstrap correlations with maths. Figure 2 shows that order working
memory accuracy [r=.17,95% CI (−.11, .41)]wasthe onlymeasure
that was not robustly related to maths at the longitudinal level, of all
the measures that had previously been related to maths at the end of
children’s second year of school.3
3.4 | Regression modelling
The regression analyses regarding the relationship between the pre-
dictor variables and maths performance at each time point followed a
similarproceduretothatofSzűcs,Devine,Soltesz,Nobes,andGabriel
(2013). For each regression model, the variables that had a significant
bootstrap correlation with maths were entered first. Non- significant
predictors of maths in each model were then removed and each pre-
dictor which had a significant partial correlation with maths but not a
significant bootstrap correlation was entered into the model one by
FIGURE1 95%bootstrapconfidenceintervalsforzero-ordercorrelationsbetweenmeasuresandmathsachievementattheendofchildren’s
firstyearofschool.Taskabbreviations:NSA:Non-symbolicaddition.Num.Comp.:Numbercomparison.Num.Ord.:Numberordering.OPQ:
ParentalOrderProcessingQuestionnaire.WM:Workingmemory
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O’CONNOR et al.
one to examine whether they became significant. Then, the four co-
variates (age, deprivation scores, vocabulary and block design) were
entered into the model to examine whether they changed significant
predictorsand improved fit. At each time-point,the model that ex-
plained the greatest proportion of variance with only significant pre-
dictors in the model was selected.
Table 3 shows the initial and final models for measures that pre-
dicted maths at the end of children’s first year of school. The initial
model consisted of OPQ scores, order WM, daily events, number
ordering, counting and number comparison accuracy. This model ex-
plained37%ofthevarianceinmathsscores;however,thismodelcon-
tained a number of non- significant predictors of maths (order WM; β
=−.07,ns; number ordering; β=.12,ns; number comparison; β=−.03,
ns). These measures were removed and only the significant predictors
(OPQscores,dailyeventsandcountingaccuracy)wereenteredintothe
next model. When adding them to the model one by one, none of the
remaining predictors explained significant additional variance in maths
performance. Thus, this was accepted as the final model (see Table 3).
Table 4 shows the initial and final models for the measures that
significantly predicted maths at the end of children’s second year of
school.TheinitialmodelconsistedofOPQscores,dailyevents,num-
ber ordering, counting, non- symbolic addition and number comparison
accuracy.Thisinitialmodelexplained30%ofthevarianceinchildren’s
maths scores at the end of their second year of school. The non-
significant predictors (number ordering, counting and number com-
parison) wereremoved and the next model contained OPQ scores,
daily events and non- symbolic addition accuracy, which explained
27%ofthevarianceinmathsperformance.Thetwointelligencemea-
sures and deprivation scores did not explain significant additional
variance in maths performance, although age was a significant factor
whenincludedin themodelcontainingOPQscores,dailyeventsand
non-symbolicadditionaccuracy,withthismodelexplaining30%ofthe
variance in children’s maths performance at the end of their second
year of school.4
FIGURE2 95%bootstrapconfidenceintervalsforzero-ordercorrelationsbetweenmeasuresandmathsachievementattheendofchildren’s
secondyearofschool.Taskabbreviations:NLT:Numberlinetask.NSA:Non-symbolicaddition.Num.Comp.:Numbercomparison.Num.Ord.:
Numberordering.OPQ:ParentalOrder-ProcessingQuestionnaire.WM:Workingmemory
TABLE3 Initialandfinalmodelspredictingmathsachievementat
the end of children’s first year of school
βt p
Initialmodel Daily events .39 3.90 < .001
Counting .33 3.09 .003
OrderProcessing
Questionnaire
.27 2.89 .005
Symbolic number ordering .12 1.25 .214
OrderWM −.07 −.65 .520
Number comparison −.03 −.31 .759
Final model Daily events .38 4.17 < .001
Counting .32 3.49 .001
OrderProcessing
Questionnaire
.28 3.23 .002
Initialmodel:R²=.37,F(6,84)=9.33,p < .001.
Final model: R²=.39,F(3,84)=18.39,p < .001.
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As a final step, we checkedwhether the longitudinal predictors
of formal maths skills at the end of the second year of school also
remained significant if the effect of formal maths skills at the end of
the first school year were taken into account. We did this by adding
formal maths skills at T1 as a predictor to the final regression model
presented in Table 4. This analysis addressed the question of whether
these longitudinal predictors of maths also predicted growth in maths
skills during the second year of school. The model is presented in
Table5.Thismodelexplained41%ofthevarianceinT2formalmaths
skills with formal maths skills at T1, the order processing questionnaire
and non- symbolic addition as significant predictors. The effect of the
daily event ordering task was no longer significant, and the effect of
age was also reduced to a non- significant trend.
4 | DISCUSSION
Children’s ability to process both numerical order (counting, num-
berordering) andnon-numericalorder (OPQ, dailyeventsand order
working memory) at the start of their first school year were robustly
related to their maths achievement at the end of their first year. These
relationships were significant, even after controlling for age, depriva-
tion scores and verbal and nonverbal intelligence. Multiple regression
analyses revealed that, after controlling for the effect of counting
ability (forwards and backwards), the order processing questionnaire
and the daily events task still remained significant predictors of maths
ability. The longitudinal analysis (i.e., predicting maths performance
at the end of the second school year) showed that children’s numeri-
cal ordering ability (counting forwards and backwards and symbolic
number ordering) at the start of formal education was robustly related
to their maths achievement at the end of their second year of school.
ScoresontheOPQanddailyeventstaskaccuracywerealsorobustly
related to maths at the longitudinal level. The regression analyses re-
vealed that early non-numerical ordering abilities (OPQ scores and
daily events task accuracy) were significant predictors of children’s
maths achievement more than 1 year later even when the significant
effects of counting ability, and non- symbolic addition were controlled.
When the effect of T1 formal maths skills was controlled, only the
OPQ and the non-symbolic addition task explained additional vari-
ance in T2 formal maths skills, whereas the effect of the daily events
task was no longer significant. This suggests that the effect of the
daily events task was the strongest during the first school year, and it
related to maths abilities in the second year of school via its links with
early formal maths skills. By contrast, everyday order processing abili-
ties remained significantly related to formal maths skills throughout
the first two years of school.
These results strongly support the notion that ordinality is import-
antto the development ofearlymaths skills (e.g.,Attout & Majerus,
2015;Attoutetal.,2014;Lyonsetal.,2014).Ourdetailedanalysesof
the components of the formal maths tests also showed that ordinality
was important to all aspects of maths, including counting, calculation,
andtheunderstandingofnumberfactsandmeasures.Ourresultsalso
extend previous findings by showing that, even at the very earliest
stages of formal schooling, children’s domain- general ability to pro-
cess order, as demonstrated in familiar everyday tasks and to a lesser
extent, their ability to order daily events, plays an important role in
the successful development of more mature maths skills. This extends
work with adults (Morsanyi et al., 2017; Sasanguie et al., 2017; Vos
et al., 2017) that showed strong relationships between non- numerical
ordering tasks and mathematics abilities. The domain- general ability
to use order information measured by the daily events task must be
based on long- term memory representations of familiar sequences,
and our findings indicate that it is distinct from the ability to process
ordinalinformationheldinshort-termmemory.Indeed,whilewerep-
licatedAttoutetal.’s(2014)findingsofaconcurrentrelationbetween
non- numerical order WM and children’s maths skills, performance on
theOPQandthedailyeventstask were in fact betterpredictorsof
maths skills both concurrently and longitudinally.
Our results are novel in suggesting that there are two distinct
domain- general ordering abilities that support maths development.
Attoutetal.(2014)showthattheabilitytoholdorderedunfamiliarse-
quences in working memory is important, and make a strong case for
whysuchanabilitymaybecrucialforcalculationabilities.Inaddition,
TABLE4 Initialandfinalregressionmodelspredictingmaths
achievement at the end of children’s second year of school
βt p
Initialmodel OrderProcessing
Questionnaire
.28 2.77 .007
Non- symbolic addition .26 2.60 .011
Daily events .25 2.38 .020
Counting .19 1.80 .075
Symbolic number ordering .11 1.07 .289
Number comparison .04 .35 .728
Final model Daily events .35 3.67 < .001
OrderProcessing
Questionnaire
.32 3.36 .001
Non- symbolic addition .30 3.04 .003
Age .20 2.06 .042
Initialmodel:R²=.30,F(6,81)=6.71,p < .001.
Final model: R²=.30,F(4,81)=9.53,p < .001.
TABLE5 Regression model predicting formal maths achievement
at the end of children’s second year of school taking into account the
effect of formal maths achievement at the end of the first school
year
βt p
T1 maths .41 3.92 <.001
Daily events .16 1.62 .109
OrderProcessing
Questionnaire
.19 2.03 .045
Non- symbolic addition .26 2.93 .004
Age .17 1.95 .054
R²=.41,F(5,81)=12.13,p < .001.
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O’CONNOR et al.
our results indicate that representing and processing familiar ordered
sequences in long- term memory may be fundamental for the emer-
gence of very early maths skills, when children are learning to repre-
sent and use numbers as an intrinsically ordinal sequence. The idea
that such domain- general abilities underpin early maths skills is con-
sistent with Rubinsten and Sury’s (2011) claim that processing order
information forms part of the cognitive foundations of mathematics.
Such a domain- general ability is likely to be in operation well be-
fore children learn mathematics, and indeed a considerable body of
research indicates that children rapidly acquire representations of re-
peated event sequences over multiple time scales during the preschool
years(Fivush&Hammond,1990;Nelson1986,1998).Acquiringand
using ordered representations of repeated events forms a crucial part
of children’s learning about the world, and indeed has been argued to
befoundationalincognitivedevelopment(Nelson,1998).Ourfindings
provide the first evidence that suggests that the same processes also
support emerging maths abilities.
Oneimportantandunresolvedissue,though,iswhetherthereisa
domain- general representational format for representing ordered in-
formation in long- term memory, and specifically whether such repre-
sentationsarespatialinnature.Ourdatadonotallowustoanswerthis
question,butwenotethatFriedman(1977,1990)hasarguedthat4-
to 5- year- olds have spatialized representations of familiar events (and,
indeed, our daily event ordering task and our number ordering task
required children to understand the mapping of temporal order to spa-
tial order; although see Tillman, Tulagan, & Barner, 2015, for evidence
that 4- year- olds do not do this mapping spontaneously). Friedman and
Brudos (1988) claimed that 4-year-olds use a common representa-
tional system for coding both spatial and temporal order information,
raising the possibility that the ability to represent items in this way is
then utilized in the context of mathematics as well. Such an idea is
broadly consistent with other claims regarding the way temporal order
and numbers are represented (e.g., see Bonato, Zorzi, & Umiltà, 2012,
forreviewofresearchon the“mentaltimeline”and“mentalnumber
line”).WenotethatBerteletti,Lucangeli,andZorzi(2012)havemade
what could be interpreted as a contrasting claim, namely that children’s
conception of numerical order develops first and is then generalized to
othernon-numerical sequences.Itis important topointout that the
non- numerical sequences that they studied are those acquired later
than the number sequence during formal education (the alphabet and
months of the year), rather than familiar event sequences which are
acquired very early in development. Moreover, the issue that Berteletti
et al. are concerned with is whether the items in sequences in ques-
tion are spaced linearly (by contrast to log spacing), rather than the
more basic issue of whether they share a spatialized representational
format. We note that children’s performance on our number line task
did not relate to performance either on the daily event task or on the
OPQ,norevenonthenumberorderingtask,suggestingthatthepre-
cision of children’s placing of numbers on a line measures something
different from the ability to represent and process either numerical or
non- numerical ordinal information.
Despite focused research on this issue, there is much that is not
yet known about the commonalities between temporal, numerical,
and spatial representation; we would suggest that our findings provide
new impetus for considering such commonalities, particularly those
between time (understood here as event order) and number, and how
such commonalities may play a role in the acquisition of maths skills.
Another important contribution of the current work is that it
provided the first evidence for a link between parentally reported
everyday ordering abilities and formal maths skills. Whereas clinical
observations of individuals with developmental dyscalculia have de-
scribed everyday order processing difficulties, this study was the first
to show that this link is also present in the case of a sample of young,
typically developing children. Indeed, the OPQ longitudinally pre-
dicted growth in formal maths skills during the second year of school.
This finding could have great practical importance, as it offers the
possibility to screen children for vulnerability to develop mathemat-
icsdifficulties even beforethey start theirformaleducation. Indeed,
our questionnaire was designed for 4- year- old children; in many coun-
tries,this would be 2–3 years before the children start theirformal
education in maths. The questionnaire that we developed to measure
children’s everyday order processing abilities had good psychometric
properties, and it only took a few minutes to complete, which makes
it very convenient to use. Nevertheless, future work could further
improve the psychometric properties and the predictive value of this
questionnaire.
Ourstudyexaminedanumberofotherpredictorsofmathsskills
used in previous studies. As we have pointed out, we replicated
Attout etal.’s (2014) finding that order WM was related to maths
skills in the first year of school, but in our sample, order working
memory at the start of formal schooling did not longitudinally pre-
dict maths performance at the end of the second year of schooling.
Regarding other predictors of maths performance, Lyons et al. (2014)
found that number comparison and number line performance were
the best predictors of maths performance in the first school year. We
also found a robust relationship between number comparison perfor-
mance and maths skills both at the cross- sectional and longitudinal
levels, which is also in line with several other studies that showed
a strong relationship between number comparison and maths skills
atthe start of formal education (e.g., Attout etal., 2014; Holloway
&Ansari, 2009;Mundy&Gilmore,2009; Rousselle& Noël, 2007).
Giventhewell-establishedlinkbetween this taskandmathsability,
and the fact that it involves symbolic number processing, it is striking,
though, that number comparison did not explain additional variance
in maths skills, once the effect of counting skills and everyday order-
ing abilities were controlled.
Regarding number line performance, several studies found a re-
liable relationship between this task and maths achievement in chil-
dren from as young as 3 years old (e.g., Berteletti, Lucangeli, Piazza,
Dehaene, & Zorzi, 2010; Booth & Siegler; 2006, 2008; Link et al.,
2014; Siegler & Booth, 2004). Studies typically use a paper- and- pencil
version of this task, and it is possible that the link between maths skills
and performance on the number line task would have been stronger
had we used the typical presentation format. Nevertheless, the task
showed good reliability, and children’s estimations were not very far
from the correct positions of target numbers. Performance on this task
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O’CONNOR et al.
was also related to children’s block design scores, which supports the
validity of the tasks. There was also a non- significant trend (p=.118)
toward a relationship between number line performance and formal
maths skills at T2.
Number ordering performance was significantly related to math
abilities both in Year 1 and Year 2. Nevertheless, surprisingly, non-
numerical ordering tasks were more strongly related to maths abilities
than number ordering. This raises the question of whether our version
ofthe task was ideally suited to measure number ordering skills. As
we noted earlier, other researchers used computer- based verification
tasks to measure number ordering skills in young children (e.g., Lyons
et al., 2014) that were analogous to our daily events task, albeit involv-
ingnumbers.However,inapilottest,ourparticipantsfoundthisver-
sion of the task too challenging, possibly because they were younger
than the participants in all the other studies. Some researchers (e.g.,
Attout& Majerus, 2015; Attoutetal.,2014;Vogeletal.,2015)pre-
sented children with dyads of numbers rather than triads in their num-
ber ordering task. The dyad version was successfully performed by
childrenasyoungas5–6years old(Attoutetal.,2014).However,an
issue with this version of the task is that Vogel et al. (2015) reported
no reverse distance effects on the task, which have been consistently
found by researchers who used number triads in their ordering task.
Thus, it is possible that the two versions of the number ordering task
(i.e., using dyads vs. triads) do not rely on exactly the same cognitive
processes. In particular,it is less certain that participants must rely
on order information per se in the dyad task than in the triad task.
For these reasons, we employed a production version of the number
ordering task.
We believe that this task was appropriate for our sample, given
that we found stronger correlations between number ordering and
maths skills than other researchers who looked at this relationship in
thecaseofyoungchildren(e.g.,Attoutetal.,2014;Lyonsetal.,2014;
Vogeletal., 2015). Indeed, the typical finding in the case of young
children is a weak/non- significant relationship.5 By contrast, we found
that number ordering was significantly related to all aspects of maths
at both T1 and T2. Furthermore, we found a moderate relationship be-
tween the daily events task and the number ordering task, suggesting
that both tasks were assessing some of the same skills. Regarding the
predictive value of production vs. verification tasks, whilst we did not
use the verification version of the number ordering task, our number
comparisontaskwasaverificationtask.Althoughchildrenperformed
better on that task than on the daily events task (i.e., a verification task
that measured ordering ability), performance on the number compar-
ison task was less strongly related to maths than the daily events task
at both T1 and T2.
There is evidence that, as children get older, number ordering skills
become increasingly strongly related to maths abilities (see Lyons et al.,
2014). Regarding non- numerical ordering skills, the developmental
pattern of their links with maths abilities has not been investigated so
far. Some recent studies (e.g., Morsanyi et al., 2017, Sasanguie et al.,
2017; Vos et al., 2017) have demonstrated that non- numerical order-
ing skills remain strongly related to arithmetic skills even in the case
of adults, although these links are not quite as strong as the relations
between numerical ordering skills and maths. Thus, it is plausible to
assume that at some point in development (most likely during the first
years of school) number ordering skills become more strongly related
to maths skills than non- numerical ordering. Nevertheless, this ques-
tion requires further investigation.
Non- symbolic addition performance was a significant predictor of
children’s later maths achievement, and growth in formal maths skills
during the second year of school, although it was not related to maths
performance at the end of the first school year. The task was designed
in such a way that children could not perform above chance if they
onlyrelied on simple perceptualstrategies(see Gilmoreetal.,2010;
Rousselle&Noël,2007;Soltész,Szűcs,&Szűcs,2010).Unsurprisingly,
young children found this task difficult. Whereas the finding that per-
formance on this task predicted maths performance is in line with
studies that found a link between non- symbolic estimation skills and
mathematicsperformance(seeChen &Li, 2014,fora meta-analysis),
it is important to note that the non- symbolic addition task has further
cognitive requirements, including memory, spatial attention and inhi-
bition, which are also important for maths development.
Indeed,one limitationofthe currentstudyis thatitdid not con-
sider some domain- general factors that are likely to play a role in nu-
mericaldevelopment.Although IQ and orderworking memorywere
measured in the current study, other general cognitive skills were not
considered. There is much evidence to suggest that other aspects of
workingmemoryprocesses(Passolunghi,Cargnelutti&Pastore,2014;
Passolunghi,Vercelloni,& Shadee, 2007; Szűcsetal., 2013;Vander
Ven, Van der Maas, Straatemeier, & Jansen, 2013) and executive func-
tions(Gilmoreetal., 2013;Passolunghi&Siegel,2001;Soltészetal.,
2010;Szűcsetal.,2013)arerelatedtomaths.Inparticular,itwouldbe
interesting to investigate verbal and spatial working memory and inhi-
bition skills together with the ordering tasks, as these skills might play
aroleinorderingperformance(e.g.,vanDijck,Abrahamse,Majerus,&
Fias, 2013; van Dijck & Fias, 2011; Morsanyi et al., 2017).
Anotherlimitationthatcould benotedisthatformal mathsskills
werenotassessedatthestartofthefirstschoolyear.Indeed,although
we used a broad range of tasks to measure basic maths abilities (in-
cluding non- symbolic measures, counting skills, and measures that
required the knowledge of symbolic numbers, such as the number
line task, and the number ordering task), it is possible that children
had already possessed some of the formal maths skills (e.g., addition
and subtraction) that were assessed at the end of the first school year.
Thus, although our findings demonstrated that early, non- numerical
ordering skills were strongly related to formal maths skills at the end
of the first school year, it is unclear whether early ordering abilities
predicted growth in formal math abilities during the first school year.
This question might be explored in future studies.
Finally, we have already discussed the possibility of using every-
day ordering abilities as early indicators of potential vulnerability to
mathsdifficultiesinyoungchildren.Anotherpossiblefuturedirection
is to develop non- numerical training exercises that could be used to
helpyoungchildrento improvetheirorderingabilities. Oneinterest-
ing question is whether the effects of training in non- numerical or-
dering might generalize to number ordering skills, and numerical skills
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O’CONNOR et al.
ingeneral.In fact,there isa possibilitythatorderingskillsmightplay
an important role in the development of other academic skills as well,
as Perez, Majerus, and Poncelet (2012) found that order WM capac-
ity longitudinally predicted reading development in the case of young
children. The same authors (Perez, Majerus, & Poncelet, 2013) also
reportedthatadultswithdyslexia displayedadeficitin orderWM.It
is possible that the link between domain- general order processing and
other academic skills is specific to short- term memory mechanisms,
but our findings suggest that it might be useful to examine whether
such a link also extends to the sort of ordering processing skills mea-
sured in our study.
In conclusion, the currentstudy has shown that children’s abil-
ity to process order, at the earliest stage of formal schooling, is an
important predictor of maths achievement concurrently and 1 year
later.In particular, it seems that non-numericalorderingability(for
familiar tasks and daily events) is a stronger predictor of children’s
maths ability than numerical order at the early stages of education.
Althoughonthebasisofthecurrentfindingsitisnotpossibletoes-
tablish whether early non- numerical ordering abilities predict growth
in formal maths skills during the first school year, such evidence was
found in the second year of school, at least in the case of the parental
report of children’s ordering skills. General orderingability may be
a suitable target for intervention for young children, and measuring
ordering ability could potentially be used to identify children who are
at risk of developing maths difficulties, even before they start formal
education.
ENDNOTES
1 The typical task in the literature that is used to measure number ordering
ability is a computer- based task in which children are shown dyads or triads
of numbers and have to judge whether the order is correct or incorrect/
ascending or descending. We piloted a computer- based number ordering
task with children from this age group using triads (i.e., comparable to our
daily events ordering task) and found that they struggled to perform the
task, even after a short training that was provided using cards representing
the numbers. By contrast, they were able to complete the computer- based
version of the daily events task after a training session with cards represent-
ing the events.
2 We selected this task, rather than non- symbolic comparison, due to the in-
consistency of the evidence supporting a link between non- symbolic com-
parisonand maths in developmental studies (DeSmedt,Noël,Gilmore, &
Ansari,2013),whichmaybe,inpart,duetoalackofanagreedmeasurement
oftaskperformanceusedinthesestudies(e.g.,Inglis&Gilmore,2014;Price,
Palmer,Battista,&Ansari,2012).Incontrast,thenon-symbolicadditiontask
has been found to be a longitudinal predictor of maths achievement, as well
as being related to mastery of both number words and symbols, which un-
derlies much of earlymaths learning (Gilmore etal., 2010). Furthermore,
otherevidence(Gilmore,Attridge,DeSmedt,&Inglis,2014;Iuculano,Tang,
Hall,& Butterworth,2008)has shown thatperformance on non-symbolic
addition and comparison tasks is correlated, suggesting that both tasks are
measuring the same underlying construct, whereas non- symbolic compar-
ison performance has been found to be unrelated to symbolic comparison
performance (e.g., Sasanguie, Defever, Maertens, & Reynvoet, 2014).
3Although in our main analyses we considereddifferent types of for-
mal maths skills together, the standardized tests that we used included
several different types of problems (see Methods section). We pres-
ent zero- order correlations between the measures that were robustly
related to maths at each time point and the different components of the
formal maths tasks (see Supplementary Tables 1 and 2). Typically, the
best predictors of maths at each time point (in particular, the counting
task and the daily events task) were significantly related to all aspects
ofmaths.Interestingly,symbolicnumberorderingwasalsorelatedto
all aspects of maths at T1 and T2, although it was not included in the
final regression models (see below), which suggests that its effect on
maths was mediated by other tasks.
4Additionalregressionanalyseswereperformedtoinvestigatewhether
the results of the cross- sectional and longitudinal regression models
were the same for predicting only the arithmetic/calculation measures
at T1 and T2. We conducted these analyses to demonstrate that order-
ing abilities were not simply related to a composite measure of maths
achievement (which included various basic components of early maths
ability, including some that were closely related to ordering). The same
threepredictors(OPQ,dailyeventsandcounting)thatsignificantlypre-
dicted maths achievement at T1 also predicted arithmetic scores at T1
(thesethreepredictorsaccountedfor31%ofthevarianceinarithmetic
scores). Three of the four significant longitudinal predictors of maths
atT2(OPQ,non-symbolic additionand dailyevents)alsosignificantly
predictedcalculationscoresatT2(accountingfor19%ofthevariance
incalculationscores).Agewasnotfoundtobeasignificantlongitudinal
predictor of calculation abilities. (Detailed results of these analyses can
be found in Supplementary Tables 3 and 4.)
5 The sample in Vogel et al. (2015) consisted of children in 1st grade in
Canada,whowereagedbetween6and7yearsold.Theauthorsfailedto
find a relationship between the size of the numerical distance effect or
meanreaction timesfortheorderjudgementtaskand maths. InAttout
et al. (2014), the children were between 5 and 6 at T1; 6 and 7 at T2
and 7 and 8 at T3. There were significant associations between numerical
ordering and maths at T2 and T3, but not at T1. Lyons et al. (2014) found
that number ordering ability was not a significant predictor of math in
grades 1 and 2 (between 6 and 8 years old) but was a significant predictor
ofmathsfromgrade3onwards(fromage9).
ORCID
Patrick A. O’Connor http://orcid.org/0000-0003-0936-0478
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SUPPORTING INFORMATION
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supporting information tab for this article.
How to cite this article:O’ConnorPA,MorsanyiK,
McCormackT.Youngchildren’snon-numericalorderingability
at the start of formal education longitudinally predicts their
symbolic number skills and academic achievement in maths.
Dev Sci. 2018;e12645. https://doi.org/10.1111/desc.12645
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APPENDIX
ParentalOrder-ProcessingQuestionnaire
Please circle the number which you feel best applies to your child for
each question
(1=verymuchdisagree;7=verymuchagree)
My son/daughter:
Iseasilyconfusedbychangesinroutine 1- - - - 2- - - - 3- - - - 4- - - - 5- - - - 6- - - - 7
Understands how the seasons of the year follow each other (e.g., that
autumn always comes after summer)
1- - - - 2- - - - 3- - - - 4- - - - 5- - - - 6- - - - 7
Caneasilyrecalltheorderinwhichpasteventshappened 1- - - - 2- - - - 3- - - - 4- - - - 5- - - - 6- - - - 7
Isabletoplanasequenceofactivitiesindependently 1- - - - 2- - - - 3- - - - 4- - - - 5- - - - 6- - - - 7
Finds it difficult to learn new activities which involve a sequence of
actions which have to be performed in a particular order (e.g.,
putting together the parts of a toy in the right order)
1- - - - 2- - - - 3- - - - 4- - - - 5- - - - 6- - - - 7
Would be able to recall the order of typical daily events 1- - - - 2- - - - 3- - - - 4- - - - 5- - - - 6- - - - 7
Understands that some things always have to be done in a particular
order (e.g., putting on a school shirt before putting on a tie)
1- - - - 2- - - - 3- - - - 4- - - - 5- - - - 6- - - - 7
Finds it difficult to understand how the days of the week follow each
other (e.g., knowing that Wednesday comes after Tuesday)
1- - - - 2- - - - 3- - - - 4- - - - 5- - - - 6- - - - 7
