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International Journal of Disability, Development andEducation
ISSN: 1034-912X (Print) 1465-346X (Online) Journal homepage: https://www.tandfonline.com/loi/cijd20
Developmental Dyscalculia and Down Syndrome:Indicative Evidence
Monica Cuskelly & Rhonda Faragher
To cite this article: Monica Cuskelly & Rhonda Faragher (2019) Developmental Dyscalculiaand Down Syndrome: Indicative Evidence, International Journal of Disability, Development andEducation, 66:2, 151-161, DOI: 10.1080/1034912X.2019.1569209
To link to this article: https://doi.org/10.1080/1034912X.2019.1569209
Published online: 07 Feb 2019.
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Developmental Dyscalculia and Down Syndrome: IndicativeEvidenceMonica Cuskelly a,b and Rhonda Faragher b
aSchool of Education, University of Tasmania, Launceston, Tasmania, Australia; bSchool of Education,University of Queensland, Brisbane, Queensland, Australia
ABSTRACTThis study investigated the hypothesis that developmental dyscal-culia is part of the behavioural phenotype of Down syndrome. Onehundred and fifty-one individuals with Down syndrome acrossthree age groups contributed data. These age groups were:6–8 years (n = 41); 13–15 years (n = 70); and 20–22 years (n = 40).Data were collected using the Stanford-Binet (Fourth Edition) andage equivalent scores on the Pattern Analysis and Quantitativesubscales were used in the analyses. A repeated measure ANOVAshowed a significant difference between performances on the twosubtests with an interaction with age group. While performance onPattern Analysis was better than on Quantitative for all groups thedifference was most marked for the oldest group. Approximately66% of the participants had a higher age equivalent score on thePattern Analysis subscale than on the Quantitative subscale. Itappears to be plausible that developmental dyscalculia contributesto the behavioural phenotype of Down syndrome and furtherinvestigation of this proposition is warranted.
KEYWORDSArithmetic; behaviouralphenotype; developmentaldyscalculia; Down syndrome;mathematics; non-verbalability; numeracy; patternanalysis
Introduction
Individuals with Down syndrome experience a number of cognitive difficulties that contributeto problemswithmeeting the demands of the school curriculum. One area inwhich problemsare almost universally reported is in the arithmetic aspects of mathematics (e.g. Faragher &Clarke, 2014; Lanfranchi, Berteletti, Torrisi, Vianello, & Zorzi, 2015; Turner & Alborz, 2003).Understanding the nature of these difficulties is a necessary precursor to the explorationand identification of potential interventions. This paper reports the results of an investigationtodetermine if thedifficulties reported in the literature couldbeunderstood as developmentaldyscalculia andwhether this may form part of the behavioural phenotype of Down syndrome.
Behavioural Phenotypes
Behavioural phenotypes are an external expression of the genotype. They have become thefocus of substantial research in the area of developmental disability as they provide cluesabout the mechanisms that may underlie development. While there was initially some
CONTACT Monica Cuskelly [email protected]
INTERNATIONAL JOURNAL OF DISABILITY, DEVELOPMENT AND EDUCATION2019, VOL. 66, NO. 2, 151–161https://doi.org/10.1080/1034912X.2019.1569209
© 2019 Informa UK Limited, trading as Taylor & Francis Group
debate about the definition of a behavioural phenotype, the field has settled on thedefinition provided by Dykens (1995). This definition is a probabilistic one which allowsthat not every individual with a particular aetiology will display the characteristic of interest;however, there is a likelihood that an individual will display that behaviour or characteristicand this likelihood is greater than in those who do not have the syndrome. An example ofa behavioural phenotype is the difficulty with expressive language in contrast to receptivelanguage experienced by those with Down syndrome (see Abbeduto, Warren, & Conners,2007; Martin, Klusek, Estigarribia, & Roberts, 2009, for reviews).
If a behavioural phenotype is established it may identify areas of the brain that areimplicated in the expression of that behaviour/characteristic, thus providing informationabout brain–behaviour relationships. It may also provide some guidance with respect tointervention (see Daunhauer & Fidler, 2011; Lemons, Powell, King, & Davidson, 2015). If, forexample, it is established that a particular group is likely to present with developmentaldyscalculia, this may influence the ways in which educators go about teaching individualswith that originating condition. As pointed out by King, Powell, Lemons, and Davidson(2017) there is a dearth of information about mathematics performance of individuals withDown syndrome and thus about appropriate teaching approaches for students with thiscondition.
Vanvuchelen, Feys, and De Weerdt (2011) suggested that if the probabilistic definition isaccepted, then understanding the sensitivity (how commonly the characteristic occurs inthe population with a particular aetiology) and specificity (how commonly the characteristicis observed in other syndromes) of a phenotypic characteristic is an important endeavour.The study reported here deals only with the first of these, that is, the sensitivity of thephenotypic characteristic.
Developmental Dyscalculia
Developmental dyscalculia (hereafter dyscalculia) is ‘a specific learning disability affectingthe development of arithmetic skills’ (Kucian & von Aster, 2015, p. 2) that is ‘characterised byproblems processing numerical information, learning arithmetic facts, and performingaccurate or fluent calculations’ (American Psychiatric Association, 2013, p. 67). Researchon the nature of dyscalculia is relatively recent and a comprehensive understanding of thecondition is yet to emerge. It is likely to have a neurological basis with the intraparietalsulcus implicated (Butterworth, Varma, & Laurillard, 2011; Dehaene, 2011).
Research into human cognition of mathematics indicates two systems. The first is a non-verbal system evident in infants which comprises an Object Tracking System (OTS) and anApproximate Number System (ANS) (Lanfranchi et al., 2015). The second is a verbal/symbolicsystem which develops later (evident at around 3 to 4 years of age), contributes to precisionwith respect to numbers (e.g. the storage and retrieval of number facts), and is dependent oneducation and cultural context (Dehaene, 2011).
Wilson and Dehaene (2007) proposed sub-types of dyscalculia each implicating differentareas of the brain: number sense, verbal memory and spatial. The number sense typeinvolves impairment of the ability to ‘quickly understand, approximate, and manipulatenumerical quantities’ (p. 213). In reviewing the literature, their conclusion is that the sourceof dyscalculia could be in either or both non-verbal systems (OTS or ANS) or in theconnection between the non-verbal and verbal systems.
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Dyscalculia is thought to occur in 3–6% of the general population (Rotzer et al., 2009).Diagnosis is generally partially based on the difference between performance ona standardised mathematics achievement test and expected performance, and requiresthe use of a timed test (see further comment on this in the Discussion). Expectedperformance is often determined by tests of intellectual ability, with the assumptionthat mathematical ability will be at approximately the same level. Several studies haveshown that individuals with a number of genetic syndromes associated with intellectualdisability have more difficulty with numerical processing than typically developing peerswhen difference in intellectual ability is controlled for (e.g. De Smedt et al., 2007;O’Hearn & Landau, 2007). The APA (2013) definition of dyscalculia includes the stipula-tion that the individual does not have intellectual disability. This is not a positionaccepted by all researchers (see, for example, Dennis, Berch, & Mazzocco, 2009) anddefinitions for other conditions have been changed in the past in response to researchand theoretical arguments (see, for example, definitions of autism spectrum disorder(American Psychiatric Association, 2000, 2013)). Other researchers have posited a linkbetween dyscalculia and intellectual disability, including Down syndrome. Lanfranchiand colleagues wrote ‘Difficulty in estimating numerosity of single sets is a criticalweakness associated with DS [Down syndrome] that could explain at least a part oftheir difficulties in arithmetic, like it has been seen in developmental dyscalculia’(Lanfranchi et al., 2015, p. 1033). This is an issue taken up again in the Discussion.
Mathematics and Down Syndrome
Difficulties in learning arithmetic appear to be pervasive in individuals with Down syn-drome, although research is sparse (Faragher, 2017). These difficulties are greater thanmight be expected on the basis of other functional capacities (e.g. Lanfranchi et al., 2015),including their performance in other aspects of mathematics (Monari Martinez & Pellegrini,2010). Lanfranchi et al. (2015) reviewed difficulties with mathematics in individuals withDown syndrome and noted two views in the literature which they characterised as: (1)developmental (difficulties stem from general low functioning) and (2) difference (poorerperformance than matched MA, including less efficient numerosity discrimination, withinsubitising range). The evidence to date suggests that it is difficulty in the non-verbal aspectsof number, implicated in dyscalculia – the OTS and the ANS – that underpins the difficultiesdisplayed by individuals with Down syndrome (Belacchi et al., 2014; Lanfranchi et al., 2015).
Hypothesis
If dyscalculia is part of the behavioural phenotype of Down syndrome then performance onmeasures of arithmetic should be poorer than on measures of non-verbal intellectual ability.We have focussed on non-verbal ability as verbal and memory abilities are compromised inindividuals with Down syndrome (see Godfrey & Lee, 2018; Næss, Halaas Lyster, Hulme, &Melby-Lerväg, 2011 for reviews). To operationalise this hypothesis with respect to the dataused in the study, we hypothesise that there will be a significant difference betweenparticipants’ age equivalent scores on the Pattern Analysis and Quantitative subscales ofthe Stanford–Binet Intelligence Scale: Fourth Edition (SB:IV, Thorndike, Hagen, & Sattler, 1986a,
INTERNATIONAL JOURNAL OF DISABILITY, DEVELOPMENT AND EDUCATION 153
1986b), with the age equivalent of the Quantitative subscale being lower than that of thePattern Analysis subscale.
Method
Participants
Data were drawn from a dataset held by the Down Syndrome Research Programme at TheUniversity of Queensland. This data set has been described in detail in Couzens, Cuskelly,and Haynes (2011). In essence, the datawere drawn from three groups: a population cohort,recruited at birth and contributing data to a longitudinal study into adulthood; a secondgroup who have also provided data longitudinally, but are not a population cohort; anda number of individuals who have participated in stand-alone studies. In total, the data setheld data from 209 individuals on 545 occasions.
In order to obviate the problem of some individuals providing data on multiple occasions,we created three groups based on age: 6–8 years (initial n = 64), 13–15 years (n = 106), and20–22 years (n = 73). For the purposes of this study, individuals could only be represented inone of these age groups and could only contribute a single set of data (i.e. age equivalentscores at one of the three age groups). As some participants had data across more than onegroup and/or on more than one occasion within an age group, the following decision ruleswere created: if an individual had contributed on more than one occasion across one agegroup, data were taken from the assessment that was closest to the midpoint of the group; ifdata were present in both the 6–8 years and 13–15 years groups the data from 6 to 8 yearswere retained; if data were present in both the 13–15 years and 20–22 years periods the datafrom 13 to 15 years were retained; if data were present in both the 6–8 years and 20–22 yearsgroups the data from20 to 22 years were retained; and finally, if datawere available in all threeperiods then the 13–15 years data were retained. These decisions were based on (1) maximis-ing the number of participants in each age group and (2) the view that the 13–15 years periodwas themost interesting as individuals would have had substantial opportunity to be exposedto teaching of mathematics and would still be likely to be engaged with this aspect ofacademic activity, while post-school activities were unlikely to be calling on mathematics inany structured way. The final groups comprised 41 individuals in the 6–8 age group (Meanchronological age = 7.44 yrs, SD = 0.82), 70 individuals in the 13–15 years group (Meanchronological age = 14.25 yrs, SD = 0.79) and 40 adults in the 20–22 years group (Meanchronological age = 21.15 yrs, SD = 0.19).
Instrument
Two subscales of the SB: IV were used to test the hypothesis. The Pattern Analysis subscalewas used as the measure of non-verbal cognitive ability (see, Bird, Cleave, & McConnell,2000; Swartwout, Garnaat, Myszka, Fletcher, & Dennis, 2010) and the Quantitative subscalewas used as the measure of arithmetical ability. The SB-IV had been the measure used in themultiple data collection tranches of the longitudinal study of Down syndrome and so wasthe instrument with the most useable data. In addition, as the two subscales came from thesame instrument the age equivalents were able to be compared in the knowledge that theycame from the same normative group, thus reducing problems due to mismatched data
154 M. CUSKELLY AND R. FARAGHER
sources. Age equivalent scores were used for the analysis (rather than standard scores) asthey provided more variation across the participants.
For both subscales, a basal is established by correctly answering all items on two con-secutive levels and a ceiling is reached when the individual makes three errors in four itemsacross consecutive levels. The initial item presented to the respondent on each subscale isdetermined on the basis of performance on a routing subtest. If a basal is not reached on thefirst two items presented, the assessor drops back through the levels until a basal is estab-lished. Each item correctly answered attracts one point. Correct answers are totalled (includingall items before a basal as correct). These raw scores can then be entered into a table in thetechnical manual (Thorndike et al., 1986b) to provide an age-equivalent score, i.e. the age atwhich the raw score falls at the 50th percentile.
The Pattern Analysis subscale measures fluid ability which is the ability to solve novelproblems. Items move through a form board, matching three-dimensional cube patterns, tomatching two-dimensional patterns using three-dimensional cubes. The Quantitative sub-scale is a measure of crystalised ability and assesses a range of mathematical competencies.Itemsmove through counting,matching, adding, completing a sequence andword problems.
These two subtests were completed as part of a cognitive assessment using the six coresubtests of the SB-IV and were presented using standardised procedures as laid out in thetest manual. All tests were administered by trained psychologists, experienced in workingwith individuals with intellectual disability.
Ethical clearance for this project was provided by the Behavioural & Social SciencesEthical Review Committee of The University of Queensland (#2,015,001,345).
Results
A correlational analysis was run to ascertain the association between chronological age andperformance on the subscales using age equivalent scores as the measure of performance.This analysis was also run using age group rather than age at testing. This second analysisused Spearman correlation. Significant positive correlations were found for both age attesting and age group and each of the subtests at p < .001, although correlations were onlymoderate (Pattern Analysis r = .46 (rho= .45); Quantitative r = .34 (rho= .31)). Correlationwasalso used to examine the association between the two aspects of performance. PatternAnalysis and Quantitative age equivalent scores were significantly correlated at p < .001across each of the three age groups (6–8 years r = .53; 13–15 years r = .56; 20–22 yearsr = .76) as well as for the combined sample (r = .68).
A mixed repeated measures Analysis of Variance (ANOVA) with age equivalent PatternAnalysis and Quantitative scores as the repeated measures and age group as the indepen-dent variable was run. Therewas a significantmain effect for the repeatedmeasures analysisF(1, 148) = 41.47, p < .001 ηp
2 = .22, power = 1, with performance on the Pattern Analysissubtest being significantly better than on the Quantitative subtest (see Table 1). There wasalso a significant between-subjects effect for age group, F (2,148) = 18.77, p < .001, ηp
2 = .20,power = 1 and a significant interaction F = (2, 148) = 8.08 (1), p = .001 ηp
2 = .09, power = .93.As Levene’s test was significant, the Tamhane post hoc test was used. The size of thedifference between the two subtests was greatest for the 20–22 year age group whichdiffered significantly from the other two age groups (6–8 years p = .018; 13–15 yearsp = .004) which did not differ from each other.
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Approximately 66% of the participants had a higher age equivalent on the PatternAnalysis subscale than on the Quantitative subscale and this proportion was similaracross the three age groups: 66%, 63%, 75% for the 6–8, 13–15 and 20–22 year groups,respectively. We examined the scores of the individuals for whom this pattern did nothold true to ascertain if there were floor or ceiling effects at play. Only two individuals(0.01%) were at floor for the Pattern Analysis subtest and none were at ceiling for theQuantitative subtest and so this did not explain the failure of this predominant patternto be displayed in the third of participants who had either no difference in the AE scoreson the two subtests or had a higher AE score on the Quantitative subtest than on thePattern Analysis subtest. Mean differences were calculated for the two groups afterremoval of scores of no difference (n = 5). The mean difference in age-equivalent scoresfor those who did better on Pattern Analysis than on the Quantitative subtest was1.46 years (SD = 1.26) and for the opposite pattern the mean difference was 0.84 years(SD = 0.94).
Discussion
The hypothesis was supported, which provides some endorsement of the view that dyscal-culia may form part of the Down syndrome phenotype. Further investigation is required toascertain if this early evidence is replicated in other groups with Down syndrome. Asevidence of a phenotypic effect requires that the characteristic of interest is not a productof intellectual disability generally, two considerations must be made: firstly comparisonstudies including others with intellectual disability are conducted, and secondly, regardlessof level of cognitive functioning overall, a relative weakness in arithmetic skills might beevident. These considerations are discussed in turn.
In a recent systematic review by King et al. (2017), the authors concluded that thereappeared to be little difference in the mathematical performance of children with Downsyndrome and others with intellectual disability; however, the studies included in the reviewdid not examine differences in performance between expected and actual performance. Inaddition, the authors noted the methodological challenges of matching groups that mayhave masked differences. Not all studies have found no differences between those withDown syndrome and other groups with intellectual disability. As an example, Paterson,Girelli, Butterworth, and Karmiloff-Smith (2006) found differences between children withDown syndrome and Williams syndrome in early childhood and in subsequent develop-mental trajectories with respect to number.
Table 1. Mean (standard deviation) and lower and upper age equivalent scores (in years) for thePattern Analysis and Quantitative subscales of the Stanford-Binet (Fourth edition) for three agegroups and the total sample of individuals with Down syndrome.
Age group
Stanford-Binet subtest 6–8 years (n = 41) 13–15 years (n = 70) 20–22 years (n = 40) Total (n = 151)
Pattern Analysis 3.81 (0.89) 5.29 (1.60) 6.29 (2.64) 5.15 (2.01)Range 1.00–6.08 2.00–9.67 1.00–11.08 1.00–11.08Quantitative 3.23 (1.23) 4.91 (1.76) 4.81 (2.10) 4.43 (1.87)Range 1.00–5.0 1.00–9.08 1.00–8.50 1.00–9.08
156 M. CUSKELLY AND R. FARAGHER
As demonstrated by Paterson et al. (2006), the mathematical difficulties experiencedby those with developmental disorders vary in form, and thus, possibly in origin. Theheterogeneity displayed across diagnostic groups is also apparent in those diagnosedwith dyscalculia (see Kaufmann et al., 2013). This complexity is deepened by issues ofmeasurement.
The second consideration when exploring the possibility of a phenotypic differenceconcerns the overall cognitive profile. We did not examine other academic skills and socannot ascertain if similar differences would be present between non-verbal ability andreading, for example. Other studies have reported a relative weakness in the area of arithmeticin contrast to other academic skills. For example, early literacy performance is typicallyreported to be stronger than early number performance (Buckley, 1985; King et al., 2017). Inaddition, there is evidence that dyscalculia can co-occur with other difficulties in academicskills (e.g. Orraca-Castillo, Estévez-Pérez, & Reigosa-Crespo, 2014), so evidence of difficulties inother areas of learning does not exclude dyscalculia.
Diagnosis of dyscalculia in individuals with Down syndrome is likely to require thedevelopment of specific diagnostic approaches/tests. One common approach to diag-nosis is to employ timed tests for the determination of mathematical accomplishment.This approach is taken as there is more than one way to arrive at a correct answer toa problem (e.g. one can just ‘know’ that a set of dots is 3 in number, a process known assubitising, or one can count them) and some individuals with dyscalculia may arrive atthe correct answer if given sufficient time to employ these alternative approaches, whichindicate different and less efficient cognitive processes. Individuals with intellectualdisability, including those with Down syndrome, are slower at problem-solving thanmental age-matched typically developing individuals (de Sola et al., 2015) and thustimed tests may reflect more general cognitive difficulties rather than problems specificto mathematics.
There are a number of areas of functioning that contribute to performance inmathematicsand that are known to be areas of deficit for individuals with Down syndrome. These includeworking memory, and language. There is uncertainty about the respective contributions ofthe symbolic and ANS system to difficulties with arithmetic processing and of the overlapbetween them (Szkudlarek & Brannon, 2017), and this lack of certainty adds to the complexityof unravelling the issues with respect to difficulties in performance of arithmetic tasks forthose with Down syndrome. Developing our understanding of how these are implicated inthe observed difficulties in the arithmetic areas of mathematics will be a necessary element ofidentifying potential educational approaches.
The APA definition of dyscalculia includes the stipulation that the individual does not haveintellectual disability. As pointed out by Brankaer, Ghesquière and De Smedt (2014) thisdefinition reflects an assumption that the difficulties with processing arithmetical and numer-ical information in those with intellectual disability have a different basis from that under-pinning dyscalculia in typically developing individuals. As noted above, there is substantialheterogeneity within those with the diagnosis of dyscalculia (Brankaer, Ghesqui`Ere, De Wel,Swillen, &De Smedt, 2016) and so these various aetiological possibilities should not necessarilypreclude the application of the diagnosis to those with Down syndrome.
Phenotypic expression related to a particular aetiology may change with development(Karmiloff-Smith et al., 2012); therefore, the identification of a phenotypic characteristic atone age does not necessarily mean that this characteristic will be apparent at all ages. While
INTERNATIONAL JOURNAL OF DISABILITY, DEVELOPMENT AND EDUCATION 157
there were differences between performance on the test of arithmetic and non-verbalability at all age groups, the study was not longitudinal and the difficulty with arithmeticdisplayed by individuals in each group may have had different bases. It is also clear thatinitial difficulties may interact with personal responses (e.g. avoidance of an activity that isunrewarding) and environmental responses (e.g. schooling that puts no emphasis onmathematical competence) that cement these difficulties as a life long pattern.
Limitations
This study was an opportunistic, retrospective study rather than one which collecteddata specifically identified as fit for the purpose of identifying dyscalculia in the parti-cipants. As a consequence, the data collected do not reflect the typical data used todiagnose dyscalculia. As noted above, the usual instruments are unlikely to be appro-priate for this purpose in individuals with Down syndrome (or others with intellectualdisability); therefore, before such a study can be conducted, instruments capable ofaddressing the confounds of time taken to solve problems and engagement with taskson which failure is the usual experience will need to be developed. While some of theparticipants were drawn from a population sample, some were from less rigorouslydesigned studies. The differences between these latter individuals who were volun-teered by their parents to participate in the original studies and the populations fromwhich they were members are unknown.
A second limitation relates to the academic experience of the participants. A substantialnumber of individuals will have received the majority of their education in a segregatededucational setting. There is evidence that children with Down syndrome do better academi-cally in inclusive rather than segregated settings, with mathematics being one area in whichthis advantage is apparent (deGraaf, vanHove, &Haveman, 2013). One possible reason for thisis that they are exposed to academic material in the regular classroom that is not available tothem in segregated settings.
Implications
This finding, if supported by further studies, should alter the prospects available to studentswith Down syndrome when studying mathematics. Attitudes towards teaching mathematicsto these students may change, thus ensuring access to appropriate schooling experiences.Strategies that have been found to be effective with students with dyscalculia such as theprovision of external supports – for example, the explicit teaching of the use of electroniccalculators – may allow effective engagement with aspects of the mathematics curriculumsuch as algebra. While limitations may continue to exist, they would not be a barrier to otheropportunities.
Conclusion
In this contribution to the special issue on mathematics learning and Down syndrome, weare being intentionally provocative in order to stimulate interest in this area of humancognition. Research in dyscalculia is a growing field and much is yet to be resolved ingeneral conceptions; however, there is emerging agreement that dyscalculia is a disruption
158 M. CUSKELLY AND R. FARAGHER
of the non-verbal system of mathematics cognition. In pursuing the hypothesis of a linkbetween Down syndrome and dyscalculia as proposed by Faragher (2017), we have foundsupporting evidence in a retrospective analysis of cognitive data.
We hope to stimulate research in this area to further test this hypothesis as theimplications for learning are profound. Mathematics is more than arithmetic and withthe aid of a calculating device to remove the computational load, learners with dyscal-culia can learn a great deal of mathematics. These opportunities may therefore beopened to learners with Down syndrome.
Acknowledgments
The research reported here would not have been possible without the commitment of the familiesof the children with Down syndrome involved in the Down Syndrome Research Programme at the[removed for blind review] and the ongoing commitment of the individuals with Down syndromeas they have become adults. The Down Syndrome Research Programme is supported by theMichael Cameron Fund, and we acknowledge and thank the fund for their support.
Disclosure statement
No potential conflict of interest was reported by the authors.
Funding
This work was supported by the Michael Cameron Fund and is part of the Down SyndromeResearch Program at The University of Queensland.
ORCID
Monica Cuskelly http://orcid.org/0000-0001-9986-9985Rhonda Faragher http://orcid.org/0000-0003-0245-9934
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