This study probed the cognitive mechanisms that underlie order processing for number symbols, specifically the extent to which the direction and format in which number symbols are presented influence the processing of numerical order, as well as the extent to which the relationship between numerical order processing and mathematical achievement is specific to Arabic numerals or generalisable to other notational formats. Seventy adults who were bilingual in English and Chinese completed a Numerical Ordinality Task, using number sequences of various directional conditions (i.e., ascending, descending, mixed) and notational formats (i.e., Arabic numerals, English number words, and Chinese number words). Order processing was found to occur for ascending and descending number sequences (i.e., ordered but not non-ordered trials), with the overall pattern of data supporting the theoretical perspective that the strength and closeness of associations between items in the number sequence could underlie numerical order processing. However, order processing was found to be independent of the notational format in which the numerical stimuli were presented, suggesting that the psychological representations and processes associated with numerical order are abstract across different formats of number symbols. In addition, a relationship between the processing speed for numerical order judgements and mathematical achievement was observed for Arabic numerals and Chinese number words, and to a weaker extent, English number words. Together, our findings have started to uncover the cognitive mechanisms that could underlie order processing for different formats of number symbols, and raise new questions about the generalisability of these findings to other notational formats.
AndraszewiczS.ScheibehenneB.RieskampJ.GrasmanR.VerhagenJ.WagenmakersE. J. (2015). An introduction to Bayesian hypothesis testing for management research. Journal of Management, 41(2), 521–543.
2.
AttoutL.FiasW.SalmonE.MajerusS. (2014). Common neural substrates for ordinal representation in short-term memory, numerical and alphabetical cognition. PLOS ONE, 9(3), Article e92049.
3.
AttoutL.MajerusS. (2018). Serial order working memory and numerical ordinal processing share common processes and predict arithmetic abilities. British Journal of Developmental Psychology, 36(2), 285–298.
4.
BahnmuellerJ.MaierC. A.GöbelS. M.MoellerK. (2019). Direct evidence for linguistic influences in two-digit number processing. Journal of Experimental Psychology: Learning, Memory, and Cognition, 45(6), 1142.
5.
BarthH.KanwisherN.SpelkeE. (2003). The construction of large number representations in adults. Cognition, 86(3), 201–221.
6.
BrannonE. M. (2002). The development of ordinal numerical knowledge in infancy. Cognition, 83(3), 223–240.
7.
BulthéJ.De SmedtB.de BeeckH. O. (2014). Format-dependent representations of symbolic and non-symbolic numbers in the human cortex as revealed by multi-voxel pattern analyses. NeuroImage, 87, 311–322.
8.
CampbellJ. I.EppL. J. (2004). An encoding-complex approach to numerical cognition in Chinese-English bilinguals. Canadian Journal of Experimental Psychology/Revue Canadienne de Psychologie Expérimentale, 58(4), 229.
9.
CampbellJ. I. D.KanzC. L.XueQ. (1999). Number processing in Chinese-English bilinguals. Mathematical Cognition, 5(1), 1–39.
10.
CaoB.LiF.LiH. (2010). Format-dependent processing of numerical magnitude: Electrophysiological evidence from Chinese numerals. Biological Psychology, 83(1), 47–55.
11.
CaplanJ. B. (2015). Order-memory and association-memory. Canadian Journal of Experimental Psychology, 69(3), 221.
12.
ChambersC. D. (2019). The seven deadly sins of Psychology: A manifesto for reforming the culture of scientific practice. Princeton University Press.
13.
CipolottiL.ButterworthB. (1995). Toward a multiroute model of number processing: Impaired number transcoding with preserved calculation skills. Journal of Experimental Psychology: General, 124(4), 375–390.
14.
CohenD. J. (2010). Evidence for direct retrieval of relative quantity information in a quantity judgment task: Decimals, integers, and the role of physical similarity. Journal of Experimental Psychology: Learning, Memory, and Cognition, 36(6), 1389.
15.
CohenD. J.BarkerK. A.WhiteM. R. (2019). The effects of depression on number perception and its implications for theories of numerical cognition. Journal of Numerical Cognition, 5(1), 105–121.
16.
CohenD. J.FerrellJ. M.JohnsonN. (2002). What very small numbers mean. Journal of Experimental Psychology: General, 131(3), 424.
17.
Cohen KadoshR. (2008). Numerical representation: Abstract or nonabstract?The Quarterly Journal of Experimental Psychology, 61(8), 1160–1168.
18.
Cohen KadoshR.Cohen KadoshK.KaasA.HenikA.GoebelR. (2007). Format-dependent and-independent representations of numbers in the parietal lobes. Neuron, 53(2), 307–314.
19.
Cohen KadoshR.WalshV. (2009). Numerical representation in the parietal lobes: Abstract or not abstract?Behavioral and Brain Sciences, 32(3–4), 313–328.
20.
DaroczyG.WolskaM.MeurersW. D.NuerkH. C. (2015). Word problems: A review of linguistic and numerical factors contributing to their difficulty. Frontiers in Psychology, 6, 348.
21.
DehaeneS.AkhaveinR. (1995). Attention, automaticity, and levels of representation in number processing. Journal of Experimental Psychology: Learning, Memory, and Cognition, 21(2), 314.
22.
DehaeneS.BossiniS.GirauxP. (1993). The mental representation of parity and number magnitude. Journal of Experimental Psychology: General, 122(3), 371.
23.
DehaeneS.CohenL. (1995). Towards an anatomical and functional model of number processing. Mathematical Cognition, 1(1), 83–120.
24.
DehaeneS.Dehaene-LambertzG.CohenL. (1998). Abstract representations of numbers in the animal and human brain. Trends in Neurosciences, 21(8), 355–361.
25.
DehaeneS.PiazzaM.PinelP.CohenL. (2003). Three parietal circuits for number processing. Cognitive Neuropsychology, 20(3–6), 487–506.
26.
de HeviaM. D.AddabboM.NavaE.CrociE.GirelliL.CassiaV. M. (2017). Infants’ detection of increasing numerical order comes before detection of decreasing number. Cognition, 158, 177–188.
27.
De SmedtB.GilmoreC. K. (2011). Defective number module or impaired access? Numerical magnitude processing in first graders with mathematical difficulties. Journal of Experimental Child Psychology, 108(2), 278–292.
28.
De SmedtB.NoëlM.-P.GilmoreC.AnsariD. (2013). How do symbolic and non-symbolic numerical magnitude processing skills relate to individual differences in children’s mathematical skills? A review of evidence from brain and behavior. Trends in Neuroscience and Education, 2(2), 48–55.
29.
DienesZ. (2011). Bayesian versus orthodox statistics: Which side are you on?Perspectives on Psychological Science, 6(3), 274–290.
30.
DienesZ.McLatchieN. (2018). Four reasons to prefer Bayesian analyses over significance testing. Psychonomic Bulletin & Review, 25(1), 207–218.
31.
DowkerA.NuerkH.-C. (2016). Linguistic influences on mathematics. Frontiers in Psychology, 7, 1035.
32.
EtzA.VandekerckhoveJ. (2018). Introduction to Bayesian inference for psychology. Psychonomic Bulletin & Review, 25(1), 5–34.
33.
FaulkenberryT. J.LyA.WagenmakersE.-J. (2020). Bayesian inference in numerical cognition: A tutorial using JASP. Journal of Numerical Cognition, 6(2), 231–259.
34.
FiasW.LammertynJ.CaessensB.OrbanG. A. (2007). Processing of abstract ordinal knowledge in the horizontal segment of the intraparietal sulcus. Journal of Neuroscience, 27(33), 8952–8956.
35.
FischerM. H.ShakiS.CruiseA. (2009). It takes just one word to quash a SNARC. Experimental Psychology, 56(5), 361–366.
36.
FranklinM. S.JonidesJ.SmithE. E. (2009). Processing of order information for numbers and months. Memory & Cognition, 37(5), 644–654.
37.
Ganor-SternD.TzelgovJ. (2008). Across-Format automatic numerical processing. Journal of Experimental Psychology: Learning, Memory, and Cognition, 34(2), 430.
38.
GelmanA.LokenE. (2014). The statistical crisis in science. American Scientist, 102, 460–465.
39.
GilmoreC.GöbelS. M.InglisM. (2018). An introduction to mathematical cognition. Routledge.
40.
GöbelS. M. (2018). How culturally predominant reading direction and number word construction influence numerical cognition and mathematics performance. In BerchD. B.GearyD. C.KoepkeK. M. (Eds.), Language and culture in mathematical cognition (pp. 229–256). Academic Press.
41.
GöbelS. M.ShakiS.FischerM. H. (2011). The cultural number line: A review of cultural and linguistic influences on the development of number processing. Journal of Cross-cultural Psychology, 42(4), 543–565.
42.
GoffinC.AnsariD. (2016). Beyond magnitude: Judging ordinality of symbolic number is unrelated to magnitude comparison and independently relates to individual differences in arithmetic. Cognition, 150, 68–76.
43.
GoldfarbL.HenikA.RubinstenO.Bloch-DavidY.GertnerL. (2011). The numerical distance effect is task dependent. Memory & Cognition, 39(8), 1508–1517.
44.
GottardoA.KohP. W.ChenX.JiaF. (2017). Models of English and Chinese word reading for adolescent Chinese–English bilinguals. Reading and Writing, 30(7), 1377–1406.
45.
HarmsC.LakensD. (2018). Making “null effects” informative: Statistical techniques and inferential frameworks. Journal of Clinical and Translational Research, 3(2), 382.
46.
HawesZ.NosworthyN.ArchibaldL.AnsariD. (2019). Kindergarten children’s symbolic number comparison skills relates to 1st grade mathematics achievement: Evidence from a two-minute paper-and-pencil test. Learning and Instruction, 59, 21–33.
47.
HollowayI. D.AnsariD. (2009). Mapping numerical magnitudes onto symbols: The numerical distance effect and individual differences in children’s mathematics achievement. Journal of Experimental Child Psychology, 103(1), 17–29.
48.
HutchisonJ. E.LyonsI. M.AnsariD. (2019). More similar than different: Gender differences in children’s basic numerical skills are the exception not the rule. Child Development, 90(1), e66–e79.
49.
ItoY.HattaT. (2003). Semantic processing of Arabic, Kanji, and Kana numbers: Evidence from interference in physical and numerical size judgments. Memory & Cognition, 31(3), 360–368.
50.
JacobS. N.NiederA. (2008). The ABC of cardinal and ordinal number representations. Trends in Cognitive Sciences, 12(2), 41–43.
51.
JASP Team. (2018). JASP (Version 0.10.0) [Computer software].
52.
JeffreysH. (1961). Theory of probability (3rd ed.). Oxford University Press.
53.
KaufmannL.VogelS. E.StarkeM.KremserC.SchockeM.WoodG. (2009). Developmental dyscalculia: Compensatory mechanisms in left intraparietal regions in response to nonsymbolic magnitudes. Behavioral and Brain Functions, 5(1), 35.
54.
LakensD.MacLatchieN.IsagerP. M.ScheelA. M.DienesZ. (2020). Improving inferences about null effects with Bayes factors and equivalence tests. Journals of Gerontology: Series B, 75(1), 45–57.
55.
LeeM. D.WagenmakersE. J. (2014). Bayesian cognitive modeling: A practical course. Cambridge University Press.
56.
LeFevreJ. A.BisanzJ. (1986). A cognitive analysis of number-series problems: Sources of individual differences in performance. Memory & Cognition, 14(4), 287–298.
57.
LewandowskyS.MurdockB. B.Jr. (1989). Memory for serial order. Psychological Review, 96(1), 25.
58.
LonnemannJ.YanS. (2015). Does number word inversion affect arithmetic processes in adults?Trends in Neuroscience and Education, 4(1–2), 1–5.
59.
LovelaceE. A.SnodgrassR. D. (1971). Decision times for alphabetic order of letter pairs. Journal of Experimental Psychology, 88(2), 258.
60.
LukasS.KrinzingerH.KochI.WillmesK. (2014). Number representation: A question of look? The distance effect in comparison of English and Turkish number words. The Quarterly Journal of Experimental Psychology, 67(2), 260–270.
61.
LyonsI. M.BeilockS. L. (2011). Numerical ordering ability mediates the relation between number-sense and arithmetic competence. Cognition, 121(2), 256–261.
62.
LyonsI. M.BeilockS. L. (2013). Ordinality and the nature of symbolic numbers. Journal of Neuroscience, 33(43), 17052–17061.
63.
LyonsI. M.PriceG. R.VaessenA.BlomertL.AnsariD. (2014). Numerical predictors of arithmetic success in Grades 1–6. Developmental Science, 17(5), 714–726.
64.
LyonsI. M.VogelS. E.AnsariD. (2016). On the ordinality of numbers: A review of neural and behavioral studies. Progress in Brain Research, 227, 187–221.
65.
MarsmanM.WagenmakersE.-J. (2017). Bayesian benefits with JASP. European Journal of Developmental Psychology, 14(5), 545–555.
66.
McCloskeyM. (1992). Cognitive mechanisms in numerical processing: Evidence from acquired dyscalculia. Cognition, 44(1–2), 107–157.
67.
MerkleyR.AnsariD. (2016). Why numerical symbols count in the development of mathematical skills: Evidence from brain and behavior. Current Opinion in Behavioral Sciences, 10, 14–20.
68.
MorsanyiK.O’MahonyE.McCormackT. (2017). Number comparison and number ordering as predictors of arithmetic performance in adults: Exploring the link between the two skills, and investigating the question of domain-specificity. The Quarterly Journal of Experimental Psychology, 70(12), 2497–2517.
69.
MoyerR. S.LandauerT. K. (1967). Time required for judgements of numerical inequality. Nature, 215(5109), 1519.
70.
NoelM. P.SeronX. (1993). Arabic number reading deficit: A single case study or when 236 is read (2306) and judged superior to 1258. Cognitive Neuropsychology, 10(4), 317–339.
71.
NuerkH. C.WegerU.WillmesK. (2005). Language effects in magnitude comparison: Small, but not irrelevant. Brain and Language, 92(3), 262–277.
72.
NuerkH.-C.WillmesK.FischerM. H. (2015). Multi-digit number processing: Overview, conceptual clarifications, and language influences. In Cohen KadoshR.DowkerA. (Eds.), The Oxford handbook of numerical cognition (pp. 106–139). Oxford University Press.
73.
OrrantiaJ.MuñezD.MatillaL.SanchezR.San RomualdoS.VerschaffelL. (2019). Disentangling the mechanisms of symbolic number processing in adults’ mathematics and arithmetic achievement. Cognitive Science, 43(1), 12711.
74.
OrtegaA.NavarreteG. (2017). Bayesian hypothesis testing: An alternative to null hypothesis significance testing (NHST) in psychology and social sciences. In Bayesian Inference Intech. https://doi.org/10.5772/intechopen.70230
75.
PiazzaM. (2010). Neurocognitive start-up tools for symbolic number representations. Trends in Cognitive Sciences, 14, 542–551.
76.
PicozziM.de HeviaM. D.GirelliL.CassiaV. M. (2010). Seven-month-olds detect ordinal numerical relationships within temporal sequences. Journal of Experimental Child Psychology, 107(3), 359–367.
77.
PinelP.DehaeneS.RiviereD.LeBihanD. (2001). Modulation of parietal activation by semantic distance in a number comparison task. Neuroimage, 14(5), 1013–1026.
78.
PinelP.Le Clec’HG.Van de MoorteleP.-F.NaccacheL.Le BihanD.DehaeneS. (1999). Event-related fMRI analysis of the cerebral circuit for number comparison. Neuroreport, 10(7), 1473–1479.
79.
PoncinA.Van RinsveldA.SchiltzC. (2020). Units-first or tens-first: Does language matter when processing visually presented two-digit numbers?Quarterly Journal of Experimental Psychology, 73(5), 726–738.
80.
RestleF. (1970). Speed of adding and comparing numbers. Journal of Experimental Psychology, 83(2), 274–278.
81.
RubinstenO.DanaS.LavroD.BergerA. (2013). Processing ordinality and quantity: ERP evidence of separate mechanisms. Brain and Cognition, 82(2), 201–212.
82.
RubinstenO.SuryD. (2011). Processing ordinality and quantity: The case of developmental dyscalculia. PLOS ONE, 6(9), Article e24079.
83.
SantosC. D. S.FingerI. (2020). Bilingualism and numerical cognition. Veredas: Revista de estudos lingüísticos. Juiz de Fora, MG, 24(1), 340–360.
84.
SasanguieD.De SmedtB.ReynvoetB. (2017). Evidence for distinct magnitude systems for symbolic and non-symbolic number. Psychological Research, 81(1), 231–242.
85.
SasanguieD.LyonsI. M.De SmedtB.ReynvoetB. (2017). Unpacking symbolic number comparison and its relation with arithmetic in adults. Cognition, 165, 26–38.
86.
SasanguieD.Van den BusscheE.ReynvoetB. (2012). Predictors for mathematics achievement? Evidence from a longitudinal study. Mind, Brain, and Education, 6(3), 119–128.
87.
SavelkoulsS.WilliamsK.BarthH. (2020). Linguistic inversion and numerical estimation. Journal of Numerical Cognition, 6(3), 263–274.
88.
SchneiderM.BeeresK.CobanL.MerzS.Susan SchmidtS.StrickerJ.De SmedtB. (2016). Associations of non-symbolic and symbolic numerical magnitude processing with mathematical competence: A meta-analysis. Developmental Science, 20(3), 1–16.
ShuH.McBride-ChangC.WuS.LiuH. (2006). Understanding Chinese developmental dyslexia: Morphological awareness as a core cognitive construct. Journal of Educational Psychology, 98(1), 122.
91.
SimonsohnU. (2015). Small telescopes: Detectability and the evaluation of replication results. Psychological Science, 26(5), 559–569.
92.
SternbergS. (1969). The discovery of processing stages: Extensions of Donders’ method. Acta Psychologica, 30, 276–315.
93.
SuandaS. H.TompsonW.BrannonE. M. (2008). Changes in the ability to detect ordinal numerical relationships between 9 and 11 months of age. Infancy, 13(4), 308–337.
94.
TurconiE.CampbellJ. I.SeronX. (2006). Numerical order and quantity processing in number comparison. Cognition, 98(3), 273–285.
95.
van den BerghD.Van DoornJ.MarsmanM.DrawsT.Van KesterenE. J.DerksK.. . . WagenmakersE. J. (2020). A tutorial on conducting and interpreting a Bayesian ANOVA in JASP. Lannee Psychologique, 120(1), 73–96.
96.
VogelS. E.GrabnerR. H.SchneiderM.SieglerR. S.AnsariD. (2013). Overlapping and distinct brain regions involved in estimating the spatial position of numerical and non-numerical magnitudes: An fMRI study. Neuropsychologia, 51(5), 979–989.
97.
VogelS. E.HaighT.SommerauerG.SpindlerM.BrunnerC.LyonsI. M.GrabnerR. H. (2017). Processing the order of symbolic numbers: A reliable and unique predictor of arithmetic fluency. Journal of Numerical Cognition, 3(2), 288–308.
98.
VosH.SasanguieD.GeversW.ReynvoetB. (2017). The role of general and number-specific order processing in adults’ arithmetic performance. Journal of Cognitive Psychology, 29(4), 469–482.
99.
WagenmakersE.-J.LeeM.LodewyckxT.IversonG. J. (2008). Bayesian versus frequentist inference. In HoijtinkH.KlugkistI.BoelenP. A. (Eds.), Bayesian evaluation of informative hypotheses (pp. 181–207). Springer.
100.
WagenmakersE.-J.LoveJ.MarsmanM.JamilT.LyA.VerhagenJ.. . . BoutinB. (2018). Bayesian inference for psychology. Part II: Example applications with JASP. Psychonomic Bulletin & Review, 25(1), 58–76.
101.
WagenmakersE.-J.MarsmanM.JamilT.LyA.VerhagenJ.LoveJ.. . . EpskampS. (2018). Bayesian inference for psychology. Part I: Theoretical advantages and practical ramifications. Psychonomic Bulletin & Review, 25(1), 35–57.
102.
WagenmakersE. J.VerhagenJ.LyA.MatzkeD.SteingroeverH.RouderJ. N.MoreyR. D. (2017). The need for Bayesian hypothesis testing in psychological science. In LilienfeldS. O.WaldmanI. D. (Eds.), Psychological science under scrutiny: Recent challenges and proposed solutions (pp. 123–138). John Wiley & Sons.
103.
WongB.BullR.AnsariD. (2018). Magnitude processing of written number words is influenced by task, rather than Format. Acta Psychologica, 191, 160–170.
104.
WoodcockR. W.McGrewK. S.MatherN. (2001). Woodcock-Johnson III Tests of achievement. The Riverside Publishing Company.
105.
YangJ.ShuH.McCandlissB. D.ZevinJ. D. (2013). Orthographic influences on division of labor in learning to read Chinese and English: Insights from computational modeling. Bilingualism (Cambridge, England), 16(2), 354.
106.
ZebianS. (2005). Linkages between number concepts, spatial thinking, and directionality of writing: The SNARC effect and the reverse SNARC effect in English and Arabic monoliterates, biliterates, and illiterate Arabic speakers. Journal of Cognition and Culture, 5, 165–190.
Supplementary Material
Please find the following supplemental material available below.
For Open Access articles published under a Creative Commons License, all supplemental material carries the same license as the article it is associated with.
For non-Open Access articles published, all supplemental material carries a non-exclusive license, and permission requests for re-use of supplemental material or any part of supplemental material shall be sent directly to the copyright owner as specified in the copyright notice associated with the article.
