# Infant Number Knowledge: Analogue Magnitude Reconsidered

Alexander Green, MPhil Candidate, Department of Philosophy, University of Warwick

Following Stanislas Dehaene’s The Number Sense (1997) there has been a surge in interest in num­ber know­ledge, espe­cially the devel­op­ment of num­ber know­ledge in infants. This research has broadly focused on answer­ing the fol­low­ing ques­tions: What numer­ic­al abil­it­ies do infants pos­sess, and how do these work? How are they dif­fer­ent from the numer­ic­al abil­it­ies of adults, and how is the gap bridged in cog­nit­ive development?

The aim of this post is to provide a gen­er­al intro­duc­tion to infant num­ber know­ledge by focus­ing on the first two of these ques­tions. There is much evid­ence indic­at­ing that there are two dis­tinct sys­tems by which infants are able to track and rep­res­ent numer­os­ity — par­al­lel indi­vidu­ation and ana­logue mag­nitude. I will begin by briefly explain­ing what these numer­ic­al capa­cit­ies are. I will then focus my dis­cus­sion on the ana­logue mag­nitude sys­tem, and raise some doubts about the way in which this sys­tem is com­monly under­stood to work.

Firstly, con­sider par­al­lel indi­vidu­ation. This sys­tem allows infants to dif­fer­en­ti­ate between sets of dif­fer­ent quant­it­ies by track­ing mul­tiple indi­vidu­al objects at the same time (see Feigenson & Carey 2003; Feigenson et al 2002; Hyde 2011). For example if an infant were presen­ted with three objects, par­al­lel indi­vidu­ation would allow the track­ing of the indi­vidu­al objects ({object 1, object 2, object 3}) rather than allow­ing the track­ing of total set-size ({three objects}). There are two fur­ther points of interest about par­al­lel indi­vidu­ation. Firstly, par­al­lel indi­vidu­ation only rep­res­ents numer­os­ity indir­ectly because it track indi­vidu­als rather than total set-size. Secondly it is lim­ited to sets of few­er than four individuals.

Secondly, con­sider ana­logue mag­nitude. This sys­tem allow infants to dis­crim­in­ate between set sizes provided that the ratio is suf­fi­ciently large (see (Xu & Spelke 2000), (Feigenson et al 2004), (Xu et al, 2005)). More spe­cific­ally, ana­logue mag­nitude allows infants to dif­fer­en­ti­ate between dif­fer­ent sets provided that the ratio is at least 2:1. Interestingly the pre­cise car­din­al value of the sets seems to be irrel­ev­ant as long as the ratio remains con­stant (i.e. it applies equally to a case of two-to-four as twenty-to-forty). Thus the lim­it­a­tions of the ana­logue mag­nitude sys­tem are determ­ined by ratio, in con­trast to the par­al­lel indi­vidu­ation sys­tem whose lim­it­a­tions are determ­ined by spe­cif­ic set-size.

So how does ana­logue mag­nitude work? I will argue that the most recent answer to this ques­tion is incor­rect. This is because con­tem­por­ary authors rightly reject the ori­gin­al char­ac­ter­isa­tion of ana­logue mag­nitude (the accu­mu­lat­or mod­el), yet fail to reject its implications.

The accu­mu­lat­or mod­el of ana­logue mag­nitude is intro­duced by Dehaene, by way of an ana­logy with Robinson Crusoe (1997, p.28). Suppose that Crusoe must count coconuts. To do this he might dig a hole next to a river, and dig a trench which links the river to this hole. He also cre­ates a dam, such that he can con­trol when the river flows into the hole. For every coconut Crusoe counts, he diverts some giv­en amount of water into the hole. However as Crusoe diverts more water into the hole, it becomes more dif­fi­cult to dif­fer­en­ti­ate between con­sec­ut­ive num­bers of coconuts (i.e. the dif­fer­ence between one and two diver­sions of water is easi­er to see than between twenty and twenty-one).

Dehaene sup­poses that ana­logue mag­nitude rep­res­ent­a­tions are giv­en by a sim­il­ar icon­ic format, i.e. by rep­res­ent­ing a phys­ic­al mag­nitude pro­por­tion­al to the num­ber of indi­vidu­als in the set. Consider the fol­low­ing example: one object is rep­res­en­ted by ‘_’, two objects are rep­res­en­ted by ‘__’, three are rep­res­en­ted by ‘___’, and so on. Under this mod­el, ana­logue mag­nitude is under­stood to rep­res­ent the approx­im­ate car­din­al value of a set by the use of an iter­at­ive count­ing meth­od (Dehaene 1997, p.29). This partly reflects the empir­ic­al data: sub­jects are able to rep­res­ent dif­fer­ences in set size (with longer lines indic­at­ing lar­ger sets), and the import­ance of ratio for dif­fer­en­ti­ation is accoun­ted for (because it is more dif­fi­cult to dif­fer­en­ti­ate between sets which dif­fer by smal­ler ratios).

More recently this accu­mu­lat­or mod­el of ana­logue mag­nitude has come to be rejec­ted, how­ever. This mod­el entails that each object in a set must be indi­vidu­ally rep­res­en­ted in turn (the first object pro­duces the rep­res­ent­a­tion ‘_’, the second pro­duces the rep­res­ent­a­tion ‘__’, etc). This sug­gests that it would take longer for a lar­ger num­ber to be rep­res­en­ted than a smal­ler one (as the quant­ity of objects to be indi­vidu­ally rep­res­en­ted dif­fers). However there are empir­ic­al reas­ons to reject this.

For example there is evid­ence sug­gest­ing that the speed of form­ing ana­logue mag­nitude rep­res­ent­a­tions doesn’t vary between dif­fer­ent set sizes (Wood & Spelke 2005). Additionally, infants are still able to dis­crim­in­ate between dif­fer­ent set sizes in cases where they are unable to attend to the indi­vidu­al objects of a set in sequence (Intriligator & Cavanagh 2001). These find­ings sug­gests that it is incor­rect to claim that ana­logue mag­nitude rep­res­ent­a­tions are formed by respond­ing to indi­vidu­al objects in turn.

Despite these obser­va­tions, many authors con­tin­ue to advoc­ate the implic­a­tions of this accu­mu­lat­or mod­el even though there isn’t empir­ic­al evid­ence to sup­port these. The implic­a­tions that I am refer­ring to are that ana­logue mag­nitude rep­res­ents approx­im­ate car­din­al value and that it does so by the afore­men­tioned icon­ic format. For example, con­sider Carey’s dis­cus­sions of ana­logue mag­nitude (2001, 2009). Carey takes ana­logue mag­nitude to enable infants to ‘rep­res­ent the approx­im­ate car­din­al value of sets’ (2009, p.127). As a res­ult, the above icon­ic format (in which infants rep­res­ent a phys­ic­al mag­nitude pro­por­tion­al to the num­ber of rel­ev­ant objects) is still advoc­ated (Carey 2001, p.38). This char­ac­ter­isa­tion of ana­logue mag­nitude is typ­ic­al of many authors (e.g. Feigenson et al 2004; Slaughter et al 2006; Feigenson et al 2002; Lipton & Spelke 2003; Condry & Spelke 2008).

Given the rejec­tion of the accu­mu­lat­or meth­od, this char­ac­ter­isa­tion seems dif­fi­cult to jus­ti­fy. Analogue mag­nitude allows infants the abil­ity to dif­fer­en­ti­ate between two sets of quant­ity, but there seems no reas­on why this would require any­thing over and above the rep­res­ent­a­tion of ordin­al value (i.e. ‘great­er than’ and ‘less than’). Consequently the claim that ana­logue mag­nitude rep­res­ents approx­im­ate car­din­al value seems to be both unjus­ti­fied and unne­ces­sary. Given this there also seems to be no jus­ti­fic­a­tion for the Crusoe-analogy icon­ic format because this doesn’t con­trib­ute any­thing oth­er than allow­ing ana­logue mag­nitude to rep­res­ent approx­im­ate car­din­al value which, as we have seen, is empir­ic­ally undermined.

In this post I have dis­cussed the abil­it­ies of par­al­lel indi­vidu­ation and ana­logue mag­nitude, in answer to the ques­tion: what numer­ic­al abil­it­ies do infants pos­sess, and how do these work? Parallel indi­vidu­ation allows infants to dif­fer­en­ti­ate between small quant­it­ies of objects (few­er than four), and ana­logue mag­nitude allows dif­fer­en­ti­ation between quant­it­ies if the ratio is suf­fi­ciently large. I have also advanced a neg­at­ive argu­ment against the dom­in­ant under­stand­ing of ana­logue mag­nitude. Many authors have rejec­ted the iter­at­ive accu­mu­lat­or mod­el without reject­ing its implic­a­tions (ana­logue mag­nitude as rep­res­ent­ing approx­im­ate car­din­al value, and its doing so by icon­ic format). This sug­gests that the lit­er­at­ure requires a new under­stand­ing of how the ana­logue mag­nitude sys­tem works.

References:

Carey, S. 2001. ‘Cognitive Foundations of Arithmetic: Evolution and Ontogenisis’. Mind & Language. 16(1): 37–55.

Carey, S. 2009. The Origin of Concepts. New York: OUP.

Condry, K., & Spelke, E. 2008. ‘The Development of Language and Abstract Concepts: The Case of Natural Number.’ Journal of Experimental Psychology: General. 137(1): 22–38.

Dehaene, S. 1997. The Number Sense: How the Mind Creates Mathematics. Oxford: OUP.

Feigenson, L., Carey, S., & Hauser, M. 2002. ‘The Representations Underlying Infants’ Choice of More: Object Files versus Analog Magnitudes’. Psychological Science. 13(2): 150–156.

Feigenson, L., & Carey, S. 2003. ‘Tracking Individuals via Object-Files: Evidence from Infants’ Manual Search’. Developmental Science. 6(5): 568–584.

Feigenson, L., Dehaene, S., & Spelke, E. 2004. ‘Core Systems of Number’. Trends in Cognitive Sciences. 8(7): 307–314.

Hyde, D. 2011. ‘Two Systems of Non-Symbolic Numerical Cognition’. Frontiers in Human Neuroscience. 5: 150.

Intriligator, J., & Cavanagh, P. 2001. ‘The Spatial Resolution of Visual Attention’. Cognitive Psychology. 43: 171–216.

Lipton, J., & Spelke, E. 2003. ‘Origins of Number Sense: Large-Number Discrimination in Human Infants’. Psychological Science. 14(5): 396–401.

Slaughter, V., Kamppi, D., & Paynter, J. 2006. ‘Toddler Subtraction with Large Sets: Further Evidence for an Analog-Magnitude Representation of Number’. Developmental Science. 9(1): 33–39.

Wagner, J., & Johnson, S. 2011. ‘An Association between Understanding Cardinality and Analog Magnitude Representations in Preschoolers’. Cognition. 119(1): 10–22.

Wood, J., & Spelke, E. 2005. ‘Chronometric Studies of Numerical Cognition in Five-Month-Old Infants’. Cognition. 97(1): 23–29.

Xu, F., & Spelke, E. 2000. ‘Large Number Discrimination in 6‑Month-Old Infants’. Cognition. 74(1): B1-B11.

Xu, F., Spelke, E., & Goddard, S. 2005. ‘Number Sense in Human Infants’. Developmental Science. 8(1): 88–101.