Infant Number Knowledge: Analogue Magnitude Reconsidered

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 (ie 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 (ie 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, ie 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 (ie ‘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.