Commentary on the article: From sense of number to sense of magnitude The role of continuous magnitudes in numerical cognition

Transcription

1 Commentary on the article: From sense of number to sense of magnitude The role of continuous magnitudes in numerical cognition 1. Target Article Authors: Leibovich, Katzin, Harel and Henik 2. Word counts: Word count abstract: 60 Word count main text: 911 Word count references: 309 Word count total: Commentary Title: Infants and number: give credit where credit is due. 4. Author Names: Sophie Savelkouls; Sara Cordes 5. Institution: Boston College 6. Institutional Mailing address: Boston College Department of Psychology 300 McGuinn Hall 140 Commonwealth Ave. Chestnut Hill, MA Institutional Telephone number: addresses: 9. Home page URL:

2 Abstract Leibovich et al. overlook numerous human infant studies pointing to an early emerging number sense. These studies have carefully manipulated continuous magnitudes in the context of a numerical task revealing that infants can discriminate number when extent is controlled, that infants fail to track extent cues with precision, and that infants find changes in extent less salient than numerical changes.

3 In presenting their case for an acquired number sense dependent upon continuous magnitudes, Leibovich and colleagues overlook a substantial literature revealing that continuous magnitudes across sets of multiple objects - are relatively difficult to track (Barth, 2008). Importantly, the authors rely primarily upon data from adults to support their developmental model, a population shown to invoke advanced estimation strategies while concurrently shying away from more intuitive strategies (Siegler & Booth, 2005). As such, adult data are more likely to reflect learned notions and strategies with regards to number, not intuitions. Evidence from early human development is required to appropriately address questions of inherent abilities. Notably, however, the authors disregard numerous findings with preverbal infants contradicting their claims of the primacy of continuous magnitudes. In particular, these studies reveal infants: (1) can discriminate number when continuous extent is controlled, (2) are relatively poor discriminators of continuous extent, and (3) are more likely to attend to changes in number compared to changes in continuous extent. Similarly to Leibovich et al., we review evidence only pertaining to large, non-symbolic numerosities (>4). First, while it is true that infant abilities to detect numerical changes can be influenced by continuous extent cues (e.g., Cantrell, Boyer, Cordes, & Smith, 2015), habituation studies reveal infants are capable of tracking number independent of these cues. The presence of purely numerical abilities early in development is consistent with a number sense account. Leibovich et al. discount infant habituation methods, arguing that inherent confounds between number and extent prevent studies from isolating numerical abilities. The authors fail to acknowledge a number of sophisticated designs that have taken advantage of the initial habituation phase to dissociate number and continuous extent (Xu & Spelke, 2000). In these designs, infants are

4 habituated to a series of exemplars of a single numerosity while extent cues are systematically varied; in subsequent test trials, the degree of change in extent across novel and familiar number test trials is equated. As such, these designs allow for one and only one dimension of discrimination: number. Under these circumstances when extent is not a reliable cue for discrimination infants have repeatedly succeeded in detecting changes in number (e.g. Lipton & Spelke, 2003, 2004; Xu & Arriaga, 2007; Xu & Spelke, 2000). If early numerical abilities were entirely dependent upon continuous magnitudes, these robust findings would not be possible. Second, infants track continuous extent with relatively poor precision. If it is the case that infants rely upon continuous magnitudes when comparing sets, then infants should be at least as good at detecting changes in continuous extent as they are at detecting changes in number. This is not the case. Studies that have examined infant abilities to track continuous extent across multiple items have consistently found that infants are remarkably limited in their ability to discriminate extent. In particular, 6-7 month olds require as much as a 4-fold change in cumulative area (Brannon, Abbott, & Lutz, 2004; Cordes & Brannon, 2008) or individual element size (Cordes & Brannon, 2011) and a 3-fold change in contour length (Starr & Brannon, 2015) in order to successfully detect a change in these continuous variables. In contrast, parallel studies isolating numerical abilities reveal that infants of this age reliably discriminate strikingly smaller changes in number (2-fold; Xu & Spelke, 2000). That is, infants notice 2-fold changes in number (when extent is deconfounded), but require the size of individual items in an array to quadruple in size in order to detect a change in extent. Although infants are capable of tracking continuous extent, they are remarkably poor at doing so.

5 Third, infants find numerical changes to be more salient than extent changes. If infants relied upon continuous extent representations for tracking quantity in the world, this would not only entail a precise ability to track changes in extent, but a greater proclivity to notice these changes over changes in number. Looking-time studies directly pitting changes in number against changes in continuous extent however, demonstrate that infants more readily attend to numerical changes. Cordes and Brannon (2009) pitted number against contour length and cumulative surface area and found that in both cases, 7-month old infants dishabituated to changes in number but not to changes in continuous extent. Additionally, Libertus, Starr, and Brannon (2014) found that even when a 1:3 ratio change in number was pitted against a 1:10 ratio change in cumulative area in a change detection paradigm, infants still did not show a preference for continuous extent, despite this dramatically larger change in the cumulative area. If numerical intuitions rested upon continuous magnitudes, infants would more readily notice changes in extent. They do not. In sum, we argue that a careful reading of the infant literature provides strong support for an intuitive number sense. The finding that infants can track continuous extent in numerical tasks does not undermine evidence from other studies revealing that infants attend to number independent of extent. Yet preverbal infants robust successes on numerical tasks do destabilize claims of an acquired number sense. Moreover, data revealing that infants are relatively poor discriminators of continuous extent and that infants find numerical changes more salient than extent changes weaken assertions that magnitude-tracking underlies numerical processing, posing a significant challenge to the neo-piagetian model posed by Leibovich et al. Whether,

6 over the course of development into adulthood, representations of number and magnitude become more closely intertwined, remains to be determined; however we believe that the infant data firmly point to an early and intuitive number sense.

7 References Barth, H. (2008). Do mental magnitudes form part of the foundation for natural number concepts? Don t count them out yet. Behavioral and Brain Sciences, 31(6), Brannon, E. M., Abbott, S., & Lutz, D. J. (2004). Number bias for the discrimination of large visual sets in infancy. Cognition, 93(2), B Cantrell, L., Boyer, T. W., Cordes, S., & Smith, L. B. (2015). Signal clarity: an account of the variability in infant quantity discrimination tasks. Developmental Science, 18(6), Cordes, S., & Brannon, E. M. (2008). The difficulties of representing continuous extent in infancy: Using number is just easier. Child Development, 79(2), Cordes, S., & Brannon, E. M. (2009). The relative salience of discrete and continuous quantity in young infants. Developmental Science, 12(3), Cordes, S., & Brannon, E. M. (2011). Attending to One of Many: When Infants are Surprisingly Poor at Discriminating an Item s Size. Frontiers in Psychology, 2, Libertus, M. E., Starr, A., & Brannon, E. M. (2014). Number trumps area for 7-month-old infants. Developmental Psychology, 50(1), Lipton, J. S., & Spelke, E. S. (2003). Origins of Number Sense: Large-number discrimination in Human Infants. Psychological Science, 14(5),

8 Lipton, J. S., & Spelke, E. S. (2004). Discrimination of Large and Small Numerosities by Human Infants. Infancy, 5(3), Siegler, R. S., & Booth, J. L. (2005). Development of numerical estimation. In Handbook of mathematical cognition (pp ). Starr, A., & Brannon, E. M. (2015). Evidence against continuous variables driving numerical discrimination in infancy. Frontiers in Psychology, 6, Xu, F., & Arriaga, R. I. (2007). Number discrimination in 10-month-old infants. British Journal of Developmental Psychology, 25(1), Xu, F., & Spelke, E. S. (2000). Large number discrimination in 6-month-old infants. Cognition, 74(1),