The authors present a provocative theoretical model describing how the development of the number concept unfolds based on children's experiences of the correlations between continuous magnitudes and number in their environment. This theory rests on the assumption that infants cannot readily represent number because they cannot individuate visually presented objects early in life and that continuous magnitudes do not suffer from this constraint and are hence easier to represent. We question this assumption based on three lines of empirical evidence: (1) parallels between infants' abilities to represent numerical information presented in the visual and auditory modalities, (2) infants' ability to match numerical information across sensory modalities, and (3) infants' greater facility in discriminating number compared with continuous magnitudes in the visual modality.
First, audition develops already in utero (Hepper & Shahidullah Reference Hepper and Shahidullah1994), and infants are born with sophisticated auditory abilities such as the ability to discriminate their mother's voice from a female stranger's voice (DeCasper & Fifer Reference DeCasper and Fifer1980). With respect to numerical skills, 6-month-old infants are able to discriminate sequences of tones when the numbers differ by a 1:2 ratio (e.g., 8 tones vs. 16 tones) but fail when they differ only by a 2:3 ratio (Lipton & Spelke Reference Lipton and Spelke2003). Auditory numerical stimuli obviate the need for individuation because tones are typically presented sequentially. However, infants' ability to discriminate auditory numerical stimuli is remarkably similar to their ability to discriminate simultaneously presented visual arrays of objects (i.e., 6-month-old infants can discriminate visually presented numerosities that differ by a 1:2 ratio, but not a 2:3 ratio [Xu & Spelke Reference Xu and Spelke2000]). If, as hypothesized by Leibovich et al., visual object individuation only emerges around 5 months of age and visual number discrimination relies on the ability to individuate objects, it is unclear why infants' number discrimination abilities in the visual and auditory modalities are subject to the same thresholds. Similarly, based on their hypothesis, one would expect that visual number discrimination should be improved if visual stimuli are presented sequentially and visual object individuation is no longer an obstacle. However, 6-month-olds are unable to discriminate a unimodal visual sequence when numbers differ by a 2:3 ratio, similar to what is found for simultaneously presented visual stimuli (Jordan et al. Reference Jordan, Suanda and Brannon2008b).
Second, infants are able to match numerical information across sensory modalities from birth (Feigenson Reference Feigenson2011; Izard et al. Reference Izard, Sann, Spelke and Steri2009). For example, newborns look longer at a visual stimulus that contains the same number of objects as a sequence of tones they hear (Izard et al. Reference Izard, Sann, Spelke and Steri2009). Similar to the ratio-dependent discrimination observed with unimodal stimuli, newborns' cross-modal matching of numerical information is ratio dependent; that is, they are able to match the number of sounds to the right number of objects when the correct and incorrect numbers of objects differ by a 1:3 ratio, but not when they differ by a 1:2 ratio. These findings suggest that infants are able to match numerical information across sensory modalities and across simultaneous and sequential presentation formats. If infants were only able to represent continuous magnitudes as hypothesized by Leibovich et al., it is unclear which continuous dimensions infants would match across sensory modalities and how they would do so without a reference point to determine which value in a given dimension is more or less. For example, should the average tone duration be matched to an average object size, and if so, which duration should correspond to which size? Without familiarization to a range of tone durations and object sizes, it seems impossible for infants to create the reference frame that is necessary to match continuous dimensions. Relying on number is the most parsimonious explanation for the observed patterns in infants' behavior.
Third, infants show greater facility in discriminating number than continuous magnitudes. For example, 6-month-old infants need a 1:4 ratio difference to detect a change in cumulative surface area and a 1:3 ratio difference to detect a change in cumulative perimeter, but only a 1:2 ratio difference to detect a change in number (Cordes & Brannon Reference Cordes and Brannon2008; Starr & Brannon Reference Starr and Brannon2015). Furthermore, when a change in cumulative surface area is directly pitted against a change in number, infants prefer to look at the change in number (Libertus et al. Reference Libertus, Starr and Brannon2014). This preference cannot be attributed to detecting a change in individual element sizes as Leibovich et al. argue (see sect. 3) because individual element size changed by the same ratio as change in number and change in cumulative surface area; that is, when a 1:3 ratio change in number was pitted against a 1:3 ratio change in cumulative surface area, the size of individual elements in both cases changed by a 1:3 ratio. When a 1:3 ratio change in number was pitted against a 1:5 ratio change in cumulative surface area, the size of the individual elements changed by 1:3 and 1:5 ratios, respectively. Despite the greater change in individual element size that accompanied the change in cumulative surface area, infants looked significantly longer at the change in number that was accompanied by a smaller change in individual element size. Therefore, changes in individual element size cannot explain why infants would attend more to a change in number than a change in cumulative surface area. The most parsimonious explanation for the observed findings is that infants are more sensitive to changes in number than continuous magnitudes and not vice versa as Leibovich et al. suggest.
Taken together, these three lines of research suggest that it is most parsimonious to assume that the concept of number is present early in development and that its acquisition does not rest on the acquisition of visual object individuation and experiences with correlations between number and continuous magnitude representations.
The authors present a provocative theoretical model describing how the development of the number concept unfolds based on children's experiences of the correlations between continuous magnitudes and number in their environment. This theory rests on the assumption that infants cannot readily represent number because they cannot individuate visually presented objects early in life and that continuous magnitudes do not suffer from this constraint and are hence easier to represent. We question this assumption based on three lines of empirical evidence: (1) parallels between infants' abilities to represent numerical information presented in the visual and auditory modalities, (2) infants' ability to match numerical information across sensory modalities, and (3) infants' greater facility in discriminating number compared with continuous magnitudes in the visual modality.
First, audition develops already in utero (Hepper & Shahidullah Reference Hepper and Shahidullah1994), and infants are born with sophisticated auditory abilities such as the ability to discriminate their mother's voice from a female stranger's voice (DeCasper & Fifer Reference DeCasper and Fifer1980). With respect to numerical skills, 6-month-old infants are able to discriminate sequences of tones when the numbers differ by a 1:2 ratio (e.g., 8 tones vs. 16 tones) but fail when they differ only by a 2:3 ratio (Lipton & Spelke Reference Lipton and Spelke2003). Auditory numerical stimuli obviate the need for individuation because tones are typically presented sequentially. However, infants' ability to discriminate auditory numerical stimuli is remarkably similar to their ability to discriminate simultaneously presented visual arrays of objects (i.e., 6-month-old infants can discriminate visually presented numerosities that differ by a 1:2 ratio, but not a 2:3 ratio [Xu & Spelke Reference Xu and Spelke2000]). If, as hypothesized by Leibovich et al., visual object individuation only emerges around 5 months of age and visual number discrimination relies on the ability to individuate objects, it is unclear why infants' number discrimination abilities in the visual and auditory modalities are subject to the same thresholds. Similarly, based on their hypothesis, one would expect that visual number discrimination should be improved if visual stimuli are presented sequentially and visual object individuation is no longer an obstacle. However, 6-month-olds are unable to discriminate a unimodal visual sequence when numbers differ by a 2:3 ratio, similar to what is found for simultaneously presented visual stimuli (Jordan et al. Reference Jordan, Suanda and Brannon2008b).
Second, infants are able to match numerical information across sensory modalities from birth (Feigenson Reference Feigenson2011; Izard et al. Reference Izard, Sann, Spelke and Steri2009). For example, newborns look longer at a visual stimulus that contains the same number of objects as a sequence of tones they hear (Izard et al. Reference Izard, Sann, Spelke and Steri2009). Similar to the ratio-dependent discrimination observed with unimodal stimuli, newborns' cross-modal matching of numerical information is ratio dependent; that is, they are able to match the number of sounds to the right number of objects when the correct and incorrect numbers of objects differ by a 1:3 ratio, but not when they differ by a 1:2 ratio. These findings suggest that infants are able to match numerical information across sensory modalities and across simultaneous and sequential presentation formats. If infants were only able to represent continuous magnitudes as hypothesized by Leibovich et al., it is unclear which continuous dimensions infants would match across sensory modalities and how they would do so without a reference point to determine which value in a given dimension is more or less. For example, should the average tone duration be matched to an average object size, and if so, which duration should correspond to which size? Without familiarization to a range of tone durations and object sizes, it seems impossible for infants to create the reference frame that is necessary to match continuous dimensions. Relying on number is the most parsimonious explanation for the observed patterns in infants' behavior.
Third, infants show greater facility in discriminating number than continuous magnitudes. For example, 6-month-old infants need a 1:4 ratio difference to detect a change in cumulative surface area and a 1:3 ratio difference to detect a change in cumulative perimeter, but only a 1:2 ratio difference to detect a change in number (Cordes & Brannon Reference Cordes and Brannon2008; Starr & Brannon Reference Starr and Brannon2015). Furthermore, when a change in cumulative surface area is directly pitted against a change in number, infants prefer to look at the change in number (Libertus et al. Reference Libertus, Starr and Brannon2014). This preference cannot be attributed to detecting a change in individual element sizes as Leibovich et al. argue (see sect. 3) because individual element size changed by the same ratio as change in number and change in cumulative surface area; that is, when a 1:3 ratio change in number was pitted against a 1:3 ratio change in cumulative surface area, the size of individual elements in both cases changed by a 1:3 ratio. When a 1:3 ratio change in number was pitted against a 1:5 ratio change in cumulative surface area, the size of the individual elements changed by 1:3 and 1:5 ratios, respectively. Despite the greater change in individual element size that accompanied the change in cumulative surface area, infants looked significantly longer at the change in number that was accompanied by a smaller change in individual element size. Therefore, changes in individual element size cannot explain why infants would attend more to a change in number than a change in cumulative surface area. The most parsimonious explanation for the observed findings is that infants are more sensitive to changes in number than continuous magnitudes and not vice versa as Leibovich et al. suggest.
Taken together, these three lines of research suggest that it is most parsimonious to assume that the concept of number is present early in development and that its acquisition does not rest on the acquisition of visual object individuation and experiences with correlations between number and continuous magnitude representations.