The mechanism of numerosity perception is often depicted as a solitary creature, living alone in the intraparietal sulcus (IPS) (its presumed birthplace), with nothing to feed it but number, and serving just the function for which it was born – giving the rest of the brain an approximation of “how many.” Although copies of this mechanism are thought to exist in the brains of an astonishing range of organisms – even those with no homologue to IPS – it is consistently predicted to have the same basic properties (noise and ratio dependence) everywhere one finds it, regardless of its age, the genes in its cells, or its history of activity.
This portrait of the numerosity mechanism – only slightly exaggerated – is in dire need of revision, and we thank Leibovich et al. for pointing toward a major issue for any future theory. As the authors note, our perception of numerosity is influenced by the spatial characteristics of the set (Frith & Frith Reference Frith and Frith1972; Gebuis et al. Reference Gebuis, Gevers and Cohen Kadosh2014). Similarly, our perception of space is influenced by numerical aspects of the set (de Hevia et al. Reference de Hevia, Girelli, Bricolo and Vallar2008). Two major – but still unsettled – issues arise from these findings. The simpler issue is describing the relation between two groups of processors: those that register the number of a set (numerosity detectors) and those that register the non-numerical spatial characteristics of the set (e.g., spatial frequency). The harder issue is to explain how this relation does or does not change over time (for a review of evidence, see McCrink & Opfer Reference McCrink and Opfer2014); this is the purpose of a developmental theory.
Although the authors have succeeded in describing the rich interactions that exist between space and number, the cornerstone of their article – their developmental theory – suffers from three major weaknesses.
First and foremost, the argument is structurally flawed. According to the authors, the number of items in a set and the non-numerical spatial properties of a set are so correlated that it is “impossible” (sect. 5.1, para. 3), “nearly impossible” (abstract), or at least really hard to tell (the authors seem to be of more than one mind on this issue) whether judgments of numerical magnitude are judgments about number or non-numerical correlates of number. Logically, then, the very existence of a stage in which organisms cannot distinguish number from magnitude is unfalsifiable. Also unfalsifiable is the very existence of a stage in which subjects can distinguish number from magnitude. This is a serious weakness. A theory that begins by pronouncing itself unfalsifiable is a non-starter.
The second challenge to their developmental theory involves the role of inhibitory control and correlational learning. “Number sense,” we are told, “develops from understanding the correlation between numerosity and continuous magnitudes” (sect. 8, para. 6). Inhibitory control then allows children to ignore irrelevant continuous magnitudes, with number words aiding this inhibitory process by emphasizing the cardinality of a set over continuous magnitudes. But this argument has a built-in contradiction: If children do not already have a sense of number, what exactly are they inhibiting when number and non-numerical magnitude conflict? The same issue arises for correlational learning. If children do not already have a sense of number, how can they track the correlation between number and continuous magnitude? Logically, number must be perceived before learning to select numerical over non-numerical cues and before learning to track what correlates with number. Empirically, this is also what the evidence indicates. Of the nine cross-modal mapping studies reviewed by Cantrell and Smith (Reference Cantrell and Smith2013), six of them found evidence of cross-modal mapping in infants (who notoriously lack inhibitory control). Therefore, we agree that inhibition and correlational learning improves the quality of numerical comparison, but their causal argument for the developmental sequence is logically untenable.
The final challenge for the developmental theory comes from the collapse of stage theories in general. Like all stage theories, the authors' theory depicts the development of numerical competence as proceeding in an invariant sequence of broadly applicable, age-related achievements. Empirically, development is seldom this orderly. Against Piaget's theory, for example, children who appeared “pre-operational” using one conservation task were found to conserve just fine on another, and the types of errors that a child would make on one conservation task seldom appeared later on the same task or on different tasks (Siegler Reference Siegler1981).
Do the stages proposed by the authors fare any better than Piaget's stages? We think not. According to their theory, children with normal visual acuity learn to correlate number with continuous magnitude only after they represent number, which occurs when they learn number words. If so, one would not expect pre-linguistic infants with normal vision to associate numerosity with magnitude, because that would violate the order of the stages. This idea has been tested directly in pre-linguistic infants with visual acuity near adult levels (de Hevia & Spelke Reference de Hevia and Spelke2010; Lourenco & Longo Reference Lourenco and Longo2010). In one such experiment, Lourenco and Longo (Reference Lourenco and Longo2010) found that 9-month-old infants easily learned an arbitrary numeric rule (less numerous sets of 2 objects are white, more numerous sets of 4 objects are black) and generalized this rule to new sets (e.g., sets of 5 and 10). Critically, infants generalized the rule to sets of a new size as well. Moreover, infants who were given a size-based rule at habituation generalize the learned rule to sets of a discrete number. Thus, pre-linguistic infants, who are not supposed to be in the stage where they can learn this sort of thing, applied a learned rule involving “more than” and “less than” across spatial and numerical dimensions, even when trained in only one dimension. These results contradict the authors' stage theory. However, they accord with a developmentally continuous theory that infants have spatial-numeric associations that arise for reasons (such as partially overlapping neural architecture) that have nothing to do with visual acuity.
In summary, although we applaud the authors for bringing attention to the findings of spatial-numeric associations in early development, we do not think the field needs another unfalsifiable and logically contradictory stage theory.
The mechanism of numerosity perception is often depicted as a solitary creature, living alone in the intraparietal sulcus (IPS) (its presumed birthplace), with nothing to feed it but number, and serving just the function for which it was born – giving the rest of the brain an approximation of “how many.” Although copies of this mechanism are thought to exist in the brains of an astonishing range of organisms – even those with no homologue to IPS – it is consistently predicted to have the same basic properties (noise and ratio dependence) everywhere one finds it, regardless of its age, the genes in its cells, or its history of activity.
This portrait of the numerosity mechanism – only slightly exaggerated – is in dire need of revision, and we thank Leibovich et al. for pointing toward a major issue for any future theory. As the authors note, our perception of numerosity is influenced by the spatial characteristics of the set (Frith & Frith Reference Frith and Frith1972; Gebuis et al. Reference Gebuis, Gevers and Cohen Kadosh2014). Similarly, our perception of space is influenced by numerical aspects of the set (de Hevia et al. Reference de Hevia, Girelli, Bricolo and Vallar2008). Two major – but still unsettled – issues arise from these findings. The simpler issue is describing the relation between two groups of processors: those that register the number of a set (numerosity detectors) and those that register the non-numerical spatial characteristics of the set (e.g., spatial frequency). The harder issue is to explain how this relation does or does not change over time (for a review of evidence, see McCrink & Opfer Reference McCrink and Opfer2014); this is the purpose of a developmental theory.
Although the authors have succeeded in describing the rich interactions that exist between space and number, the cornerstone of their article – their developmental theory – suffers from three major weaknesses.
First and foremost, the argument is structurally flawed. According to the authors, the number of items in a set and the non-numerical spatial properties of a set are so correlated that it is “impossible” (sect. 5.1, para. 3), “nearly impossible” (abstract), or at least really hard to tell (the authors seem to be of more than one mind on this issue) whether judgments of numerical magnitude are judgments about number or non-numerical correlates of number. Logically, then, the very existence of a stage in which organisms cannot distinguish number from magnitude is unfalsifiable. Also unfalsifiable is the very existence of a stage in which subjects can distinguish number from magnitude. This is a serious weakness. A theory that begins by pronouncing itself unfalsifiable is a non-starter.
The second challenge to their developmental theory involves the role of inhibitory control and correlational learning. “Number sense,” we are told, “develops from understanding the correlation between numerosity and continuous magnitudes” (sect. 8, para. 6). Inhibitory control then allows children to ignore irrelevant continuous magnitudes, with number words aiding this inhibitory process by emphasizing the cardinality of a set over continuous magnitudes. But this argument has a built-in contradiction: If children do not already have a sense of number, what exactly are they inhibiting when number and non-numerical magnitude conflict? The same issue arises for correlational learning. If children do not already have a sense of number, how can they track the correlation between number and continuous magnitude? Logically, number must be perceived before learning to select numerical over non-numerical cues and before learning to track what correlates with number. Empirically, this is also what the evidence indicates. Of the nine cross-modal mapping studies reviewed by Cantrell and Smith (Reference Cantrell and Smith2013), six of them found evidence of cross-modal mapping in infants (who notoriously lack inhibitory control). Therefore, we agree that inhibition and correlational learning improves the quality of numerical comparison, but their causal argument for the developmental sequence is logically untenable.
The final challenge for the developmental theory comes from the collapse of stage theories in general. Like all stage theories, the authors' theory depicts the development of numerical competence as proceeding in an invariant sequence of broadly applicable, age-related achievements. Empirically, development is seldom this orderly. Against Piaget's theory, for example, children who appeared “pre-operational” using one conservation task were found to conserve just fine on another, and the types of errors that a child would make on one conservation task seldom appeared later on the same task or on different tasks (Siegler Reference Siegler1981).
Do the stages proposed by the authors fare any better than Piaget's stages? We think not. According to their theory, children with normal visual acuity learn to correlate number with continuous magnitude only after they represent number, which occurs when they learn number words. If so, one would not expect pre-linguistic infants with normal vision to associate numerosity with magnitude, because that would violate the order of the stages. This idea has been tested directly in pre-linguistic infants with visual acuity near adult levels (de Hevia & Spelke Reference de Hevia and Spelke2010; Lourenco & Longo Reference Lourenco and Longo2010). In one such experiment, Lourenco and Longo (Reference Lourenco and Longo2010) found that 9-month-old infants easily learned an arbitrary numeric rule (less numerous sets of 2 objects are white, more numerous sets of 4 objects are black) and generalized this rule to new sets (e.g., sets of 5 and 10). Critically, infants generalized the rule to sets of a new size as well. Moreover, infants who were given a size-based rule at habituation generalize the learned rule to sets of a discrete number. Thus, pre-linguistic infants, who are not supposed to be in the stage where they can learn this sort of thing, applied a learned rule involving “more than” and “less than” across spatial and numerical dimensions, even when trained in only one dimension. These results contradict the authors' stage theory. However, they accord with a developmentally continuous theory that infants have spatial-numeric associations that arise for reasons (such as partially overlapping neural architecture) that have nothing to do with visual acuity.
In summary, although we applaud the authors for bringing attention to the findings of spatial-numeric associations in early development, we do not think the field needs another unfalsifiable and logically contradictory stage theory.