The question of whether human quantitative abilities are rooted in an early emerging (if not innate) sense of discrete number has defined the research agenda in this area for 30 years. This superb piece of scholarship by Leibovich et al. brings the ongoing debate into sharp focus. Their conclusion, based on a thorough review of behavioral and neural evidence, is that there is no clear indication of strictly numerical processing, and indeed, it may be impossible to isolate number from its quantitative correlates. This is a stark conclusion given the intense efforts directed toward developing an account of innate number processing; however, we agree that the evidence seems to compel this conclusion. Mix et al. (Reference Mix, Huttenlocher and Levine2002a; Reference Mix, Huttenlocher and Levine2002b) raised similar concerns nearly 15 years ago, and research generated in the intervening period has done little to alter this view. We agree with the authors that the time has come to move on to different questions, but what questions might these be?
To begin, let us take stock of what we know. It would be wrong to dismiss the innovative literature that attempted to disentangle number from its correlates. Though its explicit aims may be unattainable, this body of work nonetheless contributed a large corpus of findings on which to build. It reveals, for example, that human beings are sensitive to quantitative information from a very early age – perhaps as early as the first hours of life (Antell & Keating Reference Antell and Keating1983). This fact is so commonly accepted that it may seem unimpressive. However, immediate apprehension of quantity at birth was not the only plausible developmental scenario. One might have predicted infants need months of exposure to different quantities before they notice subtle distinctions, just as they seem to need months of exposure to detect categories (e.g., Rakison & Poulin-Dubois Reference Rakison and Poulin-Dubois2001). It is actually rather astonishing that human beings come into the world prepared to process quantitative information much as they seem prepared to process physical properties like solidity and support (Baillargeon Reference Baillargeon1994) or basic percepts like color and shape (Ricciuti Reference Ricciuti1965). Without the pioneering research on preferential looking and habituation, we might not know this. Second, there is general agreement that infants and young children – in fact, people of all ages as well as some animals – process large sets differently than they process small sets (i.e., ≤3). Though this was first demonstrated in adults in the late nineteenth century (cf. Mandler & Shebo Reference Mandler and Shebo1982), there was virtually no evidence in children prior to the late 1970s. The notion that large sets are processed in a ratio-dependent manner was well known in the psychophysical literature, but the application of these ideas to quantitative perception, particularly among non-verbal infants, is a powerful advance. Finally, to our knowledge, no one has ever claimed that infants perceive discrete number to the exclusion of other quantitative cues. There is a long tradition of research examining infants' sensitivity to continuous dimensions of quantity, such as complexity, density, surface area, and brightness (e.g., Gao et al. Reference Gao, Levine and Huttenlocher2000; see Mix et al. Reference Mix, Levine, Newcombe and Henik2016 for a review) and despite attempting to isolate number from these percepts, there has been no argument about whether infants can use these alternate cues. It is widely accepted that they can, and this is an important point of agreement.
The main point of contention has been whether infants and young children isolate discrete number from a perceptual stream that obviously includes other cues. As Leibovich et al.'s review elegantly demonstrates, this question has not only gone unanswered, but also may be unanswerable. However, rather than sounding the death knell of research on early number, this conclusion raises many equally important questions that have been largely overlooked. One question is how infants might profit from the natural correlations among quantitative variables to discover how these relations work. Cantrell and Smith (Reference Cantrell and Smith2013) proposed that numerical representations are strongest and most likely to be noticed when quantitative cues are correlated (e.g., more items also have more contour, more area, more complexity, etc.). When these dimensions vary randomly, the number representations themselves are more fragile and less likely to sustain attention. On this view, infants might discover number by noticing more intense perceptual stimulation and eventually realizing this intensity is correlated with other percepts. Indeed, when the consistency of these correlations was varied, infants demonstrated greater acuity for both number and area when the correlations were maximally correlated (Cantrell et al. Reference Cantrell, Boyer, Cordes and Smith2015a). This line of inquiry demonstrates the insight gained by viewing quantitative perception as a complex system of interrelated signals, rather than a collection of isolated streams. Another important direction for future research is to coordinate the literatures on object individuation, object tracking, short-term memory, and number perception. These processes are inherently interrelated. One cannot perceive number without separating items to be enumerated and either tracking or remembering them (Mix et al. Reference Mix, Huttenlocher and Levine2002a; Reference Mix, Huttenlocher and Levine2002b; Reference Mix, Levine, Newcombe and Henik2016). Likewise, keeping track of several separate objects is tantamount to representing number (Xu & Carey Reference Xu and Carey1996). Though experiments on one of these topics usually incorporate stimulus variations from the others (e.g., Ross-Sheehy et al. Reference Ross-Sheehy, Oakes and Luck2003), the developmental relations among these processes have not been fully explored. Yet, understanding these relations may be critical as unitization could be a major mechanism for isolating number (Mix et al. Reference Mix, Levine, Newcombe and Henik2016). Related to this, the role of verbal counting in early quantification is ripe for re-examination as count words may provide the attentional scaffolding that allows children to inhibit responses to irrelevant quantitative cues. Studies on the developmental relations involving number sense and verbal number acquisition have begun to amass (see Merkley & Ansari Reference Merkley and Ansari2016 for a review) and indicate that knowledge of the count words leads to improved acuity for perceptual quantification. More studies are needed. For example, researchers might ask whether unitization emerges before or after acquiring verbal count words.
In summary, Leibovich et al.'s conclusions urge the field to regroup and consider novel ways of conceptualizing quantitative development. We are excited to see where these new directions take us.
The question of whether human quantitative abilities are rooted in an early emerging (if not innate) sense of discrete number has defined the research agenda in this area for 30 years. This superb piece of scholarship by Leibovich et al. brings the ongoing debate into sharp focus. Their conclusion, based on a thorough review of behavioral and neural evidence, is that there is no clear indication of strictly numerical processing, and indeed, it may be impossible to isolate number from its quantitative correlates. This is a stark conclusion given the intense efforts directed toward developing an account of innate number processing; however, we agree that the evidence seems to compel this conclusion. Mix et al. (Reference Mix, Huttenlocher and Levine2002a; Reference Mix, Huttenlocher and Levine2002b) raised similar concerns nearly 15 years ago, and research generated in the intervening period has done little to alter this view. We agree with the authors that the time has come to move on to different questions, but what questions might these be?
To begin, let us take stock of what we know. It would be wrong to dismiss the innovative literature that attempted to disentangle number from its correlates. Though its explicit aims may be unattainable, this body of work nonetheless contributed a large corpus of findings on which to build. It reveals, for example, that human beings are sensitive to quantitative information from a very early age – perhaps as early as the first hours of life (Antell & Keating Reference Antell and Keating1983). This fact is so commonly accepted that it may seem unimpressive. However, immediate apprehension of quantity at birth was not the only plausible developmental scenario. One might have predicted infants need months of exposure to different quantities before they notice subtle distinctions, just as they seem to need months of exposure to detect categories (e.g., Rakison & Poulin-Dubois Reference Rakison and Poulin-Dubois2001). It is actually rather astonishing that human beings come into the world prepared to process quantitative information much as they seem prepared to process physical properties like solidity and support (Baillargeon Reference Baillargeon1994) or basic percepts like color and shape (Ricciuti Reference Ricciuti1965). Without the pioneering research on preferential looking and habituation, we might not know this. Second, there is general agreement that infants and young children – in fact, people of all ages as well as some animals – process large sets differently than they process small sets (i.e., ≤3). Though this was first demonstrated in adults in the late nineteenth century (cf. Mandler & Shebo Reference Mandler and Shebo1982), there was virtually no evidence in children prior to the late 1970s. The notion that large sets are processed in a ratio-dependent manner was well known in the psychophysical literature, but the application of these ideas to quantitative perception, particularly among non-verbal infants, is a powerful advance. Finally, to our knowledge, no one has ever claimed that infants perceive discrete number to the exclusion of other quantitative cues. There is a long tradition of research examining infants' sensitivity to continuous dimensions of quantity, such as complexity, density, surface area, and brightness (e.g., Gao et al. Reference Gao, Levine and Huttenlocher2000; see Mix et al. Reference Mix, Levine, Newcombe and Henik2016 for a review) and despite attempting to isolate number from these percepts, there has been no argument about whether infants can use these alternate cues. It is widely accepted that they can, and this is an important point of agreement.
The main point of contention has been whether infants and young children isolate discrete number from a perceptual stream that obviously includes other cues. As Leibovich et al.'s review elegantly demonstrates, this question has not only gone unanswered, but also may be unanswerable. However, rather than sounding the death knell of research on early number, this conclusion raises many equally important questions that have been largely overlooked. One question is how infants might profit from the natural correlations among quantitative variables to discover how these relations work. Cantrell and Smith (Reference Cantrell and Smith2013) proposed that numerical representations are strongest and most likely to be noticed when quantitative cues are correlated (e.g., more items also have more contour, more area, more complexity, etc.). When these dimensions vary randomly, the number representations themselves are more fragile and less likely to sustain attention. On this view, infants might discover number by noticing more intense perceptual stimulation and eventually realizing this intensity is correlated with other percepts. Indeed, when the consistency of these correlations was varied, infants demonstrated greater acuity for both number and area when the correlations were maximally correlated (Cantrell et al. Reference Cantrell, Boyer, Cordes and Smith2015a). This line of inquiry demonstrates the insight gained by viewing quantitative perception as a complex system of interrelated signals, rather than a collection of isolated streams. Another important direction for future research is to coordinate the literatures on object individuation, object tracking, short-term memory, and number perception. These processes are inherently interrelated. One cannot perceive number without separating items to be enumerated and either tracking or remembering them (Mix et al. Reference Mix, Huttenlocher and Levine2002a; Reference Mix, Huttenlocher and Levine2002b; Reference Mix, Levine, Newcombe and Henik2016). Likewise, keeping track of several separate objects is tantamount to representing number (Xu & Carey Reference Xu and Carey1996). Though experiments on one of these topics usually incorporate stimulus variations from the others (e.g., Ross-Sheehy et al. Reference Ross-Sheehy, Oakes and Luck2003), the developmental relations among these processes have not been fully explored. Yet, understanding these relations may be critical as unitization could be a major mechanism for isolating number (Mix et al. Reference Mix, Levine, Newcombe and Henik2016). Related to this, the role of verbal counting in early quantification is ripe for re-examination as count words may provide the attentional scaffolding that allows children to inhibit responses to irrelevant quantitative cues. Studies on the developmental relations involving number sense and verbal number acquisition have begun to amass (see Merkley & Ansari Reference Merkley and Ansari2016 for a review) and indicate that knowledge of the count words leads to improved acuity for perceptual quantification. More studies are needed. For example, researchers might ask whether unitization emerges before or after acquiring verbal count words.
In summary, Leibovich et al.'s conclusions urge the field to regroup and consider novel ways of conceptualizing quantitative development. We are excited to see where these new directions take us.