The equality of quantity

doi:10.1016/j.tics.2007.01.006

Trends in Cognitive Sciences

Volume 11, Issue 5, May 2007, Pages 185-187

https://doi.org/10.1016/j.tics.2007.01.006 Get rights and content

Do disparate dimensions of magnitude share an underlying mental representation? Two recent papers offer suggestive evidence that participants’ discrimination thresholds are identical across domains. Brannon, Lutz and Cordes showed that six-month-old infants’ area discriminations match their number discriminations. VanMarle and Wynn demonstrated the same pattern for six-month-olds’ discrimination of temporal duration. These parallels across infants’ responses to number, area and time raise questions about the fundamental nature of quantity processing.

Introduction

Adults, infants and animals represent number, area and time. We approximate the number of items in a set, the size or spatial extent of a stimulus and the duration of an event, and we use these quantifications to guide our learning and behavior. Intriguingly, our concepts of how many, how much and how long seem intertwined, at least at the metaphorical level. We refer to ‘starting back at square one’ (using spatial extent to refer to time), talk about an expensive item ‘setting us back’ (using spatial extent to refer to number) and ‘count down’ to a big event (using number to refer to time). Does this cross-referencing merely reflect human rhetorical flourish or does it belie a more fundamental connection between thinking about number, area and time? Two recent studies of infants offer exciting new evidence.

Section snippets

Analog number representations

Setting the stage for this new work is research showing that adults and infants represent number in an analog format. When comparing numerical values, adults’ speed and accuracy are determined by the ratio between quantities rather than by absolute value [1]. This means that the threshold for noticing a numerical change in a non-symbolic quantity is constant across changes in modality and changes in numerical scale. In adults, this threshold is ∼7:8 [2].

Infants rely on the same system of analog

Parallels in thresholds

A pair of developmental studies now suggests that these number representations converge with representations of area and time. First, Brannon and colleagues investigated infants’ representations of the area of a visual stimulus [5]. Six-month-olds were habituated to either a single small or a single large cartoon face, and then were tested with alternating trials of both a small and a large face. The two faces differed by a 1:4, 1:3, 1:2 or 2:3 ratio in total area. Infants discriminated all but

In what sense shared?

The possibility of unifying dimensions of quantity into a common format has historical precedent in the work of Piaget, who believed that children's concept of discrete number was borne from a concept of continuous spatial extent. This theory also finds support in proposed models that represent number and time, or number, space and time, via shared machinery 7, 8. According to the strongest view, a single mental device might represent all three dimensions. There are other possibilities. Number,

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Number symbols are processed more automatically than nonsymbolic numerical magnitudes: Findings from a Symbolic-Nonsymbolic Stroop task
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Are number symbols (e.g., 3) and numerically equivalent quantities (e.g., •••) processed similarly or distinctly? If symbols and quantities are processed similarly then processing one format should activate the processing of the other. To experimentally probe this prediction, we assessed the processing of symbols and quantities using a Stroop-like paradigm. Participants (N_Study1 = 80, N_Study2 = 63) compared adjacent arrays of symbols (e.g., 4444 vs 333) and were instructed to indicate the side containing either the greater quantity of symbols (nonsymbolic task) or the numerically larger symbol (symbolic task). The tasks included congruent trials, where the greater symbol and quantity appeared on the same side (e.g. 333 vs. 4444), incongruent trials, where the greater symbol and quantity appeared on opposite sides (e.g. 3333 vs. 444), and neutral trials, where the irrelevant dimension was the same across both sides (e.g. 3333 vs. 333 for nonsymbolic; 333 vs. 444 for symbolic). The numerical distance between stimuli was systematically varied, and quantities in the subitizing and counting range were analyzed together and independently. Participants were more efficient comparing symbols and ignoring quantities, than comparing quantities and ignoring symbols. Similarly, while both symbols and quantities influenced each other as the irrelevant dimension, symbols influenced the processing of quantities more than quantities influenced the processing of symbols, especially for quantities in the counting rage. Additionally, symbols were less influenced by numerical distance than quantities, when acting as the relevant and irrelevant dimension. These findings suggest that symbols are processed differently and more automatically than quantities.
Small-range numerical representations of linguistic sounds in 9- to 10-month-old infants
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Citation Excerpt :
The precision of infants' numerosity representations in the small numerosity range, which is the focus of the present work, has been previously investigated in the visual modality, in cross-modal studies, as well as with auditory stimuli (see Cantrell & Smith, 2013, for a review). Although it is generally stated that this representation is independent of the format of the stimuli (e.g., Feigenson, 2007), researchers found mixed results particularly inherent to the auditory modality. Several studies using visual stimuli attested that infant discriminations of exclusively small sets can be more precise than predicted by the Weber's law.
Coordinated studies provide evidence that very young infants, like human adults and nonhuman animals, readily discriminate small and large number of visual displays on the basis of numerical information. This capacity has been considerably less studied in the auditory modality. Surprisingly, the available studies yielded mixed evidence concerning whether numerical representations of auditory items in the small number range (1 to 3) are present early in human development. Specifically, while newborns discriminate 2- from 3-syllable sequences, older infants at 6 and 9 months of age fail to differentiate 2 from 3 tones. This study tested the hypothesis that infants can represent small sets more precisely when listening to ecologically relevant linguistic sounds. The aim was to probe 9- to 10-month-olds' (N = 74) ability to represent sound sets in a working memory test. In experiments 1 and 2, infants successfully discriminated 2- and 3-syllable sequences on the basis of their numerosity, when continuous variables, such as individual item duration, inter-stimulus duration, pitch, intensity, and total duration, were controlled for. In experiment 3, however, infants failed to discriminate 3- from 4-syllable sequences under similar conditions. Finally, in experiment 4, infants were tested on their ability to distinguish 2 and 3 tone sequences. The results showed no evidence that infants discriminated these non-linguistic stimuli. These findings indicate that, by means of linguistic sounds, infants can access a numerical system that yields precise auditory representations in the small number range.
Discrimination of ordinal relationships in temporal sequences by 4-month-old infants
2020, Cognition
Citation Excerpt :
If, as some theories suggest, different quantity dimensions (particularly number, space and time) are represented via a common representational code (ATOM theory: Bueti & Walsh, 2009; Walsh, 2003), we would expect that the same processing constraints apply to these dimensions. This prediction is already partially supported by the finding that the same ratios are required across development to successfully discriminate unequal values for the dimensions of number, size, and time (Brannon et al., 2006; Feigenson, 2007), and that human newborns and infants spontaneously relate quantities across these dimensions and expect them to vary in congruent ways (de Hevia, Izard, Coubart, Spelke, & Streri, 2014; de Hevia & Spelke, 2010; Srinivasan & Carey, 2010). More crucially, the hypothesis that the same computational mechanisms and constraints characterize the processing of number and size is supported by the finding of a common asymmetry signature in ordinal processing for both of size-based (Macchi Cassia et al., 2012) and numerical sequences (de Hevia, Addabbo et al., 2017).
The ability to discriminate the ordinal information embedded in magnitude-based sequences has been shown in 4-month-old infants, both for numerical and size-based sequences. At this early age, however, this ability is confined to increasing sequences, with infants failing to extract and represent decreasing order. Here we investigate whether the ability to represent order extends to duration-based sequences in 4-month-old infants, and whether it also shows the asymmetry signature previously observed for number and size. Infants were tested in an order discrimination task in which they were habituated to either increasing or decreasing variations in temporal duration, and were then tested with novel sequences composed of new temporal items whose durations varied following the familiar and the novel orders in alternation. Across three experiments, we manipulated the duration of the single temporal items and therefore of the whole sequences, which resulted in imposing more or less constraints on infants’ working memory, or general processing capacities. Results showed that infants failed at discriminating the ordinal direction in temporal sequences when the sequences had an overall long duration (Experiment 1), but succeeded when the duration of the sequences was shortened (Experiments 2 and 3). Moreover, there was no sign of the asymmetry signature previously reported for number and size, as successful discrimination was present for infants habituated to both increasing and decreasing sequences. These results suggest that sensitivity to temporal order is present very early in development, and that its functional properties are not shared with other magnitude dimensions, such as size and number.
The spatial representation of numbers and time follow distinct developmental trajectories: A study in 6- and 10-year-old children
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Space-number and space-time associations have been a timely topic in the cognitive sciences for years, but evidence from developmental populations is still scarce. In particular, it remains to be established whether space-number and space-time mappings are anchored onto the same spatial frame of reference across development. To explore this issue, we manipulated visual and proprioceptive feedback in a Number Comparison task (Experiment 1) and a Time Comparison task (Expriment 2), in which 6- and 10-year-old children had to classify numerical and temporal words by means of a lateralised response with or without a blindfold (visual manipulation), and with hands uncrossed or crossed over the body midline (proprioceptive manipulation). Results revealed that 10-year-old children were more efficient in associating smaller numbers and past events with the left key, and larger numbers and future events with the right key, irrespective of the visual and proprioceptive manipulations. On the contrary, younger children did so only in the Time Comparison task, but not in the Number Comparison task. In the latter task, 6-year-olds associated small/large numbers with the left/right side of space only in the presence of visual feedback, but not when blindfolded. Taken together, our findings unveil that in school-aged children the spatial representation of number and time develop on different spatial frames of reference: while space-time associations exclusively rely on external coordinates at age 6, space-number associations shift from mixed internal and external coordinates at age 6 to more adult-like external coordinates by age 10.
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Trends in Cognitive Sciences

UpdateResearch FocusThe equality of quantity

Introduction

Section snippets

Analog number representations

Parallels in thresholds

In what sense shared?

Cognition

Cognition

Trends Cogn. Sci.

J. Exp. Child Psychol.

Infant Behav. Dev.

Exact and approximate arithmetic in an Amazonian indigene group

Science

Discrimination of large and small numerosities by human infants

Infancy

The development of area discrimination and its implications for number representation in infancy

Dev. Sci.

Update
Research Focus
The equality of quantity