Non-symbolic arithmetic in adults and young children

doi:10.1016/j.cognition.2004.09.011

Cognition

Volume 98, Issue 3, January 2006, Pages 199-222

https://doi.org/10.1016/j.cognition.2004.09.011 Get rights and content

Abstract

Five experiments investigated whether adults and preschool children can perform simple arithmetic calculations on non-symbolic numerosities. Previous research has demonstrated that human adults, human infants, and non-human animals can process numerical quantities through approximate representations of their magnitudes. Here we consider whether these non-symbolic numerical representations might serve as a building block of uniquely human, learned mathematics. Both adults and children with no training in arithmetic successfully performed approximate arithmetic on large sets of elements. Success at these tasks did not depend on non-numerical continuous quantities, modality-specific quantity information, the adoption of alternative non-arithmetic strategies, or learned symbolic arithmetic knowledge. Abstract numerical quantity representations therefore are computationally functional and may provide a foundation for formal mathematics.

Section snippets

Experiment 1

In Experiment 1, adults performed numerical comparison and addition tasks on large sets of elements, presented either as visual arrays of dots or as a mixture of arrays of dots and sequences of tones. For the visual comparison task, subjects were presented with two arrays of dots in sequence and judged which array contained more elements. For the crossmodal comparison task, subjects were presented with an array of dots preceded or followed by a sequence of tones and judged whether the array or

Experiment 2

A second experiment tested adults' capacity for non-symbolic subtraction by comparing performance in comparison tasks to performance in subtraction tasks with visual arrays. For the subtraction task, subjects again were presented with three visual arrays of dots. They were asked to subtract the second array from the first and to compare this difference to the number of elements in the third array. For the comparison tasks, subjects compared two dot arrays, the first of which was equal in number

Experiment 3

To provide a direct comparison of addition and subtraction tasks and to present both tasks in a more natural context, we developed a non-symbolic arithmetic paradigm using animated sequences of events similar to those used in studies of human infants and non-human primates (Hauser et al., 1996, Sulkowski and Hauser, 2001, Wynn, 1992). For the addition task (Fig. 2A) we presented one visual quantity, occluded it, and then presented a second quantity that moved behind the occluder to join the

Experiment 4

In 1 Experiment 1, 2 Experiment 2, 3 Experiment 3, participants were adults with years of mathematical training. It is possible, therefore, that they assigned verbal numerical labels to each visual or auditory set and performed the addition and subtraction operations symbolically. Subjects' reports do not support this explanation.¹

Experiment 5

Experiment 5 replicated the findings of Experiment 4 with additional controls that allowed us to test for strategies based on summing the circumferences of the dots rather than the number of dots, and with a test of knowledge of relevant symbolic arithmetic facts.

Participants. Eighteen 5-year-old children (range 5;0–5;6, mean 5;1) were recruited from the same population as Experiment 4.

Method. The method was the same as in Experiment 4 except as follows. Children were given 4 practice

General discussion

Three experiments provide evidence that adults can mentally add two arrays, or subtract one array from the other, and then compare the sum or difference to a third array. Addition was as accurate as comparison, was equally accurate when the two addends appeared in the same vs. different modalities and formats, and showed the ratio signature of large number representations. Subtraction produced a performance deficit relative to addition, but this difference may be due to variability in the

Acknowledgements

We thank Mary C. Potter and C.R. Gallistel for comments on an earlier version of this manuscript. This research was supported by National Institute of Health grant MH56037 to N. Kanwisher, National Institute of Health grant R37 HD23103 to E. Spelke, National Science Foundation ROLE grant REC-0087721 to N. Kanwisher and E. Spelke, a National Academy of Education/Spencer Foundation Postdoctoral Fellowship to H. Barth, and a McDonnell centennial fellowship to S. Dehaene.

References (37)

H. Barth et al.
The construction of large number representations in adults
Cognition
(2003)
R.M. Church et al.
Alternate representations of time, number, and rate
Cognition
(1990)
S. Dehaene et al.
Abstract representations of numbers in the human and animal brain
Trends in Neurosciences
(1998)
C.R. Gallistel et al.
Preverbal and verbal counting and computation
Cognition
(1992)
J. Gibbon et al.
Toward a neurobiology of temporal cognition: Advances and challenges
Current Opinion in Neurobiology
(1997)
M.D. Hauser et al.
Spontaneous representations of small numbers of objects by rhesus macaques: Examinations of content and format
Cognitive Psychology
(2003)
T. Kobayashi et al.
Baby arithmetic: One object plus one tone
Cognition
(2004)
T.J. Simon
Reconceptualizing the origins of number knowledge: A ‘non-numerical’ account
Cognitive Development
(1997)
G.M. Sulkowski et al.
Can rhesus monkeys spontaneously subtract?
Cognition
(2001)
O. Zur et al.
Young children can add and subtract by predicting and checking
Early Childhood Research Quarterly
(2004)

E.M. Brannon et al.

The evolution and ontogeny of ordinal numerical ability

E.M. Brannon et al.

Numerical subtraction in the pigeon: Evidence for a linear subjective number scale

Psychological Science

(2001)

S. Carey

Evidence for numerical abilities in young infants: A fatal flaw?

Developmental Science

(2002)

M.W. Clearfield et al.

Number versus contour length in infants' discrimination of small visual sets

Psychological Science

(1999)

M.W. Clearfield et al.

Amount versus number: Infants' use of area and contour length to discriminate small sets

Journal of Cognition and Development

(2001)

S. Dehaene

Subtracting pigeons: Logarithmic or linear?

Psychological Science

(2001)

S. Dehaene et al.

Is numerical comparison digital? Analogical and symbolic effects in two-digit number comparison

Journal of Experimental Psychology: Human Perception and Performance

(1990)

S. Dehaene et al.

Sources of mathematical thinking: Behavioral and brain-imaging evidence

Science

(1999)

Cited by (243)

Characterizing exact arithmetic abilities before formal schooling
2023, Cognition
Children appear to have some arithmetic abilities before formal instruction in school, but the extent of these abilities as well as the mechanisms underlying them are poorly understood. Over two studies, an initial exploratory study of preschool children in the U.S. (N = 207; Age = 2.89–4.30 years) and a pre-registered replication of preschool children in Italy (N = 130; Age = 3–6.33 years), we documented some basic behavioral signatures of exact arithmetic using a non-symbolic subtraction task. Furthermore, we investigated the underlying mechanisms by analyzing the relationship between individual differences in exact subtraction and assessments of other numerical and non-numerical abilities. Across both studies, children performed above chance on the exact non-symbolic arithmetic task, generally showing better performance on problems involving smaller quantities compared to those involving larger quantities. Furthermore, individual differences in non-verbal approximate numerical abilities and exact cardinal number knowledge were related to different aspects of subtraction performance. Specifically, non-verbal approximate numerical abilities were related to subtraction performance in older but not younger children. Across both studies we found evidence that cardinal number knowledge was related to performance on subtraction problems where the answer was zero (i.e., subtractive negation problems). Moreover, subtractive negation problems were only solved above chance by children who had a basic understanding of cardinality. Together these finding suggest that core non-verbal numerical abilities, as well as emerging knowledge of symbolic numbers provide a basis for some, albeit limited, exact arithmetic abilities before formal schooling.
Value computation in humans
2022, Evolution and Human Behavior
Things afford positive, neutral, or negative long-run effects on the replicative probability of the focal individual's genes. At the most general level, values are internal estimates of those effects. Value information steers physiology and behavior in the right direction: approach apple, avoid lion. Thus, value computation is of paramount biological importance. Task analysis suggests there are many prerequisites for valuing things aptly. Here, I focus on two: the need to compute value accurately, and the need to properly feed and integrate value information into the various systems that use value information (e.g., emotion systems). For example, the subjective food value imputed to an apple needs to reflect the nutrient content of the apple (accuracy); the intensity of gratitude aroused if someone gave you an apple needs to reflect the food value imputed to the apple (integration). Here, I evaluate these hypotheses with two preregistered studies. Consistent with the integration hypothesis, there are close correspondences between (i) the food values that participants impute to each of 40 food items (Study 1; goods) and (ii) the social values and the social emotions (including: gratitude, anger, shame, and pride) that result when those food items occur as constituents of broader social events. Similar correspondences are observed when participants evaluate each of 28 diseases and injuries (Study 2; bads). Consistent with the accuracy hypothesis, exploratory analyses indicate that the food values, the social values, and the social emotions elicited by the food items all track the nutrient content of those food items. Valuation is inherently a computational process. For this reason, a computational–functionalist perspective is distinctively suited to spur progress in our understanding of human values.
Numerosity sense correlates with fluent mathematical abilities
2022, Acta Psychologica
Citation Excerpt :
This negative finding might have resulted from the lack of mathematical fluency related to number facts; children aged 3–7 years find these difficult because they are not yet able to respond automatically and quickly. In contrast, numerosity sense, which is something we are born with and which is barely affected by education, is easy to process, and even very young children can achieve fluency (e.g., Barth et al., 2005, 2006, 2008; Flombaum et al., 2004; Li et al., 2018; Libertus et al., 2011; McCrink & Wynn, 2004; Pica et al., 2004; Starr et al., 2013; Wynn, 1992). Similar results come from children who have been formally educated, especially those aged 8 to 11 years (grades 3 through 6; e.g., Zhang et al., 2016; Wang et al., 2016).
Although a great deal of research has shown a relationship between numerosity sense and mathematical ability, some studies have failed to do so. The main source of this inconsistency could be the varied ways of measuring mathematical abilities. The current investigation explored several types of mathematical ability, from basic number processing and arithmetic computation to numerical reasoning and arithmetic learning. We hypothesized that the correlation between numerosity sense and mathematical ability depends on mathematical fluency. A total of 415 college students (178 males and 237 females, mean age = 20.42 years, range = 18.58–22.92 years) were recruited to complete seven mathematical tasks and two numerosity tasks, as well as other tasks that measured cognitive covariates. The results showed that after controlling for age, gender, and related general cognitive factors, numerosity sense still predicted substantial variation in parity judgment, visual digit comparison, and computation, but it did not predict variation in numerosity estimation, auditory digit comparison, number series completion, or digit associate learning. The results suggest that numerosity sense correlates with mathematical abilities that accompany fluency.
Enumeration takes time: Accuracy improves even after stimuli disappear
2022, Cognition
Numerical cognition is widespread among animal species and present from birth in humans. Laboratory studies show that longer stimulus duration improves numerical discrimination accuracy (Inglis & Gilmore, 2013; Wood & Spelke, 2005). Inglis and Gilmore (2013) suggested that longer durations allow subjects to resample the stimulus image multiple times, resulting in a more accurate final estimate. The current study tested this “multiple sampling” model alongside two competing models – “serial foveal accumulator” (Cheyette & Piantadosi, 2019) and “longer processing time” – to determine how stimulus duration relates to accuracy. Adult subjects completed a fully within-subject, 2AFC task in which they judged which of two arrays had more dots; accuracy was the dependent measure. Experiment 1 revealed higher accuracy in the 500 ms stimulus duration condition compared to the 100 ms condition, replicating previous results. Experiments 2 and 4 extended this finding. When stimulus duration was held constant at 100 ms, adding a 400 ms delay between stimulus offset and mask onset improved accuracy. Accuracy was similar in the 500 ms stimulus duration condition (E1) and the 100 ms duration + 400 ms mask delay conditions (E2 and E4), indicating that post-stimulus processing time improves accuracy, rather than stimulus duration per se. This contrasts with both the multiple sampling model and the serial foveal model, although we cannot exclude the possibility that sampling continues in iconic memory. Numerosity was a significant factor in all four experiments, suggesting a serial component in the enumeration process. These findings shed light on the cognitive processes underlying nonverbal number representation in humans.
Operational momentum during children's approximate arithmetic relates to symbolic math skills and space–magnitude association
2022, Journal of Experimental Child Psychology
Operational momentum (OM) refers to the behavioral tendency to overestimate or underestimate the results of addition or subtraction, respectively. The cognitive mechanism of the OM effect and how it is related to the development of symbolic math abilities are not well understood. The current study examined whether individual differences in the OM effect are related to symbolic arithmetic abilities, number line estimation performance, and the space–magnitude association effect in young children. In this study, first-grade elementary school children manifested the OM effect during approximate addition and subtraction. Individual differences in the OM effect were not correlated with number line estimation error. Interestingly, children who showed a greater degree of the OM effect performed not worse, but better on the symbolic arithmetic task. In addition, the OM effect was correlated with the space–magnitude association (size congruity) effect measured with the Numerical Stroop task. More specifically, the OM bias was correlated with the ability to inhibit interference from competing information on the incongruent trials of the Numerical Stroop task. Our results suggest that the inaccuracy of numerical magnitude representations is not the source of the OM effect. Given that children with better math ability showed a greater OM bias, a stronger OM effect may reflect better intuition in arithmetic operations. Altogether, we carefully interpret these findings as suggesting that a greater OM effect reflects superior intuition or fundamental knowledge of arithmetic operations and a more adult-like maturation of the reorienting component of the attentional system.
Canonical representations of fingers and dots trigger an automatic activation of number semantics: an EEG study on 10-year-old children
2021, Neuropsychologia
Over the course of development, children must learn to map a non-symbolic representation of magnitude to a more precise symbolic system. There is solid evidence that finger and dot representations can facilitate or even predict the acquisition of this mapping skill. While several behavioral studies demonstrated that canonical representations of fingers and dots automatically activate number semantics, no study so far has investigated their cerebral basis. To examine these questions, 10-year-old children were presented a behavioral naming task and a Fast Periodic Visual Stimulation EEG paradigm. In the behavioral task, children had to name as fast and as accurately as possible the numbers of dots and fingers presented in canonical and non-canonical configurations. In the EEG experiment, one category of stimuli (e.g., canonical representation of fingers or dots) was periodically inserted (1/5) in streams of another category (e.g., non-canonical representation of fingers or dots) presented at a fast rate (4 Hz). Results demonstrated an automatic access to number semantics and bilateral categorical responses at 4 Hz/5 for canonical representations of fingers and dots. Some differences between finger and dot configuration's processing were nevertheless observed and are discussed in light of an effortful-automatic continuum hypothesis.

View all citing articles on Scopus

View full text

Non-symbolic arithmetic in adults and young children

Abstract

Section snippets

Experiment 1

Experiment 2

Experiment 3

Experiment 4

Experiment 5

General discussion

Acknowledgements

Cognition

Cognition

Trends in Neurosciences

Cognition

Current Opinion in Neurobiology

Cognitive Psychology

Cognition

Cognitive Development

Cognition

Early Childhood Research Quarterly

The evolution and ontogeny of ordinal numerical ability

Numerical subtraction in the pigeon: Evidence for a linear subjective number scale

Psychological Science

Evidence for numerical abilities in young infants: A fatal flaw?

Developmental Science

Number versus contour length in infants' discrimination of small visual sets

Psychological Science

Amount versus number: Infants' use of area and contour length to discriminate small sets

Journal of Cognition and Development

Subtracting pigeons: Logarithmic or linear?

Psychological Science

Is numerical comparison digital? Analogical and symbolic effects in two-digit number comparison

Journal of Experimental Psychology: Human Perception and Performance

Sources of mathematical thinking: Behavioral and brain-imaging evidence

Science