Michelle van Diën – 413154md
Problem 5. Numerical brain: From number sense to counting the numbers
Feigenson, L., Dehaene, S., & Spelke, E. (2004). Core systems of number. Trends in Cognitive Sciences, 8(7),
307–314. doi:10.1016/j.tics.2004.05.002
What representations underlie the ability to think and reason about number? Certain numerical concepts, such
as the real numbers, are only ever represented by a subset of human adults, but other numerical abilities are
widespread and can be observed in adults, infants and other animal species. Review recent behavioral and
neuropsychological evidence that these ontogenetically and phylogenetically shared abilities rest on two core
systems for representing number. Performance signatures common across development and across species
implicate one system for representing large, approximate numerical magnitudes, and a second system for the
precise representation of small numbers of individual objects. These systems account for our basic numerical
intuitions, and serve as the foundation for the more sophisticated numerical concepts that are uniquely human.
Review evidence that two distinct core systems of numerical representations are present in human infants and
in other animal species, and therefore do not emerge through individual learning or cultural transmission.
Systems are automatically deployed, are tuned only to specific types of information, and continue to function
throughout the lifespan.
Systems are limited in their representational power. Neither system supports concepts of fractions,
square roots, negative numbers, or exact integers. The construction of natural, rational and real numbers
depends on arduous processes that are probably accessible only to educated humans in a subset of cultures,
but which nevertheless are rooted in the two systems that are our current focus and that account for humans’
basic ‘number sense’.
Core system 1: Approximate representations of numerical magnitude
Core system 1 in infants
Even in infancy, children exhibit numerical knowledge.
Limits on infants’ representations of number:
1. Infants’ numerical discriminations are imprecise and subject to a ratio limit: 6-month-old infants successfully
discriminate 8 vs. 16 and 16 vs. 32 dots, but fail with 8 vs. 12 and 16 vs. 24 under the same conditions.
2. Numerical discrimination increases in precision over development: 6-month-olds can discriminate
numerosities with a 1:2 but not a 2:3 ratio, whereas 10-month-old infants also succeed with the latter, and
adults can discriminate ratios as small as 7:8.
3. Numerical discrimination fails when infants are tested with very small numerosities in tasks controlled for
continuous variables: infants fail to discriminate arrays of 1 vs. 2, 2 vs. 4, and 2 vs. 3 dots, even though these
differ by the same ratios at which infants succeed with larger numerosities.
Infants’ approximate number representations are not limited to visual arrays. Infants’ discrimination
depends on abstract representations of numerosity. These abstract representations support number-relevant
computations. Infants recognize ordinal relationships between numerosities, and form expectations about the
outcomes of simple arithmetic problems such as 5 + 5.
Models representing numerosity as a fluctuating mental magnitude, akin to a ‘number line’, shared
across modalities. Two competing mathematical formulations of the number line. Highly similar behavioral
predictions.
1. The linear model with scalar variability represents the number line as a series of equally spaced distributions
with increasing spread.
2. The logarithmic model with fixed variability represents successive numerosities on a logarithmic scale subject
to a fixed amount of noise.
Both models: larger numerosities are represented by distributions that overlap increasingly with nearby
numerosities. This variability increases the likelihood of confusing a target with its neighbors, yielding infants’
ratio-dependent performance.
Core system 1 in older children and adults
Older children and adults share this system for representing large, approximate numerosities. When shown
arrays of dots or sequences of sounds under conditions that prevent counting, adults discriminate numerosities
when continuous variables are controlled, their discrimination is subject to a ratio limit, and the ratio limit is
Problem 5. Numerical brain: From number sense to counting the numbers
Feigenson, L., Dehaene, S., & Spelke, E. (2004). Core systems of number. Trends in Cognitive Sciences, 8(7),
307–314. doi:10.1016/j.tics.2004.05.002
What representations underlie the ability to think and reason about number? Certain numerical concepts, such
as the real numbers, are only ever represented by a subset of human adults, but other numerical abilities are
widespread and can be observed in adults, infants and other animal species. Review recent behavioral and
neuropsychological evidence that these ontogenetically and phylogenetically shared abilities rest on two core
systems for representing number. Performance signatures common across development and across species
implicate one system for representing large, approximate numerical magnitudes, and a second system for the
precise representation of small numbers of individual objects. These systems account for our basic numerical
intuitions, and serve as the foundation for the more sophisticated numerical concepts that are uniquely human.
Review evidence that two distinct core systems of numerical representations are present in human infants and
in other animal species, and therefore do not emerge through individual learning or cultural transmission.
Systems are automatically deployed, are tuned only to specific types of information, and continue to function
throughout the lifespan.
Systems are limited in their representational power. Neither system supports concepts of fractions,
square roots, negative numbers, or exact integers. The construction of natural, rational and real numbers
depends on arduous processes that are probably accessible only to educated humans in a subset of cultures,
but which nevertheless are rooted in the two systems that are our current focus and that account for humans’
basic ‘number sense’.
Core system 1: Approximate representations of numerical magnitude
Core system 1 in infants
Even in infancy, children exhibit numerical knowledge.
Limits on infants’ representations of number:
1. Infants’ numerical discriminations are imprecise and subject to a ratio limit: 6-month-old infants successfully
discriminate 8 vs. 16 and 16 vs. 32 dots, but fail with 8 vs. 12 and 16 vs. 24 under the same conditions.
2. Numerical discrimination increases in precision over development: 6-month-olds can discriminate
numerosities with a 1:2 but not a 2:3 ratio, whereas 10-month-old infants also succeed with the latter, and
adults can discriminate ratios as small as 7:8.
3. Numerical discrimination fails when infants are tested with very small numerosities in tasks controlled for
continuous variables: infants fail to discriminate arrays of 1 vs. 2, 2 vs. 4, and 2 vs. 3 dots, even though these
differ by the same ratios at which infants succeed with larger numerosities.
Infants’ approximate number representations are not limited to visual arrays. Infants’ discrimination
depends on abstract representations of numerosity. These abstract representations support number-relevant
computations. Infants recognize ordinal relationships between numerosities, and form expectations about the
outcomes of simple arithmetic problems such as 5 + 5.
Models representing numerosity as a fluctuating mental magnitude, akin to a ‘number line’, shared
across modalities. Two competing mathematical formulations of the number line. Highly similar behavioral
predictions.
1. The linear model with scalar variability represents the number line as a series of equally spaced distributions
with increasing spread.
2. The logarithmic model with fixed variability represents successive numerosities on a logarithmic scale subject
to a fixed amount of noise.
Both models: larger numerosities are represented by distributions that overlap increasingly with nearby
numerosities. This variability increases the likelihood of confusing a target with its neighbors, yielding infants’
ratio-dependent performance.
Core system 1 in older children and adults
Older children and adults share this system for representing large, approximate numerosities. When shown
arrays of dots or sequences of sounds under conditions that prevent counting, adults discriminate numerosities
when continuous variables are controlled, their discrimination is subject to a ratio limit, and the ratio limit is
