# What is a Weber Fraction?

### From Panamath

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== Model Representations of the ANS == | == Model Representations of the ANS == | ||

- | In modeling performance on tasks that engage the ANS, it is necessary first to specify a model for the underlying approximate number representations. It is generally agreed that each numerosity is mentally represented by a distribution of activation on an internal “number line.” These distributions are inherently “noisy” and do not represent number exactly or discretely <ref>{{Cite book | last1 = Dechaene | first1 = Stanislas | title = The Number Sense : How the Mind Creates Mathematics | url = http://www.oup.com/us/catalog/general/subject/Psychology/CognitivePsychology/?view=usa&ci=9780195132403 | publisher = Oxford University Press | isbn = 978-0-19-513240-3}}</ref><ref>{{Cite journal | author = Gallistel, C., & Gelman, R. | year = 2000 | title = Non-verbal numerical | + | In modeling performance on tasks that engage the ANS, it is necessary first to specify a model for the underlying approximate number representations. It is generally agreed that each numerosity is mentally represented by a distribution of activation on an internal “number line.” These distributions are inherently “noisy” and do not represent number exactly or discretely <ref name = "Dechaene1997">{{Cite book | last1 = Dechaene | first1 = Stanislas | title = The Number Sense : How the Mind Creates Mathematics | url = http://www.oup.com/us/catalog/general/subject/Psychology/CognitivePsychology/?view=usa&ci=9780195132403 | publisher = Oxford University Press | isbn = 978-0-19-513240-3}}</ref><ref name = "GallistelGelman2000">{{Cite journal | author = Gallistel, C., & Gelman, R. | year = 2000 | title = Non-verbal numerical |

cognition: from reals to integers | url = http://eebweb.arizona.edu/faculty/dornhaus/courses/materials/papers/Gallistel%20Gelman%20numbers%20counting%20cognition.pdf | journal = Trends in Neurosciences | volume = 4 | issue = 2 | pages = 59-65}} </ref>. This means that there is some error each time they represent a number; and this error can be thought of as a spread of activation around the number being represented. | cognition: from reals to integers | url = http://eebweb.arizona.edu/faculty/dornhaus/courses/materials/papers/Gallistel%20Gelman%20numbers%20counting%20cognition.pdf | journal = Trends in Neurosciences | volume = 4 | issue = 2 | pages = 59-65}} </ref>. This means that there is some error each time they represent a number; and this error can be thought of as a spread of activation around the number being represented. | ||

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+ | == The Mental Number Line == | ||

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+ | The mental number line is often modeled as having linearly increasing means and linearly increasing standard deviation <ref name = "GallistelGelman2000"/>. In such a format, the representation for e.g., cardinality seven is a probability density function that has its mean at 7 on the mental number line and a smooth degradation to either side of 7 such that 6 and 8 on the mental number line are also highly activated by instances of seven in the world. In Figure 1a I have drawn idealized curves which represent the ANS representations for numerosities 4-10 for an individual with Weber fraction = .125. You can think of these curves as representing the amount of activity generated in the mind by a particular array of items in the world with a different bump for each numerosity you might experience (e.g., 4 balls, 5 houses, 6 blue dots, etc). Rather than activating a single discrete value (e.g., 6) the curves are meant to indicate that a range of activity is present each time an array of (e.g., 6) items is presented <ref name = "Nieder2009">{{Cite journal | doi = 10.1146/annurev.neuro.051508.135550 | author = Nieder, A., & Dehaene, S. | year = 2009 | title = Representation of number in the brain | url = http://www.annualreviews.org/doi/abs/10.1146/annurev.neuro.051508.135550 | journal = Annual Review of Neuroscience | volume = 32 | issue = 1 | pages = 185-208}} </ref>. That is, an array of e.g., ''six'' items will greatly activate the ANS numerosity representation of 6, but because these representations are noisy this array will also activate representations of 5 and 7 etc with the amount of activation centered on 6 and gradually decreasing to either side of 6. | ||

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+ | == Neuronal Associations of the Mental Number Line == | ||

== References == | == References == | ||

{{reflist}} | {{reflist}} |

## Revision as of 14:57, 27 March 2011

## Contents |

## Model Representations of the ANS

In modeling performance on tasks that engage the ANS, it is necessary first to specify a model for the underlying approximate number representations. It is generally agreed that each numerosity is mentally represented by a distribution of activation on an internal “number line.” These distributions are inherently “noisy” and do not represent number exactly or discretely ^{[1]}^{[2]}. This means that there is some error each time they represent a number; and this error can be thought of as a spread of activation around the number being represented.

## The Mental Number Line

The mental number line is often modeled as having linearly increasing means and linearly increasing standard deviation ^{[2]}. In such a format, the representation for e.g., cardinality seven is a probability density function that has its mean at 7 on the mental number line and a smooth degradation to either side of 7 such that 6 and 8 on the mental number line are also highly activated by instances of seven in the world. In Figure 1a I have drawn idealized curves which represent the ANS representations for numerosities 4-10 for an individual with Weber fraction = .125. You can think of these curves as representing the amount of activity generated in the mind by a particular array of items in the world with a different bump for each numerosity you might experience (e.g., 4 balls, 5 houses, 6 blue dots, etc). Rather than activating a single discrete value (e.g., 6) the curves are meant to indicate that a range of activity is present each time an array of (e.g., 6) items is presented ^{[3]}. That is, an array of e.g., *six* items will greatly activate the ANS numerosity representation of 6, but because these representations are noisy this array will also activate representations of 5 and 7 etc with the amount of activation centered on 6 and gradually decreasing to either side of 6.

## Neuronal Associations of the Mental Number Line

## References

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