# What is a Weber Fraction?

(Difference between revisions)
 Revision as of 15:13, 27 March 2011 (view source)Panamath (Talk | contribs) (→Model Representations of the ANS)← Older edit Revision as of 15:13, 27 March 2011 (view source)Panamath (Talk | contribs) (→Model Representations of the ANS)Newer edit → Line 1: Line 1: + [[Image:mentalnumberline.jpg|thumb|left|alt=Figures 1a - 1c: The Panamath task, and representations of numerosities on the mental number line.| Figures 1a - 1c: The Panamath task, and representations of numerosities on the mental number line.]] + == Model Representations of the ANS == == Model Representations of the ANS == - [[Image:mentalnumberline.jpg|thumb|left|alt=Figures 1a - 1c: The Panamath task, and representations of numerosities on the mental number line.| Figures 1a - 1c: The Panamath task, and representations of numerosities on the mental number line.]]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 {{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}}{{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 {{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}}{{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}} .  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}} .  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.

## 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.

## References

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