What is a Weber Fraction?

From Panamath

(Difference between revisions)
Jump to: navigation, search
(Model Representations of the ANS)
(Model Representations of the ANS)
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 <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
+
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.

Revision as of 15:13, 27 March 2011

Figures 1a - 1c: The Panamath task, and representations of numerosities on the mental number line.

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

Template:Reflist


Cite error: <ref> tags exist, but no <references/> tag was found
Personal tools