Frehers Process in the Philosophical Work, Wisdom Ancient
Flawiusz Józef - Przeciw Apionowi, Pisma chrześcijańskie i pokrewne, Józef Flawiusz
Family Guy S15E01 HDTV.XviD-AFG, Napisy
Fakty i Mity Nr.40 11.10.2012, FAKTY i MITY [PDF]
Fundusze UE 2007-2013 dla mikro,
Fendt-instrukcja-obslugi-ciagniki-816-818-822-824-Favorit, Ciągniki FENDT
Forumlarz - EEA1, - Dokumenty
Fiat Seicento, Do AuTa, Fiat
Free Service Manuals and Schematics, RTV, Schematy
Fertigkeitstraining Ubungsbuch A2 Deutsch als Fremdsprache LO - Praca Zbiorowa, Nauka języków(1)
FiasFischerChapter Campbell, Number processing
[ Pobierz całość w formacie PDF ]1
Spatial Representation of Numbers
Wim Fias
1
and Martin H. Fischer
2
1
Ghent University, Belgium
2
University of Dundee, Scotland UK
WORD COUNT: 7,160
Please contact
Wim Fias
H. Dunantlaan 2
B-9000 Ghent, Belgium
Phone: +32 9 264 64 11
Fax: +32 9 264 64 96
Email: wim.fias@UGent.be
2
1. Introduction
Intuitively, we think of number processing as an abstract and non-spatial
cognitive activity. Apart from those skills necessary for mental symbol manipulation, no
spatial processing seems to be involved in numerical operations. A closer inspection,
however, shows that spatial and number processing are intimately connected. A link
between mathematical abilities and spatial skills has been anecdotally reported in the
past. Great mathematicians like Einstein explicitly emphasized the role of visuo-spatial
imagery for the development of their mathematical ideas (cf. Hadamard, 1945/1996).
About 15% of normal adults report visuo-spatial representations of numbers (Galton,
1880a,b; Seron et al., 1992). This suggests that the integration of number representations
into visuo-spatial coordinates is not a rare phenomenon. The reported spatial layouts were
predominantly oriented from left to right, were mostly automatically activated, were
stable in time and had emerged in childhood.
More systematic studies have supported these anecdotal reports by demonstrating
a tight correlation between mathematical and visuo-spatial skill. In the clinical field,
learning disorders establish a similar association between visuo-spatial and mathematical
disabilities (e.g., Rourke & Conway, 1997). Evidence from brain imaging provides
further support for a link between numbers and space. Tasks that require either number
processing or spatial transformations both tend to activate structures within the parietal
lobes (Milner & Goodale, 1995, Dehaene et al., 2003). Using transcranial magnetic
stimulation in healthy participants, Gőbel et al. (2001) showed that stimulation of the left
and right parietal cortices leads to decreased performance in both visuo-spatial search and
number comparison tasks. This suggests that the processing of numerical magnitudes
3
and of visuo-spatial information are functionally connected. Patient studies further
confirm the close link between visuo-spatial processing and basic number processing. A
particular example is Gerstmann syndrome, which is characterized by the co-occurrence
of left-right confusion, finger agnosia and dyscalculia (e.g., Dehaene & Cohen, 1997).
Thus, there appears to be a convincing case for a link between numbers and space.
None of the above reports does, however, force the conclusion that truly numerical
representations or processes are associated with spatial representations. The observed
correlation could instead reflect the involvement of shared peripheral support structures.
For example, visuo-spatial working memory is engaged in symbol manipulation during
mental arithmetic (Lee & Kang, 2002). In this chapter we will report evidence that
semantic representations of number magnitude are indeed spatially defined and can be
conceptualized as positions on an oriented “mental number line”. The idea of a linear
analog representation of numbers in the mind has been proposed a while ago (e.g., Moyer
& Landauer, 1967; Restle, 1970) to account for some basic performance patterns in
numerical cognition. More recently, this useful metaphor has been augmented by
postulating that the hypothetical mental number line also has a spatial orientation. We
will also show that this spatial cognitive representation of numbers should not be
considered as fixed and unchangeable, by demonstrating that the characteristics of spatial
number coding are largely determined by numerical and spatial parameters specific to the
task at hand. Moreover, the spatial coding of numbers is not under strategic control but
rather occurs automatically.
4
2. Mental representation of number magnitude is spatially coded: The SNARC
effect
Mental chronometry involves the timing of behavioral responses in simple
cognitive tasks. Using this approach, Dehaene et al. (1990) asked their participants to
indicate with a left or right key press whether a visually presented probe number was
smaller or larger than a previously announced reference number. For example, randomly
drawn probe numbers from 1 to 99 (but excluding 55) would be compared against the
fixed reference number 55. The decision speed in this
number comparison task
with
fixed reference was recorded and analyzed as a function of the probe number’s
magnitude and the response side. Participants who had to press the left key to indicate a
‘smaller’ response and the right key to indicate a ‘larger’ response were faster than those
who had to respond left for ‘larger’ and right for ‘smaller’ probe numbers. This response
side effect suggested that number magnitude is represented on a left-to-right oriented
mental number line, with small numbers on the left and larger numbers further on the
right side. In a seminal paper, Dehaene et al. (1993) explored this observation further.
Dehaene et al. (1993) asked their participants to decide with a left or right key
whether a single number was odd or even. In the basic version of this
parity task
, the
digits from 0 to 9 appeared repeatedly in a random order in central vision, and different
response rules (odd number - left button, even number – right button; or: even number -
left button, odd number – right button) were tested in counterbalanced blocks. In this
way, each participant’s response speed as a function of number magnitude could be
evaluated. Statistical analysis of the reaction times (RT) revealed that small numbers
5
were faster responded to with the left key, whereas large numbers consistently showed a
right key advantage. Dehaene et al. (1993) named this association of numbers with spatial
left-right response coordinates the SNARC effect for Spatial-Numerical Association of
Response Codes.
The SNARC effect is of key importance for the current issue of spatial coding of
numbers. It unequivocally demonstrates that numerical magnitude information is
spatially coded in most people. The SNARC effect as an index of the spatial attributes of
number representations has led to several studies into the nature of the mental number
line. Below, we will review these studies and their implications. But first we discuss the
measurement of the SNARC effect.
Figure 1(a) shows that the SNARC effect can be expressed as a statistical
interaction between number magnitude and response side. But because the SNARC effect
reflects an association between the position of a number on the mental number line and
the position of a response key, we can assess this spatial association more effectively
with a statistical regression analysis (Fias et al., 1996). Specifically, the difference in
RTs (dRT) for right minus left key responses will be positive for small numbers and
negative for larger numbers (see Figure 1b). The most straightforward way to capture
this negative correlation between numbers and space statistically is to regress dRT on
number magnitude for each participant and to then test the slope coefficients against zero
(Lorch & Myers, 1990; Footnote 1).
Insert Figure 1 about here
There are several advantages related to this regression-based analysis of the
SNARC effect. First, the presence of a SNARC effect is judged by a main effect (Does
[ Pobierz całość w formacie PDF ]
