The Number Comparison Model

1
http//ccnl.psy.unipd.it
Representation of numerical magnitude The
numerosity model
Marco Zorzi DPG - University of
Padova ICN - University College London
2

• We have many ways of representing numbers ..
• What form of representation supports comparison
  of number size and simple whole-number
  arithmetic?

3

• Number comparison - one of the fundamental
  numerical abilities
• Number comparison requires the generation of an
  abstract representation corresponding to each
  numeral (McCloskey et al., 1985)
• Number comparison is performed on the basis of
  an analog magnitude code (e.g. Moyer Landauer,
  1967 Dehaene Cohen, 1995), and with magnitudes
  represented as segments of a "number line"
  (Gallistel Gelman, 1992)
• However, the details of the comparison process
  operating on these codes is largely unspecified.

4
Classic RT Effects
(Found with arabic numerals, patterns of dots,
and written-word numerals)
The distance (split) effect the latency is an
inverse function of the numerical distance
between the two numerals.
The number size effect for a given split, pairs
of small numbers are compared faster than pairs
of large numbers.
R20.43
R20.26
Data from Butterworth, Zorzi, Girelli,
Jonckheere, 2001, QJEP (corrected for naming time
of the larger)
5
Compressive effects The response latencies in
number comparison are well represented by the
equation proposed by Welford (1960)
Larger RT a k log
Larger-Smaller which is the best predictor of RT
data, accounting for about 50 of the variance
(e.g., Moyer Landauer, 1973 Dehaene, 1989
Butterworth et al., 2001) How can we account for
the compressive (logarithmic) effect?
6
A Fechnerian encoding of numerical magnitude?
The mapping from numerals to the number line is
non-linear, because the number line is
compressive (i.e., it obeys Weber-Fechner
logarithmic law). (Dehaene, 1992 Dehaene,
Dupoux Mehler, 1992) Thus, the subjective
difference between two numbers will depend on
their positions on the line, that is, the
subjective difference between N and N1 will be
smaller as N increases.
Compressed number line the representation of a
number n is a gaussian centered according to a
logarithmic scale and with fixed variance
7
Or a noisy mapping to the number line?
The mapping from numerals to the number line is
linear, but the variability of the mapping
increases in proportion to the magnitude (scalar
variability) (Gallistel Gelman, 1992) Thus,
the discriminability of the two numbers decreases
as their mean numerical value increases, not
because they are subjectively closer together,
but because the variability (noise) in the
mapping is scalar.
Linear number-line with scalar variability the
representation of a number n is a gaussian
centered at n with variance proportional to n
8
HOWEVER The reaction time data for the
judgement of physical magnitudes across a wide
range of domains (e.g., line length, pitch,
weight) are well represented by the same
logarithmic equation - the Welford function. How
much of the effects found in number comparison
should be attributed to the representation of
numerical magnitudes rather than to the
implementation of a decision process?
9

• The representation of numerosity
• numerosities are discrete, not analogue (a set
  of things with an exact whole number of members)
• bigger numbers include smaller numbers -
  cardinality
• coded as the number of units activated (also
  known as a thermometer representation)
• the representation is linear, not compressive
• there is no scalar variability (no noise)

10

• Advantages of numerosity representation
• It readily maps onto lower level perceptual
  processes (e.g., object identification) and
  enumeration procedures (e.g., subitizing,
  counting) each magnitude increment corresponds
  to the enumeration of a further element in the
  to-be-counted set.
• Larger numbers are more similar to each other
  than smaller numbers, without assuming a
  logarithmic compression, since large numbers
  share more active nodes (can be formalised in
  terms of the cosine of the angle formed by the
  two vectors).

11
The Number Comparison Model
Five
Decision System (select larger)
Numerosity Representation
12
Decision system A general dynamic mechanism
based on lateral inhibition previously used for
modeling response interference effects (Zorzi
Umiltà, 1995, PsyRes Tagliabue, Zorzi, Umiltà,
Bassignani, 2000, JEPHPP)
Response criterion Difference between response
nodes gt0.5, similar to the Lateralized Readiness
Potential - LRP (Coles et al., 1988 Gratton et
al., 1988)
13
R20.77
R20.42
R20.52
The model RTs account for 42 of the variance in
the human RT data (N72)
Fit of the Welford function R20.82
14
Numerosity representations support number
comparison in a straightforward way. Is
numerosity involved in arithmetic fact retrieval?

15
Previous computational models used number line
(barcode) representations an input number
activates its own representation and its
immediate neighbours (McCloskey Lindemann,
1992 Anderson, Bennet, Spoehr, 1994)
Magnitude Low Medium High
BSB (Viscuso, Anderson and Spoehr, 1989) Does not
account for RT problem-size effect MATHNET
(McCloskey Lindemann, 1992) RT problem-size
effect produced by implausible manipulation of
fact frequency
16
Can we do anything better with numerosity ?
Please be patient .. .. and wait for Stoianovs
talk this afternoon ..
17
Number Priming
Recent data show a symmetric priming effect for
N-1, N-2, N-3 vs. N1, N2, N3. Is this
incompatible with numerosity representations?
Does numerosity predict an asymmetric effect?
Data from Reynvoet, Brysbaert Fias, QJEP, in
press
18
NO! We get a symmetric priming effect Results
from a dynamic network trained on a transcoding
task (priming with numerosity). Fit to human
data R20.48(n42)
Note the time to access numerosity is a
nonlinear function of number size (as in human
data, Brysbaert, 1995, JEPGEN)
19
However, we do predict an asymmetric effect in
the case of negative priming
.. dont miss Priftiss (second) talk later ..
20

• Summary
• The core property of the numerosity model is the
  representation of cardinal meaning (cardinality)
• Numerosity representations support
• number comparison
• calculation

21
What about the popular mental number line ? Are
we claiming that there is no such thing?
Not at all! There is a form of numerical
representation with strong visuo-spatial
properties - encoding ordinal meaning.

• Some tasks and/or effects tap this kind of
  representation, e.g.
• the SNARC effect
• number bisection task
• .. dont miss Priftiss talk ..

22
Collaborators Sue Becker, McMasters Brian
Butterworth, London Konstantinos Priftis,
Trieste Ivilin Stoianov, Padova Carlo Umiltà,
Padova Funding European Commission - RTN Royal
Society of London
http//ccnl.psy.unipd.it
23
(No Transcript)