February 9th, 2010
Psychology 485
 Introduction
• Different levels of numerical competence,
• Why learn?
 How
are numbers learned and
processed?
 What is learned?
 Clever
Hans
 Oskar Pfungst
• Showed Clever
Hans was
responding to
subtle cues
 Different
levels of competence
• Numerosity Discriminations
• Counting
• Understanding number as a concept
 Arithmetic
 More
or Less
 Obvious advantages
• The more resources
the better
OR
 Each
item in a set is
‘tagged’
 Final ‘tag’ is cardinal
number of the set
 Numerons (tags) don’t
need to be in any
language
 Why count?
• Keep track of offspring,
kin, predators, social
hierarchies
 Abstract concept
• e.g. Having a concept of the number 8:
 “eightness” is a property of all sets with eight items
 Understand
of number
the mathematical properties
• 8 is:
 the sum of 7 and 1
 the sum of 5 and 3
 the product of 2 and 4
 Subitizing
• Rapid, accurate and confident judgements of
number
• Set sizes 1 to 4
 Counting or Estimating
• Increased time, or decreased accuracy for set sizes
greater than 4
• Amount of time needed increase per item
 Demo
 Object-file
system
• a separate “file” for each
•
•
•
•
item
Immediate representation
of number of “occupied”
files
Limited capacity
Good for small sets
Explains subitizing
 Analog-Magnitude
system
• Number is represented by
a physical magnitude that
is proportional to the
number of individuals in
the set
• Accumulator (pulse
generator)
 Analog-Magnitude
system
• Discriminability is
proportional to ratio
• Easy to discriminate
 1 vs 2
 3 vs 8
• Harder to discriminate
 7 vs 8
 15 vs 16
• Consistent with
Weber‘s law
 Scalar
Expectancy Theory
Pacemaker
(Pulse Generator)
Accumulator
Reference
Memory
Working Memory
Ratio
Comparator
Decision or
Response
 Meck
& Church (1983)
 Rats trained to:
• Press one lever after 2 x 1-second tone pulses
• Press another lever after 8 x 1-second pulses
 Total
duration and number are redundant
cues
 Test for control by time and number
 Control
by number
• Present 2 or 8 pulses over span of 4 seconds
 Control
by time
• Present 4 pulses in 2 or 8 second span
 Time
and number controlled response
equally
 Equal
responding at geometric mean (not
arithmetic)
 Time
and number processed
simultaneously
• Cognitive economy/simplicity
• Less mechanisms to be “built in”
 Many
species have been shown to make
more/less discriminations
 Can be difficult to study
• Many confounds (time, surface area, volume, etc)
 Sequential
(not simultaneous) numerosity
discriminations
• Shows animals “keeping track” of values
 Capaldi
and colleagues
• Trained rats with patterns of reward/no reward
at end of runway
 NRRN or RRN – count to 2
 Rats run fast for reward, slowly for no reward
 Children
don’t usually understand
concept of “zero” until 3 or 4 years old
 Can be difficult to teach
 In animals
• Alex, the African Grey Parrot
• Ai, chimpanzee
 Was taught the term “none” to compare
• Presented with 2 blocks that are same size
• Asked “which block larger?”
• Taught to say “none”
 Spontaneously
numerosities
transferred “none” to
• Presented with 3 sets: 2, 3, 6
• Asked which set contained 5 blocks
• Answered “none”
 Further
tests showed he applied term to
absence of quantity
• Shown empty tray, asked “How many?”
size
 Taught
arabic number symbols
 Shown numbers 0, 1, 4, 7, 9
• Asked to select the lowest number
• Chooses zero
 Can
match number of dots on screen to
arabic numeral
• Shown three dots, will select symbol “3”
• Shown no dots, will select symbol “0”
 Expectancy Violation
method
• Non-verbal method
• Good for children & animals
?
 Method
used with dogs, children,
monkeys
 Look longer at unexpected outcomes
• 1+1=3
 Expected
or
1+1=1
outcomes are “boring”