Introduction Different levels of numerical competence, Why
learn? How are numbers learned and processed? What is learned?
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Clever Hans Oskar Pfungst Showed Clever Hans was responding to
subtle cues
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Different levels of competence Numerosity Discriminations
Counting Understanding number as a concept Arithmetic
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More or Less Obvious advantages The more resources the better
OR
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Each item in a set is tagged Final tag is cardinal number of
the set Numerons (tags) dont need to be in any language Why count?
Keep track of offspring, kin, predators, social hierarchies
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Abstract concept e.g. Having a concept of the number 8:
eightness is a property of all sets with eight items Understand the
mathematical properties of number 8 is: the sum of 7 and 1 the sum
of 5 and 3 the product of 2 and 4
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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 Demo
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Object-file system a separate file for each item Immediate
representation of number of occupied files Limited capacity Good
for small sets Explains subitizing
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Analog-Magnitude system Number is represented by a physical
magnitude that is proportional to the number of individuals in the
set Accumulator (pulse generator)
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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 Webers law
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Scalar Expectancy Theory Pacemaker (Pulse Generator)
Accumulator Working Memory Reference Memory Ratio Comparator
Decision or Response
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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
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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
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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
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Many species have been shown to make more/less discriminations
Can be difficult to study Many confounds (time, surface area,
volume, etc)
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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
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Children dont 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
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Was taught the term none to compare size Presented with 2
blocks that are same size Asked which block larger? Taught to say
none Spontaneously transferred none to numerosities 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?
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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
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Expectancy Violation method Non-verbal method Good for children
& animals ?
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Method used with dogs, children, monkeys Look longer at
unexpected outcomes 1 + 1 = 3 or 1 + 1 = 1 Expected outcomes are
boring