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February 9 th, 2010 Psychology 485

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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) don’t 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 Weber‘s 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 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

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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”

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