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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) 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

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