1. February 9 th, 2010 Psychology 485

2.  Introduction Different levels of numerical competence, Why learn?  How are numbers learned and processed?  What is learned?

3.  Clever Hans  Oskar Pfungst Showed Clever Hans was responding to subtle cues

4.  Different levels of competence Numerosity Discriminations Counting Understanding number as a concept  Arithmetic

5.  More or Less  Obvious advantages The more resources the better OR

6.  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

7.  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

9.  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

10.  Object-file system a separate “file” for each item Immediate representation of number of “occupied” files Limited capacity Good for small sets Explains subitizing

11.  Analog-Magnitude system Number is represented by a physical magnitude that is proportional to the number of individuals in the set Accumulator (pulse generator)

12.  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

13.  Scalar Expectancy Theory Pacemaker (Pulse Generator) Accumulator Working Memory Reference Memory Ratio Comparator Decision or Response

14.  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

15.  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

16.  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”

18.  Many species have been shown to make more/less discriminations  Can be difficult to study Many confounds (time, surface area, volume, etc)

19.  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

20.  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

21.  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?”

22.  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”

23.  Expectancy Violation method Non-verbal method Good for children & animals ?

24.  Method used with dogs, children, monkeys  Look longer at unexpected outcomes 1 + 1 = 3 or 1 + 1 = 1  Expected outcomes are “boring”