understanding number discriminate –based on number and nothing else categorize –generalize...

Understanding Number

• Discriminate– based on number and nothing else

• Categorize– generalize across situations – label

• Conceptualize– serial learning (1, 2, 3, 4…)

• transitive inference• Scanning in pigeons vs. chimps

– operations (2 + 1 =3)

Early Counting Experiments

• Clever Hans– answers math questions

Clever Hans

Early Counting Experiments

• Clever Hans– answers math questions

• Spontaneous Counting– not impressive

Early Counting Experiments

• Clever Hans– answers math questions

• Spontaneous Counting- not impressive

• Mechner (1958)– press lever “X’ times– pigeons can tell the difference between 45 and 50 pecks– time passed and not number?

• Other examples of responding based on counts which seem to generalize

Generalize AcrossModalities

Discriminate Number (not Time)

• Acquisition

– RRRN

– NRRRN

R = RewardN = Nonreward

Categorize (Generalize)

• Acquisition

– CPPN

– PPN

• Shift

– PCCN

– CCN

P = Pellets

N = Nonreward

C = Corn Pops

Categorize (Label)

Answers Correctly:“How many blocks?”

Not Subitization

• Perceptual not cognitive

• Estimating small quantities at a glance without counting

• Hard with similar distractors

How Many?

Squares Ovals

Conceptualize (Serial Learning)

• Transitive inference

• Simultaneous chaining

Three-Item Transitive Inference

Train A > B B > C

Test A > C

But…………..A is 100% reinforced and C is 0%

Five-Item Transitive Inference

Train A > B B > C C > D D > E

Test B > D

But…………the are alternatives!

Successful “Species”

Children Monkeys Pigeons Fish “Inebriated” University Students

Why?

• Fewer unreinforced responses directed at B because its partner A is always reinforced (Disappointment Hypothesis)

• B reminds you of A and C, D reminds you of C and E. A and C have much more strength than D and E (Value Transfer Hypothesis).

• So, transitive inference does not imply the learning of a mental sequence

Simultaneous Chaining

Ordered Series of Responses

A-B-C-D-E

Response Chain: Each response serves as a discriminative stimulus

Serial Representation: Mental chunking

Serial Order Learning

• Simultaneous Chaining Task– All stimuli appear simultaneously in random locations

– Press numbers or objects in a specified order

– Stimuli disappear when pressed and trial finishes with reinforcement

– Trial immediately ends if a mistake is made

• Tests Trials With Missing Items – Mental Line?

– S-R sequence?

Pigeons (Test Trials)

Chimpanzees (Test Trials)

(Brannon & Terrace, 1998)

Basic paradigm: Touching stimuli in numerical order produces a reward.

Primates Rosencrantz and Macduff

Test with novel stimuli

About 75% correct

Number Operations

Sheba

• Taught to look in three different locations and choose a number for the total, 0+2+1=3

• Maybe just counting not adding?

• Numbers always added to less or equal to 4

But can’t do2+2.

Nonverbal Counting Models