1. 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)

2. Early Counting Experiments Clever Hans –answers math questions

3. Clever Hans

4. Early Counting Experiments Clever Hans –answers math questions Spontaneous Counting –not impressive

6. 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

7. Generalize Across Modalities

8. Discriminate Number (not Time) Acquisition –RRRN –NRRRN R = Reward N = Nonreward

10. Categorize (Generalize) Acquisition –CPPN –PPN Shift –PCCN –CCN P = Pellets N = Nonreward C = Corn Pops

11. Categorize (Label) Answers Correctly: “How many blocks?”

12. Not Subitization Perceptual not cognitive Estimating small quantities at a glance without counting Hard with similar distractors

13. How Many? SquaresOvals

14. Conceptualize (Serial Learning) Transitive inference Simultaneous chaining

15. Three-Item Transitive Inference Train  A > B  B > C Test  A > C But…………..A is 100% reinforced and C is 0%

16. Five-Item Transitive Inference Train  A > B  B > C  C > D  D > E Test  B > D But…………the are alternatives!

17. Successful “Species”  Children  Monkeys  Pigeons  Fish  “Inebriated” University Students

18. 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

19. Simultaneous Chaining Ordered Series of Responses A-B-C-D-E  Response Chain: Each response serves as a discriminative stimulus  Serial Representation: Mental chunking

20. 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?

21. Pigeons (Test Trials)

22. Chimpanzees (Test Trials)

23. (Brannon & Terrace, 1998) Basic paradigm: Touching stimuli in numerical order produces a reward.

24. Primates Rosencrantz and Macduff

25. Test with novel stimuli About 75% correct

26. Number Operations

27. 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

28. But can’t do 2+2.

29. Nonverbal Counting Models