This document discusses experiments on numerical abilities such as numerosity discrimination, quantity discrimination, counting, and cardinality. It summarizes various experimental procedures used to study these topics in animals, including simultaneous and sequential stimulus presentation, conditional discrimination tasks comparing "many" vs "few", and experiments testing numerical abilities using different stimulus features. Quantitative models are also discussed that aim to account for performance in numerosity discrimination and sequential timing tasks.

1. EXPERIMENTS ON NUMBER
* NOT NUMEROSITY

2. •The nature of Numeric abilities
• Quantity and Numerosity (How much?)
•Cardinality and counting (How many?)
• Experiments and quantitative models
EXPERIMENTS ON NUMBER

3. Psychological Representations
Metric Topological

4. Perceptual Cues - External
Geometric / Global Featural / Local
Modular central processes
Development
Open-field search

5. Perceptual Cues - Internal
Stimulus OrientationVs. Features
Vertical, gravity-dependent search Legge et al, 2008
Bentes, 2009
Kelly & Spetch, 2004
Non-Euclidean geometry

6. PerceptualCues- Internal
Responsecoding
Manabe et al, 1995
Vocalization
Blough, 1964
Movement

7. Absolute vs. Relative Cues
Vector sum model
(Cheng, 1989)
Spetch & Edwards,1988 Metric distance errors
(Cheng, 1992)

8. Perceptual scaling
Weber´s
Law
Psychometric
functions

9. Temporal discrimination
4 , 5.7 , 8 , 11.3 , 16
?
1 , 1.4 , 2 , 2.8 , 4 sec
?
Ext.
4 or 16 sec
1 or 4 sec
Training
Testing
1 sec
16 sec
4 sec
4 sec
The bisection procedure
(Church & Deluty, 1977)
(Machado & Keen, 1999)
The Scalar Property
Associative
connections
Behavioral States
Peck
Red
Peck
Green
Scalar Expectancy Theory (SET) Learning-to-Time (LeT)
Pacemaker
Accumulator
Memory R
Choose Red
Comparator
Memory G
Choose Green
Associative
connections
Behavioral States
Peck
Red
Peck
Green
Scalar Expectancy Theory (SET) Learning-to-Time (LeT)
Pacemaker
Accumulator
Memory R
Choose Red
Comparator
Memory G
Choose Green
Pacemaker
Accumulator
Memory R
Choose Red
Comparator
Memory G
Choose Green

10. Numerosityperception
TheneuralbasisoftheWeber-Law
From Dahene, 2003

11. Numerosityperception
Distance Effect in6-monthold infants
From Xu & Spelke, 2000

12. From Sekuler & Mierkiewz, 1977
Numerosityperception
Developmental changes in Distance Effect

13. Numerical abilities
 Numerosity discrimination Relative magnitude perception
 Subitizing Discrimination of small quantities from perceptive patterns in the set
 Estimation Assignement of numerical labels to a large array (no enumeration)
 Counting Sequencial differential responding (labelling) to individual objects,
allowing absolute, cardinal, number discrimination
 Number Discrimination transfer across sensory modalities or presentation modes

14. Quantitydiscrimination
Absolute proportions
 Relative quantity of color (Emmerton, 2001)
 Simultaneous Red/Green discrimination training
 (1)Two solid color bars (continuous, complementary variaton of colors)
 (2)Two arrays of colored rectangles (regularly/irregularly distributed)
 Variation of color proportions on the two stimuli
 Non-linear (ln) discrimination function for % correct choices
(1) (2)

15. Quantitydiscrimination
Relative proportions
 Three sets of intermixed stimulus pairs
 S+/S- -> complementary variations in color proportions
 S+/0.5 -> variations of correct color bar only
 0.5/S- -> variations of incorrect color bar only

16. NumerosityDiscrimination
Simultaneous stimulus arrays
Successive, “go/no-go” procedures (Honig & Stewart, 1989, 1993)
• Discrimination learning
o Uniform arrays of N colored dots for 20”; same size S+/S- colors
o Tests in extinction: constant N; varying proportion S+/S- (100% -> 50%)
o Same results with different N, sizes, shapes, and natural categories
• Peak-shift effects ( discrimination of relative differences)
o S+: equal Red/ Blue; S-: Blue>Red;Tests with different Blue/Red
proportions
o More responding in Red>Blue for same proportions w/ diff. N
o Same results with differently oriented figures as stimuli

17. Conditional discrimination procedures (Emmerton et al., 1997)
• “Many” (6-7) Vs. ”Few” (1-2)
 Peck Sample array on center key -> turn off sample -> Comparison keys
 Many -> Red,Right key; Few -> Green,Left key
 Test: new sample arrays + intermediate numerosities (3, 4, 5)
 Controls for confounding stimulus dimensions
 Total area and brightness <-> number/size of elements in array
 Different shapes (contours): outline and filled-in shapes of different sizes
 Test results independent of variations of stimulus features
 Easier conditional discrimination between smaller numerosities (1 -4)
NumerosityDiscrimination
Simultaneous stimulus arrays

18. Test results from balanced total area
Condition, for same and variable size
dots within each sample array
NumerosityDiscrimination
Simultaneous stimulus arrays

19. Simultaneous discrimination procedures (Emmerton et al., 1998)
• Choose the array with fewer dots, at different densities
• S+/S- paired combinations of 1-7 elements, in two training series 1-2 , 2-3 , 3-5 , 5-6
• 4 combinations of high/low density for each training pair w/ multiple dots 1-3 , 2-4 , 3-7 , 5-7
• Discrimination performance
• Better accuracy in choosing smaller numerosity when difference is greater (e.g, 3-7>3-5)
• Better performance when : a) S- (larger) had closely spaced elements in the 1-2 or 1-3
b) S+ (smaller) had closely spaced elements in the multiple cases
• Explanation: sequential visual scanning mechanism
• Spaced items increase probability of missing elements (false alarm to choose few)
NumerosityDiscrimination
Simultaneous stimulus arrays

20. Note: single dot
varies in size and
location
NumerosityDiscrimination
Simultaneous stimulus arrays

21.  Control of temporal parameters
 Duration of stimulus presentation
 Interstimulus intervals
 Total duration / rate of presentation
 Delay to reporting response (memory)
NumerosityDiscrimination
Sequential stimulus presentation

22.  Alsop & Honig (1991)
 Up to 9 random Blue/Red flashes in center key
 Relative frequency report on side keys
 Control for total number of stimuli and ISI´s
 Recency effects: later flashes and time from last event
 Saliency effects: stimulus duration biases choice
NumerosityDiscrimination
Sequential stimulus presentation

23.  Keen & Machado (1999)
 Consecutive Red/Green sequences on separate keys
 Counterbalanced order of color/number sequences
 Report least frequent sequence on Red/Green keys
 Accuracy: frequency difference and total number (4-28)
 Temporal effects: recency and primacy (for less than 8)
 Model of cumulative stimulus effects on response strength
NumerosityDiscrimination
Sequential stimulus presentation

24.  Frequency discrimination under time constraints (Roberts et al., 1995)
 Series of Red-light flashes ; report relative freq./durat. on side keys
 Tests under different delays to choose (DMTS)
 General results: freq.discrimination in all conditions; recency effects
 Number discrimination group: 2 or 8 flashes distributed over 4”
 Recency effect: choose “small” at 8 flashes when delay increases
 Time discrimination group: 4 flashes spread over 8” or 2”
 Rate effects: Choose “long” at 2 sec sequence when delay increases (as if more)
NumerosityDiscrimination
Sequential stimulus presentation

25.  Concurrent processing ofTime and Frequency (Roberts et al., 1994, 1998)
 Free operant baseline: FI reinforcement 20”/20 flashes
 Rate manipulations ->Tests in peak-procedure (pecking-rate; 100”)
 Time is a more salient dimension than number when both available
 Differential training to number of events or series duration
NumerosityDiscrimination
Sequential stimulus presentation

26. THENATUREOF NUMBER
Comparative/developmental framework
 Basic, precurrent abilities Vs. Arithmetic operations
 Coordination /synthesis of two logic structures:
 Class relations (Enumeration, cardinality)
 Abstraction of physical differences among objects
 Extensive properties of object sets – None, some, all
 Inclusive relations and compositions (parts and whole)
 Asymetric relations (Ordinal position, seriation)
 Sequencial behaviors / one-to-one correspondence
 Independence from perceptive configurational cues
 Reversibility of the ordered relations

27.  Development of true numerical competence
 Conservation of quantity -> conservation of number
(Continuous/discontinuous) (one-to-one cardinal equiv.)
 Cardinal correspondence Vs. Ordinal correspondence
(arbitrary one-to-one) (positional one-to-one)
 Cardinal values <-> ordinal positions coordination
 Additive and multiplicative operations (composition)
THENATUREOF NUMBER
Comparative/developmental framework

28. CountingandCardinality
ControlbyabsoluteNumber
 The “concept” of Number
 Abstraction of individual object properties (features and organization)
 Active, ordered responding to the individual objects (counting)
 Conservation and transfer to novel stimulus sets (functional classes)
 Novel, reversible relations between behavior and stimulus sets
 Ordinal position; additive/multiplicative compositions (operations)
 Fundamental counting criteria (Gallistel, 1978)
1. One-to-one principle - Differential behavior to each individual element in a set
2. Stable-order priciple - Fixed, ordered sequence of “tagging” behaviors
3. Cardinal principle - Last “label” represents the absolute quantity of the set

29.  Best experimental evidence come from non-avian species (e.g., Boysen´s chimpanzees)
 Social birds: Irene Pepperberg´s research with the African Gray Parrot “Alex”
 Extensive training to vocalize words for objects, shapes, and colors
 Specific training to use numerical labels: classes of shapes (e.g.,3-corner); number of objects(2-6)
 Training to respond to “How many?” questions with number and object labels (e.g., 4 keys)
 Training to tell the number of a subset of objects in heterogeneous arrays (e.g., How many cork?)
 Training to tell the number of objects with several features (e.g., How many green trucks?)
 Transfer to known objects that had not been used in numerical training.
 Counting or Subitizing ?
 Small number ranges (1-4) -> perceptive, configural recognition of numerosity in a set
 Large number ranges -> stimulus abstraction and differential responding
 Alex performance seems to reject subitizing
 equal accuracy on 1-6 range, feature conjunction, complex objects, random scattering, even distribution of
errors
 But: no evidence for ordinal use of numerical labels
CountingandCardinality
ControlbyabsoluteNumber

30.  Counting one´s own responses (Zeier, 1966; Machado et al, 2004,2008)
 Response run N on first-key -> peck second-key
 Successive increases number required pecks
 Upper limit of reliable pecking number in pigeons (8 pecks)
 Discriminating different response numbers
 Produce different number of pecks to different symbols
CountingandCardinality
ControlbyabsoluteNumber

31.  Symbol key
 Specific number of pecks
 Enter key
 Reinf. for exact number
 Time-out for wrong resp.
• Xia, Siemann, & Delius (2000)
Above chance level for
each number of pecks
Errors more frequent
in adjacent numerosities
Thousands of trials
CountingandCardinality
ControlbyabsoluteNumber

32. ExperimentalSeries 1
Numberdiscriminationandtransfer
Trial-unique
stimulus sets:
• 6 shapes
• 5 sizes
• 16 colors
• 16 locations
Within session
balancing for:
• Total area
• Total contour
• Density
Counting:
peck each element
until all disappear

33. Group I Group II
N=2 N=2 N=2 N=2
2 vs 8 4 vs 16 2 vs 8 4 vs
Discrimination
Learning I
Sequential stimuli Simultaneous arrays
Generalization
Tests
Bisection functions for intermediate Numbers
Response latencies, rate, and location gradients
Discrimination
learning II
Simultaneous arrays Sequential stimuli
Mixed training Random Sequential and Simultaneous trials
Matching to
sample tests
Seq.2:Sim(2,8)
Seq.8:Sim(8,2)
Idem for 4-16 Seq.2:Sim(2,8)
Seq.8:Sim(8,2)
Idem for 4-16
ExperimentalSeries 1
Numberdiscriminationandtransfer

34.  Some open methodological issues
 Pre-training of response vsriability/repertoire
 Total sample items / Numeric distances
 Controls for counting as single Indep.Variable
(e.g., conditions with no sample pecking)
 Novel, irregular clip-art stimuli on MTS tests
 Alternate forms of ruling out response control
ExperimentalSeries1
Numberdiscriminationandtransfer

35. Group I Group II
N=2 N=2 N=2 N=2
Many-to-one
learning
2,4 vs 8,16 2,16 vs 4,8 2,4 vs 8,16 2,16 vs 4,8
Reassignement
training
Reversed response location Novel response location
Transfer tests Probes of non-ressigned stimuli
ExperimentalSeries 2
Number classes and conservation

36. Quantitative Modeling
 Keen & Machado, 1999
Cumulative effects of stimulus ocurrences
(in sequencial numerosity discriminations)
 SF = β1 * nf after n first stimuli
 SF = β1 * nf after n last stimuli
 SF = (β1*nf) . Exp (-α*nl)
After nl ocurrences of the last stimulus
 P (last) = SF / (SF + Sl) on choice
 How does it apply to:
• Simultaneous stimulus arrays
• Conditional discriminations
• Judgements absolute number

37. Quantitative Modeling
 Davison-Nevin,1999