C82LEA Biology of learning and memory Number. What abilities are involved in numerical competence?...

C82LEA Biology of learning and memory Number

What abilities are involved in numerical competence?

1) Relative numerosity discrimination

2) Absolute number discrimination

3) Ability to count

4) Ability to do arithmetic

1) Relative numerosity discrimination

The ability to discriminate between sets of items on the basis of the relative number of items that they contain.

First to try was Koehler c. 1913

Emmerton, Lohmann & Niemann 1997

manyfew

trained pigeonsto discriminatebetween "few"(1/2 items)

and "many" (6/7 items)

few

few

many

many

few

few

many

many

.. but are the birds ignoring number, and instead using some other feature of the display?

eg

light=few

dark=many

few

few

many

many

.. but are the birds ignoring now

DARK=few

and

LIGHT = many

3

4

5

how well do they transfer to new numbers?

if they really understand few versus many, they should...

1/2 3 4 5 6/7

novel displays

2) Concept of absolute number

understanding that 4 bananas and 4 elephants have something in common...

... i.e. number is not intrinsically related to what you are counting

Koehler again... Jakob the raven could choose a pot with five spots from an array, even when size of spots varied 50-fold

Matsuzawa (1985): chimp called Ai had to select one of six response keys (labelled 1-6) when shown arrays of red pencils, with 1-6 pencils per array. Achieved > 90% accuracy.

1 2 3 4 5 61 4

But this is not necessarily the same as counting....

Animals could be learning about specific perceptual pattern-- perceptual matching.

But this is not necessarily the same as counting....

Animals could be learning about specific perceptual pattern-- perceptual matching.

Argued no, as Ai could transfer her ability to arrays of different types of item

4

the perceptual matching problem...

often number is confounded with other factors such as time(for items presented serially) and space (for items presentedsimultaneously). Are animals using number or these other cues?

e.g. smaller number of items also takes up less space.

Is it the size of the display controls the response, not number ??

with visual arrays there is always going to be something.. so hard to rule out

but people have tried in various ways e.g. Pepperberg, 1994

another perceptual matching argument...

Are the animals subitising? “ The perception at a glance ofthe number of items present, without counting them successively;the maximum number of items that can be counted in this way is five ”

HOW MANY?

HOW MANY?

The original claim was that subitizing is different from counting because there is little increase in reaction time per item for low numbers of items

whereas when dealing with numbers bigger than six, you have to count each one, and because it takes a finite amount of time to count each item the RT increases with numberof items

This implies that you do not need to count displays of five items or less -- the number is perceived immediately

But is this true?

However, there is an effect ofdisplay size with displays ofless than five items -- it takeslonger to perceive “twoness”than “oneness”, and so on

This suggests that even withsmall displays we are using a counting process

Meck and Church (1983): serially presented items.

Rats trained with two signals – 2 or 8 pulses of white noise.

after 2 were rewarded for left lever response

after 8 rewarded for right lever response

Respond LEFT Respond RIGHT

Each pulse 0.5 sec -- “2 pulse” lasted for 2 seconds,“8 pulse” for eight seconds.

Were animals were responding on the basis of the total time,not number of pulses?

To investigate this, they devised a test in which both stimuli lasted4 seconds:

Respond LEFT Respond RIGHT

If rats were responding on the basis of stimulus duration, this task should be impossible

but they continued to respond correctly

To investigate this, they devised a test in which both stimuli lasted4 seconds:

Respond LEFT Respond RIGHT

The rats were also tested with pulses of light -- and continued to respond appropriately (Church & Meck, 1984).

This is more evidence against perceptual matching

Can you think of any other confounds?

.. or can make animal respond a fixed number of times – no array involved

Davis & Bradford (1991)

Access to a plank with food pellets on it

Experimenter nearby talking to rat

Each rat had designated number of pellets to eat – if he ate more the experimenter shouted “No!” or clapped loudly.

When they ate the right number or fewer than the target they were rewarded by “praise and petting” (and also a little more food)

got it right even when no longer rewarded for correct responses

transferred to sunflower seeds -

Further evidence from Capaldi & Miller, 1988

Rats trained in a runway, sometimes with food at the end. If the rats expect food they run fast!

Trained with following sequences of reinforced (R) trials and nonreinforced (N) trials -- RRRN and NRRRN.

Learn to anticipate final N trial and run slow....

N

R

R

R

N

after extensive training....

NRRRN trial

... and on an RRRN trial

R

R

R

N

after extensive training....

NRRRN trial

... and on an RRRN trial

Learning that three rewards mean no more...?

not e.g. length of time in apparatus...

... and were trained with rat pellets; but if one or more ofthe rewards in the sequence were changed to, for example, cocoa pops, they still did well

What abilities are involved in numerical competence?

3) Ability to count

Gelman & Gallistel (1978) argued that counting involves mappingnumerosity (the property of the display -- e.g. two items) ontoa label that represents that numerosity. We usually use numberwords or symbols as labels, but presumably animals use nonverbal labels, which we can call numerons.

The process of counting involves three principles:

i) one-to-one principle: each item is assigned only one numeron

1 4 = 4!!32

The process of counting involves three principles:

i) one-to-one principle: each item is assigned only one numeron

ii) stable-order principle: numerons must always be assigned inthe same order

1 4

31 2

= 4!!32

= 2!!

The process of counting involves three principles:

i) one-to-one principle: each item is assigned only one numeron

ii) stable-order principle: numerons must always be assigned inthe same order

iii) cardinal principle: the final numeron assigned applies to thewhole display

1 4

31 2

2 31

= 4!!32

= 1!!

= 2!!

Not just about knowing correct number labels

Implies knowledge about order of these labels

e.g. 1 2 3 4

..about how these labels are ordered in relation to quantity

e.g. 4>3 2>1 --- ordinal scale

and that the size of the difference between each item is the same

e.g. 4-3= 3-2 --- interval scale

Representation of number in the chimpanzee? Biro & Matsuzawa 2000

Ai trained to touch arabic numerals in ascending order

Representation of number in the chimpanzee? Biro & Matsuzawa 2000

Ai trained to touch arabic numerals in ascending order

But some argued that it was just rote learning of a particularstimulus-response sequence... - no requirement to know anything about the quantitative relation between numbers

Representation of number in the chimpanzee? Brannon & Terrace, 2000

Chimps (Benedict, Rosencrantz & MacDuff) trained to orderarrays of 1-4 items in ascending, descending, or random order

same size same surface area vary size clip art

Representation of number in the chimpanzee? Brannon & Terrace, 2000

Chimps (Benedict, Rosencrantz & MacDuff) trained to orderarrays of 1-4 items in ascending, descending, or random order

same size same surface area vary size clip art

mixed clip art vary size and shape vary size, shape, colour

They could learn ascending and descending orders, but not the arbitrary order 1-3-2-4

Representation of number in the chimpanzee? Brannon & Terrace, 2000

Chimps (Benedict, Rosencrantz & MacDuff) trained to orderarrays of 1-4 items in ascending, descending, or random order

Representation of number in the chimpanzee? Brannon & Terrace, 2000

Then they were tested with novel displays of 5-9 items

8 6

75

The chimps taught an ascending order could generalizeimmediately to the higher numbers

.... but those taught a descending order could only generalize after further training

8 6

75

implies (limited) understandingof the ordering of quantities

Alex again... (Pepperberg, 2000)

1 orange chalk, 2 orange wood, 4 purple wood, 5 purple chalk

How many purple wood? (4)

Alex again...

4) Ability to do arithmetic

To perform the operations of addition, subtraction etc. To someextent this can be done by rote learning (e.g. times tables); but true mathematical competence would allow these operations tobe generalised to new situations in a way that implies a conceptof number.

4) Ability to do arithmetic

To perform the operations of addition, subtraction etc. To someextent this can be done by rote learning (e.g. times tables); but true mathematical competence would allow these operations tobe generalised to new situations in a way that implies a conceptof number.

It is worth asking yourself exactly what this means; is it an all-or-none skill? Or is it a matter of degree? And if the latter, might animals have a limited concept of number?

Maths in the chimpanzee? Boysen & Berntson, 1989

A chimp called Sheba was trained to label arrays with counters,and then with arabic numerals:

Maths in the chimpanzee? Boysen & Berntson, 1989

A chimp called Sheba was trained to label arrays with counters,and then with arabic numerals:

Maths in the chimpanzee? Boysen & Berntson, 1989

A chimp called Sheba was trained to label arrays with counters,and then with arabic numerals:

1 1

12

22 3

3

3

Maths in the chimpanzee? Boysen & Berntson, 1989

..and then with arabic numerals:

Maths in the chimpanzee? Boysen & Berntson, 1989

She also performed well when items swapped for everyday objects

She was given extensive training with numbers 0-4

She was given extensive training with numbers 0-4

In the final test a number of oranges were hidden in the lab, in anyof three hiding places. Sheba had to find all the oranges, and then pick the arabic numeral that represented the sum of all theoranges that were hidden. After 12 training sessions (of around20 trials per session) she was performing at about 85% correct.

Answer = 3

Potential problems.....

you could argue she memorized all the ways of adding 0,1,2,3,4

to a total of 4...

0+0 0+1 0+2 0+3 0+4 1+1 1+2 1+3 2+2

but....!

She could also perform accurately when the experimenters hidcards with numbers written on them, rather than oranges

-- and she performed above chance right away

implies understanding of the interval scale – if she understoodonly bigger than she would have chosen 4 as often as 3

1

Answer = 3

2

In another experiment ( Boysen & Bertson,1995) chimp A was given a choice between two amounts of candy. Whichever chimpA chose was given to a second chimp, B, and chimp A got to eat the other one.

A chooses

B

A

In another experiment ( Boysen & Bertson,1995) chimp A was given a choice between two amounts of candy. Whichever chimpA chose was given to a second chimp, B, and chimp A got to eat the other one. It was thus in chimp A’s interest to choose the smaller quantity, so it could eat the larger quantity. They were completely unable to solve this task -- unless the candy was substituted by numerals.

A chooses

A chooses

B

BA

A

1

3

In another experiment ( Boysen & Bertson,1995) chimp A was given a choice between two amounts of candy. Whichever chimpA chose was given to a second chimp, B, and chimp A got to eat the other one. It was thus in chimp A’s interest to choose the smaller quantity, so it could eat the larger quantity. They were completely unable to solve this task -- unless the candy was substituted by numerals!

A chooses

A chooses

B

BA

A

1

3

Is this evidence they can't count?Or just that they can't resist a treat..

correct motivation critical for good performance

General references

Pearce, J.M. (1997). Animal Learning and Cognition. Lawrence Erlbaum Associates. Chapter 7.

Shettleworth, S.J. (1998). Cognition, Evolution and Behaviour. Oxford University Press. Chapter 8 and pp.228-229

Wynne, C.D.L. (2001). Animal Cognition. Macmillan. Chapter 5 pp.101-111.

http://www.pri.kyoto-u.ac.jp/ai/video/video_library/index.html

Specific references

Biro, D., & Matsuzawa, T. (2000). Numerical ordering in a Chimpanzee: Planning, executing and monitoring.

Boysen S.T., & Berntson, G.G. (1989). Numerical competence in a chimpanzee. Journal of Comparative Psychology, 103, 23-31.

Boysen S.T., & Berntson, G.G. (1995). Responses to quantity: perceptual versus cognitive mechanisms in chimpanzees. Journal of Experimental Psychology: Animal Behavior Processes, 21, 82-86.

Brannon, E.M., & Terrace, H.S. (2000). Representation of the numerosities 1-9 byrhesus macaques. Journal of Experimental Psychology: Animal Behavior Processes, 26, 31-49.

Capaldi, E.J., & Miller, D.J. (1988). Counting in rats: Its functional significance and the independent cognitive processes that constitute it. Journal of Experimental Psychology: Animal Behavior Processes, 14, 3-17.

Church, R.M., & Meck, W.H. (1984). The numerical attribute of stimuli. (pp.445-464) In Roitblat, H.L., Bever, T.G., & Terrace, H.S. (Eds.) Animal Cognition. Lawrence Erlbaum Associates.

Davis, H, Bradford, S.A. (1991) Numerically restricted food intake in the rat in a free-feeding situation. Animal Learning & Behavior, 19, 215-222. Emmerton, J, Lohmann, A., & Niemann J. (1997). Pigeons' serial ordering ofnumerosity with visual arrays. Animal Learning & Behavior, 25, 234-244.

Gelman, R., & Gallistel, C.R. (1978). The child’s understanding of number. Cambridge, MA: Harvard University Press.

Matsuzawa, T. (1985). Use of numbers by a chimpanzee. Nature, 315, 57-59.

Meck, W.H., & Church, R.M. (1983). A mode control model of counting and timing processes. Journal of Experimental Psychology: Animal Behavior Processes, 9, 320-334.

Pepperberg, I.M. (1994). Numerical competence in an african gray parrot. Journal of Comparative Psychology, 108, 36-44.

Pepperberg, I.M. (2000). Ordinality and inferential abilities of a grey parrot. Journal of Comparative Psychology, 120, 205-216.