C82NAB numberx

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

many

few

trained pigeons

to discriminate

between "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

6

1

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 presented

simultaneously). 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

Sample of Trials






1.

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




How many
purple wood
?


(4)




2.

1 yellow block
,
2 gray block
,
4 yellow wool
,
6 gray wool





How many
yellow block
?


(1)




3.
1 rose wood
,
2 blue nail
,
3 blue wood
,
5 rose nail


How many
rose nail
?




(5)


Sample of Trials






4.

2 gray truck
,
3 gray key
,
4 orange key
,
5 orange truck




How many

gray key
?


(3)




5.

1 blue box
,
3 green box
,
4 blue cup
,
6 green cup





How many
green cup
?


(6)




6. 1

purple rock
,
2 green rock
,
3 purple plastic key
,
4 green plastic key



How many green rock?




(2)


another perceptual matching argument...


Are the animals subitising?

“ The perception at a glance of

the 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 number

of 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 of

display size with displays of

less than five items
--

it takes

longer to perceive “twoness”

than “oneness”, and so on


This suggests that even with

small 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 lasted

4 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 lasted

4 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
--

RRR
N

and
N
RRR
N
.


Learn to anticipate final
N

trial and run slow....

N

R

R

R

N

after extensive training....





N
RRR
N

trial








... and on an
RRRN trial

R

R

R

N

after extensive training....





N
RRR
N

trial








... and on an
RRR
N

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 of

the 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 mapping

numerosity

(the property of the display
--

e.g. two items) onto

a label that represents that numerosity. We usually use number

words 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!!

3

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 in

the same order





1

4

3

1

2

= 4!!

3

2

= 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 in

the same order




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

whole display

1

4

3

1

2

2

3

1

= 4!!

3

2

= 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 particular

stimulus
-
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 order

arrays 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 order

arrays 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 order

arrays 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

7

5

The chimps taught an ascending order could generalize

immediately to the higher numbers


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

further training


8

6

7

5

implies (limited) understanding

of 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)

1

2
3





5


6


what number red?




this means he knew about
naming quantities
, and
identifying

numbers

–

but not about applying numbers to quantities...

Alex again...

1


2
3





5

6


what colour bigger?




and was performing better than chance right from the start


--

so could relate written numbers with quantities

4) Ability to do arithmetic


To perform the operations of addition, subtraction etc. To some

extent this can be done by rote learning (e.g. times tables); but
true mathematical competence would allow these operations to

be generalised to new situations in a way that implies a concept

of number.








4) Ability to do arithmetic


To perform the operations of addition, subtraction etc. To some

extent this can be done by rote learning (e.g. times tables); but
true mathematical competence would allow these operations to

be generalised to new situations in a way that implies a concept

of 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

1

2

2

2

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 any

of three hiding places. Sheba had to find all the oranges, and

then pick the arabic numeral that represented the sum of all the

oranges that were hidden. After 12 training sessions (of around

20 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 hid

cards with numbers written on them, rather than oranges





--

and she performed above chance right away


implies understanding of the interval scale
–

if she understood

only 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 chimp

A 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 chimp

A 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

B

A

A

1

3

In another experiment ( Boysen & Bertson,1995) chimp A was

given a choice between two amounts of candy. Whichever chimp

A 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

B

A

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 by

rhesus 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 of

numerosity 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.



M
eck, 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.