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.
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