Numbers Flashcards
Numbers is a type of
Comparative psychology
Associative learning works in all animals – and it is clever!
Doesn’t just glue things that occur together – sensitive to correlations – can track causal relationships
…and in man they can do complex stuff: connectionist networks – language, pattern recognition
but can associations do everything? or are there other learning mechanisms involved?
What abilities are involved in numerical competence?
1) Relative numerosity discrimination
2) Absolute number discrimination
3) Ability to count
4) Ability to do arithmetic
Relative numerosity discrimination
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
trained pigeons
to discriminate
between “few”
(1/2 items)
and “many”
(6/7 items)
.. but are the birds ignoring number, and instead using some other feature of the display?
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.
Perceptual matching
But this is not necessarily the same as counting….
Animals could be learning about specific perceptual pattern
perceptual matching problem again…
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 ??
Pepperberg, 1994
with visual arrays there is always going to be something like this.. so hard to rule out but people have tried in various ways
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 ”
If you are counting then RT should increase with every item
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
Each pulse 0.5 sec – “2 pulse” lasted for 2 seconds,
“8 pulse” for eight seconds.
But were animals responding on the basis of the total time,
rather than number of pulses?
To investigate this, they devised a test in which both stimuli lasted
4 seconds:
If rats were responding on the basis of stimulus duration,
this task should be impossible
but they continued to respond correctly
Church & Meck, 1984
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 (1986)
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 -
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….
After extensive training
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 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:
one-to-one principle: each item is assigned only one numeron
ii) stable-order principle: numerons must always be assigned in
the same order
cardinal principle: the final numeron assigned applies to the whole display
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
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
Chimps (Benedict, Rosencrantz & MacDuff) trained to order
arrays of 1-4 items in ascending, descending, or random order
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
Then they were tested with novel displays of 5-9 items
The chimp taught an ascending order could generalize
immediately to the higher numbers
…. but that taught a descending order could only generalize after
further training
implies (limited) understanding
of the ordering of quantities
Pepperberg, 2000
what number green?
this means he knew about naming quantities, and identifying
numbers – but not about applying numbers to quantities…
what colour bigger?
and was performing better than chance right from the start
– so could relate written numbers with quantities
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:
She also performed well when items swapped for everyday objects
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.
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
One final warning….
sometimes animals are more competent than they appear
contextual variables
before deciding an animal is unable to do something need to be sure you have designed the experiment so it can perform to its best ability…
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 unchosen one.
In chimp A’s interest to choose the smaller quantity, so it could eat
the larger quantity. Completely unable to solve this task
substituted by numerals.
In chimp A’s interest to choose the smaller quantity, so it could eat
the larger quantity. Completely unable to solve this task
– unless the candy was substituted by numerals.
Is this evidence they can’t count?
Or just that they can’t resist a treat..
correct motivation critical for good performance
