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Studies on Number Concepts
Psych 1090
Lecture 5
We looked at number concepts at the beginning of the
course…
Now we’ll look at them in quite a bit more details
Now from the standpoint of trying to figure out exactly what
types of number concepts animals have
What constitutes numerical competence?
Number can be a descriptive category
chose one set from competing arrays chose with respect to “more”/”less” match-to-sample with respect to quantity respond to one sequential series of events
Most of these do not involve exact number….
Even if subjects discriminate a specific amount, such as ‘threeness” when given a
variety of choices
The subjects The subjects
may have only a may have only a recognition of patternrecognition of pattern
may actually be may actually be “subitizing”….“subitizing”….
Subitizing, as we learned earlier,
is a perceptual mechanism
generally depends on canonical arrays
-- think dice, dominoes
is used when time constraints exist is usually approximate for larger quantities
-- but can be sequential
If a subject is choosing “more” versus “less”
Unless the items vary considerably
Subjects may be responding to
mass or contour
brightness or density
In fact, some work on children (Spelke, Brannon) has shown
exactly that….
when the samples were controlled for mass and
contour,
the purported numerical sensitivity disappeared
What might you prefer?
Three small bites, or one huge piece?
In terms of match-to-sample
Again, unless object differ….
Various mechanisms having very little to do necessarily with
number can kick in
And even if the objects do differ somewhat
Pattern recognition can be used to solve the problem
And, as we mentioned in lecture 2
Even if the patterns are sequential,
Most experiments just examined responding to differences
Such as two versus four….
Whereby the subjects could again respond only with
respect to “more” versus “less”
Numerical research in animals has a long history….
Current thinking suggests that humans and animals share processes that involve subitizing….
Various proposals exist for the mechanisms involved….e.g., object files, accumulators….
And we need to spend a bit of time differentiating
these mechanisms
So that we can understand how they differ from true
counting
One study to claim that children recognize exact
numbers—and that actually claimed addition and subtraction
Involved a looking-time paradigm
The idea was that children become habituated to a set
And then respond only if there is a change in the set
so they are shown something like this….
Until they get bored and stop looking…
Then they are shown
To see if they are startled by the change
The initial studies with children did not test for
size or mass differences
a change in the object
pattern (i.e., sideways figures)
The argument was that the only thing changed was
numberand hence the children (at 5
mos) had to be responding to a difference in number
but the infants might simply have realized something
changed
without knowing WHAT
In fact, some researchers in the UK have done some brain
studiesthat suggest that it really is just
boredom
And then some attention mechanism that is independent
of the type of change being made
So the mechanism wasn’t necessarily counting or even
number sensitivity
And we’ll discuss the related animal study in a bit,
Which did, of course, add some controls
Another set of ideas suggests that subjects without
languagebe they human or nonhuman
use one noncounting mechanism for small groups, 4
And then estimate larger quantities
The big question is then what kind of mechanism is
used for the smaller amounts
Two main competitors exist:
Accumulator models
Object file models
The accumulator model would seem to work very well for
sequential tasks…
The model, set up to explain both counting and timing,
was first presented by Mech and Church
Here there is a sort of gating mechanism
in which the brain tags
each event as it ‘fills up’ or ‘clicks by’…
The mechanism works very well for small numbers
Usually less than four
After which it becomes approximate
So that, for example, if the number of events is six
You get a curve with the center about 6
And with tails on either end
perc
enta
ge
0 1 2 3 4 5 6 7 8 9 10 11 12
60
The system doesn’t work as well for simultaneous arrays
Because it would have to posit that each object in the array
was being scanned at a constant rate
So researchers suggested an “object file”
in which the subject hold a representation of the items as a
sort of pattern array in short-term memory
Somewhat like lambs gamboling around in a meadow
So that about the maximum that can be
kept in short-term memory is about 4
And these are compared with a mental memory of various sets
This, too, is an approximate method
And the results for larger numbers look like the same
curve we saw before
Its advantage is that it does not require sequential presentation
The difference between these mechanisms has to
do with what the differences for curves look like for a given number….
For larger numbers on the accumulator model we expect more errors and more severe
errors as the numbers increase
For the object file,
the system just breaks down for the larger sets
and a general estimation process takes over
Now remember, these are not supposed to be counting
mechanisms
But rather something different from the perceptual system of
subitizing
Which really does need some kind of pattern array for larger
numbers
So let’s look at the monkey paper first…
We saw some of the video in lecture 2 for some of the sets
of trials
And the journal article discusses the others…
Note in the studies on monkeys prior to this one
that monkeys failed at discriminating large sets
So if they saw three apples being put into one box
and eight apples being put into another
they chose the boxes at chance
suggesting that after about 4, they simply couldn’t keep track of
what was happening
even with respect to sequential ‘more’ versus ‘less’
So, the first experiment in the paper we read…
As in the earlier experiments
The animals simply reacted to some level of difference
Although with respect to a slightly larger system
The second experiment tested whether the
monkeys were responding to the amount of ‘stuff’
Note that monkeys had previously chose number over mass if the total mass was the
same….
But that didn’t track contour
Of course, this again marks difference in general….
If you saw lots of little things disappear
Wouldn’t you be surprised if one big thing appeared?
Unless you were looking a stuff like piles of sand?
In introducing the next experiment
the researchers first describe an experiment with children
Stating that the children discriminate 3-1=2 versus
2+1=2
But let’s think about this…
so you expect some change
And here
you also expect change
So the reaction is to the expectation of change of some
sort
But again, not necessarily with respect to number, just pattern
The interesting issue was why older children didn’t
discriminate
3-1=3 versus 3-0=3
issue shouldn’t just be too much stuff…
Maybe the order of testing?
Now, on this next experiment
It couldn’t just be change….
Because then the monkeys would be perfectly happy with
4
So is this some actual evidence for number?
Better, but it still could be mass or contour expectation
In the next experiment
The monkeys had to update their information twice
In addition to having to deal with the larger numbers
and this time they fell apart completely
So now, the question was whether it really was the number of updates that blew the task for
the monkeys…
This experiment has to do with separating out object files and
accumulators….
If the monkey used an accumulator for a 1+1+1=3
versus 1+1+1=2
it should succeed, because the values are still quite small
But if it failed, on a quantity it could do with a single update
That might suggest object files…
Remember, in reality, neither object files nor accumulators
require actual counting
But rather some attention to ‘stuff’ for the former and actions for the latter…
And both are approximations…
The argument for object files holds because the animals
have to keep a running total
as they make a comparison with a mental representation
And that’s more difficult
So their failure on the sequences suggest that they
are using object files
But, again, nothing here really argues for counting
Now, researchers like Brannon have looked at monkeys’ abilities with far greater
numbers…And claim that there exists a
basic, nonverbal number concept across all primates
at least in terms of comparisons…
One of the arguments is that discrimination between two numbers has to do with their
ratios
Obviously, it’s easier to see the difference between 3:8 versus
7:8 in terms of real ‘stuff’
And also with the Arabic numerals if they represent real ‘stuff’
Of course, one really finds that kind of difference only when there’s a time constraint….
We’ll see that there isn’t that much of a difference for Alex
But for almost all his tests, he has unlimited time to make his
discriminations
Now, remember that it took the monkeys about 50,000 trials to learn to make the orderings of 1
through 9….
Which suggests that the concept was not one that was easily
acquired…
But that is less important than transfer…
So the monkeys saw arrays like this
And had to
choose the one
with fewer things first
With smallish sets or big ratios, it’s not that tough…
But you could imagine how difficult it would be for, say, 8 vs 9 blobs..
The data reflect these issues very nicely…
Overall (including lots of easy trials) the monkeys did quite
well…
But, as you might expect, the monkeys
were at chance when
the ratios were very close…
As were humans
Brannon thus argues that the mechanisms are the same…
Some us of ‘mental magnitudes’…
I don’t disagree at all with the results or the argument that basic
processes are at work…
I do not, however, think that mental representations exist in the
monkeys
Unless 50,000 trials have sensitized the animals….
But one still has to put this into the perspective of the Carey and Hauser work on sequential
number….
Was it the ratios in C&H that were the problem?
If so, then we are looking at distinct mechanisms for more
versus less
and actual number recognition….
which is not at all surprising….
and ties into idea on counting….
“Counting” is a very specific behavior:
Produce a standard sequence of number tags (but maybe not out loud)
Apply a unique tag to each item to be counted
know that the last number tag used tells the quantity of interest
Most researchers argue that true counting can exist only with language ….even for humans…
And, of course, most animals do not have language….
However, a few apes, dolphins, and parrots have acquired elements of human communication systems, including number labels…..
So, let’s start talking about creatures who have been trained to label quantities
Matsuzawa’s Ai learned to associate up to 10 blobs with
the Arabic numeral 10
Like humans, she was fast and accurate on 1-4
Presumably subitizing
And, like humans for 5-9,
She took longer for each addition item
She was also less accurate, which isn’t really true for adult
humans (unless time is restricted)
but can be true for children
Interestingly, Ai (and also Sally’s chimps) did not show
much savings for learning the larger numbers in order
Such data suggest that they hadn’t really set up a
representation of number as a sequence
In fact, Ai seemed to do best on the largest number as
compared to the next largest number
Which suggested some idea of using the last number as
“many”But also confusing that with the
next smaller one..
Now, we don’t know if this was exactly true for Alex….
First, we didn’t teach him numbers in order…we did 3 and 4 at the same time, then 5, then 2
then 6 then 1Specifically because we didn’t
want him to have the advantage of knowing that each symbol was
“one more”
I didn’t want that to be some kind of cue, but wanted the
symbol to be a total representation
Second, Alex wasn’t simply associating a visual symbol with
an array
but rather talking…
Which meant that he had to learn how to get all his
articulatory muscles and air patterns into a specific
configuration….
And think about “f” and “v” without lips….
But in all cases, the question was whether the nonhuman was
actually counting, or doing something else
And, in most cases, the only way to tell was to do a timed
experiment….
If an animal could use a perceptual strategy for large
quantities
Its accuracy, unlike that of humans,
Would not decrease as the amount of time allowed to do the
task decreased
Which is what Matsuzawa tested
So one screen had a random array of dots
And, very importantly, the other screen had a random array of
numerals
So that there couldn’t be just a 1:1 association of symbols
But the symbols had to have meaning
What I did find odd was that not every numeral was present on every trial
At least according to the figure presented in the paper…
I don’t see a 2 in either one, or an 8 or 9 on the right…
The data are quite
interesting
Unlike humans, Ai’s errors on all numbers other than 9 didn’t
differ much between brief exposure and unlimited
exposure
Humans erred less overall, but particularly on 9
And response times were really telling
Ai really seemed to take lots of time when it was available to distinguish 8 from 7 and 9
But seemed to hit 9 as the upper limit, as tho’ it were ‘lots’
And she did more looking back for larger numbers
Not surprisingly, Ai had trouble initially distinguishing some symbols….1-7, 6-9, 3-8,
2-4Alex had some of the same
issues til we used very distinct 7s, and we do not plan to use
9
But he confused 2 with 5, which makes sense, too
The researchers argue that Ai did not count for the brief times
but rather estimated
The humans could have clumped… quickly saw patterns of small sets and added the sets
What Ai was doing may be something similar
The stability of Ai’s response time during brief exposure suggests
that she did some form of estimation
She would have been more accurate had she been able to
subitize
Counting would have led to her using all the time possible and
high error rates for 5-8
But, of course, if she were clumping,
that process would be consistent with no real time change for 5-8
It would be really exciting to see how Ai did on the task given Alex to separate out
processes…
Remember, in other studies when humans weren’t given enough
time
Researchers found that Researchers found that humans bottomed out at humans bottomed out at
about the same level as the about the same level as the pigeons–-pigeons–-
about 4about 4
And that Trick and Pylyshyn went about this in another
way
showing confounded sets of colors and objects as well as
numbers
Number used by Number used by AlexAlex
Num
ber
of
ob
ject
sN
um
ber
of
ob
ject
s1 2 3 4 5 6
1
2
3
4
5
6
7
8
6
9
7
1
2
1
1
1 1
4
1
1
8
But we also have to look at Boysen’s chimps
Who also were trained with human symbols
Much like Ai
Initially, Sheba learned to touch the set of numbers and then the
Arabic number
that corresponded to a small array of candies
She was then trained on comprehension before going
on to larger numbers
This procedure was unlike that given Alex, who was not given
any training on comprehension
Comprehension is critical…
Children who can look at an array and tell you that there
are “four” marbles….
But if you give them a big bowl and ask for “four”
Young children will just grab a bunch…
Boysen’s Sheba initially had trouble when four was
introduced; The table presented does not
show how long it took in terms of trials per day
But Sheba eventually did well on both “0” and “4”
Sheba was then tested on production using random
‘junk’ items in the lab
But now the Arabic numbers were always
placed in ordinal sequence….
Thus Sheba could be working on numbers of cards….
As we mentioned in lecture 2, I had to check out Alex’s
comprehension
But I wanted to be extremely careful that he was responding only with respect to the objects
and nothing else
Number Alex producesNumber Alex producesN
um
ber
of
ob
ject
sN
um
ber
of
ob
ject
s1 2 3 4 5 6 n ?
1
2
3
4
5
6
8
9
10
8
8
1
1
1
10
1
1 1
2
none 5 1
Alex’s results demonstrate competence comparable to that
of nonhuman primates and young children
And we have previously discussed his spontaneous use of “none” to represent absence of
quantity
But what about something like addition?
Boysen’s chimps could walk around the room, look at
different collections, then point to the Arabic number that
represented the sum
Sally argues that it is counting
The reason being that Sheba had to remember the representations as she went around the room….
But it is possible that, particularly given the small
numbers involved in this study
That Sheba was clumping the arrays
That ability, in and of itself, is still pretty exciting…
And in later work, Boysen extended the material up to about 6 (and maybe 8, tho’ that is not in peer-reviewed
journals)
And we began a study to And we began a study to replicate the Boysen & Berntson replicate the Boysen & Berntson
work…work…
Number Alex producesNumber Alex producesN
um
ber
of
ob
ject
sN
um
ber
of
ob
ject
s1 2 3 4 5 6 n ?
1
2
3
4
5
6
8
7
8
7
4
1 1
7
1
2
none
1
1 1
3 5
Remember, Alex did replicate Sheba’s abilities,
But remember that Sally gave Sheba as much time as
needed
And we initially time-limited Alex…
And he couldn’t do 5+0…
Data were for first trials only, and he was correct only 50%…
When he errs, he gets a total of 4 chances…and almost every time
he said “6”….
Was he engaging in a counting-like strategy for “5”??
Doesn’t seem to be using an accumulator or object files or
subitizing, because he’s too accurate
If we gave him 10 instead of 2 s, he was 100% accurate…
Was he subitizing 4, and seeing 6 as “lots”?
It would also be interesting to see what Sheba would
have done in a time-limited situation,
like that of Alex or Ai
But Sheba did something else that was extremely important
Whether or not she was counting the objects or just
summing them in some manner
Boysen replaced the objects with the Arabic numerals
And replicated the study
Even on the first few trials
Sheba was above chance
The critical test for Sheba was not 0+n
Which was just choose “n”
But 2+2, because just avoiding addend wouldn’t
work
Nor would choosing the next number in the series
But she had to have a representation
What would be interesting for both Alex and Sheba would be
to have to update their sets more times…
Like the monkeys….
What would they do with 2+2+2?
Now, other work on numbers on chimps and parrots involve their full understanding of ordinality
Do they really understand that their symbols represent a
linear order….
that 2<4 and that 2>1?
As it turns out, Boysen had to train her chimps;
Even Sheba, who had done addition
needed about 500 trials to get all the ordering done correctly
We did this with Alex, too….
First, we trained him to identify Arabic numbers by telling us their colors
So, he saw a tray like this…
1
6
4 5
2
3
The task was not simple, and it took awhile
Partly, I believe, because of something called “mutual
exclusivity”….
If six blobs were “six”, why was one squiggle also “six”?
But eventually he got to respond above chance
And he had previously learned how to respond to
“What color bigger/smaller?” for two objects
So now we showed him two differently colored
Arabic numbers…
And he was correct at a statistically significant rate
We also asked him about the same Arabic numerals, and he
said “none”
And used numbers that themselves were bigger or
smaller
Too much data to discuss here
But basically he was able to do all the various combinations
Including sets with one Arabic number and a collection of items
The take-home messages:
Given that parrots and primates evolutionary history dates from the dinosaurs….
Number concepts are likely to be relatively widespread
across species
And we haven’t talked about other critters…
Maybe numerical competence involves giving the subject the appropriate tools to express
latent abilities….
Certainly, enculturation is important, given
evidence from untrained humans in Peru
Future directions for number work---
training larger numbers
determining if Alex will comprehend new number labels more quickly than old ones
completing subtraction studies
sequential sounds in younger birds and transfer to simultaneous visual
But I hope what came through was that the
critical issue in determining these
abilities has to do with experimental design
Sci-Am show on number work in chimps