studies on number concepts psych 1090 lecture 5. we looked at number concepts at the beginning of...

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