Of Minds and Language (42 page)

There is a very large literature now on whether babies or even preschoolers count or not. An ability that counts as one in the domain is arithmetic, or more precisely, natural number arithmetic on the positive integers. First of all, the meaning of a counting list does not stand alone. There is nothing about the sound “tu” that dictates that it follows the sound “won” and so on. Instead the requirements are that a list of count words follow:

(1) the
one-to-one principle
. If you are going to count, you have to have available a set of tags that can be placed one-for-one, for each of the items, without skipping, jumping, or using the same tag more than once;

(2) the
stable order principle
. Whatever the mental tags are, they have to be used in a stable order over trials. If they were not, you could not treat the last tag as

(3) the
cardinal value
, which is conserved over irrelevant changes.

The relevant arithmetic principles are ordering, add, and subtract. Counting itself is constrained by three principles. If you want to know if the last tag used in a tagging list is
understood as a cardinal number
, it is important to consider whether a child relates these to arithmetic principles; it helps also to determine how the child treats the effects of adding and subtracting.

It helps to see that count words behave differently than do adjectives, even if they are in the same position in a sentence. In
Fig. 15.2
, one can see that it is acceptable to say that each of the round circles is
round
or
a circle
, but one cannot say that each of the five circles is
five
or a
five circle
. The other thing we know is this: if we put several objects in front of 2-year-olds who are just beginning to speak, they are likely to label the object kind. Hence it is not clear that they are going to say “One,” when there is one object. Of interest is whether it is possible to switch the child from interpreting the setting as a labeling one or one for counting. If we can switch attention, and therefore show the setting is ambiguous for the child, we might pick up some early
counting knowledge data. We accomplished this with a task that I call
What's on the Card?
(Gelman 1993).

Fig. 15.2. A set of circles that can be labeled as five circles, black circles, or five black circles. Further, each can be called a black circle but not a five circle. This is because ‘five' only refers to the set as a whole and not the individuals.

We tested three age groups of children: those who ranged in age from 2 years 6 months to 2 years 11 months; 3 years 0 months to 3 years 2 months; and 3 years 3 months to 3 years 6 months. The following example of a protocol illustrates both the procedure and how our youngest children responded.

Experimenter:
See this card? What's on this card?

Child:
A heart
.

Experimenter (feedback):
That's right. There is one heart on the card
.

Next two trials first show two hearts and then three hearts in a row:

Experimenter (with the 2-heart card):
See this card? What's on this card?

Child (has now shifted and taken up the instruction to shift domain mindset):
Two hearts
.

Experimenter:
Show me
.

Child:
One, two
.

Experimenter:
So what's on the card?

Child:
Two
.

And then we get a similar pattern for three hearts. There are several points to make about the procedure. As expected, the child first answered a wh- question with a label reply. However, when offered the option to treat differently subsequent examples that showed an increasing number of the item, the child took the bait. This was so for subsequent blocks of trials with new sets of cards, each set depicting different item kinds. Indeed, the youngest age group counted and indicated the cardinal value on 91 percent of their trials.

Thus, they understood our hint that they treat the display as opportunities to apply their nascent knowledge of the counting procedure and its relation to cardinality.

What about addition and subtraction? A rather long time ago I started studying whether very young children (2 1/2 years to 5 years) keep track of the number-specific effects of addition and subtraction. In one series of experiments, I used a magic show that was modeled after discussions with people in Philadelphia who specialized in doing magic with children. The procedure is a modification of a shell game. It starts with an adult showing a child two small toys on one plate vs. three on another plate. One is randomly dubbed the winner, the other the loser. The adult does not mention number but does say several times which is the winner-plate and which is the loser. Henceforth both plates are covered with cans and the child is to guess where the winner is. They pick up a can, and if it hides the winner plate they get a prize immediately. If they do not see a winner, they are asked where it is, at which they pick up the other one and then get a prize. The use of a correction procedure is deliberate: it helps children realize that we are not doing anything unusual, at least from their point of view. This set-up continues for ten or eleven trials, at which point the children encounter a surreptitiously altered display either because items were rearranged, or changed in color, kind, or number (more or less).

The effect of adding or subtracting an object led to notable surprise reactions. Children did a variety of things; such as put their fingers in their mouth, change facial expression, start searching, and even asking for another object (e.g., “I need another mouse”). That is, they responded in a way that is consistent with the assumption that addition or subtraction is relevant, and they know how to relate them. When we do this experiment on 2-year-olds, with 1 vs. 2 and then transfer to 3 vs. 4, we get a transfer of the greater-than or less-than relationship. That is, we have behavior that fits the description of the natural number operations.

Oznat Zur developed a new procedure that involved 4- to 5-year-olds playing a game that involved putting on different hats. Each hat signaled a new game for the child and either a repeat or variation of a condition. For example, children played at being a baker by selling and buying donuts. To start, a child was given nine donuts to put up on the bakery shelf and asked how many he had. Then someone came into the store with pennies and said, “I have two pennies.” The child then handed over two donuts, at which point an adult experimenter asked him to predict, without looking or counting, how many were left. After making a prediction, the child counted to check whether it was right. This sequence of embedded predictions and checks continued. The children did very well. Their answers were almost all in the correct direction. And many of their
answers fell within a range of n ± 1 or 2. Further, the results were replicated in a class, the members of whom were about the same age but did not have an opportunity to play a comparable game before the experiment (Zur and Gelman 2004).

In yet another experiment, Hurewitz, Papafragou, Gleitman, and Gel-man (2006) asked children ranging in age from 2 years 11 months to the late 3-year-old range to place a sticker either on a two- or four-item frame on one set of trials, or
some
vs.
many
on another set of trials. The children had an easier time with the request that used numerals as opposed to quantifiers. The word “some” gave them the most difficulties in this task, a finding that challenges the view that beginning language-learners find it harder to use numerals as compared to quantifiers.

I will conclude now with two contrasting numerical concepts: the successor principle and rational numbers. The successor principle captures the idea that there is always another cardinal number after the one just counted or thought about. This is because addition is closed under the natural numbers. As expected, when Hartnett and Gelman (1998) asked children ranging in age from about 6 years to 8 years of age if they could keep adding 1 to the biggest number that they could or were thinking about, a surprising number indicated that they could. Even when we suggested that a googol or some other very large cardinal number was the biggest number there could be, we were challenged by the child, who noted it was possible to add another 1 to even our number.

The successor principle is seldom taught in elementary school, whereas notions about fractions are. However, when it comes to moving on to considering rational numbers, and the idea that one integer divided by another is a rational number, we run into another example of a HoW domain. This perhaps is not surprising since there is no unique number between a pair of rational numbers. Formally, there is an infinite number of rational numbers between any two pairs of this kind of number. There is more to say about this, but I think that starts to give you the flavor that we really have moved into a different domain and that we may have a case of a conceptual change.

To end this presentation, I illustrate the kind of errorful but systematic patterns of responses we have obtained from school-aged children asked to place in order, from left to right, a series of number symbols, each one of which is on a separate card. Keep in mind that these children were given practice at
placing sticks of different lengths on an ordering cloth; they were even told that it was acceptable to put sticks there of the same length but different colors and to move sticks, and then the test cards, until they were happy with their placement order. Careful inspection of the placements reveals that the children invented natural number solutions. For example, an 8-year-old started by placing each of three cards left to right as follows: 1/2, 2/2, 2 1/2, etc. The following interpretation captures these and all further placements. The child took the cards as an opportunity to apply his knowledge of natural number addition:

(1 + 2 = 3), (2 + 2 = 4), (2 + 1 + 2 = 5).

Other children invented different patterns but all invented some kind of interpretation that was based on natural numbers.

One might think that students would master the placement of fractions and rational number well before they enter college. Unfortunately, this is not the case. When Obrecht, Chapman, and Gelman (2007) asked whether undergraduates made use of the law of large numbers when asked to reason intuitively about statistics, they determined that students who could simply solve percent and decimal problems were reliably more able to do so. Those who made a lot of errors preferred to use the few examples they encountered that violated the trend achieved by a very large number of instances. This continues, unfortunately, through college. I will leave you with that. If you want to know now why your students are horrified and gasp when they are faced with a graph, it is probably because they do not understand rational numbers and measurement.

To conclude, humans benefit from core domains because these provide a structural leg-up on the learning problem. We already have a mental structure, albeit skeletal, to actively search the environment for relevant data – that is, data that share the structure of innate skeletal structures – and move readily onto relevant learning paths. The difficulty about non-core, HoW domains is that we have to both construct the structure and find the data. It is like having to get to the middle of a lake without a rowboat.

H
IGGINBOTHAM
: There has been some interesting work in recent years by Charles Parsons on intuitions of mathematical objects – not intuitive judgment,
but intuitions of the number 3.
²
What he observes is that, from some fairly simple premises, you start off making a stroke. You can envisage that it is possible that you can always add 1. If you have two sequences of strokes, then one of them is an initial segment of the other, and therefore if you took one off each one, they would be different. Now that is already all of the Peano axioms, except induction, and the question would be, when they have that, to check it by saying, “Look, here's this notation system. Can you reach any number that way?” If you can ask that question and get an answer, then you'll get the intuit, because Parsons is deliberately ambivalent or merely suggestive on this point.

G
ELMAN
: Believe it or not, we haven't studied anything that is relevant. Before that, however, I do want to point out that I left out names of my collaborators on the study wherein young children correctly identified 2 and 4 but erred with the same arrays when their task was to identify
some
and
all
, one of whom is in the room, Lila Gleitman, and two of our post-docs at the time, Anna Papafragou and Felicia Hurewitz, who is the senior author of the paper that just came out.
³
As to your question, we ran another interview, where we said, “I am going to give you a dot-making machine that makes dots on paper and never breaks or runs out of paper. This is how many we have now. What happens if we push it (the button)? Will that be more dots on the paper?”. Many children understood that the successive production of dots would never stop save for physical limits on themselves, i.e., “that that would never stop … [except] if you died, had to eat or go to sleep.” This is an example of the nonverbal intuition about the effect of an iterative process.