INTUITIVE MATHEMATICS

Many of the cultural accomplishments Homo sapiens have made since the invention of civilization can be attributed to their ability to deal with numbers and the relation between numbers. The tendency for people to quantify objects and events is ubiquitous. Although advanced mathematics, such as algebra and calculus, is acquired only through formal education, all cultures have notational systems and techniques of adding and subtracting at least small quantities. Children over the world seem to acquire some basic numerical concepts at about the same time, independent of formal training (Geary, 1994). This pattern has led some researchers to propose that humans’ quantitative abilities are not simply the product of increased general intelligence; rather, mathematical cognition has been selected over the course of evolution and is governed

by

relatively domain-specific mecha- nisms (Geary,

1995, 1996).

David Geary

(1995,

1996) has proposed that several quantitative skills are good candidates for biologically primary abilities (see Exhibit

6.1).

Recall from chapter

5

Geary’s proposal of biologically primary and secondary abili- ties. Biologically primary abilities are those that have been selected over the course of evolution to deal with problems our ancestors faced. These abilities are universal and develop in a predictable pattern for all but the most deprived individuals. Children

will

spontaneously exercise these skills (i.e., their use is intrinsically motivated), and most will achieve expert-level functioning. In other words, although some children may acquire the

skills

before others, individual differences in eventual attainment

will

be small.

Biologically secondary abilities, in contrast, are built on the primary abilities and are the product of culture, today often of formal schooling. Keep in mind, however, that despite the claim of primacy for the biologically primary abilities, they still develop and require experience before they can be used effectively.

162 THE ORIGINS OF HUMAN NATURE

EXHIBIT 6.1.

Potential Bioloaicallv Primarv Mathematical Abilities

.

Numerosity-The ability to accurately determine the quantity of small sets of items or events without counting. In humans, accurate

numerosity judgments are typically limited to sets of four or fewer items.

= Ordinality-A basic understanding of “more than” and “less than” and, later, an understanding of specific ordinal relationships. For example, understanding that 4 > 3, 3 > 2, and 2 > 1. For humans, the limits of this system are not clear, but it is probably limited to quantities c 5.

-

Counting-Early in development there appears to be a preverbal counting system that can be used for the enumeration of sets up to three, perhaps four, items. With the advent of language and the learning of number words, there appears to be a pan-cultural understanding that serial-ordered number words can be used for counting, measurement, and simple arithmetic.

9 Simple Arithmetic-Early in development there appears to be a sensitivity to increases (addition) and decreases (subtraction) in the quantity of small sets. This system appears to be limited to the addition or subtraction of items within sets of three. uerhaus four, items.

Note. From “Reflections of Evolution and Culture in Children’s Cognition: Implications for Mathematical De- velopment and Instruction,” by D. C. Geary, 1995, American Psychologist, 50, p. 36. Copyright 1995 by American Psychological Association. Reprinted with permission.

Numerosity

The first biologically primary ability proposed

by

Geary is numerosity, which refers to the ability to determine quickly the number of items in a set without counting. When referring to “determining the number of items in a set,” this does not necessarily mean understanding what “two” versus

“three” means but rather being able to discriminate consistently between the number of items within small arrays. This is seemingly a basic ability, based possibly more on simple perceptual skills than on abstract quantitative cognition, and perhaps we should not be surprised that it is found early in life. For example, it has been observed in many mammal and bird species (Davis & Perusse,

1988),

including cats, chimpanzees, and an African grey parrot.

However, there is evidence that numerosity judgments are made very early in humans and that they involve more than simple perceptual abilities.

For example, infants within the first week of life can discriminate between arrays containing up to three (and sometimes four) items (Antell & Keating, 1983; Starkey, Spelke, & Gelman,

1990;

van Loosbroek & Smitsman, 1990).

Moreover, infants can make these judgments when arrays are stationary or in motion, when sets consist of the same as opposed to different types of items, and when contrasts must be made intermodally. Infants’ ability to make numerosity judgments between two different sensory modalities may reflect not merely more general perceptual processing but also a degree of

abstraction. This was demonstrated in a series of studies in which

6-

to

9-

month-old infants were simultaneously shown arrays of two or three objects and heard either two or three drum beats. Infants looked significantly longer at the visual array corresponding to the number of drum beats (Starkey et al.,

1983, 1990;

see also footnote

3).

This finding is particularly interesting because 3-year-old children have difficulty making audiovisual matches, that is, selecting visual arrays consist- ing of the same number of discrete auditory signals (Mix, Huttenlocher, &

Levine,

1996). The

lack of continuity between the performance of 6-month-

old

infants and 3-year-old children may reflect how this knowledge is repre- sented and accessed. The infants’ knowledge was represented implicitly and assessed via a looking-time procedure. In contrast, the preschool children’s knowledge was assessed via an explicit task. It is likely that cross-modal judgments early in life can be made only through implicit cognition and that such knowledge does not become available to conscious awareness until much later in development. (This is similar to the discrepancy discussed earlier between the knowledge of solidity shown between 4-month-old and 2-year-old children; Berthier et al.,

2000;

Hood et al., 2000.) Recall the results of Clements and Pemer’s (1994) study discussed in chapter

5

on implicit versus explicit performance on a false-belief task.

In

that study, children “passed” the false-belief task earlier when a relatively passive, implicit measure (looking time) was used as opposed to a verbal (explicit) measure. Consistent with the Clements and Perner findings and our discus- sion of implicit and explicit cognition in chapter

5

is the suggestion that intermodal (and thus abstract) numerosity judgments may be available to infants and young children only implicitly, and not until much later is such knowledge available to conscious awareness.

Ordinality

Ordinality

refers to a basic understanding of more-than and less-than relationships and seems to develop later in infancy, after an understanding of numerosity has been attained. In one study, 16-month-old infants were conditioned to touch the side of a screen containing either the smaller or larger array of dots (Strauss & Curtis, 1984). For example, an infant might be repeatedly shown arrays of three and four dots and reinforced consistently for touching the array with three dots. After training was completed, infants were presented arrays with different numbers of dots, in the present example, two versus three dots.

If

they had learned merely to respond to the absolute number of dots in an array, they should continue to point to the array with three dots in the transfer phase; however, if they instead had learned an ordinal relationship (select the array with the smaller number of dots), they

1 64 THE ORIGINS OF HUMAN NATURE

should touch the array consisting of two dots. Infants did the latter, suggesting they had learned an ordinal relationship.

Research with mammals and some birds has convincingly shown that they are able to learn the concept of ordinality (Boysen,

1993;

Gallistel, 1990; Pepperberg,

1994).

However, such evidence has typically been derived only after many hours, in some cases months and years, of training. The extensive training required for nonhuman animals to display a concept as basic as ordinality suggests that it may not be a very salient aspect of the environment for them. However, a seminaturalistic study has shown that rhesus monkeys do understand the more-than and less-than relationships when small numbers are involved (Hauser, Carey, & Hauser,

2000). In

this study, rhesus monkeys (Macaca mulatta) living on an island had been habituated to humans. The monkeys watched (one at a time) as researchers placed pieces of apples, one at a time, under two distinctive opaque contain- ers. The number of pieces of apple placed in the two containers differed, ranging from one piece in one container and none in the other, to eight pieces in one and three in the other. The researchers then walked away and noted which container the monkey first approached. When the number of apple pieces in the two boxes varied

by

only one, and the total number of apple pieces in the most numerous box

did

not exceed four, the monkeys consistently approached the box with more pieces in it. That is, for contrasts of

0

versus

1, 1

versus

2 , 2

versus

3,

and

3

versus

4,

the monkeys consistently approached the box with the larger cache of apples. When the number of pieces placed in the boxes was larger, the monkeys were less likely to consistently “approach the larger” first. These findings suggest that monkeys do have a “natural” understanding of ordinality, at least for quantities up to three or four, and that extensive training is not necessary for them to demonstrate this ability. These findings also demonstrate that human’s ordinality skills are evolutionarily quite old and not unique to our species.

Counting

Young children around the world enumerate small sets of items, using variants of the number of words (e.g., “one,” “two,” “three”) available in their culture. Although such counting is not observed until children are able to talk, several theorists have proposed that this ability is based on implicit, skeletal principles evident in infancy (Geary,

1994;

Gelman &

Gallistel,

1978).

Although many children can count almost as soon as they can talk, they do not typically practice “mature” counting (i.e., similar to that practiced by adults in their community) until late in the preschool years.

According to Gelman and Gallistel ( 1978), counting involves five principles:

1.

The one-one principle: Each item in an array is associated with one and only one number name (e.g., “two”).

2.

The stable-order principle: Number names must be in a stable, repeatable order.

3.

The cardinal principle: The final number in a series represents the quantity of the set.

4.

The abstraction pnciple: The first three principles can be ap- plied to any array or collection of entities, physical (e.g., chairs, jelly beans) or nonphysical (e.g., minds in a room, ideas).

5.

The order-irrelevant principle: The order in which things are counted is irrelevant.

Gelman and Gallistel referred to the first three principles as the “how-to”

principles of counting and proposed that children as young as

2.5

years of age demonstrate knowledge of them under some circumstances. For example, children as young as

3

years of age will use the one-to-one principle in counting arrays of five or less, and most will use a stable number sequence.

However, children sometimes use an idiosyncratic list of number words (e.g.,

“one, two, six”), but they use this list consistently (see Geary,

1994;

Gelman

& Gallistel,

1978).

One technique that has been useful in determining what features young children believe are necessary for proper counting is to have them observe a puppet count an array of objects and ask them whether the puppet was correct or not. Using this procedure with

3-, 4-,

and 5-year-olds, Briars and Siegler (

1984)

concluded that children’s understanding that one-one correspondence and stable order were necessary for accurate counting in- creased over the preschool years

(30%,

90%, and 100% for the

3-, 4-,

and 5-year-olds, respectively). However,

60%

of the 5-year-olds also viewed other features as essential, such as beginning to count at an end rather than in the middle of an array and pointing to each object only once. In other words, young children learn the critical features of counting by

4

years of age but infer, from watching others, additional features that are characteristic of, but not necessary for, proper counting.

Simple Arithmetic

Geary’s proposal that addition and subtraction of small quantities reflect a biologically primary ability has received support from research with infants. In an initial study, developmental psychologist Karen Wynn (1992) showed 5-month-old infants a sequence of possible and impossible events that involved the addition or subtraction of elements. For example, infants sat in front of a stage and watched as one object was placed on it. A screen was then raised, covering the object. This was followed by a hand placing 166 THE ORIGINS OF HUMAN NATURE

a second object behind the screen. The hand then left empty. The arithmetic logic here is that there are now two objects behind the screen, that is,

1

+

1

=

2.

In the possible event, the screen was then lowered, and the two objects that the infants had previously seen placed on the stage were indeed there. For the impossible event, when the screen was lowered, only one object was there. Following the logic we described earlier in this chapter that infants should increase their looking time when a result violates their expectation, we would assume that infants would look longer at the impossi- ble than the possible event only if they have some basic conception of addition. This was, in fact, Wynn’s finding, and it has been replicated several times

(T. S.

Simon, Hespos, & Rochat,

1995;

Uller, Carey, Huntley-Fenner,

& Klatt, 1999; see Wakeley, Rivera, & Langer,

2000,

for an exception).

As

with the research evidence of object permanence in young infants discussed earlier, there have been alternative interpretations of Wynn’s results. Such patterns may be the product of more general perceptual or attention mechanisms and not the product of an “arithmetic module” per se (Haith & Benson, 1998;

T. J.

Simon, 1997). For example, rather than reflecting infants’ abstract understanding of integers (i.e., there should be 1 or

2

objects behind the screen), performance on such tasks may be based on representations of the actual objects (e.g., V versus V V), suggesting that decisions are based more on perceptual than conceptual relations (Uller et al., 1999; see Mandler,

2000).

There have been at least two sets of studies demonstrating simple arithmetic abilities in nonhuman primates. One study used a paradigm similar to Wynn’s to assess understanding of subtraction in free-living rhesus monkeys habituated to humans (Sulkowski & Hauser,

2001).

Individual monkeys were shown food objects placed on two side-by-side stages. Screens were then raised, hiding the food. In some conditions, food objects were then removed while the monkey watched, altering the number of objects behind the screen. The humans then backed away, and the subject was allowed to approach the stages.

If

monkeys can subtract, they should be able to keep track of how many food objects are behind the two screens and approach first the stage with the greatest number of food items.

This

is exactly what the monkeys

did,

so long as the number of items initially placed behind the screen

did

not exceed three.

The other set of studies demonstrating arithmetic abilities in nonhu- man primates was performed with an enculturated chimpanzee. For example, comparative psychologist Sally Boysen and her colleagues taught the chim- panzee Sheba the Arabic numerals 1-8. When a number was shown on a video screen (e.g.,

2))

Sheba would have to point to an array containing two objects to receive a reward (see Boysen, 1993). In one set of experiments (Boysen & Berntson, 1989)) one to four orange slices were placed at two of three sites in the laboratory. Sheba’s task was to inspect the sites and

return to a home base and select the Arabic numeral that corresponded to the sum of the orange slices. So, for example, if one site had two orange slices, a second had three, and the third had none, Sheba would have to select the Arabic numeral

5

to be correct. This she did on the very first trial, requiring virtually no training.

In

a second experiment, the arrays of oranges were replaced by Arabic numerals. So instead of finding two and three orange slices at two sites as in the example above, she would find only the numerals

2

and

3.

Sheba performed significantly above chance on this task beginning with the first experimental session. Sheba’s simple arithmetic performance is comparable to the simple counting strategies observed for

3-

and 4-year-old children (Starkey & Gelman,

1982)

and suggests that simple, humanlike addition abilities can be used

by

at least some chimpanzees.

Basic quantitative abilities are important not only for humans but for many species. Humans, however, have made especially great use of mathematics, and Geary’s claim that there are a handful of biologically primary quantitative abilities finds support in the infancy and child develop- ment literatures. We are also intrigued by the evidence that some monkeys and chimpanzees can develop counting and simple arithmetic skills, suggest- ing that the origins of humanlike mathematics may extend back to our simian ancestors. We should emphasize that despite the primacy of these abilities, they do not appear,

de

^nooo,

^fully

formed when first seen. Rather, they develop over infancy and childhood and, interestingly, there may be a discontinuity in their development, with early expression of these skills being implicit in nature, whereas the more mature forms are expressions of explicit cognition. Despite the unresolved issues, we believe that the evi- dence is strong that humans are prepared to deal with quantitative relations, that such preparation exceeds that shown

by

other species, and that it is displayed early in ontogeny.