### Featured Book

*Numbers and the Making of Us*

*Numerical Ability in Children and Animals*

Perhaps the most convincing evidence of our limited innate numeric ability comes from cognitive research on children.

Since humans, including infants, stare longer at unexpected or new events, child development psychologist Karen Wynn hypothesized that infants would stare longer at impossible outcomes. She wanted to test whether 5-month-old infants have the ability to add 1+1 and found that when one item was added to another, but the outcome was only one item the infants were perplexed and stared longer. In fact, they stare *significantly* longer at the impossible outcome. It is now recognized that infants can differentiate three items consistently.

A study by psychologists Fei Xu and Elizabeth Spelke demonstrated that infants can recognize quantitative differences in large sets if the ratio between compared sets is significant. Infants six months old can recognize the difference between 8 objects and 16 objects. However, when they tested the infants’ ability to recognize the disparity between arrays of 8 and 12 dots, instead of 8 and 16 dots, reducing the ratio of quantities to 2:3 (8:12) from 1:2 (8:16), the results shifted dramatically. The infants’ staring patterns reflected no appreciation of the disparity between 8 and 12 dots. This is compelling evidence confirming that infants can recognize differences between large sets of quantities when the ratio is a least 1:2 which suggests that humans have an innate ability for approximating large quantities.

Recognizing the difference between one and two doll-like figures, or between 8 and 16 dots on a screen only implies that infants are drawn to visual disparities. Recent studies have attempted to study cross-modal recognition of quantities by infants. Psychologist Veronique Izard and her colleagues have demonstrated that newborn babies can recognize some differences between quantities on an abstract, cross-modal basis. After Infants heard a sequence of four syllables, they stared longer at a screen that displayed four images than one that displayed either eight or twelve images. The findings of Izard and her colleagues support the claim that humans are born with the ability to approximate large sets of items, and that this ability is tied to more than one of our senses.

A team of psychologists at the University of Chicago studied the numerical gestures of children 3-5 years old. The children were more at home using their fingers to describe small quantities than using number words. Fingers can represent the number of items in a small set, whereas number words are arbitrary and must be memorized. 3-year olds learn number words but have only a very basic understanding of what they mean. By the age of four, children learn the successor principle, which refers to the awareness that each number in a counting sequence is one more than the previous number. Another milestone on the way to arithmetic thought is the cardinal principle. When children acquire this principle, they recognize that the last number in a counting sequence represents the quantity of the entire set.

The use of words to represent numerical concepts greatly enhances our ability to manipulate numeric quantities but getting to this stage is hard work and is acquired over the course of many years. When children learn numbers, Everett emphasizes, it can’t be a process of learning names for concepts they already have. It must be a process of “concepting” those names: words like “four” and “five” and “six” function as place holders for ideas children eventually grasp. With instruction and practice, they start with concepts they know intuitively (such as that 2 is one more than 1) and learn to construct others by analogy (such as that 8 is one more than 7). Much as a fishing rod is a tool we use to acquire fish, number words are tools we use to acquire number concepts.

What about the numeric abilities in animals? A study 1971 found that rats could be trained to approximate numbers. When rats were rewarded for pushing a lever a certain number of times, they tended to push the lever about the same number of times from then on. Many species are capable of discriminating quantities in approximate ways. In one experiment, when lionesses were played a recording of a lioness roaring, they tended to approach the source of the sound. However, when the sound of three lionesses was played, the lionesses tended to retreat from the source, an ability to differentiate that has obviously improved their survival odds. Some non-primates can exactly distinguish quantities in small sets. New Zealand robins have been shown to discriminate sets of up to four items, above that only if the ratio is at least 1:2, such as choosing between four and eight.

In some species, quantity discrimination seems to be based on continuous variables, so it’s often unclear whether quantity estimations are based on the greater size, density, or movement of the larger quantity. In one study, salamanders were given a choice between two containers of fruit flies. They consistently chose the container with the largest number of flies; but when the choice was between two selections of live crickets, the salamanders’ choice was based on the amount of movement among the observed insects.

For several decades, researchers have been studying the cognitive abilities of primates. It turns out that they share our ability to exactly recognize small quantities and our approximate recognition of larger quantities. In an experiment similar to one conducted with children, psychologists found that rhesus monkeys are capable of distinguishing quantities between 1 and 4 where the monkeys always chose the greater amount. For larger quantities, the monkeys were able to pick the greater amount only if there was a significant numerical difference between them. In other words, chimp selections were characterized by the same ratio effect observed in many other animals and in anumeric humans.

There is a clear relationship between number systems and subsistence strategies. Recent evidence confirms a correlation between simple numbers systems and hunter-gatherer cultures, and complex number systems and agricultural societies. Patience Epps of the University of Texas and her team of linguists studied hunter-gatherers in Australia and found that more than 80 percent of the languages spoken by this group have an upper numeric limit of just 3 or 4.

Large agricultural societies necessitated elaborate number systems, and both evolved together. Elaborate mathematical practices developed in Mesopotamia, China, and in Central America only after agricultural revolutions enabled food surpluses that allowed for mathematically trained classes of people in those regions. Unsurprisingly, numerals are common to all early forms of writing. Chinese writing from 3,000 years ago contain numerals showing such things as the number of enemy prisoners and the number of animals hunted. In Mesoamerica, some of the earliest forms of writing are lines and dots used for calendars. In Egypt, the oldest known hieroglyphics usually depict information about quantities of goods.

Numerals and higher number limits enabled new forms of agriculture and trade because they allowed for the exact discrimination of all relevant numeric quantities. The ease of representing precise quantities with lines and dots on multiple surfaces provided a foundation for more elaborate numerical representation, which led to writing and subsequent inventions in both the sciences and the arts. As societies evolved mathematics did as well: the facilitation of mathematical problem-solving meant the facilitation of architecture and science, and led eventually to modern cities and technology.

How did we ever manage to create names for quantities if we weren’t biologically “wired” for numbers? Everett suggests that *“when some number-inventors happened upon the realization that words could be used to distinguish quantities like five from six, it enabled them to establish a new way of thinking about quantities that others began to adopt. Through that adoption, numbers spread.”* And the rest, as they say, is history.