Learning to Count
Although infants are born with the same rudimentary number sense observed in rats and chimpanzees, they possess two arithmetic capabilities that quickly separate them from other animals. One is the ability to count. The other is to use and manipulate symbols that represent numeric quantities.
Recognizing the number of objects in a small collection is part of innate number sense. It requires no counting because the numerosity is identified in an instant. Researchers call this process subitizing (from the Latin for “sudden”). But when the number in a collection exceeds the limits of subitizing, counting becomes necessary.
No one knows when and how humans first developed the idea of counting beyond the innate sequence of “one, two, and many.” Perhaps they began the way young children do today: using their fingers. (This system is so reliable that many adults also do arithmetic with their fingers.) Our base- 10 number system suggests that counting began as finger enumeration. The Latin word digit is used to mean both numeral and finger. Even evidence from brain scans lends further support to this
When a person is performing basic arithmetic, the greatest brain activity is in the left parietal lobe and in the region of the motor cortex that controls the fingers (Dehaene, Molko, Cohen, & Wilson, 2004). Part of the parietal lobe and the section of the motor cortex that controls finger movement is highly activated when a person is doing arithmetic.
This raises an interesting question. Is it just a coincidence that the region of the brain we use for counting includes the same part that controls our fingers? Or is it possible that counting began with our fingers, and the brain later learned to do counting without manipulating them? Some researchers speculate that if our human ancestors’ first experience with numbers was using their fingers, then the region of the brain that controls the fingers would be the area where more abstract mental arithmetic would be located in their descendants (Devlin, 2000).
Assuming fingers were our first counting tools, we obviously ran into a problem when counting collections of more than 10 objects. Some cultures resorted to using other body parts to increase the total. Even today, the natives of the Torres Straits Islands in New Guinea denote numbers up to 33 by pointing to different parts of their body, including fingers, arms, shoulders, chest, legs, and toes. Naming the body part evokes the corresponding number. Thus, the word six is literally “wrist,” and nine is “left breast.” They use sticks for numbers larger than 33 (Ifrah, 1985). But this process is hopeless for numbers beyond 30 or so. Eventually, some cultures used a physical tally system, such as making notches on a bone or stick. Notched bones have been discovered that date back about 40,000 years. According to the fossil record, this is about the same time that humans started to use symbolic representations in rock carvings and cave paintings (Devlin, 2000).
Finger counting and physical tallies show that these cultures understood the concept of numerosity, but that does not imply they understood the abstract concept of number. Archeologists, such as Denise Schmandt-Besserat (1985), suspect that the introduction of abstract counting numbers, as opposed to markings, appeared around 8,000 BC and were used by the highly advanced Sumerian society that flourished in the Fertile Crescent of what is now Iraq and Syria. They used tokens of different shapes to represent a specific quantity of a trade item, such as a jar of oil or loaf of bread. They used symbolic markings on clay tablets to keep running totals of items in commerce. It was not really a separate number system, but it was the first use of a symbol system that set the stage for the functional, abstract numbers we use today.
Our present numbering system was developed over two thousand years by the Hindus, and attained its present form in about the sixth century. In the seventh century, it was introduced to Europe by Persian mathematicians and thus became known as the “Arabic system.” This ingenious invention now enjoys worldwide acceptance for several reasons.
- Each number has its own word, and the number words can be read aloud. Saying a number, such as 1776 (one-thousand seven-hundred and seventy-six), clearly reveals the numeric structure of units, tens, hundreds, and thousands.
- The numerical system is not just symbols but also a language, thereby allowing humans to use their innate language fluency to handle numbers.
- It is concise and easily learned.
- We can use it to represent numbers of unlimited magnitude and apply them to measurements and collections of all types.
- It reduces computation with numbers to the routine manipulation of symbols on a page.
In fairness, I should mention that the original idea of denoting numbers by stringing together a small collection of basic symbols to form number words came from the Babylonians around 2000 BC. But the system was cumbersome to use because it was built on the base 60, and thus did not gain wide acceptance. Nonetheless, we still use it in our measurements of time (60 seconds make one
minute, etc.) and geography (60 seconds make one degree of latitude and longitude).
Dehaene, S., Molko, N., Cohen, L., & Wilson, A. J. (2004, April). Arithmetic and the brain. Current Opinion in Neurobiology, 14, 218–224.
Devlin, K. (2000). The math gene: How mathematical thinking evolved and why numbers are like gossip. New York: Basic Books.
Ifrah, G. (1985). From one to zero: A universal history of numbers. New York: Penguin Books.