What is Number Sense?
Tobias Danzig (1967) introduced the term “number sense” in 1954, describing it as a person’s ability to recognize that something has changed in a small collection when, without that person’s knowledge, an object has been added or removed from the collection. We have number sense because numbers have meaning for us, just like words and music. And as in the case of learning words, we were born with it or, at the very least, born with the ability to acquire it without effort at a very young age. Mathematician Keith Devlin (2000) refined the definition by suggesting that number sense consisted of two important components: the ability to compare the sizes of two collections shown simultaneously, and the ability to remember numbers of objects presented successively in time.
Because we are born with number sense does not necessary mean that we all can become great mathematicians. But it does mean that most of us have the potential to be a lot better at arithmetic and mathematics than we think. If this is true, then why do so many students and adults say they “can’t do math”? We will answer this fascinating question later.
Why Do We Have Number Sense?
Number sense became an innate ability in humans and other animals most likely because it contributed to their survival. Animals in the wild must constantly assess dangers and opportunities in their environment. To do so, they need cerebral systems than can rapidly compute the magnitude of any challenge. As primitive humans went searching for food, they also had to determine quickly whether the number of animals they spotted represented an opportunity or a danger, whether they were moving too fast, were too big to capture, or were just too far away. A mistake in these calculations could be fatal. Consequently, individuals who were good at determining these magnitudes survived and contributed to strengthening their species’ genetic capabilities in number sense.
Piaget and Number Sense
Contemporary research on number sense dramatically undermines Jean Piaget’s constructivist views of 50 years ago. He asserted that newborns enter the world with a clean cognitive slate (remember tabula rasa?) and, by gathering information from their environment, they gradually construct a coherent understanding of the world around them. Piaget (1952, 1954) conducted experiments and concluded that children younger than 10 months of age had no recognition that physical objects are permanent, what he called “object permanence.” He also believed that children did not possess number sense and that they were unable to grasp the concept of number conservation—the idea that rearranging items in a collection does not change their number—until about five years of age. Furthermore, Piaget and his fellow constructivists suggested that children do not develop a conceptual understanding of arithmetic until they are seven or eight years of age.
Piaget’s work had great influence on educational thought because his conclusions were based on experimental psychology. Many educators interpreted Piaget’s work as meaning that the child is not ready for arithmetic until the age of six or seven. Teaching arithmetic earlier would be counterproductive because it might result in distorted number concepts. The child’s frustration in learning even simple arithmetic operations too soon would only generate feelings of anxiety about mathematics. According to Piaget, it was better to start teaching logic and the ordering of sets because these ideas are necessary for acquiring the concept of number. These notions are still prevalent in many of today’s preschools.
Does this persistent Piagetian approach make sense? Researchers today recognize that many of Piaget’s experimental procedures with children were flawed, thus leading to erroneous conclusions. We noted earlier how birds and rats easily recognize a certain number of objects as well as their spatial configuration. And chimpanzees spontaneously choose the larger of two numerical quantities of food. Why, then, would human children have to wait until the age of four or five to gain the same arithmetic capabilities of other animals? We already know the answer: They don’t. Human infants are at least as gifted as animals in arithmetic, and their ability to acquire number concepts grows rapidly within their first year of life.