Numbers: Their Tales, Types, and Treasures.
Chapter 2: Numbers and Psychology
2.5.HOW WE LEARN TO COUNT
We have seen that children initially have only an approximate concept of large numbers. In order to develop the idea of exact large numbers, they have to break the limitations of their core-knowledge systems. The approximate-number system tells them that the numbers 12 and 13 are essentially indistinguishable, that they are the same. But they can track small numbers up to three or four exactly; here a difference by one creates a completely different sensual impression. Then, at the age of three or four, when they learn verbal counting, they also learn to combine these two concepts. Even if they cannot count very far, they understand that every number word designates an exact cardinality and does not apply any longer when a single object is removed from, or added to, the collection. There is no evidence for this type of human learning in animals.
Verbal counting according to a systematic numeral system is unique to humans living in a highly developed culture. Learning to count is nevertheless a complicated process with several stages, which is still a matter of ongoing research in mathematical cognition. It is of particular interest, and particularly rewarding, for parents to observe their own children in their individual approach to counting and their understanding of the number concept. Parents should help their children through that process because children who master all the hurdles early often have fewer, or even no, difficulties with mathematics in school. Based on well-known research results by the American mathematics-education professor Karen Fuson (1943–) in 1988, we will first consider the typical steps in the acquisition of verbal tools and number concepts.
When children learn to talk, roughly at the age of two, they also learn number words, which are first used without any understanding of cardinality. They learn to recite the sequence of number words “one-two-three-four-five…” like the words of a rhyme, as a single, whole word. Indeed, there are several nursery rhymes that are great for learning the number-word sequence:
One, two, three, four, five,
Gradually, children become more fluent in reciting the sequence of number words, but they do not yet use it to count. Next they begin to understand that the chain can be broken into individual words arranged in a particular order. They can start using the number-word sequence for counting as soon as they understand the rule “exactly one number word for exactly one object” (bijection principle).
Perhaps by the age of three or four, they can name the successor of a number—for example, say which word comes after six—without going back to the beginning and reciting the whole sequence starting from one. They begin to associate smaller/less with numbers that come earlier in the sequence and larger/more with numbers that come later in the sequence. This must also mark the beginning of the association of numbers with positions on a mental number line with a built-in direction. They are able to understand simple arithmetic and associate increase in number with “going forward” on the number line, and “taking away” with “going backward” on the number line or in the number-word sequence. So far, however, only the ordinal aspect of numbers is understood (ordinal principle).
Now comes the big step, perhaps happening at the age of four or five: As a result of increased experience in the game of counting, children begin to understand that numbers indicate not only a position in the counting sequence but also the (cardinal) number of the objects counted. So the number four not only is the fourth position reached during counting but also indicates that a group of four objects has been counted, and that this group also contains one, two, and three objects (cardinal principle). During this time, their skill in handling number words also improves: They can name the successor and the predecessor of a number, can recite the number sequence starting with any number, and are partially successful in counting backward.
But there are big individual differences, and there are three-year-old children who can count better than some five-year-olds. At the age of four, children of our culture can typically count to 10 and are learning to count up to 20. Beginning at about five years of age, they learn to understand the systematic and repetitive structure of the number words between 20 and 100. At this time, they do not need to memorize every single counting word and its position in the sequence, but they need to understand the rule according to which the number words are generated. Understandably, it always takes longer to learn how to count backward. With insight into the general structure of counting words comes the insight that the sequence of counting words never ends. For every counting word, one can produce a next one, just by following the general rule.
The integration of the ordinal and the cardinal aspect of numbers, the realization that the last-recited number word tells us about the numerosity of the set, is not achieved by all children without problems. This might well be a source of dyscalculia in school. When these children count a set, they still cannot answer the question “how many?” because they associate the last counting word like a name only, with the last-counted object and not with the collection of all counted objects. (As reported by Karen Fuson, they would point to the last toy car and say, “This is the five cars,” instead of “This is the fifth car.”1) When they answer the question “how many?” just by counting the objects again, this indicates that they consider number in the sense of the bijection principle. They just represent the set of toy cars by the corresponding number of counting tags. Instead of “five cars,” it is “one-two-three-four-five cars.” The number words are used just like tally marks on a counting stick.
When children understand the importance of the last counting word for the set as a whole, and that every number of the counting sequence describes the cardinality of the set of already-counted objects, they can start counting from every point of the number sequence. When asked to add five and three objects, they do not need to count every group separately starting with one; instead, they understand that the first result “five” denotes the numerosity of the first group, and they would continue counting the second group with “six, seven, eight” and give the answer eight.
Later, an understanding of differences in number is developed, and numerical relationship between the whole and its parts is understood. Starting from any number, the child now can count forward or backward without difficulty, and, accordingly, the child begins to develop an understanding for the “same distance” between subsequent numbers. And this development, normally reached during the first school year, also paves the way for more sophisticated strategies of calculation than simple counting. Failing to learn the relationship between the whole and its parts, and how a group can be decomposed into subgroups, can be another source of dyscalculia. A child needs to understand that a group of five can be decomposed (for example, into a group of two and another group of three). This is an important prerequisite for understanding computational strategies in arithmetic (for example, the observation that 8 plus 5 equals 8 plus 2 plus 3).