Numbers: Their Tales, Types, and Treasures.

Chapter 2: Numbers and Psychology

2.6.LOGICAL FOUNDATIONS FIRST?

According to an earlier psychological model by Piaget, certain logical faculties must have developed before it makes sense to teach numbers to children. Learning and understanding concepts (including the idea of number) develops through the active and constructive cognitive processes that our brain constantly performs. The goal of these processes is to harmonize the internal mental representations with sensual impressions.

According to Piaget, the cognitive development of children is not gradual but is marked by certain qualitative changes in cognitive abilities and logical understanding, which indicate that a new stage has been reached. At the age of six or seven, a child should reach the so-called concrete operational stage and have the logical faculties necessary for a working knowledge of numbers. According to Piaget, it makes little sense to teach numbers to children before that. The required logical insight would include an understanding of the concept of a set. One needs the ability to recognize eventual similarities between objects and to group them into sets of similar things that belong together, like a group of marbles or a group of persons. Piaget calls this process “classification.” Next, one must be able to order objects, from first to last. One can order things simply by their position on the table, or from shortest to longest, or from smallest to largest, or according to some other criterion. This skill is called “seriation,” in Piaget's terminology. A sense of invariance must be developed together with these faculties. For example, if the objects of a set are rearranged with larger distance between them, so that the set appears larger, the set will nevertheless contain the same number of objects. So the appropriate training for children to help them learn numbers would be to engage them in exercises involving invariance, classification, and seriation. An understanding of number would arise from their synthesis only when these concepts are fully mastered. This “logic-first” approach was clearly influenced by the strictly logical structure of mathematics. It was very influential in the second half of the twentieth century and was one of the motivating factors for the introduction of the “new math” movement in schools during the 1960s. Suddenly, children had to learn set theory instead of traditional counting. But, unfortunately, the logic of a child's development is quite different from the logic of mathematics, and this approach was later abandoned.

Later research has shown that, actually, the traditional way of learning mathematics wasn't a bad idea at all. The developmental stages are not as homogeneous and stringent as claimed by Piaget. Some abilities are actually acquired much earlier or are not really necessary in the initial use of numbers. The ordinal aspect of understanding numbers develops before the cardinal aspect, and verbal counting with ordinal numbers has a much higher importance than was admitted by Piaget.

But Piaget was right in that the child's own mental activity is the central component of learning. Numbers need not be explained; children use their own cognitive processes to develop abilities and understanding. The process of learning numbers is a very complex process of learning the series of number words, counting by associating an object in a row with a number word, and finally understanding the aspect of quantity and the whole-part relationship. It is not “logic-first” but “counting-first,” through which a logical understanding is achieved with time.