What Do Preschool Children Know About Number?

Over the last few decades, developmental researchers have attempted to study mathematical cognition as they seek to understand cognitive changes from infancy to aging since mathematics poses a very interesting set of questions in terms of the fact that mathematical knowledge takes on several forms and its concepts tend to be abstract, complex and sophisticated. Studies of counting, conservation, quantitative comparison, arithmetic, and other aspects of mathematical thinking now provide a rich insight on cognitive development, one in which the development of problem solving, reasoning, memory, perception, and motivation is examined in the context of acquiring knowledge and skills that are culturally relevant and important in the daily lives of children. By exploring differences related to gender, disability, brain injury, and genius, as well as probing the effects of schooling and other characteristics of culture as they have strived to develop tools for educational assessment and for instructional intervention.

Before reaching this stage however, one of the primary points of debate has been whether this abstract knowledge of mathematics is an innate quality that infants are born with or whether babies start out with a conceptual blank slate and mathematical abilities are progressively developed in the their minds by observing, internalizing and abstracting regularities about the world. According to the “constructivism” theory of renowned researcher Jean Piaget, the latter seems to be true and he posited that children evolve gradually through characteristic stages of thinking, known familiarly as the sensorimotor, pre-operational, concrete operations, and formal stages of thinking (Piaget, 1952). He examined how cognitive growth takes place, a proposed a model that allows for a continual “folding in” of more complex understandings. Keeping this in mind, Piaget and his many collaborators seemed to believe that young children were unable to grasp or understand arithmetic, a theory that he substantiated with his claims of their failure to follow “object permanence”. For instance, they found that ten-month old babies failed to reach for a toy that was hidden under a cloth, which they assumed meant that the babies believed that the toy had ceased to exist when it was out of sight, suggesting that if they were ignorant of the fact that objects continue to exist even when they are out of sight, then it wouldn’t be possible for them to comprehend the more evanescent and abstract properties of number.

Further, Piaget found that children even upto the ages of four or five were unable to pass the “number conservation” test. This test that has sort of become a major bone of contention between researchers from either camp was one of the first to demonstrate according to Piaget that children are unable to conserve number. In this test, first children were shown equally spaced rows of six glasses and six bottles and asked whether there were more glasses or more bottles. At this stage, children seemed to rely on the one-to-one correspondence and replied that it was the same thing. In the next stage, the glasses were spread out so that it became longer than the row of bottles, and the children were aslked the same question. However, this time, the children systematically responded that there were more glasses than bottles, even though the number wasn’t affected by the manipulation, which made researchers conclude that they do not “conserve number”.

Piaget also suggested that while children do eventually learn to conserve number as they grow older, they do not gain the basic conceptual understanding of arithmetic. Upto the age of seven or eight they are easily confused by simple numerical tests and fail to understand the most basic concept of set theory, that is, that a subset cannot have more elements than the original set from which it is drawn, which is something that many mathematicians believe provides the foundation for arithmetic.

Piaget’s pessimism about children’s ability to grasp mathematics and his idea to start teaching logic and ordering of sets as a prerequisite to the acquisition of the notion of number, brought up an interesting question for the education system. Further, it also suggested the disturbing idea that human children before the age of four or five lag behind even rats, pigeons and chimps in their numerical abilities given that these species have been shown to be able to readily recognize certain numbers of objects inspite of varying spatial configuration as well as be able to spontaneously choose the larger of two numerical quantities (Dehaene, 1997).

This led to the belief that certainly some part of this constructivism theory must be wrong at some level and though it is apparent that children have much to learn about mathematics, it is not that are entirely devoid of genuine mental representations of numbers, even at birth. A line of reasoning that was used to fault Piaget’s studies was that the research methods were not tailored to the age and understanding of the young children and the question arose as to whether children really understood what they were being asked, as when they were placed in experimental settings similar to the animals where there was no use of dialogue and the context was improved to something they could identify with, their numerical abilities were considerably better.

For example, in an adaptation of the “number conservation” task, Mehler and Bever (1967) showed that the results changed drastically dependent on the context and the children’s level of motivation. In the first task, they performed the original version of the task with marbles and found that children failed to conserve the number when the manipulation was performed. However, when the task was repeated with M&M candies and the children were asked to pick up one of the rows and consume it right away (thus overcoming the language comprehension barriers and increasing their motivation to choose the row with the maximum candy), they found that a majority of the children selected the larger of the two numbers, even when the length conflicted with it. An interesting observation in these studies was that while three and four year olds were able to conserve number in the second task, younger children even aged 2, were able to conserve number on both the marble and M&M’s task implying that the ability to conserve number declines between the age of two to three and four years. Since it is extremely evident that cognitive abilities of the older children certainly are not less well developed than those of the two year olds, perhaps there is a problem of interpreting the experimenter’s questions

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