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Developmental Factors in Problem Solving
There are a minimum of three different components that are involved in a child’s acquisition of problem solving skills. The three components that will be covered here are: (1) spatial perception, (2) experience, and (3) language. Even though all three of these components may fall under a single classification, we choose to separate them because, though they are dependent upon one another, they are at the same time independent. This paradox can be illustrated with the following model.
As children perceive the environment, they make the appropriate spatial associations. This provides the child with experience, which can later be used to form more complex spatial associations. Thus, while experience needs spatial perception for it to exist, so does spatial perception need experience in order for it to exist at higher levels, as will be shown later. Since language is built upon experience (e.g., the different words used by the Eskimos for the word snow), so too can some forms of problem solving experience be built upon language.Spatial Perception
Piaget and Inhelder (1948) state that a child experiences spatial relationships at two different levels, the perceptual level and the representational level. Briefly, perceptual space is the knowledge of objects which results from direct contact with them, while representational
space is a continuation of perceptual space whereby an object is generated mentally in spite of its absence.
One of the first mathematical discoveries that the child comes upon is the topological properties of objects in space (Choat 1978). For example, the object may be closed, open, flat, curved, full of holes, etc. The recognition of these properties and of their relations, provide the child with the spatial concepts important in future problem solving. From this it is conceivable that the child’s first exposure to the world and math is geometrical, in that it is geometry that attempts to explain spatial relationships. This is why play performs such a vital role in the acquisition of mathematics and problem solving skills. Through play the child learns to recognize and categorize objects, which are the roots of classification and set theory. This suggests that experience itself plays a major role in the acquisition of problem solving skills.
Educators, such as Polya (1957) and Adams (1974), emphasize the importance of experience in advanced problem solving. Spearman (1927), however, iterated that, unless a child has formed prior concepts (which are achievable through experience) and has the necessary intellectual abilities, then he will be unable to diagnose and solve problems related to a new learning task. Piaget (1972) also proposed that experience is essential to the formation of logical and mathematical concepts, defining two types of experience:
empirical, and logico—mathematical. Empirical experience is derived directly through perceptual space and results in knowledge obtained by acting on objects and abstracting from them. Logico—mathematical experience is acquired when physical objects are internalized into symbolically manipulable operations.
The effects of experience of future problems solving skills are best put for by J. Hunt (1964). His theory is that poor children who develop in a deprived environment will consequently suffer from stunted intellectual growth, which will prevent them from attaining an adequate level of achievement in school. In the author’s words (p153):
Cultural deprivation may be seen as failure to provide an opportunity for infants and young children to have experiences required for adequate development of those semi— autonomous central control process’ demanded for acquiring skill in the use of linguistic and mathematical symbols.
In brief, a deprived environment does not provide a diversity of experience necessary for the acquisition of future problem solving skills. This may help to explain why the academic performance of poor children is generally inferior to that to rich children. The wealthier parent can provide the child with more toys, for example, which, in turn, can provide the child with more diversified experience.
In spatial representation and concept formation, a logical sequence of events can be seen to unfold. First, for a concept to be acquired, the child must interact with and abstract knowledge from his actions. These abstractions are represented cognitively via symbols. Secondly, once the concept has been formed, then it is necessary that a symbolism exist which will also allow the individual to communicate to, and receive feedback from others. This symbolism is obviously language itself. Thus, while the learner reacts with the physical elements of his environment, the existence of language, in addition to being a means of communication, is necessary for giving the individual an instrument of representation and data discrimination.
In relation to spatial skills, Choat (1978) states that distance itself seems to have little meaning for the young who cannot yet understand what is meant by “next to” and “between.” This seems to imply that in order for the child to understand relationships that exist in their world, it is important that the child have some grasp of knowledge to make the appropriate associations. Donaldson (1979) put forth a similar conclusion in her “Naughty Teddy” experiments.
Bruner (1964) hypothesized that language provides a means not only of representing experience, but transforming it also. He contends that once a child has succeeded in internalizing language as a cognitive instrument, then it becomes possible for the child to represent and transform systematically the regularity of experience with greater flexibility and power. The author also reasons that language comes from the same basic root out of which symbolically organized experience grows, and he emphasizes a need for the preparation of experience and mental operations before language can be used. In other words, once language is applied, it becomes possible to reach higher levels of thought; i.e., when experience has been coded into language, surplus meaning can be read into experience. Until then, language and experience are independent of each other.
Siegel (1982) conducted several experiments in which she addressed two aspects of early quantity concepts, namely, linguistic factors, and perceptual factors. Her first experiment concerned concept language. She wanted to assess the sequence of the development of elementary quantity concept, and the understanding of language which describes quantity. The results of this study indicate that, for the preschool child, concepts of numerical equality are learned before the relational terminology, In a different experiment, Siegel also concluded that in young children concepts emerge on a nonverbal, probably perceptual level, even before language has any relationship to them. This is supportive of Bruner’s “mental operation before language” concept.
Further experiments by Siegel indicate that there is an increase with age in the degree to which language plays a role in the child’s understanding of quantity. She demonstrated that perceptual and non—quantitative factors precede the use of language. As the child develops, there is a movement away from a perceptual matching strategy to a conceptual, numerical strategy. That counting, considered a higher mathematical skill, is tied to language, supports the idea that language is necessary for the development of certain mathematical skills. Furthermore, estimation errors are probably the result of a failure to employ language skillfully or even at all.
Siegel’s findings are supportive of Hunt’s hypothesis on cultural deprivation and its role in the acquisition of math skills. If it is correct to assume that middle—class parents are generally better educated than the parents of lower income families, then we can deduce that middle income homes provide children with a richer verbal atmosphere, which will result in a greater success rate in the acquisition of problem solving skills.
Siegel’s findings may seem to contradict those of Choat and McDonaldson; however, the latter investigated language and its role in the interpretation of problems, whereas Siegel examined language and concept formation.
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