Verbal arithmetic

Abstract: The study of early mathematical development provides important insights into young children's emerging academic competencies and, potentially, a basis for adapting instructional methods. We presented nonverbal forms of two- and three-term arithmetic problems to 4-year-olds to determine (a) the extent to which certain informationprocessing demands make some problems more difficult than others and (b) whether preschoolers use arithmetic concepts spontaneously when solving novel problems. Children's accuracy on simple arithmetic problems (a + b and a - b) was strongly related (r^sup 2^ = .88) to representational set size, the maximum number of units that need to be held in working memory to solve a given problem. Some children also showed spontaneous use of procedures based on the arithmetic principle of inversion when solving problems of the form a + b - b. These results highlight the importance of identifying information-processing and conceptual characteristics in the early development of mathematical cognition. The study of mathematical development prior to formal schooling provides important insights into young children's emerging competencies and, potentially, a basis for adapting instructional methods for later schooling (Ginsburg, Klein, & Starkey, 1998). Much of the work on this topic has been focused on the development of early mathematical activities such as counting (e.g., Gelman & Gallistel, 1978) and quantitative comparison (e.g., Mix, 1999). Relatively little research has been devoted to preschoolers' arithmetic, despite a growing body of research on arithmetic in older children (Geary, 1994) and the fact that even infants show some sensitivity to transformations involving addition and subtraction (Simon, 1997; Wynn, 1992). Because competence in arithmetic is an important goal of early schooling, research on the development of arithmetic-related skills prior to and during early schooling is important for understanding and optimizing the transitions children undergo as academic knowledge is acquired. Part of the difficulty in studying arithmetic in preschoolers is that they tend to do very poorly when problems are presented in the conventional, symbolic format used with older children and adults. When problems are presented in a manner that is less abstract and less verbal, however, performance tends to be higher (Hughes, 1986). Levine, Jordan, and Huttenlocher (1992), for example, found that 4to 6-year-old children performed much better on problems presented using a nonverbal format than they did on problems presented verbally. Their nonverbal method consisted of several steps. First, an experimenter presented an array of objects and then asked the child to construct an identical array. Next, the experimenter covered his or her array with a cardboard box and, as the child watched, changed the quantity of the hidden array by adding or removing objects through holes in the box. Finally, the child was asked to make his or her array match the experimenter's concealed array. At no point was the child required to verbalize a quantity. Using this method, these same researchers found that social class differences among 5-year-olds on verbal arithmetic problems were eliminated when nonverbal problems were used Jordan, Huttenlocher, & Levine, 1992) and that children as young as 2.5 years calculated solutions with some success when problems were presented nonverbally (Huttenlocher, Jordan, & Levine, 1994). Thus some of the difficulty that young children have in doing arithmetic stems from the words and symbols that are often used when arithmetic problems are presented. The use of nonverbal methods of presentation may facilitate performance by helping children establish an appropriate mental model of the transformations involved in adding and subtracting (Huttenlocher et al., 1994). Even when nonverbal methods are used, however, the performance of preschoolers is far from stellar. …

Abstract: 12-and 13-year-olds were tested with two types of tasks to test their understanding of applications of the multiplication and division of positive numbers: (i) writing down calculations required to solve verbal problems, and (ii) making up stories to fit given calculations. Selected pupils were interviewed to investigate further the thinking processes involved. The results indicate (a) the pervasive nature of certain numerical misconceptions, (b) the effects of structural differences among the items; particularly whether multiplication can be conceived as repeated addition or not, and whether division has the structure of partition, quotition or rate, (c) specific effects of context attributable to such aspects as relative familiarity, and (d) various interactions between these three sets of factors.

Abstract: As any first-grade teacher can attest, beginning schoolchildren differ in their ability to solve basic mathematics problems. Some children advance their arithmetic skills with apparent ease, whereas others struggle to grasp certain concepts and, even with prolonged instruction, experience some confusion. The purpose of this study was to examine one of the factors that may account for individual differences in the acquisition of arithmetic concepts and skills-the presence of more general cognitive developmental abilities that may be prerequisites for mastering the arithmetic operations. Specifically, the study was designed to investigate the relationship between several Piagetian abilities and an information processing capacity, and first-grade children's performance on verbal addition and subtraction problems.