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Article Excerpt
SUMMARY
Although many students who enter kindergarten are cognitively ready to meet the demands of the kindergarten mathematics curriculum, some students arrive without the early abstract reasoning abilities necessary to benefit from the instruction provided. Those who do not possess key cognitive abilities, including understandings of conservation, insertions into series, and the oddity principle, are at a disadvantage when attempting to master mathematical concepts and skills that require early abstract thought. Recognizing the need to address this gap, this study examined the effects of an intervention designed to teach children conservation, insertions into series, and the oddity principle. The study included 78 kindergartners enrolled in a culturally, linguistically, and socioeconomically diverse metropolitan school district. Students were randomly divided among one of three groups: cognitive intervention, numeracy instruction, and art instruction. Instruction for each group was matched in number, timing, and extent of sessions. The study found that kindergartners who received the cognitive intervention scored significantly higher on measures of cognitive ability than those in the comparison group who participated in the art instruction or those who received numeracy instruction. On the Woodcock-Johnson III Applied Problems scale, those in the cognitive intervention scored significantly higher than those who received art instruction. Those in the cognitive intervention and those in the numeracy intervention performed similarly. These results suggest that it is possible to provide instruction that enhances the cognitive abilities of kindergartners who do not possess key reasoning abilities. In addition, there is evidence that promoting early abstract thought can enhance kindergartners' mathematical abilities.
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Today's climate of standards-based reforms and accountability holds teachers and schools responsible for the achievement of their students. Because of the No Child Left Behind Act (NCLB; 2001), states receiving federal funds are required to develop and administer assessments that enable them to report student progress on an annual basis. As a result of this emphasis, there appears to be a tendency for schools to introduce mathematical skills and concepts at much earlier ages, which may, in some cases, mean prematurely introducing mathematics skills and concepts that are beyond children's cognitive capabilities. The expectation is that children will achieve the standards once instruction has been provided. "The rhetoric of higher standards and achievement may be appealing, but the reality is not" (Neuman, 2003, p. 287). Neuman asserted that the law makes the erroneous assumption that all children enter school "on a level playing field" (p. 287). In reality, disparity exists among students entering kindergarten.
Although many children enter kindergarten with key reasoning abilities that promote their academic success, some children do not. Children who are not as cognitively advanced may face serious difficulties as they try to navigate a curriculum filled with concepts and skills beyond their reach. Many mathematics concepts and skills require children to draw upon early abstract abilities, including the oddity principle, insertions into series, and number conservation. The oddity principle is the ability to identify the only item in a group that differs from all others on some dimension. Children who have not mastered the oddity principle may have difficulty learning basic kindergarten skills. For example, kindergartners are expected to differentiate among a penny, nickel, dime, and quarter; "sort and classify objects according to similar attributes (size, shape, and color)" (Virginia Board of Education, 1995, ([paragraph]) K.19); and "identify representations of plane geometric figures (circle, triangle, square, and rectangle), regardless of their position and orientation in space" (Virginia Board of Education, 1995, ([paragraph]) K.14). Children who have not developed the ability to identify an item that differs from the others on some dimension may struggle as they try to perform tasks related to these expectations.
The same is true of insertions into series, which is the ability to relate an item to others in an increasing or decreasing series and insert the item in its proper place in that series. This is an important cognitive ability that comes into play when kindergarten students compare the size (larger/smaller) of plane geometric figures (Virginia Board of Education, 1995). Number conservation also plays a role in children's success in mathematics. Number conservation is the understanding that the number of items in a group cannot change unless one or more is added or subtracted. This skill enables kindergartners to determine whether one set of objects has the same, fewer, or more objects than another set (Virginia Board of Education, 1995).
Fortunately, some aspects of cognitive functioning can be improved to enhance learning. Studies by Pasnak, McCutcheon, Campbell, and Holt (1991) and Pasnak, Hansbarger, Dodson, Hart, and Blaha (1996) found that when kindergartners were provided extensive instruction on the oddity principle, number conservation, and insertions into series, they scored higher in these reasoning abilities as well as measures of mathematics concepts and verbal comprehension, as measured by the Stanford Early School Achievement Test. Similar results were found with preschool children. When preschool children were taught oddity and insertion, their reasoning abilities in these areas improved (Ciancio, Sadovsky, Malabonga, Trueblood, & Pasnak, 1999). Ciancio, Rojas, McMahon, and Pasnak (2001) also found that preschool students could be taught the oddity principle and insertions into a series, and they showed subsequent gains in numeracy as measured by the McCarthy Scales of Children's Abilities.
Background
The Oddity Principle
Children who apply the oddity principle are able to identify the one object in a group that differs from all of the other objects in the group in one characteristic. The comprehension of relations involved in employing the oddity principle marks the transition from understanding events primarily in terms of perceptual thought to understanding based on early abstract thought. The ability to recognize similarities and differences, to sort reasonably well, and to categorize objects hierarchically into basic, subordinate, and superordinate classes are usually relatively well developed prior to age 4 (Gelman & Wellman, 1991; Mervis, Johnson, & Mervis, 1994; Waxman, 1994), but mastery of the oddity principle depends on more advanced relational responding. When confronted with a group of objects that are all identical except that one differs in size, preschool children try to solve the problem on the basis of some quality of an object, rather than on the relation between objects. For example, if shown one large and three small safety pins, a preschool child may correctly identify the large pin as different and not belonging with the others. However, if shown one small and three large pins, the same child will not be able to identify the small one as unlike the others and instead may persistently select the large pins, one after the other. The child is selecting large pins because they have the quality of being "big," rather than responding to oddity. Another child might do the opposite, always selecting the small size. Similar difficulties arise when oddity involves dimensions of shape, function, color, orientation, texture, or any other dimension (Pasnak, 1987; Pasnak et al., 1986). The difficulty is not a communication problem and cannot be resolved without extensive instruction (Chalmers & Halford, 2003). It arises from an immature stage in the cognitive development of all children. The child does not adequately recognize the relation between the objects in the group and instead searches for an absolute quality like "big" or "little" to govern choices. This is pervasive across all dimensions and is difficult to overcome. Initial progress depends upon identifying the item that differs most from the others--a response rule that is only very slowly replaced by purely relational responding (Chalmers & Halford, 2003). A child who has learned to apply the oddity principle to one dimension will have substantial difficulty in applying it to different dimensions and to different types of problems within dimensions (Pasnak et al., 1986). Much like the preschoolers studied by Zelazo and Frye (1998), children who are in the process of developing their understanding of the oddity rule need a great deal of instruction before they can abstract oddity in a new dimension.
Unidimensional Seriation
Unidimensional seriation is arranging things in order by size or some other ordinal dimension. This is a very fundamental form of reasoning that is expressed in several different ways at different levels of cognitive development and has long been thought to be important (Inhelder & Piaget, 1959/1964; Leiser & Gillieron, 1990).
Many preschool children develop the ability to form a series of objects in the natural course of maturation and experience. However, inserting new interior items into an already constructed series is much more difficult. When confronted with this task, nearly all 3-year-olds and many 4-year-olds make the error of placing the new object at one end of the series or the other and are unable to find the appropriate place for it in the middle of the series by relating it to neighboring objects (Leiser & Gillieron, 1990; Malabonga, Pasnak, & Hendricks, 1994; Southard & Pasnak, 1997; Young, 1976). Progress in recognizing that misplacement of the object is an error and making corrections to such errors is not very predictable (Southard & Pasnak, 1997). It is necessary to comprehend clearly the relations between objects in a series in order to make accurate insertions. This necessity for relational thinking is the reason for the difficulty in making insertions and the importance of being able to do so easily. It marks the transition from perceptually based thinking to early abstract thought. This is the transition that is needed for a child to deal with such concepts as the number line and ordinality, which are among the earliest understandings of numeracy presented in kindergarten and early elementary school, and nearly all 5-year-olds have made it (Malabonga et al., 1994). Children who cannot make an ordered series with concrete objects to compare lack the understanding necessary to put abstract numerical symbols like 6, 9, 11, 15 in order; they do not understand that there is anything inappropriate in an ordering like 6, 7, 10, 8, 9. Memorization, rather than understanding, is the only way a child can deal with such problems, and the child's efforts are as unsuccessful as they are uninsightful.
Conservation
Children who grasp number conservation understand that the number of items in a group cannot change unless one or more is added or subtracted. Many school curricula offer some instruction in some early forms of conservation. However, the instruction is too brief to benefit a child whose understanding of conservation is not already emergent. This is unfortunate because the concepts involved in conservation are critical to an understanding of numeracy. A kindergartner who does not understand conservation does not understand that adding one object is necessary to increase the number of objects in a group from 13 to 14, even though the addition,...
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