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The relationship between volume conservation and a volume algorithm for a rectangular parallelepiped Feghali, Issa Nehme

Abstract

This study was designed to investigate the relationship between the level of conservation of displaced volume and the degree to which sixth grade children learn the volume algorithm of a cuboid, "Volume = Length X Width X Height (V = L x W x H)," at the knowledge and comprehension levels. The problem is a consequence of an apparent discrepancy between present school programs and Piaget's theory concerning the grade level at which this algorithm is introduced. While some school programs introduce the algorithm as early as grade 4, Piaget (1960) claims that it is not until the formal operational stage that children understand how they can find volume by multiplying the boundary measures. Very few children in grade 4 are expected to exhibit formal operations. In such a predicament there seems to be a need for research in order to justify our present school curriculum or to suggest modifications. Subjects of three suburban schools in British Columbia were classified as nonconservers (N = 57), partial conservers (N = 16) and conservers (N = 32) using a judgement-based test of volume conservation. The subjects were then divided into two experimental groups and one control group by randomizing each conservation group across the three treatments. One of the experimental groups (N = 36) was taught the volume algorithm using an approach (Volume Treatment) which resembles that of school programs used in North America. Activities of this treatment included comparison, ordering, and finding the volume of cuboids by counting cubes and later by using the algorithm "V = L x W x H." The other experimental group (N = 39) was taught the algorithm using a method that emphasized multiplication skills (Multiplication Treatment). This treatment included training on compensating factors with respect to variations in other factors and was supplemented by a brief discussion of the volume algorithm. The control group (N = 30) was taught a unit on numeration systems. Four different tests were used: Volume Conservation (11 items), Volume achievement (27 items), Multiplication Achievement (20 items) and the computation section (45 items) of the Stanford Achievement Test. The pretests were: Volume Conservation, Volume Achievement, and Computation. The posttests and retention tests were: Volume Conservation, Volume Achievement, and Multiplication Achievement. Data from the posttests and retention tests were analyzed separately using a 3x3 fully crossed two-way analysis of covariance. Subjects in the volume treatment showed they were able to apply the volume algorithm to computation and comprehension questions regardless of their conservation level. On the posttest and retention test, subjects of this group showed a 65 per cent performance level. For the grade 6 students in the study, conservation level was not a significant factor in learning the volume algorithm at the - computation and comprehension levels. On the posttest, subjects of the multiplication treatment performed significantly (F = 10.33, p < 0.01) better than those in the other groups on the Multiplication achievement Test. Subjects of the volume treatment did significantly (F = 12.24, p < 0.01) better than those in the other groups on the Volume Achievement Posttest. It seems appropriate, therefore, to teach the volume algorithm of a cuboid using a method that includes students* active involvement in manipulating physical objects. There was, generally, an improvement of the students' conservation levels regardless of their volume achievement scores or treatments. The transition from a lower to a higher level of conservation was found a) independent of treatments between the pretest and each of the posttest ( x² = 0.93, df = 2) and retention test (x² = 0.97, df = 2) and b) independent of volume achievement scores between the pretest and each of the posttest (biserial r = 0.13) and retention test (biserial r = 0. 09) . In an addendum to the Conservation Test students were asked to write reasons for their judgements in items involving equal and unequal volumes. Those written reasons were more explicit on the items of unequal volumes than of equal volumes.

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