The relationship of unmanipulated self-reports of children's internalized representation of numbers to mathematics achievement

The purpose of this research was to examine children's unmanipulated self-reports of their internalized representation of numbers and the relationship of the spatio-organizational patterns that are represented by the children's drawings to children's ability to solve basic addition problems. Also of interest were possible changes that occurred in children's spatio-organizational patterns as a result of age, mathematics achievement or gender. It was hypothesized that children whose drawings demonstrated more structured spatio-organizational patterns would achieve a higher number of correct answers on a timed test of basic addition problems. It was also hypothesized that the structure of the spatio-organizational patterns that children drew would be influenced by age, gender and mathematics achievement.

The results of this exploratory study of children’s unmanipulated internalized constructs of number provided some interesting results. The children were asked to image specific numbers of dots for numerals from 4 through 13 and then to draw a representation of their images. The representations were categorized according to the structure of spatio-organizational patterns. The analyses revealed that the patterns had more structure for older children. Multiple regression analyses also indicated that the correctness of the cardinality of the number of dots imaged was the most frequently occurring variable that had a significant effect on the Imagery Scores. Less than five of more than 450 students expressed any difficulty with the imagery task and then only as it related to one of the ten numerals they were asked to image.

The students were asked to image at the foundational level of imagery--reproductive imagery (Piaget & Inhelder, 1971). Because the research task developed for the students did not involve anticipatory images, those requiring transformations or movement, these imaging tasks were not influenced by the children's IQ or mathematics achievement. According to Piaget and Inhelder, children's ability to use anticipatory images indicates that children are developing an operational understanding and use of imagery. The children in this study were not asked to do anticipatory imaging. This may account for the negative relationship of the Imagery Scores to the fifth-grade students’ math percentile scores and the positive relationship between Imagery scores and mathematics percentile scores for the primary level students. The imagery tasks requested of the students were not of sufficient difficulty to relate to any mathematical operations or logio-mathematical thinking for older children.

The ability of children to produce reproductive images which have varying degrees of spatio-organizational patterns was demonstrated by this study. Future studies need to address the higher level of anticipatory images. If students were asked to image a specific number of dots and then to image adding another quantity of dots to the original image, would the spatio-organizational patterns used by children in this transformation process change or transform the image? Are there specific spatio-organizational patterns that more easily allow children to develop anticipatory images that use mathematical operations? Are there children who have developed static reproductive images, and as a result, have created internalized constructs that inhibit their future understanding and development of higher level mathematical concepts?