Children's meanings of fractional number words

This study was conducted within the framework of the constructivist theory of knowledge. The purpose of the study was to formulate experiential models of children's meanings of fractional number words. Fives types of subdivisional as well as fractional tasks involving continuous and discrete quantities were planned and presented to each of four children in one-to-one teaching interviews. The interview sample consisted of four children from the sixth grade. They were selected by their mathematics teacher, who labelled them "under achiever", "typical", "high achiever", and "gifted" according to her records. During the interviewing processes, seven types of tasks other than those planned were used to investigate each of the four children's meanings of fractional number words. All interactive processes between the investigator and the children were transcribed and edited from video tapes of the interviews. Because of the availability of children, the interviewing times were varied among them, but the recorded activities of every participant were between 5 and 6 hours. The recorded communications and children's activities as well as collected drawings that children made during the interviewing process were used as the source of data for the study. Each of the four children's meanings of fractional number words were analyzed according to two major themes: part-whole conceptions and subdividing schemes. Four case studies were conducted using the records of the teaching interviews and form the basis for experiential models.;According to the four experiential models of children's meanings of fractional number words, the part-whole conceptions were distinguished as: no part-whole sense, part-in-whole sense, integrative part-of-whole sense, and subdivisional part-of-whole sense. Four types of subdividing schemes were identified: splitting schemes, segmenting schemes, coordinations of segmenting and splitting schemes, and partitioning schemes. Qualities of subdividing schemes are differentiated according to whether they were interiorized or internalized, i.e., their levels of disassociation from sensorimotor materials. Accordingly, four types of meanings of fractional number words were found--splitting or embedded patterns, initial unit fractions or many-in-many patterns, additive fractions, and nested fractions.;The four experiential models were further taken together to propose a hypothetical model of children's meanings of fractional number words. Conceptual operations used in accounting for whole number knowledge, such as the integration operation, progressive integration operation, part-to-whole operation, and measurement operation, and the type of units produced by these operations, were used to account for different levels of part-to-whole conceptions occurring in children's usages of fractional number words. Subdivided units were taken as the type of units used in the prefractional stage, whereas the constructs of fractional units and of subdivisible fractional units were taken as the type of units for which fractional numbers were their specified numerosities.