Deepening children's understanding of rational numbers, a developmental model and two experimental studies

grantor: University of TorontoThe present thesis concerns the design, implementation and assessment of an experimental program to teach the difficult topic of rational numbers. Based on Case's theory of cognitive development, it was hypothesized that a core conceptual structure for a global understanding of rational number is formed by the coordination of children's intuitive understanding of proportion and their numerical splitting schemas. In order to support this coordination, the experimental curriculum introduced rational number through the teaching of percent in linear measurement. Props such as cylindrical beakers filled with water, allowed students to make proportional judgements of fullness of these containers relative to the whole using the language of percents. Thus, the traditional sequence of rational number instruction was altered so that decimals and fractions were taught later grounded in students understanding of percents. Two formal teaching studies were conducted. In the first, the participants were an intact grade 4 class who were new to rational number and in the second the curriculum was implemented with grade 6 students who had 4 years of previous instruction in rational number prior to the intervention. Results showed that all of the students made large and statistically significant gains from pre to posttest. Qualitative analyses of this data revealed that the students had acquired a number of new competencies including the ability to (1) move flexibly among representations, (2) resist misleading cues, (3) order numbers by magnitude, and (4) invent their own procedures. In a further analysis, the posttest results of the experimental students were compared to the performance of normative groups on the same measure. These groups consisted of students from grades 4, 6, and 8 and from a postgraduate teacher training program. Both the Grade 4 and Grade 6 experimental students achieved better scores than the grade 8 students and equal scores to the preservice teachers. The experimental students also were less reliant on whole number strategies when solving novel problems, and made more frequent reference to proportional concepts in justifying their answers.Ph.D