| Problem solving and computation involving division of fractions are difficult for adults and children alike, and regarded by educators to be among the most difficult topics in the elementary mathematics curriculum (Ma, 1999). One explanation for the problem is that learning how to divide fractions is often taught devoid of meaning. The lack of sense making in carrying out algorithms procedurally without making connections to concrete or other types of representations contributes to perplexity. This study explores how children build ideas about the division of fractions in the absence of instruction of algorithms.; This research has two components. The first and primary aspect of this research focuses on a segment of a yearlong classroom-based study of the development of mathematical ideas about fractions, prior to the formal introduction of algorithms, with a group of twenty-five nine to ten year old children from a suburban public elementary school in New Jersey. The second replicates the work with an intervention by this researcher with a class of thirteen ten to eleven year old students at a private parochial school in Piscataway, New Jersey.; For the first component, data came from videotapes of four one-hour classroom sessions recorded by three cameras, each positioned in a different area of the classroom; children's written work; and researcher's notes. Detailed narratives of the videotapes were coded to flag elements that would illustrate children's justification and reasoning, how and to whom children's ideas were expressed, what types of representations were created and how researchers interacted with children.; For the second component, the database consists of students' written work and researchers' notes.; Analysis of the data indicated that children in both groups built and successfully justified models that illustrated the division of a natural number by a unit-fraction and by a non-unit fraction. They supported their arguments with pictorial representations and verbal explanations and built generalizations about the division of a natural number by a fraction. |
