Interpreting students' explanations of fraction tasks, and their connections to length and area knowledge

Much research on fractions has concentrated on the sub-constructs of measure, quotient, operator and ratio from Kieren's model of coordinated fraction knowledge (Kieren, 1980). In the primary school, partitioning, equivalence and unit-forming also can be used to describe children's approaches to fraction tasks (Kieren 1988, 1992, 1993, 1995). Given the approaches used to teaching fractions, other areas of the curriculum such as multiplicative thinking, measurement and spatial knowledge could affect students' understanding of fractions. In one-to-one interviews, 88 Grade 6 students were asked 65 questions designed to ascertain their understanding of fraction, measurement, geometry and/or visualisation, and multiplication concepts. The students' answers and explanations were recorded on a record sheet at the time of interview and audio- and video-recording enabled later detailed analysis. The associations between four categories based on Lehrer's key concepts (2003) for spatial measurement (attribute, additivity, units, and proportionality) and the measure sub-construct of fractions were analysed. The measure sub-construct was assessed using number lines, fraction comparison tasks, and length and area diagrams. From detailed examination of students' explanations, insights into misconceptions were gained. Gap thinking in fraction pair size comparisons was discovered to be triggered at the same time as equivalence understanding began. The limitations of a part-whole double count approach to fractional area diagrams was noted. Further, Kieren's four-three-four model (1988, 1992, 1993, 1995) describing coordinated fraction knowledge for analysing students' fraction understanding at the upper primary school level was evaluated. Use of the model enabled descriptions of students' responses to tasks to be placed in a framework of understanding which connected these three underlying concepts and the four sub-constructs.