A FRACTION IS NOT A PIECE OF PIE - ASSESSING EXCEPTIONAL PERFORMANCE AND DEEP UNDERSTANDING IN ELEMENTARY-SCHOOL MATHEMATICS

The purpose of this study was to investigate how exceptional performan ce representing deep understanding of mathematical concepts might be a ssessed in the classroom or in larger scale assessment contexts. The s tudy focused on several types of assessment based on cognitive analysi s of fractions: problem solving, justification, and explanation tasks. Additional assessments, including propositional and procedural knowle dge measures, provided validation data. Data were collected from 540 f ifth grade students. To score at the highest levels, students had to s how that they understood that fractions are numbers that can express r elations between quantities, or at least that they had taken major ste ps toward such a conceptualization. Overall, student performance was c onsistent with that obtained in many other research and assessment stu dies. Fewer than 10% of the students performed adequately on the expla nation task. More than 60% of the students failed to express any fract ion principle or concept in their explanations, and 54% expressed seri ous misconceptions. A small number of students, however, displayed exc eptional understanding. The effective-additional reason for optimism: students who received seven days of instruction on fractions in measur ement contexts performed better on the explanation task than students who received traditional part-whole instruction. This result, in addit ion tasks were strongly associated with the expression of conceptual o r principled knowledge in explanations, supported the use of the expla nation task to assess understanding of concepts and principles.