The purpose of this study was to investigate how exceptional performance representing deep understanding of mathematical concepts might be assessed in the classroom or in larger scale assessment contexts. The study focused on several types of assessment based on cognitive analysis of fractions: problem solving, justification, and explanation tasks. Additional assessments, including propositional and procedural knowledge measures, provided validation data. Data were collected from 540 fifth grade students. To score at the highest levels, students had to show that they understood that fractions are numbers that can express relations between quantities, or at least that they had taken major steps toward such a conceptualization. Overall, student performance was consistent with that obtained in many other research and assessment studies. Fewer than 10% of the students performed adequately on the explanation task. More than 60% of the students failed to express any fraction principle or concept in their explanations, and 54% expressed serious misconceptions. A small number of students, however, displayed exceptional understanding. The effectiveness of a short instructional intervention provided an additional reason for optimism: students who received seven days of instruction on fractions in measurement contexts performed better on the explanation task than students who received traditional part-whole instruction. This result, in addition to the finding that high levels of performance on all tasks were strongly associated with the expression of conceptual or principled knowledge in explanations, supported the use of the explanation task to assess understanding of concepts and principles.
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