“How do I understand thee? Let me count the ways.” Well, that's not what Elizabeth Barrett Browning said, but if she were a mathematician, teacher, or mathematics educator referring to any topic in mathematics, she might have. And if she were an assessment specialist she might have noted that there are countless ways to explore and document those understandings.

This section provides two detailed explorations of one mathematics topic, fractions. Broadly speaking, it addresses two main issues: what does it mean to understand fractions (at, say, the sixth-grade level), and what is the potential of various kinds of assessments to reveal those understandings?

Let us start with procedural fluency. Certainly, one expects sixth graders to be fluent in adding, subtracting, multiplying, and dividing fractions. They should be able to convert fractions to decimals, and place both fractions and decimals on the number line; consequently, they should be able to compare the magnitudes of various fractions. They should have a sense of magnitude, and be able to answer questions like: “Which of the numbers 0, 1, or 2 is the sum ⅞ + 12/13 closest to?”

A next level of performance consists of being able use one's knowledge of fractions and be able to explain why what one has done makes sense. Here are some items from the Mathematics Framework for California Public Schools [California 2006, pp. 77–78]: