Research has shown that misconceptions in mathematics develop early and are carried from year to year. If these preconceived ideas are ignored, they may obstruct learning and get in the way when new ideas are introduced. The diagnostic assessment strategies presented by the authors gauge students' prior knowledge and uncover their misunderstandings. Teachers can then modify instruction to build on each individual student's existing understandings while addressing their identified difficulties. The heart of the book consists of 25 probes: brief, easily administered activities to determine your students' thinking on core mathematical concepts. The probes are two-level: 1) for elicitation of common misunderstandings and understandings, and 2) for elaboration of student thinking.