Recent research in cognitive psychology has drawn attention to the important role that students' personal understandings and representations of subject matter play in the learning process. This chapter briefly reviews some of this research, and contrasts the kind of learning that results in an individual's changed conception or view of a phenomenon with the more passive, additive kind of learning assessed by most traditional achievement tests. To be consistent with a view of learning as an active, constructive process, educational tests are required that focus on key concepts in an area of learning, and that take into account the variety of types and levels of understanding that students have of those concepts. In these tests, scoring responses right and wrong is likely to be less appropriate than using students' answers to infer their levels of understanding. This will require not only imaginative new types of test items, but statistical models that permit inferences about students' understandings once their responses have been observed. Psychometric approaches are sketched to construct measures of achievement from such tests.