Herbert P. Ginsburg, Luz S. Lopez, Swapna Mukhopodhyay, Takashi Yamamoto, Megan Willis, and Mary S. Kelly
There is widespread agreement that mathematics should be taught as a thinking activity (see, for example, National Council of Teachers of Mathematics, 1989). Doing this requires that evaluators and teachers obtain information concerning students' thinking activities, their efforts at understanding, and their procedural and conceptual difficulties. Yet, too often, teachers appear to understand little of what mathematical thinking is all about; evaluators provide teachers with assessments that fail to illuminate thinking and understanding; and teachers themselves seem to possess few sound methods for obtaining information concerning thinking, particularly in the classroom, the setting where it is most important to do so.
Given this situation, it is essential to develop methods for assessing children's understandings of a variety of key mathematical topics, including whole number arithmetic. Note that we have referred to methods and understandings in the plural. This reflects our belief that, even for a subject as apparently simple as arithmetic, understanding is extraordinarily complex and many methods are necessary to assess it (or them). Some assessment methods may be useful for evaluators (school psychologists, math specialists, assessment specialists, and so on) while other methods entirely may be appropriate for teachers to use in the everyday classroom. Similarly, understanding is not a single thing but a multitude of processes and functions, the essence of which we are only now beginning to glimpse.