This introductory section establishes the context for what follows in this volume. In broad-brush terms, it does so by addressing three P’s: pragmatics, philosophy, and policy.

Alan H. Schoenfeld opens with a discussion of pragmatics in Chapter 1. A simple question frames his contribution: Who wants what from mathematics assessments? As he shows, the issue is far from simple. It is true that at some level everyone involved in mathematics testing is interested in the same basic issue: What do students know? However, various groups have very different specific interests. For example, a teacher may want to know what his or her particular students need to know in order to improve, while a state superintendent may be most interested in trends and indications of whether various performance gaps are being closed. Other groups have other interests. Those interests are not always consistent or even compatible. Understanding who has a stake in assessment, and what different groups want to learn from it, is part of the big picture—the picture that is elaborated throughout the volume, as representatives of different stakeholder groups describe the kinds of information that they need and that carefully constructed assessments can provide. Schoenfeld also looks at the impact of assessment. Tests reflect the mathematical values of their makers and users. In the United States, tests are increasingly being used to drive educational systems—to measure performance aimed at particular educational goals. This is the case at the national level, where various international assessments show how one nation stacks up against another; at the state level, where individual states define their intended mathematical outcomes; and at the individual student level, where students who do not pass state-mandated assessments may be retained in grade or denied a diploma. Schoenfeld addresses both intended and unintended consequences of assessments.

Judith A. Ramaley’s discussion in Chapter 2 addresses the second P, philosophy. As indicated in the previous paragraph, testing reflects one’s values and goals. The issue at hand is not only “What is mathematics,” but “Which aspects of mathematics do we want students to learn in school?” Is the purpose of schooling (and thus of mathematics instruction in school) to provide the skills needed for successful participation in the marketplace and in public affairs? Is it to come to grips with fundamental issues of truth, beauty, and intellectual coherence? To use some common jargon, these are consequential decisions: the answers to questions of values shape what is taught, and how it is taught. As Ramaley notes, the Greeks’ two-thousand-year-old philosophical debates lie at the heart of today’s “math wars.” But, as she also observes, science brings philosophy into the present: questions of “what counts” depend on one’s understanding of thinking and learning, and of the technologies available for assessing it. Discussions of what can be examined, in what ways, bring us firmly into the twenty-first century.