To establish a common point of departure with Jim Milgram’s chapter, this chapter is framed around the two basic questions with which his chapter began:

• • What does it mean for a student to be proficient in mathematics? (What should students be learning?)

• • How can we measure proficiency in mathematics? (How can we tell if we are succeeding?)

My main emphasis is on the first question, because much of the rest of this volume addresses the second.

In the introduction to this volume and in the first chapter, I pointed to the fact that the “cognitive revolution” (see [Gardner 1985], for instance) produced a significant reconceptualization of what it means to understand subject matter in different domains (see also [NRC 2000]). There was a fundamental shift from an exclusive emphasis on knowledge—what does the student know?—to a focus on what students know and can do with their knowledge. The idea was not that knowledge is unimportant. Clearly, the more one knows, the greater the potential for that knowledge to be used. Rather, the idea was that having the knowledge was not enough; being able to use it in the appropriate circumstances is an essential component of proficiency.

Some examples outside of mathematics serve to make the point. Many years ago foreign language instruction focused largely on grammar, vocabulary, and literacy. Students of French, German, or Spanish learned to read literature in those languages—but when they visited France, Germany, or Spain, they found themselves unable to communicate effectively in the languages they had studied. Similarly, years of instruction in English classes that focused on grammar instruction resulted in students who could analyze sentence structure but who were not necessarily skilled at expressing themselves effectively in writing. Over thepast few decades, English and foreign language instruction have focused increasingly on communication skills—on mastering the basics, of course (e.g., conjugating verbs, acquiring a solid vocabulary, mastering grammar) and learning the additional skills that enable them to use what they have learned.

A similar evolution took place in mathematics. The knowledge base remains important; it goes without saying that anyone who lacks a solid grasp of facts, procedures, definitions, and concepts is significantly handicapped in mathematics. But there is much more to mathematical proficiency than being able to reproduce standard content on demand. A mathematician’s job consists of at least one of: extending known results; finding new results; and applying known mathematical results in new contexts. The problems mathematicians work on, in academia or in industry, are not the kind of exercises that get solved in a few minutes or hours; they are problems that may take days, weeks, months, or years to solve. Thus, in addition to possessing a substantial amount of specialized knowledge, mathematicians possess other things as well. Good problem solvers are flexible and resourceful. The have many ways to think about problems— alternative approaches if they get stuck, ways of making progress when they hit roadblocks, of being efficient with (and making use of) what they know. They also have a certain kind of mathematical disposition—a willingness to pit themselves against difficult mathematical challenges under the assumption that they will be able to make progress on them, and the tenacity to keep at the task when others have given up. As will be seen below, all of these are aspects of mathematical proficiency; all of them can be learned (or not) in school; all of them can help explain why some attempts at problem solving are successful and some not.