The teaching of mathematics is undergoing radical changes, as is evident in the recommendations set forth by NCTM's new sets of Standards.The emphasis is no longer on transmitting an established body of knowledge but on making students good problem solvers and critical thinkers, confident in their mathematical ability.

With these new goals in mind, Raffaella Borasi worked closely with two high school girls who did not like math and who rarely had achieved any degree of success in the subject. Over the course of ten lessons, the students and Riffle engaged in a personal inquiry into the nature of mathematical definitions. Together they explored the roles and uses of definitions, which made the students better appreciate the more humanistic, contextualized dimensions of mathematics.

This inquiry enables the students to experience the excitement and satisfaction of acting as "real mathematicians" engaged in original mathematical explorations-a novel role for high school students. And it enables Raffaella to capture the processes and thinking involved in the students' inquiry, and to document the learning and changes that the experience brought about in her students.

In analyzing this experience, Raffaella provides a natural setting in which to raise and discuss more general issues about the learning and teaching of mathematics. Thus, the description of what happened in the classroom is intertwined with important reflections about the experience's wider implications. By providing a concrete example of how school mathematics can be conceived differently, Raffaella enables mathematics teachers to reconceive their own teaching practices and encourage them to begin implementing innovative changes in their own classrooms.