This account of a sixth-grade teacher introduces the reader to the annotated vignettes that are used throughout this document to elaborate the visions of teaching, the evaluation of teaching, and professional development. Narratives - drawn from actual school and university classrooms with a range of teachers and students in a variety of contexts are annotated throughout in italics. The narratives are meant to be like video clips. They provide brief but vivid glimpses into diverse settings and help to build depth into the images created by this document. As such, they are intended to animate the standards: they illustrate points discussed in the text and make the issues multidimensional. Although the vignettes do exemplify some specific worthwhile practices, they do not suggest one "correct" approach to teaching mathematics. In the following introductory vignette, the comments on the right foreshadow issues that are examined throughout this volume. These annotations are intended to help orient the reader to the sections that follow.

The Professional Development section discusses teachers' responsibilities for their own professional development and takes the position that good mathematics teaching should be modeled in teachers' professional development experiences.

Both the Evaluation and the Teaching sections describe ways in which teachers can analyze their own teaching.

The Evaluation section addresses the need for teachers to receive support from mathematics specialists.

The Teaching section deals with orchestrating the discourse in the learning environment. Both the Teaching and the Evaluation sections stress the commitment to helping EVERY student develop mathematical power.

Three days remain until the beginning of school. Sharon Robinson is sitting in her classroom, leafing through materials from the summer school class she took as part of her master's program at the nearby college. She really liked the course. It included a stimulating mix of new ideas, opportunities to experiment, and time for the teachers enrolled in the course to think and talk with one another. Former classroom teachers themselves, the two professors who team-taught the course seemed sensible and realistic, and yet they clearly had a different orientation to mathematics teaching.

The eight-week course had been exactly what Sharon needed, for she had finished the school year in June feeling vaguely dissatisfied with her mathematics teaching. After five years of teaching, she had become able to manage her classroom effectively, to cover the required curriculum, and to incorporate some neat supplementary activities. Sharon received an excellent evaluation from her principal, Mrs. Bowdoin. The principal, however, had little mathematics background and her comments were always focused on management issues. These did not address Sharon's growing questions about her mathematics teaching.

Her sixth graders typically did quite well on the district mathematics test. Still, Sharon was troubled about her students' participation in, and success with, mathematical reasoning and problem solving. For example, she noticed that her boys talked and were much more active than her girls. She is also aware - and concerned - that many of her African American and Hispanic students did not go on to take algebra in the eighth or ninth grade. She wants to do something to affect these patterns of participation. In general, she felt that her students lacked both confidence and skills with mathematical reasoning and problem solving. If she gave her students word problems to do, for example, they often gave up easily or came up with answers that made no sense at all.

The Teaching section elaborates on the dimensions of teaching - tasks, discourse, environment - that are involved in changing one's approach; it makes plain that this document offers a vision, not a recipe, for creating new practices.

Evaluation should support and enhance a teacher's professional growth. Both the Evaluation and the Support sections discuss the role administrators play.

How teachers can support one another's professional growth is a continuing theme in this volume. All sections stress the importance of teachers paying attention to students' knowledge and their ways of thinking about mathematics.

The Teaching section elaborates criteria for fruitful mathematical tasks.

Teachers who strive to change their mathematics teaching in the directions outlined by these standards are in the position of creating and reflecting on new practices.

Tom Flood, Sharon's colleague, wanders into the room, and she describes her quandary to him - how to start the new year? She wants to begin to shape the classroom in this new direction. She wants to promote more conjecturing and problem solving. She also wants to learn more about her students - how they think, what they know and can do, how they feel about mathematics. On the basis of these concerns, Sharon is considering using the following problem:

Take 12 tiles and build rectangles with them. How many rectangles can you make using all 12 tiles?

Now try 16. How many can you make using all 16?

Now find another number of tiles that will let you make MORE rectangles than you can make with 12 tiles.

She explains that she likes this problem because it also gives students an opportunity to make some nice connections - between number theory and geometry, for instance.

"It is a nice problem," nods Tom. "You will be able to learn a lot about what they already know-like about factors, about what a rectangle is, about how to work on mathematics problems."

"Yes!" Sharon breaks in. "When we did a problem like this in my summer school class, an important part of it was the question of how you knew when you had all the possible rectangles and the idea of proving that to the other people in the group. I had never thought about that as an important issue at all. Like with 12 tiles, you know you are done when you have built a 1 x 12, a 2 x 6, a 3 x 4 - and the opposites of those, like 12 x 1, 6 x 2, and so on. You can prove that you have finished because you have taken all the numbers that divide 12 - all its factors - and made rectangles with them." The words are tumbling out of Sharon's mouth. "And I thought - just as you said - that I could learn a lot about what they know and about their dispositions toward mathematics. I wanted to do more than just set the tone for the year, although that is a good idea, too. I hope I manage that."

"But why did you choose 12 - and then why 16?" asks Tom.

Knowledge of students' understandings and ways of thinking helps teachers to construct worthwhile mathematical tasks. This is explored in the Teaching section.

"Well, 12 makes six rectangles, but 16 makes only five. I think that will surprise them. And I'm not sure they will consider the 4 x 4 shape a rectangle. I bet they think a square is not a rectangle. I would like to get them to examine that," explains Sharon.

Tom thinks about this for a moment. "So is that why you ask about 16 tiles before you ask the last question - about finding one that will make more rectangles than 12? Aren't you the clever one!"

Sharon laughs. The two discuss the activity further and find themselves having fun considering ways to extend the problem or have the students generate extensions - some to pursue perhaps now and others to return to later in the year. They keep a list:

• What other numbers of tiles will make an odd number of rectangles? Why is that?

• What numbers of tiles will make the fewest rectangles? Why is that?

• What numbers of tiles can produce exactly three rectangles? Or six? Is there a pattern?

Sharon and Paul continue to brainstorm, and they come up with still more things to explore later. They consider the possibility of exploring, in some related ways, triangular arrays, or pursuing similar investigations in three dimensions.

That collegial interactions contribute a great deal to teachers' professional development is discussed in all sections of this document. The Teaching section also deals with teachers' study of their own practice.

The patterns of discourse described in the Teaching section are significantly different from the traditional patterns in mathematics classrooms.

Tom has to meet with a parent. Before he leaves, he indicates that he will also try the same problem with his class and suggests that they compare notes afterward. He admits that he, too, has been thinking about making some changes in the ways he approaches mathematics in his class.

Sharon jots down a few more notes from their work together. Her mind turns to the problem of how she can get her sixth graders used to a different pattern of discourse, one in which answers will be determined to be right by whether they make sense and can be proved or explained - not by whether she says, "Good job!" She realizes that her students are used to the teacher being the one who tells you if you're right - that when other kids are talking, it has nothing to do with you.

Sharon wonders how her students will respond to her asking them, "What do you all think about what So-and-so just said?" She remembers how one of the teachers in the summer workshop told how when she first asked questions like that, kids just stared at her, quite confused. She wonders whether they will even be able to explain their answers.

The importance of teachers working together and learning from one another is emphasized throughout all sections.

Sharon sits back with a sigh and stretches. There is a lot to do. Still, she is excited at the prospects that lie ahead. And she is glad that Tom wants to work on this too - it will really help to have someone to talk to. But she needs to take a break from all this thinking. With one last gulp of strong lukewarm coffee, she rises and returns to the task of putting up her bulletin boards.