The teacher of mathematics should create a learning environment that fosters the development of each student's mathematical power by-

providing and structuring the time necessary to explore sound mathematics and grapple with significant ideas and problems;

using the physical space and materials in ways that facilitate students' learning of mathematics;

providing a context that encourages the development of mathematical skill and proficiency;

respecting and valuing students' ideas, ways of thinking, and mathematical dispositions;

and by consistently expecting and encouraging students to-

work independently or collaboratively to make sense of mathematics;

take intellectual risks by raising questions and formulating conjectures;

display a sense of mathematical competence by validating and supporting ideas with mathematical argument.

Elaboration

This standard focuses on key dimensions of a learning environment in which serious mathematical thinking can take place: a genuine respect for others' ideas, a valuing of reason and sense-making, pacing and timing that allow students to puzzle and to think, and the forging of a social and intellectual community. Such a learning environment should help all students believe in themselves as successful mathematical thinkers.

What teachers convey about the value and sense of students' ideas affects students' mathematical dispositions in the classroom. Students are more likely to take risks in proposing their conjectures, strategies, and solutions in an environment in which the teacher respects students' ideas, whether conventional or nonstandard, whether valid or invalid. Teachers convey this kind of respect by probing students' thinking, by showing interest in understanding students' approaches and ideas, and by refraining from ridiculing students. Furthermore, and equally important, teachers must teach students to respect and be interested in one another's ideas.

Demonstrating respect for students' ideas does not imply, however, that teachers or students accept all ideas as reasonable or valid. The purpose of valuing students' ideas and ways of thinking is not just to make students feel good but to foster the development of their understanding of, and power with, mathematics. Therefore, the central focus of the classroom environment must be on sense-making. Mathematical concepts and procedures - indeed, mathematical skills - are central to making sense of mathematics and to reasoning mathematically. Teachers should consistently expect students to explain their ideas, to justify their solutions, and to persevere when they are stuck. Teachers must also help students learn to expect and ask for justifications and explanations from one another. Teachers' own explanations must similarly focus on underlying meanings; something a teacher says is not true simply because he or she "said so."

Emphasizing reasoning and justification implies that students should be encouraged and expected to question one another's ideas and to explain and support their own ideas in the face of others' challenges. Teachers must help students learn how to do this: Students need to learn how to question another's conjecture or solution with respect for that person's thinking and knowledge. They also need to learn how to justify their own claims without becoming hostile or defensive.

Serious mathematical thinking takes time as well as intellectual courage and skills. A learning environment that supports problem solving must allow time for students to puzzle, to be stuck, to try alternative approaches, and to confer with one another and with the teacher. Furthermore, for many worthwhile mathematical tasks, tasks that require reasoning and problem solving, the speed, pace, and quantity of students' work are inappropriate criteria for "doing well." Too often, students have developed the idea that if they cannot answer a mathematical question almost immediately, then they might as well give up. Teachers must encourage and expect students to persevere, to take the time to figure things out. In discussions, the teacher must allow time for students to respond to questions and must also expect students to give one another time to think, without bursting in, frantically waving hands, or showing impatience.

Students' learning of mathematics is enhanced in a learning environment that is built as a community of people collaborating to make sense of mathematical ideas. It is a key function of the teacher to develop and nurture students' abilities to learn with and from others - to clarify definitions and terms to one another, consider one another's ideas and solutions, and argue together about the validity of alternative approaches and answers. Classroom structures that can encourage and support this collaboration are varied: students may at times work independently, conferring with others as necessary; at other times students may work in pairs or in small groups. Whole-class discussions are yet another profitable format. No single arrangement will work at all times; teachers should use these arrangements flexibly to pursue their goals.

Vignettes

5.1 A class of primary students has been working on problems that involve separating or dividing. The teacher, Laurie Morgan, is trying to give them some early experience with multiplicative situations at the same time that she provides them with contexts for deepening their knowledge of and skill with addition and subtraction. These students can add and subtract, but their understanding of multiplication and division is still quite informal. They have begun to develop some understanding of fractions, connected to their ideas about division. They have not yet learned any conventional procedures for dividing.

The teacher has selected this problem because it is likely to elicit alternative representations and solution strategies as well as different answers. It will also help the students develop their ideas about division, fractions, and the connections between them.

The teacher allows time for the children to develop their solutions independently, with a few others, and then in the whole group. By asking who would like to share their solution, she encourages the students to take intellectual risks.

Today Mrs. Morgan has given them the following problem:

If we make 49 sandwiches for our picnic, how many can each child have?

After they have worked for about twenty minutes, first alone and then in small groups, Mrs. Morgan asks if the children are ready to discuss the problem in the whole group. Most, looking up when she asks, nod. She asks who would like to begin.

Two girls go to the overhead projector. They write:

Students expect to have to justify their solutions, not just give answers.

The teacher solicits other students' comments about the girls' solution without labeling it right or wrong. She expects the students, as members of a learning community, to decide if an idea makes sense mathematically.

Students respectfully question one another's ideas. The girls "revise" their solution because they have been convinced by the boy's explanation. There is no sense here that being wrong is shameful.

One explains, "There are twenty-eight kids in our class, and so if we pass out one sandwich to each child, we will have twenty-two sandwiches left, and that's not enough for each of us, so there'll be leftovers."

The teacher and students are quiet for a moment, thinking about this. Then Mrs. Morgan looks over the group and asks if anyone has a comment or a question about this solution.

One boy says that he thinks their solution makes sense, but that "nine minus eight is one, not two, so it should be twenty-one, not twenty-two." He demonstrates by pointing at the number line above the chalkboard. Starting at nine, he counts back eight using a pointer. The two girls ponder this for a moment. The class is quiet. Then one says, "We revise that. Nine minus eight is one." Mrs. Morgan is listening closely, but does not jump into the interchange.

Another child remarks that he had the same solution as they did - one sandwich.

The students work together to solve the problem. Sometimes they build on the solutions offered by classmates. The teacher gathers insights about students through close listening and observation. At times, she takes responsibility for pushing students' thinking along.

The teacher expects the students to reason mathematically.

"Frankie?" asks Mrs. Morgan, after pausing for a moment to look over the students. She remembers noticing his approach during the small-group time. Frankie announces, "I think we can give each child more than one sandwich. Look!" He proceeds to draw twenty-one rectangles on the chalkboard. 'These are the leftover sandwiches," he explains. "I can cut fourteen of them in half and that will give us twenty-eight half-sandwiches, so everyone can get another half."

"I agree with Frankie," says another child. "Each child can have one and a half sandwiches."

"Do you have any leftovers?" asks the teacher.

"There are still seven sandwiches left over," says Frankie.

"What do the rest of you think about that?" inquires the teacher.

Students seem willing to take risks by bringing up different ideas.

The teacher expects students to clarify and justify their ideas.

Several children give explanations in support of Frankie's solution. "I think that does make sense," says one girl, "but I had another solution. I think the answer is one plus one-half plus one-fourth."

"I don't understand," Mrs. Morgan says. "Could you show what you mean?"

The teacher has posed a task that gives the students an opportunity to develop their understanding of "sample" and "population," as well as to build their ability to use statistics to reason about real-world situations.

The task itself promotes collaboration. It is not clear how to proceed, and everyone's contributions are needed in order to come up with an approach.

The teacher is providing time for students to grapple with the problems.

These students seem to have developed the disposition to question one another, and they respond to one another's queries as a matter of course.

Helping a group of students develop a sense of community is facilitated by having some sense of shared purpose.

The students seem accustomed to thinking through problems together.

5.2 Mr. Cohen's class of high school students is working in small groups on projects that involve collecting, organizing, and interpreting data. Before beginning, they had a discussion about different possibilities for their projects. They decided on questions that they would like to pursue, such as finding the average number of hours per week that high school students work.

One group, having read an article in the newspaper on changes in the popularity of first names in the last sixty-five years, has decided to investigate the most common boy's and girl's first names among students their age in the city. Is John still the most common boy's name, as it was in 1925, 1950, and 1975? They are curious about what is happening with girls' names, since the popular names seem to change more often. Because theirs is an ethnically diverse community, they also wonder how that affects the pattern of names.

Mr. Cohen works his way around to the different groups, listening, making suggestions, and verifying that the group members are listening and working together. The group that is working on the names study has decided to sample the high school population in the city and is discussing the best way to go about this.

John: Let's choose three of the high schools and then write to them and ask for a list of the students enrolled in the school. We can take our sample from those lists at random.

Jenny: But how will we pick the three schools? And why is three a good number to pick?

John: It seemed like enough out of all the schools in this city if we were careful to include one of the schools that has more kids from different backgrounds, because we want to make sure our sample has lots of different kinds of names, just like there are around here.

Anna: I think we should try to figure out about how many high school kids there are in the whole city and then pick a size for our sample based on that.

John (nodding): I guess that makes sense. How are we going to figure that out, though?

Maria: And then how big would our sample have to be to be big enough? We want to be pretty sure that our sample tells us something about all the kids in high school here.

5.3 Mr. Davies, who teaches seventh grade, wants to begin encouraging a more collaborative environment in his classroom. However, he finds that his students seem anxious about having the right answer. They do not reflect on whether or not their answers make sense - as long as they match the answer key. They do like to work in groups, but it seems that the attraction is primarily social.

For example, one day when they were working on a set of ratio and proportion exercises and story problems, he told them to form small groups to discuss the work. The groups were loud, and in each group someone seemed to dominate while others sat by passively. When their answers differed, they usually erased their answers to match one another's. Typically the answer considered "right" was the answer given by Julio or Evie - or whomever they considered smartest in math.

Teachers can help one another focus on and enhance the learning environment.

Tasks affect both the environment and the discourse. Different tasks require different types of instructional strategies.

Teachers give off many cues about what is valued. When teachers are trying to change their approach, sometimes the messages students receive can get mixed.

Discouraged, Mr. Davies consults with the teacher next door, who offers to come observe during her physical education period the next day. As they talk, she suggests that the tasks he is using in the groups may be contributing to the problems. Together they design a couple of tasks dealing with ratio and proportion that they think might get the students thinking more and also provoke better discussion in the groups. When she observes him, she also notices that he frequently praises students for having the right answer and prods those who are not doing their "own work." She points this out to him gently, confessing that this gives her a hard time, too. He says, ruefully, "I guess we have some habits that give the kids some mixed messages at times."

Summary: Environment

The learning environment is a key element in fostering the goals of the Curriculum and Evaluation Standards for School Mathematics. Creating an environment that supports and encourages mathematical reasoning and fosters all students' competence with, and disposition toward, mathematics should be one of the teacher's central concerns. The nature of this learning environment is shaped by the kinds of mathematical tasks and discourse in which students engage. Teachers' skills in developing and integrating the tasks, discourse, and environment in ways that promote students' learning are enhanced through thoughtful analysis of their instruction, which is the focus of the last section of the standards for teaching.