The preservice and continuing education of teachers of mathematics should provide multiple perspectives on students as learners of mathematics by developing teachers' knowledge of-

research on how students learn mathematics;

the effects of students' age, abilities, interests, and experience on learning mathematics;

the influences of students' linguistic, ethnic, racial, and socioeconomic backgrounds and gender on learning mathematics;

ways to affirm and support full participation and continued study of mathematics by all students.

Elaboration

Learning is an active, dynamic, and continuous process that is both an individual and a social experience. Children are naturally inquisitive and have a desire to learn. Their early experiences reflect the excitement of discovery. In school, however, limitations of time, place, and perceptions often constrain what is natural as children encounter environments that are not responsive to them as learners.

The study of general principles of teaching and learning is insufficient for teachers of mathematics because it does not include consideration of the nature of mathematics and of current research on children's mathematical thinking and its implications for instruction. Children build a variety of perceptions of mathematics as they learn. Some of these perceptions are confused or incomplete; others are remarkably effective. Teachers need opportunities to examine children's thinking about mathematics so that they can select or create tasks that can help children build more valid conceptions of mathematics. Developing multiple perspectives on students as learners of mathematics enables teachers to build an environment in which students may learn mathematics with appropriate support and acceptance.

Professional development programs, both preservice and in-service, should incorporate current theories and research from mathematics education and the behavioral, cognitive, and social sciences as they relate to mathematics learning. For example, central to current theories is the view of the learners as active participants in learning. Learners construct their own meaning by connecting new information and concepts to what they already know, building hierarchies of understanding through the processes of assimilation and accommodation. Mathematics is learned when learners engage in their own invention and impose their own sense of investigation and structure.

The implications of such research and theory building to teaching are continually unfolding as new results from research and practice provide new insights and directions for our understanding. Programs for teachers should help them develop habits of mind that include becoming active researchers in their own classrooms as well as users and interpreters of research as it relates to their everyday teaching. Teachers must be able to interpret research related to instructional issues in order to determine how these issues can be addressed in their teaching. A sampling of current instructional issues includes-

• the role of number sense, counting, and manipulative materials;
• language and its impact on early mathematics learning;
• the implications of children's informal mathematics concepts about size, shape, and space as well as number and chance;
• the role of calculators and computers;
• probabilistic and proportional reasoning;
• the role of variable and function;
• the inclusion of discrete mathematics.

These are but a few of the many instructional issues that will continue to be investigated as we strive to improve the teaching and learning of mathematics.

With the help of the available technologies, it is possible to bring the study of children's mathematical thinking "alive" in new ways. Videotapes may be used to portray developmental sequences in learning or to demonstrate assessing students' developmental levels on the basis of specific learning tasks such as those prescribed by learning theorists or designed for new research on students' learning. Indeed, computer-controlled videodisk options make it possible to develop interactive learning environments that teachers may explore to better understand children's thinking in different classroom environments.

In addition, clinical experiences such as interviewing children one-on-one or in groups allows teachers at any level to appreciate what can be learned by talking to students. In conjunction with seminars, courses, or other professional development activities, practicing teachers can learn about current research on children's understanding of mathematics concepts and can validate their knowledge of their own students in order to build deeper understanding of the research and its implications. In follow-up seminars, teachers have opportunities to report and discuss their findings. Changed perceptions about what their students can and cannot do affects teachers' attitudes and beliefs about their students and about their teaching strategies. The importance of teachers' knowledge of how students learn mathematics cannot be minimized. Such knowledge provides direction for the kinds of learning environments that teachers of mathematics create, the tasks they select, and the discourse that they foster.

Teacher expectations have significant impact on what happens to children in school. Teacher expectations are founded on knowledge and beliefs about who their students are and what they can do. Teachers' understanding of the impact of students' age, abilities (both mental and physical), interests, and experience on their learning of mathematics are all important ingredients in building perceptions of students as individuals.

How does mathematics appear to an eight-year-old? A twelve-year-old? A fifteen-year-old? Teachers must simultaneously be able to perceive mathematics through the minds of their students while they perceive the minds of their students through the mathematics in which they are involved. Such a perspective requires a thorough knowledge of children's developmental characteristics that emphasizes children's patterns of intellectual, social, and emotional growth. Beyond a general, comprehensive overview, teachers of the middle grades and secondary levels need a more detailed understanding of adolescence. Such understandings at all levels must be interwoven with teachers' own developing knowledge about how children learn mathematics.

Teachers' beliefs about students often are tied to their perceptions of students' intellectual abilities. Yet the research on ability grouping calls into question current tracking practices. For example, the research indicates that heterogeneous groups make sense in elementary schools. Further, the results of homogeneous grouping in secondary school do not justify its strong support among teachers (Kulik and Kulik 1982; Oakes 1985; Slavin 1986). Teachers need knowledge about, and experience with, using alternative strategies such as cooperative and team learning that permit them to work well in heterogeneous environments.

Another problem with tracking in mathematics is that students who have difficulty during their elementary and middle school years with traditional paper-and-pencil computation, both arithmetic and algebraic, often are limited in their access to advanced mathematics. However, computational competence is not always a valid measure for success at advanced levels of mathematics. Hypothesizing, approximating, estimating, reasoning, problem solving, and communicating are skills and abilities not often tapped or promoted through traditional computational work. In affirming and encouraging full participation by every student, issues surrounding false scope and sequence barriers that establish inappropriate prerequisites must be considered in professional development activities with teachers.

Teachers also need to understand the importance of context as it relates to students' interest and experience. Instruction should incorporate real-world contexts and children's experiences and, when possible, should use children's language, viewpoints, and culture. Children need to learn how mathematics applies to everyday life and how mathematics relates to other curriculum areas as well.

The ability to recognize and enfold mathematical aspects of ethnic and cultural identity helps in providing an impetus for further study of mathematics. Providing students from underserved and underrepresented groups who lack the observed presence of role models with other means of motivation and incentive for study is one way to do this. References to the contributions made to the discipline by members from underrepresented groups can partially meet this need.

It is important to note that culture-sensitive does not mean a focus on the traditional arts, foods, and folklore of a group. Instead culture-sensitive means sensitivity to "relatively subtle aspects of interactional etiquette [that] are likely to go unrecognized by non-minority teachers." (Erickson and Mohatt 1982)

Language and its role in students' understanding and doing mathematics needs attention in programs for the development of teachers. Students may lack appropriate vocabulary and syntax to express themselves mathematically but still be able to learn and demonstrate sophisticated knowledge of mathematics. In some circumstances, students' understanding of the language used to communicate mathematics may be incomplete or incorrect; these misunderstandings can create subtle barriers to success in the mathematics classroom. Teachers' knowledge of their students' cultural backgrounds and the implications of this knowledge for their teaching is crucial in recognizing the impact of language on learning. Beyond this, teachers have a responsibility to help students grow in the correct and appropriate use of mathematical language.

Increasing attention has been given to girls' lack of participation in mathematics. Reasons have included a range of hypotheses, such as girls' lack of self-confidence in their mathematical abilities, their association of mathematics with males, and a belief that mathematics is distant from everyday concerns. Earlier considerations of this problem have focused on how to change girls' perception of, and involvement in, mathematics. However, current work indicates that females make sense of information and learn in ways that are significantly different from the traditional approach to teaching mathematics. Programs for mathematics teachers need to provide access to the literature that explicates the problem of engaging girls in the study of mathematics and identifies successful intervention strategies.

Encouragement and compliments must be specific to be effective, and it is best to praise girls for their ability...not just for their effort. Don't say, "That's okay, you tried." or "Your work is very neat" even though it may be of poor academic quality. (Franklin 1990)

The general issue of discrimination in the classroom is of concern. Such discrimination is often subtle and not intentional, yet it exists. Testing for differential treatment of students is one aspect of addressing the problem of reaching all students. Are there gender, cultural, or racial differences a teacher's interactions with students in the class? Teachers need help in learning to monitor classroom interactions; a colleague observing or videotaping a class can be of assistance in doing this. Recording instances of positive and negative feedback, disciplinary and social interactions, as well as the name of each student who does and does not receive attention, can provide insights into unconsciously biased behaviors. If inequities are identified, then strategies need to be developed to help a teacher address these concerns. For example, a teacher may decide to keep a list of the names of students in the class and check off those she or he calls on during an instructional period, thus becoming more aware of the participation of all children. Such strategies need to be discussed in professional development activities for teachers.

Grouping of students, classroom climate, choice of materials, topics, activities, testing, and teaching strategies all have impact on how effectively all students consider themselves as involved and active members of the classroom. Each component must be addressed with regard to students' age, abilities, and interests and to their academic, ethnic, racial, cultural, and gender differences. A genuine respect for, and understanding of, students as individuals and as participants in a community of learning is essential to promoting the kinds of experiences that involve all students in mathematics.

Vignettes

Dr. Williams has access to a few of the videotapes that have been produced as part of a research project an students' understanding of the concept of average.

By solving the problem themselves, the teachers gain personal awareness of possible strategies before watching a sixth-grade student solve the same problem.

3.1 Dr. Williams, the district mathematics coordinator, and several grade 4-8 teachers have formed a mathematics study group that meets monthly. They have been reading about and discussing ways to investigate their students' thinking. In the course of this discussion they began to explore the issue of teaching statistics-particularly the concept of mean. Dr. Williams has brought in a tape of an interview with a sixth-grade student.

Dr. Wiliams: Before we look at this tape, I want to pose a problem. You should solve the problem and pay particular attention to the strategy you use.

You have nine bags of different kinds of potato chips and you know that the average cost for a bag of chips is $1.38. What might be the actual prices of each of the nine bags of chips?

The teachers work individually and then compare strategies with one another. They then spend time discussing their various strategies together. Following this, Dr. Williams plays the videotape, which the teachers discuss in light of their own strategies.

Mary: I noticed that Sara, the student in the tape, seemed comfortable with the problem. It made some sense to her because she is familiar with the cost of bags of potato chips.

Irv: Yes, right away she said, "Well, I know they can't all be the same price. That's not the way it is in the store." Unlike her, I immediately decided to do a "quick and easy" solution and make each bag the same price-but that certainly isn't real!

The teachers focus on the role of the researcher. How does someone work with a student to gather information about her thinking strategies?

The teachers have an opportunity to work with students, using research tasks from the project considered in the study group.

These teachers are engaged in experiences, such as considering the effects of age on student learning, that can affect their knowledge of research and their dispositions to listen to children and use the information gained to make instructional decisions.

Dr. Williams: What did you notice about the way the interviewer worked with Sara?

Don: Well, he posed the problem and explained the materials that were available, pointing out the pictures of the bags and the markers she could use to record prices. Then he became an observer. Occasionally he asked a question for clarification, but he really didn't involve himself in her work. Gee, it would be difficult for me to know when to intervene and when to keep silent. I would want to jump in with suggestions!

The teachers and Dr. Williams continue the discussion, commenting on the student's response to the problem. Finally, Dr. Williams provides a synthesis from this research project, detailing other problems used in the research and providing an overview of the results of the research that focuses on the developmental differences that were found between the fourth-, sixth-, and eighth-grade students who were interviewed.

The teachers agree to pose the same problem to some of their students and bring the results to the next study group meeting.

The teacher looks for ways to engage all students and selects a task that will push his students' thinking further.

3.2 Darren Hensh and Cora Horatio have been long-time colleagues and often consult each other about their teaching concerns. In a recent discussion Darren brought up a situation that arose when teaching mathematics to his third graders (Ball 1990).

Darren: Up to this point the work my class has done with division has included dividing things into groups of equal size and talking about leftovers. I felt they were ready to consider fractions by dividing the leftovers up, too. So I posed this problem:

You have a dozen cookies, and you want to share them with the other people in your family. If you want to share them all equally, how many cookies will each person in your family get?

The teacher knows his students and can use this knowledge to plan activities that are mathematically interesting and enhance the discourse in the classroom.

The teacher is sensitive to children's different family backgrounds, recognizing that the appropriateness of the context of a problem is an important factor to consider when designing and selecting problems.

Cora: Hey, that sounds like a clever problem! Your divisor would vary nicely among the students, providing for a range of interesting solutions, some simpler than others. I often use family size as a context for problems I do with my sixth graders.

Darren: You do? You know, I like the diversity it provides, but I am questioning my choice because I didn't anticipate the interactions it stimulated. As I listened to their discussions, I realized that who to count in a family is not cut and dried, particularly for young children. For example, Enrico wondered if he should be counting the baby that is due soon. Angela included her mother's boyfriend as one of her family members, which confused her friend Jill. And ... oh, yes,.... Micha's parents are divorced, and his father lives in another state. But he included him in his count. His two partners argued with him, saying that he really couldn't count him if he didn't live with him.

All the students found solutions that were satisfactory, but somehow, I felt uneasy about the nature of their arguments. They seemed intrusive and possibly made some of the students uncomfortable. I am doubting the wisdom of choosing such a problem.

Cora: I know what you mean. They all have very different conceptions of family, so it can be awkward. But I don't think you should avoid issues like this. Not only is family size a realistic and engaging context for such problems, but it is important for children to become aware of the different ways people define family and to respect these differences.

Darren: I agree. I am pleased to have the diverse range of views in the class. It was their tendency to impose their own conceptions on one another that I found troubling.

Cora: When I've used family size with students we begin with a class discussion about who to count. Yes, it is a sensitive issue, but I have found that the children naturally respect one another's lifestyles when approached as part of "finding a definition" that works for all of us. It's very interesting to listen to their own conceptions of family. I always learn so much about each student. Beginning with a discussion like that should help them recognize that there are many ways to think about the family unit.

The behavior of fifth graders in male-female interactions is explored.

3.3 The student teachers in Dr. Dreyfus's seminar group are discussing a teaching dilemma described in an article on how teachers manage to teach (Lampert 1985):

"The children in my classroom seem to be allergic to their peers of the opposite sex. Girls rarely choose to be anywhere near a boy, and the boys actively reject the girls whenever possible. This has meant that the boys sit together at the table near one of the blackboards and the girls at the table near the other."

Boys are receiving more attention and hence more help as a result of the teachers behavior management strategy.

No algorithms for such decisions exist. The teacher is shown struggling with a real teaching dilemma.

Issues of gender equity must be addressed in light of other pedagogical concerns, such as the role of discourse. These students are gaining a sense of the complexity of teaching and of ways in which they need to monitor their own behaviors in the classroom.

Student teachers are developing sensitivity to all students in their classrooms and seeking ways to manage the ongoing dilemmas of teaching.

"But whole-class discussions are a central part of my mathematics classes," Maureen responds. "By the time the students are in ninth grade the pattern has been established. The boys dominate the discussion and many girls are reluctant to contribute. I find myself asking the girls less challenging questions or changing the tone of my voice in order to encourage them to participate. So in a sense I'm countering one imbalanced situation with another. But I'm hoping to gradually change that."

"I don't think we can expect changes overnight," Rika reminds her peers.
As we leave the seminar, other students are sharing strategies for the teacher's dilemma and raising questions about similar issues from their student teaching.