The preservice and continuing education of teachers of mathematics should provide them with opportunities to-

examine and revise their assumptions about the nature of mathematics, how it should be taught, and how students learn mathematics;

observe and analyze a range of approaches to mathematics teaching and learning, focusing on the tasks, discourse, environment, and assessment;

work with a diverse range of students individually, in small groups, and in large class settings with guidance from and in collaboration with mathematics education professionals;

analyze and evaluate the appropriateness and effectiveness of their teaching;

develop dispositions toward teaching mathematics.

Elaboration

I recently saw a young man in a video store; he reminded me that he had been in my class during my second year of teaching third grade. He was telling me that he is now a teacher. I told him that I had learned so much since that time; that I regretted my lack of experience when he was in my class. He really surprised me. He said that he had enjoyed my class even then because I loved math and that he had not had another teacher that truly loved math until he took trigonometry in high-school. (A middle-grades teacher)

This standard addresses issues that are at the heart of teaching. The goal of teacher education is to "light the path" for those who follow, providing directions on how to plan and teach mathematics. It is the practice of teaching, the growing sense of self as a teacher, and the continual inquisitiveness about new and better ways to teach and learn that serve teachers in their quest to understand and change the practice of teaching.

The nature and kinds of teaching experiences that should be part of the preservice and continuing education of teachers of mathematics are varied and numerous. For teacher candidates, this involves opportunities to work one-on-one or with small groups of students in clinical settings that permit them to focus on interviewing or microteaching. They need a sequenced program that provides them with opportunities to be in classroom settings for a variety of purposes and with increasing levels of responsibility. Finally, they need long-term placements that permit them to become the teachers of students under the guidance and support of both a cooperating practitioner and a mathematics educator.

During the first few years, teaching is an intensely focused experience that centers on the students for whom the teacher is responsible and on the teacher's growing sense of self as a teacher of mathematics. Colleagues and supervisors can function informally and formally as resources during this time of transition between the structured and guided preparation to teach and the comfort provided by a few years' successful experience with teaching. Indeed, beginning teachers often welcome and seek the advice of more experienced teachers to give guidance and provide some diversity in models of how to teach.

Experienced teachers have different needs. They have general frame that surrounds their picture of teaching and understand the ebb and flow of the learning process as it proceeds daily, weekly, and monthly throughout the school year. They are better able to anticipate timing, overall organization and management, and student response. Their repertoire of instructional methods has "filled out", and they often can successfully anticipate what works and does not work in the classroom. Nevertheless, they might find times and opportunities when they turn to colleagues and supervisors to assist them in assessing their teaching and making changes. In addition, when teaching new material or trying out new methods of teaching, teachers are in a position of self-evaluation regarding what works and does not work for them.

Good mathematics teaching is enhanced by conversations with colleagues and supervisors who know mathematics and have been successful in teaching mathematics. Preservice teachers should have opportunities to teach with exemplary mathematics teachers. They should be supervised by teacher education faculty who know mathematics and are experienced mathematics teachers themselves. Practicing teachers also should involve colleagues or teacher educators with backgrounds in mathematics teaching when they are exploring new ways to teach or seeking feedback on current teaching strategies. Mathematics has its own content and pedagogy. Only those knowledgeable about the associated special issues and experienced in the field should serve as mentors or supervise teachers' clinical and filed-based learning experiences.

Essentially, being a teacher of mathematics means developing a sense of self as such a teacher. Such n identity grows over time. It is built from many different experiences with teaching and learning. Further, it is reinforced by feedback from students that indicates they are learning mathematics, from colleagues who demonstrate professional respect and acceptance, and from a variety of external sources that demonstrate recognition of teaching as a valued profession. Confident teachers of mathematics exhibit flexibility and comfort with mathematical knowledge and commitment to their own professional development within the larger community of mathematics educators.

Vignettes

The university faculty and cooperating teacher work together to help preservice teachers develop as teachers of mathematics.

5.1 In her mathematics methods class, Dr. Palmer has been trying to help prospective elementary teachers learn to "listen mathematically" to children. They have been reading case studies of young children, watching videotapes, and reading theoretical pieces on how children learn mathematics. This week Dr. Palmer assigned students the task of examining closely some aspect of students' understanding in their field classrooms. This afternoon she is meeting with Mr. Konook and five prospective teachers who work in his fourth-grade class to discuss their observations.

Damon immediately brings up a conversation he had with one of the students that afternoon. "The last question on the board was to show which was more, 2/6 or 1/3. Tia wrote

The student teacher is learning aspects of informal assessment of children's thinking.

When I asked her to explain, she drew this picture:

Because she didn't draw each piece the same size, her picture of 2/6 was indeed larger than 1/3."

"That's interesting," Lisa said. "Latalya got the same answer but for a different reason. She drew this picture:

and said that 2/6 is more than 1/3 because one-third is the same as one out of three and two-sixths is the same as two out of six. We have three dogs, one of them is black. We have six hamsters, two of them are black. We have more black hamsters than black dogs."

"Wait," Maura said, "I don't understand. One of three dogs and two of six hamsters are both one-third."

"But she is right," Peter exclaimed, "because she does have more black hamsters than black dogs."

The discussion raises teaching issues that focus on how children learn and confront the prospective teachers with subtleties of the mathematics.

The instructor pushes the prospective teacher's thinking by asking questions, not by suggesting answers.

The instructor asks questions that encourage the prospective teachers to call upon their own mathematical knowledge to assess the depth of student understanding.

The university instructor uses her students' observations of children's partial understandings to highlight difficult concepts and enrich her students' understandings of the topic.

Lisa explained, "I wouldn't have understood how Latalya got her answer if I hadn't asked. I just assumed that she didn't understand the problem. If that were a question on a test, her answer would not reflect what she does know about fractions."

"But we know that /3 is equivalent to2/6 when we are talking about two equal-sized things. Don't we want them to see that? Isn't this just confusing them?" asked Maura.

Kamisha added, "If Latalya has twenty cats and two of them are black, should she say that 2/20 was the same amount as 2/6?"

"What do you think she would do?" Dr. Palmer asked Kamisha.

"It seems like she is only using the numerators to determine which of them is more."

"What difference does that make?" asked Dr. Palmer.

"She's not thinking about how much of the denominator it is."

"What do you mean? Lisa asked. "She seems to understand that the denominator is the number of things in the set."

"But when she is comparing two fractions of different-sized sets she is only considering the number of parts and not how much of the total set they are."

"That is one of the things that makes fractions complex," Dr. Palmer commented. "Not only are we interested in the numerator and denominator, but we also want to know about their relationship to one another."

"Perhaps she is thinking about that, but she is considering two different-sized sets," Lisa suggested.

The thinking of each prospective teacher is encouraged.

Ways to think about fractions come out naturally in the discussion. The prospective teacher is disposed toward understanding the students' thinking rather than merely judging it as right or wrong.

"Could you say more about that, Lisa?" Dr. Palmer asked.

"One of the things Latalya is comparing is twice as big as the other and all the pieces are the same size. So two of the larger set is more than one of the smaller set." Lisa went to the board and drew the boxes around the circles in Latalya's drawing to show that she set of hamsters was twice as large as the set of dogs.

"Her 2/6 is more than her 1/3," she concluded.

The classroom teacher's analysis of the task and rationale for using it model the process of selecting and posing worthwhile tasks. He also uses the open-ended situation to assess children's thinking.

"Tia is doing something different," Peter volunteered. "She is comparing two equal-sized sets. She has just divided them up unequally. She may not realize that each thing should be divided into equal pieces, although she seems to understand the part-whole relationship."

Turning to Mr. Konook, Damon suggested, "Maybe the task should have already had the pictures drawn with it; then these problems wouldn't have come up."

"Actually," explained Mr. Konook, "I intentionally put the problem on the board as a symbolic statement and asked them to show how they got their answers. I wanted to see what they understood about comparing fractions. Listening to your descriptions of the conversations you had with these students has been very helpful."

Students are encouraged to consider subtle aspects of teaching fractions.

"Wow," exclaimed Damon, "I was thinking that you would want to give the students a model to use to help them get the answer and avoid the confusion. But it seems like having them come up with their own models and allowing for some confusion revealed what the students do and don't understand. These two students got the same answer, but for very different reasons."

"It seems like thirds and sixths are hard to work with," Damon noticed. "They are much more obscure than halves and fourths."

"That's a good point. Why did you decide to use thirds and sixths in this problem?" Lisa asked Mr. Konook.

The instructor pushes the students to think about the next step in developing children's understanding of fractions. The entire activity focuses an analysis as an ongoing part of teaching. Dispositions to be a reflective teacher are enhanced.

"You're right, Damon, about thirds and sixths being more foreign to young children. Up until this point we've mostly worked with quarters and halves. The students have a lot of informal knowledge about halves and a little bit about fourths. We used their knowledge of how to write ½ to determine the meaning of the numerals within a fraction. They were able to connect this knowledge to their knowledge of quarters pretty readily because they all knew that there are four quarters in a dollar. They are not as familiar with thirds and sixths, so I was interested in how they would use their understanding of halves and fourths in working with thirds and sixths."

Dr. Palmer encouraged her students to consider with Mr. Konook what the students understand about thirds and sixths. She followed this by asking them how they would further develop these ideas in the next day's lesson.

The instructor focuses on ways to help the teachers examine and revise their assumptions about mathematics, how it is taught, and how students learn it..

The instructor models good teaching of mathematics as she engages the prospective teachers in learning mathematics.

Dr. Foote wants to help these students learn some mathematical content and, at the same time, reconsider their assumptions about what it means to learn and teach mathematics, particularly as it deals with the crucial pedagogical skills of analysis, questioning, and creating tasks.

This fall, Dr. Foote decides to try a different approach. She designs a three-phase learning cycle. First, to give her students a new experience with being a learner of mathematics, she has them grapple with the topic of permutations, since this is a topic that many students are unlikely to have studied, or if they have, they are unlikely to understand its conceptual basis.

Exploring the problem of how many different license plates can be made in their state (which has license plates with three letters and three numbers), Dr. Foote concentrates on helping her students learn mathematics in a way that is quite different from their earlier experiences. She does not show them what to do. Rather, she asks probing questions about their thinking and about the conclusions they have drawn. She deliberately seeks neither to approve nor to disapprove of their answers, expecting them to prove their answers through argument and justification.

This is an opportunity for the prospective teachers to examine and revise their notions about how children think and what they know. Teaching someone else addresses the prospective teachers' assumptions about how mathematics should be taught.

Even though the prospective teachers have experienced and observed the use of new teaching strategies, it will take time before they can assimilate such changes into their own personal views of what constitutes good teaching.

The student recognizes that her own knowledge is incomplete. At the same time, she assumes that the role of the teacher is to answer questions-it is hard for her to consider that questions may be used to guide the investigation.

The prospective teachers' focus during this time is on the ways Dr. Foote interacts with the children, the tasks she selects, and the ways the children respond. At the end of this session, she encourages the prospective teachers to ask the children any questions they may have. Following this, in the next class session, she and the prospective teachers discuss the teaching and learning that occurred.

As the final phase, the students become teachers-they are expected to try to help someone else explore the concept of permutations. "Someone else" may be children, roommates, parents, and so on. They make preliminary preparations during class, meeting together in small groups.

The discussion that follows this phase is enlightening. As Dr. Foote suspects, the students begin to realize that there is more here than meets the eye. In particular, the students struggle with wanting to tell their learners instead of helping them build their own understandings. As one student notes:

"I tried to teach my roommate something about permutations. It just didn't work. I thought I understood permutations enough myself, but hen she began asking me questions, I was lost. I couldn't answer some of them, which made me think that I really didn't understand as well as I thought I did."

The project director has selected a problem that is not typical of those that appear in most textbooks. She expects it to raise many questions among the teachers.

5. 3 Ms. Costa, the director of a local state-supported teachers' project, wants to encourage participating teachers to examine their assumptions about the nature of mathematics and how it should be aught. In particular, she wants them to appreciate the difference between understanding how to solve a problem and merely being able to apply an algorithm. At a meeting she poses this problem:

Bob: There isn't enough information.

Tony: Are the people in the community only married to others in the community?

Connie: We don't know how many people there are. Are there the same number of men and women?

Ms. Costa: You raise some good issues and questions. What are some conditions that aren't explicitly stated but that we would assume in order to make sense of the problem?

Tom: Men and women in the community who are married are only married to others in the community.

Connie: There are the same number of men as women.

Susan: I don't think that's true. The number of married men should be the same as the number of married women but the totals could be different.

In order to solve the problem the teachers make explicit their assumptions and determine which are necessary to its solution.

The instructor allows participants time to grapple with the problem and discuss their ideas with each other.

The members of each group select and use a strategy that makes sense to them and eventually analyze a range of strategies for approaching the problem.

The instructor invites the teachers to reflect on what they know about the traditional mathematics curriculum and the ways of presenting new kinds of problems.

The teachers continue to list several assumptions, selecting those with which they agree and recognizing others as misleading.

The teachers work on the problem in small groups while Ms. Costa walks around the room. She notices that some people are still clarifying issues raised in the initial discussion while others have agreed on conditions and are trying to determine what part of the community is married.

A variety of approaches emerge from the small-group discussions. The participants later share these with the class. They are encouraged to generalize: How are the solution strategies alike and how are they different? Do they all produce the same result? What if the fractions in the original problem are changed?

Ms. Costa: Now suppose this sort of problem is included as part of the school mathematics curriculum. I'd like to demonstrate a possible way to present the problem and solution to students.

Ms. Costa then walks the teachers through a rule-based approach to solving the problem. She develops a three-step procedure and explains exactly what calculations should be done at each step.

Step 1. Find the L.C.N. (least common numerator). [The L.C.N. for 2 and 3 is 6.]

Step 2. Change fractions to equivalent fractions with same L.C.N. [2/3 = 6/9 and 3/5 = 6/1 0.)

The final step is to use the "Add Add" method of combining fractions. In this procedure we combine two fractions by adding the numerators to get the new numerator and the denominators to get the new denominator.

Step 3. Add Add the two fractions. [6/9 + 6/10 = 12/19.]

Ms. Costa: These three simple steps will allow you to solve all "marriage type" problems.

Ms. Costa visually surveys the participants to observe their reactions Some participants seem surprised, others confused, and still others are nodding in agreement with the answer and procedure.

The instructor expects reactions. Many of the teachers may be experiencing some dissonance between how they solved the problem and how students are likely to be taught to solve such problems.

Several "good" arguments are provided to justify the "tell, show, and do" model of problem solving. The instructor asks the teachers to discuss the differences they observed.

The instructor helps a teacher recall that although the steps may make sense now, they probably were not the focus during the group's initial efforts.

The goal is for the teachers to realize that "easier" is not necessarily better when it comes to truly understanding mathematics.

Ms. Costa: What do you think? It didn't take very long and could be accomplished quite easily in the school curriculum. Of course you would have to spend time memorizing the three steps and practicing them. How does the method compare to the methods you used earlier to solve the problem?

Joe: It's quick, but it doesn't explain what you're doing.

Marissa: It was really the same as my group did before, only more formal. We found the same numerator. Then the total number of each is the denominators. But we just counted everything to get the totals.

Ms. Costa: Your group tried to make sense of the three steps. Do you think you could have done that without having had time to work on the problem on your own and comparing the various methods we came up with before?

Tony: I don't think so but it sure seems easier for the kids to follow the three steps.

Ms. Costa: Well, I could have started today's lesson by presenting the problem and giving you the three steps to solve it. We could have spent time practicing the steps and doing similar problems. You could have been quite successful in solving problems of that type without understanding much about why you were doing the things you did. In the short run you would appear to be successful, but in the long run where would you be?

Cross-Referencing Standard 5 with Vignettes in Section II

The vignettes presented in the second section, "Standards for the Evaluation of the Teaching of Mathematics," reflect teaching episodes documenting evaluation as professional development and assessing the teaching of mathematics. As might be expected, these vignettes demonstrate teachers' applications of the components detailed in this standard as well. The chart below provides a cross reference from the subtopics of this standard to the vignettes presented in the second section. For example, vignette 1.2 provides an example of a teacher examining and revising his assumptions about how mathematics should be taught.

As part of the preservice and continuing education of teachers of mathematics, the vignettes from the second section may be helpful as part of teachers' efforts to think about their own teaching in light of the subtopics of this standard.