The challenges of instructional leadership school renewal.

Two major publications addressed the issue of reforming American mathematics education in 1989. These reports proposed wide-ranging, radical changes in mathematics learning and teaching. Everybody Counts: A Report to the Nation on the Future of Mathematics Education written by the National Research Council in 1989, concluded that ineffective mathematics education posed a potential threat to America's economic security in a technological world. Curriculum and Evaluation Standards for School Mathematics, written by the National Council of Teachers of Mathematics (1989), echoed similar views and drafted recommendations intended: (I) to expand school mathematics away from "shopkeeper's arithmetic"; (2) to include mathematics meaning congruent with the needs of the 21st century; and (3) to establish instructional beliefs and practices based on the epistemological foundations of constructivism.

The need and direction for instructional reform in mathematics was clearly documented in 1989. However, there was not a large body of research about how educational reform in mathematics was experienced in practice. A few studies indicated that new classroom cultures and learning/teaching theories can be implemented successfully (Cobb & Bowers, 1999; Cobb & Yackel, 1996; Hiebert, Carpenter, Fennema, Fuson, Human, Murry, Olivier & Wearne, 1996). Despite some research on the subject of implementing NCTM Standards, there was still a need to learn more about how these instructional reforms are led by educators and received by school communities.

The intention of the research reported in this paper was to share the processes a school community and school leaders (principal, assistant principal, teacher-leaders) in a K-4 elementary school created to reform mathematics instruction. In particular, the case study reports the complex and dynamic nature of a shift from behaviorist teaching methods toward constructivist methods. The case study exposed the changing nature of classroom environments, where teachers created learning opportunities for students to make sense of doing significant mathematics through social interaction, dialogue, and mathematical modeling. The research revealed transformation in leadership strategies and changes in teachers' roles and relationships.

Methods

The study was guided by qualitative design and constructivist inquiry from Guba and Lincoln (1994), Lincoln and Guba (1985). The methodology attempted to investigate the dynamics of school change within the assumption "that the issue at hand turns in some way on the ways in which individuals conceive of or construe their world." (McCracken, 1988. p. 59) Overall, the study tells the story of educators' efforts to lead instructional reform in mathematics education according to NCTM Standards (1989, 1991, 1995) and constructivist learning theory (Cobb & Bowers, 1999; von Glasersfeld, 1995) in an elementary K-4 school.

Setting and Participants

The K-4 elementary school is located in an affluent and racially/culturally diverse suburb bordering a large Midwestern U.S. city. The school enrolls 525 students (65% African-American, 30% white and 5% other racial or multiracial groups). The school has 25 classroom teachers, five special area teachers (art, music, physical education, and library), three learning disability teachers, six tutors for language arts, two kindergarten literacy tutors, an assistant principal and a principal. The average class size is 22 students.

Data collection included interviews, observations, surveys, and documents from educators, parents, students and support staff. Data collection and data analysis were guided by Lincoln and Guba's (1985) constant comparative method. Patterns and themes that emerged can be summarized as: 1) Teachers as Leaders and Change Agents, 2) Role of Principals, and 3) Role of Parent and Students. The following sections describe the main features of these themes.

Teacher Leaders and Change Agents

Reforming mathematics teaching and learning at this K-4 school represented a major shift from a behaviorist perspective on teaching and learning mathematics to a constructivist perspective. This shift was nonlinear and uncertain. Behaviorist methods of instruction and assessment focus heavily on transmission of knowledge from teacher's head to student's head through direct instruction. Constructivism values the autonomous construction of meaning where the learner is an active builder of understanding. Building understanding is dependent on context, experience and interaction within the environment. Constructivist methods on instruction value problem-solving, reasoning, communication and mathematical representation (NCTM, 1989, 1991, 1995 and 2000). Constructivist classrooms differ from conventional classrooms in several ways. Such classrooms recognize students and teacher as a community of learners. Student-student and teacher-student interactions and dialogue are valued as ways of knowing and negotiating me aning (Bauersfeld, 1988; Cobb, Wood & Yackel, 1990). Third, these classrooms emphasize putting mathematics within a context relevant to the learner, so students can make sense of numbers, computations, and formulae (Wheatley & Reynolds, 1999).

Teachers who emerged as leaders were initially involved in learning more about key mathematical concepts. They acknowledged their need to expand understanding about topics like probability, algebra, geometry, metric measurement, and problem-solving strategies. Furthermore, they reconceptualized their ideas about how students learn mathematics. Guiding students toward constructing mathematical understanding required different pedagogy and different classroom environments.

All this relearning and redesigning took much time and effort. Personal commitments were made and risks were taken. How the new methods and new classroom environments were viewed by parents was uncertain. Teachers knew their principal and assistant principal supported, encouraged, and respected their actions toward change. Much time and money was spent on professional development meetings and professional sharing with middle, high school, and college mathematics educators. They recognized the message from the NCTM Standards (1989, 1991, 1995). However, would the new teaching and learning environment get results? The biggest concern centered on computation skills and memorization of basic facts. Teachers wanted to believe that students would learn to compute in the context of good, relevant mathematical problems, but this was unproven and risky. Teachers had difficulty blending the need for fluency in basic facts with problem-solving. Thus, they realized that they had to teach both computation and problem-sol ving in order for students to understand and apply basic operations and concepts to problem-solving situations. Doing both required far more time. The traditional 45 minutes for daily mathematics instruction was not enough time. Teachers concurred with the principals that time for mathematics lessons had to be doubled to 90 minutes each day.

The following vignettes provide a glimpse of some changes for students and teachers in a constructivist mathematics classroom. Each vignette demonstrates some changes from traditional classroom practices. These changes in classroom interactions occurred in about 75% of the classrooms. Such changes were embraced and refined through action-research by about 40% of the school classroom teachers.

Fourth grade students are sitting in a circle on the floor. One student immediately distances himself from the circle and leans against the radiator and listens as the teacher poses the following problem:

Teacher: You are a contestant in the "Big Spinner Game." To win the game, the tip of the needle must land on the letter "B." Which of these two spinners would you select to win this game? [In the middle of the circle is a large 18" x 24" tablet with newsprint. She draws two large circles. One circle is divided into thirds. Each section is labeled "A" "B" "C". The second circle is divided into fourths. Two of the sections are labeled "B" and the remaining sections are each labeled, "A" and "C."

T: John, are you feeling all right? [John is sitting outside the circle.]

John: Yeah.

T: Why don't you join us? Or are you more comfortable over there?

John: I'm more comfortable over here. I'm just kind of tired.

T: O.K. but don't get burned by the radiator.

John: I won't.

T: Now here's the situation. You are at a carnival. One game at the carnival is a game where you can win a television-VCR if the needle of the spinner lands on the letter "B". Landing on the letter "B" wins the TV-VCR. The rules of the game are simple. Before you play the game, you have to choose which spinner to use. You get a choice of spinners. Everybody understand so far?

Students: Yes.

T: What do you think my question is about this problem?

Sally: Which spinner would we choose to make sure we would win on just one spin.

T: That's right. Let me write your question on the paper. [Teacher writes the question.]

T: OK. Now just to make sure we all understand the problem. Who can retell the problem? How about you, Ted? Now, everybody listen to make sure this is how you see the situation. Make sure important information isn't left out.

Ted: Well, we're at the carnival and we want this TV set...

John: and VCR combination! [students giggle]

Ted: Yeah, right. At this game we have to select a spinner to use. Landing on the letter "B" wins the game.

John: You have to do it the first time-no second chances at spinning.

Ted: Yes, only one spin.

T: Is that how everybody understands the problem?

Students: Yes.

T: OK, which spinner is best to use for this game?

Students: [silent and thinking]

T: Yes! I hear thoughts, smell brain cells burning. What's the problem here?

Alice: It isn't easy. It's tricky.

Bonnie: Yeah, you repeated the "B" in the circle with the fourths.

John: [now, a little closer to the circle] The parts aren't the same size either.

T: What were you expecting, a "no-brainer" activity from me

Students: NO!

John: [now laying on the floor within the circle] I would take this spinner [pointing to the spinner divided into fourths].

Bonnie: No, you shouldn't take that one, the sections are smaller than the ones on this spinner [pointing to the spinner divided into thirds].

Alice: But there are more B's on this spinner.

John: Yeah, but B's means more chance to win.

Bonnie: But you are counting "B's", you aren't looking at the part. 1/3 is bigger than 1/4.

John: But there are more B's here. See, 1/4 and 1/4 is two fourths.

Ted: No, that's not 2/4, it's one half.

Students: [pause, a little discussion among themselves]

T: Um, 2/4, 1/2, which is it?

John: It's 2/4 because the spinner is divided into four equal parts. Two of those equal parts have "B's" on them. 2/4 "B". [John, now sitting up, folds arms across chest.]

Bonnie: [quietly] I think it is the same, John. You can say 2/4 or 1/2 "B". It's the same, look. Here's one half of the circle [tracing one half of the circle with her finger]. Now here's two fourths [tracing sections of circle again]. 2/4 and 2/4 equal one whole circle.

Aaron: 3/4 needs 1/4 to be a whole. And 2/3 needs 1/3 to be a whole. Since they both need just one piece to be a whole, they must be equal and either spinner will win the game. [Students are quiet and puzzled.]

John: I don't know about that.

T: I don't know either. You folks have me most confused. Lam going to have to think about this. I never thought about it this way. I think we will have to continue this discussion sometime tomorrow. [Her way of ending lessons when she is unsure about how the students are thinking and how she needs to continue the instruction.]

The next vignette describes a conversation between fourth grade students and two teachers about "time." In this situation, the teacher-leader (TL) was modeling a lesson for her mentee. The intent of the lesson was to investigate what students already knew about "time" and to use their understanding to create lessons. The students were required to dialogue with each other and to explain/defend/justify their ideas. The unanticipated result was a perturbation of teachers' understandings about students' prior experiences and present thoughts.

Setting: Fifteen potentially underachieving fourth grade mathematics students are sitting in a circle, on the floor, with two teachers. One teacher-leader (TL) is the mentor of the other teacher (T). The teachers are preparing students for the state mathematics test. Each student has a small, yellow plastic clock.

TL: So, what do you know about time?

S I: There is twenty-four hours in a day. Seven days in a week. 30 or 31 days in a month. Four weeks in a month. Twelve months in a year.

TL: Anything else that needs to be added? Anybody know other things about time?

S II: He left out ten years in a decade.

S I: Oh, year. And one hundred years in a century. Thousand years in a .....I forget.

TL: Anyone know? [Students, thinking, are silent.]

TL: Well, I'm not sure either. So, let's go on. What about this thirty or thirty one days in a month? Why those numbers?

S II: I think it has something to do with the earth or sun.

S III: Yeah, it does. Um, let me see if I remember. It's how long it takes the earth to spin around.

S IV: No, that's the time for a day-one complete spin-12 hours a day, facing the sun, 12 hours of night, facing away from the sun equals one complete day. But I can't remember about months. Maybe it has something to do with orbit about the sun.

S V: Yeah, it has to do with orbit. Like where in the orbit the earth is.

S VI: Yeah, whether it is close or far from the sun.

S V: It never gets close to the sun or it would burn up!

S VIII: But it has to get close sometimes because we have summer and winter. [Students are silent.]

TL: Hum, I don't know about this orbit. Let's go back to one complete spin. Does everyone agree that one day is measured when earth spins completely around one time?

Students: Yes.

TL: And that one full day is measured in hours?

Students: Yes.

TL: And that one spin takes 24 hours?

S VIII: Not always.

TL: Oh, tell me when it doesn't take 24 hours?

S VIII: Well, sometimes it goes faster. Sometimes it goes slower. But most of the time it takes 24 hours. [Teacher leans forward and responds to student's comment.]

T: That's not right! Where did you get that idea?

TL: Let him finish. I am interested. There is fast time and slow time?

S VIII: Yes.

TL: I never thought about time this way. What does everyone else think?

S I: Time does go fast and slow. Like waiting for Christmas or my birthday, times goes slow. Vacations go fast.

S IV: Yeah, I agree with you. Sometimes school goes fast and slow, especially lunch! [Students laugh and smile to show agreement.]

TL: So, let me understand. You're saying that time is measured in hours but sometimes those hours go fast or slow. So what about the earth's spin? Does the earth spin faster one day than another? [Students think about the question.]

S I: I think it spins slower in spring and summer than in winter and fall because the days are longer in spring and summer.

S III: Yeah, that's right. My bedtime is 8:30. In the summer I hate going to bed when it is still light outside. In the winter; it is dark. That's because of fast and slow time.

TL: Well, if hours go fast and slow, do months also go fast and slow?

S IX: Yes, look at the months, some are thirty days, some are thirty-one and February is 28 days.

S X: Sometimes it is 29 days.

S IX: Yeah, but only once every four years.

TL: This is most interesting. Fast and slow hours, days and now months. I guess I always thought that an hour was always the same Length of time. It didn't speed up or slow down. Now you are telling me that February is "fast" for most of the time but every fourth year, February slows down and takes one extra day.

S XI: That's right because that's "Leap Year." One extra day is added to the calendar. Instead of 365 days, there's 366 days that year.

TL: What makes this happen?

S XII: Probably the earth just stands still in its orbit.

TL: Who then "jump starts" earth moving again? [Students are quiet.]

Later that day, the teacher-leader and her mentee met after school to debrief the lesson and to plan instruction.

Mentee: Why did you let them continue with that nonsense about "fast and slow" time? Where did they get that notion? It was so painful to listen to. Eventually, you should have told them how wrong they were.

TL: I couldn't tell them they were wrong because I didn't have a way to change their thinking that would make anymore sense to them. I didn't find the situation "painful." I found it intriguing how they built such elaborate misunderstandings to make sense of time.

Now, we've got to figure out how to dismantle these perceptions and help them construct it differently. Before I do this, I have to revisit science concepts and make sure I understand orbits, rotations, revolutions and seasons. Then, I suggest we create experiences to let them see that time is measured in a regular constant rhythm. Erasing misunderstandings will probably be more difficult than building new.

T: You've got that right. Just think, we've got to explain "leap year!" [Both teachers laugh.]

Two levels of mentoring took place simultaneously. On one level, there was ongoing communication between teachers and principals about changes in teaching, learning and assessment suggested from the NCTM Standards (1989, 1991, 1995). On another level, there were collegial relationships and sharing between teacher-leaders (selected by principals because of their interest and commitment to reform) and new teachers. Sometimes meetings were scheduled with principals, teacher-leaders and new teachers to talk about learning theory and teaching/learning philosophy. Other meetings were scheduled for teacher-leaders and principals to share new methods with the entire teaching staff, middle school, secondary and university mathematics educators. New teachers were always assigned a teacher-leader as a mentor for their first year. These multi-layers of connectivity and interactions significantly facilitated teacher professional development.

Also, two other factors contributed to a gradual change of teachers' beliefs and classroom practices.

The way we are teaching now is a lot harder. Many people aren't ready. The knowledge of math that is required now is more difficult. There is an element of belief in what we are doing. You really have to believe in it. People have to experience in order to change their whole schema. (teacher-leader)

The first factor affecting teachers' beliefs and practices was actively listening to students' explanations of mathematical solutions. Teachers were surprised by the diversity of strategies and procedures students invented to solve problems. "Student voice" often revealed prior experiences students used to make mathematical meaning. "Student voice" was a powerful catalyst for influencing teachers' instruction. Teachers frequently designed mathematical problems from individual student experiences. Often students and their families, friends, classmates, teachers, and principals played a role in the mathematics problems created by the teachers and students. Perhaps one of the most important aspects of "student voice" was teachers' realization that what they taught was not always what students learned or understood. Teachers realized the importance of knowing what the student understood or did not understand in organizing and planning instruction.

You know, I was thinking on my way to work this morning, how differently I teach now. You wouldn't believe how I used to teach. Before, I used to 'sugar coat' the curriculum, make a big show, entertain the kids to keep them from being bored. I never thought about the substance of my lessons or if they helped students learn. I just assumed they did. But now I listen to the kids before designing lessons. I am always thinking about how I can improve so the student will really learn. I feel so much more fulfilled than before. I could never teach that way again. (teacher-leader)

The second factor that led to changes in mathematics instruction was the improvement of teachers' content knowledge through interactions with their students and their colleagues at the secondary and college level. Teachers engaged in critical self-reflections and self-evaluations about their teaching. They recognized the dialectical relationship between teaching and learning as ongoing, recursive, and in constant flux. They valued risk-taking arid the "virtue of not knowing" (Duckworth, 1987).

I think people tend to think, you just come up with all these ideas [for lessons]. People don't realize that the idea just doesn't pop into your head. You are thinking about them [mathematical ideas and processes] all the time. (teacher-leader)

Role of Principals

As we mentioned earlier, this research study started in 1997 and ended in 2000. However, the reform really started in 1989 and continues to evolve as of today. There was a passion for reforming mathematics instruction from both the principal and assistant principal. Both building leaders were committed to putting the NCTM Standards (1989, 1991, 1995) in practice at this K-4 school. They sought and received about $300,000 in grant monies from 1990 - 2000 to support the reform efforts. These monies were used for professional meetings, consultations with secondary and university mathematics educators, instructional books/software, and concrete materials for classrooms.

Early in the reform process, the two principals realized the need to I earn and relearn mathematical content so that they would have a foundation for the curricular, instructional, and assessment changes. They opened communications and scheduled meetings with a middle school and a high school mathematics teacher, a professor of mathematics education, and a middle school science teacher.

The secondary teachers and university professor consulted with the principals and teachers at this school. They visited classrooms each month. They taught with elementary teachers. They shared ideas and led professional development seminars with elementary teacher-leaders and principals. They conducted collaborative dialogues with the school educators to build better understanding of mathematical topics such as probability, metric measurement, algebra, and geometry.

The traditional connections among university, secondary, and elementary educators, and between educators and elementary parents, quickly assumed a different pattern. Traditionally, elementary educators worked with secondary and university educators in a linear, hierarchical, and top-down pattern about how to teach mathematics to young children. Seldom had teachers and parents experienced collaboration among educators from such a range of instructional levels. However, the principals believed that they and the teachers need to "step up" as adult learners in order to acquire the extra knowledge needed to reform elementary mathematics. Reforming mathematics education needed to be a K-College process. Elementary, secondary and college teachers needed to work together.

This connection of secondary, university, and elementary educators was accepted and appreciated by some teachers, but not by all. It signaled the first possibility of tension and division among the school faculty. While some teachers welcomed new knowledge and new professional relationships as a sign of growth and respect, other teachers felt threatened and resentful of secondary teachers meddling or "forcing" ideas on elementary trained "experts."

Fortunately, when the interactions among secondary and elementary educators occurred willingly, hierarchical ideas did not dominate the experiences. Secondary and college teachers were fascinated with the creative ideas of elementary teachers. They came to respect and admire the elementary teachers' abilities to create innovative ways of teaching mathematics concepts. So, in most instances, teachers involved in these collaborations were open to change and saw it as a "two-way street." That is, elementary educators learned more mathematics content from secondary teachers; secondary teachers learned how to open the instructional process to more active and interactive experiences from the elementary teachers. The school principals learned much mathematics content and they celebrated the successful secondary-university-elementary partnership. Secondary mathematics teachers also benefited from the interactions and professional relationships with the elementary teachers and principals. "When I leave this building, I feel better than when I came in. I can take ideas from classroom observations and teacher interactions with me into my own classroom" (teacher).

From 1995 through 1999, two secondary mathematics teachers visited the K-4 building about two or three days each month. Secondary mathematics teachers were also involved in professional workshops three or four times each school year from 1995 through 1999. But some high school and middle school teachers viewed their colleagues' connections to elementary educators as strange. "Why are you spending all that time at that elementary school? What would be so interesting at an elementary school?" (secondary teacher, not involved in the collaboration)

Role of Parents and Students

The steadiest and firmest support for mathematics instructional reform came from parents. From the beginning parents were informed by the school principals of the need to change instruction and the kinds of reform required. Parents seemed open to change and they reacted positively when they started to see how enthusiastically and intelligently their children were doing relevant mathematics problems.

There has been a big improvement in math this year. She really had a problem in math last year. But now, I see her level of interest is very high in math and she is growing in problem-solving. Before, she would say that she "hates math" but now she likes it. She is more confident in her problem-solving and reasoning. (fourth grade parent)

Some parents expressed a willingness to teach calculation skills at home so the teachers could concentrate on the application of mathematics skills and authentic problem-solving. Most parents had never been taught mathematics in this manner; they were rarely taught such concepts at such an early age. They were eager to see their K-4 children doing "complex" mathematics. Parents were surprised that their children loved problem-solving, talked about mathematics problems at home, and often demonstrated their solutions for them.

The problem is we do not know how to translate 7/16 into cents. The price per share means how much we paid and how much the stock went up. We do this every Wednesday...(fourth grade student)

Most parents were also open to the use of calculators in K-4 classrooms. They seemed to understand that technology was changing the way people did mathematics in the "real world," and they wanted their children to know how to use calculators.

She is going to teach long division...She lets us use calculators because my past teachers never let me use calculators and the problems were really hard. My parents are really glad she lets us use calculators because they say it is required for high school and college math. (fourth grade student)

Overall, the parents trusted and liked the principals. They were willing to take risks with these educational leaders toward changing an instructional area that they seemed to know was ineffective. Some parents joined a parent mathematics support group. They met every two months with the principal to discuss instructional changes and worked with the assistant principal to provide mathematics manipulatives and family mathematics activities to all parents.

The most controversial and sensitive role for parents and students in this reform process emerged around the state's mandated fourth grade mathematics test. The state mathematics test began about half-way through the reform process in 1996. Results from the state test were used to compare schools within the local district and throughout the state. From 1996 to 1998 (the first three years of the state testing) this school's results were about the same as those of the four other district elementary schools. About 67% of the school's fourth grade students passed the state mathematics test for the first three years. These results were about 12 percentage points higher than the state passage rate (about 55%). However, this school was not ahead of the other four elementary schools in the local school system, and, in the third year of the state testing the school actually was ranked last among the local district elementary schools. These results were surprising to parents and community members because the school had spent so much time and money to reform and improve mathematics instruction. Even though the principals and teachers suggested to parents that the state test was an inappropriate instrument for measuring students' knowledge, many parents became skeptical about instructional reform. Doubts were most noticeable among district administrators and school board members. Constructivist theory was questioned and attacked at local board of education meetings. Same district administrators told the principals at this school that they should avoid the "C word" (constructivism). In the midst of these controversies, the principals and teacher-leaders became even more cognizant that instructional practices guided by constructivist theory required more instructional time. If the goal was to teach all children to understand and do mathematics, then the time structure for mathematics instruction had to be changed (Pourdavood, Cowen, Svec, Skitzki & Grob, 1999).

To meet the challenge of state testing, principals and teacher-leaders created many extra hours of instruction for about 40 third and fourth grade students who seemed to be at risk of failing the fourth grade state mathematics test. These students were nicknamed "Scholars." They attended Saturday morning school, early morning tutoring (three days each week), and after school tutoring (three days each week). After school instruction focused on basic computation skills. Early morning and Saturday school instruction focused on teaching mathematics concepts in a problem-solving context according to constructivist learning theory.

In 1999, after one year of this expanded instruction, passage rate on the fourth grade state mathematics test jumped from 67% to 90%. The same 90% passage rate was repeated in 2000 school year. Many educators and community leaders were especially impressed with the 80% to 90% passage rate of African-American students for two consecutive years. Parental confidence and student pride were renewed.

Putting NCTM Standards and constructivist learning theory into practice required reculturing the teaching, learning, and assessment environment and expanding instructional time for all students, especially students who learn more slowly and students whose socioeconomic conditions or family situations lacked necessary resources for school success.

Discussion

New roles and relationships grew out of the collaborative decision-making between teacher-leaders and principals. Some teachers began to realize that teacher leadership was necessary for instructional reform. By putting research into practice and developing instructional models, teachers began guiding and mentoring other teachers. Their efforts also built trust and respect with principals and helped principals understand what changes were working and what strategies needed revision.

Conflict was both beneficial and disruptive. It moved the reform forward by helping to create a critical mass of teachers who supported instructional change. Conflict created a situation where there was no "middle ground." Teachers either implemented the reform or they did not. Conflict also did damage. Support or non-support of the reform was viewed by some teachers as support or non-support for the principals. Disagreements about changes in rules, roles, and relationships caused a few teachers to resist instructional change and make attacks on professional abilities of principals and teacher-leaders.

Conflict and disequilibrium did not stifle reform. Change continued. Pedagogy dominated "teacher-talk". Teacher-leaders shared ideas about their instruction. They designed different learning environments to accommodate new practices. Teachers asked different types of questions, valued student experiences and supported student risk-taking. They designed lessons, listened to students' understandings and "rebuilt" their own understanding of mathematics. Some teachers began to see the inadequacies of their prior assumptions about learning. They realized what they taught was often not what students learned. The reform impacted some individuals personally. "It is hard to change people's minds. You couldn't change mine. It just so happened in my own pursuit, for my own reason, I came upon it and changed my mind" (teacher-leader). Success depended on teachers' abilities to learn and take risks. Teachers' change occurred from within. However, their changing beliefs and practices were significantly aided by their inte ractions with students, reflections on classroom practices, and dialogue with peers and principals.

It is hard to change. I was in tears a lot. I would go home not knowing if what I was doing was right. I was afraid that at the end of the year the kids would not understand. It is scary to take risks. You have to say, 'Look, it is working, there are so many good things.' (teacher-leader).

Interactive mathematics instruction evolved into the development of a dialogic community. Professional sharing eliminated teacher isolation. Collegial relationships and professional discussions encouraged many teachers to take risks and learn more about mathematics (Pourdavood & Fleener, 1998). Teachers and principals became learners, decision-makers, and instructional leaders. Trust, professional respect, and pride in students' abilities to do contextual mathematics emerged from collaborations and common interests among some members of a transforming school community.

From all these efforts, presently it is estimated that about 70% of the classroom teachers are beyond traditional behaviorist teaching, learning, and assessment practices. Three teachers, including an assistant principal, earned their doctorates in various fields of education during the ten-year reform process. About 40% of the teaching staff seems transformed and proud of their ongoing professional growth in mathematics instruction. Perhaps they will serve as models for mathematics reform at this school and other schools throughout the community and nation.

REFERENCES

Bauersfeld, H. (1988). Interaction, construction and knowledge: Alternative perspectives for mathematics education. In T. Cooney & D. Grouws (Eds.), Effective mathematics teaching (pp. 27-66).

Cobb, P., Wood, T., & Yackel, E. (1990). Classroom as learning environments for teachers and researchers. Journal for Research in Mathematics Education, 4, 25-146.

Cobb, P., & Yackel, E.(1996). Constructivist, emergent, and sociocultural perspectives in the context of developmental research. Educational Psychologist, 31, 175-190.

Duckworth, E. (1987). The having of wonderful ideas and other essays on teaching and learning. New York: Teacher College Press.

Guba, E.G., & Lincoln, YS. (1994). Comparing paradigm in qualitative research. In N.K. Denzin & YS. Lincoln (Eds.), Handbook of qualitative research. Thousand Oaks, CA: Sage Publications.

Hiebert, J., Carpenter, T.P., Fennema, E., Fuson, K., Human, P., Murry, H., Olivier, A., & Wearne, D. (May, 1996). Problem-solving as a basic for reform in curriculum and instruction: The case of mathematics. Educational Researcher, p. 12-21.

Lincoln, Y.S., & Guba, E.G. (1985). Naturalistic inquiry. Beverly Hills, CA: Sage Publications.

McCracken, G. (1988). The long interview. Newbury Park, CA: Sage Publications.

National Council of Teachers of Mathematics. (1989). Curriculum and evaluations standards for school mathematics K-12. Reston, VA: NCTM.

National Council of Teachers of Mathematics. (1991). Professional standards for teaching mathematics. Reston, VA: NCTM.

National Council of Teachers of Mathematics. (1995). Assessment standards for school mathematics. Reston, VA: NCTM.

National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: NCTM.

National Research Council. (1989). Everybody counts: A report to the nation on the future of mathematics education. Washington, D.C.: National Academy Press.

Pourdavood, R.G., Cowen, L.M., Svec, L.V., Skitzki, R., & Grob, S. (1999). A paradoxical path to reform. Ohio Department of Education, Columbus, Ohio.

Pourdavood, R.G., & Fleener, Mi. (1998). The ecology of a dialogic community as a socially constructed process. Teaching Education, 9(2), HYPERLINK http://www.teachingeducation.com.vo19-2/pourdavood.htm.

von Glasserfeld, E. (1995). A constructivist approach to teaching. In L. Steffe, & J. Gale (Eds.), Constructivism in Education (pp. 3-15). Hillsdale, New Jersey: Lawrence Eribaum Associates.

Wheatley, G.H., & Reynolds, A.M. (1999). Coming to know numbers. Mathematics Learning, Tallahassee, FL.