# Passion for teaching/learning mathematics: a story of two fourth grade African American students.

Abstract

The intent of this study was to investigate the experiences and reflections of two African American children, their teachers, and their parents about mathematics learning and what these experiences imply for educators as they attempt to reform mathematics education to help all students gain a deeper mathematical disposition. A qualitative design employing in-depth interviews, participant observations, and the examination of relevant documents was used to explore the mathematical environments of two African American children. As the stories of these two children emerge, it is hoped that a better understanding about mathematics teaching and learning, grounded in the experiences of people of color, can be added to scholarship, thereby strengthening the chorus of "other" voices increasingly present in this study.

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Current research studies indicate that our elementary grade students are performing better in mathematics and science proficiency tests of basic skills (Mullis, Dossey, Campbell, Gentile, O'Sullivan, & Latham, 1994; National Science Foundation [NSF], 1996). Also the results of the Third International Mathematics and Science Study (TIMSS) illustrate that the U.S. fourth grade students performed significantly higher in mathematics and science when they were compared to their counterparts from other industrial countries. However, research results show that our students do not possess deep conceptual understanding of these mathematical skills (National Council of Teachers of Mathematics [NCTM], 2000a; 2001a). The results are even more disturbing when they focus on minority students' achievements and their mathematical self-perceptions. The NCTM (2001b) states that "Mathematical power must be considered the right of, and the expectation for, every child" (p. 1).

In spite of the recent improvements regarding minority students' mathematical dispositions and narrowing the achievement gap between white and other minorities (I.e., African American, Hispanic, Native American, etc.), considerable disparity still exists. This incompatibility between white and minority students' attitudes and performances in mathematics carry economic and social justice implications (Bigelow, Harvey, Karp, & Miller, 2001; Campbell, 1996; Frankenstein, 1995; Ladson-Billings, 2001; Moses & Cobb, 2001; NCTM, 2000a, 2000b, 2000c; NCTM, 2001a, 2001b, 2001c; Ogbu, 1987; Secada, 1992). One might conjecture multiple reasons for these differences such as inadequacy of resources, unqualified teachers, disconnected and repetitive curriculum, as well as lack of a relationship between minority students' cultural backgrounds and their classrooms social norms and sociomathematical norms.

Problems of diversity have existed since the inception of this country, in fact they are an integral part of the history and creation of the nation as it stands. Often certain individuals have been perceived and treated differently than others in American society. Some of the questions regarding teaching and learning mathematics include: can the old patterns of race, class, and gender discrimination continue to exist in the face of changing population patterns? Can we (meaning mostly those who make policy decisions--legislators, superintendents, principals, teachers, and even parents) continue to support these old patterns; or can we transform how we (and others) feel and what we all believe for the sake of educating our children and sustaining the nation? "If the mathematics is deemed worthwhile to learn, then students in poor communities should have the same access to that mathematics as those in affluent communities" (NCTM, 2001b, p. 10).

We ought to examine the patterns of teaching and learning that exist for various groups in our schools. These patterns may not be the same for all groups, or the resulting performance levels and students' attitudes and beliefs toward mathematics would not be so different. It seems the perceptions by educators, parents, and students alike are very powerful indicators of what is taught and learned (Moses & Cobb, 2001). Thus, the examination of these perceptions would be an important point of research. This study was an attempt to examine such perceptions in one small microcosm that reflects connectivity and relationship among these groups.

Theoretical and Pedagogical Considerations

The theoretical and pedagogical assumptions of this study are grounded in social constructivist epistemology. Social constructivist theory posits that all knowing is inherently grounded in social and cultural activities (Cobb & Yackel, 1996). The notion of experience and context is crucial to social constructivist epistemology. This perspective assumes that an individual child or adult, who helps to build the shared meaning of the mathematical activity of the group and comes to understand the mathematical thinking of the others, and who is also able to contribute to the mathematical understanding of the group, would be bound, in such a social setting to come to perceive of him/herself as a more capable learner (or even teacher) of mathematics.

Assumptions about the nature of knowing, how it is acquired, and how to facilitate its acquisition greatly influence one's practices as he/she attempts to construct new models of mathematics instruction and assessment. One of the major tenets of social constructivist pedagogy is that methods of instruction should reflect the teacher's respect for the students and their ability to reason and figure things out for themselves. Also, students come to view and rely on the teacher as a guide/aide, rather than as the sole authority in the classroom. According to this perspective, allowing students to make sense of their activities provides them with opportunities to build their mathematical dispositions, and thus encourages students to seek more and greater understanding of mathematics.

According to Brooks & Brooks (1993), to become constructivist educators, we should start by honoring the learning process. They make an important point when they explain that:

For Brooks & Brooks (1993), each new construction of knowledge depends on the cognitive abilities to accommodate discrepant data and perceptions and the "stockpile" of experiences an individual has at any given time. Likewise, Yackel (1993) observes that "from the constructivist perspective, meaning is not inherent in written or spoken language. It is constructed by each individual on the basis of his or her own experience and involves individual interpretation" (p. 34). This viewpoint supports the data that knowing is not an object to be acquired, but an activity during which meaning is constructed. Similarly, Wood, Cobb, & Yackel (1991) shed more light on this topic when they explain that, "it is useful to see mathematics as both cognitive activity constituted by social and cultural processes and a sociocultural phenomenon that is constituted by a community of activity recognizing individuals" (p. 402). The importance of the student/teacher interaction was emphasized by Cobb, Wood, & Yackel (1991) when they discussed research conducted in a second grade classroom. They reported, "the emotional tone of the classroom" (p. 174), and stated that "the children were enthusiastic, persistent, and experienced joy when they solved a personally challenging problem" (p. 174). They concluded that emotional considerations are very important to the successful implementation of constructivist pedagogy. The authors go on to discuss the role of the teacher as a promoter of acceptance and trust in the classroom. They made the observation in the second grade classroom they studied:

Yackel (1995) endeavored to understand how students talk, act, and think to promote their mathematical communication. She also endeavored to investigate teacher interaction with students and how it promotes learning for all in the classroom. She concluded that:

This epistemological and pedagogical awareness is central at the school as a transforming learning community.

Research Question

The intent of this study was to examine one central question: what do the experiences and reflections of two African American children (and their parents and teachers) about mathematics learning imply for educators as they attempt to reform mathematics education to help all students gain a deeper mathematical disposition?

Research Design and Methodology

The observational study was grounded in constructivist inquiry (Guba & Lincoln, 1989, 1994; Lincoln & Guba, 1985). The investigation at hand was about children, specifically African American children; and how they saw themselves according to the multiple realities that existed within the context of mathematics teaching and learning in a K-4 elementary school. The data collected during the investigation, "[was] not easily handled by statistical procedures" (Bogdan & Biklen, 1982, p. 2), because it was rich in the description of people, places, events, and conversations. One of the aims of this investigation was to understand the actions and perceptions of participants from their own frames of reference; according to their own experiences; and not according to a perceived fixed truth or reality that exists outside of and independent of human activity (Anderson, Herr, & Hihlen, 1994; Banks, 1993).

Data sources include: (1) in-depth interviews with two students (10 for each), their parents (four for each), the school administrators (two for each), and their teachers (four for each), (2) participants observations (10 for each classroom), (3) field notes, and (4) examinations of relevant documents (I.e., student's journals and portfolios, school news letters, test results, lesson plans, teacher notes, etc.).

Interview tapes were transcribed. All transcriptions, field notes, and other related documents were subjected to regular and ongoing analysis and interpretation. The participants (I.e., the school principals, teachers, parents, and to some extent, students) were allowed to read and help revise the interpretations. Although the data analysis was ongoing, synthesis across the data sources occurred when the data collection was completed. The data is presented and interpreted as follows: voices of the participants (I.e., the principals, the teachers, the students, and the parents), physical structure of the classroom and the social norms, mathematics instruction, dialogue with students, and the development of students' mathematical dispositions. Then the discussion follows.

Story-Construction: Voices of the Participants

The study was conducted in a suburban school of a Mid-Western U.S. City (for more information about the school and its community see the introduction and the first paper in this issue). The K-4 elementary school enrolls 550 students. Sixty percent of the students are African American, 34% are white, and 6% are others (I.e., Asian, Middle Eastern, Hispanic, etc.).

The key participants in the study were Dr. Adams (principal), Dr. Kelly (former assistant principal who is now principal in another school in the same district), two fourth grade teachers (Ms. Patterson and Dr. Rahman), two African American students (Ciara and Malik), and the students' parents (Mrs. Thomas and Mrs. Williams). Pseudo names were used to secure anonymity of the participants.

The Principals

Dr. Adams began his career in the district as an English teacher in the middle school. After he earned his M.S. and Ed.D. in school administration, he became an assistant principal at the middle school and then came to the school as principal in 1988. He discussed his experience and explained how he came to be involved in the mathematics reform at the school:

Dr. Adams has made it quite evident that his opinions are and what he is going to help change the system, such as extended instructional time for mathematics before school, after school, and Saturday school. His vision is apparently long standing and deep and he seems truly committed to it.

Dr. Kelly was assistant principal at the school when the study was conducted. She has a somewhat different story to tell. She finished her degree in political science early on, stopped to have a family, and came back to teaching in mid-career, after her family moved to this region. She completed her M.S. (1989) and Ph.D. (1995) degrees in educational administration. She has strong opinions herself, though they spring from different origins than those of Dr. Adams.

It seems these two administrators, coming from different backgrounds, were heading in the same direction. They had both been sparked by earlier experiences that shaped their ideas about how "things should be" and offered cause for change. They were both ready to do something to evoke change.

In this mode, the principals and a few teachers began to search out and win grants. Dr. Kelly got a small grant that helped fund the first attempt at writing a new mathematics curriculum. It turned out to be a thick binder called Bumbershoots. Many staff members participated in this project. They eventually got a state Venture Capital Grant of $125,000 over five years in 1994, to improve mathematics instruction. They wanted to reconstruct mathematics teaching and learning at the school. They read and researched new theories and methods, consulted with mathematics educators in the district high school and a local university. They also studied the NCTM Standards (1989, 1991, & 1995) thoroughly. They were determined to find a better way to help students learn mathematics. As they read, wrote and talked they became convinced that children were able to tackle more complex mathematical problems that it was ever thought they could. Children were able to solve problems and communicate their solutions to their groups and to the whole class. Students were encouraged to restate in their own words what the questions were asking. They were expected to draw pictures or model their solutions and express their thinking and reasoning in writing. Children were allowed to manipulate objects to enhance understanding. The teaching staff began to realize that it was indeed worth it to listen to how children were thinking about mathematics concepts and how they understood those concepts. This approach of actively listening to the student's voice continues to this day at the school. Teachers attend and conduct workshops and conferences to learn more and improve understanding and methods. They also team-teach and take opportunities to observe one another's teaching methods.

The Teachers

The teachers at the school have a general outline that contains mathematics topics such as length, area, volume, time/money, and chance. They realize that most of the mathematics that takes place in the world involves these numeric and measurement systems. Teachers meet to share ideas and problems they have written in various categories and work towards a "community of learner" in their classrooms, where the teachers create the next step in the learning process based on what the children have expressed they understand from the day before.

Mrs. Paterson and Dr. Rahman are two fourth grade teachers that are highly involved in the reform process. They actively participate in inservice meetings, sit on committees, and read and comment on acticles shared by other staff and administrators. They are both considered "teacher leaders" in the reform effort, and try to help other teachers see the benefits of engaging students in mindful problem solving. They use materials that involve greater student participation.

Mrs. Paterson grew up outside of Washington D.C., and went to school there. She earned her teaching degree there. She married and taught until she "retired" to raise a family. She came back to teaching when her children were both in school and realized things were different. "I could feel a need for it [my teaching] to be different and right about that time math manipulative seemed to be surfacing ..." When Mrs. Paterson came to the school in 1990, the educators at the school were just beginning to try to reform mathematics. She had taken some little workshops conducted by a "passionate teacher" she had met at the local university. She recommended that the administrators at the school contact the university teacher for advice on how to make instructional activities more child-centered. With all of this emphasis on mathematics reform at the school, Mrs. Paterson decided to get her master's degree in Curriculum and Instruction with a specialization in Mathematics.

Mrs. Paterson values what she calls a "concrete, experiential foundation" in mathematics so that children won't be "terrorized by math before sixth grade" as one of her children was.

Dr. Rahman came to the school seven years ago. She taught in another school for ten years, but was encouraged to come to the school by Dr. Kelly, who had met her in the doctoral program in which they were both enrolled. Dr. Kelly was impressed by her obvious intelligence and sensitivity to children. When Dr. Rahman did come to the school, this brought the total number of Ph. D. staff members there to three. She is a person who is passionate about mathematics instruction and teaching youngsters. She taught first grade in her previous assignment and has taken charge of a fourth grade class at the school.

These two teachers believe talking to (not at) and listening to the children are valuable teaching tools. They talk about passion and love for teaching and learning. They both feel obligated to put in some genuine time and work with children toward shared meaning and understanding.

The Students

Our original question involved how African American students' attitudes and perceptions might be influenced due to the learning climate at the school. In the search for possible explanations, we explored the above question through observations and discussions with two fourth grade African American students (one boy and one girl). We observed each of the two classrooms systematically (10 times in each) for one whole year. We conducted 10 one-on-one interviews with each student during the course of our study. What follows are descriptions of the students in their particular settings, their activities, and their thoughts.

Ciara is a fourth grader in Dr. Rahman's class. She is very athletic and full of smiles and enthusiasm. She informed us that she loves to play basketball and is an excellent volleyball player. Her teacher describes her as an all around good student. She particularly loves math. "When Dr. Rahman says it is time for math, I get happy. People yell at me." She tackles problems with relish and is a leader during mathematics discussions and activities. These are some thoughts she shared with us:

Ciara: I like to collect bugs. I don't kill them. I have to keep them away from my Mother! I have 10 teddy bears. My favorite teachers are Mr. Singerman [The third grade teacher], Dr. Rahman, and Ms. Price. [The kindergarten teacher]. They are very nice and kind. I don't like a teacher that is always nice like if people break something and she say 'that's okay' and the class said they were mad and she said 'don't be upset.' I like a teacher that is both ways, nice and even. They are fair and they taught things that were interesting.

Clearly, fairness is a very important thing in Ciara's life. It is very high on her list of how she judges people and events. She seems to include herself in her moral code. She does not even "like to kill bugs". It seems that she believes everyone should "behave decently" in a situation. She strives to do her best and is conscious of her role as a student.

Malik is a fourth grader in Mrs. Paterson's class. He is also an athlete. He told us he was good at "football, basketball, and kinda [good at] hockey, golf, bowling, baseball, rugby, and lacrosse." He also likes mythology and history, especially African American History. "I liked when we learned about Harriet Tubman and the Undersground Railroad, and the escape of the slaves." His teacher describes him as a "remarkable kid. He does a really good job of explaining his thinking. He can show you how to solve a problem visually or in words." These were some thoughts he shared with us as he described himself:

Malik: I think I'm funny ... and creative. I am really funny at recess and lunchtime ... [pause] and I'm a person who tries to help other people who have problems. Mr. Singerman [third grade teacher] is one of my favorite teachers. He got me into computers. I like science. I've learned about rocks. Pumice, it floats, and marble, that's what they use to make statues. I'm good at math ... I would rank myself a 10. I go back and correct my work. My grade for math was an A. This year was easy ... the work she [Mrs. Paterson] gives me is not that hard. I'm caught up with my work, and I don't get into that much trouble.

Malik is a serious, hardworking young man. He thinks deeply for someone of his tender years, and already feels some responsibility for others. He is the older of two brothers, in a nuclear family of four. He demonstates a position of moral decency that he has internalized and his attitude reflects that. He is very confident and does not like to let things go until they are done to his satisfaction.

The Parents

When we asked the parents of the two students to express their thoughts on education, their hopes, dreams, and fears for the children we received a variety of answers. We observed that one theme ran through all of their responses: the fear of bring misunderstood, mainly in the sense of being thought less of by people who are not black. This sentiment is identified as "devaluing" by Steele (1992). In the words of Ciara's mother, Mrs. Thomas, "Our lifestyle, I feel, is very much different from a white person's, because we have struggled, we have really struggled." She recalled an experience she had with her nephew in another school system.

Next was the testimony of Malik's mother, Mrs. Williams. She is a teacher for a large metropolitan school district. She is a bright, energetic woman, who is able to express her great enthusiasum for the education of her own two children; as well as the children she is teaching. She seems to understand what the school is trying to do. She appreciates all of the opportunities her son has been given to experience mathematics learning at the school.

Physical Structure of the Classrooms and the Social Norms

Mrs. Paterson's classroom is very colorful. The eyes are pulled from one thing to another. She has a large bulletin board in the back of the room that is decorated with over a dozen book jackets with the words, "wonderful exciting books". She has two ant farms (with real, live ants), several big dead beetles, a plastic worm, and many books on science. On every window shade is a poster extolling the virtues of reading. There is a display devoted to the stock market. Her class is following three stocks in particular. There is a blue desk with two red and blue chairs, set up, fully equipped with a beginner's chess board and pieces. The room is friendly, interesting, inviting, and relaxing.

During a mathematics lesson we observed, students were called together on the rug in the front of the classroom where Mrs. Paterson joined them. Previously they had been gathering data from the dictionary and other places about the frequency of letters in common words. They had each been assigned to do a letter count on ten words. They reported out their data to see if there were any patterns, and if so what might be some of the reasons for them. Mrs. Paterson was recording their reporting on a white marking board on the floor near the children. After about half of the class had reported, students noticed that the letters with the highest frequency were vowels and the one with the very highest frequency was the letter "a", closely followed by the letter "e". After all the tallies were in, the students began a discussion about the construction of written language and the function of vowels and consonants. The students did most of the talking and made most of the discoveries on their own. It was a student-centered lesson.

The first thing you see when you walk into Dr. Rahman's room is the piano. It sits directly across from the door. She plays it after lunch sometimes as a little treat. She may let a student who wants to try something have a turn. There is a big rocking chair in the room for story telling. Near the chair is the reading corner. There are lots of big colorful pillows for students to relax on. The students also meet here to have discussions and lessons. Students' works are everywhere on bulletin boards around the room.

During a mathematics lesson we observed, students were trying to figure out how much ribbon they should buy to frame a picture they were giving to the art teacher. Students were gathered around the overhead, clipboards in hand, working intently. The overhead was on a low table, almost in the middle of the room, so Dr. Rahman could sit on a student chair and facilitate dicussion better. The way the students were gathered around, and in the darkened room, they had the appearance of being gathered around "the campfire" light of the overhead. There was a feeling of shared purpose and interest. Dr. Rahman said, "How do we know what to do first?" From there, students offered suggestions which worked out to mean that they should measure all four sides and add them up. Then they had to figure out the price of the ribbon. The students were expected to restate in their own words, what the question was or how they were understanding the problem. They were expected to model (draw pictures), and communicate the solutions with the classroom community. While one student was defending his/her solution the obligation of the class community was to listen very carefully and ask questions for claification.

Dr. Rahman asked her students what they might do if they wanted to add a second border of ribbons inside of the frame. For this part of the problem, they were asked to work at their tables (groups of four) to solve the problem, including drawing and explaining.

During the small group work, students were responsible for their own learning. If they had differences of opinion, it was their obligation to come up with a consensus. Also, they were expected to offer support if someone in their group needed help. Dr. Rahman communicated with her students that the only time they could ask the teacher for help was if the whole group had the same problem. Students were completely in control of their small group problem solving. They seemed perfectly capable of solving the problem with very little assistance from Dr. Rahman.

Mathematics Instruction

What follows are two lessons that demonstrate the degree to which methods in these two classrooms were designed to encourage students' dialogue and inquiry-based mathematics instruction. The first lesson took place in Dr. Rahman's classroom. She was continuing a series of lessons about fractions that had begun earlier. The students had reacquainted themselves with the notion that things can be divided into portions called halves, thirds, fourths, eighths, etc. They had previously discussed why these divided wholes were called by their respective names. They had also worked with renaming fractions, as 2/4 equal 1/2 and 2/6 equal 1/3; and they had experience drawing these different fractional configurations and learned to call them euivalent fractions. At this point Dr. Rahman determined that the students were ready to tackle the idea of combining fractions with different denominators.

T: [She presented a situation, writing the information on the overhead as she explained.] Today, I have a problem that I hope you can help me solve. I had my family over for pizza last night and I ordered a lot and they ate a lot! I ordered the rectangle kind because they are bigger. Each of the pizzas had been cut into 12 slices. [She drew 3 rectangles, cut them each into 12 equal pieces and labeled each one.] When we were done this is what I had left: 1/6 veggie, 1/3 cheese, 1/4 pepperoni. [She wrote the fraction for each under the drawings, respectively.] Tonight, I'm having some of my friends over to work on a project for our street club. I told them I had some leftover pizza that we could eat and they said that would be fine. If I invite three friends and each of us has four pieces of pizza will I have enough or will I have to order more? If I do order more, how much should I order? [The students got into their groups and busily began to draw the portions of pizza as they had been described. The classroom was noisy and students were talking to one another. The teacher was circulating a round for observation, listening to student discussion, and making notes.]

S1: Each pizza was 12 slices right?

S2: Well, look 1/12 is the same as 1/6, look, look! [The student highlights the portion that is 1/6 and drew a line to show others the new configuration.]

S3: Okay, that's right, 1/6 is the same as 2/12. So, the veggie is 2/12 ... can we make everything into twelfths? What about the cheese pizza?

S2: Well, look ... Wait a minute ... we're got 1/3, right? If you take 12 and divide it into three parts, it gives you ... four pieces in each group, so the cheese has got four pieces! [The student was excited as he explained.] Four pieces makes 4/12! Look! Four out of twelve!

S4: Okay, well, that means you got the 2/12 and the 4/12 ... that's 6/12, right?

S1: Yeah, yeah ... that's right! [He referred to the teacher's drawing to confirm S4's solution.] So that's the cheese and the veggie ... we have to do the pepperoni.

S3: Watch this ... You take the twelve for the pepperoni ... when you cut the whole thing into fourths ... you cut it this way ... each section is ... okay, three, yeah, three pieces. Yeah, four times three is twelve, so that's right! Look!

S1: That's the pepperoni, right?

S2: Yeah! So the three pieces is 3/12 ... so ...

S4: I know! 2/12 plus 4/12 ... and 3/12 ... that's 9/12! It's 9/12!

S3: Okay, yeah, that's right ...

S1: So, how many does she need? What'd she say? How many people?

The students worked diligently to figure out how much pizza was actually left and if there would be enough for the new meal. After about 30 minutes, many groups began to come to the conclusion that there would not be enough pizza. Dr. Rahman started the whole class discussion.

T: We will have to stop our group work now, so we can share our results. Who can tell us how much pizza we need? Is there a group ready to explain? [Many groups were eager to share their solutions. One group of four students was selected and brought their papers up to the front to show to the class. One student began.]

S5: First we thought how could they be alike, you know, how to make the fractions alike. They all have to be the same if you're going to add them up so we thought about twelfths. We thought they could all be twelfths, so we found out that the 1/6 of the veggie pizza is the same as 2/12, that was easy. So that was the first part.

S6: Yeah. Then we found out that if you take the cheese pizza and make it so some of it is twelfths ... you can cut it like this and you can make the whole thing into thirds ... so like, all the sections have four pieces. So there are four pieces left, that's 1/3 of the pizza, see? Each part is 4 pieces so, that makes four pieces of cheese pizza that are left.

S6: Okay, yeah, the cheese is four pieces ... and then we had to find the pepperoni. So we thought of the same thing, 'cause you can do a lot of stuff to get twelve ... like three times four is twelve and four times three is twelve ... so we cut the pepperoni the other way, and you can see four sections like this ... so the pepperoni is three pieces.

S7: Yeah! And we just put them all together. We just added up the twelfths. So that's 2 plus 4 plus 3, so all together that's ... nine pieces! You've got nine pieces of pizza for everybody!

T: I see. Are there any questions for our group presentation? [No response.] I think I see what you did! Wow! I think I'm impressed! You did some hard thinking! Do we agree? [There was much nodding of heads, although not from everyone. Some were still working it out, looking at their own papers and changing their thinking.] Thank you, boys and girls for your great work! Wonderful job! [The student group proudly took their seats.]

At this point, Dr. Rahman saw this was important to make sure all the students understood how the group got the total of 9/12, so she stopped here to restate the group's results.

T: Let's look at this again and check it out. [She referred to the pictures of pizzas on the overhead, using other colors to show the larger divisions, and renaming the fractions as she explained. There was discussion and writing among the students in their groups.] Do we all see something about this? [Many more students seemed to agree.] Great! Now, let's look at the next part. Does anyone know how much pizza I need for tonight?" [Many hands went up.]

S: [many voices.] Sixteen, Sixteen!

T: How do you know that? [She called on a student.]

S8: If there are three people plus you, that's four! And four times four is sixteen! You each get four pieces, that's sixteen!

T: Do you agree? [Referring to the class]

S: Yes! Yes! [Many voices]

T: Well, I think I agree, too! We have to stop soon so let's see if we can finish this up quickly ... I observed that one group presented their results and they concluded that I had 9 slices of pizza left over. Will this be enough?"

S: No! No! [Many voices]

T: How many more do I need?

S: Seven! [Many voices]

S11: You have to buy another pizza!

S12: Maybe you could get a small!

T: Yes, I guess I will! Or maybe I will just eat less ... I need to lose weight! [Everyone laughed.]

In the above episode, Dr. Rahman encouraged students' dialogue concerning mathematical representations. Students had some control over classroom instruction and their own learning. She was not the sole authority evaluating and controlling student solutions. She saw her role as a facilitator and guide.

The next lesson took place in Mrs. Paterson's room. It was a lesson about area and perimeter. Again, the students had some prior experience with these two topics. On this day, Mrs. Paterson offered a situation to help students explore the notion that spaces with the same area can have different perimeters.

T: Here's a picture of my dog Lester! I just got him from my son! Isn't he cute! [She showed students a photograph of a basset hound puppy.] I don't want him to run around all over my yard. You know how dogs are! He will ruin my flower beds and my vegetable garden! So, my husband and I have to build a smaller exercise place for him inside the yard so he can run around and not mess up everything. The problem is I don't exactly know how I want it to be. I know I can give him 36 square feet of space, that's all the room I can give him so everything else still fits. I would like for the area to have a large perimeter so he can run a good bit, but I'm not sure how to make it look. I would like for you to get into your groups and find all the possible ways you can make a space with an area that is 36 square feet. I also want you to find the perimeter of each space and see if there are any differences. Then, maybe I can decide how I want to make Lester's exercise place look and where to put it in the yard. Here is a sheet of grid paper for each of you. Discuss this with your group members and help each other find all the different areas and perimeters. [Students get into their groups and begin to work.]

S1: Different areas? I thought it was the same area ... didn't she say 36 square feet?

S2: Yeah, but I think you can do it different ways! Remember multiplication! You can get 36 by six times six ... look! [She drew a square on the grid paper.] Six up and six across, that's 36!

S1: Oh, yeah ...

S3: And there is three times twelve ... so you can make it look like this [He drew a rectangle with these dimensions.] That's 36. [Other students in the group quickly began to make the figures mentioned by their group members and tried to find other areas of 36 square feed.]

S4: Here's four by nine! [He drew the rectangle.]

S2: Don't we have to find the perimeters, too?

S1: Yeah, but ... don't mix me up! I want to find all the areas first!

S2: Okay, we'll find all the areas. Look, there's six by six, three by twelve, and four by nine, and ...

S3: Does five times anything make 36?

S2: I don't think so ...

S1: No, no. Nothing by five is 36.

S4: Look! You can do two by eighteen! It's long and skinny!

S2: Okay so we've got two by eighteen, three by twelve, four by nine, nothing by 5 five, six by six ... seven?

S1: Nope!

S2: Eight by something?

S3: I don't think so ... eight by four is ... 32 ...

S1: Nope!

S2: Nine by ...

S4: We got that one. It's nine times four ... but you can put it the other way ...

S2: So is that all? [They agreed that that's all they can think of.] Now let's do perimeters. [They counted around each area they found.]

S3: Look! Six by six is ... 24!

S1: Yeah ... six, plus six, plus six ... 24!

S4: Nine by four is ... two fours is eight ... and two nines is eighteen ... so that's ... 26! It's more than six by six!

S1: Two by eighteen is ... 40! Wow! he can run far!

S2: But he can't run in a circle. Maybe he wants to run in a circle sometimes.

S3: Three by twelve is 30.

S2: Is that all? [They all began to check]

S1: I think so ...

S4: Yeah ... yeah, that's all we got.

S2: Maybe she wants the long skinny one or maybe she wants the square one.

S3: Which one takes up the most room?

S1: Where's her other stuff? The garden and stuff?

Students all around the room began to find the combinations as this group had. Mrs. Patterson sees that most groups found areas and perimeters. She called all the students to sit on the carpet in a semi-circle for discussion.

T: Well, what did you find out?

S: [Several voices] Forty's the biggest! Seventy! Seventy-four! Forty is he biggest! No, It's not!

T: Okay, okay! We'll see! Why don't you guys come up and share what you found out about the areas. [She pointed to a group of four students.]

S5: We found all the areas! There's two by eighteen, three by twelve,... [The student pointed to each area as she named them.]

S6: Yeah! And four by nine ... and six by six.

S7: Plus, you could have one by 36, couldn't you? [He looked at Mrs. Patterson.]

S: [Several voices] Yes! Yes! We got that!

T: Sure you could! That's great! Any more? [Students checked their papers. They agreed there were no more.] Okay, group! That was exceptional! [The group sat down.] Could we get another group to share what they found out about the perimeters? [She pointed to another group.]

S8: The biggest perimeter is one foot by 36 feet! If you count all the way around it's 74! But it's real long and skinny.

S9: Okay. If you got six by six, the perimeter is 24. We just added it up.

S10: Two by eighteen is 40 and three by twelve is 30 ... and four by nine is 26.

S11: The second longest is the two by eighteen one. It comes up to 40. That's it.

T: Is that all? Does anyone have anything to add? My Goodness! Did you realize there could be all those different perimeters from the same area? There are five different possibilities! You've given me a lot to think about! I'm going to take your results home and tell my husband. I'll let you know what we decide!

This lesson offered students rich experiences that deepened their understanding concerning the relationship between area and perimeter. Students came to realize that an area can stay the same while the perimeter changes. Students gained confidence in their abilities to think and solve contextual problems.

There was evidence that lessons in these two classrooms were created and presented to allow for student thinking and reasoning. The two teachers did not follow any particular mathematics textbook. They worked hard to create and present mathematical situations so that students could make relevant sense of the activities.

Dialogue with Students

In one after school session we gathered Ciara, Malik and two other students, Brittany and Kent, from another school. Brittany and Kent experienced mathematics learning in a mostly conventional mathematics classroom, where the main focus of mathematical activity was following the textbook and doing worksheets. Their teacher would give the directions on how to find answers to a given worksheet assignment. Then, students would follow the same procedure presented by the teacher for answering further problems on the worksheet.

We wanted to know whether or not the school mathematics program at the school that promulgates constructivism as a viable theory of learning, affects mathematics problem solving of participating students. We anticipated that putting these four students together (two from the school and two from another school) within the context of mathematics problem solving would provide us with opportunities to observe their social interaction and see them in juxtaposition to one another. We started by posing a problem involving fractions. We gave the children this problem:

We told the students they could solve the problem anyway they were comfortable. However, they were expected to explain the solution to us (in the below conversation, R stands for reseacher). Malik and Ciara started drawing pictures. Brittany started making a table. Kent was quiet and was thinking. After awhile, Malik and Ciara began to talk with one another.

Malik: Okay, here's my pizza, this is my part, 2/12. That's not much [slicing the pizza into 12 pieces and covering 2/12 of that], Brittany gets 1/4 ...

Ciara: There are no fourths on this pizza, just twelfths [Referring to Malik's drawing].

Malik: Well, we could do this [he started with his 12 slices and then paired them with different colors] I know 2/12 is the same as 1/6, so we color this and that's your part [Referring to Ciara's 1/6].

Ciara: We get the same. I think they get more ... [Referring to Brittany and Kent].

Brittany: I don't get it ... [She gave up her chart and listened to Ciara's and Malik's dialogue.]

Ciara: Okay, here is a 12 slice pizza. [She started all over again making a circle with 12 slices.] You can have 1/4 of this 12 slices. [She divided 12 pieces into four groups of three and colored each fourth with different colors.] so, you have three pieces. Now, did you get it?

At this time Kent was carefully listening to the dialogue but could not figure out his pieces of pizza.

R: Can someone help Kent to figure out his pieces of the pizza?

Brittany: Yeah, [She started with 12 slices and then grouped them into three equal piles and colored them.] Well, I learned from Ciara that mine was three pieces out of 12. I did the same thing for Kent and he gets fours pieces, which is one piece more than what I have.

R: Does it make sense to you Kent?

Kent: Yeah, with picture it is easier.

R: Can someone tell me if there are any slices left for me?

Students: [Almost all together] Yeah, there would be one slice left for you.

Kent: It is not much.

This observation and others conducted during the study made it clear to us the importance of students' interactions in mathematical situations. Because of their classroom experiences, it was natural for Ciara and Milak to negotiate mathematical meaning with one another. Kent and Brittany experienced a different classroom culture. At first, they were hesitant to talk with their group as if it was forbidden to talk. Then Brittany broke the silence by saying "I don't get it". She was indirectly communicating with her group that she needed their support. Kent, however, was still uncertain about what was expected of him or how he needed to interact with his peers. Because of the drastic difference between social norms of his regular classroom and the current small group culture, he needed more experience. Another plausible explanation for lack of active participation for these two students might be their limited knowledge about fractions.

The Development of Students' Mathematical Dispositions

We present two short vignettes to demonstrate the attitudes and mathematical dispositions of Ciara and Malik. We intend to communicate something about their character and circumstances as each of them lives out a "typical" school experience. Then we discuss how the culture and circumstances of the school, as reflected through its administrators and teachers, seems to have influenced the students' beliefs and attitudes toward mathematics.

Ciara walked down the hall with her friend Tiffany and the primary researcher. The researcher wanted to meet her again and get to see her on a regular school day. The morning bell had just rung and students were hurrying to class. She was talking to Tiffany and amiably allowed the researcher to listen in:

Ciara: Today we have art and P.E.! Plus we get to finish that probability stuff we started with Dr. Adams [the principal] yesterday. It was fun.

Tiffany: I don't think it's much fun.

Ciara: Well, listen, Dr. Adams is always pretending he doesn't know the answer and acting weird about it and stuff. It's pretty funny. Like yesterday when he came in and asked what the probability of two people having the same head size was. Then he got a tape measure and started measuring Jeremy's head. Dr. Adams sure has a big head! He says it's so it can hold more brains!

Tiffany: Yeah, that was pretty funny! Maybe he'll do something funny today, too.

One of the researchers made an arrangement with Ciara to meet with her and talk further. They met a lunch time. The researcher asked her what had gone on in her mathematics class.

R: What did you find out about probability?

Ciara: We measured everyone's head in our class. Out of 24 people, two sets of two had the same head size. That means in our class, there were four out of 24 chances of people having the same head size. Of course, that's only our class. It could be different in another class. We would have to measure some more.

R: What about Dr. Adam's head?

Ciara: Oh, it was bigger than everyone's, including the teacher's. But that's because of his brains, remember? [laughing] You wanna hear some more about probability? Listen, it's like you're doing chances, you have two dice and you roll. What are your chances of getting a two or a six?

The researcher repeated Ciara's question, trying to tune into what she was saying.

R: What are the chances of getting a two or a six from a pair of dice?

Ciara: Look! You can only get a 2 one way, one plus one.

R: How about six? How many chances are there to get total of six?

Ciara: Well, [She pauses] There's one plus five, that's one way. [counts on fingers] Then ... two plus four, three plus three, five plus one, four plus two,... I think there are five ways, Yes! Five ways!

R: Wow! What else do you know?

Ciara: Sometimes we do percents ... say the regular price is $30.00. So, you save 10%, that's $3.00. So, then the sale price would be $27.00, and that's it!

R: Now, say you have a big screen TV that costs $2000.00. What's 10% of $2000.00? Can you figure it out?

Ciara: No ...

R: Well, what's 10% of $200.00?

Ciara: $20.00.

R: And 10% of $2000.00?

Ciara: $200.00! Then I subtract it from that and I come up with ... I think it's $1800.00. I like math better than anything in the world because I'm good at it ... From 1 to 10, I would give myself a 7.

R: Why a 7?

Ciara: Oh, because Ben and Eileen know everything. I know most stuff but they're better than me.

Ciara seemed to have a good grasp on some important mathematics. She seemed excited and interested and was willing to go on without the researcher's support once she saw the pattern in the percent problem. She showed great confidence in demonstrating her mathematics understanding. We wondered what could be contributing to these attributes in her, and we could not help but following them back to the experiences in mathematics class and her relationship with her teacher and her classmates. We asked Ciara to describe a "regular" mathematics activity in her class.

Ciara: We spend most of our time in class working in groups. I don't think anybody is the leader at the table. We take turns and stuff. I listen to the other students, what they have to say first, and then I say what I have to say. Like, sometimes they get it right and sometimes they get it right and sometimes I get it right.

It seemed that the social norms established by the teacher and students were pivotal for creating learning opportunities for all students. Sharing and respecting one another are certainly considerations for Cieara when she talked about interactions. Even when she talks about individual problem-solving, there is an element that shows she is working by a set of norms respected and accepted by the members of the classroom community.

Ciara: When we work by ourselves ... first we have to think it over and put down an answer. You can't just write anything. You have to think. Then you have to go up there [in front of the class] and tell it to others how you got it.

She expressed some norms on the conditions necessary to ask for help. One must really try hard before asking. Ciara seemed to accept that norm as a fair expectation.

We observed Malik in Mrs. Paterson's room. He and Brandon (his partner) were at the computers.

Brandon: Wait, wait! You are going too fast! You're gonna miss it!

Malik: No, I'm not! Just give me a minute ... there! Now you write it down.

The boys had found the city they were keeping track of. They were monitoring the amount of sunlight that Rio de Janeiro was receiving each day. We went over to say good morning to Mrs. Paterson. She bragged happily about what her class was doing.

Paterson: There are students monitoring the amount of sunlight in cities all over the world. They're checking put the idea that in the Northern Hemisphere, as we approach spring, the Southern Hemisphere is approaching winter. It's really amazing to watch the kids realize this by looking at the daylight graphs. They see that the daylight is increasing in places like Bangor, Maine, and Amarillo, Texas, which is what we expect. But what's really amazing to them is that the days are also getting shorter in Rio and Sydney! No matter how much we say that to them, it's still surprising for them to see. I love to have them talk about their ideas on this after we've done it for a few weeks.

Mrs. Paterson is a creative teacher. She has many ways to generate interesting experiences for students that encourage them to create and manipulate useful data that has relevance in life. In Mrs. Paterson's class, we observed the pattern of classroom social interactions. First, students worked on the problems individually. Second, students worked in their small groups. And third, they were provided with opportunities to share their multiple solutions with the class. The problem-centered classroom was observed ten times throughout the year in Mrs. Paterson's class. We asked Malik about his reflections on one of his class activities.

Malik: The activities are called 'pay the principal' and 'how long have you been in school?' We do them in the morning, along with other stuff. Well, we don't really pay him [principal]. On the first day of school, we started out with $20,000.00 and you keep withdrawing a dollar a day ... and then you have to find the interest every month. You figure out how much you put into his account. And, 'How long have you been in the school?'... Well, we figure out the real amount of time we've been at school. [Each day] it's six hours and 10 extra minutes. We figure out everyday how long we've been in school. We added what we had to the day before. We learned about fractions ... like total hours and total minutes. Because of the extra ten minutes we had to divide the hours into minutes all the time. So we talked about fractions a lot. We add, subtract, and multipy them.

Malik was able to extend his understanding of the relationship between time and money to fractions. He was able to explain his ideas with confidence. In both classrooms, it became clear to us that mutual respect and shared meaning created by students and teachers were crucial for learning mathematics.

As we discussed earlier, social norms of the classrooms significantly impacted students mathematical dispositions. In addition, the principals and the teachers' passion seemed to have positive impact on students' attitudes and beliefs towards mathematics. Many of the statements expressed by the principals and the teachers at the school reflect evidence of passion about teaching and learning.

Discussion

Educators, politicians, and parents all express great concern about low levels of mathematics performance by American students; especially African American students, who generally perform lower than their white counterparts. As D'Ambrosio (2001) asserted:

Bigelow, Harvey, Karp, & Miller (2001) observed, it is possible that these poor performances are due in part to lack of meaningful mathematical experiences that many students, particularly African American students have. The NCTM, Principles and Standards for School Mathematics (200b), emphasizes mathematical dispositions as one of the major attributes for all students to possess. At this school radical attempts are being made to implement the spirit of NCTM Standards with the goal of "mathematics for all children."

The findings of the study suggest that the culture and structure of the school positively influenced students' beliefs and attitudes towards mathematical learning. In addition, the teachers and the administrators' passion for reforming mathematics created a synergistic transforming learning community which provided students with opportunities to gain deeper mathematical dispositions.

One of the main features of inquiry-based mathematics is understanding students' understanding via active listening (Burns, 1992; Romberg, Shafer, & Webb, 2000; Wheately & Reynolds, 1999). Mrs. Paterson and Dr. Rahman are convinced that social interaction and mathematical dialogue are crucial for conceptual understanding of mathematics. They understand the meaning of the relationship between the two. They know that they must organize their instructional methods around what they genuinely observe from their students.

The importance of constructivist teaching and the centrality of students' voices provide teachers with opportunities to develop and modify their instruction. One of our concerns is continued educational opportunities for these students once they leave the school and transition to more traditional schools. Have administrators in the new settings taken into consideration the learning experiences of these students? Will any adjustment provisions be made? Are there or should there be any transition strategies employed to sustain learning for these students? How do teachers with constructivist epistemology, communicate effectively with teachers from more conventional backgrounds? How do administrators accomplish this same task according to their own ideologies, for the good of all students? These are important questions that must be answered if we are to continue to have a chance to educate tomorrow's American children.

We end our discussion with an anecdote that is hopeful. One day after school, as the children were leaving for home, we overhead the comments of a group of second graders, including a white boy, a black boy, and a black girl. They asked the assistant principal, Dr. Kelly, if she was coming into their room for a mathematics lesson the next day. She thought for a moment and said: "I think I will be able to fit in." "Yes!" was the unanimous reply by the students, who were obviously pleased at the possibility. So it seems to us that here we have a dream come true: everyone being pleased at the prospect of their white female, non-mathematics oriented, but converted, socially conscious, constructivist teacher/administrator coming in to experience an ordinary mathematics lesson with a group of enthusiastic, multicultural, mixed class, and gender, American children. Perhaps there is hope after all.

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Belvia K. Martin, Shaker Heights School District

Roland G. Pourdavood, Cleveland State University (CSU)

Nicole Carignan, University of Quebec at Montreal (UQAM)

The intent of this study was to investigate the experiences and reflections of two African American children, their teachers, and their parents about mathematics learning and what these experiences imply for educators as they attempt to reform mathematics education to help all students gain a deeper mathematical disposition. A qualitative design employing in-depth interviews, participant observations, and the examination of relevant documents was used to explore the mathematical environments of two African American children. As the stories of these two children emerge, it is hoped that a better understanding about mathematics teaching and learning, grounded in the experiences of people of color, can be added to scholarship, thereby strengthening the chorus of "other" voices increasingly present in this study.

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Current research studies indicate that our elementary grade students are performing better in mathematics and science proficiency tests of basic skills (Mullis, Dossey, Campbell, Gentile, O'Sullivan, & Latham, 1994; National Science Foundation [NSF], 1996). Also the results of the Third International Mathematics and Science Study (TIMSS) illustrate that the U.S. fourth grade students performed significantly higher in mathematics and science when they were compared to their counterparts from other industrial countries. However, research results show that our students do not possess deep conceptual understanding of these mathematical skills (National Council of Teachers of Mathematics [NCTM], 2000a; 2001a). The results are even more disturbing when they focus on minority students' achievements and their mathematical self-perceptions. The NCTM (2001b) states that "Mathematical power must be considered the right of, and the expectation for, every child" (p. 1).

In spite of the recent improvements regarding minority students' mathematical dispositions and narrowing the achievement gap between white and other minorities (I.e., African American, Hispanic, Native American, etc.), considerable disparity still exists. This incompatibility between white and minority students' attitudes and performances in mathematics carry economic and social justice implications (Bigelow, Harvey, Karp, & Miller, 2001; Campbell, 1996; Frankenstein, 1995; Ladson-Billings, 2001; Moses & Cobb, 2001; NCTM, 2000a, 2000b, 2000c; NCTM, 2001a, 2001b, 2001c; Ogbu, 1987; Secada, 1992). One might conjecture multiple reasons for these differences such as inadequacy of resources, unqualified teachers, disconnected and repetitive curriculum, as well as lack of a relationship between minority students' cultural backgrounds and their classrooms social norms and sociomathematical norms.

Problems of diversity have existed since the inception of this country, in fact they are an integral part of the history and creation of the nation as it stands. Often certain individuals have been perceived and treated differently than others in American society. Some of the questions regarding teaching and learning mathematics include: can the old patterns of race, class, and gender discrimination continue to exist in the face of changing population patterns? Can we (meaning mostly those who make policy decisions--legislators, superintendents, principals, teachers, and even parents) continue to support these old patterns; or can we transform how we (and others) feel and what we all believe for the sake of educating our children and sustaining the nation? "If the mathematics is deemed worthwhile to learn, then students in poor communities should have the same access to that mathematics as those in affluent communities" (NCTM, 2001b, p. 10).

We ought to examine the patterns of teaching and learning that exist for various groups in our schools. These patterns may not be the same for all groups, or the resulting performance levels and students' attitudes and beliefs toward mathematics would not be so different. It seems the perceptions by educators, parents, and students alike are very powerful indicators of what is taught and learned (Moses & Cobb, 2001). Thus, the examination of these perceptions would be an important point of research. This study was an attempt to examine such perceptions in one small microcosm that reflects connectivity and relationship among these groups.

Theoretical and Pedagogical Considerations

The theoretical and pedagogical assumptions of this study are grounded in social constructivist epistemology. Social constructivist theory posits that all knowing is inherently grounded in social and cultural activities (Cobb & Yackel, 1996). The notion of experience and context is crucial to social constructivist epistemology. This perspective assumes that an individual child or adult, who helps to build the shared meaning of the mathematical activity of the group and comes to understand the mathematical thinking of the others, and who is also able to contribute to the mathematical understanding of the group, would be bound, in such a social setting to come to perceive of him/herself as a more capable learner (or even teacher) of mathematics.

Assumptions about the nature of knowing, how it is acquired, and how to facilitate its acquisition greatly influence one's practices as he/she attempts to construct new models of mathematics instruction and assessment. One of the major tenets of social constructivist pedagogy is that methods of instruction should reflect the teacher's respect for the students and their ability to reason and figure things out for themselves. Also, students come to view and rely on the teacher as a guide/aide, rather than as the sole authority in the classroom. According to this perspective, allowing students to make sense of their activities provides them with opportunities to build their mathematical dispositions, and thus encourages students to seek more and greater understanding of mathematics.

According to Brooks & Brooks (1993), to become constructivist educators, we should start by honoring the learning process. They make an important point when they explain that:

We construct our own understandings of the world in which we live. We search for tools to help us understand our experiences. To do so is human nature ... Each of us makes sense of our world by synthesizing new experiences into what we have previously come to understand. Often, we encounter an object, an idea, a relationship, or a phenomenon that doesn't quite make sense to us. When confronted with such initially discrepant data or perceptions, we either interpret what we see to conform to our present set of rules for explaining and ordering our world, or we generate a new set of rules that better account for what we perceive to be occurring. Either way, our perceptions and rules are constantly engaged in a grand dance that shapes our understanding. (p. 4)

For Brooks & Brooks (1993), each new construction of knowledge depends on the cognitive abilities to accommodate discrepant data and perceptions and the "stockpile" of experiences an individual has at any given time. Likewise, Yackel (1993) observes that "from the constructivist perspective, meaning is not inherent in written or spoken language. It is constructed by each individual on the basis of his or her own experience and involves individual interpretation" (p. 34). This viewpoint supports the data that knowing is not an object to be acquired, but an activity during which meaning is constructed. Similarly, Wood, Cobb, & Yackel (1991) shed more light on this topic when they explain that, "it is useful to see mathematics as both cognitive activity constituted by social and cultural processes and a sociocultural phenomenon that is constituted by a community of activity recognizing individuals" (p. 402). The importance of the student/teacher interaction was emphasized by Cobb, Wood, & Yackel (1991) when they discussed research conducted in a second grade classroom. They reported, "the emotional tone of the classroom" (p. 174), and stated that "the children were enthusiastic, persistent, and experienced joy when they solved a personally challenging problem" (p. 174). They concluded that emotional considerations are very important to the successful implementation of constructivist pedagogy. The authors go on to discuss the role of the teacher as a promoter of acceptance and trust in the classroom. They made the observation in the second grade classroom they studied:

As a constructivist teacher, she did not refrain from carrying out certain activities characteristic of traditional teachers and relinquish her authority. Rather, she expressed that authority in action by initiating the mutual construction of certain obligations and expectations. In doing so she influenced the children's beliefs about both the nature of the activity of doing mathematics and their own and the teacher's role in the classroom. Above all else, the obligations and expectations that were established constituted a trusting relationship. The teacher trusted the children to resolve their problems and they trusted her to respect their efforts. This trust is, in our opinion, the most important features of constructivist teaching (p. 174)

Yackel (1995) endeavored to understand how students talk, act, and think to promote their mathematical communication. She also endeavored to investigate teacher interaction with students and how it promotes learning for all in the classroom. She concluded that:

It is necessary for teachers to understand that students' activity is reflexively related to their individual contexts, and what the teacher contributes, as do the children, to the interactive constitution of the immediate situation as a social event. (p. 158)

This epistemological and pedagogical awareness is central at the school as a transforming learning community.

Research Question

The intent of this study was to examine one central question: what do the experiences and reflections of two African American children (and their parents and teachers) about mathematics learning imply for educators as they attempt to reform mathematics education to help all students gain a deeper mathematical disposition?

Research Design and Methodology

The observational study was grounded in constructivist inquiry (Guba & Lincoln, 1989, 1994; Lincoln & Guba, 1985). The investigation at hand was about children, specifically African American children; and how they saw themselves according to the multiple realities that existed within the context of mathematics teaching and learning in a K-4 elementary school. The data collected during the investigation, "[was] not easily handled by statistical procedures" (Bogdan & Biklen, 1982, p. 2), because it was rich in the description of people, places, events, and conversations. One of the aims of this investigation was to understand the actions and perceptions of participants from their own frames of reference; according to their own experiences; and not according to a perceived fixed truth or reality that exists outside of and independent of human activity (Anderson, Herr, & Hihlen, 1994; Banks, 1993).

Data sources include: (1) in-depth interviews with two students (10 for each), their parents (four for each), the school administrators (two for each), and their teachers (four for each), (2) participants observations (10 for each classroom), (3) field notes, and (4) examinations of relevant documents (I.e., student's journals and portfolios, school news letters, test results, lesson plans, teacher notes, etc.).

Interview tapes were transcribed. All transcriptions, field notes, and other related documents were subjected to regular and ongoing analysis and interpretation. The participants (I.e., the school principals, teachers, parents, and to some extent, students) were allowed to read and help revise the interpretations. Although the data analysis was ongoing, synthesis across the data sources occurred when the data collection was completed. The data is presented and interpreted as follows: voices of the participants (I.e., the principals, the teachers, the students, and the parents), physical structure of the classroom and the social norms, mathematics instruction, dialogue with students, and the development of students' mathematical dispositions. Then the discussion follows.

Story-Construction: Voices of the Participants

The study was conducted in a suburban school of a Mid-Western U.S. City (for more information about the school and its community see the introduction and the first paper in this issue). The K-4 elementary school enrolls 550 students. Sixty percent of the students are African American, 34% are white, and 6% are others (I.e., Asian, Middle Eastern, Hispanic, etc.).

The key participants in the study were Dr. Adams (principal), Dr. Kelly (former assistant principal who is now principal in another school in the same district), two fourth grade teachers (Ms. Patterson and Dr. Rahman), two African American students (Ciara and Malik), and the students' parents (Mrs. Thomas and Mrs. Williams). Pseudo names were used to secure anonymity of the participants.

The Principals

Dr. Adams began his career in the district as an English teacher in the middle school. After he earned his M.S. and Ed.D. in school administration, he became an assistant principal at the middle school and then came to the school as principal in 1988. He discussed his experience and explained how he came to be involved in the mathematics reform at the school:

I was always interested in the philosophy of learning in actual teaching/learning process, and approaching it by what goes on in the classroom, what goes on with the students. One thing I did that really made an impression on me was to run summer school. I ran the program for five years. That really opened my eyes. It might be the thing that told me that something was wrong, because we had so many kids who were struggling, and there were too many of them ... probably 95% of those were African American kids. Some African American leaders and academics these days say that African American kids have to learn to operate within the system. That's true to some extent because I can't predict that the system is going to change or to what extent. The thing is, the system is meant to favor families who have money. It's meant to favor ... and actually perpetuate, I think, the gaps that exist in intellectualism and achievement ... The present system favors those kids who already have advantages, and it will favor African Americans who have advantages too because there ae some and it will continue to favor them. I think that's what still drives my interests. In my opinion, there is something intrinsically, internally wrong with the way we approach educating children. There is something wrong with the basic structure and the culture in the system. I still believe that. It's what's got me excited about trying to develop new instructional and assessment practices. There is so much research and evidence that we should stop just quizzing and testing kids and ranking them.

Dr. Adams has made it quite evident that his opinions are and what he is going to help change the system, such as extended instructional time for mathematics before school, after school, and Saturday school. His vision is apparently long standing and deep and he seems truly committed to it.

Dr. Kelly was assistant principal at the school when the study was conducted. She has a somewhat different story to tell. She finished her degree in political science early on, stopped to have a family, and came back to teaching in mid-career, after her family moved to this region. She completed her M.S. (1989) and Ph.D. (1995) degrees in educational administration. She has strong opinions herself, though they spring from different origins than those of Dr. Adams.

I was originally put into this job to be in charge of instructional improvement, which is typically unlike what most assistant principals are charged to do ... so, for me this administrative position has been a real learning experience because I've had very diverse kinds of learning experiences, which I think does me well professional. My heart though, is in instructional reform ... I am beginning to think that the whole systems needs to be redesigned, realigned, and redrafted around something that mirrors real life. For example, you and I don't stop and think when we stop at a traffic light, 'Oh, this is social studies, this is government at work.' We don't do that in education [either], so I'm much more interested in giving children real world problems within the confines of a school building. My interest in instructional reform, I think comes from my degree in political science and history; and my experience being a female in a southern college, at the University of Virginia, and being raised in the Kentucky era where you believed one person can make a difference and ... it was part of your obligation as a citizen to have social responsibility and accountability for others and help improve others lots in life ... so when I was in college we were registering black voters ... we lost some of our funding from state over that ... I had planned to be a neurosurgeon and took microanatomy classes and was happily cutting up brains and everything until somebody said, 'you need math and you need science more that just biology'. At that point, doors to my career in medicine shut ... so I majored in political science and history and I took constitutional law ... which really sealed the fact that there needed to be something done to improve people's lots in life; and I believed it could be done through education. Also, it seemed like math and science held a key. You and I do not have the same chance of participating in most well paying jobs in this nation unless we can dialogue pretty well in mathematics and science. Now I'm not talking about rocket science jobs, I am talking about my job ... so, basically I think my love for the instructional reform is the importance of science and math; the wanting to make sure that our children have equal chances ... that everyone has the same [instructional] opportunity regardless of gender, race, or religion.

It seems these two administrators, coming from different backgrounds, were heading in the same direction. They had both been sparked by earlier experiences that shaped their ideas about how "things should be" and offered cause for change. They were both ready to do something to evoke change.

In this mode, the principals and a few teachers began to search out and win grants. Dr. Kelly got a small grant that helped fund the first attempt at writing a new mathematics curriculum. It turned out to be a thick binder called Bumbershoots. Many staff members participated in this project. They eventually got a state Venture Capital Grant of $125,000 over five years in 1994, to improve mathematics instruction. They wanted to reconstruct mathematics teaching and learning at the school. They read and researched new theories and methods, consulted with mathematics educators in the district high school and a local university. They also studied the NCTM Standards (1989, 1991, & 1995) thoroughly. They were determined to find a better way to help students learn mathematics. As they read, wrote and talked they became convinced that children were able to tackle more complex mathematical problems that it was ever thought they could. Children were able to solve problems and communicate their solutions to their groups and to the whole class. Students were encouraged to restate in their own words what the questions were asking. They were expected to draw pictures or model their solutions and express their thinking and reasoning in writing. Children were allowed to manipulate objects to enhance understanding. The teaching staff began to realize that it was indeed worth it to listen to how children were thinking about mathematics concepts and how they understood those concepts. This approach of actively listening to the student's voice continues to this day at the school. Teachers attend and conduct workshops and conferences to learn more and improve understanding and methods. They also team-teach and take opportunities to observe one another's teaching methods.

The Teachers

The teachers at the school have a general outline that contains mathematics topics such as length, area, volume, time/money, and chance. They realize that most of the mathematics that takes place in the world involves these numeric and measurement systems. Teachers meet to share ideas and problems they have written in various categories and work towards a "community of learner" in their classrooms, where the teachers create the next step in the learning process based on what the children have expressed they understand from the day before.

Mrs. Paterson and Dr. Rahman are two fourth grade teachers that are highly involved in the reform process. They actively participate in inservice meetings, sit on committees, and read and comment on acticles shared by other staff and administrators. They are both considered "teacher leaders" in the reform effort, and try to help other teachers see the benefits of engaging students in mindful problem solving. They use materials that involve greater student participation.

Mrs. Paterson grew up outside of Washington D.C., and went to school there. She earned her teaching degree there. She married and taught until she "retired" to raise a family. She came back to teaching when her children were both in school and realized things were different. "I could feel a need for it [my teaching] to be different and right about that time math manipulative seemed to be surfacing ..." When Mrs. Paterson came to the school in 1990, the educators at the school were just beginning to try to reform mathematics. She had taken some little workshops conducted by a "passionate teacher" she had met at the local university. She recommended that the administrators at the school contact the university teacher for advice on how to make instructional activities more child-centered. With all of this emphasis on mathematics reform at the school, Mrs. Paterson decided to get her master's degree in Curriculum and Instruction with a specialization in Mathematics.

I'm tired of people saying kids can't do it; that it's a fifth grade thing or a seventh grade thing! They can do it if you just give them faith, and even if they don't understand everything, at least introduce it to the kids ... I think we just can't keep teaching the way we've been teaching. It's not working. Things have to be different. We have to take risks and try ... children have to build up their own understanding. I do a lot less teaching and a lot more sharing with other kids because they learn from each other a lot more. I present with with a situation ... and they learn from each other and they get power from each other.

Mrs. Paterson values what she calls a "concrete, experiential foundation" in mathematics so that children won't be "terrorized by math before sixth grade" as one of her children was.

Dr. Rahman came to the school seven years ago. She taught in another school for ten years, but was encouraged to come to the school by Dr. Kelly, who had met her in the doctoral program in which they were both enrolled. Dr. Kelly was impressed by her obvious intelligence and sensitivity to children. When Dr. Rahman did come to the school, this brought the total number of Ph. D. staff members there to three. She is a person who is passionate about mathematics instruction and teaching youngsters. She taught first grade in her previous assignment and has taken charge of a fourth grade class at the school.

I found it very exciting to be in this building because I've always loved math and been really involved in my other school and trying to improve math instruction. For my master's degree, that was one of the things I worked on, goals to improve math instruction. I think the thing I really like here is the passion and commitment to improve math instruction ... the thing that is different at [name of the school] is the dialogue. At first I didn't think they valued the hands-on, but I see it's sort of a combination. You want to see what children's understanding of what they're thinking really is, and the dialogue is really so powerful. [It is] something I had never done before, really listen to children to see what they are thinking. I think that's really been a big thing. It's not just coming in and doing all of these hands-on activities, but really looking at what the children understand is first. Then pull out the money or counters and do things like that, but first find out where they are ... The piece I have learned now is with the constructivism. Hearing what they are thinking and talking about ... so they learn by talking things through and listening to one another. I didn't know the value [of this] as much as I do now.

These two teachers believe talking to (not at) and listening to the children are valuable teaching tools. They talk about passion and love for teaching and learning. They both feel obligated to put in some genuine time and work with children toward shared meaning and understanding.

The Students

Our original question involved how African American students' attitudes and perceptions might be influenced due to the learning climate at the school. In the search for possible explanations, we explored the above question through observations and discussions with two fourth grade African American students (one boy and one girl). We observed each of the two classrooms systematically (10 times in each) for one whole year. We conducted 10 one-on-one interviews with each student during the course of our study. What follows are descriptions of the students in their particular settings, their activities, and their thoughts.

Ciara is a fourth grader in Dr. Rahman's class. She is very athletic and full of smiles and enthusiasm. She informed us that she loves to play basketball and is an excellent volleyball player. Her teacher describes her as an all around good student. She particularly loves math. "When Dr. Rahman says it is time for math, I get happy. People yell at me." She tackles problems with relish and is a leader during mathematics discussions and activities. These are some thoughts she shared with us:

Ciara: I like to collect bugs. I don't kill them. I have to keep them away from my Mother! I have 10 teddy bears. My favorite teachers are Mr. Singerman [The third grade teacher], Dr. Rahman, and Ms. Price. [The kindergarten teacher]. They are very nice and kind. I don't like a teacher that is always nice like if people break something and she say 'that's okay' and the class said they were mad and she said 'don't be upset.' I like a teacher that is both ways, nice and even. They are fair and they taught things that were interesting.

Clearly, fairness is a very important thing in Ciara's life. It is very high on her list of how she judges people and events. She seems to include herself in her moral code. She does not even "like to kill bugs". It seems that she believes everyone should "behave decently" in a situation. She strives to do her best and is conscious of her role as a student.

Malik is a fourth grader in Mrs. Paterson's class. He is also an athlete. He told us he was good at "football, basketball, and kinda [good at] hockey, golf, bowling, baseball, rugby, and lacrosse." He also likes mythology and history, especially African American History. "I liked when we learned about Harriet Tubman and the Undersground Railroad, and the escape of the slaves." His teacher describes him as a "remarkable kid. He does a really good job of explaining his thinking. He can show you how to solve a problem visually or in words." These were some thoughts he shared with us as he described himself:

Malik: I think I'm funny ... and creative. I am really funny at recess and lunchtime ... [pause] and I'm a person who tries to help other people who have problems. Mr. Singerman [third grade teacher] is one of my favorite teachers. He got me into computers. I like science. I've learned about rocks. Pumice, it floats, and marble, that's what they use to make statues. I'm good at math ... I would rank myself a 10. I go back and correct my work. My grade for math was an A. This year was easy ... the work she [Mrs. Paterson] gives me is not that hard. I'm caught up with my work, and I don't get into that much trouble.

Malik is a serious, hardworking young man. He thinks deeply for someone of his tender years, and already feels some responsibility for others. He is the older of two brothers, in a nuclear family of four. He demonstates a position of moral decency that he has internalized and his attitude reflects that. He is very confident and does not like to let things go until they are done to his satisfaction.

The Parents

When we asked the parents of the two students to express their thoughts on education, their hopes, dreams, and fears for the children we received a variety of answers. We observed that one theme ran through all of their responses: the fear of bring misunderstood, mainly in the sense of being thought less of by people who are not black. This sentiment is identified as "devaluing" by Steele (1992). In the words of Ciara's mother, Mrs. Thomas, "Our lifestyle, I feel, is very much different from a white person's, because we have struggled, we have really struggled." She recalled an experience she had with her nephew in another school system.

... the teacher called him stupid and another teacher called him a screwball. He was basically giving up, but he didn't want to ... he couldn't pick up the math. His mother told him if he didn't get it this year [he had already failed] he would just have to take his GED. He said 'Mommy, please take me out. I am trying, but I just need help.'

Next was the testimony of Malik's mother, Mrs. Williams. She is a teacher for a large metropolitan school district. She is a bright, energetic woman, who is able to express her great enthusiasum for the education of her own two children; as well as the children she is teaching. She seems to understand what the school is trying to do. She appreciates all of the opportunities her son has been given to experience mathematics learning at the school.

Math is his favorite subject and he likes literature. He has a great mythology collection ... he likes his stuff! I have been trying to read ... to keep up with him so we can converse about it!... I hope he has a successful life. I hope he gets married to a wonderful person. I hope he can become a CEO or president of a company, or better, have a business of his own, not working for anybody. I want him to go to college and learn accounting. So he knows the ins and outs of business. I don't want him to rely on other people to tell him what's going on. I want him to be able to know for himself. Mrs. Williams like many other middle-class African American, does not trust a "traditional" financial advisor to take care of her business, nor does she want her son to do so. She would prefer that he learns his own way and make his own judgments.

Physical Structure of the Classrooms and the Social Norms

Mrs. Paterson's classroom is very colorful. The eyes are pulled from one thing to another. She has a large bulletin board in the back of the room that is decorated with over a dozen book jackets with the words, "wonderful exciting books". She has two ant farms (with real, live ants), several big dead beetles, a plastic worm, and many books on science. On every window shade is a poster extolling the virtues of reading. There is a display devoted to the stock market. Her class is following three stocks in particular. There is a blue desk with two red and blue chairs, set up, fully equipped with a beginner's chess board and pieces. The room is friendly, interesting, inviting, and relaxing.

During a mathematics lesson we observed, students were called together on the rug in the front of the classroom where Mrs. Paterson joined them. Previously they had been gathering data from the dictionary and other places about the frequency of letters in common words. They had each been assigned to do a letter count on ten words. They reported out their data to see if there were any patterns, and if so what might be some of the reasons for them. Mrs. Paterson was recording their reporting on a white marking board on the floor near the children. After about half of the class had reported, students noticed that the letters with the highest frequency were vowels and the one with the very highest frequency was the letter "a", closely followed by the letter "e". After all the tallies were in, the students began a discussion about the construction of written language and the function of vowels and consonants. The students did most of the talking and made most of the discoveries on their own. It was a student-centered lesson.

The first thing you see when you walk into Dr. Rahman's room is the piano. It sits directly across from the door. She plays it after lunch sometimes as a little treat. She may let a student who wants to try something have a turn. There is a big rocking chair in the room for story telling. Near the chair is the reading corner. There are lots of big colorful pillows for students to relax on. The students also meet here to have discussions and lessons. Students' works are everywhere on bulletin boards around the room.

During a mathematics lesson we observed, students were trying to figure out how much ribbon they should buy to frame a picture they were giving to the art teacher. Students were gathered around the overhead, clipboards in hand, working intently. The overhead was on a low table, almost in the middle of the room, so Dr. Rahman could sit on a student chair and facilitate dicussion better. The way the students were gathered around, and in the darkened room, they had the appearance of being gathered around "the campfire" light of the overhead. There was a feeling of shared purpose and interest. Dr. Rahman said, "How do we know what to do first?" From there, students offered suggestions which worked out to mean that they should measure all four sides and add them up. Then they had to figure out the price of the ribbon. The students were expected to restate in their own words, what the question was or how they were understanding the problem. They were expected to model (draw pictures), and communicate the solutions with the classroom community. While one student was defending his/her solution the obligation of the class community was to listen very carefully and ask questions for claification.

Dr. Rahman asked her students what they might do if they wanted to add a second border of ribbons inside of the frame. For this part of the problem, they were asked to work at their tables (groups of four) to solve the problem, including drawing and explaining.

During the small group work, students were responsible for their own learning. If they had differences of opinion, it was their obligation to come up with a consensus. Also, they were expected to offer support if someone in their group needed help. Dr. Rahman communicated with her students that the only time they could ask the teacher for help was if the whole group had the same problem. Students were completely in control of their small group problem solving. They seemed perfectly capable of solving the problem with very little assistance from Dr. Rahman.

Mathematics Instruction

What follows are two lessons that demonstrate the degree to which methods in these two classrooms were designed to encourage students' dialogue and inquiry-based mathematics instruction. The first lesson took place in Dr. Rahman's classroom. She was continuing a series of lessons about fractions that had begun earlier. The students had reacquainted themselves with the notion that things can be divided into portions called halves, thirds, fourths, eighths, etc. They had previously discussed why these divided wholes were called by their respective names. They had also worked with renaming fractions, as 2/4 equal 1/2 and 2/6 equal 1/3; and they had experience drawing these different fractional configurations and learned to call them euivalent fractions. At this point Dr. Rahman determined that the students were ready to tackle the idea of combining fractions with different denominators.

T: [She presented a situation, writing the information on the overhead as she explained.] Today, I have a problem that I hope you can help me solve. I had my family over for pizza last night and I ordered a lot and they ate a lot! I ordered the rectangle kind because they are bigger. Each of the pizzas had been cut into 12 slices. [She drew 3 rectangles, cut them each into 12 equal pieces and labeled each one.] When we were done this is what I had left: 1/6 veggie, 1/3 cheese, 1/4 pepperoni. [She wrote the fraction for each under the drawings, respectively.] Tonight, I'm having some of my friends over to work on a project for our street club. I told them I had some leftover pizza that we could eat and they said that would be fine. If I invite three friends and each of us has four pieces of pizza will I have enough or will I have to order more? If I do order more, how much should I order? [The students got into their groups and busily began to draw the portions of pizza as they had been described. The classroom was noisy and students were talking to one another. The teacher was circulating a round for observation, listening to student discussion, and making notes.]

S1: Each pizza was 12 slices right?

S2: Well, look 1/12 is the same as 1/6, look, look! [The student highlights the portion that is 1/6 and drew a line to show others the new configuration.]

S3: Okay, that's right, 1/6 is the same as 2/12. So, the veggie is 2/12 ... can we make everything into twelfths? What about the cheese pizza?

S2: Well, look ... Wait a minute ... we're got 1/3, right? If you take 12 and divide it into three parts, it gives you ... four pieces in each group, so the cheese has got four pieces! [The student was excited as he explained.] Four pieces makes 4/12! Look! Four out of twelve!

S4: Okay, well, that means you got the 2/12 and the 4/12 ... that's 6/12, right?

S1: Yeah, yeah ... that's right! [He referred to the teacher's drawing to confirm S4's solution.] So that's the cheese and the veggie ... we have to do the pepperoni.

S3: Watch this ... You take the twelve for the pepperoni ... when you cut the whole thing into fourths ... you cut it this way ... each section is ... okay, three, yeah, three pieces. Yeah, four times three is twelve, so that's right! Look!

S1: That's the pepperoni, right?

S2: Yeah! So the three pieces is 3/12 ... so ...

S4: I know! 2/12 plus 4/12 ... and 3/12 ... that's 9/12! It's 9/12!

S3: Okay, yeah, that's right ...

S1: So, how many does she need? What'd she say? How many people?

The students worked diligently to figure out how much pizza was actually left and if there would be enough for the new meal. After about 30 minutes, many groups began to come to the conclusion that there would not be enough pizza. Dr. Rahman started the whole class discussion.

T: We will have to stop our group work now, so we can share our results. Who can tell us how much pizza we need? Is there a group ready to explain? [Many groups were eager to share their solutions. One group of four students was selected and brought their papers up to the front to show to the class. One student began.]

S5: First we thought how could they be alike, you know, how to make the fractions alike. They all have to be the same if you're going to add them up so we thought about twelfths. We thought they could all be twelfths, so we found out that the 1/6 of the veggie pizza is the same as 2/12, that was easy. So that was the first part.

S6: Yeah. Then we found out that if you take the cheese pizza and make it so some of it is twelfths ... you can cut it like this and you can make the whole thing into thirds ... so like, all the sections have four pieces. So there are four pieces left, that's 1/3 of the pizza, see? Each part is 4 pieces so, that makes four pieces of cheese pizza that are left.

S6: Okay, yeah, the cheese is four pieces ... and then we had to find the pepperoni. So we thought of the same thing, 'cause you can do a lot of stuff to get twelve ... like three times four is twelve and four times three is twelve ... so we cut the pepperoni the other way, and you can see four sections like this ... so the pepperoni is three pieces.

S7: Yeah! And we just put them all together. We just added up the twelfths. So that's 2 plus 4 plus 3, so all together that's ... nine pieces! You've got nine pieces of pizza for everybody!

T: I see. Are there any questions for our group presentation? [No response.] I think I see what you did! Wow! I think I'm impressed! You did some hard thinking! Do we agree? [There was much nodding of heads, although not from everyone. Some were still working it out, looking at their own papers and changing their thinking.] Thank you, boys and girls for your great work! Wonderful job! [The student group proudly took their seats.]

At this point, Dr. Rahman saw this was important to make sure all the students understood how the group got the total of 9/12, so she stopped here to restate the group's results.

T: Let's look at this again and check it out. [She referred to the pictures of pizzas on the overhead, using other colors to show the larger divisions, and renaming the fractions as she explained. There was discussion and writing among the students in their groups.] Do we all see something about this? [Many more students seemed to agree.] Great! Now, let's look at the next part. Does anyone know how much pizza I need for tonight?" [Many hands went up.]

S: [many voices.] Sixteen, Sixteen!

T: How do you know that? [She called on a student.]

S8: If there are three people plus you, that's four! And four times four is sixteen! You each get four pieces, that's sixteen!

T: Do you agree? [Referring to the class]

S: Yes! Yes! [Many voices]

T: Well, I think I agree, too! We have to stop soon so let's see if we can finish this up quickly ... I observed that one group presented their results and they concluded that I had 9 slices of pizza left over. Will this be enough?"

S: No! No! [Many voices]

T: How many more do I need?

S: Seven! [Many voices]

S11: You have to buy another pizza!

S12: Maybe you could get a small!

T: Yes, I guess I will! Or maybe I will just eat less ... I need to lose weight! [Everyone laughed.]

In the above episode, Dr. Rahman encouraged students' dialogue concerning mathematical representations. Students had some control over classroom instruction and their own learning. She was not the sole authority evaluating and controlling student solutions. She saw her role as a facilitator and guide.

The next lesson took place in Mrs. Paterson's room. It was a lesson about area and perimeter. Again, the students had some prior experience with these two topics. On this day, Mrs. Paterson offered a situation to help students explore the notion that spaces with the same area can have different perimeters.

T: Here's a picture of my dog Lester! I just got him from my son! Isn't he cute! [She showed students a photograph of a basset hound puppy.] I don't want him to run around all over my yard. You know how dogs are! He will ruin my flower beds and my vegetable garden! So, my husband and I have to build a smaller exercise place for him inside the yard so he can run around and not mess up everything. The problem is I don't exactly know how I want it to be. I know I can give him 36 square feet of space, that's all the room I can give him so everything else still fits. I would like for the area to have a large perimeter so he can run a good bit, but I'm not sure how to make it look. I would like for you to get into your groups and find all the possible ways you can make a space with an area that is 36 square feet. I also want you to find the perimeter of each space and see if there are any differences. Then, maybe I can decide how I want to make Lester's exercise place look and where to put it in the yard. Here is a sheet of grid paper for each of you. Discuss this with your group members and help each other find all the different areas and perimeters. [Students get into their groups and begin to work.]

S1: Different areas? I thought it was the same area ... didn't she say 36 square feet?

S2: Yeah, but I think you can do it different ways! Remember multiplication! You can get 36 by six times six ... look! [She drew a square on the grid paper.] Six up and six across, that's 36!

S1: Oh, yeah ...

S3: And there is three times twelve ... so you can make it look like this [He drew a rectangle with these dimensions.] That's 36. [Other students in the group quickly began to make the figures mentioned by their group members and tried to find other areas of 36 square feed.]

S4: Here's four by nine! [He drew the rectangle.]

S2: Don't we have to find the perimeters, too?

S1: Yeah, but ... don't mix me up! I want to find all the areas first!

S2: Okay, we'll find all the areas. Look, there's six by six, three by twelve, and four by nine, and ...

S3: Does five times anything make 36?

S2: I don't think so ...

S1: No, no. Nothing by five is 36.

S4: Look! You can do two by eighteen! It's long and skinny!

S2: Okay so we've got two by eighteen, three by twelve, four by nine, nothing by 5 five, six by six ... seven?

S1: Nope!

S2: Eight by something?

S3: I don't think so ... eight by four is ... 32 ...

S1: Nope!

S2: Nine by ...

S4: We got that one. It's nine times four ... but you can put it the other way ...

S2: So is that all? [They agreed that that's all they can think of.] Now let's do perimeters. [They counted around each area they found.]

S3: Look! Six by six is ... 24!

S1: Yeah ... six, plus six, plus six ... 24!

S4: Nine by four is ... two fours is eight ... and two nines is eighteen ... so that's ... 26! It's more than six by six!

S1: Two by eighteen is ... 40! Wow! he can run far!

S2: But he can't run in a circle. Maybe he wants to run in a circle sometimes.

S3: Three by twelve is 30.

S2: Is that all? [They all began to check]

S1: I think so ...

S4: Yeah ... yeah, that's all we got.

S2: Maybe she wants the long skinny one or maybe she wants the square one.

S3: Which one takes up the most room?

S1: Where's her other stuff? The garden and stuff?

Students all around the room began to find the combinations as this group had. Mrs. Patterson sees that most groups found areas and perimeters. She called all the students to sit on the carpet in a semi-circle for discussion.

T: Well, what did you find out?

S: [Several voices] Forty's the biggest! Seventy! Seventy-four! Forty is he biggest! No, It's not!

T: Okay, okay! We'll see! Why don't you guys come up and share what you found out about the areas. [She pointed to a group of four students.]

S5: We found all the areas! There's two by eighteen, three by twelve,... [The student pointed to each area as she named them.]

S6: Yeah! And four by nine ... and six by six.

S7: Plus, you could have one by 36, couldn't you? [He looked at Mrs. Patterson.]

S: [Several voices] Yes! Yes! We got that!

T: Sure you could! That's great! Any more? [Students checked their papers. They agreed there were no more.] Okay, group! That was exceptional! [The group sat down.] Could we get another group to share what they found out about the perimeters? [She pointed to another group.]

S8: The biggest perimeter is one foot by 36 feet! If you count all the way around it's 74! But it's real long and skinny.

S9: Okay. If you got six by six, the perimeter is 24. We just added it up.

S10: Two by eighteen is 40 and three by twelve is 30 ... and four by nine is 26.

S11: The second longest is the two by eighteen one. It comes up to 40. That's it.

T: Is that all? Does anyone have anything to add? My Goodness! Did you realize there could be all those different perimeters from the same area? There are five different possibilities! You've given me a lot to think about! I'm going to take your results home and tell my husband. I'll let you know what we decide!

This lesson offered students rich experiences that deepened their understanding concerning the relationship between area and perimeter. Students came to realize that an area can stay the same while the perimeter changes. Students gained confidence in their abilities to think and solve contextual problems.

There was evidence that lessons in these two classrooms were created and presented to allow for student thinking and reasoning. The two teachers did not follow any particular mathematics textbook. They worked hard to create and present mathematical situations so that students could make relevant sense of the activities.

Dialogue with Students

In one after school session we gathered Ciara, Malik and two other students, Brittany and Kent, from another school. Brittany and Kent experienced mathematics learning in a mostly conventional mathematics classroom, where the main focus of mathematical activity was following the textbook and doing worksheets. Their teacher would give the directions on how to find answers to a given worksheet assignment. Then, students would follow the same procedure presented by the teacher for answering further problems on the worksheet.

We wanted to know whether or not the school mathematics program at the school that promulgates constructivism as a viable theory of learning, affects mathematics problem solving of participating students. We anticipated that putting these four students together (two from the school and two from another school) within the context of mathematics problem solving would provide us with opportunities to observe their social interaction and see them in juxtaposition to one another. We started by posing a problem involving fractions. We gave the children this problem:

I have a 12/slice pizza. Malik can have 2/12. Brittany can have 1/4. Ciara can have 1/6. Kent can have 1/3. Does everyone get an equal share? Who gets the most? The least? Is there any left?

We told the students they could solve the problem anyway they were comfortable. However, they were expected to explain the solution to us (in the below conversation, R stands for reseacher). Malik and Ciara started drawing pictures. Brittany started making a table. Kent was quiet and was thinking. After awhile, Malik and Ciara began to talk with one another.

Malik: Okay, here's my pizza, this is my part, 2/12. That's not much [slicing the pizza into 12 pieces and covering 2/12 of that], Brittany gets 1/4 ...

Ciara: There are no fourths on this pizza, just twelfths [Referring to Malik's drawing].

Malik: Well, we could do this [he started with his 12 slices and then paired them with different colors] I know 2/12 is the same as 1/6, so we color this and that's your part [Referring to Ciara's 1/6].

Ciara: We get the same. I think they get more ... [Referring to Brittany and Kent].

Brittany: I don't get it ... [She gave up her chart and listened to Ciara's and Malik's dialogue.]

Ciara: Okay, here is a 12 slice pizza. [She started all over again making a circle with 12 slices.] You can have 1/4 of this 12 slices. [She divided 12 pieces into four groups of three and colored each fourth with different colors.] so, you have three pieces. Now, did you get it?

At this time Kent was carefully listening to the dialogue but could not figure out his pieces of pizza.

R: Can someone help Kent to figure out his pieces of the pizza?

Brittany: Yeah, [She started with 12 slices and then grouped them into three equal piles and colored them.] Well, I learned from Ciara that mine was three pieces out of 12. I did the same thing for Kent and he gets fours pieces, which is one piece more than what I have.

R: Does it make sense to you Kent?

Kent: Yeah, with picture it is easier.

R: Can someone tell me if there are any slices left for me?

Students: [Almost all together] Yeah, there would be one slice left for you.

Kent: It is not much.

This observation and others conducted during the study made it clear to us the importance of students' interactions in mathematical situations. Because of their classroom experiences, it was natural for Ciara and Milak to negotiate mathematical meaning with one another. Kent and Brittany experienced a different classroom culture. At first, they were hesitant to talk with their group as if it was forbidden to talk. Then Brittany broke the silence by saying "I don't get it". She was indirectly communicating with her group that she needed their support. Kent, however, was still uncertain about what was expected of him or how he needed to interact with his peers. Because of the drastic difference between social norms of his regular classroom and the current small group culture, he needed more experience. Another plausible explanation for lack of active participation for these two students might be their limited knowledge about fractions.

The Development of Students' Mathematical Dispositions

We present two short vignettes to demonstrate the attitudes and mathematical dispositions of Ciara and Malik. We intend to communicate something about their character and circumstances as each of them lives out a "typical" school experience. Then we discuss how the culture and circumstances of the school, as reflected through its administrators and teachers, seems to have influenced the students' beliefs and attitudes toward mathematics.

Ciara walked down the hall with her friend Tiffany and the primary researcher. The researcher wanted to meet her again and get to see her on a regular school day. The morning bell had just rung and students were hurrying to class. She was talking to Tiffany and amiably allowed the researcher to listen in:

Ciara: Today we have art and P.E.! Plus we get to finish that probability stuff we started with Dr. Adams [the principal] yesterday. It was fun.

Tiffany: I don't think it's much fun.

Ciara: Well, listen, Dr. Adams is always pretending he doesn't know the answer and acting weird about it and stuff. It's pretty funny. Like yesterday when he came in and asked what the probability of two people having the same head size was. Then he got a tape measure and started measuring Jeremy's head. Dr. Adams sure has a big head! He says it's so it can hold more brains!

Tiffany: Yeah, that was pretty funny! Maybe he'll do something funny today, too.

One of the researchers made an arrangement with Ciara to meet with her and talk further. They met a lunch time. The researcher asked her what had gone on in her mathematics class.

R: What did you find out about probability?

Ciara: We measured everyone's head in our class. Out of 24 people, two sets of two had the same head size. That means in our class, there were four out of 24 chances of people having the same head size. Of course, that's only our class. It could be different in another class. We would have to measure some more.

R: What about Dr. Adam's head?

Ciara: Oh, it was bigger than everyone's, including the teacher's. But that's because of his brains, remember? [laughing] You wanna hear some more about probability? Listen, it's like you're doing chances, you have two dice and you roll. What are your chances of getting a two or a six?

The researcher repeated Ciara's question, trying to tune into what she was saying.

R: What are the chances of getting a two or a six from a pair of dice?

Ciara: Look! You can only get a 2 one way, one plus one.

R: How about six? How many chances are there to get total of six?

Ciara: Well, [She pauses] There's one plus five, that's one way. [counts on fingers] Then ... two plus four, three plus three, five plus one, four plus two,... I think there are five ways, Yes! Five ways!

R: Wow! What else do you know?

Ciara: Sometimes we do percents ... say the regular price is $30.00. So, you save 10%, that's $3.00. So, then the sale price would be $27.00, and that's it!

R: Now, say you have a big screen TV that costs $2000.00. What's 10% of $2000.00? Can you figure it out?

Ciara: No ...

R: Well, what's 10% of $200.00?

Ciara: $20.00.

R: And 10% of $2000.00?

Ciara: $200.00! Then I subtract it from that and I come up with ... I think it's $1800.00. I like math better than anything in the world because I'm good at it ... From 1 to 10, I would give myself a 7.

R: Why a 7?

Ciara: Oh, because Ben and Eileen know everything. I know most stuff but they're better than me.

Ciara seemed to have a good grasp on some important mathematics. She seemed excited and interested and was willing to go on without the researcher's support once she saw the pattern in the percent problem. She showed great confidence in demonstrating her mathematics understanding. We wondered what could be contributing to these attributes in her, and we could not help but following them back to the experiences in mathematics class and her relationship with her teacher and her classmates. We asked Ciara to describe a "regular" mathematics activity in her class.

Ciara: We spend most of our time in class working in groups. I don't think anybody is the leader at the table. We take turns and stuff. I listen to the other students, what they have to say first, and then I say what I have to say. Like, sometimes they get it right and sometimes they get it right and sometimes I get it right.

It seemed that the social norms established by the teacher and students were pivotal for creating learning opportunities for all students. Sharing and respecting one another are certainly considerations for Cieara when she talked about interactions. Even when she talks about individual problem-solving, there is an element that shows she is working by a set of norms respected and accepted by the members of the classroom community.

Ciara: When we work by ourselves ... first we have to think it over and put down an answer. You can't just write anything. You have to think. Then you have to go up there [in front of the class] and tell it to others how you got it.

She expressed some norms on the conditions necessary to ask for help. One must really try hard before asking. Ciara seemed to accept that norm as a fair expectation.

We observed Malik in Mrs. Paterson's room. He and Brandon (his partner) were at the computers.

Brandon: Wait, wait! You are going too fast! You're gonna miss it!

Malik: No, I'm not! Just give me a minute ... there! Now you write it down.

The boys had found the city they were keeping track of. They were monitoring the amount of sunlight that Rio de Janeiro was receiving each day. We went over to say good morning to Mrs. Paterson. She bragged happily about what her class was doing.

Paterson: There are students monitoring the amount of sunlight in cities all over the world. They're checking put the idea that in the Northern Hemisphere, as we approach spring, the Southern Hemisphere is approaching winter. It's really amazing to watch the kids realize this by looking at the daylight graphs. They see that the daylight is increasing in places like Bangor, Maine, and Amarillo, Texas, which is what we expect. But what's really amazing to them is that the days are also getting shorter in Rio and Sydney! No matter how much we say that to them, it's still surprising for them to see. I love to have them talk about their ideas on this after we've done it for a few weeks.

Mrs. Paterson is a creative teacher. She has many ways to generate interesting experiences for students that encourage them to create and manipulate useful data that has relevance in life. In Mrs. Paterson's class, we observed the pattern of classroom social interactions. First, students worked on the problems individually. Second, students worked in their small groups. And third, they were provided with opportunities to share their multiple solutions with the class. The problem-centered classroom was observed ten times throughout the year in Mrs. Paterson's class. We asked Malik about his reflections on one of his class activities.

Malik: The activities are called 'pay the principal' and 'how long have you been in school?' We do them in the morning, along with other stuff. Well, we don't really pay him [principal]. On the first day of school, we started out with $20,000.00 and you keep withdrawing a dollar a day ... and then you have to find the interest every month. You figure out how much you put into his account. And, 'How long have you been in the school?'... Well, we figure out the real amount of time we've been at school. [Each day] it's six hours and 10 extra minutes. We figure out everyday how long we've been in school. We added what we had to the day before. We learned about fractions ... like total hours and total minutes. Because of the extra ten minutes we had to divide the hours into minutes all the time. So we talked about fractions a lot. We add, subtract, and multipy them.

Malik was able to extend his understanding of the relationship between time and money to fractions. He was able to explain his ideas with confidence. In both classrooms, it became clear to us that mutual respect and shared meaning created by students and teachers were crucial for learning mathematics.

As we discussed earlier, social norms of the classrooms significantly impacted students mathematical dispositions. In addition, the principals and the teachers' passion seemed to have positive impact on students' attitudes and beliefs towards mathematics. Many of the statements expressed by the principals and the teachers at the school reflect evidence of passion about teaching and learning.

Discussion

Educators, politicians, and parents all express great concern about low levels of mathematics performance by American students; especially African American students, who generally perform lower than their white counterparts. As D'Ambrosio (2001) asserted:

Children of color as a group have not realized the same level of mathematical success as European American students in our classrooms and often under represented in higher-level mathematics courses and professions requiring significant mathematical competence ... Many of these children simply do not realize that they are mathematically capable and they do in fact possess a long and rich mathematical heritage. (P. 310)

Bigelow, Harvey, Karp, & Miller (2001) observed, it is possible that these poor performances are due in part to lack of meaningful mathematical experiences that many students, particularly African American students have. The NCTM, Principles and Standards for School Mathematics (200b), emphasizes mathematical dispositions as one of the major attributes for all students to possess. At this school radical attempts are being made to implement the spirit of NCTM Standards with the goal of "mathematics for all children."

The findings of the study suggest that the culture and structure of the school positively influenced students' beliefs and attitudes towards mathematical learning. In addition, the teachers and the administrators' passion for reforming mathematics created a synergistic transforming learning community which provided students with opportunities to gain deeper mathematical dispositions.

One of the main features of inquiry-based mathematics is understanding students' understanding via active listening (Burns, 1992; Romberg, Shafer, & Webb, 2000; Wheately & Reynolds, 1999). Mrs. Paterson and Dr. Rahman are convinced that social interaction and mathematical dialogue are crucial for conceptual understanding of mathematics. They understand the meaning of the relationship between the two. They know that they must organize their instructional methods around what they genuinely observe from their students.

The importance of constructivist teaching and the centrality of students' voices provide teachers with opportunities to develop and modify their instruction. One of our concerns is continued educational opportunities for these students once they leave the school and transition to more traditional schools. Have administrators in the new settings taken into consideration the learning experiences of these students? Will any adjustment provisions be made? Are there or should there be any transition strategies employed to sustain learning for these students? How do teachers with constructivist epistemology, communicate effectively with teachers from more conventional backgrounds? How do administrators accomplish this same task according to their own ideologies, for the good of all students? These are important questions that must be answered if we are to continue to have a chance to educate tomorrow's American children.

As the gap between 'haves' and 'have nots' grow in our stratified society and as the rate of poverty among school-aged youth increases, we must take steps to address the quality of education in the schools that serve students from low-income communities ... we cannot and will not have excellent mathematics education in the nation unless and until we have quality mathematics education in every classroom. (NCTM, 2001b, p. 22)

We end our discussion with an anecdote that is hopeful. One day after school, as the children were leaving for home, we overhead the comments of a group of second graders, including a white boy, a black boy, and a black girl. They asked the assistant principal, Dr. Kelly, if she was coming into their room for a mathematics lesson the next day. She thought for a moment and said: "I think I will be able to fit in." "Yes!" was the unanimous reply by the students, who were obviously pleased at the possibility. So it seems to us that here we have a dream come true: everyone being pleased at the prospect of their white female, non-mathematics oriented, but converted, socially conscious, constructivist teacher/administrator coming in to experience an ordinary mathematics lesson with a group of enthusiastic, multicultural, mixed class, and gender, American children. Perhaps there is hope after all.

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Belvia K. Martin, Shaker Heights School District

Roland G. Pourdavood, Cleveland State University (CSU)

Nicole Carignan, University of Quebec at Montreal (UQAM)

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Author: | Carignan, Nicole |
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Publication: | Focus on Learning Problems in Mathematics |

Date: | Jan 1, 2005 |

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