Culture, social interaction, and mathematics learning.

Abstract

The research report investigates connections between five 3rd garde students (one African American boy, one African American girl, one Caucasian girl, one Asian boy, and one Middle Eastern boy) belief, attitudes, and practices about mathematics and their parents' expectations. In addition, the study examines the relationships between the social interaction of these five students on mathematics learning and their mathematical dispositions. In search of a viable understanding of the above situations, constructivist inquiry was used. In this paper we discuss students' attitudes and beliefs towards themselves and towards mathematics, parents' expectations, classroom social norms and sociomathematical norms, and the teacher's role for establishing classroom culture.

Culture, Social Interaction, and Mathematics Learning***

In any culture, people share language, place, traditions, and ways or organizing, interpreting, conceptualizing, and giving meaning to their physical and social world. National Council of Teachers of Mathematics (NCTM, 2001a) acknowledges the importance of culturally relevant mathematics by stating that "social condition, social tradition or culture, and social goals influence student learning ... However, a student's social traditions or culture may either coincide or conflict with classroom norms for student activity, student conduct, and student-teacher interactions" (p. 7). In the case of conflict between a student's culture and classroom social norms, the student may or may not accept the classroom social norms and may or may not be able to adapt to the school environment. Therefore, that conflict may impact the student's learning mathematics. On the other hand, a sensitive teacher may observe the student's cultural patterns and develop activities that accommodate the student's learning. In this type of situation, the student and the teacher come to a mutual understanding and agreement. NCTM (2000a, 2000b, 2000c; 2001a, 2001b, 2001c) suggests recognizing and valuing students' cultural heritage significantly influences students' mathematical learning.

In this study, we use social interaction to mean norms and values negotiated and established implicitly and/or explicitly by the members of a local community (i.e., teacher, students, and parents). In this sense, understanding students' backgrounds is crucial for mathematics teaching and learning. As D'Ambrosio (2001) observed, "An important component of mathematics education today should be to reaffirm, and in some instances to restore, the cultural dignity of children" (p. 308).

There is a body of research focusing on the notion of ethnomathematics. According to this perspective, contributions made by people from various non-European cultures (i.e. Arabs, Asians, Hispanics, African Americans, etc.) for mathematics advancement have not been given the attention they deserve (Ascher, 1994; Bishop, 1991; D'Ambrosio, 2001; Frankenstein, 1995; Nunes, 1992; Orey & Rosa, 2001; Secada, 1992; Zaslavsky, 1973). Zaslavsky (1973) asserted that the study of ethnomathematics would benefit all students who "learn to respect and appreciate the contributions of peoples in all parts of the world" (p. 309).

This research study focuses on five 3rd grade students' (one African American boy, on African American girl, one Caucasian girl, one Asian boy, and one Middle Eastern boy) beliefs and practices towards mathematics learning as they interacted with each other in problem solving, reasoning, communication, and mathematical representations. In addition, the study explored their parents' beliefs and values about education and attempted to examine the relationship between parents' expectations of their children's mathematics education and students' mathematics learning.

The five students were selected within a larger classroom community of 28 students. They had been working together from the beginning of the school year and were comfortable expressing their feelings and views among themselves. The selection of this particular group of students was based on their diverse social and cultural backgrounds, their willingness to participate in the study, their parents' interests and support for the study, and their different levels of mathematical understanding.

The researchers were not seeking to generalize the relationship of culture and mathematics learning of these five students, instead, they were searching for a better understanding and interpretation of these five students' cultures and their mathematics learning. The ideas, issues, and situations presented in this study by the participants may help mathematics educators who face similar challenges in their classrooms.

The research questions are: (1) How may students' attitudes and beliefs towards mathematics be influenced by their parents' expectations? And, (2) How may social interactions among pupils influence their mathematical understanding? In this paper, we discuss our theoretical and philosophical considerations, the research design and methodology, the students' attitudes and beliefs towards mathematics and their parents' expectations, and the teacher's role for establishing social norms and sociomathematical norms of the classroom.

Theoretical and Philosophical Considerations

From our perspective, the notion of mathematical learning and understanding involves people's construction, deconstruction, and reconstruction of their knowing through the process of cultural participation, social interaction, and contribution to the local activities of the community. In this sense, mathematics is the study of patterns and relationships where people learn by doing (Cobb & Yackel, 1996; Fleener, 2002; Wheatley & Reynolds, 1999). Construction of knowledge is inherently cultural and experiential (Bauersfeld, 1988; Lave, 1988; Lave & Wenger, 1991; Lerman, 2000; Rogoff, 1990; Saxe, 1991). Thus, we refute the notion of mathematical knowing and understanding as innate (i.e. "people are born with it" or "some people have it and some people don't").

Current research in cognitive science, neural science, and the history of mathematics suggests the limitation of innate mathematics knowledge (Lakoff & Nunez, 2000; Nunez, 1992). Lakoff & Nunez (2000) observed that infants less than two months old can recognize the difference between two objects and one object. This part and only this part of mathematical recognition is innate. But this is limited mathematical knowledge and understanding. What would happen after that, they suggest, is closely connected to the evolution of brain mechanisms, sociocultural interactions, and communication. Since mathematical ideas are embedded in society, mathematics is always emerging. Furthermore, research shows that the formulation of the mathematical and physical laws or our understanding of them is not fixed. Instead, the history of mathematics indicates that people invented and reinvented mathematical and physical laws. Once these inventions are compatible with the present patterns of the physical universe, they become stable, consistent, and globally accepted within the culture and across the cultures (Lakoff & Nunez, 2000).

Research Design and Methodology

The research methodology is grounded in constructivist inquiry (Guba & Lincoln, 1989, 1994; Lincoln & Guba, 1985). This perspective sees the relationship between the knower and the known as reflexive, recursive, and interactive. This view on understanding and interpreting the classroom activities influenced our mode of inquiry. The nature of this interpretation is dialectic and participatory. The research team consisted of three university professors and a classroom teacher.

The observational study was conducted in an elementary school (K-4) located in a Mid-Western city in the United States. This suburban school enrolls 550 K-4 students. Sixty percent of the population is African American, 34% is white, and 6% is other (i.e. Middle Eastern, Asian, Latino, etc.). The five pupils were selected within a third grade classroom. The selection was based on: 1) their cultural diversity, 2) their willingness to participate, 3) their parents' willingness to participate, 4) their parents' educational background, 5) their achievements based on the school's record of their performance, 6) the students' self-representation in terms of their attitudes and beliefs towards mathematics and mathematics learning, and 7) gender consideration (i.e., two girls and three boys). Data collection occurred in two phases, the preliminary phase and the active phase. The preliminary phase went from May 1 to May 15, 2002. The active phase started from mid-May and ended mid-June, 2002.

Data sources included audiotapes of the five students' small group cooperative learning on mathematics problem solving; interviews with parents and students; audiotapes of students' group of interviews; field notes; and school documents (i.e. school bulletin boards, newsletters, students' portfolios, etc.). The purpose of the preliminary phase was to get to know the students and collect data about their attitudes and beliefs towards mathematics learning. The active phase included transcripts of audiotapes of everyday classroom observations and small group interactions (at least two hours each day Monday through Friday), audiotapes of one-on-one interviews with students after each observation, students' group interviews, parents' interviews, and field notes. The researchers (three university teachers and one classroom teacher) met four times during the course of the study (approximately two hours each time) for discussions and exchange of ideas and information.

Data collection and data analysis occurred simultaneously. The researchers communicated with each other systematically during the active phase of the student observations about the study procedures and their understanding/interpretation of the data. Although the data collection and data analysis were interactive, ongoing, and evolving among the researchers, synthesis across the multiple data sources occurred when the data collection was completed. In what follows we provide a summery of initial data (see Table 1) about students' and their parents' attitude, beliefs, and expectations towards mathematics. We changed the five students' names to secure anonymity.

John's Case

John came to the United States with his parents from China a year and a half ago. He was nine years old. His father was a visiting professor at one of the local universities. He liked mathematics. He liked problem solving and multiplication "because when you solve a problem, you feel good about yourself." He did not like addition and subtraction "because they are too easy." John liked small group cooperative learning "because we first do our own and then we tell others how we do it and then if the others agree, it's good and then the others will tell us how they do the problem. So, everybody can say an idea and we can think about it." John had several good friends in his classroom; however, in his home environment, "I only have one friend because I don't know the others." He said he liked coming to the school because of his friends and his teacher. "The teacher cares about you. If children tease you and laugh about you, teacher would stop them." John liked doing his homework assignments without help from his parents. He asked his parents for help only if he did not understand his assignments. We asked John "what are your parents' expectations about your school work?" He said with a strong voice, "to be the best."

We talked with John's father. He earned his Ph.D. in Germany. He taught (during the course of our study) statistics as a visiting professor. He said his son learned English within nine months. He believed a good school is a place where "communication is open among administrators, teachers, and parents, where there is not discrimination and no racism." He thought the school was a very good one. "The school area has been recommended by my chair [the local university]. My child talks about the school at home." We asked John's father about similarities and differences between schools in China and schools in the U.S. from his perspective. Here is what he said:

     In China, classes are larger and in order to help our children we
     should be close to the teacher. It is more demanding in China than
     here. They have a lot of homework. We help our child at home. I
     teach him math about half an hour each day and my wife teaches him
     Chinese about half an hour. What teacher can do is limited. The
     parents should do more.

We asked him about the role of the teacher in the classroom:

     Dr. [the teacher] is very good with newcomers. She communicates
     with us very openly and I like how she helps my son. In Chinese
     education, the role of teacher is the most important for students'
     learning. Students would respect their teachers and believe that
     teachers are always true. Dr. [the teacher] encourages him to get
     to a higher level of thinking. She helps him to reach to his
     potential.

We asked him about homework assignment and his reaction to it:

     In regards to homework assignment, I see two very different
     approaches [between China and the U.S.]. In China, homework is very
     challenging and tough. And they do a lot of homework. Children have
     to work late at night doing their homework. As Chinese we need to
     help our child a lot because I'm concerned about the future. If we
     stay in the United States I have no problem with the current level
     of expectations. My son can follow that easily. But if we go back
     to China, he will be in a big problem. It will be very difficult
     for him to survive educationally. We keep teaching what he should
     know if we were in China such as challenging math problems,
     history, and Chinese. I hope he is the best in the class. I
     encourage him. 'You are very clever, very smart. You should be the
     best.

John's father believed that mathematics learning requires a lot of effort. "It is not innate. You are not born with it. You have to work hard for it." He liked the diverse learning climate in his son's class. "It is good. It gives the students a better understanding of each other. However, I am worried about behavior. I hope my son will have some friends that will behave and they will challenge each other educationally."

Our interviews with John and his father indicated that the student's attitudes and beliefs towards mathematics and mathematics learning were closely related to his cultural and social experiences. For example, John liked mathematics because in his society, values and expectations towards mathematics are very high. In our interviews with John, he clearly stated several times that he was good at mathematics and he could solve any mathematics problems and homework assignments on his own with some effort. He said that he only goes to his parents for help if he needs it. His parents believed that all students could learn mathematics if they put forth enough effort. Furthermore, his father believed it is parents' expectations about and involvement with education that make a difference in a child's attitudes and beliefs towards education. He also stated that parents and teachers have mutual responsibility for educating children. He was very supportive of the teacher.

Hamid's Case

Hamid was a nine year old boy from Jordan. He could speak Arabic and English. He came to the school this year. His father taught Physics as a visiting professor at a local university. He liked math but he did not like very complicated computation. "I like our 'snap shots' problem solving, but I don't like very complicated ... you have to compute a lot ..." He likes small group work "cause if one person can't do all of it ... Sometimes we do extras, that's so we could solve each other's problems in math ... I like it in small group because if it was the whole class, one would have decide one, then another would have decide one, then they would all argue. It is easier in small group. Sometimes I don't understand what they are saying." Due to a language barrier, Hamid was mostly quiet during small group interaction and whole class discussion. We asked Hamid about his friends at the school and at home. (R stands for Researcher):

Hamid: Well, actually, I don't have any friends.

R: You don't have any friends? Why not?

Hamid: Because no one wants to be friends with me.

R: Why?

Hamid: I don't know.

R: Do you go to the playground and want to talk with them but they don't want to talk with you?

Hamid: No, I actually play ... team sport of some kind ... and that's like a team. You have to play with a team ... something like that.

R: Do they play with you as a team?

Hamid: Yeah.

R: In your small group, it seems to me that they like to talk with you. Why do you think that they don't like to talk with you?

Hamid: I didn't say that they didn't want to talk with me. I say that they didn't want to be friends with me.

It seemed socializations and having good friends was Hamid's main concern. We asked Hamid about homework assignments, his parents' expectations about his schooling, and his friends at his home environment.

Hamid: I do my homeworks on my own. My father helps me when I have question. It is very important to my parents that I work hard on my school work. Because they always want me to learn something.

R: Do you have a lot of friends at home?

Hamid: No, I just play with my dad.

R: What are your parents' expectations about your school work?

Hamid: What do expectation mean?

R: Expectation means that they want you to do about your school work and how they want you to do it.

Hamid: It's close to high.

We talked with Hamid's father who volunteered to be one of our participants. He was an atomic physicist. As a visiting professor, he taught in one of the local universities. When he was in school, "I was not the first in the class. I was the second or third in the class." We asked Hamid's father about his definition of a good school. "Maybe this is something relative. There is no absolute standard to tell if it is good or not. But what I hear about this school, it is better than other schools around." We asked him his definition of a "bad" school.

     In those schools people don't care about the kids. In terms of
     education, my opinion is that if school is not good we can help him
     at home, teach him at home. The main concern is communication,
     language, and safe environment at school.

Hamid's father's main concern was language and communication. He wanted to see his child happy and surrounded by good friends. That was Hamid's main goal too. His father was grateful to the school district for providing special teachers to Hamid and other second language students. "They hired teachers especially for second language students." In terms of parent involvement, he said they spend half an hour with him for his homework. "Comparing one hour for homework at home in Jordan, we try to fill up the gap with giving him more homework assignments at home. His mother spends more time with him because I spend more time with my job." He believed one of the main roles of the teacher is to guide students to understand each other. For example, "let everyone every once in a while talk about himself or herself culturally, so that students can accept him and he can accept students. The students cannot do that alone. It is teacher's role to provide the opportunity for them to understand each other."

Like John's father, Hamid's father was concerned about the lack of challenging problems as homework assignments. He also mentioned the safety and security of his child. But his biggest concern was his child's level of communication and cultural exchange. He mentioned that in Jordan, his son was number two in his class in terms of achievement. But because of the language barrier in the U.S., his achievement level decreased. He believed mathematics learning depends on one's experience and practice. By that he meant, "if you are not good at math, you can spend more time studying." Like John's father, Hamid's father believed mathematics knowing is not innate, instead it depends on one's effort and experience with mathematics.

In summary, Hamid likes mathematics. He said it was easier for him to communicate with other students and the teachers in terms of mathematics. To him, finding friends and not being isolated was very important. His parents reacted in a similar way. To his father, the single most important thing was his child's cultural exposure and social interaction with his classmates. He believed the academic part could be learned gradually. He said parents and teachers could help the child, but socialization was fundamental.

Darcy's Case

Darcy was a nine-year-old African American girl. She liked mathematics, particularly geometry "because I like to see how many shapes there are and I like to figure out." She liked problem solving when it was easy but "sometimes when it's too hard, I like being with somebody to help me." She said she does her homework assignments on her own most of the time, and her school work was very important to her parents. "They expect me to do good work and I shouldn't make many mistakes."

Darcy was confident in her ability to do her assignments on her own. She wanted to become either a writer or a teacher. She liked writing stories and reading story books. She also liked science "because I am always curious."

Darcy's mother agreed to participate in our study. She worked at the local electric company as an analyst. Her thoughts on education were:

     When I look at those progress reports, I do notice that there are
     different groups that have lower scoring, and African American is
     one of them. I don't know the reason for this. It's not one reason.
     If there are opportunities for additional sessions with teachers or
     older children that can help them, that's fine. If it's a matter of
     the teacher connecting with the parent to see where there might be
     shortcomings, that's fine. I don't like to make generalities about
     things. There is no one reason why a group is performing lower than
     another group. I don't think anybody should be classified in a
     negative way. If there's a better way to reach them, you try to
     reach them. You can't cast them off and toss them aside.

We asked her about the teacher's role in her child's education:

     The teacher's role is very important to create the kind of climate
     that students have opportunities to intermingle, interact, and
     communicate. If they are taught one way at home, then they get a
     different exposure when they get to school. It is based a lot on
     the teacher, because I'm African American. I do know it can be. I
     have a daughter who is bright and she doesn't have difficulties in
     school. However, I know that there are some children who do. I
     wouldn't call them learning disabilities. They're just a little bit
     slower at picking things up. It all depends on the teacher's
     perceptions of them, how they get treated, therefore how well they
     do in school.

Darcy's mother believed that the single most important factor for a student's academic success is the teacher. Her expectations were higher than the number of assignments her daughter did. "I always want her to do her best and whatever that best may be. But I'm kind of partial because I believe the potential is high, so I like A's. I can't say I frown on B's." She believed both ability and effort make a child succeed. She believed learning in a diverse environment helps the child grow positively.

     It's the best situation you can have because children should be
     involved with different cultures at an early age. They're going to
     be interacting with all kinds of folks when they're older ... It
     also teaches you how to open your mind, express your feelings,
     learn some acceptance, agree or disagree if necessary on some
     things and you get an understanding of a different point of view.
     If you have a majority of folks in a certain culture that stick
     together all the time and you have the group of the minority, they
     stay together all the time, then that defeats the purpose.

Darcy's mother believed the greatest impact on children's education occurs during their early childhood. She believed the biggest challenge for African American students is the change in perception. According to her, this change of perception does not happen in a vacuum. She believed everyone should take it very seriously, including teachers, administrators, and all staff at a school. "If they [students] have a feeling when they walk into school that they have people who are supportive of them and look at them just like any other student, then naturally they can have higher success."

Darcy's mother's main concern was equity and tolerance in educational settings. She put strong emphasis on the teachers and administrators' roles in creating a fair and equitable climate.

Dave's Case

Dave was a nine-year-old African American boy. He was born in Houston, Texas and moved here with his parents. He liked mathematics because "I'm real good at math. I just like it." He likes problem solving and group work but "some people talk too much and I don't like listening to them all the time. She [name of student] acts like she's the boss." He liked multiplication and thought using a calculator was "like cheating."

Dave did not like homework assignments. "I don't like my homework. But I can really do it." When he did his homework, he did it alone. He would ask his mom or dad only when he needed help. Dave liked to write. During one of our interviews he showed us a piece he wrote:

     I like The Watsons Go to Birmingham because it was a realistic
     story about life for African Americans in 1963. Black history is
     important to me because of my third grade teacher. She is black and
     made me realize how scary it was to be a black person in 1963.
     Kenny the storyteller has an important role in the story because
     of Byron who in the beginning bullys him. My favorite part was Nazi
     parachutes attack America and get shot down over the Flint River by
     Captain Byron Watson and his flamethrower of death (I've been
     working on that).

Dave liked to read and write. He was very interested in Black history. Dave's father was a minister. He could speak several languages such as Latin, Hebrew, Greek, and Spanish. He has several children, and Dave is his adopted child. His wife was the director of a child care center. Due to the nature of his job, Dave's father had to travel a lot. He believed the role of the teacher was to be able to recognize and respond to the level where children are and be able to push them that much further. He was very supportive of the teacher and the school's administrators. "I have a lot of respect for the leadership in this school here. They've done an admirable job. This school is the most interesting and best performing in many ways. I'm impressed with the teaching corps here too. They seemed engaged and knowledgeable."

We asked Dave's father about his definition of a good school. He said a good school is where a child can come regardless of his/her ability and reach their full potential. "I don't like putting children through a factory or a machine ... I want the creativity that I believe God has put into each one of us."

Dave's father believed the parent is the primary educator, not the school. "No one should have more power in shaping the direction of the child than the home. Now, unfortunately, every child doesn't have a home that can shape what they will become, but there was no one more powerful in my life than my dad and my mom."

He talked about the importance of parents' involvement in educating their children. He believed the role of the teacher was both of a facilitator and a director. The connection between Dave's interest in reading and writing stories became clear after our conversation with his father. During our interview, his father emphasized the importance of writing and communicating in proper English language.

Amy's Case

Amy was an eight-year-old Caucasian girl born in Ohio. Amy liked mathematics, but she was not confident in her ability. "I like it but sometimes it can get a little tricky and hard to do the math. So I need a lot of help." She liked geometry. She liked shapes and making figures with the shapes.

R: Do you like when you work together with your group and you talk with other students about how to solve problems?

Amy: Well, I really haven't had a chance to do that because really when you've been trying to work together, usually some people always go ahead and I can't really catch up.

Amy like to use a calculator when she did problem solving because "it helps me to figure out things faster." She usually would spend half an hour for her homework assignments at home alone, unless she had questions. She thought her parents valued education very highly and expected her to be good "in the things that I know I can do."

R: Do you believe in yourself that you can do well in math?

Amy: Maybe.

Amy's father agreed to participate in our study. He earned his Master's degree in aerospace engineering and worked at NASA. His wife worked in a bank as a system engineer with a mathematics background. We asked him about his opinion of Amy's mathematics practices. "Well, my sister was not good at math. My mother didn't do quite as well early on, but she recovered." We asked Amy's father about his exposure to other cultures.

     I am a classical guitar aficionado. There is a fellow that was born
     and raised and educated in Iran. One, in Vietnam up through
     college, another in India and then me. We are all rallying around
     this instrument of Spanish roots. Work is really a dynamic
     multicultural experience. It was important and I wish that I would
     have had a little bit of that early on.

His expectations for educating his daughter included watching his daughter progressing more quickly through some of the basic skills such as reading and basic mathematics. He was satisfied with the school. "I have always been welcome to come and observe. That has always been the case here." When asked his view about the role of parents in education, he responded:

     We look to the schools to do a lot of the ground work for us but in
     the end the education of our children comes out of us ... So for us
     the education is really on our shoulders and we look to the school
     to help with the basic addition and subtracting.

Like Dave's father, he had a lot of respect for the teachers and for the school administrators. He said his wife and he alternate helping Amy with her homework assignments. He believed mathematics success depends on individual efforts 90%. He stated that children working together from different backgrounds is a real positive event.

Amy was less confident in problem solving. She mentioned several times that she was not very good at it and needed a lot of help. Her father reflected on this issue in a similar manner during our interviews. According to his view, "[his] sister was not good at math ... [his] mother didn't do quite as well early." It seemed that the perceptions of the father were consistent with Amy's performance. To some degree she accepted that perception as a reality. However, in the later part of the study, she became more confident about tackling non-routine problems. Our observations indicated that social interactions, small group cooperative learning, and the teacher's role contributed to her change of attitudes and beliefs about herself as a mathematician.

An important aspect of the teacher's role in the classroom was her effort to create a classroom climate conducive to learning for all children. In what follows, we discuss the classroom social norms and sociomathematical norms.

Social Norms and Sociomathematical Norms of the Classroom

There were 28 students in this third grade class. Students were sitting around tables in groups of four or five. The class was colorful and the classroom rules and obligations were posted on the classroom walls. During mathematics activities, the classroom was very noisy and students were able to walk around their tables to exchange ideas and information.

The classroom social norms established by the teacher and the students included: 1) students were responsible for their own learning, 2) they were obligated to listen to each other and ask questions, 3) they needed to support one another so that all members of the small group could understand and defend their solutions, and 4) they could ask the teacher for help if the group had the same question. During mathematics problem solving, the students were allowed to use calculators if they needed them. Mathematics manipulatives such as base-ten blocks, color tiles, pattern blocks, tangrams, Cuisennaire rods, geoboards, etc. were available for mathematical modeling. The teacher's role during mathematics activities was to walk around, observe and listen to students' discussions, asking guiding questions if needed, and facilitate whole-class discussions. These patterns of social norms were observed throughout the course of our study.

Sociomathematical norms refer to the ways students negotiated their mathematical thinking and reasoning. For example, they were expected to explain their thinking and understanding of the problem and communicate their solutions to their small groups and to the whole class. The teacher used a problem-centered approach. There were three components in her problem-centered mathematics class: 1) the tasks were presented by the teacher or students, 2) individual problem solving and small group interactions, and 3) whole-class discussion. Students were encouraged to restate the question in their own words, draw pictures (model), write their thinking and reasoning, and show their results. The following description demonstrates two episodes of social interaction during mathematics learning for these five students. The first episode is a task presented by the teacher. The second episode displays problems posed by the students and solved by their peers.

Teacher: Okay, today we're going to solve a problem about time and money. Your job is to figure out the problem on your own for about 10 minutes. When 10 minutes is up, I let you discuss your solution to your group, then as a whole class, we will see how we solved the problem. [Teacher distributed a worksheet that presented the problem.] Ellen earns money baby-sitting. She wants to save her money to buy a portable CD player that costs $128.00. Ellen charges $4.50 per hour to baby-sit during the day and $5.50 per hour after 8:00 p.m. Mr. Holmes hires Ellen to watch his two grandchildren every Saturday in May* from 3:30 p.m. until 11:00 p.m. How much money does Ellen earn in one Saturday? *Change to current calendar month.

After about 15 minutes of individual problem solving, the students were ready to explain their solutions. Hamid offered an explanation. (See figures 1-5 for the students' solutions to the Time and Money problem.)

Hamid: We know that she charges $4.50 before 8:00 p.m. and $5.50 after 8:00 p.m. So that means she charges two different prices, yes?

Students: Yeah.

Hamid: I'm doing this! 3:30 to 7:30 is four hours; from 7:30-8:00 p.m. is thirty minutes; so before 8:00 p.m. there are four hours and thirty minutes.

[FIGURE 1 OMITTED]

Amy: Yes, you have to explain how to break $4.50 in half.

Hamid: Half of $4.50 is $2.50. [He paused.] No! That's half of $5.00. Half of $4.50 is $2.25. From 8:00 p.m. to 1:00 p.m. is three hours. So it'll be seven hours and thirty minutes. I multiplied $4.50 by 4 and got $18.00. Then I added $2.25 to it. I got $20.25. That is for 4 hours and 30 minutes. [Students were actively listening to Hamid's explanation.] Then I multiplied $5.50 by 3, I got $16.50. I added $20.25 and $16.50, I got $36.75. This is per Saturday. Then I multiplied $36.75 by 4. I got $147.00. That's what Ellen earns in May. I subtracted $128.00 from $147.00 and I got $19.00 to spare. (See figure 1.)

Amy: But I think she will have $16.80 left over after she buys the CD player.

Darcy: No, Amy! $36.75 times 4 is $147.00 You came up with $134.80. You need to subtract $128.00 from $147.00, not $134.80!

John: I agree. (See figures 2 & 3.)

[FIGURE 2 OMITTED]

Dave: I don't get it. [John tried to reach Dave and offered help.]

John: I have an idea. First we find the hours before 8:00. That is four and a half hours. Ellen baby-sits for seven hours and thirty minutes each Saturday, agree?

Dave: Agree. [Nodding his head]

John: First you have to know that about the different prices. I did the "time-jump." Four and a half hours at $4.50 per hour and 3 hours at $5.50 per hour after 8:00 p.m. From 8:00 p.m. to 11:00 p.m., agree?

Dave: Yes! I did the time-jump too. I'm multiplying $4.50 by 4, then I add $2.25. Not I multiply $5.50 by 3 ...

[FIGURE 3 OMITTED]

John: Yeah, there is four hours and a half. I multiplied 4 and $4.50, I got $18.00. There's a half hour, so it's half of $4.50 which is $2.25. Now, I add $2.25 from the half hour to the $18.00 from the 4 whole hours, I got $20.25. I multiplied $5.50 by 3 hours, I got $16.50. Then I add it up $2.025 and $16.50, I got $36.75. So she gets $36.75 every Saturday. Now did you get it?

Dave: Yeah, I think so.

John: But I did not answer the second part of the problem. But I agree with Hamid and Darcy. (See figures 4 & 5.)

[FIGURE 4 OMITTED]

[FIGURE 5 OMITTED]

Sociomathematical norms established by the classroom community were followed systematically by the students in the classroom. The teacher would walk around to facilitate students' problem solving and negotiated meaning. She would interrupt the class or small groups only if there was a need for clarification of the problem.

The second episode of classroom social interaction on mathematics focuses on student-created problem posing and problem solving situations. The teacher started with a case scenario:

Teacher: Summer is coming and it's time to think about having fun! We are going to open our own amusement park so everybody at [name of the school] can enjoy themselves! Our park should have rides, games, and plenty of good things to eat. Let's think of how to set up everything and how much stuff will cost. Then create a problem involving time, money, area, and/or perimeter. Use addition, subtraction, multiplication, division, and/or fractions to help create your problem!

The teacher distributed the worksheet problem to all students in the class. The whole activity took five days of mathematics for over one hour each day. We discuss a portion of this project by focusing on the five students' communication in both problem posing, and problem solving. On the second day of this group activity, Darcy posed her problem.

Darcy: There were three people going to the park. One was a teenager, one was ten, and one was an adult. Seniors were $5.00, adults where $20.00, babies free, kids $10.00, and teens $15.00. How much did it cost the three people to get into the fair? How much did it cost? $45.00.

John: But you solved it or us! You should come up with a question.

Amy: Here's my problem ... we spent $32.00 on admission, $15.00 on rides, $25.00 on food, and $12.00 on souvenirs. How much money did we spend?

John: So that's $84.00! I did it in my head! [After some figuring, they all agreed.]

Hamid: Ok. Here's my problem. I want to spend $4.00 each on four games $6.00 each on four rides, $13.00 each on two dinners, $9.00 each on three souvenirs. If I have $50.00, how much more money do I need? [Students spent a bit of time on their own figuring out the problem.]

Dave: I got $93.00.

Hamid: But this is what I want to spend.

Dave: Oh!

Amy: I got $63.00.

Dave: I got $77.00. [John was quickly calculating the amount.]

John: I got $43.00 because if you want to spend $93.00 and you have $50.00 you need $43.00 more. [The other students agreed.]

John's problem was perplexing and required mathematical thinking. He presented his problem:'

John: Armonie arrived at the [name of the school] Amusement Park at 11:34 a.m. He went on the roller coaster for one half hour and 12 minutes. Then he watched a movie for one hour and 26 minutes. After that he went to a shooting gallery. It took him four minutes to shoot and win a prize. He ate some food for 21 minutes. He met a friend. They talked for seven minutes. Then he went home. It took 16 minutes. How much time did he spend out? When did he get home?

The small group was very quiet and everyone was trying to figure out the problem. John was observing students very carefully. After ten minutes or so, Amy explained her solution:

Amy: First I took the hours and put them together and then I took all the minutes and added them together ... the minutes are 56 minutes. And then I added the one whole hour to the other hour and I got two hours and 56 minutes ... but I didn't figure out what time it was yet ...

Darcy: I got three hours and 27 minutes! [Dave was quietly listening.]

Hamid: When he got home? Wait a minute ...

John: [After waiting for a couple of more minutes, he offered help.] This is how you do it! First you separate the hours and minutes ... that's one hour there and 42 minutes over here. Then I added the other minutes ... that's 26 more minutes ... all together then it is 176 minutes. So there are two hours and 56 minutes.

Amy: So, I was right. [Smiling]

John: Yes, but you need to answer the second part of the problem!

Amy: He got back at 4:05? [She was not sure.]

Hamid: If we add one hour to 11:45, that's 12:45, then one more hour that's 1:45, and 56 minutes more ... [He was thinking how to connect 56 minutes to 1:45.]

Darcy: 2:35.

John: You are six minutes short. He got back home at 2:41 p.m.

John's problem was perplexing, interesting, and it required fairly complex mathematical ideas and procedures. Students wanted to find out the solution to the problem posed by their peers. Although it needed some more time for Amy to reach a viable solution, she was satisfied with her effort to solve part of the problem. Darcy and Hamid were convinced by John's last comment. However, Dave remained somewhat perplexed. Perhaps he needed more time and experience with time and measurement.

Discussions

NCTM (1989, 2000a) Standards, current research (Lubienski, 2000; Romberg, Shafer, & Webb, 2000) state that problem solving is fundamental for school mathematics. More research, however, is needed about students' problem posing. Our observation of classroom interactions indicated that students were eager to solve their peers' problems and welcomed the challenges posed by their peers. They sensed that they had some control of teaching/learning milieu. When they presented a problem and explained their solutions to one another, each child played the role of the teacher. Their peers critiqued the problems they posed.

In addition, current research demonstrates the importance of culture in mathematics learning. Its presence and influence can be observed in every classroom in our society. As Ladson-Billings (2001) posits:

     Culture refers to the deep structures of knowing, understanding,
     acting, and being in the world. It is the bases for all human
     thought and activity and cannot be suspended as human beings
     interact with particular subject matters or domains of learning.
     Its transmission is both explicit and implicit. (p. 9)

Revisiting our first research question regarding the impact of parents' expectations on students' attitudes and beliefs about themselves and towards learning mathematics, we found that the parents' beliefs and values had remarkable influence on the students' attitudes toward mathematics learning.

The values placed by John's father toward excellence put a lot of pressure on the child's belief that he must be the best in the class. The child put forth much effort during problem solving and contributed significantly during the group projects. He came to class prepared every day. He solved all homework assignments, and he posed new problems to his group. He liked to challenge his group in problem solving. His parents encouraged him in problem writing. "The school homework assignments are not enough. He needs to be challenged in solving more complicated problems. Otherwise he won't be able to survive academically if we go back to China."

Hamid was mostly quiet and less talkative during the small group problem solving. His comments were short and sometimes incomplete. The language barrier probably had much to do with his ability to communicate. He knew more than he could express. Socialization was his main concern. His father also expressed his concern regarding this matter. He believed that mathematics learning could occur through the help of parents and teachers. However, the socialization and cultural exchange could occur by a sensitive teacher's encouragement. The notion of socialization and cultural exchange was missing in Hamid's experience. He did not have any friends at home either. According to him, his only interaction in his home environment was with his parents. He said in his home, he communicated with his parents in Arabic only.

Although Amy had grown substantially in her ability to tackle rather complex mathematical problems, her home environment and her parents' expectations impacted her attitudes and stated beliefs about this ability. According to her father, female members of his family did not necessarily perform well in mathematics. Therefore, it was assumed that Amy would probably follow a similar pattern. Perhaps Amy, as a female, internalized that notion. It seemed that she did not truly expect herself to be very good at mathematics. This perception was challenged as she experienced a degree of success while working with her small group.

Dave seemed to be less interested and involved in working with the group. One plausible explanation might be the differences in expectations from his home environment and what was required of him to interact in the group. According to his father, the ability to read, write and speak proper English was paramount. Therefore, Dave was more interested in story writing than problem solving alone. When he was faced with more sophisticated aspects of the problem, he became less confident. Dave may benefit from increased opportunity to problem solve.

Darcy was very quiet but also confident in her ability to do problem solving. She liked mathematics and science very much. Her mother's expectation for her to do well in mathematics was evident in our conversations. Both parents worked intensively with her studies and made sure that her assignments were done thoughtfully. Her mother was very sensitive to the issues of gender and race. She wanted her child to be able to rise above the stereotypes connected with these issues. Darcy's performance reflected her parents' attitudes and beliefs towards education.

Revisiting our second research question about social interactions and mathematics learning among the five students, we observed that small group cooperative learning remarkably improved students' abilities to communicate their thinking and reasoning. Furthermore, there was a reflexive relationship between the students' attitudes, beliefs, and practices towards mathematics during the preliminary phase and active phase of the study. These attitudes and beliefs dialectically impacted the group dynamics.

All of the students liked small group cooperative learning, except to some degree, Dave. He was not very excited about group work. "Some people always slow you down. Some people are very bossy." He seemed to be frustrated by some members of the group. Although Dave came up with inadequate solutions several times, he was very confident. He liked people to listen to him, but he was less patient to pay attention to other people. As a result, they had to repeat their solutions to him several times.

Amy reflected on the value of small group work. She thought it was important. "Without group work, I am lost. I need a lot of help. I don't know what to do, but with my group, I can always ask if I am doing right or doing wrong." It was apparent that she was the most active member in her small group. She always initiated the discussion by asking questions such as: "How did you solve this problem? I don't understand this. How did you get this? I solved it this way. Is it correct?" She had less confidence in her ability to tackle non-routine problems independently. However, small group interactions and communication with her peers helped her to build confidence. "I like math. I like the group work. Mostly I like my group. They help me a lot. Math is a lot easier now." She contributed to the group processing by reading and verbalizing the group solutions. Her ability to use clear communication contributed to her learning mathematics. Perhaps because of her language skills, she improved in her problem solving in the latter part of the study.

Hamid liked interacting but because of the language barrier, he often could not interact effectively with is peers. He said, "I like to solve problems alone." He worked very hard to find solutions and made valuable contributions. There were times when he did not quite understand the problems. As a result he did not always find viable solutions. In spite of this difficulty, with the help of the teacher and his small group, he was able to modify his solutions.

John, who was meticulous and precise with is problem solving, would rather work in a group only after he solved the problem on his own. He wanted to make sure that his solution was accurate. In many respects, he was the leader of his group. However, with all of his confidence and ability, he did not dominate the group discussions. He listened to others, acknowledged their contributions, and offered support as needed.

The role of the teacher for establishing social norms and sociomathematical norms was clearly expressed by all parents and the participating students. Darcy's mother expressed her belief that it is the teacher's role in the classroom that can make a difference in children's attitudes and beliefs about themselves. She stated that a teacher's expectations towards African American students can have a positive or negative impact on a child's education. She said her child's interest and curiosity have increased because of her teacher in this class. "I observed Dr. [teacher's name] class several times. I am impressed by the way she engages all students. Her calm voice, her inviting attitude, and her ability to understand her students' backgrounds make her classroom environment relaxing and risk-free. My child likes her very much."

The classroom teacher facilitated whole-class discussions where students had opportunities to defend their solutions. Students were encouraged to take risks and present their solutions in front of the class even when they were not sure of them. Perhaps one of the most important aspects of the classroom teacher's success as a competent and caring teacher was her understanding of and sensitivity to students' cultural backgrounds. For example, she understood Hamid's main concern by observing him carefully during classroom social interactions and listening to him during recesses. She gradually provided him with leadership roles in the classroom. He became one of the leaders in his class the following school year. He improved his communication skills remarkably and has found many good friends. The teacher's awareness about students' cultural backgrounds, reminds us of D'Ambrosio's (2001) assertion:

     We can help students realize their full mathematical potential by
     acknowledging the importance of culture to the identity of the
     child and how culture affects how children think and learn. We must
     teach children to value diversity in the mathematics classroom and
     to understand both the influence that culture has on mathematics
     and how this influence results in different ways in which
     mathematics is used and communicated. (p. 309)

In addition, the classroom teacher touched John's life in progress by providing him with opportunities to adapt to the school environment and the classroom social norms. She also helped Dave and Darcy connect their writing interests to their cultural and social heritage. For example, during the course of the study, these two students wrote several compositions about African American history. Furthermore, she provided Amy with problem-solving and problem-writing opportunities so that she became confident in her own ability to do important mathematics.

We close our discussion with a remark made by D'Ambrosio (2001), "It is time for educators to improve their understanding of the role that culture has played and continues to play in shaping mathematical development. It is time for educators to empower their students with this knowledge" (p. 310).

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Roland Pourdavood, Cleveland State University

Nicole Carignan, University of Quebec at Montreal

Belvia K. Martin, Shaker Heights School District

Michael Sanders, Cleveland State University

***An earlier draft of this paper was presented at the Research Council on Mathematics Learning (RCML) in Tempe, AZ, 2003.

Students' Attitudes towards Mathematics and Their Parents' Expectations:

                            John            Hamid              Darcy
                                                            Born in the
Length of time in US     1.5 years         1 year           US

Parents' education/      Ph.D., Visiting   Ph.D., Visiting  Electric
profession               professor,        professor,       company
                         statistics        Physics          analyst
Students' attitudes      Likes problem     Likes problem    Likes
toward mathematics       solving,          solving,         geometry
                         Dislikes          Dislikes         Dislikes
                         addition and      complicated      hard
                         subtraction.      computations     problems
Parents' beliefs:        Requires efforts  Requires         Innate and
learning of mathematics                    efforts          requires
                                                            efforts
Students' main concern   To be the best    To find a good   Making many
                                           friend           mistakes
Parents' main concern    Student's         Language and     Equity in
                         behavior and      communication    mathematics
                         low                                for African-
                         expectations                       American
Parents' views on role   Challenge         Communication    Being caring
of teacher               students          among students   and
                         educationally                      competent
Students' attitudes      Likes             Likes            Likes
toward cooperative       cooperative       cooperative      cooperative
learning                 learning          learning         learning
Students' beliefs about  Likes school      Likes school     Likes school
the school               and the teacher   and the teacher  and the
                                                            teacher
Parents' beliefs about   Very good         Supports the     Supports the
the school               school but        school           school and
                         worry about       leadership       the teacher
                         expectations

                              Dave              Amy
Length of time in US     Born in the US    Born in the US

Parents' education/      Minister          NASA engineer
profession
Students' attitudes      Likes             Likes it but
toward mathematics       multiplication    thinks it is hard
                         and thinks using
                         a calculator is
                         cheating
Parents' beliefs:        Innate and        Innate and
learning of mathematics  requires efforts  requires efforts
Students' main concern   Not being         Slow in
                         dominated by      problem solving
                         the group
Parents' main concern    Being             Helping
                         articulate in     students basic
                         language          number
                                           operations
Parents' views on role   Establishing      Helping
of teacher               expectations      students to learn
                                           basic skills
Students' attitudes      Somehow           Likes
toward cooperative       dislikes          cooperative
learning                                   learning
Students' beliefs about  Does not like     Likes school,
the school               homework          the teacher, and
                                           her group
Parents' beliefs about   Supports the      Supports the
the school               school and the    school and the
                         teacher           teacher

Table 1: Summary of students' and their parents' attitudes, beliefs, and
expectations.