Social constructivism in practice: case study of an elementary school's mathematics program.

Abstract

This research investigated implications for the implementation of social constructivist epistemology on teaching/learning of mathematics in a K-4 public school with particular focus on African American fourth grade students. In addition, the study examined the impact of social constructivist theory on the structure and culture of the school. Constructivist inquiry was used to make sense of the data. In this paper, there is a discussion of lessons learned from this study with particular emphasis on structural changes, cultural changes, politics of reforming mathematics education, and the impact of social constructivist teaching on African American students' achievement.

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There is much research about how students learn mathematics and how mathematics ought to be presented to young children (Burns, 1992; Campbell, 1996; Cobb, Wood, & Yackel, 1991; Cobb & Yackel, 1996; Fennema, Franke, Carpenter, & Carey, 1993; Romberg, Shafer, & Webb, 2000; Simon, 1995; Wheatley & Reynolds, 1999; Yackel, 1995). In addition, research documents indicate that mathematics instruction does not provide students with opportunities to acquire deep mathematical understanding (National Council of Teachers of Mathematics [NCTM], 2000a).

A National Science Foundation (NSF, 1996) report indicates fourth grade students in the United States performed better on mathematics proficiency tests of basic skills compared to previous test results. However, the report exposes students' lack of conceptual understanding of those basic skills. Furthermore, current research shows that minority students (i.e., Native Americans, Hispanics, and African Americans) perform well below the national average (Ladson-Billings, 2001; NCTM, 2000b, 2000c; 2001a, 2001b, 2001c).

This study communicates the complexity of reforming mathematics education with particular focus on instruction for underachieving African American students. Moses & Cobb (2001) posit that if all students can learn mathematics, then they ought to be provided with opportunities to learn mathematics. Therefore, mathematics literacy is a right rather than a privilege for a few. This study may offer some ideas about how to raise mathematics achievement among African American students who have scored low compared to the average national scores on mathematics achievement tests.

This project investigated the following questions: (1) Can social constructivist epistemology be implemented to raise achievement levels of African American students? (2) What impact does social constructivist theory have on the structure and culture of school? And (3) What are the social and political implications for reforming mathematics education in a K-4 elementary school? The research tells the story of a K-4 elementary school that struggled to reform mathematics education (1990-2003) by implementing social constructivist theory and pedagogy in classrooms. After eight years (1990-1998) of focusing on restructuring and recapturing mathematics classrooms, fourth grade students' scores on state mathematics tests dramatically improved for a five year period: 1999-2003. The test results attracted local and statewide attention because African American and white students achieved at about the same high level. In what follows, we describe some history of the reforms in this K-4 school. Then, we discuss our theoretical framework, the design of the study, and lessons learned.

A Bit of History

The school enrolls 525 students: 60% African American, 34% white, and 6% other racial/ethnic groups. This school is one of five K-4 elementary schools in a Midwest school district that is racially and economically diverse. Eighty-five percent of the teaching staff holds a Master's Degree. The principal, former assistant principal, and two teachers have doctorates in education. The building studied enrolls more students than the other four K-4 buildings in the district and also enrolls the highest percentage of African American students. All the other elementary schools have an African American student population less than 50 percent. About one-half (approximately 150 students) of the African American students in this school come from middle class families. The other half (150 students) of the African American students are from lower middle class or economically disadvantaged families. Ninety percent of the potentially at-risk students are lower middle class or economically disadvantaged African American students.

In 1990, school principals (principal and assistant principal, who is now principal in another K-4 elementary school in the same school district) and some teachers led an effort to introduce social constructivist theory in most classrooms. State testing began in 1995. Most of the questions on the fourth grade mathematics test were multiple choice (35 out of 40). Only five questions asked students for written explanations for solving problems. Nearly all test questions asked students to apply skills in the context of relevant mathematics situations.

Educators could not easily obtain copies of the test booklets or their students' test responses. Educators were told that the state was not equipped to release classroom sets of students' test booklets. To obtain a specific students test booklet, educators/parents had to write a request and pay to have the test booklet copied (booklets averaged $8.00-$9.00 each). Obtaining a copy of a students booklet was not easy. For example, in July 1999, the principals requested 25 booklets of underachieving students. They wanted to analyze students' problem-solving strategies. State officials were reluctant to provide the testing booklets when informed that information from the test was being used for this paper.

State Official: We can't release these booklets for research. Students booklets are confidential.

Principal: But I am their principal who happens to be conducting a research study about my underachieving students' ability to perform on the proficiency test. How am I going to improve instruction without knowing how these kids did? That is what the state wants, to improve student learning and get all these kids through the test, right?

State Official: Well, I need a letter from your superintendent that he is aware of this research study and will guarantee that you have taken precautions about confidentiality of subjects. (Phone conversation, mid-July, 1999)

Educators at this school faced the political pressures of state tests. The publication of test results disturbed the climate of this learning community. In 1998, only 69% of fourth grade students passed the mathematics tests, which was clearly below the 80% average passage rate of other elementary schools in the district. Because of test results, some parents, teachers, school board members, and central office administrators questioned the credibility of the mathematics reform. School principals and teachers searched for ways to blend social constructivist practices with preparation for the state mathematics test.

      It's a focused curriculum. Focused on big ideas and on the skills
      necessary to take those big ideas and apply them to real world
      situations. The problems that we give the kids are highly
      contextual. So if you are asking me what makes this different from
      what you might see in other classrooms, I would think that in
      other classrooms, mathematics is driven by textbooks. There is no
      math textbook here (former assistant principal and current
      principal of another school in the same school district).

The reform tacked the challenge of closing the achievement gap between white and African American students on the state test. In September 1998 the school principals extended instructional time with "at-risk" students. About 45 third graders and about 40 fourth graders (40% of each grade level) attended extended instructional programs. Instruction was designed according to constructivist learning theory. Main mathematical topics were emphasized as outlined in NCTM Standards (1989, 1991, 1995). Instruction was intentionally active and interactive with learning experiences that stressed writing, speaking, illustrating, building, and role-playing mathematical problems. Before students took the test in March, fourth grade scholars received about 240 extra hours of mathematics tutoring. Parents were urged to enroll their children in these extended programs. As more and more parents noticed and heard about the popularity and effectiveness of these tutoring programs, enrollment grew and most parents praised the school's efforts. Potentially at-risk students were selected and invited based on scores from teacher-made tests, classroom performance, and scores on a third grade Standford Achievement Test (SAT). In addition to closing the achievement gap, the reform confronted the challenge of teaching high level mathematics to all students so that students and educators could meet the goals of a postmodern society.

      I think it's a shift in the way you think about yourself as a
      teacher. I think the biggest challenge is getting people to want
      to devote more time and more energy to improving mathematics
      education (principal).

Parents actively supported the framework of social constructivist teaching theory for mathematics learning even when test scores were low. Parent support, confidence, and enthusiasm increased when 1999 mathematics scores showed a dramatic improvement (69% passage rate to a 90% passage rate). Mathematics scores continued to improve over the next four years, and the achievement gap continued to narrow.

Theoretical and Philosophical Assumptions

The theory and philosophy of this study were influenced by social constuctivist epistemology. Social constructivist theory posits that learning and knowing are build via active and interactive activities in a classroom (Bauersfeld, 1988; Cobb & Yackel, 1996). This theory values time for discourse among members of the learning community and time for building or drawing models of mathematical situations (Fennema, Franke, Carpenter, & Carey, 1993; Simon, 1995). Furthermore, the theory recognizes prior and present experiences, relevancy of context, and the value of multiple perspectives (Guba & Lincoln, 1994).

Social constructivist theory assumes a teaching/learning environment beyond the rote and routine learning of basic mathematics skills, not in place of learning basic skills. Social constructivist teaching practices emerged as an important feature for students' understanding of mathematics beyond the limited memorization of basic facts and mechanical procedures. In this sense, classroom practices focus on: dialogue, prior knowledge, mathematical modeling, multiple solutions, students' preconceptions, problem solving, problem posing, and the importance of context for building understanding (Romberg, Shafer, & Webb, 2000).

Methods and Study Design

The study was grounded in the constructivist inquiry of Guba & Lincoln (1989, 1994) and Lincoln & Guba (1985). It was an observational study. Participant-observation, interviews, and review of public documents were used to collect information and to interpret the data. Interview data included the transcription of audio tapes from interviews with teachers, principals, assistant superintendents, superintendents, community officials, parents, and students. Field notes were used when the primary researcher observed and participated in the teaching/learning process. Data were collected over a five-year period (1999-2003). Additional data sources were triangulated and negotiated among the researchers (one university teacher and two school principals) for trustworthiness of data analysis. Based on multiple data sources, several themes emerged, and categories were developed. In what follows, we discuss each of these themes.

Lessons Learned

Major mathematics reform at the elementary level probably requires structural and cultural changes. Implementing social constructivist practices in elementary mathematics classrooms may call for changing conventional teaching, learning, and assessment practices.

      I taught, years ago, in a very traditional classroom. And, I
      taught exactly how my advising teaching taught. She introduced me
      to a college professor who was doing math differently. I had been
      a very apprehensive math student, but she was teaching the [math]
      and showing it through games, manipulative, and stuff. It was
      fascinating to me. I lapped up everything I could. I'd tell the
      kids that I didn't understand this [mathematics] when I was their
      age. I didn't have a clue what mathematics was. It's kind of neat!
      (fourth grade teacher).

Most parents and teachers did not learn mathematics in a setting where instruction focuses on the development of mathematical concepts. Many teachers and principals may need to relearn mathematics.

      If you are going to apply it [social constructivism], if you're
      going to build and draw models, role play it, and have time to
      correct and discuss misunderstandings, change your mind and
      rebuild things, and come back and revisit things, and try to put
      it all together, that takes time (principal).

Some parents and concerned citizens may also need to view mathematics education differently. Most school schedules (day and year) do not provide enough time for the effective implementation of social constructivist practices. Classrooms are often not equipped with resource materials that support active, interactive instruction. Some schools which serve lower middle class and economically disadvantaged students may need more technology to support changes in mathematics education. In what follows, we discuss structural changes, cultural changes, politics of reforming mathematics education, and African American students' engagement in a social constructivist setting.

Structural Changes

Teaching all students to understand and apply key mathematical ideas within a social constructivist framework necessitates extending instructional time. Some students need extended time to learn and understand mathematics. Mathematics classroom instruction at this K-4 elementary school was extended from 45 minutes to 90 minutes for all students each day.

Students who were still underachieving at the 3rd and 4th grade level were invited to participate in an extended school day, week, and year. Morning and afternoon tutoring was provided (75 minutes before school for four days each week, 45 minutes after school for three days each week). Teachers and principals tutored the students. Tuition-free summer school was offered for six weeks, and three and a half hours of instruction was provided on each Saturday from September to middle of March. Students who participated in extended time initiatives had about 240 more hours of instruction prior to taking the state mathematics test. About 90% of participating students in extended time programs were African Americans whose parents drove them to school early each Saturday morning and early to school for four days a week. Some parents did not have access to transportation on Saturday mornings, so the program coordinator drove a car to pick up about six students every Saturday morning.

Most instruction during these extended sessions was conducted within a social constructivist framework. Computers and mathematics software programs were used extensively to support classroom instruction. Social constructivism in practice demanded more instructional time because it went beyond basic calculation skills towards an emphasis on understanding how calculation skills are used in real-life contexts.

      Seeing how you ask probing questions and how you get students to
      talk about what they are seeing is my favorite part. It's really
      not only teaching me about math, but also teaching me how I can be
      a better teacher (first year teacher's reflection on teacher-
      leader's interaction with students involved with extended time
      programs).

The school did not adopt a constructive pedagogy completely. The Kumon Mathematics Program was in use as a tutoring tool and was available to all students who were at-risk based on low scores from teacher-made basic computation tests. The Kumon Mathematics Program uses a behaviorist approach for learning basic arithmetic skills. The program costs about $30.00 per student. About eight of the schools' teachers used Kumon materials for 45 minutes three afternoons each week from January of the students' third year until April of the students' fourth year. Overall, the program costs the district about $17,000 per year. Kumon was used in after school tutoring sessions and offered free of charge to students. The researchers were uncertain that all students could effectively learn basic facts within a social constructivist framework. Also, the researchers were aware that there was little time to teach both basic skills and conceptual understanding in order to improve students' achievement on the state tests. The Kumon Program was done after school and away from regular classroom time so that teachers could focus on learning mathematics within a problem-solving and problem-finding environment.

Cultural Changes

Using a learning theory that moved away and beyond the behaviorism created disequilibria for some teachers, some district administrators, some parents, and some members of the school district's boards of education. Mathematical processes and mathematical ideas started to be viewed as less definite and more probabilistic than most parents and teachers had learned when they were young. Most teachers learned more about mathematics and the application of mathematical skills and concepts. Some teachers began to view mathematics as a creative process. They taught students to create different patterns, to understand probability, and to design different strategies for solving mathematical problems.

      It's the leadership of the building, number one. There is a real
      commitment to the fact that all students can learn. There is not a
      student at this building that can't learn. I see an emphasis on
      the extra time that the students need to learn as a big priority
      at this school. They are doing whatever it takes here to get it
      done. A lot of parents almost come to expect it. I also see a
      focus on curriculum a lot more intense here than at some of the
      other buildings [district schools] in terms of what the most
      up-to-date research says about how we're teaching. Some of those
      practices are put into play here at this building (new assistant
      principal of the school).

Many parents learned to appreciate, enjoy, and view mathematics as a creative art. Many parents also respected their child's use of discourse, illustration, and concrete models to build understanding about important mathematical concepts.

Perhaps the most interesting lesson learned from this reform effort was the importance of "student voice" in mathematical discourse. Discourse and "student voice" provided opportunities for students' self-reflection and opportunities for teachers to understand what students did not know and what to do next. This recursive relationship between "student voice" and classroom discourse was pivotal for reforming instruction and curriculum.

      Mathematics is hard and sometimes it can be fun but it is hard. I
      want to learn mathematics but also I want to have fun while I do
      mathematics. I have to think a lot while I am problem solving at
      [name of school]. We have to explain our reasoning. We have to
      draw pictures of money when we are solving problems. We do a lot
      of discussions while we are solving fraction problems. I like
      fractions and I like graphing. It is fun to put stuff on the chart
      and color it in to measure ... There is not one way to answer a
      question, there is always another way you can do it. In problem
      solving you have to use your mind, you have to go way back when
      you explain, you have to use all your knowledge (fourth grade
      student).

Guided by "student voice" and classroom dialogue, teachers realized that textbook instruction and standardized courses of study might be contextually limited. It seemed that value placed on "student voice" and student discourse provided a more open, respectful, and democratic environment for the mathematics classroom. "You don't know what they're thinking unless you ask them. I only get it [instructional decisions] from the voice of the students. They greatly impact the instructional path that I take" (former assistant principal of the school).

Social Constuctivism and African American Students

In 2001, 100% of the elementary school's white students and 92% of the school's African American students passed the state's fourth grade mathematics test. The difference between African American scores and white scores was much smaller than the differences between these two racial groups' scores at the local level which was 98% passage rate for white and 70% passage rate for African American. One might conjecture that social constructivist teaching practices within an extended instructional program can effectively raise African American achievement in mathematics. The researchers suggest that extended time is required because social constructivist practices such as dialoguing, role-playing, building, illustrating, and writing do not fit into the framework of conventional school schedules. The principals also conjecture that the Kumon program which is a behaviorist approach to learning mathematics was a necessary component for building a basic foundation for such students.

In what follows, we demonstrate a small group problem solving situation during a Saturday morning class in spring 2003. Forty-five students participated in the Saturday program. The students were divided into small groups of six, seven, eight, nine, and ten students. Some teacher-leaders and principals agreed to work with these students in their small groups. The following episode is about a fourth grade teacher's interaction with six African American students. The teacher was sitting in a low chair and the six students were sitting on the carpet floor in a half circle. The teacher posed a problem about fractions.

Teacher: Here is the problem. If Olivia ate one and six eighths of some candies [she wrote on a small, white erasable board O = 1 6/8]. Brianna ate three eighths [she wrote B = 3/8]. Latasha ate one and one fourth [she wrote L = 1 1/4]. Cierra ate half [she wrote C = 1/2]. Marcus ate four eighths [she wrote M = 4/8] and Wesley ate three fourths [she wrote W = 3/4]. I want you to figure out how much candy they all ate [students were listening very attentively]. We want to combine all the candies they ate together. Now, your job is to draw nice and neat pictures and explain to us how you figured out the solution. Go for it!

Students started tackling the problem individually and quietly by drawing pictures of candies and cutting them into halves, fourth, and eighths. The teacher carefully observed students' illustrations. After about 15 minutes, she asked each student to demonstrate his/her solution. Students and teachers established norms that required students to listen to each other's explanations, to agree or disagree with each other, and to ask questions if they did not understand their peer's solutions. The teacher would facilitate the activity by observing, listening, and asking guiding questions.

Teacher: Okay, who wants to draw a picture on the board and show how many candies Olivia ate? Latasha!

Latasha: [She went closer to the white board and drew one whole candy and divided another candy first in halves, then in fourths and eighths.] Olivia ate one whole candy and six eights. [She colored one whole candy with blue marker and six eighths of another whole candy with red marker.]

Teacher: Do you all agree with Latasha's solution? [Students nodded their heads in agreement.] Now, I want a volunteer to show how many candies Brianna ate.

Marcus: [He went down to the board and picked up a blue marker.] Brianna ate three eights. I divided a whole candy into eight equal pieces and colored three pieces. [He drew a picture for his solution. Students agreed with his solution.]

Teacher: Brianna, you are very quiet. Are you sleepy?

Brianna: No!

Teacher: Can you show me how many candies Latasha ate?

Brianna: [With a strong voice] yeah. [She picked up a brown marker and drew one whole candy and one fourth of another candy. Then she colored 1 1/4]. My answer is one and one fourth.

Wesley: I can show that another way. [He went close to the white board and drew one and two eighths and colored them.]

Teacher: Do you all agree with Wesley's solution?

Students: Yeah [almost unanimously].

Teacher: Okay, who wants to go next? Cierra?

Cierra: I ate half of a candy and that is the same as Marcus' four eighths because one half equals four eighths.

Teacher: Wow! I am impressed. Can someone show me three fourths in another way?

Brianna: Three fourths is the same as six eighths because I started with a whole candy and I divided it in half then I divided it in fourths because the whole piece is four, and I colored three pieces of four pieces. Then I divided the four pieces into eight pieces and I found they are equal. [Students agreed with Brianna's solution.]

Teacher: Now, who can figure out the most and least candy bar eaten?

Olivia: I ate the most and Cierra ate the least.

Latasha: I disagree, because Cierra's half candy is four eighths and Brianna ate three eighths.

Olivia: Oh, that's right, Brianna ate the least.

Teacher: Olivia, you ate a lot. Brianna was not very hungry. Now, who can show how many candies were eaten all together?

Latasha: I think all of the candies eaten were five.

Teacher: Do you all agree with Latasha's solution?

Students: Yeah, no, yeah.

Teacher: Okay, we have differences of opinion. Now how many of you agree with Latasha's solution, raise your hands. [Two students raised their hands.] Okay, how many of you do not agree with Latasha's solutions, raise your hands. [One student raised her hand.] How many of you aren't sure? [Two students raised their hands]. Okay! Cierra, you don't agree with Latasha's solution. Would you come up here and show us how you figured out your solution?

Cierra: [Quietly went closer to the white board and explained her solutions.] Well, I know that I ate half and Marcus ate half, so that's one. Then I know that Olivia ate one whole and six eights, and Latasha ate one whole and two eighths, that is three whole. [She drew her half and Marcus's half as one whole with two different colors, and shaded Olivia and Latasha's as three whole with a different color.] I know Brianna ate three eights and Wesley ate six eighths, that is one more whole candy and one eighth left over. [Students were carefully listening.]

Teacher: Do you all agree with Cierra's solution?

Marcus: I don't get it. Where did you get one eighth?

Latasha: I know where I went wrong. It is five whole and one eighth. I didn't count the fraction correctly. Six eighths and three eighths would be one and one eighth.

Marcus: But three eighths and six eighths are nine sixteenths! [He made sixteen pieces and colored nine of them.]

Brianna: But you don't have to make sixteen pieces. You need to count three pieces of a whole and sixth pieces of another whole. That makes it one and one eighth left over.

Marcus was puzzled and needed more time and experiences with fractions. The teacher recognized that she needed more one-on-one interaction with Marcus regarding the relationship between parts and wholes.

The above episode demonstrates the social norms established and accepted by teachers and students. The students were expected to listen and respectfully challenge each other's solutions. They were obligated to provide support for each other and communicate their shared meaning with one another. The teacher's role was to pose challenging problems and help students reach their zone of potential development. The climate of the classroom was conducive to learning. The teacher encouraged students' risk-taking in a problem solving situation. She valued and reflected on Marcus's situation of "not having figured it out yet." Due to time constraints, she continued her dialogue with Marcus after the class period.

The next episode illustrates student-principal interactions in the Saturday morning program where the focus was on key ideas in measurements and fractions. In this episode, one of the principals found that the students in her Saturday class had limited understanding of the passage of time. Ten African American students and the principal sat together in a circle on the floor. A large pad of newsprint, markers, a dry erase floor easel, a box of play money, and a large "Judy" clock were available materials. In this setting, she designed the following problem to address students' preconception of elapsed time.

Principal: OK, I have a tricky problem for you to solve today.

Student A: You can't trick us. We know everything!

Student B: Yeah, you haven't been able to trick us yet.

Principal: But, remember, I am not paid to teach you what you already know. I get paid to 'mess with your minds,' confuse you. So, you better watch out because today I am going for 'big bucks!' Now, here is the problem. Watch me as I illustrate it on the white board. [Principal draws a happy face on the board. Children giggle at the hair.] Ok, the hair looks a little strange but what should you be doing?

Student C: Listening to the problem and seeing what important information is in it.

Principal: [Continuing to draw and dictate problem] Jason wants to earn money to buy a bike. To earn money for the bike, Jason takes care of pets. He charges $4.00 an hour. [She draws four-one dollar bills under a picture of dogs and cats] Mr. Johns wants Jason to take care of his puppy from 1:30 p.m. until 4:00 p.m. How much money will Jason earn to take care of Mr. Johns' little puppy?

Student C: How much is the bike?

Principal: I don't know. What do you think? Is it a part of the information needed to solve the problem?

Student D: No, not really. The question did not ask about the bike, only about how much money would Jason earn from Mr. Johns.

Principal: Does everyone agree with [name of student D]'s answer?

Student E: Yeah, I guess you don't need the cost of the bike. But the price is important because Jason doesn't know how long he will have to watch dogs in order to have enough money.

Principal: That's a good thought. I understand that you want to know the price and leaving it out bothers you. But, do we need the cost of the bike?

Student E: No.

Principal: Can we go on to solve this time and money problem? [Students nod their heads, answer, 'yes'].

Principal then asks students to retell the problem and identify the important information needed to solve the problem. Students are given about ten minutes to solve the problem. Students are encouraged to work together, compare their solutions. After the ten minutes, the principal reconvenes the discussion:

Principal: OK, who wants to tell us how much money Jason earned? Who wants to share their strategy and thinking?

Student F: I will.

Principal: Ok, [student's name] move up here, next to the board and show us how you solved the problem. [Student moves next to the white board easel.]

Student F: First, the answer is $14.00.

Principal: Does everyone agree with this answer. [Most students agree, Student G does not agree.]

Student G: I got $10.00.

Principal: Any other solutions?

Student H: I got $11.00.

Principal: Other answers? [There are no other answers.]

Principal: Ok, so what do we know? Can all the answers be correct?

Student A: No, because this is math and there is usually only one right answer.

Principal: [laughing], Not always but maybe in this case. We will have to find out if we have more than one 'right' answer. You guys know how I feel about 'one right answer' in math. It isn't about the answer but the way you think about the answer.

Students: Yeah, we remember.

Principal: Ok, [calls student F's name], share your thinking with us. How did you arrive at your solution?

Student F: Well, Jason makes $4.00 an hour--just like your picture on the board. He works for 3 and 1/2 hours. Three hours times 4 is 12.

Principal: Why did you multiply by four?

Student F: [Drawing three circles on the board] These are hours. For every hour, Jason earns four dollars [student puts $4.00 inside each circle.] You can add 4 + 4 + 4, or you could just multiply 3 times 4 and get the answer which is 12.

Student B: But you said the answer was 14.

Student F: I am not finished. Then, Jason works more than 3 hours. He works 30 minutes more-until 4:00. So, then thirty minutes is 1/2 of an hour, so I halved the four dollars. That is two dollars. I added the two dollars to 12 and got 14. So that's that. [Student F returns to place in the circle. There is a silence as students mull over the information.]

Principal: Well, what are you thinking? I smell smoke, so some of you are really thinking about [Student F's name] solution.

Student G: Well, I am changing my answer. I agree with [student F's name]. I forgot to add the money for the half hour.

Principal: OK, you changed your answer. You found a mistake, good. What about you, [calls student H's name]? What do you think?

Student H: Well, now I am not sure. When I did it, I thought that I had the answer, now, I don't know.

Principal: Why don't you show us your solution? [Student H moves to the easel.]

Student H: I think Jason earned $10.00 because he worked 2 1/2 hours, not 3 1/2.

Principal: How did you figure it was 2 1/2 hours? [Student H reaches for the clock and sets the time for 1:30.]

Student H: [Moving hands of the clock] 1:30 to 2:30 is one hour. 2:30 to 3:30 is another hour. 3:30 to 4:00 is thirty minutes or 1/2 hour. So Jason earned $10.00. I figured the money just like [student F's name] except he had Jason working one more hour. I am not sure if I counted the hours right.

Principal: What do all of you think? We have a difference of one hour. How should we solve this difference of hours Jason worked?

Student I: [Calls student F], show me how you counted the hours.

Student C: Yeah, use the clock, like [student H]

Student F: Ok, watch. [Student F takes the clock and moves the hands.] Begin at 1:30-that's one hour, 2:30, that's 2 hours, 3:30-3 hours, 4:00-thirty minutes more. 3 1/2 hours. [Again, students are silent.]

Student H: But you counted 1:30 as one hour. I don't think you can do that.

Student F: What do you mean?

Student H: Well, when we come to school at 9:00. We haven't been there even one hour. Jason shows up at 1:30 but he has not worked yet. He begins to work. When he has worked until 2:30, then he has worked for one hour. Just like school. If your teacher says that in one hour, we will go to art, you know that at 10:00 you have art. You don't go to art at 9:00. [Now other students are listening to student H's argumentation.]

Student D: Maybe [student H] is right. One hour has to go by before you can say that you worked one hour. [Student H is getting more support from classroom community.]

Student E: Yeah, I know that my baby-sitter doesn't get paid when she comes to the house. She has to work first.

Principal: Your thinking is interesting. Let's see if we can act this problem out.

Students: Yeah, this will be fun!

Principal gets several sheets of paper. She places the papers in a line. There is a 20 inch distance between each paper. She instructs the students to stand and gather around papers.

Principal: Ok, let's let each paper stand for the time that Jason worked. What time did Jason begin the job?

Students: 1:30. [Principal writes 1:30 on one paper.]

Principal: And one hour later?

Students: 2:30. [Principal writes 2:30 on the next paper in line.]

Principal: And one hour later?

Students: 3:30. [3:30 is written on the next paper.]

Principal: And one hour later?

Students: 4:30. [4:30 is written down.]

Student E: But Jason only worked until 4:00. You need to change it.

Principal: Thank you, I made a mistake. See, I was trying to trick you. Ok, who wants to be Jason? [Several students volunteer.] Ok, [calls student C's name], you be Jason and let's act out [student F] solution. I will be Mr. Johns. Let me get some play money for this math drama. [Principal takes a handful of one dollar bills from the box on the floor.] OK. Jason, thank you for coming over to watch Little Rover. Here's $4.00. [Principal hands Student F four dollars.]

Students: No, that's not right.

Principal: What do you mean?

Student A: Don't give him money, yet. He hasn't watched Rover. He just got there.

Principal: Let's just continue to use [student F's] strategy. Let me continue to pay Jason, then we will look at [student H's] strategy and then we will discuss them. Ok, Jason, move to 2:30. [Student steps on next paper.] Good, here's your next four dollars. Now move to 3:30. Ok, here's another four dollars. Now, what do we do? Jason did not work until 4:30?

Student I: Have [student C] stop in the middle, the space between 3:30 and 4:30. That will show 1/2 hour.

Principal: Does that make sense to everybody?

Students: Yes.

Principal: So how much money do I give Jason?

Student A: $2.00.

Principal: Why?

Student D: Because 2 dollars is half of four dollars. [Principal hands 'Jason' 2 more dollars.]

Principal: OK, 'Jason,' count your money.

Student F: I have $14.00.

Principal: $14.00. You all saw it. It must be the right answer, right?

Student H: We didn't do my answer.

Principal: That's right. Why don't you be Jason? Begin at 1:30. [Student stands next to the paper labeled 1:30.]

Principal: Hi, 'Jason' Welcome. Little Rover is happy that you've come to watch him. Here's your $4.00.

Student H: No, I haven't worked an hour yet. I don't get paid right now. See, you have to work to get paid. Watch, I show up at 1:30. Mr. Johns leaves. I play with Rover for 60 minutes. [Student moves to the paper marked 2:30.] 2:30 is one hour past 1:30. 60 minutes has gone by. Now I can be paid 4 dollars. [Principal gives four dollars to the student.] Then, I play another sixty minutes. 2:30 and 60 minutes goes by. That makes 3:30, another hour. Now, pay me again because I worked another hour. [Principal gives student another four dollars.]

Principal: How many hours has this 'Jason' worked so far?

Students: Two. You only paid him for two hours.

Student H: OK. Now watch. I don't play with Rover until 4:30 because Mr. Johns comes back at 4:00. From 3:30 to 4:00 is only 30 minutes. One half hour. I don't work a whole hour. I don't get another four dollars. I get half of four dollars. Half of four dollars is two. So now, you can give me two more dollars. [Principal gives students two more dollars.]

Student H: Now, I am going to count my money. 2, 4, 6, 8, 10 dollars. Yup, Jason made ten dollars.

Principal: OK. We have two different solutions for this problem. What do you think?

Student F: I think I want to change my answer. [Student H] is right. You have to work for an hour to get paid for an hour. I think I know how I made the mistake about hours. I went like this. [Student holds up his hand.] 1:30 [holds up one finger.] That's one, 2:30 [holds up another finger.] That two, 3:30 [holds up a third finger.] That's 3 and 3:30 to 4:00 is half.

Principal: That's a good demonstration. How would you change your counting of the hours?

Student F: Maybe what I could do would be to say quietly, 1:30 but don't count it. I would begin counting with 2:30. Time has to go by before you count it.

In the above episode, we observed three components of problem-centered inquiry: communication, questioning strategy, and use of manipulative (I.e., a large pad of newsprint, markers, a dry erase floor easel, a box of play money, and a large "Judy" clock). The problem presented by principal was interesting. It was within the students' zone of potential construction. Students wanted to find out a viable solution. Communication between the principal and students was focused on mathematical argumentations and justifications. The principal did not dominate the conversation. Instead, she provided students with opportunities to explain their thinking and reasoning. Students were expected to listen carefully to one another and ask challenging questions. When the principal asked questions, she intended to understand students' thinking and guided them toward making sense of mathematics. She provided students enough time for reflections and modifications of their solutions. The learning climate supported students' inquiry.

Another important characteristic of the problem-centered inquiry approach was the principal's and students' role playing. Role playing and use of manipulative provided the classroom community a context that was relevant to them. Within the context, they were able to model and interpret the mathematical situation. Students were encouraged to restate the problems and communicate their solutions verbally and pictorially.

African American Students' Attitudes towards Extended Time

The principal, the assistant principal, a few teachers, and the program coordinator, who is also coordinator of the Kumon program, invite parents of potentially at-risk students to a meeting every August before school begins and two other meetings during the school year before the state mandated mathematics test is administered in March. The details of extended instructional time are explained to all parents. These details include explanations of social constructivist teaching practices and ideas for how parents could help their sons and daughters at home.

Parents were skeptical and uncertain during the first one or two years of the program starting in 1996.

      The challenge is huge, especially for African American parents.
      Not all, but some, because when you're talking about having a kid
      come in early morning, extended during the day, and on Saturdays,
      of course I get a lot of complaints-saying, 'I think this is
      overwhelming for my child.' 'I don't think Johnny needs to come in
      as much.' I have to sit and sort of put the confidence in the
      parents and maybe sometimes I say something that, being African
      American, knowing that mom has to work and mom gets off work, a
      lot of times mom doesn't have time to sit and help the child. It's
      a big cultural difference. We built trust among ourselves. The
      parents have trust with me. They understand I have five kids who
      went through the district. They all graduated. They all went to
      college. Now, I have three grandkids here at this school. One just
      finished his fourth grade. He was the top of his class, going
      through the same program. I have to explain these to the parents
      that if the opportunity is there and the school is offering this
      help, we, as African Americans, have to take advantage of this. We
      can't just sit back and say, 'Hey, let's let the school do it.
      We'll just send our kids there and we won't do anything else.' No!
      It doesn't work that way. You also, as a parent, have to help in
      order for us to get this done! I go through this and sometimes
      it's frustrating to me. And, I know it's frustrating to parents,
      but I have to get their confidence (program coordinator & Kumon
      Coordinator).

The first two years were frustrating and required formal and informal communication between the school and home. However, because the scores rose so high (90% passage rate) in two consecutive years (1999, 2000), parents' pride and confidence in the school increased dramatically. Parents had frequent and easy access to principals and the program coordinator whenever they had questions about teaching schedules, schools, concepts being taught, and a student's progress. Parents were encouraged to call at any time if they had questions or concerns and all calls were returned within the same school day. The program coordinator made a "reminder call" to all homes on Thursday and Friday before each Saturday morning session.

Ninety percent of the African American students eagerly participated in extended time programs where they learned much in a constructivist teaching/learning environment. They also learned much through Kumon mathematics program. The study evinced that students enjoyed learning and responded well to social constructivist teaching practices which encouraged dialogue, risk-taking, problem-solving, writing, illustrating, concrete modeling and the use of technology. Students also responded eagerly to the Kumon program. Parents were surprised by student achievement. In 2001, 40% of the 43 African American students who were considered "at-risk" in third grade, scored "advanced proficient" on the state mathematics test. The study suggests social constructivist practices combined with Kumon program were effective in raising mathematics achievement of African American students.

      I'll tell you what keeps me going, is I can't stand the statistics
      [the achievement gap between white and African American students].
      I looked at my classroom list and I see more and more problems and
      quite frankly, I think, jeez, these are the same children who are
      going to be taking care of me when I'm older. And if I can't
      change this trend what is going to happen to these kids? (fourth
      grade teacher)

This fourth grade teacher's statement reminds us of Campbell's (2001) observation of a classroom interaction. She shared her experience working with teachers and principals in urban settings for instructional changes and school reform:

      There are many times when this effort is frustrating and the
      obstacles seem daunting. That is why it is important to have a
      shared sense of purpose with conviction. For me the source of that
      conviction is simple. It is the eyes of the children, children who
      have and more important who know they have mathematical power.
      (p.49)

Campbell's statement echoed throughout this study from the voices of competent and caring teachers, administrators, parents, and most importantly, young children.

The problem of African American underachievement in mathematics is filled with complexity, frustration and anger. American public education has been unable to effectively raise African American achievement in mathematics (Bigelow, Harvey, Karp & Miller, 2001; Frankenstein, 1995; Ogbu, 1987; Secada, 1992). Failure to educate African American students in mathematics limits their access to jobs in the technological society. This type of failure has economical and social justice implications (Lappan, 2001; Moses & Cobb, 2001; NCTM, 2000a, 2000b, 2001a).

Politics and Reform

The study may offer some ideas about how to raise mathematics achievement among African American students. "The agenda is primarily social and political ... I think that the Emmy, the Oscar, the Grammy will go to the person who can combine the social agenda and the political agenda with the academic agenda" (principal).

The political environment that surrounded mathematics reform at this elementary school was challenging and somewhat explosive. Parents were informed regularly about goals and processes for changing mathematics education. Most parents were supportive and encouraging. District administrators were somewhat skeptical and cautious. Politics came center stage when the state began "proficiency" tests in mathematics for fourth grade students in 1996. The scores were used to compare and rank elementary schools within the district and throughout the state. The predominately multiple choice and short answer tests were not compatible with this school's constructivist approach to teaching and assessment.

Early in the reform the fourth graders were not scoring well compared to fourth graders in the other elementary schools in the district. However, when the principal and assistant principal started early morning, Saturday, after school, and summer school classes for about one-fourth of the fourth grade students, the state mathematical test scores soared to the highest in the district and the state. Educators learned to maintain constructivist pedagogy within extra teaching and tutoring sessions, and students learned deeper mathematical concepts and scored very high.

Fourth grade students achieved the following mathematics scores over the last five years (1999-2003). In 1999, mathematics proficiency scores where 86.1% passage rate. In 2000, the result was even better at a 90.0% passage rate. In 2001 the student's passage rate was 95.5% and 58% of the class (90 students) scored "advanced." In years 2002 and 2003, the student's passage rate was 93% and 89% respectively. The five-year average was 90.7%. Because the passage rate for African American students averaged about 85% for these five years, many parents and community leaders opined that these reforms were particularly effective for African American students.

Maintaining and sustaining reform efforts at this school remain a challenge. Restructuring and recapturing processes may be expensive, time consuming, and dependent on dedicated, intelligent teachers and administrators. The changes represent a dramatic shift away from conventional mathematics education at the K-4 level.

      He's been [principal] hiring people who are wonderful teachers and
      wonderful people. But, he really, I guess he lets them know that a
      lot is expected of them. So when he is looking for a teacher, he's
      looking for someone who is going to come to school and not only
      follow his or her contract, but also do a lot more. I've noticed
      the teachers that he hired in the last couple of years; those are
      the ones who are here on Saturdays. If he's able to hire people
      that are totally dedicated like that and continue to hire teachers
      like this as people retire, the program will continue (fourth
      grade teacher).

Maintaining theses reforms within a changing and controlling bureaucratic institution may be difficult. For example, some experienced staff members retire. Other experienced staff members leave for family obligations and career advancements. It is not easy to acquire and educate new administrators and new teachers about mathematics, social constructivist learning theory, and social constructivist instruction. This type of instructional reform requires ongoing passion toward adult learning, creative instructional design, and more instructional time for some students.

In addition to the uncertainty that comes with personnel changes, this type of reform may be threatened by district decisions to purchase textbooks and instructional guides that package content, predetermine instruction, and give teachers a "crutch" that allows them to avoid active and creative instruction. The reforms at the school were achieved without standard textbooks because teachers and administrators trusted their own abilities for creating active, interactive, and contextual mathematics instruction.

Overall, the reforms represent a major commitment to teacher and administrator learning, to teacher as designer of instruction, to teacher as learner, to teacher as instructional leader, and to the education of all children even through it must happen outside the conventional school schedule. These kinds of reforms may not be sustained without money, time, and the ongoing commitment of teachers and administrators.

Discussion

Creating learning opportunities for all children to make sense of important mathematics concepts may require restructuring and reculturing schools. "If the board educational goals of increased access and achievement for all students are to be reached, it is essential that policies are put in place and actions are undertaken to enhance the teaching and learning of mathematics in poor communities" (NCTM 2001a, p. 2). Fitting social constructivist theory into traditional school schedules and school resources may produce frustration and disappointment until new teaching/learning environments and new time schedules are designed. Findings suggest that social constructivist theory helps teachers and learners move beyond the limited mathematical information gained through practices designed from behaviorist theory. Kumon program is a behaviorist approach. It builds an important foundation for some students. It perfects computational skills for some students. Nevertheless, acquiring computational skills without understanding and application seriously limits students' success in mathematics. Constructivist teaching is required for understanding and applying mathematics.

Reaching all students with mathematical understanding may also require more money, more time, more resources and a radical philosophical shift for mathematics teachers and educational leaders. We close our discussion with a remark from the principal.

There are so many unknowns. The structure [of existing school system] is not big enough for this reform. It doesn't fit. I've used comparisons like a box. The current instructional box isn't big enough to hold this change. It's not just 'thinking out-of-the box' either. That might be easier. It's that you don't have a box big enough to fit what you need. The instructional frame isn't big enough to fit the art of it, the beauty of it. You need a bigger frame. I've looked for appropriate metaphors or similes that would fit all of this and explain it to people clearly. It's not easy to explain this whole effort (principal).

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Roland Pourdavood, Cleveland State University

Lawrence V. Svec, Lomond Elementary School

Lynn M. Cowen, Onaway Elementary School

**An earlier draft of this paper was presented at the Psychology of Mathematics Education (PME-NA) in Athens Georgia, 2002.