Promoting Productive Discourse 1

EMAT 8990 DISCUSSION PAPERD. WHITE

Promoting Productive Mathematical Classroom

Discourse with Diverse Students

Dorothy Y. White

University of Georgia

Department of Mathematics Education

105 Aderhold Hall

Athens, GA 30602-7124

(706) 542-4096

Fax: (706) 542-4551

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Running head: PROMOTING PRODUCTIVE MATHEMATICAL DISCOURSE

Submitted for Publication in the Journal of Mathematical Behavior. Do not cite or quote.

The research reported in this material was supported by the National Science Foundation under Grant number MDR 8954652 and ESI 9454187. The opinions, conclusions, or recommendations expressed in these materials are those of the author and do not necessarily reflect the views of the National Science Foundation.

FOR DISCUSSION IN THE EMAT 8990 SEMINAR. DO NOT CITE WITHOUT PERMISSION OF THE AUTHOR.
Abstract

Productive mathematical classroom discourse allows students to concentrate on sense making and reasoning; it allows teachers to reflect on students' understanding and to stimulate mathematical thinking. The focus of the paper is to describe, through classroom vignettes of two teachers, the importance of including all students in classroom discourse and its influence on students’ mathematical thinking. Each classroom vignette illustrates one of four themes that emerged from the classroom discourse: (a) valuing students’ ideas, (b) exploring students’ answers, (c) incorporating students’ background knowledge, and (d) encouraging student-to-student communication. Recommendations for further research on classroom discourse in diverse settings are offered.

Promoting Productive Mathematical Classroom Discourse with Diverse Students

Productive mathematical classroom discourse can facilitate the development of children’s mathematical thinking (Davis, 1997; Kazemi, 1998; Knuth & Peressini, 2001; Lo & Wheatley, 1994; Martino & Maher, 1999; National Council of Teachers of Mathematics [NCTM], 1996, 1991; Pirie, 1996). Research on classroom discourse often cites the NCTM (1991) recommendations that mathematics teachers initiate and orchestrate discourse by posing questions that elicit, engage, and challenge students' thinking; by listening carefully to students' ideas; and by asking students to clarify and justify their ideas orally and in writing. Classroom discourse, properly managed, allows the students to concentrate on sense making and reasoning; it allows teachers to reflect on students' understanding and to stimulate mathematical thinking. Teachers can stimulate students’ growth of mathematical knowledge by asking more open-ended questions aimed at problem solving and conceptual understanding (Martino & Maher, 1999).

Productive classroom discourse requires that teachers engage all students in discourse by monitoring their participation in discussions and deciding when and how to encourage each student to participate. By actively listening to students' ideas and suggestions, teachers demonstrate the value they place on each student’s contributions to the thinking of the class. Thus, if classroom discourse is essential to the learning of mathematics, then researchers and teachers need to examine the nature and type of communication occurring in classrooms of diverse student populations. As Hart and Allexsaht-Snider (1996) specifically suggest, we need more research on teacher development programs that focus explicitly on teachers of diverse students and the sociocultural contexts of mathematics learning in their school settings.

The purpose of this paper is to describe how two teachers used classroom discourse to promote the mathematical learning of their diverse students. Through classroom vignettes, I demonstrate the importance of including all students in classroom discussions and its influence on students’ mathematical thinking. I begin with a brief overview of the mathematics education experiences of diverse students, a description of the teachers, their students, and their pedagogical practices. Next, I present four vignettes to demonstrate how the teachers used classroom discourse to promote students' mathematical learning. Each vignette illustrates one of four themes: (a) valuing students’ ideas, (b) exploring students' answers, (c) incorporating students' background knowledge, and (d) encouraging student-to-student communication. Each vignette includes a brief explanation of the importance of the type of discourse with respect to the mathematical content and implications for students’ learning. Finally, I present implications and recommendations for teacher educators and mathematics education researchers. The recommendations are designed to help those interested in the educational experiences of diverse students identify and extend the current research on effective teaching strategies in mathematics classrooms.

Educational Experiences of Diverse Students

The disparities in mathematics achievement among students are well documented (Strutchens & Silver, 2000; Tate, 1997). In national mathematics assessments, African American and Hispanic students continue to score at significantly lower levels than White and Asian American students. For example, data from the 1996 National Assessment of Educational Progress ([NAEP], Strutchens & Silver, 2000) found that the average proficiency of African American and Hispanic students at all grade levels was considerably lower than that of White students. These differences were especially substantial on tasks that called for extended responses and complex problem solving. Although African American and Hispanic students have made achievement gains in recent years, these gains have been on low-level, basic mathematics skills. As Secada (1992) noted, basic skill proficiency is not enough for “true knowledge and mastery of mathematics” (p. 630). Instead, all students "need to learn a new set of mathematics basics that enable them to compute fluently and to solve problems creatively and resourcefully" (NCTM, 2000a, p. 1).

The poor academic performance of African American and Hispanic students in mathematics is attributable, in large part, to their educational experiences in mathematics classrooms (Campbell & Langrall, 1993; Oakes, 1990; Secada, 1992). According to NCTM (2000b), "students' understanding of mathematics, their ability to use it to solve problems, and their confidence in, and disposition toward, mathematics are all shaped by the teaching they encounter in school" (p. 17). Researchers that have examined the educational experiences of African American and Hispanic students in mathematics report that these students are disproportionately placed in low-tracked mathematics classes that are largely taught by direct instruction, rely heavily on worksheets, and cover relatively little of the curriculum (Oakes, 1990; Secada, 1992). Teachers often believe that a primary goal of instruction is control of minority students, which can best be achieved in teacher-centered classrooms (Stiff, 1998). In these classrooms, teachers spend more time directing students on repetitive tasks, remedial work, and conformity to rules than on developing students’ mathematical competence and autonomous thinking. However, research supports the view that students do not learn mathematics effectively when passively listening to teacher directions and disengaged from the learning process. As Campbell (1998) suggests, "The character of the child is not the issue; the issue is the character of the instruction" (p. 50).

Descriptions of teachers successfully educating African American and Hispanic students (Author, 1997, 2000; Gutstein, Lipman, Hernandez & de los Reyes, 1997; Ladson-Billings, 1997; Malloy, 1997) can help us understand the unique features of improving instruction and learning for these students. These studies demonstrate that improving the mathematical performance of African American and Hispanic students requires a classroom climate that promotes their learning. Malloy (1997) contends that teachers can create a classroom atmosphere that is conducive to African American students' mathematical learning by: (a) allowing students to be active in their learning, (b) encouraging high levels of peer interaction, (c) encouraging group decision making, and (d) avoiding judging any student either verbally or nonverbally on the basis of the teacher’s biases. Gutstein et al. (1997) propose a three-part model of culturally relevant mathematics for Mexican American students. The three components are (a) building on students’ informal mathematical knowledge and building on students’ cultural and experiential knowledge, (b) developing tools of critical mathematical thinking and critical thinking about knowledge in general, and (c) orientations to students’ culture and experience. In classrooms with these features, students learn mathematics through a system of instruction that combines the learning of basic computation skills with higher-order conceptual reasoning. Central to this environment is the type and nature of the classroom discourse and whether it is accessible to all students.

Methodology

This investigation examined how teachers used classroom discourse to teach mathematics and whether the discourse enhanced the educational experiences of their diverse student populations. The research questions were: (1) What was the nature and focus of teachers’ classroom discourse? and (2) How did teachers use classroom discourse to attend to the mathematical needs of their diverse students? In this section, I present a summary of the teachers’ and students’ characteristics and a description of the teachers’ involvement in Project IMPACT (Increasing the Mathematical Power of All Children and Teachers). I also provide a description of the data sources and analysis used in this investigation.

Participants

The participants were two third-grade teachers and their students in a large, urban school district located just outside of Washington, DC. The teachers and students were part of a longitudinal research project entitled Project IMPACT to design, implement, and evaluate a model for mathematics instruction in schools serving children of diverse ethnic and socioeconomic backgrounds (for more information on Project IMPACT, see Author, 1997, 2000). The teachers, Ms. Davis and Ms. Tyler, were both White and in their second year of teaching. Ms. Davis taught 22 students at a language arts magnet school, and Ms. Tyler taught 27 students at a social studies/science magnet school.

The students in Ms. Davis’s and Ms. Tyler’s classes represented various ethnic/racial groups and were classified into the following categories: Asian, Black, Hispanic, and White. This racial categorization was based on the school system's policy for classifying students. Asian students were from Vietnam, Korea, Cambodia, and nations of Southwest Asia, as well as Asian Americans. Students who were African American, African, Haitian, or from the Caribbean were considered Black. Hispanic students were students who were Hispanic American, or immigrants from Central and northern South America, and Spanish-speaking European countries. Any student who was not considered Asian, Black, or Hispanic, as defined above, was classified as White. The 22 students in Ms. Davis’s class had the following racial distribution: 18% Asian, 36% Black, 36% Hispanic, and 9% White. Of the 27 students enrolled in Ms. Tyler’s class, 48% were Black and 52% were White.

Students in both classrooms were evenly mixed across gender but diverse with respect to their socioeconomic status and mathematical academic performance. More specifically, of the 17 students in Ms. Davis’s where data were available, 12 of the students received free or reduced-fee lunch. In Ms. Tyler’s class, 5 of the 23 students where data were available received free or reduced-fee lunch. Both classrooms were heterogeneous with respect to mathematical performance. Based on the Project IMPACT 113-item Mid-year Assessment, student scores ranged from 16-91 in Ms. Davis’s class, and from 32-95 in Ms. Tyler’s class.

Project IMPACT. A major component of Project IMPACT was its teacher enhancement program, which included a summer inservice program and on-site support during the academic year. As participants in the project, teachers attended a 22-day grade-specific summer enhancementinservice program. During this program, project staff addressed issues relating to (a) adult-level mathematics content; (b) teaching mathematics for understanding, including use of manipulative materials and integration of mathematical topics; (c) current reform documents and research on children’s learning of mathematics as well as teaching and learning from a constructivist perspective; and (d) teaching mathematics in culturally diverse classrooms including implications of teacher’s expectations, use of praise versus encouragement, grouping practices. Particular attention was devoted to helping teachers develop techniques for implementing productive classroom discourse, including the implications of teacher’s questions and responses for students’ mathematical thinking and participation in class discussions.

The summer inservice program provided time for teachers to experience teaching from a constructivist perspective, to practice and refine their questioning techniques, and to plan for the upcoming academic year. For 10 mornings during the inservice, the teachers taught mathematics to small groups of four to five elementary school children enrolled in summer school who had either just completed third grade or were entering third grade in the coming school year. Small debriefing groups with project staff and the other third-grade teachers in their schools followed morning teaching sessions. For their participation in the summer program, teachers received three graduate credits and a financial stipend.

During the school year following the Project IMPACT summer program, the teachers received academic support from an on-site Project IMPACT mathematics specialist assigned to each school. Throughout the school year, the teachers participated in weekly planning sessions with the mathematics specialist and the other third-grade teachers at their schools. The mathematics specialist also assisted the teachers by providing demonstration lessons and helping in preparing instructional materials.

Data Sources and Analysis

Transcripts of classroom observations, supplemented by my field notes, provided the first source of data for this study. During the academic year following the Project IMPACT summer inservice, I observed each teacher teaching mathematics on eight separate occasions from January to June. In collecting the data, I assumed the passive observer stance in which I sat off to the side in the front of the room with a pad and tape recorder. Classroom observations were audiotaped via a remote microphone worn by the teacher that allowed me to record most of her verbal interactions. In conjunction with the audiotapes, I recorded the teacher’s nonverbal actions (e.g., writing on the blackboard, distributing materials, observing students, and using manipulative materials) and her selection of students. After the last classroom observations, I individually interviewed each teacher to provide data about her perceptions of the classroom discourse, questioning patterns during mathematics instruction, and whether her views were consistent with her actual classroom practices. These semi-structured interviews provide the second source of data for the study.

A separate set of analyses was conducted for each teacher using methods of analytic induction (Bogdan & Biklen, 1992). I chose a qualitative perspective because it afforded me the opportunity to describe the teachers’ classroom discourse and questioning patterns in a naturalistic setting while attending to both the content and context of the discourse (Carlsen, 1991).

Transcripts of classroom observations and field notes were first analyzed by examining her question and response patterns. These patterns were often a series of questions and responses rather than a single exchange. For example, when a teacher asked an open-ended question and selected several students to respond, that exchange was considered one pattern. Four general questioning patterns emerged, and upon second readings of the transcripts, subsidiary patterns were identified based on the teachers' responses. These patterns were then categorized into themes based on the nature and focus of the discourse. The four themes were: (a) valuing students’ ideas, (b) exploring students' answers, (c) incorporating students' background knowledge, and (d) encouraging student-to-student communication.

Once the themes were identified and assigned to units of data, these themes were analyzed to identify the students that were involved in the interactions based on categories across students' gender and race (Irvine, 1985; Simpson & Erickson, 1983). This analys1s helped answer the second research question, how did teachers use classroom discourse to attend to the mathematical needs of their diverse students?

RESULTS

Two Classrooms

Ms. Davis taught mathematics in the morning as the first period of the day. Desks clustered in groups of four filled her large, well-lit classroom. A crescent-shaped table with chairs was positioned in one corner of the room for small-group work, and a large carpeted area in front of the chalkboard was available for students to gather. Each day began with an early-bird mathematics problem for the students to solve as they entered the classroom. Most early-bird problems involved some sort of data collection and representation in which students placed their answers on either a graph or Venn diagram. Once the class completed the problems, Ms. Davis gathered the students around the chalkboard to share their answers and solution strategies. After the early-bird activity, the class discussed the topic for the day and was assigned groups in which to work. As the children worked on the task, Ms. Davis circulated around the room to monitor their progress and to ask and answer questions. On some occasions, she would work with a small group of students while the rest of the class worked individually at their desks. When time allowed, Ms. Davis followed the small-group work with a whole-class sharing activity in which students shared their answers and how they solved different problems.

Ms. Tyler taught mathematics in the middle of the day immediately after lunch and recess. Her small, poorly lit classroom was also arranged with desks clustered in groups of four. Three corners of the room had a center for a different subject of the curriculum. There was a reading corner, a science corner, and a mathematics corner. As the lesson began, the class sat at their desks while they discussed the topic of the day. Whole-class discussions were followed by students being assigned to work on mathematical problems in groups, pairs, or individually. As the students worked, Ms. Tyler circulated around the room to monitor the student’s progress and to ask and answer questions. Whole-class sharing, in which students shared their answers and solutions strategies, followed the seatwork.