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A HIGHLY INTERACTIVE
DISCOURSE STRUCTURE
Alan H. Schoenfeld
INTRODUCTION
This somewhat speculative chapter is grounded in observations made during the detailed analysis of two very different mathematics lessons. The first is a high school mathematics/physics lesson conducted by Jim Minstrell toward the beginning of the school year. In broadest terms, the question explored by Minstrell’s class is how to determine the “best value” for some quantity when a number of measurements have been taken. The day before the lesson examined here, Minstrell had posed the question in terms of five different measurements of someone’s blood alcohol content. Eight students had also measured the width of a table, obtaining a range of different values. On this fourth day of the school year the students discuss whether some or all of the numbers should be taken into account, and how best to combine them. During the lesson, Minstrell’s questioning style invites contributions from the students. These contributions provide a significant proportion of the content of the lesson. The second lesson to be examined occurs in Deborah Ball’s third grade mathematics classroom, in the middle of the school year. Ball’s students have been discussing the properties of even and odd numbers. The previous day they had met with a class of fourth graders to discuss some of the issues they had been grappling with – for example, is the number zero even, odd, or “special”? Ball begins this day’s lesson with the request that the students reflect on their thinking and learning, using the previous day’s meeting as a catalyst for
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Social Constructivist Teaching, Volume 9, pages 131–169. Copyright © 2002 by Elsevier Science Ltd. All rights of reproduction in any form reserved. ISBN: 0-7623-0873-
reflection. The ensuing discussion takes on a life of its own, with an intermingling of discussions of content and reflections on student learning. In some ways the two lessons discussed in this chapter are worlds apart. To begin with the obvious, students in elementary and high school are very different in terms of social and cognitive development. In Ball’s class the subject matter content is elementary mathematics, and the agenda is to have students reflect on their understandings. In Minstrell’s class the subject matter is more advanced, and the agenda is to have the students sort out how best to make sense of it. Thus the agendas are radically different. Moreover, the two classroom communities are at very different points in their evolution. At the beginning of the year, Minstrell’s class has not yet been shaped as a functioning discourse community (that is, the norms of interaction have not been established and internalized). By mid-year, Ball’s class has well established sociomathematical norms. In other ways, these two lessons are very similar. Both Minstrell and Ball work very hard to have their classrooms function as communities of disciplined inquiry. A major instructional goal is for students to experience mathematics/physics as a sense-making activity – as a disciplined way of understanding complex phenomena. A long-term goal of both teachers is for their students to internalize this form of sense-making. They believe it is important for their students to see themselves as people who are capable of making sense of mathematical and real-world phenomena, by reasoning carefully about them. Part of the way that Ball and Minstrell work toward these goals is to have their classrooms function as particular kinds of discourse communities, in which inquiry and reflection are encouraged and supported. Over the course of the year, sociomathematical norms in support of such practices are established. Classroom discourse practices support students’ engagement with the content and their reflection on both the content and their understandings of it. One such discourse practice, captured as a pedagogical routine, is the focus of this paper. This chapter unfolds as follows. I begin with a brief description of the analytic enterprise that gave rise to the discussion in this chapter, the work of the Teacher Model Group at Berkeley. This discussion explains how we came to examine lessons by Minstrell and Ball, and some of what we saw – including the classroom routine that I claim is common to both teachers. I also point to some of the literature on classroom discourse practices, to establish the contrast between traditional discourse patterns and the highly interactive routine used by Ball and Minstrell. With this as context, I move to a description of the routine itself. Following the general description, I work through sections of lessons by Minstrell and Ball, showing in detail how this routine plays out in
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work, in which the theory is used to build models of specific teachers teaching specific lessons. The models serve to test the adequacy and scope of the theory. Part of the specification of the model of a teacher teaching a particular lesson is the delineation of the cognitive and interactional resources that are available to the teacher and relevant to the lesson being modeled. Here I will not describe the architecture of knowledge used in the model, save to say that TMG’s assumptions regarding the organization of memory are consistent with the standard cognitive model. Rather, I will focus on one particular kind of interaction, the classroom routine. As Leinhardt notes,
Routines are vital. They reduce the cognitive processing load for both the student and the teacher; they are easy to teach because, by second grade, students have a schema of “learn the routine for X ” – they expect them. Routines are considered efficient when they elicit an action with a minimum of time and confusion. Effective teachers have management, support, and exchange routines in place by the end of the second day in a school year. They retain 90% of these routines at midyear (Leinhardt, Weidman & Hammond, 1987). But routines are also subtle and set the tone of the class (Leinhardt, 1993, p. 15).
One classic teaching routine, a nearly ubiquitous discourse structure in classrooms in the U.S., is the “IRE sequence” – a sequence in which a teacher initiates an interaction, the student responds, and the teacher evaluates the response (see, e.g. Cazden, 1986; Mehan, 1979; Sinclair & Coulthard, 1975). This structure can be implemented with a fair amount of latitude, in that the student response and the teacher’s evaluation of it can range from a word or a phrase to lengthy expositions. However, the stereotype – grounded in reality – is that in traditional didactic mathematics lessons, short IRE sequences are ideal vehicles for fostering student mastery of procedural skills. Typically, at some point in a lesson a teacher will ask students to provide their answers to a set of assigned problems. Students will be called upon to give their answers to the problems in sequence, and the teacher will assess the responses, possibly elaborating on points of importance. Here, as an example, is part of the dialogue from a U.S. lesson on complementary and supplementary angles (U.S. Department of Education, 1997). The lesson comes from the videotape collection of the Third International Mathematics and Science Study (TIMSS). The tapes that were publicly released were chosen because of their representativeness. The teacher begins the lesson by going over a homework assignment. After reminding students that measures of complementary angles add up to ninety degrees, he calls on a series of students to give their answers to the problems. The teacher works through the first problem with a student who had not done the assignment, and then continues:
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I1. Teacher: What’s the complement of an angle of seven degrees? Ho. R1. Student: Eighty-three degrees. E1. Teacher: Eighty-three. I2. Teacher: The complement of an angle of eighty-four, Lindsay? R2. Student: Sixteen. E2/I3. Teacher: You sure about your arithmetic on that one? R2. Student: Oh. Six. E3. Teacher: Six. Six degrees. I4. Teacher: Albert, number four. R4. Student: Seventy-nine degrees? E5. Teacher: [acknowledges correctness by continuing]. I6. Teacher: Number five, Joey. R6. Student: Thirty-three. E6/I7. Teacher: Sure about that? Claudia? R7. Student: Twenty-three. E7. Teacher: Twenty-three. You’ve got to be careful about your arithmetic... Later in the lesson the teacher introduces the students to supplementary and vertical angles. The relevant information for working on the problems he assigns is that vertical angles are equal, and that supplementary angles add up to one hundred eighty degrees. After handing out a work sheet, the teacher continues: Teacher: Look at the examples on the top. Similar to your warm-up. Look at the figure [below].... Find the measure of each angle.
I8. Teacher: If angle three is one hundred twenty degrees... and angle three and angle one are vertical, what must angle one be equal to? R8. Student: One twenty. E8. Teacher: One hundred twenty degrees. I9. Teacher: What can you tell me about angles two and three? R9. Student: That they are vertical. E10. Teacher: Two and three are not vertical. One and three are vertical. Two and four are vertical. Two and three are supplementary. I11. Teacher: So, if three is a hundred and twenty, what must two be equal to? R11. Student: Sixty. E11. Teacher: Sixty. Two is sixty. I12. Teacher: What must four be equal to? R12. Student: Sixty. E12: Teacher: Okay. Little needs be said here by way of analysis. In terms of discourse, the I, R, and E labels say it all. The teacher posed a series of “short answer” questions.
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lesson was being refined (see Schoenfeld, Minstrell & van Zee, 1999, for a description of the model), members of the research group saw a videotape of Deborah Ball’s “Shea numbers” class. This lesson offered new challenges. Although Minstrell’s and Nelson’s lessons are very different, they share some very important properties. They deal with high school mathematics (and thus with high school students). And, both lessons are driven by the teacher’s agenda. In Ball’s class the students are third graders, so there are significant differences in terms of the students’ knowledge bases, and their cognitive and social development. Equally important, the lesson in question had taken unexpected twists and turns. The agenda appeared to be co-constructed by the students and teacher, in response to ongoing events. The question was, could TMG’s theoretical notions suffice to model this lesson – or was a detailed model of this lesson beyond the scope of the theory? For quite some time the issue was in doubt; in Schoenfeld, Minstrell, and van Zee (1999) the authors noted that they had, thus far, been unsuccessful in modeling Ball’s decision-making during the lesson in question. Ultimately, however, a model of the first part of the lesson, with all its unexpected twists and turns, was developed. When the structure of the lesson came to be understood, Ball’s decision-making was represented in flow-chart form. At that point, TMG made a surprising discovery. The decision procedure represented by the flow chart was remarkably similar to the decision procedure that we had attributed to Minstrell! The classroom routine represented by that decision-making structure is the focus of this chapter. I conjecture that this routine occurs with some frequency in “inquiry-oriented” classrooms, and that it helps such teachers to establish classroom communities in which disciplined inquiry is a major feature. The following section of this chapter provides a description of the routine.
A COMPLEX ROUTINE FOR SOLICITING AND
WORKING WITH STUDENT IDEAS.
Unlike the IRE sequences described in the previous section, the teaching routine described in this section has as its function the elicitation and elaboration of student ideas. The full routine is outlined as a flow chart in Fig. 1. The discussion that follows provides a brief “tour” of the flow chart. Each of the rectangles in Fig. 1, labeled [A1] through [A7], represents a possible action by the teacher. Each of the diamonds, labeled [D1] through [D5], represents a point at which the teacher makes a decision. In broadest outline, the routine operates as follows. In [A1], the teacher introduces a topic to the class. In [A2], the teacher invites comment and calls
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40111 Fig. 1.^ A Highly Interactive Routine for Discussing a Topic.
The lesson discussed here is part of a series of lessons specially designed by Minstrell as an introduction to his high school physics course. It takes place the fourth day of the course. The first two days of the course are devoted to introductory activities such as an extensive “name game” and a diagnostic test that documents the students’ initial knowledge. On the third day Minstrell begins the substantive content of the course with a non-standard problem of his own design, the Blood Alcohol Content (BAC) problem. In essence, the problem is as follows. Suppose someone has been stopped for drunk driving, and five measurements of that person’s blood alcohol content have been taken. You have the five numbers. Which of those numbers should be combined, in what way, to give the “best value” for the person’s blood alcohol content? The Blood Alcohol Content problem is a carefully chosen mechanism for introducing the content and social dynamics of the course. Minstrell has a number of high level goals for his students. He wants them to see physics as a sense-making activity – a way of making reasoned judgments about physical phenomena. He wants the students to see themselves as competent reasoners who are capable of sorting through complex issues themselves. He has, thus, chosen a problem that is meaningful to the students, and which they can engage fully. His discourse style will foster students’ growth and autonomy: rather than evaluate student comments and questions, he will consistently (by means of an interactive technique he calls “reflective tosses”) turn questions back to the students. Minstrell works to foster a classroom environment in which students feel enfranchised – an environment in which they feel it is their right (indeed, their responsibility) to raise issues and think through them carefully. Van Zee and Minstrell describe the context for the fourth lesson as follows. The students worked on the Blood Alcohol Content problem in small groups during the 3rd day of class. In addition, a student from each group independently measured the length and width of the same table. The numbers obtained for the width in centimeters were 106.8, 107.0, 107.0, 107.5, 107.0, 107.0, 106.5, and 106.0. Near the close of the 3rd day of class, Minstrell brought the students together for a brief discussion of reasons for using only some or using all of the numbers in the Blood Alcohol Count problem. For homework, the students were find the best value for the blood alcohol count and to decide whether the driver was drunk. They were also to calculate best values and uncertainties for the length and width of the table. Minstrell and students examined these issues on the fourth day of class during the discussion analyzed here. Minstrell described this as an ‘elaboration benchmark discus- sion’ in which he planned to work through a series of issues which the students had already opened and considered in small groups in class and on their own at home (van Zee & Minstrell, 1997, p. 240). The first part of this fourth lesson is devoted to “housekeeping” issues related to course administration. When those issues have been dealt with, Minstrell turns to a discussion of the Blood Alcohol Content problem. Appendix A picks
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up the transcript of the lesson at this point. The discussion of Appendix A that follows will indicate that the flow of classroom discourse corresponds, with great fidelity, to the routine described in Fig. 1. 1
First Implementation of the Routine: Lines 1–
I claim that the classroom dialogue captured in lines 1–70 of the transcript can be represented by three “passes” through the routine, in lines 1–33, 34–45, and 46–70 respectively.
First Pass: Lines 1– Minstrell provides context and background for the discussion (step [A1] of the routine) in lines 1–12 of the transcript. He follows this in lines 13–14 by a request for student input (step [A2]). S1’s response in line 15 is on target. Hence [D1] = “no,” and he moves to [D3]. S1’s comment in line 15 does call for elaboration ([D3] = “yes”), and Minstrell pursues the elaboration in lines 16–33. At this point neither abstraction nor re-framing is necessary ([D4] = “no”); the delineation of various contexts in which the highest and lowest values might be eliminated is sufficient. This completes the first pass through the routine. As the discussion has just begun, circumstances clearly warrant a contin- uation of the discussion ([D50] = “yes”). Hence Minstrell asks the students for additional comments. This first pass through the routine is represented schematically in Fig. 2.
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40111 Fig. 2.^ A Schematic Representation of Lines 1–33 of Appendix A.
Second Implementation of the Routine: Lines 68–
At this point in the lesson, the issue is how to best combine the given data. There are, of course, three classical measures of central tendency: mean, median, and mode. Rather than lay these out, Minstrell will ask the class “What the heck are we going to do with these numbers?” He has reason to expect, of course, that the students will generate the three measures of central tendency
- and if they fail to generate one, he can always “seed” the conversation with reference to it. This situation is ideal for the use of the routine, in that the order in which the students generate ideas doesn’t matter. Hence he can solicit suggestions and take them as they come. I claim that lines 68–220 of the transcript can be represented by four passes through the routine (lines 68–89, 90–109, 110–214, and 214–220). Lines 221–251 represent the adaptive move suggested above, adding an approach to the list when the students fail to generate it themselves.
First Pass: Lines 68–89. Minstrell begins in lines 68–70 by framing the problem of “best value” for class discussion (step [A1]), and continues in lines 71–72 by asking, “What’s one thing we might do with the numbers?” (step [A2]). S5’s response in line 73 is on the mark ([D1] = “no”) and calls for clarification ([D3 ] = “yes”), which
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Fig. 4. A Schematic Representation of Lines 46–70 of Appendix A.
Minstrell requests in lines 74–75. S5’s definition in lines 76–77 is correct ([D3] = “no”). Minstrell decides ([D4] = “yes”) to expand upon the defini- tion in lines 78–89. This is just the beginning of the discussion, so ([D5] = “yes”) he will pursue the discussion. This first pass is represented schemati- cally in Fig. 5.
Second Pass: Lines 90–109. Minstrell begins the second pass (lines 90–95, step [A2]) by asking if the students have “any other ideas” for computing the best value. S7’s comment in line 96 begs for clarification ([D3] = “yes”), which emerges in dialogue in lines 97–104. Minstrell provides the formal definition of the term they have been discussing ([D4] = “yes”) in lines 104–105 and ([D5] = “yes”) moves to continue the discussion in lines 105–109. This second pass is represented schematically in Fig. 6.
Third Pass: Lines 110–213. Minstrell begins the third pass in line 110, with another request (step [A2]) for “another way of giving a best value.” S8’s response, which is non-standard, raises a number of very interesting issues ([D1] = “yes”) which Minstrell pursues ([D2] = “yes”) for quite some time. A detailed examination of Minstrell’s decision to follow up on S8’s comments, and the way in which he did so, is fundamental to understanding how Minstrell’s teaching reflects his top-level goals for his students (specifically, his goal of
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Fig. 5. A Schematic Representation of Lines 68–89 of Appendix A.
The class has generated two of the three measures of central tendency (mean and mode) but failed to generate the third (median) in response to Minstrell’s invitation in line 214. Minstrell introduces it himself in line 221. This adaptive modification of the routine will be considered in the concluding discussion of this chapter.
ASPECTS OF DEBORAH BALL’S
“SHEA NUMBERS” LESSON
Appendix B provides an excerpt of the first part (roughly six minutes of class time) of a third-grade lesson taught by Deborah Ball. Here is the relevant context. This class takes place in January, mid-way through the school year. The discourse community is well established. Ball has worked with her third graders to establish a community that operates according to specific sociomathematical norms, using a vocabulary tailored to those norms. Students make conjectures , and they are expected to provide evidence in favor of those conjectures. When a student disagrees with another student’s conjecture, he or she must provide reason for the dissent: “I disagree because.. .” When a student wants to retract or alter a previously expressed opinion, he or she says “ I revise my thinking .” Ball’s class has been exploring the properties of even and odd numbers. On the basis of empirical observations they have made some conjectures, for example that the sum of two odd numbers will always be even. They have
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Fig. 7. A Schematic Representation of Lines 110–213 of Appendix A.
also dealt with some conundrums, such as the classification of zero. All of the other whole numbers are either even or odd. Is zero even, odd, or perhaps “special”? Part of Ball’s agenda is to have the students reflect on their learning, and on the processes by which they come to understand mathematics. She wants them to understand that it takes a long time to make sense of some things – for example, that last year’s third graders, now in the fourth grade, are still grappling with some of the issues that this year’s class is working through. Ball had arranged for a meeting between this year’s and last year’s classes, to discuss even and odd numbers. That meeting took place on the day before the lesson in question. Her agenda as she opens this lesson is to “debrief” the students about their impressions of the previous day’s meeting. What issues did it raise for them? She announces “I’d just like to hear some comments about what you thought about the meeting, what you noticed about the meeting, what you learned at the meeting.” As will be seen, the conversation takes some interesting twists and turns; it seems very loosely structured at first. Yet, the flow of dialogue corresponds closely to the routine discussed above: lines 1–8, 9–20C, 20D–24, 25–58, and 59–67 will be seen to correspond to five passes through the flow chart given in Fig. 1. There is much more to the analysis than can be discussed here; see Schoenfeld (1999) for details. The summary given here is derived from that analysis.
First Pass: Lines 1–8. Ball begins the lesson in line 1 by establishing the context for the discussion (step [A1]) and (step [A2]) calling on Shekira to comment on the previous day’s meeting. Shekira’s comment in line 2 is on target ([D1] = “no”) but needs clarification ([D3] = “yes”). Ball prompts for greater specificity (step [A5]) in line 3 and again in line 5. Given Ball’s reflective agenda, Shekira’s comment in line 6 does call for reframing ([D4] = “yes”). Ball does so in line 7 and ([D5] = “yes”) calls on Shea to continue the discussion. This first pass through the routine is represented schematically in Fig. 8.
Second Pass: Lines 9–20C Events move differently in lines 9–20. Ball begins (step [A2]) by asking for more comments about the meeting. Shea’s comment is not focused on the prior day’s meeting, however. Rather, Shea disagrees with Shekira about an issue of mathematical content ([D1] = “yes”). Ball decides that this issue should be
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Fig. 9. A Schematic Representation of Lines 9–20C of Appendix B.
Fig. 10. A Schematic Representation of Lines 20D–24 of Appendix B.
[A3]) in lines 26C through 58. Having done so ([D2] = “yes”) she returns to her reflective agenda. In line 59, Ball starts the next pass through the routine (step [A2], “I’d really like to hear from as many people as possible what comments you had or reactions you had to being in that meeting yesterday”). Shea himself announces in line 60 that his comment is off topic. Ball misinterprets Shea’s comment in lines 60 and 62 (see Ball, undated); she believes that he is addressing Benny’s conjecture, which had been the focus of lines 25–58. This issue, having been resolved at some length, does not warrant further discussion ([D2] = “no”), and Ball moves to obtain closure in line 65. She still wishes to pursue her reflective agenda ([D5] = “yes”), and in line 67 she asks for more comments. These two passes through the routine are represented in Fig. 11.
DISCUSSION
If the preceding analyses are right, then the two very dissimilar-looking lesson segments taught by Jim Minstrell and Deborah Ball share, at one level of analysis, the same deep structure. Does that matter? I think it does. This is where the discussion becomes conjectural. What follows is grounded in my reflections on the way I teach my undergraduate problem solving course
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Fig. 11. Representations of Lines 25–58 and 59–67 of Appendix B, respectively.
