COM METH IN MOL BIO
COM METH IN MOL BIO MATH 127
Popular in Course
Popular in Mathematics (M)
This 39 page Class Notes was uploaded by Kavon Feest on Thursday October 22, 2015. The Class Notes belongs to MATH 127 at University of California - Berkeley taught by Staff in Fall. Since its upload, it has received 9 views. For similar materials see /class/226596/math-127-university-of-california-berkeley in Mathematics (M) at University of California - Berkeley.
Reviews for COM METH IN MOL BIO
Report this Material
What is Karma?
Karma is the currency of StudySoup.
Date Created: 10/22/15
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 rst is a high school mathematicsphysics 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 ve different measuremenm 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 signi cant 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 7 for example is the number zero even odd or special Ball begins this day s lesson with the request that the students re ect on their thinking and learning using the previous day s meeting as a catalyst for Social Constructivist Teaching Volume 9 pages 131 169 Copyright 2002 by Elsevier Science Ltd All rights of reproduction in any form reserved ISBN 0762308737 131 132 ALAN H SCHOENFELD re ection The ensuing discussion takes on a life of its own with an intermingling of discussions of content and re ections 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 re ect on their understandings In Minstrell s class the subject matter is more advance 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 intenialized By midyear 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 mathematicsphysics as a sensemaking activity 7 as a disciplined way of understanding complex phenomena A longterm goal of both teachers is for their students to intenialize this form of sensemaking They believe it is important for their students to see themselves as people who are capable of making sense of mathematical and realworld 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 re ection 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 re ection 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 7 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 pattenis 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 A Highly Interactive Discourse Structure 133 practice In a concluding discussion I elaborate on a conjecture that this routine serves as a mechanism that teachers can use to help their classrooms evolve into highly interactive communities of inquiry BACKGROUND AND CONTEXT The work described here is part of an ongoing body of work conducted by the Teacher Model Group TMG at Berkeley In broadest terms the goal of the TMG is to provide a rigorous theoretical characterization of the teaching process employing an analytical framework that explains how and why teachers ake the choices they do in the midst of classroom interactions Roughly speaking the idea is that teachers decisionmaking is a function of their goals beliefs and knowledge That is a teacher enters a speci c classroom with certain contentrelated and social goals in mind for that day as well as overarching goals for the school year That teacher has certain understandings or beliefs about the nature of mathematics about appropriate teaching practices and about his or her students He or she has various kinds of knowledge as well 7 knowledge of the mathematics of pedagogy in general of the students in the class about the ways that class has unfolded in recent days and where the teacher wants it to go etc During the lesson various things come up For example a student may make a mistake and the teacher may suspect that other students need help with the same concept Or a student who has been quiet may risk a suggestion Any of a million things may happen How will the teacher respond and why According to the theory what the teacher does depends on the teacher s knowledge goals and beliefs Take the case of a student saying something incorrect How serious does the teacher consider this mistake to be Does the teacher believe mistakes should be dealt with immediately Does he or she believe in correcting mistakes or in seeking the underlying cause for them How much time does the teacher have to deal with the issue What pedagogical methods or classroom routines does the teacher have available for dealing with this situation On the basis of all of these the teacher will choose whether or not to address the issue How the issue is pursued will depend on what options the teacher perceives are available what the costs and bene ts of each option might be and what the constraints of the situation might allow This brief description merely suggests a research agenda which has unfolded over more than a decade see eg Schoenfeld 1998 1999 for details That agenda has theoretical components what do we mean by knowledge goals and beliefs How do they interact and a corresponding body of empirical 134 ALAN H SCHOENFELD work in which the theory is used to build models of speci c teachers teaching speci c lessons The models serve to test the adequacy and scope of the theory Part of the speci cation 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 1 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 secon gmde students have a schema of learn the routine for X 7 they expect them Routines are considered ef cient 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 amp 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 US is the IRE sequence 7 a sequence in which a teacher initiates an interaction the student responds and the teacher evaluates the response see eg Cazden 1986 Mehan 1979 Sinclair amp 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 7 grounded in reality 7 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 ere as an example is part of the dialogue from a US lesson on complementary and supplementary angles US Department of Education 1997 The lesson comes from the videotape collection of the Third Inteniational 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 rst problem with a student who had not done the assignment and then continues A Highly Interactive Discourse Structure 135 11 Teacher What s the complement of an angle of seven degrees Ho Rl Student Eightyrthree degrees El Teacher Eightyrthree 12 Teacher The complement of an angle of eightyrfour Lindsay R2 Student Sixteen E213 Teacher You sure about your arithmetic on that one R2 Student Oh Six E3 Teacher Six Six degrees 14 Teacher Albert number four R4 Student Seventyrnine degrees E5 Teacher acknowledges correctness by continuing 16 Teacher Number ve oey R6 Student Thirtyrthree E617 Teacher Sure about that Claudia R7 Student Twentyrthree E7 Teacher Twentyrthree 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 warmrup Look at the gure below Find the measure of each angle 18 Teacher 1f 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 19 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 111 Teacher So if three is a hundred and twenty what must two be equal to Rll Student Sixty Ell Teacher Sixty Two is sixty 112 Teacher What must four be equal to Rl2 Student Sixty E12 Teacher Okay Little needs be said here by way of analysis In terms of discourse the 1 R and E labels say it all The teacher posed a series of short answer questions 136 ALAN H SCHOENFELD When students responded correctly he con rmed the correctness of their answers When they responded incorrectly he had them recalculate in EZB and E6l7 or in the case of a factual mistake E10 he informed them of the correct answer In terms of sociomathematical norms an earlier observation made by Lampert hits the nail on the head Commonly mathematics is associated with certainty knowing it with being able to get the right answer quickly These cultuml assumptions are shaped by school experir ence in which doing mathematics means following the rules laid down by the teacher knowing mathematics means remembering and applying the correct rule when the teacher asks a question and mathematical truth is determined when the answer is mtified by the teacher Beliefs about how to do mathematics and what it means to know it in school are acquired through years of watching listening and practicing Lampert 1990 p 3 As noted in the introduction both Minstrell and Ball have very different goals for their students than the outcomes of traditional instruction described above by Lampert Rather serendipitously the TMG wound up analyzing lessons by both Minstrell and Ball The Minstrell study came about early in TMG work The rst lesson we analyzed see Zimmerlin amp Nelson 1999 was of a student teacher teaching a rather traditional lesson While we were engaged in that analysis Emily van Zee who had worked with Minstrell joined the group At that time van Zee was working on the analysis of a lesson taught by Minstrell The van Zee and Minstrell 1997 analysis focused on a questioning strategy employed by Minstrell re ective tosses TMG believed it would be useful to do a complementary analysis focusing on Minstrell s knowledge goals and beliefs In addition Minstrell s lesson was very different from the lesson we had been analyzing Minstrell was an experienced teacher while Nelson was a beginner Minstrell s lesson was nontraditional and of his own design while Nelson s was traditional And while unexpected events in Nelson s lesson had caused him to run into some dif culties unexpected events in Minstrell s lesson were dealt with smoothly Studying Minstrell s lesson would be good for theory building examining radically different cases is an important way to test the scope of an emerging theory as well as its robustness Over a period of about two years TMG re ned its understanding of the Minstrell lesson and constructed a model of Minstrell s teaching of that lesson One component of the model was an interactive routine used by Minstrell to solicit ideas and information from his students This routine which used Minstrell s re ective tosses was a powerful tool for enfranchising the students It made use of their ideas rather than information provided by the teacher to deal with the issues at hand Success in modeling the Minstrell lesson led to some con dence about the robustness of the TMG s theoretical constructs Then as the model of the A Highly Interactive Discourse Structure 137 lesson was being re ned see Schoenfeld Minstrell amp 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 differenL 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 signi cant 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 coconstructed by the students and teacher in response to ongoing events The question was could TMG s theoretical notions suf ce to model this lesson 7 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 decisionmaking during the lesson in question Ultimately however a model of the rst 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 decisionmaking was represented in owchart form At that poinL TMG made a surprising discovery The decision procedure represented by the ow chart was remarkably similar to the decision procedure that we had attributed to Minstrell The classroom routine represented by that decisionmaking structure is the focus of this chapter I conjecture that this routine occurs with some frequency 11 inquiryoriented 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 ow chart in Fig l The discussion that follows provides a brief tour of the ow chart Each of the rectangles in Fig 1 labeled Al 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 Al the teacher introduces a topic to the class In A2 the teacher invites comment and calls 138 A1 Provide context and background about the topic T A2 1 Ask class What else can you say about T Call on a student i 01 Does the response raise other issues BS ALAN H SCHOENFELD Legend Action ltgt Decision Point 02 Should these issues be pursued 33 l Have the class work through the issues no no A4 Seek Closure egt A5 yes Either provide or ask student for clarificationelaboration Is clarification called for Lno Would Expansion or reframing be useful A6 V yes Highlight particular aspects of discussion for class circumstances warrant more discussion of no A7 I Move to next item on agenda Fig I A Highly Interactive Routine for Discussing a Topic A Highly Interactive Discourse Structure 139 on a student There is always the possibility that the student s response will raise issues beyond those intended by the teacher If it does D1 yes the teacher must decide whether or not to deal with those issues That decision is represented by the righthand branch leading from Dl If the student s comment is directly responsive to the teacher s prompt D1 no then the teacher uses that response as grist for the classroom conversation First the teacher decides at D3 whether the class would pro t from the clari cation of the students comment If so the teacher may prompt the student to say more or the teacher may elaborate on what the student has said Ultimately the student s comment is clari ed to the degree deemed appropriate by the teacher The next issue faced by the teacher at D4 is whether it would be useful to expand on the student s comment bringing particular aspects of the discussion to the class s attention Having made that decision and acted accordingly the teacher then decides at D5 whether circumstances warrant a continuation of the discussion If so the teacher invites further comment If not the teacher makes a transition to the next item on his or her agenda It should be stressed that all of the teacher s decisions are highly contextdependent Whether or not the teacher decides to ask a student to elaborate on a given point may for example depend on the time left in that day s lesson the teacher s perception of the student s readiness and willingness to pursue the idea whether the class seems engaged or many other factors In the following two sections of this chapter I show how extended segments of dialogue in Minstrell and Ball s classrooms correspond to the routine described in Fig 1 Two preliminary comments are necessary First I make no claim that Ball or Minstrell either consciously or unconsciously employed the decision procedure outlined in Fig 1 Rather the claim is that the routine captures the discourse pattenis employed by the teachers 7 and that see the nal section of this chapter this kind of routine may be a useful pedagogical device for teachers who wish to have their classrooms function as speci c kinds of discourse communities Second the focus and length of this chapter preclude a detailed linebyline analysis of how and why these teachers made the choices they did Detailed analyses of Minstrell s and Ball s lessons may be found respectively in Schoenfeld Minstrell and van Zee 1999 and in Schoenfeld 1999 ASPECTS OF JIM MINSTRELL S BENCHMARK LESSON Appendix A provides an extended excerpt roughly 20 minutes of class time of a lesson taught by Jim Minstrell Here is the relevant context 140 ALAN H SCHOENFELD 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 rst 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 nonstandard 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 ve measuremenm of that person s blood alcohol content have been taken You have the ve 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 sensemaking activity 7 a way of making reasoned judgmenm about physical phenomena He wanm 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 re ective tosses tuni questions back to the stu ents Minstrell works to foster a classroom environment in which students feel enfranchised 7 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 t e width in centimeters were 1068 1070 1070 1075 1070 1070 1065 and 1060 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 nd 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 analyze 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 amp Minstrell 1997 p 240 The rst part of this fourth lesson is devoted to housekeeping issues related to course administration When those issues have been dealt with Minstrell tunis to a discussion of the Blood Alcohol Content problem Appendix A picks A Highly Interactive Discourse Structure 141 up the transcript of the lesson at this point The discussion of Appendix A that follows will indicate that the ow of classroom discourse corresponds with great delity to the routine described in Fig 11 First Implementation of the Routine Lines 70 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 Puss Lines 33 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 Sl s response in line 15 is on target Hence D1 no and he moves to D3 Sl 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 reframing is necessary D4 no the delineation of various contexts in which the highest and lowest values might be eliminated is suf cient This completes the rst pass through the routine As the discussion has just begun circumstances clearly warrant a contin uation of the discussion D50 2 yes Hence Minstrell asks the students for additional comments This rst pass through the routine is represented schematically in Fig 2 v A 13 z Fig 2 A Schematic Representation of Lines 1 33 of Appendix A 142 ALAN H SCHOENFELD Second Pass Lines 34 45 Minstrell begins the second round of discussion step A2 in line 34 84 s response in line 36 dealing with the elimination of values that are too large or small is on target D1 no and does not need clarification D3 no Following the student s comment Minstrell chooses to introduce a new vocabulary term outliers and to eXpand upon the rationale for eliminating outliers D4 yes This completes the second pass through the routine As there is still much to be said DS yes Minstrell asks for additional input This second pass is represented schematically in Fig 3 Fig 3 A Schematic Representation of Lines 34 45 of Appendix A Third Pass Lines 46 66 Minstrell asks for another one step A2 in line 46 84 s response lines 47 49 is again on target D1 no and does not need clarification D3 no As in the previous pass Minstrell chooses to eXpand upon the student s answer D4 yes in doing so he completes the pass In lines 65 66 Minstrell provides the opportunity for a continuation of the discussion When it appears that the well has run dry he moves lines 67 70 to the next item on the agenda step A8 the issue of how best to combine the numbers This third pass is represented schematically in Fig 4 A Highly Interactive Discourse Structure 143 4 Mi i 4 Fig 4 A Schematic Representation of Lines 46 70 of Appendix A Second Implementation of the Routine Lines 68 25 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 Puss 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 85 s response in line 73 is on the mark D1 no and calls for clari cation D3 yes which 144 ALAN H SCHOENFELD Minstrell requests in lines 74 75 85 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 DS yes he will pursue the discussion This rst pass is represented schemati cally in Fig 5 j Fig 5 A Schematic Representation of Lines 68 89 of Appendix A 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 clari cation D3 yes which emerges in dialogue in lines 97 104 Minstrell provides the formal de nition of the term they have been discussing D4 yes in lines 104 105 and DS 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 88 s response which is nonstandard 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 88 s comments and the way in which he did so is fundamental to understanding how Minstrell s teaching re ects his toplevel goals for his students speci cally his goal of Fig 6 A Schematic Representation of Lines 90 190 of Appendix A creating a discourse community that respects and encourages student initiative That analysis which is outside the scope of this chapter may be found in Schoenfeld Minstrell and van Zee 1999 Suf ce it to say here that Minstrell takes a substantial amount of time to explore the rami cations of the student s de nition In the process he covers some important subject matter and sends the message that a thoughtful suggestion from a student is important enough to warrant the expenditure of a signi cant amount of class time Minstrell wraps up this discussion in line 213 and DS yes invites the students to make additional suggestions This third pass is represented schematically in Fig 7 Fourth Brief Pass Lines 214 220 At this point the mean and the mode have been discussed but the median has yet to be mentioned Minstrell begins the fourth pass in line 214 with a request step A2 for another way to approach the problem 812 s response you could possibly take the number that appears most often reintroduces the mode Minstrell notes this and then moves to bring closure to the discussion Coda Introducing a Missing Element Lines 221 25 Due to the extended unplanned conversation in lines 110 213 it is much later in the class period than it typically would be at this point of the discussion 146 ALAN H SCHOENFELD 4 o o Fig 7 A Schematic Representation of Lines 110 213 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 modi cation 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 rst part roughly six minutes of class time of a thirdgrade lesson taught by Deborah Ball Here is the relevant context This class takes place in January midway 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 speci c 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 A Highly Interactive Discourse Structure 147 also dealt with some conundrums such as the classification of zero All of the other whole numbers are either even or odd ls zero even odd or perhaps special Part of Ball s agenda is to have the students re ect 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 7 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 leanied at the meeting As will be seen the conversation takes some interesting twists and tunis it seems very loosely structured at rst Yet1 the ow of dialogue corresponds closely to the routine discussed above lines 178 9720C 20D724 25758 and 59767 will be seen to correspond to ve passes through the ow 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 178 Ball begins the lesson in line 1 by establishing the context for the discussion step Al and step A2 calling on Shekira to comment on the previous day s meeting Shekira s comment in line 2 is on target Dl 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 rst pass through the routine is represented schematically in Fig 8 Second Pass Lines 9720C Events move differently in lines 9720 Ball begins step A2 by asking for more commenm 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 Dl yes Ball decides that this issue should be 148 ALAN H SCHOENFELD Fig 8 A Schematic Representation of Lines 1 8 of Appendix B worked through D2 yes and she and the class watch step A3 as Shea and Shekira come to an uneasy accord In line with her re ective agenda Ball comments in lines 2OAC about how dif cult some of these issues are Then she moves DS yes in lines 2OD F to continue the discussion This pass through the routine is represented in Fig 9 Third Pass Lines 20D 24 Ball invites further comment step A2 in lines 2ODF calling on Lin in line 20G Lin s comment is on target D1 no but invites a followup ques tion D3 yes which Ball asks in line 22 Lin s response in line 23 stands on its own D4 no Still interested in pursuing her agenda DS yes Ball asks for more comments See Fig 10 Fourth and Fifth Passes Lines 25 58 and 59 67 Lines 25 58 are extremely interesting the question being why Ball in line 26 embarked on an explicit announced detour from her re ective agenda A great deal can be said about this decision see Schoenfeld 1999 for detail That issue is beyond the scope of the current discussion Here I restrict my attention to the routine described in Fig 1 Ball calls for more comments step A2 in line 24 From her perspective Benny s response in line 25 raises issues that she wanted to address D2 yes Ball works through those issues step A Highly Interactive Discourse Structure 149 Fig 9 A Schematic Representation of Lines 9 20C of Appendix B o 7 gt Fig 10 A Schematic Representation of Lines 20D 24 of Appendix B 150 ALAN H SCHOENFELD A3 in lines 26C through 58 Having done so D2 yes she returns to her re ective 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 re ective agenda DS 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 dissimilarlooking 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 re ections on the way I teach my undergraduate problem solving course Fig I Representations of Lines 25 58 and 59 67 of Appendix B respectively A Highly Interactive Discourse Structure 151 and my understandings of Ball s and Minstrell s intentions and actions Although the three of us are very different people and teachers we do have some goals and practices in common The goals include creating learning environments in which our students experience mathematics or physics as a form of sensemaking in which the students re ect on their learning and in which they develop certain productive habim of mind The practices include the routine that has been the focus of this chapter I suspect that the routine described in Fig l 7 which is quite exible depending on the constraints the teacher imposes on it 7 can and often does play a signi cant role in the establishment and maintenance of highly interactive classroom discourse communities To establish the context for the discussion that follows let me describe an apparent paradox One of the most important goals of my problem solving course is the shaping of the classroom environment in a very particular way 7 as a community of independent thinkers engaged collaboratively in reasoned discourse l have written about this goal as follows The activities in our mathematics classrooms can and must re ect and foster the under7 standings that we want students to develop with and about mathematics That is 39 we believe that doing mathematics is an act of sensermaking if we believe that mathematics is often a ands on empirical activity if we believe that mathematical communication is important if we believe that the mathematical community gmpples with serious mathe7 matical problems collaboratively making t ntative explanations of these phenomena and then cycling back t ough those explanations inclu ing de nitions and postulates if we believe that learning mathematics is empowering and that there is a mathematical way of thinking that has value and power then our classroom practices must re ect these beliefs Hence we must work to construct learning environments in which student actively engage in the science of mathematical sensermaking Schoenfeld 1994 pp 6amp61 If I am true to my word then my problem solving class should have a pretty freewheeling atmosphere Yet when Arcavi Kessel Meira and Smith 1998 analyzed my problem solving course they found that this was decidedly not the case as the course got under way In the rst few days of the class analysis revealed 1 exercised a subtle but rm controlling hand Is this hypocritical or inconsistent with my avowed goals In a word no At the beginning of the semester the class had not yet developed the norms of respectful and substantive exchange that are necessary for the successful functioning of a freewheeling community One of my major jobs as the semester began was to encourage the students to take risks and express their opinions 7 but in such a way that they did so on solid mathematical grounds and without stepping on each other s toes The more practiced they became at this the less I needed to provide structure Ultimately 152 ALAN H SCHOENFELD the community functioned on its own see Schoenfeld 1994 With this as background let me retuni to the routine described in Fig 1 In a sense the ow chart in Fig 1 makes things look too straightforward for there is a great deal of subtlety to the implementation Teachers have a great deal of exibility in implementing the routine because the decision points D1 through D5 provide a large degree of latitude Suppose for example that the teacher consistently declines to pursue issues other than those on his or her agenda that is D2 is consistently no and that when clari cation is called for the teacher provides it The result is that classroom events although highly inter active will unfold very much according to the teacher s agenda On the 0 hand the teacher might encourage the students to pursue interesting issues when they arise D2 is consistently yes and may consistently tuni issues back to the students pointing out that clari cation is needed but leaving it to the students to provide it When the routine is implemented in this way the class room agenda is essentially coconstructed and the classroom community is largely on its way to autonomous functioning The same routine then can be made to function in different ways depending on the state of the community It is reasonable to conjecture that a teacher might use the routine in somewhat constrained ways while the norms of the classroom community are being developed and then in less constrained ways as the discourse community comes into its own I don t want to push the data too far but this seems to be the case in the two lessons examined in this chapter Minstrell s lesson is wonderfully enfranchising and interactive especially compared to traditional presentations At the same time his use of the routine in Fig l is also reasonably well constrained On the one hand Minstrell employs a large number of techniques that invite student participation in important ways The simple act of waiting for as long as nine seconds after asking a question thirteen seconds in other pars of the lesson makes it clear that his questions are not rhetorical but are meant to provoke student responses The frequency of questions is astounding More than half of Minstrell s dialogic tunis involve posing serious questions to the students and the students responses provide much of the substance of the discussion Some of that substance is clearly new 7 consider for example the exchange in lines 171 to 189 Minstrell clari es the suggested procedure and asks the class for its opinion of it In the ensuing discussion all of the ideas come from the students And when a student makes an unexpected comment in lines 1117114 Minstrell devotes a large amount of time to exploring the idea she suggests Looked at from the students perspective this is a remarkably open lesson Students are actively encouraged to participate and when they do the teacher A Highly Interactive Discourse Structure 153 picks up their ideas and runs with them 7 even if the discussion leads in unexpected directions The ideas are worked through carefully In exploring the properties of S8 s proposed average Is this the same as what we usually call average Does it provide a good summary of the data Minstrell models the kind of discourse practices he expecm in this class new ideas will be honored by being subjected to careful scrutiny Over time the students will perceive it to be their obligation to run with each other s ideas in similar ways On the other hand Minstrell remains rmly in charge of the agenda for the class His questions while often turning responsibility back to the students provide clear direction for the conversation his comments often add substance to what a student has said His one deviation from his planned agenda in lines 1107213 has signi cant value added Pursuing S8 s question honors student inquiry It provides the opportunity to explore the properties of the arithmetic average and this proposed variant of it both of which are plausible extensions of the lesson s content And it models the process of exploring new ideas The opportunity is serendipitous and the decision to pursue it spontaneous 7 but pursuing it is very much in line with the teacher s toplevel agenda Beyond this it is worth noting that Minstrell s use of the routine to discuss the best value has a builtin safety valve The students may well suggest all three standard measures of central tendency mean median and mode But if they only mention two Minstrell can always mention the third himself In sum Minstrell s use of the routine sets the students on the path to autonomy by providing a structure that will ultimately support freewheeling classroom discussion 7 and it is used in a way that is carefully scaffolded This seems entirely appropriate for one of the rst classes of the year In contrast Ball s lesson takes place midyear at which point the relevant sociomathematical norms have been well established As one indication of this we have Shea s comment to Shekira in tum 10 The comment is polite it uses the technical term disagree and his disagreement is backed up with an implicit appeal to the de nition of evenness Let us examine the rst three passes through the routine In this part of the lesson Ball plays much more of a facilitative rather than a directive role In the rst pass through the routine Ball asks questions designed to help Shekira articulate her feelings about the meeting During the second pass Ball stands aside while Shea and Shekira discuss Shekira s statement that zero could be even Her doing so is important and re ects the state of the community The conversation between Shea and Shekira is about the mathematics rather than about the meeting 7 in focusing on the properties of zero it raises other issues than those in Ball s re ective agenda By standing aside and giving Shea and Shekira room to pursue this conversation Ball not only honors student 154 ALAN H SCHOENFELD initiative but de facto gives the students a role in the day s agendasetting In the third pass Ball asks Lin the obvious question 7 in essence how are you going to deal with your current state of confusion 7 and then lets Lin s answer speak for itself These actions I would argue are entirely in line Ball s goals and with the capacity of the class to function as a productive discourse community They are consistent with what happens later in the class session when the agenda is again coconstructed the class pursues a conjecture by Shea that the number six can be both even and odd and the students largely on their own engage in extended and substantive mathematical discussions In sum the routine outlined in Fig 1 plays out very differently in the two lessons studied 7 appropriately so given the state of each discourse community at the time the routine was implemented It appears on the basis of these lessons and my re ection on my own teaching that this routine 7 tailored to circumstances 7 plays a useful role in shaping and then maintaining the productive exchange of ideas For those of us who believe that classrooms should be homes to communities of reasoned discourse it can be a useful tool CODA When we were invited to contribute to this volume Jere Brophy asked the authors to address six speci c issues I have dealt with a number of those issues tacitly in the body of this chapter but in the spirit of cooperation let me be explicit in addressing them here The questions and my responses follow What Does Social Constructivist Teaching Mean in the Areas of Teaching on Which your Scholarly Work Concentrates I hate to start off on an oppositionist note but I have some serious dif culties with the phrase social constructivist teaching For me social constructivism is a theoretical perspective that can be used to help understand what happens in classrooms 7 any classrooms As such social constructivism doesn t represent or endorse a particular kind of teaching Be that as it may I view mathematics as a particularly powerful and empowering lens through which one can make sense of the world Mathematics coheres 7 it ts together and one can make sense of it I want students to experience mathematics this wa and to come away from their mathematics instruction with a sense of themselves as competent and autonomous reasoners I may have gone a bit overboard rhetorically in the segment of Schoenfeld 1994 quoted above but I still believe the bottom line If we want students to become mathematical A Highly Interactive Discourse Structure 155 sensemakers we need to construct learning environments in which they actively engage in mathematical sensemaking Some clari cation is necessary here 7 I want to avoid extremes On the one hand a steady diet of straight didactic presentations and imitative exercises deprives students of autonomy and of the sense that they are capable of doing mathematics on their own On the other hand the equally extreme alternative a caricature of discovery leanring in which students are given interesting problems and set loose with little guidance is also untenable The sink or swim approach is no more appropriate for learning to think mathematically than it is for learning to swim What I think is appropriate is a carefully chosen combination of curriculum and pedagogy What we know in curricular terms is that it is not necessary for students to be taught everything and then to engage in imitative exercises it is possible for students to learn some things by solving problems rather than leanring things rst and then applying what they have learned to socalled problems We have also seen 7 and Ball s and Minstrell s classrooms are prime examples 7 that students are capable of much more sophisticated reasoning than we tend to give them credit for However classrooms that support that kind of reasoning do not tend to occur by spontaneous generation It is an act of great pedagogical skill to shape a classroom discourse community so that it facilitates productive exchanges among students It takes vigilance to maintain such a community 7 although paradoxically the presence of the teacher may seem diminished as the students become more autonomous and the community seems to function more on its own Yet such intellectual communities 7 classrooms in which students participate in disciplinary sensemaking that is structured and scaffolded where necessary and appropriate 7 are what I hope to see more of What is the Rationale for Using These Methods and What Forms do they Take There is a large body of research indicating that students develop their sense of the mathematical enterprise from their experience in mathematics class rooms The consequences of traditional didactic instruction are all too well known recall the quote from Lampert 1990 see also Schoenfeld 1992 Voigt 1989 Increasingly there are existence proofs of the kind discussed here where students learn to engage in disciplined inquiry Moreover there is now compelling evidence that some of the new reform curricula in mathematics are producing gains on oh yes standardized tests Schoenfeld 2001 156 ALAN H SCHOENFELD The rationale for the middle ground approach suggested above is simple Students are much more likely to develop productive habits of mind when they have the opportunity to practice those habits and to develop a disposition toward sensemaking when they are members of communities that engage success fully in such practices As suggested above crafting such communities takes a good deal of work People are not born knowing how to interact respectfully and productively they have to be taught to do so In each classroom the didactical contract needs to be established and negotiated Students often begin a course with the default assumption that this course like others will be run according to the standard rules in courses that operate differently different expectations need to be made explicit Moreover a fair amount of scaffolding is likely to be necessary In Schoenfeld 1994 for example I describe the way in which I explicitly violate the normative expectation that my job as teacher is to evaluate the correctness of the arguments they propose The second day of class a student volunteered to present a problem solution at the board As often happens the student focused his attention on me rather than on the class when he wrote his argument on the board when he nished he waited for my approval or critique Rather than provide it however I responded as follows Don t look to me for approval because I m not going to provide it I m sure the class knows more than enough to say whether what s on the board is right So turning to class what do you folks think n this particular case the student had made a claim which another student believed to be false Rather than adjudicate I pushed the discussion further How cou d we know which student was correct The discussion continued for some time until we found a point of agreement for the whole class The discussion proceeded from there When the class was done and satis ed I summed up This pr blem discussion illustmted a number of important points for the students points consistently emphasized in the weeks to come First rarely certi ed results but turned points of controversy back to the class for resolution Second the class was to accept little on fai 39 is we proved it in Math 127 was not considered adequate reason to accept a statement s validity Instead the statement must be grounded in mathematics solidly understood by this class Third my role in class discussion would often be that of Doubting Thoma That is I often asked Is that true How o we know Can you give me an example A counterexample A proof both when the students suggestions were correct and when they were incorrect A fourth role was to ensure that the discussions are respectful 7 that it s the mathematics at stake in the conversations not the students This pattern was repeated consistently and delibemtely with effect Late in the second week of class a student who had just written a problem solution on the board started to turn to me for approval and then stopped in midrstream She looked at me with mock resignation and said I know I know She then turned to the class and said OK do you guys buy it or not After some discussion they did Schoenfeld 1994 pp 62763 Reprinted with permission A Highly Interactive Discourse Structure 157 The net result of this kind of interaction was that by the end of the semester the class was challenging assertions much more regularly demanding solid rationales and often deciding autonomously whether or not an argument that had been presented was indeed correct I still played the role of Doubting Thomas on occasion and I had no hesitation in weighing in when l judged that my mathematical input was needed 7 but in many ways my intervention was needed less than at the beginning of the course It is dif cult to abstract this kind of interaction into a general rule and l distrust general rules Parenthood may not be a bad metaphor however The idea is to tum over as much to the students as one thinks they can handle responsibly 7 and to be nearby with a safety net just in case What are the StrengthsAreas of Applicability of these Teaching Methods and What are their Weaknessesareas of Irrelevance or Limited Applicability Teaching for deep understanding is hard It calls for a substantial amount of understanding and exibility on the part of the teacher 7 the willingness to explore ideas as they come up the ability to make judgmenm about what might be productive directions and what might not and the ability to provide the ri t level of support for students individually and collectively Few teachers have had the relevant kinds of experiences as students much less as teachers Nor do we at present provide opportunities for on the job training As I suggested above what is needed is a combination of particular kinds of curriculum and pedagogy assessment plays a critical facilitating or inhibiting role as well Various reformoriented curricula developed after the issuance of the 1989 NCTM Standards support some of the practices discussed here Evidence is mounting that when teachers are provided professional develop ment consistent with those curricula and assessments are aligned with them that students learn a lot more see Schoenfeld 2001 When Why and How are Social Constructivist Methods used Optimally I m not sure I can address this question partly because there s a chicken and egg problem I teach a problem solving course at the college level In many ways it s remedial 7 l have to teach some things I d hope my students would have learned long ago Because of this I focus more on thinking problem solving strategies habim of mind than I do on speci c subject matter content But I can imagine a world in which students had learned to think mathemat ically from kindergarten on 158 ALAN H SCHOENFELD Even so optimality is going to be elusive for quite some time 7 until we have much more widespread experience with teaching techniques and curricula as discussed in this chapter Optimality will also I think always be a question of values Clearly lvalue a certain kind of mathematical disposition and certain habits of mind But1 what do students absolutely have to know Will I be content if my students can regenerate some things rather than recalling them or if they know where to look them up Your answers may differ from mine And depending on the answers you give your pedagogical practices may vary When Why and How do these Methods Need to be Adjusted from Their Usual Form in Order to Match the Affordances and Limitations of Certain Students Instructional Situations etc I think the discussion of the two examples in this chapter suggests an answer though the answer may be more vague than the reader would like In some sense everything is contextdependent One has certain values and certain goals for one s students one makes certain researchbased assumptions about the kinds of environments that will help students attain those goals The rest is scaffolding But that s easy for me to say I d be tempted to leave it at that but I do have to address one peniicious misconception regarding such issues Some will argue that the kinds of practices discussed here are OK for bright kids but that slow kids need more didactic instruction To pursue that path is only to exacerbate inequities Evidence is now becoming available that reformoriented instruction works across the boards Not only do more students do well when they engage meaningfully with mathematics but fewer students bottom out Briars 2001 Briars amp Resnick 2000 Schoenfeld 2001 When and Why are these Methods Irrelevant or Counterproductive and What Methods Need to be Used Instead in these Situations This is a matter of belief and a matter how extreme one is willing to be on either the didactic or the teaching for understanding side Take a simple procedure like the one for subtraction From one perspective there s one right way to do subtraction the standard algorithm and the most effective way to teach it is to drill students on it From another perspective what counts is understanding the algorithm If you do you ll be exible and have a number of different ways to do subtractions Which is right I m reminded of a story I was told long ago by Fred Reif Fred needed a blood test The technician at his HMO said please give A Highly Interactive Discourse Structure 159 me the index finger of your left hand Fred said I play the viola and l have a rehearsal tonight Please use the right hand instead The technician then said please give me the index finger of your left hand Fred repeated his request The technician thought long and hard and then said I guess that would be OK On the one hand we can all be horri ed at the thought of a technician in an HMO who doesn t know whether it s OK to take blood from the right hand instead of the left On the other hand Fred pointed out that there are cosm involved Would you want a doctor to be doing all the blood tests That could get expensive The odds are that less than one patient in a hundred causes the kind of problem Fred did so the limited training the technician received was adequate the vast majority of the time The optimal solution lies somewhere between those two extremes 7 and what you decide is optimal depends on your values NOTE am not claiming that Minstrell consciously or unconsciously follows this routine in an explicit way 7 an more an one would claim traditional teachers consciously employ IRE sequences The claim rather is that the routine in Fig 1 serves as a remarkably accurate post hoc description of the discourse patterns in Minstrell s classroom ACKNOWLEDGMENTS I would like to thank Deborah Ball and Jim Minstrell for their willingness to make their lessons available for comment Thanks to Emily van Zee and Cathy Kessel for comments on this manuscript and to all four of the above for their colleagueship through the years REFERENCES Arcavi A Kessel C Meim L amp Smith J P 1998 Teaching mathematical problem solving An an ysis of an emergent classroom community A Schoenfeld J Kaput amp E Dubinsky Eds Research in Collegiate Mathematics Education 111 pp 1770 Washington DC Conference Board of the Mathematical Sciences Ball D L Undated Annotated tmnscript of segments of Deborah Ball s January 19 1990 class Distributed by Ball at the research prersession to the 1997 annual NC39IM meeting San Diego Briars D amp Resnick L 2000 Standards assessments 7 and what else The essential elements of standardsrbased school improvement Manuscript submitted for publication Briars D 2001 Mathematics Performance in the Pittsburgh public sc ools Presentation at a Mathematics Assessment Resource Service conference on tools for systemic improvement San Diego CA 160 ALAN H SCHOENFELD Cazden C 1986 Classroom discourse In M C Wittrock Ed Handbook of Research on Teaching 3rd ed pp 432463 New York Macmillan Lampert M 1990 When the problem is not the question and the solution is not the answer Mathematical knowing and teaching American Educational Research Journal 271 29763 Leinhardt G 1993 On teaching In R Glaser Ed Advances in Instructional Psychology Vol 4 pp 1754 Hillsdale NJ Erlbaum Leinhardt G Weidman C amp Hammond K 1987 Introduction and integration of classroom routines by expert teachers Curriculum Inquiry 172 13 7176 Mehan H 1979 Learning lessons Cambridge Harvard University Press Schoenfeld A H 1992 Learning to think mathematically Problem solving metacognition n sensermaking in mathematics In D Grouws Ed Han ook o se r h Mathematics Teaching and Learning pp 3347370 New York MacMillan Schoenfeld A H 1994 Re ections on doin and teaching mathematics In A Schoenfeld d Mathematical Thinking and Problem Solving pp 53770 Hillsdale NJ Erlbaum Schoenfeld A H 1998 Toward a theory of teachingrinrcontext Issues in Education 41 1794 on Schoenfeld A H 1999 Dilemmasdecisions Can we model teachers online decision making Paper presented at the Annual Meeting of the American Educational Research Association Montreal Quebec Canada April 19723 1999 Schoenfeld A H Ed 1999 Examining the complexity of teaching Special issue of the Journal of Mathematical Behavior 183 Schoenfeld A H 2001 Making mathematics work for our kids Issues of standards testing and equity Manuscript submitted for publication Schoenfeld A H Minstrell J amp van Zee E 1999 The detailed analysis of an established teacher carrying out a nonrtraditional lesson Journal of Mathematical Behavior 183 2817325 Sinclair J amp Coulthard R 1975 Towards an analysis of discourse The English used by teachers and pupils London Oxford University Press US Department of Education 1997 Attaining excellence TIMSS as a star1in point to xamine teaching Eighth grade mathematics essons Unites States Japan and Germany Videotape ORAD 971023R Washington DC Office of Educational Research and Improvement Voigt J 1989 Social functions of routines and consequences for subject matter learning International Journal of Educational Research 136 6477656 van Zee E amp Minstrell J 1997 Using questioning to guide student thinking Journal of the Learning Sciences 62 2277269 Zimmerlin D amp Nelson M 1999 The detailed analysis of a beginning teacher carrying out a traditional lesson Journal of Mathematical Behavior 183 2637280 A Highly Interactive Discourse Structure 161 APPENDIX A EXTENDED SEGMENT 0F JIM MINSTRELL S CLASS DISCUSSION OF MEASUREMENT l T OK So we ve talked a bit about Blood Alcohol Count and then we ve also got the 2 table measurement to include in this So those are the contexts in which we can talk 3 about measurement And let me see if I can remember where we were on Friday in 4 terms o the discussion And let s see 5 You can help me out ah but I think one of the topics that we had talked a bit about 6 was getting what Iended up calling a best value Getting the best number we can 7 What is the blood alcohol count number for this person7 What is the number for the 8 length of that table7 or for the width of that table7 9 And on Friday some people said let s take all the numbers and some said let s only 10 take some num ers o that was one of the issues that we came up with whether to 11 take all the numbers that were used in the measurement or whether to only take some 12 of the numbers 13 And some of the reasons that we listed for taking some numbers were 7 what was one 14 of them7 15 S1 Eliminate highest and lowest 16 T OK You might want to eliminate highest and lowest writes on board Is there a 17 context in which ah that s done7 A measurement context that you can think of where l that s done7 19 S2Math 20 T In some in some math situations 21 S 7 unintelligible 22 T Pardon me7 23 S7 unintelligible 24 T Y ways do that 25 S7 unintelligible 26 T Teachers always do that do they7 Teachers always eliminate the highest and lowest7 27 Ss overlapping unintelligible student comments 2 T OK Sometimes ah e m i some classes the teacher will ah or students will 29 or teachers will allow students to or whatever it is to eliminate the highest and lowest 30 S3partially intelligible sometimes scores are eliminated in diving 31 T K 39 i 39 e 39 es you eliminate the highest and lowest7 Often in the 3 measurement of diving you have all those judges eliminate the top and bottom number 33 an take the rest and do something with the rest of them 34 OKWhat s another way at going at taking some of the numbers and not all of them7 S4 7 36 S4The ones that unintelligible comment involving eliminating extreme values ALAN H SCHOENFELD T OK So we might eliminate 7 You need to let me know if I m writing too small or if you don t understand the words that I m writing down O T That s eliminate points to abbreviation on board All right Eliminate ah what I ll call an outlier If there are numbers that are just completely out of the ballpark I mean the rest of these are sort of in a ballpark in there and then there s one that is just way out of there or two that are just way out of there or something like that then you mightjust eliminate the outliers possibly OK T And keep all the measures that seem like they re pretty much in the same ballpark That s another ah decision outcome that you might come to OK another one S4 What about it said like in the law that you um had like a certi ed nurse and a doctor I think it was and you have to eliminate certain people who weren t certi ed to test the blood or to take the blood T OK Can you hear her back there S5 S5No T No All right You want to say that a little louder S4You have to eliminate the ones that weren t absolutely certi ed to take the blood test T OK In that context or essentially in geneml then you might want to take those numbers ah done by quote the experts something like that and then ah in there here was a special context like ah like for example ah some people said Oh goodness the MD is the one who should do it and others disagreed with that as being that person as being the expert some people said Oh what s important is that I take MINE m measurement because my measurement I KNOW is right but anybody else s I don t know but MINE I know is right So there are several ways of getting at experts there you might even want to question you know Who is t e expert Am I really an expert here or Is the MD really an expert there so there s some question in there when you start taking it from the experts you know 7 who s to decide who the experts are OK Any other reasons you can think of to only take some of the numbers 9s pause OK I think that s pretty much the list we had on Friday All right Now We re trying to get a best value and we might take all of the numbers or we might take some of the numbers and then it s what the heck are we going to do with those numbers OK So now we ve got some numbers there what are we going to do with those numbers What s one thing that we might do with the numbers S Avemge them OK writes avemge them on board We might average them Now what do you mean by average here S5 Add up all the numbers and then divide by whatever amount of numbers you added up All right That is a de nition for avemge In fact that s what we ll call an operational de nition An opemtional de nition is a de nition where you where you give a recipe for how to nd what it is that you re talking about And in this particular case she s saying Add the number of 7 whether you re talking about some of the numbers or all of the numbers 7 add t ose up an divide by however many there are And that s called the arithmetical avemge and to get that you add them up and divide by how many there are OK A Highly Interactive Discourse Structure agaga I I am You ve got a bunch of numbers that are t e s me This is a little complicated 163 talking while writing this on the board That s an avemge that you often use in lots of different contexts and it s an average that we ll use in here but look out because there are lots of times when that s not the best avemge to use On nding the best value that s a pretty good way to get the arithmetic or the arithmetic avemge is a pretty good way of getting a best value O Any other suggestions for what we might do So we can avemge them 7 8s pause Any other suggestions there for what we might do to get a best value I ll put up the numbers that we had from the table measurement on Friday in rst period for the length for the width and the length As you look at that army of numbers any other ideas there that come to mind as to how you might go about getting a best value from these numbers you re going to take there S7 num er OK Like what are you talking about there 7 10 All right 107 point zero 107 point 0 107 point 0 107 point 0 Is there any other number in the width column that shows up as much as 107 point zero No No OK So it s the number that shows up the most often is another way of picking one That s called the um the mode OK writes on board The number that shows up most frequently OK That s another way of getting a best value out of a collection of numbers that you re willin to keep Does that make sense Anybody confused here yet Haven t confused anybody yet Then I ve got to push a little harder 4s pause right Anybody think of another way of giving a best value S5 but I mean it might work If you see that 107 shows up 4 times you give it acoef cient of 4 and then 1075 only shows up one time you give it a coef cient of one you add all those up and then you divide by the number of coef cients you have u lost me unintelligible overlapping student comments One of those numbers It s just that the more times it shows up that makes like makes it a more um a more weight OK Let me see if I can follow what you re saying then You re saying one zero seven point zero shows up four times writing on board so let me put a multiplier in front of it sotto voce that s what a coef cient is of four and then what what am I going to Ah you average that well then you just say there are ah ve numbers and another one is Well let s go ahead and use this rst column right here OK Then ah well unintelligible So everything else only comes up on ce Wait One yeah looks like it So everything else just gets one All right So one and ah we ve got ah one oh six point eight Eight And one Oh seven point ve 164 133 ALAN H SCHOENFELD SS Oh seven point ve One oh seven point ve And one Six point ve One oh six point ve One oh six One one of six OK Now what do I do You add all that O K agagaga And you divide by muttering eight One two three four and four makes eight 7 SS Ot J Instructor has written on the board 4 1070 1106S 11075 11065 11060 S T All right What do you think of that method Ss overlapping student comments including Forget it quotToo hard Too hard Ss overlapping unintelligible student comments including It s the same T All right So actually it ends up being the same as the arithmetic avemge SS No Because 107 gets four times the value so the 107 counts more T Ah OK If you were to take the arithmetic avemge of these numbers what would you do What would be the operations that you would go through there S5 you were the one who suggested arithmetic avemge S5 You d add all the numbers together and then divide it by S Now what do you mean by adding all the numbers S5 You would add each separate number that everybody got you wouldn t just add one 107 you d add all the 1 7s T OK All right So what S5 is suggesting is for an arithmetic average is to add this number then add this number then a 39s number even though it s a repeat of that one then a d this one this one and this one Ss overlapping unintelligible student comments T No ould e as this if you did this Ss overlapping Yeah yes it would T All 39g t o if you just took the arithmetic avemge by adding each one of these numbers all eight numbers and divide it by eight then that would end up giving you the same number as this so this is kind of maybe a quickie way of gmbbing some of them but outside of that it it gives us the same answer SS Yeah It does I didn t mean it to when I did it though T OK What about this other method that ah that was mentioned of saying let s just add up the numbers that are different like 1068 and 1070 1075 1065 and 1060 tha s all our different numbers right S9 Why 1070 T Well because that s a that s a different S10 It s different T I mean there s at least one of those at least one of these at least these etcetem add those up and then take that and divide by 5 How do you like that S No S That doesn t show doesn t represent it truthftu though cause I mean there s a lot A Highly Interactive Discourse Structure Hmbl S11 H S 165 more 107s and that s be that d change K Wouldn t that give us the same answer as if we just took the arithmetic avemge overlapping no s Can you give me an instance that s a real clear example that would drive home to me as to why that would give me a different number than if Itook all of these and divided by eight We can do that with the numbers and see that it would come out different S11 If everybody got 107 except for one person who got 99 and then if you took 107 and 99 and divided it by 2 it d be a lot different Does that make sense overlapping agreement So if people go back there and measure the table 107 107 107 107 107 107 107 107 107 107 107 and somebody else gets 99 so we go over there and we say Hmm ah half way between 107 and 99 unintelligible comments overlapping comments no Now I want you to listen to yourselves because a lot of you are saying that s a ridiculous situation of course it wouldn t be halfquot 7 what is half way between 99 and 107 103 103 Yeah K Of course it couldn t be 103 right But you know what There are going to be some contexts within here in which some of you are going to fall into that very tmp right there if you re not careful OK So watch out so watch out for it Is it clear that one oh what d I d say 103 would not be a good average for 99 and then all those 107s Is that clear OK All right Ah OK So this is really not a very good way to do it Do we agree there Somehow we need to weight to weight in there the fact that 107 occurs so many times So we ve got this way of doing it or if we added them all up in there that would include all those 107s OK 4s pause Anybody confused yet 2s pause No 2s pause OK 3s pause Got to be honest 4s pause se see a different way of approaching S12You could possibly take the number that appears most often like you were saying before if everyone got 107 and then a couple of people got 99 or like one pe 99 and one person got 120 you could pretty much assume that 107 would be nearest to the correct answer and so that you could just select that OK And that s the one that we called the mode there it shows up the most frequently OK The mode There s another measure in here that ah that ah is sometimes used and nobody mentioned it but I ll I ll go ahead and throw it in here then it s what s called the median measure Anybody know what the median is alf way rson got Half Yeah If you were to take if you were to take all of the numbers 7 this is getting pretty messy there let me clean that up a bit 7 if we were to take all of these numbers and mnk them writing on board the highest one is 1075 then it s 107 107 107 107 N w l m m overlapping T Nah 107 s ALAN H SCHOENFELD and then 1068 1065 1060 do you see what I did there m hum I what s called ranked them from the biggest measure that we got to the smallest measure that we got for the width of that table and then after I mnk all of those I go for the middle number the middle number Oh Beep what do I do here The middle number right in there ey re the same number so it doesn t matter Is that zero then Cause it s right in there pointing between two numbers comments no ove it 107 s below it right in there I might even go half way between these two if they differed maybe but the median number in this case would probably be a nice 107 OK So the median is the writing on board middle number w en are ranked And ranked you know like from the top to the bottom etc so then you take the middle number when all the numbers are mnked there You have to rearmnge all the numbers and then take the middle one And that s called the median That make sense Sure OK All right Now those are some of the ah some of the ways then that we might we rst of all might take all of the numbers to get a best value or we might take some of those numbers to get a best value then what we might do with them is that we might avemge them or we might go for the number that shows up most frequently or we might go for the middle number These are all different techniques for getting a best value APPENDIX B DEBORAH BALL S CLASS FRIDAY JANUARY 19 1990 THIRD GRADE SPARTAN VILLAGE SCHOOL EAST LANSING MICHIGAN 1 Ball A Okay A few delays but Ithink we re ready to start now B I d like to open open the discussion today with um 7 I have a few questions about the meeting yesterday that I d like to ask C So to begin with I would just like everybody to put pens down there s nothing to take notes about or do right now D But I d like you to be thinking back to yesterday and to the meeting that we had on even and odd numbers and zero E And I have a few questions First 7 my rst question is 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 just what kinds of comments you have about yesterday s meetin F And could you listen to one another s comments so that we can um bene t from what other people say G See what yi what you think about other people s comments Shekim do you want to start 2 Shekim If If Iliked it because well I like talking to other classes and and when you talk to other classes sometimes it he ps A Highly Interactive Discourse Structure 167 all Mb m 7 Ball oo Ball Shea So 11 Shekim 13 Shea 14 Shekim cooqu m m u 20 Ball 21 Lin 22 Ball 23 Lin B Shekim Ball Shekim Shekim In what way It helps you to understand a little bit more as there an example of something yesterday that you understood a little bit more during the meeting ell I didn t think that new was 7 zero um 7 even or odd until yesterday they said that it could be even because of the ones on each side is odd so that couldn t be odd So that helped me understand it Hmm So y7 So you thought about something that came up in the meeting that you hadn t thought about before Okay nods Other people s comments Shea 39 st want to say something to Shekira when sh7 what she said about um that that one um 7 new has to be an odd an even number bec7 I disagree because um because what what two things can you put together to m e it Could you repeat what you said please speaks to Bernadette and asks her to listen to this Okay um I disagree with you because um if it was an even number how 7 what two things could make it Well I could show you it Moves toward the chalkboard and points to the number line above the chalkboard Um I forgot what his name was 7 but yesterday he said that this one points to the 1 on the number line and e c 39 ne is odd and this one points to the 1 on the number line is odd so this one has to be even But that doesn t mean it always is even It could be even It could be but I m not saying that is has to be even I meant that it could be You said it was A Before we take this up again I underst7 I7 I understand that this is still a problem and that we didn t a 7 we didn t settle it we re probably not going to settle it B Um there s a lot of disagreement about this issue right C And you saw that the fourth graders who have been thinking about this for a long time also disagree about it don t they D I m still kind of interested um in hearing some more comments about the meeting itself E Shekim commented that it was good to have the two classes together because she heard an idea that she hadn t thought about and it made her think about and even revise her own idea when she was in the meeting yesterday F What other comments do other people have about the meeting and what happened yesterday G Lin do you have a Um I h7 I thought that new was always going to be a even number but from the meeting I sort of got mixed up because I heard other ideas I agree with and now I don t know which one I should agree wit Um7 hm So what are you going to do about that Um I m going to listen more to the discussion and nd out comment Ball Benny Benn Ball Benny B all Bernadette all Bernadette Lin Shekim all Benny all Benny Ball Shea Bernadette Shea L1n Bernadette Shea Bernadette Shea Benny Shea Bernadette ALAN H SCHOENFELD Other people Benny Um rst I said that um zero was even but then I guess I revised so that zero I think is special because um I 7 um even numbers like they they make even numbers like two um two makes four and four is an even number and four makes eight eight is an even number and um like that And and go on like that and like one plus one and go on adding the same numbers with the same numbers And so I I think zero s special A Can I ask you a question about what you just said B And then I ll ask people for more comments about the meeting C Were you saying that when you put even numbers together you get another even number 7 Yeah 7 or were you saying that all even numbers are made up of even numbers Yes they are This i very hard to make out There has been signi cant dispute over whether Benny said yes they are or no they re not Bernadette you said something like that yesterday too W at Were you 7 were you not listening to this just now No Benny said a minute ago that when you put even numbers together you get an even num er Mm7hm But he also said I think that all even numbers are made up of other even numbers I sagree says something to Lin Two even numbers just the same Un h The same even number Yeah like our A Like eight is four plus four B Are all the even numbers 7 can you do that with all the even numbers That they d be made up of two identical even numbers ot7 not7 not7 looking toward Benny You can t Like six Six is two two Six you can t get two Six is two odd numbers to make an even to make an even number Three three 7 still looking toward Benny You need three twos to make six You can t put a four and a four or a Three twos looking toward Benny Three s 7 Three is odd Or um 7 I know that but um um I m talking about like two plus two is four and four plus four is eight and Ijust skipped the six so Ijust added the ones that that d Like the two plus two is four and four is an even number and I m just talking about the things that um like 7 Six can be an odd number A Highly Interactive Discourse Structure 54 kn kn m kn anquot m m Benny Bernadette Benny Bernadette Benny Ball Shea 169 What Ijust said 7 the um like two is plus two is four and four plus four is eight and 7 So what you re doing is you re going by twos and then what two equals from then you go from 7 all the wa u Yeah I m not going by every single number Like Okay Two four six eight A More comments about the meeting B 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 C Shea Um I don t have anything about the meeting yesterday but I was just thinking about six that it s a I m just thinking I m just thinking it can be an odd number too cause there could be two four six and two three twos that d make six Uh7 uh And two threes that it could be an odd and an even number Both Three things to make it and there could be two things to make it And the two things that you put together to make it were odd right Three and three are each odd Uh huh and the other the twos were even A So you re kind of 7 I think Benny said then that he wasn t talking about every even number right Benny B Were you saying that C Some of the even numbers like six are made up of two odds like you just suggested Uh7uh agreeing with the teacher Other people s comments