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NOVICE INSTRUCTORSAND STUDENT-CENTERED INSTRUCTION:UNDERSTANDINGPERCEPTIONS ANDRESPONSES TO CHALLENGESOF CLASSROOM AUTHORITYDale Winter PhD a & Carolyn A. Yackel PhD ba Department of Mathematics , DukeUniversity , Box 90320, Durham, NC, 27708, USAE-mail:b Department of Mathematics , IndianaUniversity , Rawles Hall, Bloomington, IN,47405, USA E-mail:Published online: 13 Aug 2007.

To cite this article: Dale Winter PhD & Carolyn A. Yackel PhD (2000) NOVICEINSTRUCTORS AND STUDENT-CENTERED INSTRUCTION: UNDERSTANDINGPERCEPTIONS AND RESPONSES TO CHALLENGES OF CLASSROOM AUTHORITY,PRIMUS: Problems, Resources, and Issues in Mathematics Undergraduate Studies,10:4, 289-318, DOI: 10.1080/10511970008965968

To link to this article: http://dx.doi.org/10.1080/10511970008965968

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NOVICE INSTRUCTORS ANDSTUDENT-CENTERED

INSTRUCTION:UNDERSTANDING PERCEPTIONS

AND RESPONSES TOCHALLENGES OF CLASSROOM

AUTHORITY

Dale Winter! and Carolyn A. Yackel2

ADDRESS: (1) Department of Mathematics, Duke University, Box 90320,Durham NC 27708 USA, amanita<Dmath. duke. edu and (2)Depart­ment of Mathematics, Indiana University, Rawles Hall, BloomingtonIN 47405 USA, cyackeUiu-math.math.indiana.edu

ABSTRACT: We analyze interviews of graduate student instructors todescribe beginning instructors' interpretations of and reactions to stu­dent behavior. The emphasis of this article is on student behavior thatthe instructors regarded as direct challenges to classroom authority.The interpretations and responses of beginning instructors are con­trasted with expert advice offered in teaching literature. We identifypriorities that influence instructors' interpretations of and responsesto these challenges. These priorities are: (1) protecting evaluative au­thority or credibility as assessor, (2) following the regulations of thecourse, (3) protecting rights to make pedagogical decisions, (4) pro­tecting credibility as a person that students can be confident learningmathematics from, and, (5) protecting a basic level of respect for selfand others. We suggest guidelines for course directors and facultymentors to help beginning instructors learn from the difficult situa­tions that they may encounter.

KEYWORDS: Classroom authority, student-centered instruction, class­room interactions, novice instructors.

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1 INTRODUCTION

In a student-centered classroom, students are often encouraged to see them­selves as important participants in the exercise of authority. From the in­structor's point of view, this participation is usually confined to receivingstudent input concerning the direction or content of each class. From thestudents' point of view, a student-centered classroom may be a stark con­trast with the more familiar situation where authority is exercised exclu­sively by the instructor. FUrthermore, students may be uncertain of exactlywhat forms of authority they are permitted to exercise.

In this article, differences between the instructor's and students' pointof view are described as "misalignments." As has been described elsewhere,[25, 26], these misalignments are a potential source of difficult interactionsbetween instructors and students. We are prompted to ask the followingresearch questions:

1. What factors appear to be important to instructors when they inter­pret and respond to such misalignments?

2. How can beginning instructors be helped to understand, respond toand learn from difficult situations in their classrooms?

The purpose of this article is to provide answers to these questions by:

1. conceptualizing difficult situations as the results of misalignments be­tween the instructor's and the students' ideas of who can exercisevarious forms of authority,

2. contrasting the interpretations of beginning instructors with "expert"viewpoints,

3. identifying priorities shaping beginning instructors' interpretations ofand reactions to difficult situations,

4. providing concrete examples of how beginning instructors interpretand react to difficult situations, and

5. suggesting guidelines for helping novice instructors make sense of,learn from and deal with difficult situations.

Much of the published discussion of difficult instructor-student interac­tions (see [3, 6, 15]) focuses on providing advice for dealing with very specificsituations. We observe ([3, p. 133]) that:

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". .. research on teaching excellence to date has been very rudi­mentary, focusing primarily on the identification of 'character­istics' or 'traits' that mark the outstanding teacher... Only re­cently have studies gone beyond vague, general terms to describethe professors, their knowledge or their actions. Few researchershave adequately addressed the need for an overriding theoreti­cal framework to explain the relationship of the traits excellentteachers are said to possess."

This paper represents an attempt to move a discussion of difficult in­structor- student interactions beyond the point of listing characteristics ofteachers who don 't usually experience difficult situations, and to begin todescribe the factors motivating instructors' actions. We hope the presentwork will contribute data and analysis that will eventually be of use for thecomposition of a theoretical framework to describe classroom interactions.

1.1 Organization of this Article

In Section 2, we describe some of the teaching references commonly availableto beginning college mathematics instructors. The advice offered in suchreferences is discussed. We offer a perspective on professional developmentbased on learning theory, and suggest ways in which such "expert advice"may not meet the needs of beginning instructors.

Section 3 contains a description of our methodology and the classroomenvironments that the instructors worked to implement.

In Section 4, five priorities that influence instructor's interpretation ofand response to these challenges are described. These descriptions are or­ganized into the following sub-sections: (1) protecting evaluative authorityor credibility as assessor, (2) following the regulations of the course, (3)protecting rights to make pedagogical decisions, (4) protecting credibilityas a person that students can be confident in learning mathematics from,(5) protecting a basic level of respect for self and others. Each of thesesub-sections includes a description of relevant details about the classroomenvironment, an excerpt from an interview, and an analysis of the contentof the interview. Analysis is included to make explicit the instructors' cri­teria for interpreting and responding to the situation that they describe,and to contrast the instructor's response with "expert" perspectives. Eachsub-section concludes with a set of guidelines summarizing the discussion.

Section 5 contains suggested perspectives on difficult classroom situa­tions and two possible-lines of future inquiry.

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2 COMPARING EXPERT ANDBEGINNER PERSPECTIVES

ON CLASSROOM INTERACTION

The purpose of this section is to use learning theory to suggest that thereferences commonly available to beginning mathematics instructors mayhave limited potential to help beginning instructors to make sense of difficultclassroom situations.

2.1 Learning Theory

There is a clear and useful analogy between instructors learning to dealwith classroom situations and students learning mathematics. Mathemat­ical problems appear to be transparent and immediately solvable to ex­pert mathematicians, who have sophisticated conceptions of the workingsof mathematics. The same problems can be impenetrable to beginning stu­dents . Likewise, teaching problems that appear transparently and immedi­ately solvable to experienced teachers may appear impenetrable to noviceinstructors.

Many instructors realize that dealing with difficult mathematical prob­lems is frequently a highly emotional process for beginning students (see [15,pp. 60-61]). Many instructors recognize that listening to students expresstheir understandings of mathematics and supporting students through theproblem solving process is as important as helping students to manipulatethe mathematical symbols , [22]. Likewise, we feel that it is important tolisten to beginning instructors express their understandings of teaching.

Research strongly suggests that beginning teachers and experiencedteachers tend to interpret classroom situations in radically different ways([3, p. 237]). There are many possible factors that could explain such dif­ferences in perception. For example, novices' feelings of being challengedmight stem from the newness of the position of "instructor" and the un­familiar pressures of being on display. (See, for example [27, pp. 64-65].)It is vital, therefore, to listen for both the facts and for the ways in whichthe beginning instructor interprets and describes the difficult situation. Wesuggest that advice to beginning instructors can only be viable if it is ex­pressed through a system of referents that he or she constructs to makesense of experiences in the classroom. That is, the advice offered must becomprehensible through the intellectual framework that beginning instruc­tors employ to interpret and describe the situation, rather than through theframework that an expert might use to describe the situation.

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One aim of mathematics instruction is to help students to construct ameaningful understanding of mathematics, rather than to provide studentswith "cook book" recipes for dealing with particular classes of mathematicalproblems. Similarly, one aim of professional development is to help instruc­tors to construct a meaningful understanding of the classroom environment,rather than to provide instructors with "cook book" recipes for dealing withparticular classroom situations.

The idea of meaningful understanding that we employ here is due toAusubel, [2], who argues that its development is only possible when threecriteria are met:

1. the material itself must have potential meaning,

2. the learner must already possess relevant concepts to anchor the newideas, and

3. the learner must voluntarily choose to incorporate the new knowledgein a nonarbitrary, nonverbatim fashion.

In the next sub-section, this learning theory is employed to analyze someof the references commonly available to college mathematics instructors.

2.2 Analyzing References for Instructors

There is very little printed material conveniently available to college math­ematics instructors regarding difficult instructor-student interactions, andeven less focusing on the student-centered classroom. The material that isavailable (see, for example, "The Torch or the Firehose?" included in [6])usually consists of sets of advice for dealing with particular instances of stu­dent behavior. One of the more comprehensive, focused and concrete sets ofadvice is included in [31. This collection of advice is based on the reactionsof distinguished and very experienced college teachers to common collegeteaching problems. We discuss how such sources can fail to meet Ausubel'scriteria through an analysis of the following excerpt from [3, p. 2431 .

• Problem: "I have problems with students not taking me seriously andfeeling they can verbally abuse me or physically intimidate me."

• Response (Professor D.): "Again, this is a Day One situation. Youhave to exude that you are in charge and are the authority. If youtry to pose as some friend of the students and just the good guy orgal, the students will walk all over you. Beyond that I would wantto know specifically what type of abuse or intimidation the person istaking about to respond more pointedly."

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Specific problems that we see here include the following. (The pointsunder '1' correspond to Ausubel's first criteria, etc.)

1. (a) Poor problem definition or lack of substantive detail. Thelack of information supplied to the experts precludes a responsethat has relevance or potential meaning for the beginning instruc­tor. This is shown when Professor D. notes that he would needto know more of the particulars of the verbal and physical abuseto speak more pointedly. Clearly, descriptions of situations andadvice can only be meaningful to beginning instructors if thereis enough information for the beginning instructor to see what isgoing on.

(b) The general discussion that exists does not interpretevents with the perspective of the beginning instructor,nor does the discussion probe the question of what be­ginners find most significant in those events. As indicatedin [3, p. 237] , with further information, experienced instructorsoften interpret difficult situations in radically different terms thanbeginners do. In this sense, Professor Do's perspective and com­ments are typical of these types of materials. They are voicedfrom the expert point of view. As noted, interpretations of dif­ficult situations can only have potential meaning to beginninginstructors if beginning instructors can relate their experiencesto the interpretations. We hope that this article will serve asa set of worked examples for how beginning instructors expressthemselves, and how meaning can be distilled from descriptionsof troubling situations.

2. Beginners may lack the necessary language and conceptionsof classroom interaction to anchor the ideas offered by ex­perts. For example, Professor D. advises instructors to "exude thatyou are in charge and that you are the authority." This sounds valid,but it is also vague. Exactly how does one exude the proper message?Dress formally? Speak loudly? Play the autocrat? Although Pro­fessor D. could undoubtedly act in the manner he has in mind, it isunclear that a beginning instructor could replicate the actions and re­sults, based on the three sentences offered above. We suggest that theterms that Professor D. uses to express his thoughts are meaningfulto him, but will probably not be similarly meaningful for beginningteachers. In terms of the Ausubelian framework, beginning instructorswill not have relevant concepts to anchor new ideas about classroom

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interactions. In one-on-one interaction, mentors can help beginninginstructors become aware of concepts that they do have by helpingthe beginning instructors isolate these concepts and then find labelsto describe them. This article provides insight into the ways that be­ginners conceptualize difficult interactions, the ways that beginnersexpress their thoughts on interactions, and a system for interpretingand categorizing such thoughts.

3. We assume that beginning instructors will be more likely tovoluntarily incorporate new ideas about classroom interac­tion if the ideas are expressed in ways that are relevant tothe instructor concerned. A familiar experience in professionaldevelopment is the case of an instructor who needs help, but is un­willing to receive help in the form offered. In particular, instructorshave been known to dismiss credible suggestions with statements suchas, "That doesn't sound like something I think I could do," or, "Thatisn't really compatible with my teaching style." Such responses can beseen as critiques of proffered advice, or they can be seen as rejectionsof suggestions because the suggestion was not expressed in terms towhich the instructor could relate. The inherent inflexibility of writtenmaterial does not allow for essential modification of wording of adviceto adapt to a particular instructor's understanding of his or her styleand classroom.

In our experience , very difficult interpersonal encounters between in­structors and students are not widely discussed, except perhaps duringhighly informal "bull" sessions. There appears to be relatively little dis­cussion of these kinds of teaching issues in general, particularly where theinstructors perceive that classroom authority is an issue. In many instances,our interviews were the first time that these graduate student instructorshad told anybody about their experiences, and the first time that they hadattempted to relate their thoughts and interpretations to anyone in a seriousand meaningful way. We found that beginners' attempts to process difficultteaching situations are usually characterized by a high level of emotional in­volvement, and a low inventory of conceptions of the classroom environmentand the significant factors in its operation.

3 METHODOLOGY

This study was initiated in the winter of 1998. In contrast to other investiga­tions that have described some of the practical difficulties of implementing

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the techniques of student-centered instruction, (see [8]), such as in-classcooperative and active learning, the goal of this investigation is to betterunderstand the nature of the interactions between instructor and student,especially those for which the instructors found it difficult to understand,process or respond to adequately.

Our data was collected via a program of clinical interviews. Our sub­jects were the graduate student instructors of the mathematics departmentat a large , research-oriented, Mid-western state university. We conducted70 interviews, a near census of mathematics graduate students with class­room teaching experience. Fewer than ten of the instructors were eitherunavailable or unwilling to participate in the study. Our specific interviewprotocols are described in Section 3.2 and Appendix A.

The graduate students interviewed had all taught in either precalcu­lus or introductory calculus service courses. These courses, closely alignedwith the so-called Calculus Reform Movement, were not directed towardsmathematics majors, and are not mathematically rigorous. However, thecourses emphasized intellectual rigor by requiring students to solve difficult,non-routine problems and to clearly describe the reasons for each step oftheir work. Students were expected to actively participate in the class, andall had access to technology (in the form of hand-held graphing calcula­tors) during each lesson. Instructors were encouraged to use alternativesto the lecture method whenever possible, and were provided with a weeklong pre-semester training workshop that described and demonstrated somealternatives to lecturing such as cooperative group work.

The focus of this article is on the situations that instructors characterizedas challenges to classroom authority. During the interviews, the instructorscommented on a much wider range of classroom situations that might becharacterized as difficult. For example, 93% of instructors reported thatstudents would chatter in class at inappropriate times. One advantage offocusing on challenges is that these experiences stood out in the instructors'minds, and instructors were able to remember and describe their feelings,interpretations and actions much more clearly than for routine annoyancessuch as classroom chatter.

3.1 The Nature of the Classrooms - Teaching Methods

In their classrooms, the graduate student instructors were attempting to im­plement ideas that have been described in [8, 18, 24]. Ideally, a class mightconsist of several mini-lectures interspersed with individual, pair or group­based activities. The mini-lectures are usually no longer than ten minutesin duration; the activities were intended to deepen students' comprehension

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of the subject matter discussed or to guide students in the "discovery" ofmathematics. During in-class group work, instructors would typically as­sign students problems from the course text to complete in groups of threeor four. While the students worked, instructors would circulate and try toclarify specific issues. Usually, the instructors provided guidance and hints,tried to correct misconceptions, and ensured that the pace of the class andtreatment of the subject matter were appropriate for the course. The classesalso used group homework, where teams of three to five students would meetonce or twice per week outside of class time to complete four challenging,non-routine problems.

3.2 Data Collection and Analysis

The individuals interviewed varied from first year students with one ortwo semesters of teaching experience to experienced instructors with tensemesters of classroom teaching experience. During the time at which thisstudy was conducted, both authors were also mathematics graduate stu­dents. One of the authors (D. W.) had worked as a course director andstudent mentor, and had personally supervised 20 teaching assistants .

Because of the sensitivity of the topics of discussion, and the diffi­culty that some of the instructors had formulating and communicatingtheir thoughts, we found the most suitable data collection method to besemi-structured, clinical interviews. Interviews were conducted with oneinstructor at a time, and with either one or two interviewers present fol­lowing the protocols described in Appendix A. When two interviewers werepresent, one took the lead as questioner, while the other simply took notesand asked occasional questions. Interviews were recorded on audiocassette,with the interviewer(s) taking notes at the same time. Interviews generallylasted between forty minutes and two hours, with one hour being typical.

The interviewers' notes were used in two ways. Firstly, the situationsdescribed by the instructors were coded and sorted into 34 categories. Theprevalence of situations in each category was noted. (These categories andthe frequency information are described in [30]; some of the categories aresimilar to those described in [23, Chapter 4].) Secondly, the notes wereused to identify particularly detailed descriptions of situations. The audio­cassettes of these descriptions were transcribed.

Our analytical methods can be characterized as the constant compara­tive method of grounded theory (see [13, 16]). In particular, we examinedthe transcripts of interviews where instructors had been able to describe hisor her understanding of classroom situations in some detail. We attemptedto find and abstract the assumptions, considerations and factors that the

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instructors had been able to explicitly identify. These were grouped and theanalytic categories emerged.

4 REPORTS OF INSTRUCTORS'INTERACTIONS WITH STUDENTS

This section contains excerpts from interviews and the analysis of each ex­cept. The excerpts are reproduced from transcripts of interviews with in­structors. These excerpts are close to verbatim, with the only changes beingthe removal of particularly distracting vocalized pauses.

This section is divided into five sub-sections, each representing one an­alytic category derived from the interview data. These analytic categoriesdescribe the major priorities of beginning instructors when dealing withdifficult classroom situations. As noted in the introduction, the term "mis­alignment" is used to describe a difference between the instructor's andstudents' point of view on who is permitted to exercise various forms ofauthority.

4.1 Category 1: Protecting evaluative authority orcredibility as assessor of students

A majority of instructor recollections could be placed in this category. Otherreferences almost unfailingly include advice for such situations (see [3, 6, 15,28]).

The interview fragment included here deals with a misalignment in whichthe instructor sees herself as able to make decisions regarding an individualstudent 's work, but the student has uncovered some kind of "evidence" toquestion the instructor's decisions. A second kind of authority misalignmentoccurs when the student simply believes that he or she can judge the valueof student work as competently as the instructor. Often this second kindof misalignment surfaces around evaluative decisions of partial credit. Themost complicated form of evaluative misalignment (see [28]) arises whenthere is a significant difference between the instructor's and student's ideasof what evaluation actually means or the criteria that should be used tomake evaluative decisions.

4.1.1 Interview Excerpt

The instructor describes a situation that occurred after the first course-wideexam. Common exams were given to all sections of the course three timesduring the semester. Each problem was graded by a team of instructors to

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try to ensure some kind of consistency in grading. As a result, individualinstructors usually did not know how their class had done until the gradingwas completed and they had a chance to review their students' papers.

Instructor: ... after the first exam had been graded the firstsemester that I was teaching and I hadn't had a chanceto look at the exams yet, a couple of students dropped byto see what their grade was. And his had been, I think Iremember that it was a 91 or something. And then afterI had looked at it , urn, a question had been graded incor­rectly, and he actually got an 89. But he'd already seen the91. .. So what happened with this kid was they were allowedto look at their exams and think, "Oh, I need, this needs tobe regraded or I disagree with how, you know, somethinghappened". So they were to write a little thing saying Ithink this problem was misgraded and you need to look atit more closely. He came back with this five page letter.(Laughs) And that wasn't something I felt at all equippedto handle on my own. I think there was a PS from' theguy: "I hope that you've considered my arguments." Youknow, almost a penitent PS, after this, after this very selfimportant, you know, diatribe, or whatever. Yeah, and hesaid, "The first person who graded it said , 'What a greatanswer.' And you came along and said, 'No. What a not­great answer." , Yeah, I definitely didn't know what to dowith him ... I brought that to ... [the course director's] at­tention. And, you know, I think [the course director] reallywanted to go put that kid in his place. But it never actuallycame to that. Which was too bad . But it probably wouldhave damaged the relationship that I had with [the student]for the rest of the semester .. . My first response [might havebeen] probably to send him off some kind of a smart reply,and that wasn't the right thing to do. I was a little incredu­lous. More than a little incredulous. Again, because that'snot something I would ever have attempted to do myself.In, in any student-teacher relationship. Whether they werea graduate student or not. That's the kind of, that's thekind of move that I don't think I could ever pull off. WhatI'm getting at is that this person, I think, clearly crossedsome line with me, because it' was a bullying and intim­idating tactic. That the explanation of why the problem

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was actually wrong wasn't enough. And that some kind ofunusually aggressive persuasion was called for on his part.

The central issue here is that the instructor felt that she had the au­thority to make evaluative decisions regarding students' work. Note thatshe believed that she had the authority to review the evaluative decisionsmade by other instructors, and to change them if she saw them as mistakenor inappropriate. The instructor felt as though the issue was the individualinstructor's legitimacy or credibility as a person who can make decisionsabout students' work. The specific argument that she recalled was onewhere the student said, "Someone else thought that this was worth twopoints... " (emphasis added). The issue in the instructor's mind was notwhether the two points are warranted, but whether she could make thatdecision.

The instructor had implemented a policy of asking the students to writesome kind of explanatory note if they believed that any part of their examwas incorrectly graded. (Such a policy was a standard suggestion given toinstructors during their training.) Students may have interpreted this policyas an invitation to participate in the exercise of evaluative authority in theclass. When the other students handed in their brief notes, the student whowent from a 91 to an 89 turned in a five page, single-spaced, typed essaywith several arguments claming that the two points should be restored. Theinstructor found this student's actions to be inappropriate and described histactics as "bullying and intimidating." The instructor also characterized thecontent of the student's paper as self-important, and the paper itself as a"diatribe." The instructor indicated that she does not find the student'sarguments compelling.

The instructor felt that she did the correct thing. She clearly felt thatshe did not overstep the boundaries of her evaluative authority. However,she also felt that she had made a mistake, not by correcting the gradingerror, but by letting the student see the uncorrected grade. On one hand shewas convinced that she acted properly, fairly and correctly. On the otherhand, she believed that she had made a mistake. This is an important pointof difference between expert and novice points of view. Whereas the expertmight view the situation as a student acting as "students always do" (see[3, p. 245]) - that is, the instructor bears no fault - this instructor tookresponsibility for the situation.

The instructor also said that she was incredulous over the extent of thestudent's behavior. She saw this as something that she would never havebeen able to do herself. She was using herself as a model for how studentsshould behave towards instructors, and she characterized the student as

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"crossing some kind of a line with her." Her conception of herself as amodel for student behavior may be what created this line. This is a commondifference between the expert and novice points of view. Whereas the experthas developed an impression of the ways in which instructor and studentdiffer in their attitudes towards and beliefs about appropriate behavior (see[22]), novices generally have not had the opportunity to develop such apicture.

To summarize this discussion:

• Beginning instructors may feel unprepared for dealing with the passionof students who are interested in obtaining higher grades for theirwork.

• Beginning instructors may lack appropriate models for interpretingand understanding student behavior.

• Beginning instructors may assume too much responsibility for situa­tions that occur in their classes.

4.2 Category 2: Protecting authority to make or en-force the regulations of the course

This category includes situations where the instructor feels that he or sheshould be able to determine how the course will work - policies on gradingconventions, class attendance, and the like - and yet the students feel thatthey should have a strong voice in making such decisions as well. Here wedescribe situations where the site of authority misalignment is the admin­istrative decision itself, rather than the implementation of the decision.

Instructor: I was trying to deal with an attendance issue. I wasseeing attendance lower than I expected. I was annoyed byit and I wanted to [do something] and I announced that Iwould be giving in-class quizzes and I also told them, whichwas probably a mistake, that the reason was to encourageattendance. . .. The students were already feeling indignantabout being in the classroom and felt incredibly indignantabout this new policy, and openly, intentionally, verballychallenged me in the classroom as I was trying to lecture,and I was trying to go on with the class. He had decidednot to let me, and he had decided not to let me do thispolicy. I argued with him for a couple of minutes in themiddle of class, and then said, "This is getting out of hand,

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I'll talk to you after class." We stayed around after classfor about, well probably over an hour with me trying toconvince him. At first I was trying to convince him thatthis was a good policy, and then I just decided that my rolewas to convince him that I had the authority to make thispolicy, because he just strongly believed that I just didn'thave the authority to make this policy and I convinced himotherwise by showing him that he was wrong. (Laughs)

Interviewer: What did you say?

Instructor: The course pack discusses authority of the teacherto make attendance decisions. It says very clearly that theprofessor has the authority to do things far harsher thanwhat I was doing. So he backed off that. But that was aproblem for us for another week or so. I could just tell thathe was irritated at me. I mean, before he was answeringquestions and participating and he sort of backed off fromthese things.

The instructor saw the major issue as his authority to set policy. At leastone of the students believed that the class should have veto power over theinstructor's decisions. The administrative structures that the university haddefined were very clear on this point (at least in regard to class attendancepolicies): the instructor could take quite severe steps to ensure that studentsattend class. An experienced instructor might simply note that the studentis unfamiliar with the university's regulations, and point the regulation outto the student (which is what this instructor eventually does). However,the situation is not so straightforward for this instructor.

At first, the instructor did not invoke the university regulations, butinstead tried to persuade the student that the policy was a good idea. How­ever, the instructor eventually realized that the real issue here was notwhether the policy was good or bad, but who had the authority to decidewhat class policy would be. The instructor recognized that on a basic level,either the student refused to acknowledge the instructor's authority to setpolicy, or the student felt that he should be a co-participant in the exerciseof such authority. The instructor eventually gave up trying to persuade thestudent, and focused on the authority he had been granted by the university(as documented in the "course pack" - a booklet that sets out the rules ofthe course and contains supplementary materials and old exams).

It is interesting to note that this instructor was flexible and calm enoughto change tactics amid the ruckus. This was manifested first in his decision

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to quit trying to convince the student and instead to table the discussionuntil after class. It came up again in the instructor's decision to abandonefforts to persuade the student to believe in the policy and instead provethat he has the authority to make policy. Although earlier the studentdismissed the instructor as a legitimate policy-maker, he was prepared torecognize the instructor's authority once proof of university endorsementwas produced.

In summary:

• Although beginning instructors are undoubtedly aware that studentshave a vested interest in class policy, they may not anticipate students'reactions to those policies. In part, this can result from the instructorusing his or her own behavior as a model for typical student behavioras discussed in Section 4.1.

• Beginning instructors may not recognize that some students will notsee them as legitimate holders of authority. Beginners may assumethat their position as "instructor" implicitly conveys this to students.

• Beginning instructors need to be made aware of the university's poli­cies.

• This instructor remained calm. Although emotional reactions are typ­ical of beginning instructors, the deft way in which this instructor wasable to understand and respond to the situation demonstrates thatall instructors (beginner or expert) are capable of acting effectively indifficult circumstances.

4.3 Category 3: Protecting authority to make decisionsabout how lessons will be constructed and run

This category includes situations where the instructor believes that he or shehas the authority to make decisions about how lessons will be constructedand run. In contrast, the students believe that they should be able todetermine what goes on in the classroom, and act to try to disrupt orfrustrate the instructor's plans.

Because of the unfamiliar atmosphere student-centered instruction cre­ates for many mathematics students, this category may be particularly im­portant for mathematics classes emphasizing student-centered instruction.Many students hold limiting beliefs about the nature of mathematics, (see[12, 20, 21]), that conflict with techniques used in student-centered class­rooms. Such beliefs have implications regarding the ways that mathematicsmust be learned and taught. Generally speaking , beliefs such as:

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"Almost all mathematics problems can be solved by the directapplication of the facts, rules, formulas and procedures shownby the teacher or given in the textbook. ([12, p. 502]"

may leave students feeling confused, frustrated and disoriented by a student­centered approach to mathematics (see [4, 5]). An expert might interpretthis misalignment as a result of the instructor's desire to innovate and thestudents' vested interest in the status quo (see [11, 22]).

4.3.1 Interview Excerpt

In this class the instructor felt that there was a core of students who evi­dently knew each other very well, simply did not like the instructor, and wereout to get him. The instructor described the class as very confrontationalfrom the beginning of the semester. On several occasions, the instructorhad addressed the class as a whole, recognized that some of the people inthe class were already very familiar with the subject matter, and told themthey were under no compulsion to attend class, if they didn 't feel it wasvaluable for them. However, the students who persistently chatted duringclass still dutifully came to class each day.

Instructor: It got to the point where I was shouting in order tobe heard. Then I'd get all ticked off and decide that no, I'dwhisper , and if they want to hear me they can quiet down.That didn 't do any good, because they apparently didn'twant to hear me. But then the real problem is the peoplewho aren't causing trouble. They get really irritated, be­cause they're not getting any good out of this class, andthen they don't like you either, and things just spiral to­wards oblivion.

Interviewer: Did they say things to you? Sometimes studentssay something like, you know, "We can't hear you. Canyou do something about those students?"

Instructor: Yeah, I got some responses like that. Like, youknow, "You need to do something about this." But theybasically felt like I was incompetent. I mean, if this werea high school class, I'd send them to the principal's office,or something. But what do you do in college? .. . On acouple of occasions I told someone specifically to shut up.That didn 't go over well, either. The person sort of satthere with a screwed up face for ten minutes and then went

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right back to what he was doing. Meanwhile, everyone elsethought I was being mean. In the end, it didn't changeanything.

A first reading of this interview fragment might suggest the problem tobe "control in the classroom." That is, the instructor is not exerting suffi­ciently aggressive psychological or social control mechanisms to contain thedisruptive students' behavior. We suggest that he did exert control mech­anisms that are as aggressive as anything suggested in printed references(see [15, p. 64] for an exception involving R. L. Moore) . The instructor toldoffending students to "Shut up" or asked them to leave the class, yet thedisruptive behavior continued. We see the root cause as a misalignment be­tween the instructor's view that he had the authority to decide the courseof classroom events and the students' views that they should be able toinfluence the course of classroom events to a large extent.

The instructor felt that the students who were persistently talking weremaking it very difficult for him to do what he saw as his job in the classroom- to communicate the meaning and uses of the mathematical subject matterto those students who wished to pay attention and try to get somethingout of the class. Secondly, the instructor felt that other students in theclass expected him to wield some sort of authority on their behalf, butthat his efforts to do so were ineffective. Lastly, the instructor felt thathis effectiveness as a teacher - certainly in the eyes of the students - wassubstantially diminished. The instructor believed that the attitudes of thestudents in the class regarding his capabilities were shaped more by thebelligerence of students who refused to cooperate, rather than by his abilitiesto plan and conduct an effective lesson.

To summarize this discussion,

• Beginning instructors (especially those who attempt student-centeredmethods) may not be aware of the intellectual or emotional obstaclesthat students may face in learning to function in such an environment.

• Beginning instructors may feel what Finkel and Monk term the "AtlasComplex," [11]. In particular, beginners may feel personally respon­sible for controlling every aspect of the classroom.

• It cannot be assumed that simply because someone is the instructor,every student will cooperate with the person's goals. Beginners mayneed to receive some indications of how to set the tone of the class­room , or how to define appropriate forms of classroom participation,and then communicate these to students (see [8, 29]).

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• It is an exceptional individual (beginner or expert) who can receivethis kind of response from students and single-handedly resolve toreinvent their approach to teaching in positive and credible way. In­structors need to be supported through difficult experiences so thatthey have the opportunity to learn from their experiences, rather thanto become cynical, jaded and uninspired.

4.4 Category 4: Protecting credibility as someone fromwhom students can be confident in learning math­ematics

One of the impor tant premises of Finkel and Monk's article, [11], is thatinstructors must find new ways to interact with students, rather than tocontrol the classroom through their subject matter knowledge. Given stu­dent beliefs about mathematics similar to those reported in [12, 20, 21], thiscan be a significant undertaking for both instructor and students. Develop­ing a meaningful understanding of mathematical concepts is a difficult andemotional process for many students ([19]). In order to supply the effortnecessary to accomplish difficult and emotionally demanding tasks, stu­dents need to (a) see value in completing the task, and (b) feel that thereis a reasonable chance that they will be able to complete the task ([17]). Inpractice (as well as in theory, see [31]) it usually falls to the instructor tohelp students in these ways. In order to do so effectively, students need tosee that the instructor is capable of helping them to complete the task oflearning mathematics. The misalignment that we see here is not so mucha misalignment of the ability to exercise authority, but a misalignment ofperceptions. In particular, the instructor will usually perceive him or her­self as able to help students learn mathematics, but students may not besimilarly convinced.

4.4.1 Interview Excerpt

In this excerpt from an interview, the instructor described a situation andhis feelings regarding a class period about a section of material that somestudents had previously learned in a particular way (a way that seeminglyemphasized procedural competence, rather than conceptual understanding),but which other students in the class had never encountered.

Instructor: In the previous situation, I had a lot of troublewith that [challenging the validity of the instructor's viewof the mathematics] ... I tried to sort of properly tell the

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class ... if you've seen it in a different way in high school,will you come and see me about it or don't worry aboutit, we can discuss how your way is different. I just didn'twant people sort of blurting out in the middle of the class,"Oh no, that's not the way we did it in high school, that'swrong." It feels like a courtroom situation, you know. Youcan say something and there'll be an objection so what yousaid is overruled, but it's still in the jury's head. You can'tget it out of there . . . You just have to say one sentence andthen that's tested the faith of my students.

Interviewer: When this was happening, how did you feel aboutit?

Instructor: My feelings before were total frustration. My classhas no faith in me anymore. I can't communicate to themanymore, because they don't believe me. I'm just standingthere talking at the wall. My feeling was I've got to getrid of the dissidents .. . and be left with the masses whoare fickle and they'll follow whoever stands up and gives aspeech. So I had to get rid of the speech givers, and havethe fickle masses again, who are unsure of themselves, andthen I'll make them sure , you know, regain their confidencein me.

In this excerpt the same instructor described a similar situation from asubsequent semester.

Instructor: I found like last semester again I had a small argu­ment in the first week about inverse functions with a coupleof students. I managed to discuss it in class, openly, andsaid, "Okay look this is what you had in high school; thisis what I'm doing," and they got the message ... It was adiscussion, but one of the students got pretty heated, andso I describe it as an argument, because after class it didbecome an argument. She was adamant that this is theway she had done it.

Interviewer: When she did get emotional or just sort of overlyinvolved in this thing, what did you do?

Instructor: Well, I just did it logically. I just wrote an equationon the board and said, "Do you agree with this?" And shesaid, "Yes." And I wrote the next equation, "Do you agree

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with this?" And she said, "Yes." Three steps later, she hadagreed with something that I had demonstrated, but it wasopposite to what she was saying, so she said, "Yeah, but no,no that's not the way." So I pretty much mathematicallyproved to her that what I was doing was correct, and shesaid, "Yeah, but I did it this way." So then she acceptedmine as correct, but she said that, "No, no," that her waywas different and better. But then I said, "Oh, well they'reactually equivalent so neither way is better," and so thenshe came across and she converted .. . When I presented itto the class, I said, "So these are the two different thingsyou can do. You can do whatever you want ... as long asit is consistent and you don 't mix up two different things."

Interviewer: When this was happening, how did you feel aboutit?

Instructor: How did I feel? .. I did manage to say, "Okay, whydon't we discuss this after class?" When I had her one-on­one, I felt a lot more comfortable . I could convince her, [itwas] just a matter of presenting the things clearly.

In the interview fragments that we present in this section, it may appearthat the instructor is attempting to control the class through his knowledgeof mathematics. We argue that this is not the case - at least in the mostobvious interpretations. In particular, we note that the instructor was care­ful not to supplant notions of the subject matter that students alreadyhad; he encouraged students to use those alternative notions as they sawfit. The ways in which the instructor attempted to convince his studentswere far subtler. The interview fragment seems to reveal a student whohad developed what theorists might call an action or process conception ofthe mathematics (see [1]). That is, the student saw the mathematics as aprocedure or action to be performed. When the instructor suggested whatappears to be a different procedure for the same mathematics, the studentclung to her notion, claiming that it was "bet ter." The instructor attemptedto help the student move to a more sophisticated view of the mathematicalconcept of the inverse (perhaps the object or schema conception describedin [1]). The instructor did not insist that the student had do things "hisway," instead he attempted to show the student that both ways were simplydifferent expressions of the same mathematical concept .

In the instructor's opinion, he had to fight an uphill battle to help thestudents who needed help because their impressions of him as a helpful, com­petent individual were undermined by the constant interruptions of students

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who believed that they knew the subject matter better than the instructor(when in reality they had a much narrower, rigid and procedure-based viewof the subject matter). Instead of trying to understand the mathematicalviewpoint that the instructor suggested, these students simply adopted theknee-jerk response of claiming that the instructor was wrong if he didn'tpresent the material with exactly the same emphasis and procedures thatthese students were familiar. This complicated the instructor's job by ne­cessitating a delicate presentation of topics some students had seen before,because many of these students seemed unable to tolerate a fresh or unfa­miliar approach to the material.

The instructor felt that the students' interjections stripped him of whathe saw as a critically important prerequisite for students to be able to learnfrom him. The students needed some level of confidence in his decisionsabout what were appropriate ways to present and engage them with thesubject matter. The instructor felt that his credibility was undermined,because there were students who said that he is wrong. This behaviorinterfered with his ability to relate the subject matter, and it interferedwith other students' learning. The students in the class not only didn'tknow the mathematics being discussed, they didn't know who to trust forhelp . It might have appeared, because of the continuing disputes, that theinstructor was incompetent in terms of instructional skills, incompetent atmaking decisions regarding how the subject matter should be treated in theclass, or simply did not know the subject matter in sufficient depth to relateit effectively. In any case, the iinage of the instructor as someone who canhelp students comprehend the subject matter may have been undermined.

The instructor felt much more comfortable dealing with this situationone-on-one with the particular student concerned, rather than in front ofthe entire class. The instructor wanted to guide the student through hisversion of the subject matter logically, clearly, and at a pace set by thestudent. He clearly thought that if the situation could simply be made lesspublic, so that there was no contest between himself and the student, withthe rest of the class as spectators, then he would not have to concentrateon defending his credibility or his approach to the mathematics. Insteadhe could concentrate on presenting his view of the subject matter in waysthat would help this student see why "his way" and "her way" were notin conflict. The instructor felt that it was important to win the studentover in some sense - to get her on his side, to "convert" her to the way ofapproaching the subject matter that he decided to use.

Even when the instructor provided indisputable mathematical proof thatboth approaches to the subject matter were equally valid, and the student

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had accepted his proof as a valid demonstration of their equivalence, thestudent still belligerently asserted that her way was different and better.The student, perhaps, saw this as a competition between herself and theinstructor, and she wanted to win. Even when the instructor showed thatthere were no winners (or that they were both winners), the student createda new criteria (although she did not explicitly say what the criteria was)and then asserted that her way is better. It is not clear that the instructorcould ever have reached a satisfactory resolution of this situation until thestudent decided to participate by ending the competition. Had the situationtaken place in front of the entire class, then the stakes of winning or losingmay have seemed much higher. The student may have tried to hold outuntil the instructor capitulated due to other pressures, such as the need toinvolve other students in the class, and the need to complete treating thesubject matter allotted for that class period.

From this discussion we note the following,

• A major point is that when the instructor took the time to reflect onthe difficult situation reported in the first interview fragment, we wasable to learn from this . In particular, in the second instance,

- The instructor thought about how to handle the situation, andmanaged it in a way that he felt more comfortable with.

- The instructor realized that he had to identify priorities (studentunderstanding, use of class time, confusion that might be causedfor other students) and then decide how to juggle them whenresponding to the student.

- The instructor has developed a concrete plan for how he wouldgo about addressing the situation.

- The instructor is proactive in his approach, sensing trouble andaddressing it before it has become a serious issue for the class asa whole

• Beginning instructors may feel that their "legitimacy" as the class­room instructor is very fragile, and see it as something that can veryeasily be damaged or lost altogether.

• Beginning instructors may not have appropriate models for students'mathematical notions, or for students relative lack of mathematicalsophistication. In particular, beginners may use their own levels ofmathematical sophistication as models for students. This may besimilar to beginning instructors' use of themselves as models of studentbehavior (see Sections 4.1 and 4.2).

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• Beginning instructors may see mathematical inquiry as the only cred­ible use of class time. In particular, they may see as irresponsible anyuse of class time that is aimed at establishing credibility or establish­ing parameters for student contributions to the class.

4.5 Category 5: Protecting a basic level of respect forself and others

This category includes instances of students behaving, either towards theinstructor or towards each other, in ways the instructor perceives as disre­spectful. We describe a scenario in which the instructor feels a student'sdisrespect is directed at him.

Instructor: One day they had to do a group homework. Andthis is my best interpretation of this. But everybody gotstarted, and this group hadn't started. They were justsitting around chatting, and making jokes. I think at onepoint he was doing some sort of mock imitation of me. Andhe's doing this. And I'm sitting down trying to explain tothis one student at this table how to get started on thisproblem. And this guy's joking and making fun of me. Ireally considered that a flagrant challenge of my authority.I don't think there's any clearer challenge of authority thanto not be doing the work the person presents to you, theperson's standing right there in your face trying to workwith you, and you're actually making fun of him.

Interviewer: What did you do?

Instructor: I just looked at him stone cold and expressionless .

Interviewer: Did it work?

Instructor: Oh yeah.

The instructor perceived this situation on two levels. Firstly, the instruc­tor saw the students' actions as disrespectful because the student was notdoing the activity that the instructor had assigned. Secondly, the instruc­tor felt that the student's imitation was inappropriate for the classroomsituation. These two levels translate directly as two distinct authority mis­alignments. Firstly, at least some of the students clearly saw themselvesas having the authority to decide how they would spend their time in theclassroom, whereas the instructor believed that the students should have

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used their time to work on the learning activities that he had assigned. Sec­ondly, at least some of the students saw the terms of the instructor-studentrelationship as something that they could determine. Under this student'srules, in-class parodies of the instructor were permitted. The instructorfelt that this was unacceptable both in terms of the instructor-student re­lationship and, more widely, in terms of showing a basic level of respect forothers.

We might be tempted to objectify the instructor's concerns by suggestingthat he was trying to maximize the value of class time for student learning.However, this does not reflect the emotional content that the situation heldfor the instructor. The problem that the instructor saw was not based onstudents wasting class time. Instead, the instructor saw the problem asstemming from the fact that he had told the students to do something, andyet this one individual is not doing it. This student's behavior was evenmore problematic for the instructor when it included what the instructorsaw as a display of disrespect. Notice that this instructor did nothing toaddress the way that the student was wasting class time - instead he focusedon the fact that the student had disobeyed him, and then begun to makefun of him. The extent to which this emotional content determined theinstructor's response can be seen by observing that instead of remindingthe student that there is a job to be completed, the instructor tried theaggressive maneuver of staring the student down.

From this discussion, we note the following:

• Classroom situations often involve a significant emotional componentfor the beginning instructor.

• It is possible for this emotional component to be an important factorin shaping the instructor's response to the situation.

• Instructors may have to take some kind of proactive steps to commu­nicate the levels of respect for others that they see as appropriate forthe classroom (see [7, 14, 29, 32]).

5 CONCLUSION: NEW THOUGHTS ON"AUTHORITY ISSUES"

We have identified five priorities that novice instructors follow when inter­preting and deciding how to respond to classroom situations. Specifically,these priorities were: (1) protecting evaluative authority or credibility asassessor, (2) following the regulations of the course, (3) protecting rights

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to make pedagogical decisions, (4) protecting credibility as a person thatstudents can be confident in learning mathematics from, (5) protecting abasic level of respect for self and others. We suggest that difficulties mayarise from the critical misalignment of the instructor's and the students'ideas about who is free to exercise authority in the classroom.

Our work with novice instructors suggests a perspective on difficult sit­uations that we have not been able to find elsewhere in the literature. Thisperspective is that difficult situations represent learning opportunities forinstructors, rather than problems that must be fixed. In our work, theinstructors learned from the situations that they experienced (see, for ex­ample , Section 4.4), and reported that they were able to handle themselvesmore satisfactorily when similar circumstances occurred again. Here sat­isfactorily means that the instructors' actions were more in accord withthe criteria that they saw as important for their classes. Our work withthese instructors strongly suggests that this learning occurs during a periodof reflection where the instructor attempts to identify concerns and decidewhich concerns are of paramount importance. As indicated in Section 2, wesuggest that course directors and faculty mentors can do the most to helpbeginning instructors by facilitating this kind of learning, rather than byproviding "answers." It is the job of the mentor to listen carefully as theinstructor relates his or her story, and then to help the instructor abstract orinterpret the situation. Instructors should be encouraged to clarify goals,decide what goals are most important, and then to take actions that areconsistent with those goals. We suggest that instructors try to stop feelingthat they need to "fix everything," and to stop assigning the same priorityto every minute thought.

We suggest that beginning instructors need clear models to follow tohelp them learn to find alternative ways of interacting with students inthe student-centered classroom, and that the development of such modelscould form a useful line of future investigation. As Fennema and Frankepoint out ([10, p. 160]), novice instructors tend to teach in the same wayas they were taught. DeLong and Winter note in [8] that many beginninginstructors have no experience of student-centered classrooms, either as in­structors or as students themselves. The implication is clear. We imaginethat teaching models might include detailed descriptions of the classroompractice, underlying beliefs of distinguished instructors (see [3]), explicitstatements of both what instructors should be able expect from students,and what students should be able to expect from instructors, their peersand the student-centered classroom environment (see [29]).

Finally, we note that although the present work focuses on the instruc-

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tors' point of view in difficult situations, we feel that it is suggestive ofother perspectives. We suggest that challenging behavior from students canbe viewed not as attempts to derail the class or to threats to the positionof the instructor, but as manifestations of a higher learning process (seeSections 4.4 and 4.5). Woods (reported in [9]) observes that students whoare asked to take greater responsibility for their learning often pass throughsome of the steps that are associated with trauma or grief (shock, denial,anger/strong emotion, resistance and withdrawal, surrender and acceptance,return of confidence, and success). The printed references commonly avail­able to mathematics instructors implicitly regard difficult situations as tem­porally localized phenomena that can be treated as discrete problems to besolved. We suggest that difficult situations in student-centered classroomsmay be windows looking into a more complicated process of student learningand development (see [22]) , as students learn to function in an environmentwhere they are asked to take greater responsibility for their learning. Tobegin to understand the nature and working of such a process, we suggestthat a clearer picture of students' expectations of mathematics classroomsand teachers is needed. Investigations into this process could form a second,useful line of future inquiry.

ACKNOWLEDGMENT

The authors would like to thank Morton Brown for his support of this work.The work that led to both [30] and the present article was partially sup­ported by grant NSF-DMS-9252503 from the National Science Foundation.

APPENDIX A:INTERVIEW PROTOCOLS

Potential interviewees were usually approached in person or contactedthrough e-mail. The interviewees were told that we were interested in find­ing out about their experiences while teaching mathematics classes, espe­cially the difficult experiences. If the potential subject agreed to participatein the study, an interview was arranged. Interviews were conducted usingthe following protocols (main questions are numbered; typical follow-upquestions are indicated below each main question).

1. What were you most concerned about before the first day of class, thefirst time you taught?

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2. One way of thinking about authority challenges is by categorizingthem each as verbal or nonverbal, and either intentional or uninten­tional. Figure 1 was shown to interviewee.) While you think aboutdifficult situations you've had in your classroom, it might help to thinkin terms of these categories. Can you give us some examples of thesorts of things that you had to deal with in your classes?

• What was the context of this incident?

• How did you feel when that was happening? What emotions didyou experience?

• What about this situation did you perceive as a challenge to yourauthority

• Which category do you see this situation as falling into? ((non)­verbal and (un)intentional)

• What was your response to this situation? What did you doabout it?

• Why do you think this happened? Why do you think the studentswere doing that?

• In retrospect, how effective was your response? What results didyou see? Would you do things the same way again?

Verbal

Nonverbal

V,U

N,U

..Unintentional

V, I

N, I

Intentional

Figure 1. Graphic presented to interviewees.

3. Going back through each scenario you've mentioned, in your opinion,was each more likely, less likely, or equally as likely to happen to afaculty member as opposed to a graduate student instructor? (Herethe interviewer reminded the interviewee of the scenarios and hadhim/her answer this question one by one.)

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4. We're trying to define classroom authority, to say what it is and howyou go about getting it. How would you define or charac terize class­room authority?

• Think about one of your own professors who you would describeas having authority. What made them have authority

• What are th e differences between a classroom where the instruc­tor has aut hority and a classroom where the instructor doesn 'thave authority?

• What do you think cont ributes to that sense of authority?

• How would you suggest t rying to get aut hority?

REFERENCES

1. Asiala , Mark , Anne Brown, David J . DeVries, Ed Dub insky, DavidMatthews, and Kar en Thomas. 1996. A Framework for Research and Cur­riculum Development in Undergraduate Mathemati cs Educat ion. CEMSIssues in Mathem atics Education. 6: 1-32.

2. Ausubel, David P. 1963 The Psychology of Meaningful Verbal Learn­ing. New York: Grune and Stratton.

3. Biaocco, Sharon, A. and Jamie N. DeWat ers. 1998. Successful CollegeTeaching: Problem-Solving St rategies of Distinguished Professors. Boston,MA: Allyn and Bacon.

4. Bookman , Jack and Lewis Blake. 1996. Seven Years of Project CALCat Duke University. Approaching a Steady Stat e? PRIMUS. 6(3): 221- 234.

5. Bookman , Jack and Charles P. Friedman. 1994. A Comparison ofthe Problem Solving Performance of Student s in Lab Based and TraditionalCalculus. CEMS Issues in Mathematics Education. 4: ' 101-116.

6. Case, Bet ty Anne (Editor). 1989. Keys to Impro ved Instruction byTeaching Assistants and Part-time Instru ctors. MAA Notes #11 . Wash­ingt on DC: Mathematical Association of America.

7. Davidson, Neil. 1985. Small Group Learning and Teaching in Math­ematics. A Selective Review of the Research. In R. E. Slavin, et al., (Eds.)Learning to Cooperate. Cooperating to Learn . New York: Plenum.

8. DeLong , Matthew and Dale Winter. 1998. Addressing Difficultieswith Student-Centered Instruction. PRIMUS. 5(4) : 340-364.

9. Felder , Richard M. and Rebecca Brent. 1996. Navigating the BumpyRoad to Student-Centered Instruction. College Teaching . 44: 43-47.

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Winter and Yackel Student-Centered Instruction

10. Fennema, Elizabeth and Megan Loef Franke. 1992. Teachers'Knowledge and Its Impact. In D. A. Grouws (Ed.) Handbook of Researchon Mathematics Teaching and Learning. New York: Macmillan.

11. Finkel, D. L. and S. G. Monk. 1983. Teaching and Learning Groups:Dissolution of the Atlas Complex . From Boulton, C. and R. Y. Garth,(Eds.), Learning in Groups . San Francisco: Jossey-Bass Inc., Publishers.

12. Garofalo, Joe. 1989. Beliefs and Their Influence on MathematicalPerformance. The Mathematics Teacher. 82(7): 502-505.

13. Glaser, B. G. and A. 1. Strauss. 1967. The Discovery of GroundedTheory. Chicago: Aldine.

14. Johnson, David W. and Roger T . Johnson. 1990. Social Skills forSuccessful Group Work. Educational Leadership . 47(4): 29-33.

15. Krantz , Steven, G. 1993. How to Teach Mathematics: A PersonalPerspective. First Edition. Providence ill: American Mathematical Society.

16. Lincoln, Y. S. and E. G. Guba. 1985. Naturalistic Inquiry . NewberryPark: Sage Publications, Inc.

17. Porter, L. W. and E. E. Lawler. 1968. Managerial Attitudes andPerformance . Homewood IL: Irwin.

18. Sallee, G. T . 1979. Teaching 200 Students in a Personal Way. Amer­ican Mathematical Monthly. 86(7): 589-590.

19. Schoenfeld, Alan H. 1985. Mathematical Problem Solving. OrlandoFL: Academic Press.

20. Schoenfeld, Alan H. 1988. When Good Teaching Leads to Bad Re­sults. The Disasters of 'Well-Taught' Mathematics Courses. EducationalPsychologist. 23(2): 145-166.

21. Smith, David A. and Lawrence C. Moore. 1990) Duke University.Project CALC. In Thomas W. Tucker (Ed.) Priming the Calculus Pump:Innovations and Resources. Washington DC: Mathematical Association ofAmerica

22. Smith, David A. 1994. Trends in Calculus Reform. In Anita Solow(Ed.) Preparing for a New Calculus. MAA Notes #36. Washington DC:Mathematical Association of America

23. Stratham, Anne, Laurel Richardson, and Judith A. Cook. 1991.Gender and University Teaching: A Negotiated Difference. Albany NY:State University of New York Press.

24. Swenson, Carl, Michael A. Brilleslyper, and Dale Winter. 1997.Instructor's Manual to Accompany: Functions Modeling Change. A Prepa­ration for Calculus. New York: John Wiley and Sons.

25. Tennant, Mark. 1991. Establishing an "Adult" Teaching-LearningRelationship. Australian Journal of Adult and Community Education.

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31(1): 4-9.26. Thompson, Thomas C. 1995. Student and Teacher Roles and Expec­

tations as Sources of Potential Conflict in the Classroom. Paper presentedat the Annual Meeting of the Conference of College Composition and Com­munication. (Washington DC, March 23-25, 1995).

27. Whalen , D. Joel. 1996. I See What You Mean. Persuasive BusinessCommunication. Thousand Oaks CA: Sage Publications.

28. Wiesenfeld, Kurt. 1996. Making the Grade . Newsweek. June 17,1996.

29. Winter, Dale and Carolyn A. Yackel. 1997. Establishing a SuccessfulClassroom Atmosphere Through an Instructor-Student Contract . Unpub­lished manuscript.

30. Winter, Dale and Carolyn A. Yackel. 1998. Graduate Student In­structors and Classroom Authority. Ann Arbor MI: Department of Mathe­matics, The University of Michigan.

31. Wlodkowski, R. J . 1991. Enhancing Adult Motivation to Learn SanFrancisco: Jossey-Bass.

32. Zucker, Steven. 1997. Teaching at the College Level. Notices of theAmerican Mathematical Society . 43(8): 863-865.

BIOGRAPHICAL SKETCHES

Dale Winter is a lecturing fellow at Duke University. He received his PhDfrom the University of Michigan under t he direction of Joe l Smoller. Hismathematical research interests include mathematical physics (especiallygravitational physics and General Relativity). Besides mathematics, physicsand education, he enjoys the novels of Primo Levi, military history andminiature reef aquaria.

Carolyn Yackel is Visiting Zorn Assistant Professor in the MathematicsDepartment at Indiana University. She completed her PhD at the Uni­versity of Michigan under the direction of Mel Hochster. Her main area ofmathematical interest is commutative algebra. On her list of extracurricularactivities are pottery and crocheting.

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