Interconnecting content and community: A qualitative study of secondary mathematics teachers

ANDREA LACHANCE and JERE CONFREY

INTERCONNECTING CONTENT AND COMMUNITY: AQUALITATIVE STUDY OF SECONDARY MATHEMATICS

TEACHERS

(Accepted 7 February 2003)

ABSTRACT. The publication of the National Council of Teachers of Mathematics initialStandards (1989) has acted as a catalyst to begin reforming the way mathematics is taughtin the USA. However, the literature regarding reform movements suggests that changingour educational systems requires overcoming many barriers and is thus difficult to achieve.Reform in mathematics education, like reform movements in other areas of education,has thus been slow to take hold. One structure that has been shown to support educa-tional reform, particularly instructional reform, has been teacher community. This paperdiscusses a professional development intervention that attempted to start a professionalcommunity among a group of secondary mathematics teachers through in-service work onmathematical problem solving and technology. The results of this study suggest that theuse of mathematical content explorations in professional development settings provides ameans to help mathematics teachers build professional communities. Together, these twocomponents – mathematical content explorations and teacher community – provided thesesecondary mathematics teachers with a strong foundation for engaging in the reform oftheir mathematics classes.

KEY WORDS: mathematical content knowledge for teachers, problem solving andteachers, professional communities, professional development and mathematics teachers,secondary mathematics teachers, technology use and teachers

PREFACE

The largest study ever undertaken of the causes of crime and delinquency has found thatthere are lower rates of violence in urban neighborhoods with a strong sense of communityand values, where most adults discipline children for missing school or scrawling graffiti.In an article published last week in the journal Science (Sampson, Raudenbush, & Earls,1997), three leaders of the study team concluded, ‘By far the largest predictor of the violentcrime rate was collective efficacy’, a term they use to mean a sense of trust, common valuesand cohesion in neighborhoods. Dr. Felton Earls, the director of the study and a professorof psychiatry at the Harvard School of Public Health, said the most important characteristicof ‘collective efficacy’ was a ‘willingness by residents to intervene in the lives of children’.(F. Butterfield, New York Times, August 17, 1997)

The study cited above made national news, its results being reported onnetwork TV and major newspapers and radio programs throughout the

Journal of Mathematics Teacher Education 6: 107–137, 2003.© 2003 Kluwer Academic Publishers. Printed in the Netherlands.

108 ANDREA LACHANCE & JERE CONFREY

country. And why not? A scientific study that claims a correlation between“collective efficacy” and reduced violent crime is good news. In a worldplagued by ethnic warfare, extreme poverty, and excessive materialism,such results give us hope that we can work with each other to effectpositive change not just for ourselves, but for our world.

INTRODUCTION

In educational systems, the power of working together as a community hasbeen recognized repeatedly. Consider the movement towards cooperativelearning as an instructional practice. Allowing children to work togetheras “communities of learners” has been promoted as an effective teachingtool for over two decades (Slavin, 1980; Johnson & Johnson, 1987).In addition, many of the educational reform movements currently beingundertaken place the creation of school community at their centers. Someof these movements look to create community among all stakeholders inthe school: administrators, teachers, students, parents and school neighbors(Sergiovanni, 1994). Other reform efforts concentrate on communityamong one segment of a school’s constituents – creating cohesion amongdifferent “houses” in a large school (Raywid, 1996) or bringing togetherparents and other community leaders to support curriculum changes insubject areas such as mathematics (Moses, Kamii, Swap & Howard, 1989).

Operating under the assumption that teachers occupy prominent posi-tions in classrooms, the development of teachers’ professional com-munities is seen as a means for promoting positive change in our schools(Westheimer, 1998). Given the broad reform of mathematics educationthat has been underway for the past ten years, the examination of teachercommunity and its relationship to reformed mathematics instructiondeserves attention. In this paper, we will discuss a professional devel-opment workshop that led to a budding professional community amongthe mathematics teachers who were its participants. Our qualitative studyof how this workshop provided the catalyst to develop this professionalcommunity among these secondary school mathematics teachers will bethe main focus of this discussion. But before getting to know how thisintervention worked, we must first explore why we believed examiningthe development of professional communities among these teachers was aworthy pursuit.

CONTENT AND COMMUNITY 109

WHY PROMOTE TEACHER COMMUNITIES INMATHEMATICS EDUCATION?

For nearly a decade, the initial publication of Curriculum and Evalu-ation Standards for School Mathematics (1989) by the National Councilof Teachers of Mathematics has acted as a catalyst for reform activity inmathematics education and provided the impetus for rethinking the waysin which mathematics is taught in this country. Yet, change has been slowto come. While new research suggesting that non-traditional methods areeffective in helping students learn and understand mathematics (Schoen,Fey, Hirsch & Coxford, 1999), it is still difficult for many teachers totransform their practice to align with reform mandates.

One structure that could motivate and support teachers to adoptStandards-based instructional practices is membership in a professionalcommunity. There is substantial research in the broader area of schoolreform that suggests that peer collaboration and support is a crucialprerequisite for teachers to be successful in restructuring their classroomsand their schools (Levine & Lezotte, 1995; Gilmore, 1995). For example,in a study of four “successful” and two “unsuccessful” schools, Little(1982) found that more successful1 schools had teachers who had continualand substantive interactions. These teachers “sustain shared expectations(norms) both for extensive collegial work and for analysis and evalu-ation of experimentation with their practices; continuous improvementis a shared undertaking in these schools, and these schools are the mostadaptable and successful of the schools we studied” (p. 338). Little foundteachers in successful schools more likely to be influenced by professionaldevelopment and more open to trying new ideas.

Examples from other studies support this same idea. In studying thecharacteristics of schools that support the professionalization and growthof teachers, McLaughlin and Yee (1988) named “collegiality” as animportant quality to support teacher achievement. Teachers they surveyedand interviewed pointed to peer interaction as an important source of feed-back and stimulation. Similarly, Rosenholtz (1989) found that in schoolswith high levels of teacher collegiality, teacher interaction focused moreon improving curriculum and practice; in environments where teacherswere more isolated, teacher interaction consisted of complaining aboutstudents.

In his discussion of the role of professional communities in teachers’professional development, Lord (1994) suggests that “critical colleague-ship” may act as a means to help teachers reform their practice. However,to attain “critical colleagueship,” teachers must be more than collegial.They must be willing to subject themselves to on-going critique:


For a broader transformation, collegiality will need to support a critical stance towardteaching. This means more than simply sharing ideas or supporting one’s colleagues inthe change process. It means confronting traditional practice – the teacher’s own and thatof his or her colleagues – with an eye toward wholesale revision (p. 192).

This type of interaction is quite different from what many teachers mightexpect from a professional community. But Lord (1994) argues that the“difference and conflict” which drive “critical colleagueship” is the keycomponent to instigating deep and meaningful change in teacher practice.

Along the same lines, Krainer (2001) suggests that not only are currentprofessional development efforts insufficient to help teachers substantiallyimprove their practice, but they focus too much on the individual. Krainer(2001) uses the case of Gisela, a secondary mathematics teacher with over20 years experience, to illustrate how the growth of a single teacher canhave little impact on the whole system. It is only when Gisela takes on anadministration post that her own professional transformation provides themotivation to help other teachers develop. From this case study, Krainer(2001) concludes that professional development activities need to aim formore of “a good balance between initiatives focusing on individual aspects,on organizational ones, and on those related to the whole educationalsystem” (p. 291).

In addition to being better for teachers, McLaughlin and Talbert (1993)point to the benefits for students from strong teacher communities. Theseresearchers claim that the “character of teachers’ professional community”is highly correlated with how teachers perceive students and student work.The authors argue that “supportive collegial communities, committed tothe success of all students, provide the necessary conditions to begin tomount a collective challenge to constraining myths (i.e. the kids can’t doit, etc.) as explanations for unsuccessful student outcomes or disappointingclassrooms” (p. 244).

In mathematics education, where we are attempting to get teachersto think about and teach mathematics in ways which they have neverexperienced as learners, the benefits of teacher community, cited in theschool change literature, have significant relevance. We need teachers tobe open to the new instructional methods promoted by the original Stand-ards (NCTM, 1989) and the recently published Principles and Standardsfor School Mathematics (NCTM, 2000). We need them to be willing tolook critically at their curriculum and make the necessary changes andimprovements. Most importantly, we need mathematics teachers to believethat with appropriate instruction, all students can learn and be successfulin mathematics. Thus, given the positive outcomes associated with teachercommunity in the context of school change, promoting such communities


among mathematics teachers may spell greater success in implementingreform of mathematics teaching.

CONTEXT AND DESIGN OF THIS INTERVENTION

In the spring of 1997, our research group approached the Mathe-matics Administrative Supervisor for the Forest (TX) Independent SchoolDistrict (FISD)2 with some ideas for professional development aimedat reforming mathematics instruction at the classroom level. Specifi-cally, the researchers presented a set of multimedia pre-calculus materials(described below) as a means for helping teachers extend and deepenboth their knowledge of mathematics and their technological skills. Theresearchers also believed that before teachers could reform their ownteaching, they must revisit and reconstruct their own understandings ofand attitudes towards mathematics (Schifter & Fosnot, 1993). Thus, ouroriginal goal in this professional development intervention was to buildteachers’ mathematical content knowledge and not necessarily to buildteacher community.

The district administrator, with whom our research group startednegotiations, was pleased to hear what resources we had to offer as sheand her staff were in the process of revamping the entire district’s mathe-matics curriculum. While reform was well underway at the elementaryand middle school level, it was somewhat stalled at the secondary level.The district personnel saw the lack of innovative curricula for secondarymathematics courses and the general resistance of secondary teachers tocurricular change as being significant barriers to reform. Thus, the admin-istrator and her staff saw a workshop aimed at improving and deepeningsecondary teachers’ understanding of mathematics as a good first step toreforming their instructional practice.

The one point upon which the administrators differed from researcherswas in the sample of teachers with whom this intervention would beconducted. The researchers had assumed that teachers would be from allover the district and would participate on a voluntary basis. In this way wecould be assured of enthusiastic participants and begin “sowing seeds” ofreform in a variety of schools. However, the district administrator and herstaff wanted to focus on the entire mathematics faculty of a single highschool along with the mathematics teachers from the middle schools thatfeed that high school.

Her reasons for this were two-fold. Firstly, working with such a groupwould be consistent with the district’s reform plan and its focus on verticalteams – teams of teachers from a given high school and the middle and


elementary schools that feed that high school. Secondly, and perhaps moreimportantly, the district administrator felt that conducting this workshopwith the faculty from a single school would be a more effective meansof encouraging change. The administrator believed that when individualteachers attend professional development workshops, they often struggleto implement change without support from peers who have had the sameprofessional development experience. Thus, the district leaders thoughtthat any changes that would take place as a result of our intervention wouldbe better sustained if an entire mathematics faculty participated together inour course.

Given what the researchers had to offer and what the district needed,the group decided that a set of mathematics teachers from the same verticalteam at the middle and high school level would make an appropriate targetfor a content-based professional development intervention. Because thedistrict had already adopted a standards-based middle school mathematicscurriculum (The Connected Mathematics Project) that was set to be imple-mented in grades 6, 7 and 8, the Algebra I and subsequent secondarymathematics courses3 seemed to be the next logical target for reform. Thecourse would use higher-level content materials (specifically the multi-media pre-calculus materials described below) with teachers so they couldgain the skills and experience with reformed content in order to engage indiscussions about what changes should be made in Algebra I and the otherhigh school mathematics courses they teach.

The district administrator’s insistence that the workshop be conductedwith the entire mathematics faculty from a given school led to our interestin developing teacher community. How would working on mathematicalcontent together affect the relationships of the teachers in this mathematicsdepartment? We thus broadened our focus to include examining the impactof this intervention on teacher community.

THE WORKSHOP AND ITS PARTICIPANTS

The intervention was planned as a three credit4 university-level mathe-matics education course aimed at high school mathematics teachers. Itconsisted of two weeks of full-time class sessions which took place June9–20, 1997, with a follow-up workshop day on September 29, 1997.Between the two-weeks of class sessions and the follow up day in thefall, teachers had to continue working on the problems that we explored inthe workshop. This included complete written solutions of the problems,with supporting explanations and computer generated tables and graphs.The multimedia pre-calculus materials mentioned above were used as the


content for the course. These materials emphasize problem solving andmodeling using families of functions and transformations as organizingconcepts. The activities themselves are derived from a pre-calculus coursetaught at the researchers’ university for the past eleven years. In their newform, the materials are presented in Netscape which allows for access tothe Internet and interactive diagrams as well as the use of multimediaresources including text, graphics, video, photographs, animations, andsound (Confrey & Maloney, in preparation).

In this professional development course, topics which overlap withmaterial covered in high school mathematics courses were emphasized.Those topics included: introduction to functions through sequences, linearfunctions, transformations, and quadratic functions. Over the two weeks,teachers worked together on the problems and on learning how to use thecomputer hardware and software as tools to solve problems.

The school selected for this intervention was Tree High School, aninner-city school that serves approximately 1720 students and employs101 teachers. The student body is somewhat diverse, consisting of approxi-mately 72% Hispanic students, 14% white students and 14% AfricanAmerican students. Approximately 52% of the students are eligible for freeor reduced cost lunch. An on-site daycare facility is available for studentswho are parents.

In all, eighteen teachers participated in the course. The core group ofthirteen of these teachers was from Tree High School. Two other teacherswere from Elm Middle School, one of three feeder middle schools toTree High School.5 The other three teachers were representatives fromtwo other high schools in the same district that applied to be the siteof the intervention but could not be accommodated because of limitedresources. The entire group consisted of eleven women and seven men.Seven teachers were from ethnic minority groups. Because this paper isfocused on the development of teacher collegiality among the faculty of aspecific mathematics department, this discussion will focus mainly on thethirteen participants from Tree High School.6

The subgroup of thirteen teachers from Tree High School had thefollowing characteristics: five were male, eight were female; six belongedto ethic minority groups (three males and three females); four of thesethirteen teachers were new hires who had not previously worked at theschool. Of these four, two were newly trained teachers who had neverbefore taught high school although one had considerable experienceteaching at the community college and college level. The other newly-trained teacher had worked for many years as a computer engineer butnever as a classroom teacher.


METHODS

Because outcomes in a study such as this are highly context dependentand because different participants in this study had different perspectiveson and experiences with this intervention, data collection and analysistook place within an interpretivist framework (Smith, 1993; Guba &Lincoln, 1989; Patton, 1990). As a paradigm for inquiry, interpretivismasserts that knowledge does not exist separate from the knower. In effect,interpretivism acknowledges that all knowledge is constructed and thatsuch constructions are influenced by the prior beliefs, knowledge andexperiences of the knower (Smith, 1993). As such, different people exper-iencing the same intervention will have different constructions of thatexperience. As a framework for research, interpretivism expects thesemultiple perspectives and encourages their solicitation and representationin research.

To uncover the various perspectives on this experience a multitude ofdata was collected. All participants, including district mathematics admin-istrators and the school’s principal, were interviewed prior to the work-shop and several months after the workshop. Each workshop participantcompleted informational surveys about their background and experiencebefore the start of the workshop and, on the last day of the workshop,filled out evaluation forms discussing their feelings about the intervention.In addition, each participant completed a portfolio consisting of samplesof work on various content problems and several written reflection pieces.All sessions of the workshop were videotaped.

The videotaped observations, documents, and interviews were analyzedthrough a process of coding and category building (Miles & Huberman,1994; Patton, 1990). Using this scheme, the researchers created a series ofcodes by which to mark the data. Codes in this case represent “bins” orcategories within which data can be “placed”. As the data were coded, theinitial codes changed, merged, or decayed until the resulting set of codescould accommodate, describe, and represent all the themes and issuespresent in the data.

DISCUSSION OF FINDINGS

Pre-Existing Conditions

Prior to the start of the workshop, each teacher participant was interviewed.Interviews were structured and lasted anywhere from 45–60 minutes inlength. The interview protocols focused on the several areas of interest


in this intervention. There were sets of questions relating to: Teachers’attitudes towards and beliefs about learning and teaching mathematics,reform in mathematics education, the use of technology in the classroom,and the faculty relationships in their department. This paper is based onthe data related to the last set of questions. A typical question in this setwas: Describe the types of interactions you typically have with other mathteachers. How satisfied have you been with these interactions?

From the description of faculty relationships given prior to the work-shop, the Tree mathematics department did not appear to be a verycohesive group. Department meetings were infrequent and generally wereonly called to deal with administrative tasks such as the administrationof standardized tests. There was little or no discussion of curricular ormathematical issues.

However, a core group of approximately six female members of thedepartment interacted on a regular basis. Three members of this core groupworked together to create a more discovery-oriented geometry curriculum,and, in addition, they often had lunch together and reported positivesocial relationships with other members of the core group. The rest of themembers of Tree’s mathematics department appeared to be somewhat onthe periphery of the core group’s interactions.

To the outside observer, it may seem that the existence of this “core”group made this mathematics faculty more prone to building a larger,department-wide community. The “core” group could act like a foundationupon which a larger community could be developed. However, the oppositeappeared to be true. In discussing the faculty’s relationships prior to theworkshop, there appeared to be an “us versus them” relationship betweenthe core group members and the other members of the department. To themembers of the core group, the other members of the department wereunwilling to collaborate or were anti-social. To the other members of thedepartment, the core group was seen as being something of a “clique” andnot welcoming of outsiders.

This phenomenon is what Noddings (1996) refers to as the “dark side”of community. While many researchers extol the virtues of community,Noddings’ work sounds a call of caution. She warns of community’s“tendencies toward parochialism, conformity, exclusion, assimilation,distrust (or hatred) of outsiders, and coercion” (p. 258). As she sees it,a commitment to community can sometimes support movements towarduniformity as opposed to those that celebrate difference. Thus, the coregroup’s existence could actually be seen as a barrier to building a faculty-wide community in this department. Indeed, four of the nine experiencedfaculty members in this department reported in pre-workshop interviews


that there was tension among individual members of the mathematicsfaculty.

Tree High School was chosen as the site of this intervention largely dueto its ability to get 100% of its mathematics faculty members to participatein the workshop. However, unbeknownst to the researchers when Tree waschosen as the intervention site, although all of Tree’s mathematics teachershad agreed on paper to participate in the workshop, several of them werenot very happy about having to attend this summer course. This added tothe tension that already existed in the department. Four members of thedepartment reported being “forced” to take the workshop by the principal,who was new to the school. Some of these teachers had summer jobs thatthey had to give up (at a significant financial loss) in order to participatein the two week intervention. Other teachers had to change travel plans. Inaddition, the majority of the teachers had little idea of what the workshopwould entail and thus were somewhat apprehensive about participating.All of the teachers reported some reservation about taking this workshop.

General Reaction to the Workshop

Despite the faculty tensions and the concerns the Tree mathematicsteachers had about participating in the workshop, the videotapes of theworkshop sessions, the workshop evaluations and the post-workshop inter-views suggest that all of Tree’s mathematics teachers had a positiveworkshop experience. The extent of the positive experience of the teacherworkshop is reflected in the workshop evaluation forms. These evaluationsconsisted of thirty-five Likert scale items (with a five point scale where 5= strongly agree and 1 = strongly disagree) and six open-ended items (seeAppendix A for some selected items from the workshop evaluation).

The Likert scale items in the evaluations covered the following sixareas: change in teachers’ beliefs about algebra (3 items), change inteachers’ feelings toward technology (10 items), change in teacher rela-tionships (4 items), change in teacher content knowledge (9 items),improvement of teachers’ understanding of student thinking (5 items), andchange in teachers’ attitudes toward reform (4 items). The overall averageresponse for all of the Likert scale items was 4.22 with a standard deviationof 0.63. When the items were analysed by topic, the average responsefor each topic was above 4.00. Such high average responses suggest anoverwhelmingly positive reaction to the workshop.

Change in Teacher Relationships

Interestingly enough, among the Likert scale items, the post course evalu-ation area that had the highest response average was the one on the change


in teacher relationships, which had an average of 4.53 with a standarddeviation of 0.75. These items focused on whether participants thoughttheir relationships with their colleagues had improved because of the work-shop experience. From their responses, we judge that the teachers believedstrongly that the workshop had led to improved faculty relationships.

The open-ended anonymous comments at the end of the evaluationsupported this Likert scale data. In response to the open-ended item thatasked teachers to describe the best part of the workshop, six of the thirteenTree teachers pointed to the opportunity to work with other members oftheir faculty. Here are some of their comments:

The big plus for me was getting to meet and get acquainted with the teachers who will bein my department – and feel comfortable with them!

Working with other teachers – especially those at Tree with whom I work on a regular basis[– was the best].

[The best aspect was] the development of a more open and purposeful department dynamic.

Interview data also revealed teachers’ perceptions that their facultyhad become closer as a result of their participation in the workshop.Approximately six weeks into the school year (1997–1998) followingthe workshop, teachers at Tree High School were interviewed abouttheir experiences in the workshop. These post-workshop interviews wereintended to parallel the interviews conducted before the workshop. Thus,they were of the same structure and duration and focused on the sameareas of interest as the pre-workshop interviews (teacher’s attitudesabout teaching mathematics, reform in mathematics education, the useof technology in the classroom, and the faculty relationships). While thepre-workshop interviews were intended to discover the existing conditionsprior to the workshop, the post-workshop interviews were intended todiscover the impact of the workshop as an intervention.

At this time, the mathematics department at Tree consisted of fourteenteachers. Twelve of those teachers had participated in the summer work-shop (one of the thirteen workshop participants from Tree took another joband left the faculty before the start of the school year). Of those twelve,four teachers were new to Tree that year and eight were experiencedteachers at the school.

Of these eight experienced teachers, four suggested that teacher rela-tionships were significantly improved over the previous year:

The rapport has been much, much better. Much deeper. Perhaps individual differences thatwere noted at the workshop have improved a lot. I know that even with the other [name ofcourse] teacher our rapport has been much better. (Respondent 1)7


Definitely – a change from last year. As I mentioned to you before, I think there are peoplewho had conversations that hadn’t spoken to each other in years. There were deep rifts andwhile they may still be strained – the bridge is there. (Respondent 9)

Another three of the returning eight teachers suggested that relationshipswere slightly better than in previous years, with the remaining returningteacher suggesting that the mathematics teachers no longer shared thesame wing and were scattered throughout the building. In previous years,the mathematics teachers in this school had classrooms located along thesame hallway, which they referred to as the “math wing”. However, thefall after the workshop, teachers’ classrooms were reassigned to allow forthe creation of ninth grade teams. Thus, not all the mathematics teacherswere on the same wing. This teacher felt that, in spite of the department’spositive experiences in the summer workshop, the lack of proximity to oneanother was a significant barrier to improving departmental relationships.

While the fact that seven of the eight experienced teachers reportedsome improvement in department relationships as a positive result, theconsideration of the pre-existing conditions in the department makes itmore impressive. While nearly half of the experienced teachers (four ofthe nine original ones) had reported tension among faculty members inthe pre-workshop interviews, not one of the faculty, new or experienced,suggested that level of tension still existed. In general, teachers’ responsesto the questions about their relationships with their colleagues was muchmore positive in post-workshop interviews than in pre-workshop inter-views. None of the teachers mentioned the existence of the “clique” ofsix teachers, as they had in the pre-workshop interviews. From teacherdescriptions, it appeared that teachers as a group were more integrated andgenerally felt more comfortable with all of their colleagues – not just aselect few.

As might be expected, the four new teachers to the departmentsuggested that they benefited significantly from spending time in theworkshop with their new peers. Here are two representative comments:

The first [positive] thing I point to [about the workshop] is meeting everybody. Forget whatthe content of the workshop was – the relationships have been invaluable. (Respondent 12)

I was very comfortable coming into this school because I already knew where everythingwas. I knew all the math people. I was comfortable with them. I felt comfortable enoughto say anything to them or ask anything of them. I can’t imagine what it would have beenlike for me if I hadn’t had that. (Respondent 3)

Despite having to deal with problems such as large class sizes andadjusting to a new school (and for one of these teachers, a new career),all four of the new teachers reported having positive interactions with the


other mathematics faculty members. They felt their experienced peers weresupportive, approachable and helpful.

What Led to This Reaction?

Teachers’ overwhelmingly positive reaction to the workshop and theirbelief that the workshop helped make department relationships strongeractually surprised us as researchers. Given the pre-existing conditions, wedid not expect such a universally positive response to our intervention.This reaction leads us to ask the question: What was it about this experi-ence that helped these teachers grow closer as a group? Based largely oninterviews with teachers and researcher field notes, three factors appearedto contribute to the growth of teacher community in this setting. Thesefactors were determined through the coding system applied in the analysisof the data.

In analyzing the transcripts of the post-workshop interviews and of thevideotapes of workshop interactions, we labeled each response or interac-tion with a different code. For example, when a teacher compared theirexperiences in other professional development workshops with his/herexperience in our workshop, we labeled that response “experience withother workshops”. Because we were using a computer program to label ourdata, at the end of the coding and labeling process, we were able to sort bylabels. When we sorted the data concerning teachers’ positive reaction toour workshop, we found that a substantial portion of the data fit into threebroad categories. Each of the categories is discussed below.

INTERACTING WITH COLLEAGUES OVER MATHEMATICS:THE REVELATION OF DIFFERENT PERSPECTIVES

When asked to explain why they thought the workshop had been sobeneficial to teacher relationships in their department, one of the mostfrequent responses given by teachers concerned their interaction in solvingmathematics problems. Throughout the workshop, teachers were askedto work on problems in pairs or small groups. Teachers were instructedto switch partners or groups frequently in order to work with a largernumber of people. For some teachers, this small group work provided themwith a mathematical focus outside of themselves about which they had tocommunicate.

Just the act of problem solving together and putting everything aside and turning to acolleague and saying: “I don’t know how to do this or I don’t quite understand the wayyou’re approaching this. Could you explain it to me?” . . . was important. (Respondent 9)


The way you all had us shift partners – even though my first thought was: “Oh God, no.I just got comfortable with this one and now I’ve got to go and get to know another one.”But it ended up being really good because when you’re both thinking about a problem,you’re not thinking about each other. And the relationship just sort of happens without youknowing it. (Respondent 11)

The act of working on mathematics together gave teachers a sense thatthey were interacting over shared territory. This was not a cocktail partywhere the individuals comfortable with small talk would excel. This wasexploration of mathematics – something with which they all had experi-ence and in which they had all invested. As Westheimer (1998) points out,community is not built in and of itself – but is built over a set of substantiveissues important to all participants. In this case, the mathematical contentgave these participants a reason and purpose for interacting. The result wasthat participants developed relationships by way of their problem solvingactivities and their discussions of mathematics.

However, even more common than the belief that the problem solvinggave teachers a substantive focus, was the feeling among these teachersthat the problem solving activities allowed them to see how each indi-vidual thought. During the workshop, after the small groups had a chanceto work together on a problem, each group was asked to report its find-ings to the large group. In these reports, teachers saw and appreciated thevarious approaches different groups took in solving problems. At times,the variation across solutions was significant.

For example, in a problem where teachers were asked to come up witha model for the growth of the chambers of a nautilus shell (commonlymodeled with a geometric sequence), teacher investigations had extremedifferences. Each group took measurements of successive chambers in areal nautilus shell, and then plotted this data on a graph or made tables ofthe data in an effort to find a mathematical model that described their data.One group of teachers used graphing tools on the computer in an effortto find a curve that approximated the plot of their data. Another groupattempted to linearize the data and used a graphing calculator to draw aline of best fit. Still another group manipulated the data through calcula-tions. They took the logs of their data in an effort to derive a series ofrational functions that would best fit the data. In listening to presentationsof these types of non-traditional approaches, teachers were challenged intheir attempts to understand the methods used by others.

In post-workshop interviews, teachers were asked to “Describe oneevent or interaction during the workshop that you feel was important orhad an impact on your relationship with the other faculty members inyour department.” Eleven of the twelve remaining Tree teachers suggestedthat hearing other people’s methods for solving problems had a significant


impact on their relationships with their colleagues. There seemed to betwo reasons why teachers felt the understanding of each other’s problemsolving approaches helped improve their relationships. The first was thatsimply by hearing the way others approached a problem, teachers felt thatthey started to understand the way others thought and thus got to knowthem better.

We were very actively involved and participated in what was going on so that it allowedus to just share. I still think of Susan as having a certain approach – from her background– to problems. And I see Carol having a unique approach. And Sam of course [was] thegenius of the group. And I had a chance to work with them but also to learn a little ofwho they are and how they think and just have fun solving problems together at that level.(Respondent 6)

A second reason why hearing other people’s perspectives seemed to besignificant for workshop participants was simply being able to see thatmathematics can be explored in a variety of ways. The discussion ofdifferent approaches to problems seemed to allow these teachers an oppor-tunity to air, acknowledge and accept the diverse perspectives from whichthey approached mathematics. Doing this in the somewhat neutral contextof a problem solving activity, as opposed to a more formal departmentmeeting, seemed to allow teachers to step back and simply appreciateothers and their perspectives.

Respondent: [The workshop] allowed for opinions to be expressed. And people listened toeach other’s opinions for the first time.Interviewer: Do you think it was listening or do you think it was airing opinions in the firstplace?Respondent: Yes, airing the opinions and then exposure to perhaps this is a different wayto approach it. This is a new idea. . . .

Interviewer: And that had never happened before?Respondent: Right . . . if it had happened, nobody was willing to listen to someoneelse’s viewpoint. And without recognizing each other’s professional value. You know, justbecause you don’t do it my way – that doesn’t mean that you’re stupid or that you don’tknow what you are doing as a teacher. (Respondent 1)

The benefit of hearing others’ perspectives on issues of mathematics cameup repeatedly in the interviews. It was almost as if the existence of thesedifferences among their faculty was a revelation for this group of teachers.Understanding the perspective from which a peer was coming seemedto bring with it a deeper appreciation and respect for the various posi-tions of that individual. Such understandings also brought with them asense of acceptance of the different individuals and their unique view-points.

I enjoyed some of the freewheeling discussions where . . . you got to know how peoplethought a lot more – when people were fairly free in asking questions or challenging what


was happening . . . just watching and listening and thinking – “Oh, OK, I didn’t know heor she thought this way or came from that point of view.” (Respondent 7)

The fact that teachers came away with a better appreciation of each other’sperspectives on mathematical ideas is not a total surprise. One of our goalsin the workshop was to promote the development of multiple perspec-tives and representations of mathematical content. The workshop itselfcontained many structures to allow different viewpoints and approachesto emerge. First and foremost was the creation of an environment whereteachers were expected to collaborate in problem solving. Group workwas the norm, and we hoped small and large group activities would allowteachers to see that not all of us think alike. In learning to appreciate thediversity among themselves, we hoped teachers would begin to understandthe diversity that exists in student thinking. Perhaps the most importantstructural feature which led to the airing and appreciation of different view-points was the work of the lead facilitator. There were five team membersimplementing this workshop. Three team members were graduate students,one was a staff member, and one was the principal investigator of the grantsupporting this workshop and the leader of the research team. The researchteam leader, who is also second author on this paper, acted as the leadfacilitator for the workshop. The three graduate students (one of whomis first author of this paper) and the staff member supported her workas facilitator by preparing materials, assisting teachers as they workedon the problems and with the computers, leading certain activities, anddealing with various logistical concerns. The three graduate students alsovideotaped all the workshop sessions and took field notes of their observa-tions of the workshop activities. It was clear from these observations that,although the teachers had friendly and positive relationships with all fivemembers of the workshop team, the lead facilitator was the person whomthe teachers held in highest regard.

From the beginning of the workshop, the facilitator made clear thatmultiple interpretations and solutions in mathematical problem solvingwere not only acceptable but also expected. As mentioned above, theNautilus problem elicited a variety of solutions from teachers. The facili-tator went around to each group, offering support and posing questions forfurther explorations. In organizing the group presentations, the facilitatortried to organise the order of the group presentations so that each methodshown complemented the next and further broadened the problem. Themethods went from being straightforward to being fairly sophisticated.

The last group to present chose a fairly complicated approach tomodeling their data. The leader of this group presented her method to therest of the teachers, who were rather awe-struck.


Penny: We did a graph of the points. . . . Graph of area versus chamber number – I think itlooks parabolic. . . . Well, anyway, I didn’t start trying to guess what the power was. WhatI did was take the log base 10 of the area.Diedre: Why did you even know to do that?Penny: Because I’ve done it before. And it’s the nature of logs. Log x4 is equal to 4 log x.So plot log base 10 of y versus log base 10 of x, then the slope of that line will give me thepower relationship. So I don’t have to guess the power – I can find it.Facilitator: Let’s check in – does everyone follow?Penny: Do you see because of the nature of logs, that slope will give me the power relation-ship between y and x? And it doesn’t have to be a whole number power. It can be y = x5.Facilitator: The trick is that log x4 = 4 log x. Because if you are plotting log x versus logx4, then the claim is you’re plotting log x versus 4 log x. So the 4 is functioning as m in theline mx. Basically, you’re giving the same function to both variables, but it’s bringing thepower down in front.Penny: And that gives me my power and once I have my power that gives me the relation-ship between y and x.Facilitator: That’s a way to find out the relationship if you think it’s a power function [like]x2, x3, etc. . . . some kind of polynomial. It’s a way to model something.Penny: But it doesn’t have to be a polynomial.Facilitator: Yeah, you can do rational functions too – but I’m worried that they’re not thatclean.Penny: And this also is wonderful because if I plot log x versus log y and get a disjointedline – that tells me I have two power relationships. I have two functions at work on differentparts of the data. That happens all the time like with magnetic fields – strength of themagnetic field drops off at different rates depending on distance away from the field. Thisis what we do in our physics labs. (Some teachers whistle like this is impressive or wayover their heads.) (Day 5 – Tape 2).8

The teachers in the audience found this approach challenging. Several ofthem asked clarifying questions and a few of them made it clear theyfound it difficult to understand. “I thought I knew logs until you starteddoing this,” said one. But despite this very sophisticated method and thediscomfort it caused many of the teachers, the facilitator made clear theimportance of bringing it into this discussion.

As a teacher I would say – I came around and saw what they were doing and gulped –I said: “We’re going to have to understand and present this method.” And I told my staffnot to disrupt this – I could have guided them to use geometric sequences and gotten you[referring to the small group] to stop. But I think you offered the group a really importantinsight – which got brought into the group. (Day 5, Tape 2).

By stressing the importance of appreciating multiple approaches to prob-lems, the facilitator not only modeled what we believed teachers shouldbe doing with students in their classrooms, but she also helped teachersappreciate the diversity and strength of perspectives existing among theirfaculty.


Certainly, the teachers recognized that the workshop was structuredto allow for varying perspectives – and appreciated the freedom andopportunity to explore mathematical content in this way.

We were never told: You have to do it this way. We were allowed all these different ideas. Imean I did it one way, Juanita did it another way, Penny did it another way, Timothy anotherway. But we were all doing the same thing and we were allowed to do that. Nobody toldus how to work anything. I like that. I like that. (Respondent 2)

Thus, the freedom to have a unique interpretation on a mathematicalproblem gave these teachers the opportunity to share their individualitywith their colleagues. These mathematical interactions became the catalystfor their sharing of themselves and listening to the perspectives of theirpeers.

NAVIGATING NEW TECHNOLOGY: WE’RE ALL IN THESAME BOAT

Giving participants the time and opportunity to interact over mathematicalcontent may also have helped these mathematics teachers face a significantbarrier to improved department relations: the strength of each individual’smathematical content knowledge. The whole issue of content knowledgeis a touchy one for teachers – especially in the state of Texas where teachercompetency testing has an interesting and not very teacher-friendly history(Shepard & Kreitzer, 1987). For a mathematics teacher, to work with one’speers on mathematics problems is to make oneself vulnerable for there isalways the chance of making mistakes and looking foolish.

At points, the teachers in this workshop did make comments thatreferred to their discomfort with some aspect of content. When the facili-tator was discussing the derivation of the formula for the sum of ageometric sequence, several of the teachers mentioned that this derivationwas new to them.

Diedre: Uh-uh. I’ve never seen this before. I mean I’ve seen the formula and I’ve taught theformula – but I didn’t really know where it came from. And I guess I never had the time tosit down and play with it and see it.Allison: No one ever explained it this way and I understand it now – why the formulais the way it is. That’s why I wanted to be real sure about what you were saying. Nobodyexplained it to me this way and I was confused about these formulas. But now I understand.Facilitator: You’ve got to fight that thing that you’re stupid because you just didn’t gettaught it. And if you didn’t get taught a tool for thinking something through . . . then whatcan you do? (Day 4, Tape 1).

Given some of the teachers’ insecurities regarding mathematical contentknowledge, the work on mathematical problem solving throughout the


workshop could easily have been a divisive force among these teachers.However, teachers were exploring mathematics content with technologythat was unfamiliar to all of them. While some teachers had reportedhaving previous experience with computers, none of the teachers had usedthe main tool of the workshop, a program called Function Probe (Confrey,1997), and none of them had seen mathematics problems presented in amultimedia format.

Adding technology to the problem-solving mix seemed to maketeachers feel as if they were covering mathematics content that was newto all participants. For many teachers, the technology seemed to level the“content knowledge playing field” and led to more relaxed and supportiveinteractions between participants. The idea that everyone was on “equalfooting” came up several times when teachers were asked what aspect ofthe workshop helped improve teacher relationships.

It wasn’t just one thing, it was the same thing that just kept happening over and over whenwe were working with the problems and working with the computer with software that wasnew to everybody. So even if you were good with another computer we all sort of startedon equal footing there – so you didn’t have to feel like a total dummy. (Respondent 11)

Equality – starting out on an equal footing – that none of us had used Function Probe andwe were all exploring it together. Putting us all on the same plane there . . . in that maybesome people came with computer literacy and many of us didn’t. And being able to get tothe same level with that. (Respondent 1)

The belief that everyone was “in the same boat” in relation to thenew technology also led to the growth of helping relationships amongthe teachers. They were eager to understand both the technology andthe content and generally were not shy about asking for help. Teacherswillingly assisted each other with both technological and content issues.When teachers made mistakes or got stumped in their presentations, otherteachers came to their rescue.

Tree teachers continued these helping relationships well after the work-shop. At the end of the workshop, teachers took home a computer in orderto continue working on workshop-related activities. In dealing with prob-lems with setting up and using these new computers, teachers were quickto call each other for assistance. In many cases, a more “expert” teacherwould make a house call to a peer who was having computer problems.Similarly, in finishing the problem solving activities and written reflectionpieces that made up the teacher portfolios for the course, several teachersworked together to complete these assignments.

Among the remaining twelve teachers at Tree High School, eightof them reported getting or giving assistance with workshop-relatedtechnology or activities on their own personal time. While some skeptics


may suggest that such helping relationships are common in professionaldevelopment workshops, one of these teachers recounted a similar type ofworkshop with participants from all over the country where other groupmembers were not very helpful.

You would be in the computer lab and you would ask some of these teachers that knewmore than you did and they felt bothered because you were slower and you were holdingthem back from finishing their work. And in this [workshop], people were more helpful. Ifyou didn’t know how to do it, somebody else came to your rescue. So it was much, muchdifferent. (Respondent 5)

At the start of school, the helping relationships that teachers had startedto build in the workshop were evident. The beginning of school year 1997–1998 was chaotic for many of Tree’s mathematics teachers. Because ofsome changes in the school’s structure, almost all of the mathematicsteachers at Tree had to change classrooms. Teachers helped each otherwith moving materials and books from one room to another. In addition,many of Tree’s mathematics teachers were teaching classes they had nottaught for many years, if at all. More experienced teachers of these coursesoffered to share lesson plans or activities with their peers. Finally, evenwith large class sizes and hectic schedules, teachers managed to find timeto check in on each other. Several reported stopping by each other’s roomsafter school to see how they had survived the day. The helping relationshipsstarted in the workshop were weathering the challenges of the beginningof the school year.

Starting school we had a great advantage in already having a working relationship witheach other. People felt like they could ask things of other people – whether it was helpwith doing something, resources, where to go – I mean we were already beyond whateverbarriers there usually are . . . I think the fact that we sat down and we were learners together[meant] that nobody had anything to hide anymore. I mean everyone knew who was whoand what was what. And I think that made for a very comfortable working relationship andI don’t think anyone felt isolated in starting – which frequently happens. (Respondent 9)

Thus, working with new technology, which was not comfortable for any ofthese teachers, led them to bond. In helping each other face the commonchallenge of developing these new technological skills, these teachers alsodeveloped stronger collegial relationships.

HANDS-ON LEARNING: BEING ENCOURAGED TO TRYTHINGS FOR OURSELVES

The majority of teachers from Tree stated that they had never been to aworkshop where they had been so active.


Most of the workshops I’ve gone to we just sit there and listen to somebody. In this work-shop, we did the work and then we exchanged ideas. I haven’t been to that many workshopswhen we’ve done that . . . Actually sitting down and doing it. Seeing what you’re doing. Iwill not learn if you tell me this is how it’s done. I have to do it. (Respondent 2)

Over the nearly 40 years I’ve been teaching school, the only workshops that I ever attendedthat I came back with more than being numb on both ends, were the ones where somebodywas down to earth enough to tell you something and then let you do it. [This workshop]was a hands-on situation. It was a combination of having a computer everyone could gettheir hands on, having the different pieces of hardware here like the motion detectors andthe overhead projection screen. . . . I was really fascinated with everything. (Respondent11)

Most of these teachers suggested that the hands-on nature of the coursemade it more enjoyable and allowed everyone to have a more positiveexperience. However, few of them attributed the hands-on nature of thecourse to improvement of faculty relationships.

Nevertheless, a strong case can be made that the hands-on learningsupported the growth of relationships in this workshop. Some of this argu-ment is intuitive. Allowing teachers to be more hands-on gave them theopportunity to have a variety of different kinds of interactions with eachother. They not only were discussing mathematics or debating the validityof a certain method – but they were struggling to find accurate measuresof the volume of a nautilus shells chambers, or they were trying to figureout how to arrange themselves to get the motion detector to produce a stepgraph. Very frequently, they were attempting to figure out how to get that“blasted” computer to perform certain tasks. In each of these contexts, thenature of their learning was different – so the nature of their interactionsand resulting relationships would be broader and perhaps deeper.

Another reason hands-on learning can be cited as a factor in helpingthese teachers improve their collegial relationships comes from work ofWestheimer and Kahne (1993). One of their five means for fostering andsustaining community is through breaking norms as a way for creatingopportunities for new relationships. In essence, this involves giving peopleexperiences different in nature or context from the course of their “normal”interactions. Such nontraditional experiences give people the chance to“recast” their relationships with each other. For teachers, workshop activ-ities such as rolling a basketball up and down a ramp in front of a motiondetector or sending goofy email messages to each other while sitting inthe same room could be classified as norm-breaking activities. The levelof humor and interaction during these activities was high, and it seemsclear that such activities gave teachers a chance to see each other in adifferent light. Given the positive effect that teachers claim the workshophad on their relationships with each other, it seems likely that the chance


to work together on hands-on activities contributed to the growth of theserelationships.

GROWING PAINS

Despite the fact that in post-workshop interviews, nearly all of Tree’smathematics teachers reported positive growth in the collegiality of theirdepartment, just as many of them mentioned school factors that may act asbarriers to further departmental growth. First and foremost was the lack ofproximity that these mathematics teachers have to each other. While manyof these teachers are just down the hall from each other, several of theteachers are in separate parts of the building. With the limited amount oftime teachers have to interact with each other during the school day, severalteachers see having classrooms in close proximity to other departmentmembers as vital to maintaining positive teacher relationships.

Some of the distance between department members is due to theschool’s creation of ninth grade teams. These teams present another poten-tial barrier to the continued growth of the mathematics department’sbudding professional community. A ninth grade team consists of a groupof teachers from each of the main subject areas who teach the same set ofninth grade students. All of the teachers on a given team have classroomsnear one another to facilitate communication about student progress andcurricular issues. Mathematics teachers on these teams would thus haveclassrooms away from the majority of their department.

While certainly positive in some respects, the creation of teamsaccording to grade level may undermine the power of content-specificdepartments. In the case of Tree’s mathematics teachers, those teacherson ninth grade teams have responsibilities to both their department andtheir team. For several of them, this means they are in either department-sponsored or team-sponsored meetings for four of their five planningperiods per week. A few complained about being “burned-out” on meet-ings and being separated from their non-team peers in the mathematicsdepartment.

Another barrier to community for Tree mathematics teachers is staffturnover. In the year after our workshop, the mathematics departmentat Tree lost five faculty members, four of whom had participated in theworkshop. Because replacement faculty members will not have had theworkshop experience, the mathematics department will need to find someway to integrate these newcomers into the larger group. Failure to do thiswill certainly limit the improvement of collegial relationships within thisdepartment.


Several teachers mentioned lack of support from administrators asbeing a barrier to a stronger department. A few teachers thought theywere on administrators’ “bad list” because they had been given undesirableteaching schedules. Some other teachers felt that school administratorsdid not take the time to find out what was really going on in mathe-matics classrooms. No administrators ever attended department meetingsand some members of the department felt they were being neglected. Inter-estingly enough, interviews with administrators revealed that they werequite pleased with the progress that the mathematics faculty had made overthe past year. They noted better communication among faculty membersand increased use of technology and innovative teaching practices inmathematics classes.

If Tree’s mathematics department is going to continue to grow, the riftthat many teachers described between the department and school adminis-trators needs to be repaired. A considerable amount of energy seems tobe expended in the department complaining about administration andbureaucratic demands – energy that could be used better to promoteimproved classroom practice. In addition, just as faculty members needpeer support to undertake classroom reform, they also need the support ofschool administration. Such support allows teachers to remain connectedto and make use of the larger structure of which they are a part.

The final barrier to continued growth of the relationships in this mathe-matics department is time. As almost any teacher anywhere will report,there simply is not enough time to do all the things teachers need or want todo. In the case of Tree’s mathematics department, almost all of the teacherspointed to time constraints as being a barrier to continued improvementof their faculty relationships. Many of the teachers are dealing with largeclasses of 30 or more students. Several of these teachers do extra curricularwork such as coaching or tutoring of students. The current schedule at Treedoes not permit all of the mathematics faculty to share the same planningperiod or the same lunch period. This makes group interaction and plan-ning difficult. If a mathematics faculty community is to be fostered andmaintained in this setting, the time will have to be found to allow teachersto continue to build it.

EPILOGUE

In the three years following our workshop, our research group continuedto work with the Tree mathematics teachers on a number of initiatives.The year following the workshop (’97–’98) the researchers and teachersworked together on an assessment measuring students’ understanding of


simultaneous equations. In the summer of 1998, teachers participated infive more days of workshop activities with researchers, focusing on othercontent topics and constructing a replacement unit for Algebra I courses(Castro-Filho, 2000). During school year 1998–1999, the mathematicsdepartment at Tree, with the help of researchers, piloted two experimentalcourses with students: one integrating math and science and a second using“mathematics as modeling” as an organizing theme (Castro-Filho, 2000;Confrey, Castro-Filho & Wilhelm, 2000). In the summer of 1999, teachersand researchers reviewed the student results of the piloted courses andmade revisions to the courses as needed (Confrey, Bell & Carrejo, 2001).

Through our observations of teacher relationships in these subsequentprojects, we watched as the Tree mathematics teachers continued to worktowards maintaining the strong professional relationships they developedin our initial workshop. As mentioned previously, one major issue for theseteachers was the constant influx of new faculty members into their depart-ment. By 1999, only 7 of the 13 original workshop participants still taughtmathematics at Tree High School (Confrey, Bell, & Carrejo, 2001).

Our continued work with the Tree mathematics teachers provided onemeans to integrate the new faculty members into the department andfamiliarize them with the issues that had been discussed in our initialworkshop. In addition, the Tree mathematics teachers continued to meeton a regular basis to discuss issues related to curriculum and studentprogress. They also instituted a monthly potluck dinner to welcome newfaculty members and to continue to improve and develop their faculty’srelationships (Wilhelm, personal communication).

CONCLUSIONS

It is clear from the data presented in this article that teacher relationshipsdid improve and intensify in the period following the workshop. However,can it be said that a professional community was developed through thisintervention? The answer to this question probably rests on how onedefines community. Westheimer (1998) suggests a “community” will havethe following five characteristics: a large degree of interaction and partici-pation, interdependence among its members, shared interests and beliefs,concern for individual and minority views, and concern for personal rela-tionships. Our data suggests that while the relationships among Tree’smathematics faculty have some of these characteristics, they do not haveall of them. Thus, by this definition, a community was not developed.

However, other studies (McLaughlin & Yee, 1988; Little, 1982;Rosenholtz, 1989; McLaughlin & Talbert, 1993) define professional com-


munity much more loosely. Professional communities in these studiesare characterized by high levels of teacher interaction and collegiality.Using this criterion and the fact that Tree’s mathematics teachers reportedthey were interacting much more frequently and substantially than everbefore, it would seem the Tree teachers’ relationships were exhibiting thehallmarks of a professional community.

Probably more important than whether or not the effect of the work-shop was to produce a professional community (by any definition) isdetermining what we have learned about professional development andprofessional communities from this experience. Knowing that faculty rela-tionships and department interactions often have a long history, we, asresearchers, did not expect that our initial two-week intervention wouldresult in a full-blown teacher community. However, we did hope that theprocess of interacting over substantive mathematical content and curricularissues would help this faculty begin the process of improving teacher rela-tionships. By studying the early formation of a potentially budding teachercommunity, we hoped to contribute to the growing understanding of howschools and teachers can begin to move toward a sense of community andthe subsequent realization of reform goals.

So what have we learned? When the teachers are back in school, anumber of constraints become evident which impede continued relation-ship building. The multitude of demands on teachers’ time and thedistance between their classrooms create barriers of space and time whichprevent teachers from communicating regularly. This causes us to ques-tion the extent to which the workshop enabled participants to continuerelationships in the school setting.

However, by combining teacher “teaming” in a professional develop-ment setting with teachers’ exploration of mathematical content, webelieve we have developed a powerful vehicle for achieving two goalsin pursuit of improving mathematics instruction. First, given the role thatteacher community and interaction has been shown to play in supportinginstructional change (Little, 1982; McLaughlin & Yee, 1988; Rosenholtz,1989), structuring professional development activities such that they alsowork as community development may increase the chances that suchactivities will support subsequent changes in curriculum and proposedteaching innovations.

In addition, given that one of the major concerns in reforming mathe-matics instruction is the depth of teacher content knowledge (Ball, 1991,1994), shared mathematical inquiries in a professional development settingnot only give teachers an opportunity to work with one another andbuild community, but they also give teachers a means to develop deeper


mathematical understandings. These two objectives, building both contentknowledge and community through professional development, are thusintegrated and mutually reinforcing.

As an illustrating example, the two teachers at Tree who consistentlyteach Algebra II had a somewhat strained relationship before the work-shop. After their experiences in the workshop, they decided to try collabo-rating more frequently. When we arrived at the school to do post-workshopinterviews, both teachers, at separate times, excitedly shared the tale ofhow they had recently debated the “real” meaning of the function concept– and enjoyed the debate. For these two teachers, the content is at the centerand focus of their relationship, which continues to grow.

Westheimer (1993) has argued that teacher community does notdevelop in and of itself. In fact, as a consultant, Westheimer is oftenapproached by school district administrators who want him to come totheir schools and “show” teachers how to build a professional community(personal communication). In response, Westheimer always asks theinquirers: What are the pressing, important issues that your teachers arefacing or are concerned about? He then advises district personnel tohave teachers start working on and facing those issues. According toWestheimer, it is only in wrestling with these issues that teacher com-munities can begin to develop (Westheimer, personal communication).In our case, the issue over which the Tree teachers engaged was themathematics, which is central to their teaching and the success of theirstudents.

There is very little in the literature discussing the development orexistence of teacher communities that addresses the notion of using mathe-matical content (or other subject content) as the “issue” around whichteachers can interact and professional communities can develop. There arestudies which discuss the growth of teacher communities around the devel-opment of a new school-wide curriculum (Sergiovanni, 1994), recognitionof community-shared values (multiculturalism, equity, etc.) (Westheimer,1998) or implementation of new instructional practices (Moses, Kamii,Swap & Howard, 1989). In addition, several studies look at the correla-tion between school success and teacher community (Little, 1982) or thedifferent levels of teachers’ interactions that promote reform (Little, 1990).However, no study seems to look at field-specific content explorations (likemathematics) as the basis for building teacher community.

Thus, this study is important in that we learned that teachers’ jointmathematical investigations CAN act as a means to build teacher com-munity. Our experience clearly illustrates that by allowing teachers toexplore rich, open-ended mathematical problems with a variety of tools,


they can learn to appreciate both the mathematics and a given individual’sperspective on the mathematics. Diversity of thought and personalityamong department members, a potentially divisive force, is thus givenroom to flourish through the exploration of these problems. This lessonhas been so powerful for our research group, that professional developmentinterventions that simultaneously develop teacher community and contentknowledge are now an integral part of our proposed model for imple-menting standards-based curricula and technology innovations (Confrey,Bell & Carrejo, 2001).

“What we want for our children, we should also want for their teachers”(Hargreaves, 1995, p. 27). In the end, we found that what supports thedevelopment of community among secondary mathematics teachers issimilar to what we would define as a “good” mathematics program forstudents: the pursuit of interesting mathematical investigations, the use oftechnological tools, and the stimulating discussion of mathematics. Allthese structures promote an environment that can “grow” professionalcommunities while at the same time develop teacher content knowledge.We believe that together, these two components – content and community –provide a strong foundation for teachers to undertake instructional reformin their mathematics classes.

ACKNOWLEDGEMENTS

This research was supported by a grant from the National Science Founda-tion (RED 9453876). All opinions and findings are those of the authorsand not necessarily those of the Foundation.

APPENDIX A: WORKSHOP EVALUATION ITEMS

Below are a sample of the post-workshop written evaluation items.

Part I: Likert scale items: The response scale is as follows: 1 = Strongly Disagree,2 = Disagree, 3 = Neutral, 4 = Agree, 5 = Strongly Agree.

1. This course changed the way I think about algebra.2. I feel more comfortable with computers since taking this course.3. This course did NOT help to improve my relationship with the other members

of my department.4. I was unimpressed with the technology used in this course.5. Viewing and analyzing videos of students working on problems increased my

understanding of how students think about mathematics.


6. Using technological tools allowed me to gain more insight into algebracontent.

7. I liked exploring the multimedia problems presented in Netscape.8. For me, this workshop has raised some concerns about the directions in which

reform in mathematics education is headed.9. This experience has helped me realize how important hands-on materials are

to student learning.10. I plan to use more technological tools in my teaching this year because of this

course.11. This course made me realize how difficult it can be to really understand what

students are thinking.12. In general, I feel this course was a positive experience for our entire math

faculty.

Part II: Open-ended questions.

1. Please describe the BEST aspects of this workshop for you.2. Please describe the WORST aspects of the workshop for you.3. In your opinion, how has this workshop affected your content knowledge?4. In what ways, if any, do you believe this workshop will impact your

relationships with the other mathematics faculty in your school?5. How do you think your experiences in this workshop will impact your

teaching in the next year?6. Would you be willing to take a similar course to this one focusing on

trigonometric and exponential functions? Please give reasons for your answer.

NOTES

1 To find successful schools, Little examined aggregate standardized achievement testdata for a three year period. She also considered nominations for successful schools fromdistrict administrators.2 All proper names relating to participants in this study are pseudonyms.3 In the United States, the traditional sequence of mathematics courses in high schooltends to be as follows: an introductory course in algebra in the first year of high school, anintroductory course in geometry in the second year of high school, and a more advancedcourse in algebra in the third year of high school. In the fourth and last year of high school,students can choose from a variety of courses including trigonometry or pre-calculus.4 A “credit hour” is the unit of measure used in universities in the United States todetermine the amount of time a given course of study is worth. Generally, one credit houris equivalent to approximately 16 hours of contact/class time with a university instructorover the course of a 16 week semester period. For each hour of contact/class time, a studentis expected to spend approximately three hours preparing for class or doing out of classassignments. In the United States, university classes are generally structured to be worththree credit hours. Thus, this workshop was structured as a three credit course. Teacherparticipants were given the option of taking the workshop for university credit.


5 In this large, urban district, students from the middle schools in the city have some voiceas to which high school they will attend. The city has several “specialized” high schools(a performing arts school and a science and technology school) to which middle schoolstudents could apply. Thus, Elm Middle School was only one of the several middle schoolswhose students went to Tree. Despite efforts to get math teachers from the other middleschools which feed into Tree, only two teachers from Elm Middle School were able toattend the workshop. Elm Middle School is only one mile from Tree High School andserves a very similar student population to Tree High School.6 Although the intent was that the two participants from Elm Middle School would alsobe included in the development of the Tree math teachers’ professional community, thisdid not happen. Both Tree and Elm teachers felt that the biggest barriers to the inclusionof middle school teachers in this community was time and distance. Since the two schoolswere on separate campuses, and the teachers’ schedules were so demanding, it was difficultfor the Tree and Elm math teachers to collaborate once the workshop was over.7 In this study, each participant was assigned a number by which data related to that indi-vidual would be identified. To preserve confidentiality, only the researchers know whichnumbers match to which participant. Quotes from post workshop interviews in this text arelabeled with the appropriate respondent’s number. This allows readers to see that a varietyof respondents’ views are represented throughout the text.8 As previously mentioned, all workshop sessions were videotaped. The videotapes werelabeled by the workshop day on which they were used and the order in which they wererecorded. On most days, three videotapes were needed to record the days events. All quotesfrom videotapes are labeled by the videotape from which they came.

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ANDREA LACHANCE

Cortland Education DepartmentState University of New YorkP.O. Box 2000Cortland, New York 13045-0900USAE-mail: [email protected]

JERE CONFREY

Department of Curriculum and InstructionUniversity of TexasCampus Mail Code D5700Austin, Texas 78712USAE-mail: [email protected]

Documents

Interconnecting content and community: A qualitative study of secondary mathematics teachers