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DOI: 10.3102/0002831210385103

2011 48: 718 originally published online 21 October 2010Am Educ Res JJulianne C. Turner, Kristen Bogner Warzon and Andrea Christensen

Beliefs During a Nine-Month CollaborationMotivating Mathematics Learning : Changes in Teachers' Practices and

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Motivating Mathematics Learning:Changes in Teachers’ Practices and Beliefs

During a Nine-Month Collaboration

Julianne C. TurnerKristen Bogner Warzon

Andrea ChristensenUniversity of Notre Dame

The authors investigated patterns of change in three middle grade teachers’beliefs and practices about motivation in mathematics during a yearlongprofessional development project with a university researcher. Four princi-ples of motivation (and corresponding instructional strategies) were intro-duced in this sequence: supporting students’ competence, belongingness,autonomy, and making learning meaningful. Teachers enacted strategiesin their classrooms, were observed and interviewed, and discussed and re-flected on results during monthly meetings. Enactment of motivation strate-gies was related to changes in mathematics instruction for some teachers.Results reflect the importance of situating motivation strategies in contentarea instruction and providing adequate support for teacher conceptualchange. The authors examine both personal and contextual explanationsfor changing instructional practices and the obstacles to such changes.

KEYWORDS: motivation, middle school, mathematics, professional development

JULIANNE C. TURNER is an associate professor in the Department of Psychology atthe University of Notre Dame, 118 Haggar Hall, Notre Dame, IN 46556; e-mail:[email protected]. Her research interests include motivation to learn in classrooms,teachers’ instructional practices, and theoretical and methodological approachesfor studying classroom interaction.

KRISTEN BOGNER WARZON is a postdoctoral research associate in the Department ofPsychology at the University of Notre Dame; e-mail: [email protected]. Herresearch interests include student motivation and academic engagement as well asthe role of the peer context in shaping these outcomes.

ANDREA CHRISTENSEN is a doctoral candidate, Department of Psychology at theUniversity of Notre Dame; e-mail: [email protected]. Her research interests includeacademic motivation, low-achieving students, and change in teachers’ beliefs andpractices.

American Educational Research Journal

June 2011, Vol. 48, No. 3, pp. 718–762

DOI: 10.3102/0002831210385103

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When teachers and administrators name their most pressing concern,they often reply ‘‘student motivation.’’ Educators readily confess that

they are confused and ill prepared to address what they regard as studentdisinterest and lack of effort (Boaler, 2002; National Research Council &Institute of Medicine [NRC & IOM], 2003). Although researchers commonlysuggest prescriptions for supporting student motivation based on theoryor experimental research, often the suggestions have negligible impact inclassrooms. Several factors may help explain the gap between motivationresearch and teachers’ practices. First, few studies have investigated howresearch-based principles might be translated for complex classroom settingsand how or whether teachers might understand or enact them (Urdan &Turner, 2005). Second, it is rare for teachers to have opportunities to collab-orate in investigations of motivation theory and classroom practices (for ex-ceptions, see Maehr & Midgley, 1996; Reeve, Jang, Carrell, Jeon, & Barch,2004; Stipek, Givven, Salmon, & MacGyvers, 1998a; Stipek, Salmon, et al.,1998; Weinstein, Madison, & Kuklinski, 1995). Third, research on motivationhas been separate from research on instruction. Even though instruction isthe primary activity in classrooms, instructional practices have not beenthe focus of much motivational research (Blumenfeld, 1992; for an excep-tion, see Wigfield & Wentzel, 2007).

This study addressed the gap between motivation research and teachers’instructional practices through a 9-month collaboration between researchersand middle grade mathematics teachers. Drawing on four research-basedmotivation principles, teachers discussed, enacted, and reflected on motiva-tion strategies as related to their daily mathematics instruction. Our goal wasto understand how and why teachers’ beliefs and practices related to moti-vation and mathematics developed and changed over time, given participa-tion in the collaboration.

In this article we first discuss how the separation of research on motiva-tion and instruction has limited the usefulness of motivation theory for class-room teaching. Next we present the four principles of motivation thatinformed our collaboration with six mathematics teachers. We analyze threeteachers’ changes in practices and beliefs over the school year based onclassroom observations, interviews, and monthly meetings with the collabo-ration group. Finally, we interpret the teachers’ changes within theories ofprofessional development and teacher conceptual change.

Motivation and Instruction

Motivation research has commonly investigated students’ general moti-vational beliefs, such as goals, expectancies, and values in different domains(e.g., Eccles, Wigfield, & Schiefele, 1998) but often has not focused on whatis taught and learned or how students are learning about mathematics(Turner & Meyer, 2009). Middleton and Spanias (1999) argued that theories

Changing Practices in Motivation and Mathematics

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of motivation have not been adequately tested in mathematics classrooms, inpart because motivation theories need to be more accurate, precise, andapplicable to the content domain. Therefore, as a basis for this collaboration,we drew on a small body of research that has addressed the importance ofinstruction for motivation to learn (e.g., Ames, 1990; Guthrie et al., 2004;Reeve et al., 2004; Stipek, Givven, et al., 1998; Stipek, Salmon, et al.,1998). For example, Stipek and colleagues (Stipek, Salmon, et al., 1998)pointed out that the literatures on achievement motivation and reform prac-tices in mathematics education converge on five motivational constructs,which are consistent with the four principles used in this study: a focus onlearning, self-confidence in mathematics, risk taking, enjoyment, and posi-tive feelings (e.g., pride).

Motivation Theory

For this study, we synthesized the theoretical literature to select fourmajor principles of motivation that were relevant to mathematics learningand instruction. They served as the focus of group discussion over the schoolyear and as rationales for the instructional strategies we designed. Principlesincluded fostering students’ competence, autonomy, and belongingness inthe mathematics classroom and emphasizing the meaningfulness of mathe-matics. These principles cut across prominent theories of motivation, sug-gesting their centrality to fostering motivation in the classroom. They arealso central in studies of teaching and learning mathematics (i.e., NRC &IOM, 2003; Stipek, Givven, et al., 1998).

Competence. Developing academic competence is both a human needand the expressed goal of schooling. Competence includes understandinghow to achieve certain outcomes and feeling efficacious in performing therequired actions (Bandura, 1997; Deci, Vallerand, Pelletier, & Ryan, 1991).Many teachers, especially in mathematics, believe that students disengagebecause they think that they aren’t ‘‘good at math.’’ It is not surprising,then, that competence is central to many current theories of motivationsuch as expectancy-value (Wigfield & Eccles, 2000), self-efficacy (Usher &Pajares, 2008, for a review), self-determination (Deci & Ryan, 1985), andachievement goals (Ames, 1992; Dweck, 2000).

To support student competence, teachers can provide feedback (vs.evaluation) and demonstrate that mistakes are informational. These practi-ces are related to increased intrinsic motivation to learn (Turner et al.,1998). In addition, Kazemi and Stipek (2001) illustrated how ‘‘pressing’’for mathematical arguments and justification rather than merely acceptingcorrect answers promoted conceptual thinking. Furthermore, by helpingstudents reflect on what they do and do not understand, and why, teacherscan make them aware of their growing competence and more inclined toincrease effort.

Turner et al.



Belongingness. Belongingness is the human need to be an acceptedmember of a group and to have strong, stable relationships with others(Baumeister & Leary, 1995). The need to feel accepted may be accentuatedduring the early adolescent years and in transition to middle school(Carnegie Council on Adolescent Development, 1989). Students routinelyreport that respectful, caring teachers are major factors in their engagement(Furrer & Skinner, 2003; Wentzel, 1997). Students’ peer relationships are alsorelated to academic achievement (Martin & Dowson, 2009; Wentzel, 1991).In interviews with middle school students, Dowson and McInerney (2003)found that students listed belonging to a group and wanting to assist or fulfillresponsibility to others as reasons for achievement (also see Patrick,Anderman, & Ryan, 2002). This contradicts the view often held by teachersthat working alone is the most efficient way to learn (e.g., Boaler, 2002). Inaddition, when students come to know each other academically and socially,more students feel like legitimate members of the classroom community(Cohen & Lotan, 1995). Therefore, belongingness helps counter the individ-ual and isolating nature of learning in many classrooms, factors associatedwith low achievement and dropping out of school (NRC & IOM, 2003).

Instructional strategies that foster working productively with others cansupport belongingness. Roseth, Johnson, and Johnson (2008) conducteda meta-analysis (encompassing 80 years of research) examining the relativeeffectiveness of cooperative, competitive, and individualistic goal structuresin promoting early adolescents’ achievement and positive peer relationships.They found that when students are working productively in cooperativegroups, they are more likely to participate, to develop positive attitudestoward others and toward content, and to exert more effort. Research hasdemonstrated that students can be taught to work productively (Johnson& Johnson, 2003), to ask good questions (e.g., King & Rosenshine, 1993),and to offer good explanations (Webb et al., 2008).

Autonomy. Autonomy refers to self-initiating and self-regulating one’sown actions (Deci et al., 1991). When actions are freely chosen, endorsedas valuable, or judged consistent with one’s own values or needs, motivationis autonomous (Ryan & Deci, 2000). Conversely, when people are threat-ened, are given deadlines or pressured evaluations, or have goals imposed,they feel controlled by others. In an extensive research program, Deci andRyan and their colleagues (e.g., Ryan & Deci, 2000) have demonstrated,both experimentally and in field settings, that autonomy enables intrinsicmotivation. Field studies have shown that autonomy-supportive teachersengender more curiosity and desire for challenge in their students (Deci,Nezlek, & Sheinman, 1981), whereas controlling teachers tend to have stu-dents who are less interested and learn less effectively, especially whenlearning requires conceptual understanding (Grolnick & Ryan, 1987).Similarly, Assor and his colleagues (e.g., Assor, Kaplan, & Roth, 2002),




demonstrated that ‘‘empathic-active’’ teachers attempted to understand stu-dents’ goals, interests, and needs and then to connect them to school tasks,fostering engagement. Based on observations in mathematics classrooms,Stefanou, Perencivich, DiCintio, and Turner (2004) suggested that ‘‘cognitiveautonomy’’ support, or treating students as ‘‘origins’’ of ideas rather than sim-ply recipients of content, fostered students’ motivation.

Although autonomy is crucial for motivation to learn, it can be a difficultconcept for teachers to understand and accept. Some teachers fear that‘‘sharing control’’ can lead to off-task behavior and less learning, and somany adopt controlling behaviors (Barrett & Boggiano, 1988). However,Reeve and colleagues (2004) demonstrated that high school teachers couldlearn autonomy-supportive strategies and found that the students in theexperimental teachers’ classrooms reported greater engagement than dida comparison group. Mathematics researchers have spoken of opportunitiesfor autonomy where students are ‘‘aware of, and draw on, their own intellec-tual capabilities when making mathematical decisions’’ (Yackel & Cobb,1996, p. 473). This view of students as developing mathematics expertsbecame one of our rationales for autonomy-supportive practices.

Meaningful learning. ‘‘Why do we have to learn this?’’ is a commoncomplaint in classrooms. Standard responses such as ‘‘It’s on the test’’ or‘‘You will need this in high school’’ are not satisfying to many middle schoolstudents. The importance of meaning and understanding has long beena central issue in research on mathematics learning. As Kaput (1989), elo-quently explained, ‘‘[The] experienced meaninglessness of school mathemat-ics devastates the motivation to learn or use mathematics and is entirelyincompatible with a view of mathematics as a tool of personal insight andproblem solving’’ (pp. 99–100; also see Turner & Meyer, 2009).

Meaningfulness is related to students’ value for learning. Brophy (1999,2008) elaborated on Eccles’s (e.g., Wigfield & Eccles, 2000) notion of valueto include experiencing the satisfaction of gaining understanding, aestheticappreciation of the content, and ‘‘awareness of the role of learning in improv-ing the quality of one’s life’’ (Brophy, 2008, p. 5). Meaningfulness has been lessemphasized in motivation research, yet we believed that it was central to fos-tering a sense of purpose in mathematics instruction, for both teachers and stu-dents. Importantly, Stipek and colleagues (Stipek, Givven, et al., 1998) foundthat students’ motivation increased when they were engaged in understandingfractions in contrast to completing exercises with ease and accuracy.

Researchers agree on some major features of meaningful instruction (e.g.,Brown & Campione, 1994; Donovan & Bransford, 2005; Newmann, 1992). Forexample, Newmann, Marks, and Gamoran (1996) mention several strategiesincluding building on students’ prior knowledge, providing opportunitiesfor students to do complex thinking by addressing the central ideas of a topicor subject, and offering opportunities for students to participate in extended

Turner et al.



conversations with teachers and peers that build shared understanding (alsosee Tharp, Estrada, Dalton, & Yamauchi, 2000). These practices contrastwith typical Initiation-Response-Evaluation (IRE; Mehan, 1985) discourse inmathematics (Nathan, Kim, & Grant, 2009) and a focus on decontextualizedprocedures (Boaler & Greeno, 2000).

Teachers’ Beliefs About Motivation and Mathematics

Teachers’ rationales for their practices do not often reflect a strong special-ized knowledge base. Rather, rationales for practices may be based on institu-tionalized routines (such as the emphasis on classroom management overinstruction), the dominance of textbooks and bureaucratic controls such asstandardized test results, or pervasive beliefs in the school culture (Talbert& McLaughlin, 1994). These conditions, coupled with their own experiencesas mathematics students (Pajares, 1992), likely influence teachers’ beliefsabout motivation and mathematics. In reference to motivation, it is commonfor teachers to rely on extrinsic reinforcers to engage students and also tobelieve that mathematics ability (Prawat, 1992; Weinstein et al., 1995) andmotivation are fixed rather than malleable (Dweck, 2000). Thus, teachersmay believe that some students ‘‘aren’t motivated’’ to learn and are unlikelyto change (Cherubini, Zambelli, & Boscolo, 2002; Talbert & McLaughlin,1994). Therefore, one goal of the study was to investigate whether and whyteachers might change their motivation beliefs and practices, given the oppor-tunity to learn about and reflect on motivational theory and strategies.

Similarly, there is a strong tradition in mathematics instruction thatfocuses on teaching sets of procedures and operations to solve problems,emphasizing correct solutions and teacher control of student learning(Thompson, 1992; Wood, Cobb, & Yackel, 1991). Interestingly, Stipek,Givven, Salmon, and MacGyvers (2001) found that upper elementary math-ematics teachers’ motivational beliefs and mathematics instruction wererelated. Teachers who held ‘‘traditional’’ views of mathematics—strong con-trol of student learning and focus on correctness and procedures—alsobelieved in using extrinsic motivators and in the stability of math ability.Those who believed that mathematics could be interesting and that mathe-matical ability is malleable also believed that instruction should help stu-dents use mathematics as a tool for thought, helping them understandconcepts and providing some student autonomy. As Stipek and colleagues(Stipek, Salmon, et al., 1998) demonstrated, practices suggested by theoryand research on motivation are consistent with those currently reco-mmended by mathematics educators (National Council of Teachers ofMathematics, 1991) rather than with more traditional instruction.Therefore, a second goal of this research was to investigate if and whyopportunities to enact the four principles of motivation might prompt teach-ers to change their model of mathematics instruction.




Professional Development and Conceptual Change

The literature on professional development is replete with examples ofthe difficulty of change—both in teachers’ practices and in school communi-ties (e.g., Weinstein et al., 1991; Weinstein et al., 1995). Even when changeoccurs, it is typically uneven and takes root slowly (Richardson, 1990).Many elements play a role in the success of professional development(Fullan, 2007; McLaughlin & Talbert, 2006), including personal beliefs, inter-personal factors such as collaboration and support for professional growthfrom peers and administrators, and the influence of the school and thenational policy context (Hochberg & Desimone, 2010).

Theories of professional development emphasize the importance offocusing on teachers’ daily activities and offering teachers opportunitiesto participate in cycles of shared and ongoing dialogue, classroom enact-ment, and reflection (e.g., Blumenfeld, Krajcik, Marx, & Soloway, 1994;McLaughlin, 1994). Three features of the research design addressed thesecriteria. First, we translated motivation principles into strategies that weredoable within the teachers’ curriculum and relevant to student needs.Second, monthly meetings provided opportunities to problem solve, reflecttogether, and model for peers. Third, classroom observations and follow-up interviews using stimulated recall allowed teachers to reflect with thefirst author on enactment and student outcomes. The intention was forteachers to develop ‘‘warranted practice’’ (Richardson, 1990), or new con-ceptions based on integrating theory with teachers’ practices.

Theories of teacher conceptual change address what motivates teachers toattempt and to sustain new practices. Gregoire (2003) has theorized that suchconceptual change is difficult to achieve and depends on the extent to whichteachers perceive professional development as a challenge that enhances effi-cacy or as a threat to efficacy. Teacher efficacy beliefs are ‘‘teachers’ situation-specific expectations that can help students learn’’ (Ashton & Webb, 1986,p. 3). Teacher efficacy might relate to general ability to teach or more specif-ically to teaching mathematics. Gregoire (2003) adds that environmental sup-port is also a factor in teacher efficacy for change. Guskey (2002) proposesthat student change increases teacher efficacy and makes teachers more likelyto continue enacting new practices. With these guidelines in mind, we selectedpractices that we believed teachers could enact with success and that had goodpotential for increasing students’ motivated behaviors, and we provided bothpeer and researcher support through discussion, reflection, and problem solv-ing. In parallel to our theory of student motivation, we intended the interven-tion to support teachers’ competence, autonomy, and sense of belonging tothe collaborative group and to be meaningful to teachers.

We investigated this research question: Which factors help explain thedifferent change patterns among three experienced elementary teacherswho participated in the collaboration?

Turner et al.



Research Design

We synthesized the motivation literature to select four major principlesof motivation that were relevant to mathematics learning and instruction.They served as the focus of group discussion and as rationales for ins-tructional strategies. Principles included fostering students’ competence,autonomy, and belongingness and emphasizing the meaningfulness of math-ematics. Researchers and teachers met for approximately 90 minutes eachmonth over one school year to discuss the rationales for the four motiva-tional principles, how they might be enacted through mathematics instruc-tion, and how their enactment influenced students’ motivation to learn.Enacting strategies provided teachers with experiences to analyze and reflecton in later meetings and interviews. Enactments also served as research datademonstrating how (or if) teachers used the strategies and changed practi-ces. Therefore, enactment of new instructional strategies and opportunitiesto reflect with others anchored our research design.

Method

Setting and Participants

Setting. The study took place in one intermediate school (Grades 5–8) ina midsized urban school corporation in the Midwest during the 2005–2006school year. The participating school had a history of poor performanceon standardized tests; during the collaboration year, approximately 50% ofstudents passed the mathematics portions of the state-administered standard-ized test, compared to the state average of approximately 77% of studentspassing. All the teachers perceived their students as unmotivated and lowachievers, and (although not stated explicitly) their beliefs may have beeninfluenced by the students’ race, ethnicity, and relatively low SES. The schoolhad just been reopened as an intermediate school after redistricting. Theprincipal was a veteran from the district but new to this school. Althoughthe researcher took opportunities to inform the principal about the teachers’efforts and notable changes in practices and student response, he offeredonly polite acknowledgment; he did not show interest in the teachers’ ef-forts. The teachers mentioned that the school had a poor reputation andthat it was not an attractive choice for teachers in the school district. Theteachers perceived a heavy emphasis from the district on raising achieve-ment scores.

Students. The student population of the participating intermediateschool was 46% African American, 31% Caucasian, 17% Hispanic, 5% multi-racial, and 1% Asian. In 2005–2006, 77% of students received free or reducedlunch.




Teachers. Six teachers (two male, four female, all Caucasian) from theintermediate school participated in the motivation and mathematics collabo-ration. The first author recruited teachers for the study initially through theprincipal of the school and then through participating teachers in the school.All teachers were volunteers (see Table 1 for teacher descriptions).

We chose to analyze the change patterns of three teachers, Marcie, Kate,and Helen. These teachers were similar in that all were trained as elementaryteachers, all were experienced, and all of them enacted instructional strate-gies during the collaboration. They differed from each other in that Marcieand Kate, as opposed to Helen, had relatively weak content knowledge inmath. As a result, these three teachers presented cases of how contentknowledge might be related to teachers’ attempts to enact different instruc-tional strategies. The three teachers also differed in teaching efficacy; Marcieand Kate appeared to have relatively high efficacy for teaching, whereasHelen did not. Marcie had low efficacy for teaching mathematics. We exam-ine how the confluence of personal and contextual factors appeared to facil-itate or constrain the teachers in changing practices. We do not focus on Pegor Garett because they were novice teachers, nor on Todd, who resisted im-plementing strategies.

Procedures

Intervention. The collaboration consisted of nine monthly meetings,held after school and led by the first author, which were focused on thefour motivation principles and related instructional strategies applied tomathematics (see Table 2). An additional meeting was added at the end ofNovember to focus on strategies for grouping. The first author arrangedfor a master teacher (from another school) to offer a workshop on cooper-ative grouping strategies (the KISS workshop). During regular meetings, thefirst author introduced the principles and provided rationales; then partici-pants discussed strategies and how they related to motivation and mathemat-ics instruction and chose strategies (from a list) to implement in theirclassrooms in the upcoming month (see Table 3 for an example). After

Table 1

Descriptions of Participating Teachers

Teacher Grade Level College Major Years of Experience

Kate 5 Elementary ed 7

Marcie 6 Elementary ed 6

Peg 7 Elementary ed; math endorsement 2

Garett 7 Social studies First-year teacher

Helen 8 Elementary ed 20

Todd 8 Mathematics 10

Turner et al.



teachers began strategy enactments, they (not the first author) led the discus-sion about their experiences, raising questions, troubleshooting, and offer-ing support, advice, and challenges.

Each teacher also participated in one-on-one initial and final interviewswith the first author, held in October or November and May (see theAppendix in the online journal). These interviews were intended to captureteachers’ beginning and changing motivational and mathematics beliefs andstated practices. Classroom observations occurred between three and fourtimes during the year; the purpose was to observe teachers enacting motiva-tional strategies in their instruction. We used a stimulated recall procedure ininterviews (usually 2–4 weeks) after each observation, in which the firstauthor and the teacher discussed an edited video of the lesson, focusingon motivation and mathematics instruction. These interviews provided anopportunity to discover how teachers interpreted the strategies, both con-ceptually and through their instruction.

Researcher’s role. The first author played several roles during the inter-vention including discussion leader, discussion participant, and researcher.These roles were necessarily overlapping and impossible to disentangle.As discussion leader, the first author strove to present the rationale for the

Table 2

Teacher Meeting Topics by Month

Month Meeting Topic Materials

September Overview of four

motivation principles

Video of Japanese and American mathematics

instruction from the Trends in International

Mathematics and Science Study (Stigler, Gonzales,

Kawanaka, Knoll, & Serrano, 1999)

October Competence Reading: Reinhart (2000)

November Belongingness Grouping strategies from a local teacher; readings

from Johnson and Johnson (2003) on forming and

using groups

December Grouping strategies

(belongingness)

January Autonomy Cathy Humphreys video (Boaler & Humphreys, 2005)

February Autonomy and

meaningfulness

March Meaningfulness

April Create lesson

incorporating four

principles

May Summarize previous

meetings, celebrate

successes




Table

3

Exam

ple

so

fS

trate

gie

sB

ased

on

Mo

tivati

on

al

Pri

ncip

les

Princi

ple

:Com

pete

nce

Why

Does

ItW

ork

?In

stru

ctio

nal

Stra

tegie

s

Help

students

feelm

ore

succ

ess

ful,

and

hav

em

ore

controlofth

eir

lear

nin

gin

mat

h.This

support

incr

eas

es

students

’co

nfidence

inth

eir

ability

tole

arn,im

pro

ve,an

dac

hie

ve.

We

allw

antto

feelth

atw

ear

egood

or

impro

vin

gat

som

eth

ing.Even

students

who

deny

they

care

aboutm

ath

orsc

hool

know

how

importan

tac

hie

vem

entis

and

they

would

like

tobe

succ

ess

ful.

1.H

ave

students

write

com

ments

orquest

ions

inth

ela

st

few

min

ute

sofcl

ass.

Addre

ssth

ese

the

nextday

.This

enco

ura

ges

students

toth

ink

aboutw

hat

they

do

and

do

notunders

tand.This

help

sth

em

see

how

they

are

gettin

gsm

arte

r.

Childre

n’s

need

for

com

pete

nce

is

fost

ere

dw

hen

teac

hers

pro

vid

ehig

h

expect

atio

ns,

consi

stentst

andar

ds

for

lear

nin

g,an

dad

equat

ehelp

and

feedbac

kto

ensu

repro

gre

ss.

When

students

feelth

atth

ey

are

impro

vin

g,th

ey

are

more

likely

to(a

)try

har

der,

(b)

val

ue

the

subje

ct,an

d(c

)fe

el

pro

ud

aboutac

adem

icac

com

plish

ments

.

2.Tell

students

every

day

how

they

are

pro

gre

ssin

gan

d

what

they

lear

ned

the

day

befo

re;so

me

students

need

tobe

convin

ced

that

they

can

and

are

lear

nin

g.This

is

afo

rmofposi

tive

feedbac

k.

Bra

insc

ience

has

confirm

ed

that

acad

em

icab

ilitie

sar

eexpan

dab

lean

d

impro

vab

le.St

udents

who

vie

wth

eir

cognitiv

eab

ilitie

sas

fixed

from

birth

or

unch

angeab

lear

em

ore

likely

to

experience

decr

eas

ed

confidence

and

perf

orm

ance

when

face

dw

ith

difficu

ltie

sor

setb

acks.

This

work

sbest

ifst

udents

experience

real

succ

ess

on

task

sw

ith

som

ech

alle

nge

involv

ed.Su

cceedin

gat

overly

eas

yta

sks

hold

slittle

val

ue

for

students

.

3.G

ive

dai

lypro

ble

mso

lvin

gopportunitie

s.To

fost

er

com

pete

nce

,m

ake

the

pro

ble

ms

very

eas

y,but

nonro

utine.After

am

onth

,st

udents

willbeco

me

com

fortab

lean

dm

ore

confidentan

dab

leto

move

to

more

chal

lengin

gpro

ble

ms.

4.H

elp

students

lear

nhow

toexpla

inw

hy

they

are

doin

gso

meth

ing;th

en

ask

them

toexpla

into

clas

s,

eac

hoth

er,

and

hav

epai

rsquest

ion

eac

hoth

er.

‘‘Ise

e

you

did

___.H

ow

did

you

com

eup

with

that

?’’

5.Anal

yze

your

quest

ionin

gst

rate

gie

s.U

sequest

ions

that

fost

er

thin

kin

g;as

kst

udents

toju

stify

thin

kin

g.

Students

do

notle

arn

from

hear

ing

rightan

swers

.T

hey

nee

dto

know

an

dbe

able

toex

pla

inw

hy.



principle to the teachers, to note the benefits of the principle for the teachersand students, to support teachers’ efforts, and to contextualize the principlesand strategies in mathematics instruction. As teachers began to take moreleadership in the discussions, and as they recounted and debated the instruc-tional strategies, the first author also took on the role of participant. Her goalswere to encourage teachers to elaborate and reflect on practices and out-comes and to challenge teachers’ statements, such as assertions that studentscould not learn to work together. Also, teachers sometimes spoke directly toher rather than to the group as a whole, denoting the power she held as a uni-versity researcher and one kind of ‘‘expert.’’ As such, the trajectory of the col-laboration reflects her interaction with the teachers throughout and reflectsdecisions she made, such as the sequencing of the principles and which videoclips to use in interviews. The third role, researcher, was always in play,although the author did not refer to the activities as ‘‘research’’ to the teachers.Rather, all the activities served dual purposes; to collaborate with teachers toenact motivational strategies and simultaneously to investigate the teachers’beliefs and practices during the 9 months.

Data Collection and Analysis

Data sources. Data included individual fall and spring interviews, obser-vations of classroom lessons, interviews following each observation, teacherdiscourse from eight teacher meetings, and, occasionally, e-mails betweenteachers and the first author. The first author conducted interviews, observa-tions, and observation interviews. All teacher meetings and interviews wereaudio- and/or video-taped and transcribed for analysis. Classroom observa-tions were video-taped but not transcribed.

Collaborative meetings. Monthly meetings were videotaped and tran-scribed. The transcribed discourse from teacher meetings (N = 8) was copiedinto an Excel file. Two to three coders coded each meeting transcript inde-pendently and then met to resolve discrepancies. Transcripts were coded intwo ways. First, text segments were coded as addressing one or more moti-vational principles (i.e., autonomy, belongingness, competence, or mean-ingfulness), thus providing insight into how teachers talked about,enacted, interpreted, and reflected on the principles and strategies.Second, using an open coding scheme (Miles & Huberman, 1994), segmentsof text were coded for content that indicated how teachers’ beliefs and prac-tices about motivation and math might be changing.

Classroom observations. One camera was placed in the back of the roomfocused on the teacher. The observer sat in the back and took notes on theclass session (e.g., content, interaction patterns). Videos were stored oncomputer to provide a record of teacher practices over the year and alsoto edit for observation interviews. Videotapes were not coded. Instead,




researchers viewed observations to determine which strategies were beingused, whether teacher behaviors changed over time, and whether (andhow) student behaviors changed in response to teacher strategy use. Wewere particularly focused on the teaching and learning behaviors we pre-dicted the strategies would engender (e.g., teachers asking open-endedquestions resulting in students explaining their thinking more) but werealso open to unexpected findings.

Interviews. All interviews were conducted in person during the schoolday. They usually lasted a whole class period (from 45 to 90 minutes).Initial (fall) and final (spring) interviews (see the Appendix) were semistruc-tured. Probes were used if teachers were uncertain about the intent of thequestion. The same questions were used for final interviews in order todetermine if teachers’ beliefs and their description of practices changed.

Postobservation interviews were based on edited versions of each teach-er’s videotaped classroom observations (fall, winter, spring). The intent was touse several events from the lesson as a springboard for discussion of the strat-egy the teacher was using and how the teacher viewed student responses interms of motivation and of learning. The events were also chosen with theparticular teacher in mind, based on the first author’s assessment of which ex-amples would challenge teachers’ thinking or induce reflection. For example,if a teacher was trying to teach math more conceptually but her questions fellshort of that goal, we might discuss that particular event. Interviews began bywatching (and sometimes ‘‘talking over’’ the video); then the video was stop-ped and questions were asked about each segment. It was not uncommon forintended topics to change, initiated by either the interviewer or the teacher.For example, if the teacher made a comment that the first author believedto be a fruitful topic in terms of motivation or mathematics, she would followup. In many cases, these diversions were ‘‘teachable moments,’’ and teachersformulated new goals, which they followed up in later lessons. In other cases,teachers appeared more interested in other topics and would redirect theinterview. Sometimes this provided information about important issues forthis teacher, and sometimes teachers appeared to evade questions.

We analyzed the three teachers’ initial and final interviews and postob-servation interviews for evidence of how they interpreted and reflected onmotivation strategies and principles and on their students’ responses.Initial and final interviews were focused more on global changes in beliefsand practices, whereas observation interviews focused more on implemen-tation of specific strategies within math content and related student behavior.

Results

The results section chronicles the three teachers’ changing views andpractices related to motivation and mathematics. This section is structured

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chronologically because teachers’ enactments and reflections are related tothe order in which motivation principles were introduced (see Table 1).The principles were introduced as follows: competence (October), belong-ingness (November), autonomy (January), and meaningfulness (March). Wechose to begin with competence because the teachers described their stu-dents as low achievers and fearful of math. Belongingness was next becausewe reasoned that the emphasis on grouping offered new opportunities forsupporting competence. We introduced the autonomy principle thirdbecause we believed that these strategies might build logically on earlier en-actments of competence strategies and because they provided additionalways to enhance students working together. We placed meaningfulnesslast because we hypothesized that it would be the most difficult for teachersto enact, and we hoped that teachers’ experience enacting other principleswould create a desire to provide students with mathematics ‘‘worth doing.’’

Introduction to the Motivation Principles

The goal of our September meeting was to assess teachers’ initial viewsof motivation and mathematics instruction and to situate the four motivationprinciples in instruction. We showed video segments from the Trends inInternational Mathematics and Science Study (Stigler, Gonzales, Kawanaka,Knoll, & Serrano, 1999). The segments were typical geometry lessons ineighth grade Japanese and American classrooms and showed very differentapproaches.1 In the Japanese geometry lesson, the teacher presented a prob-lem and asked students to work on it individually for 3 minutes. Then heasked students to choose from several ways of discussing the problemwith others, and finally invited student volunteers to share their solutionsat the board. In the American lesson, the teacher asked factual questions(e.g., ‘‘What is a supplementary angle?’’) and led students through a home-work review of exercises computing the number of degrees in supplemen-tary and complementary angles. The juxtaposition was intended to provoketeacher responses about whether and why the instruction in the two lessonsappeared motivating, especially in relation to supporting student compe-tence, autonomy, belongingness and to making mathematics meaningful.The teachers seemed to admire the Japanese lesson but felt that all facetsof it—the social collaboration, the teacher’s ‘‘inquiry’’ role, and the studentconstruction of knowledge—were impossible, even alien, to how mathemat-ics could be taught in their classrooms. Nevertheless, they judged theAmerican lesson to be boring and meaningless.

Competence Principle

At the October meeting we introduced the competence principle withan article written by a middle school math teacher (Reinhart, 2000).Reinhart (2000) described strategies (and rationales) for fostering students’




independent mathematical thinking, including giving students wait time andasking questions that prompted reflection. Reinhart explained that his defi-nition of a good teacher had changed from ‘‘one who explains things sowell that students understand’’ to ‘‘one who gets students to explain thingsso well that they can be understood’’ (p. 478). He advocated more talkingfor students and more listening for teachers. We created a strategy sheetfor teachers with other competence-supportive strategies, such as giving stu-dents informational feedback, allowing students to lead homework review,helping students set goals and monitor their progress, and having studentspose questions about the content (see Table 3).

Teachers’ Initial Beliefs and Practices

Marcie. Marcie began the collaboration as an efficacious teacher of lan-guage arts but with low efficacy for teaching math and for her own math skills.She seemed to dislike mathematics and had strong biases about how mathcould be taught. For example, she commented that ‘‘with decimals . . . I didn’tfeel that there was a . . . different way to show it other than, ‘This is what youdo.’’’ In the fall, her instruction emphasized procedures and textbook exer-cises, possibly because they appeared to give her confidence and conferredthe mathematical authority she did not feel. In her initial interview, Marciedescribed her students as particularly unmotivated in math; she believedthat their attitudes, rather than her instruction, made motivation difficult.She complained that her students’ goals in math were to finish quickly, getright answers, and ‘‘check [it] off’’ their list. In addition, she was dissatisfiedthat only a few students participated. Nevertheless, Marcie felt a deep respon-sibility for her students’ achievement. In an early meeting, she questionedwhether her instruction (i.e., ‘‘You get the problem of the day up, you dothe homework, you answer questions, and you have 15 minutes to doa new lesson’’) was helping her students ‘‘get math.’’

In her initial interview, Marcie mentioned that one of her most success-ful lessons was one in which everyone was ‘‘participating’’ (on task), indicat-ing her value of a well-managed class. At the September meeting, Marcieexpressed discomfort when the videotaped Japanese students laughed andinteracted while solving math problems. She commented that even thoughthe instruction was ‘‘good,’’ ‘‘it just looked like they were having fun . . .I’m not used to seeing that in a math room.’’ She seemed to regard enjoy-ment and purposeful mathematics as incompatible. In contrast, in languagearts, Marcie used strategies such as group discussion. When we pointed outthat she could use the same strategies in math, she wondered aloud, ‘‘Whatwould that look like in math?’’ This event foreshadowed a yearlong strugglefor Marcie: a growing desire to allow students to talk and learn more fromone another (and for themselves) combined with uncertainty about whatwould happen if she allowed her students to have more cognitive autonomy

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in math class. During the introduction to the competence principle and strat-egies at the October meeting, Marcie selected these strategies: ‘‘questioningbetter at different levels, especially in math,’’ using pair share to encouragemath talk, and inviting students to ask questions at the end of class. She wasoptimistic about the potential of the group: ‘‘I’m thinking that through thisproject I am going to get a lot of techniques to use in the classroom, includ-ing questioning.’’

Kate. Kate joined the collaboration because she was ‘‘bored’’ with hermath teaching and admitted that many math topics seemed meaningless tostudents. She worried that her instruction, which she had not questionedpreviously, was ineffective with her low-achieving fifth grade math class,many of whom had just left special education math and ‘‘really feel likethey’re dumb.’’ She noted that her students thought that ‘‘you must not besmart . . . if you actually have to think about [a problem] for anything longerthan 10 seconds.’’ Therefore, she came to the collaboration with desire tolearn how to support her students’ competence.

Kate believed that the district’s emphasis on standardized testingimpeded her goals. Watching the video of the Japanese teacher inSeptember, she seemed torn between what she saw as two mutually exclu-sive ends—teaching for understanding (which she regarded as competencesupportive) and the pressure to raise student achievement scores. Sheadmired the Japanese teacher’s open-ended questions and the time hegave students to think, but then dismissed this approach as impractical.Kate spent most of the year trying to resolve this dilemma.

The discussion of the Reinhart (2000) reading at the October meetingprompted Kate’s realization that she always rushed to her students’ rescuewhen they seemed uncertain, connoting lack of trust in their thinking. Shereflected that she needed to ‘‘switch the teaching to the students so theyare . . . learning for themselves. . . . If they are the ones . . . doing mostof the talking, then they’re going to feel more successful, more control . . .or more pride.’’

During Kate’s initial interview, she expressed dissatisfaction with herinstruction:

The common denominator is me going up to the board and doingexample after example . . . and [students] . . . not paying attention.And then it becomes a twofold thing. You’re not only trying toinstruct whatever the lesson is that day, but, you’re also having tokeep them on task, because it’s so stinkin’ borin’ that they don’tcare about it anymore.

She noticed that she had ‘‘seen some more motivation’’ when she changedher instruction so that students led homework discussions (a competencestrategy). ‘‘They LOVE that,’’ she reported.




By November, Kate had begun to reflect more on her students’ mathe-matical understanding, which prompted her to take some instructional‘‘risks.’’ During an exercise on place value (find the number of 10s in500,000), students’ behaviors indicated clear lack of understanding. Ratherthan continue this meaningless exercise, Kate ordered students out of theirdesks to the floor for a ‘‘math chat’’ in which they simply talked about theconcept. Students were astounded to discover that ‘‘doing math’’ involvedtalking and thinking rather than just paper and pencil exercises. As Katenoted, students ‘‘don’t think about thinking about math.’’ The incidentbecame a turning point as Kate she realized that she needed to challengethe students’ (and her own) conceptions of ‘‘learning mathematics.’’ Katetried to use math chat many more times that fall, but with less success.However, this experience revealed to Kate that ‘‘thinking’’ was more interest-ing (and valuable) to her students than completing exercises.

Helen. Helen was in her first year of teaching at this public school after20 years of teaching in Catholic schools. She joined the group to becomemore integrated in the teacher peer group. From the beginning, Helen ex-pressed low expectations for her students (e.g., students ‘‘have no motiva-tion’’ and will do ‘‘anything to get the answer without actually doinganything’’). Helen appeared to have low efficacy about her ability to changestudents’ motivational orientations. At her initial interview, she told the firstauthor, ‘‘I used to think that I could motivate anybody and that I was going tosolve all ills. . . . My first year I learned that that wasn’t going to happen . . . Iwas disillusioned.’’

Although Helen seemed to have low efficacy for teaching her studentsand negative views about their ability, she seemed to like and understandmathematics. She once explained the meaning of an algorithm to Kate.However, she still conceived of math as practicing a set of procedural skills.At the October meeting when the teachers were discussing how to respondto student confusion, she noted that she would teach the lesson over andover again.

Early in the collaboration (and through the year), Helen often inter-preted the principles and strategies through the lens of low expectationsfor and a negative view of students. She tended to blame students ratherthan to consider how she could support them. This might have been one ori-gin of Helen’s misinterpretation of many strategies. Helen reported imple-menting the strategy of having students keep track of their progress askeeping track of their grades. In the October meeting that focused on sup-porting competence, teachers were suggesting how they might avoid tellingstudents an answer and ask a good question instead. Helen suggested thisquestion: ‘‘Did you read what you were supposed to be doing?’’ She contin-ued, ‘‘That is what kills me: that you have to explain but they never read it inthe first place.’’ In contrast to the other teachers, Helen interpreted student

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confusion as a ‘‘gotcha’’ moment, assuming that it was their fault that theydid not understand. In October, she also complained that discipline prob-lems in her classroom kept her from trying new strategies.

Helen was easily frustrated and tended to abandon an attempted strat-egy if it was not immediately successful. She appeared not to have the effi-cacy to retry or revise strategies. For instance, in October, she reported ontwo attempted strategies. She had put students in partners so that they couldlearn by discussing the problems, but ‘‘they just messed around . . . even thegood kids.’’ In response, Helen took over instruction, relegating the students’role to copying answers. Helen had similar difficulties when she tried thecompetence strategy of having students write down questions they hadabout that day’s lesson. She reported that students wrote that they had noquestions. Although her peers suggested other ways of enacting the strategy,she nevertheless abandoned it.

Belongingness Principle

During the November meeting, a seventh grade teacher led the group dis-cussion about teachers’ successes and challenges implementing the compe-tence strategies. During the second part of the meeting, we introduced thebelongingness principle, which we translated as strategies to help studentswork productively together (Johnson & Johnson, 2003). Included in the mate-rials was a grouping strategy, called KISS, borrowed from a local teacher. KISSsummarized four criteria for successful group work: Keep in your group,Include everyone, Share, and Sixteen inch voices. Teachers were enthusedabout the KISS strategy, partly because it gave them a template for teaching stu-dents to work together. At the first author’s suggestion, the group asked thelocal teacher to present a workshop on cooperative grouping. During theKISS workshop in late November, teachers practiced using cooperative group-ing skills, such as taking different roles (e.g., captain, summarizer). The teacher-leader also demonstrated and distributed curriculum materials that she and col-leagues had used effectively in group work. The teachers in the collaborationappeared to be inspired by the enthusiastic presenter, whose experience pro-vided convincing evidence that grouping could work well in middle school.Most teachers began to implement grouping activities in December.

At the December meeting, Kate and Marcie led the discussion of teach-ers’ implementation of KISS and grouping strategies. Most teachers wereenthusiastic about how students were using the KISS rules and how theywere working together. Teachers discussed both the ‘‘how to’’ of groupingstudents and the mostly positive social and academic outcomes. Two teach-ers commented on how positive peer pressure seemed able to convinceeven disaffected students, ‘‘the outsiders,’’ to participate. Most were pleasedto find that students were adept at explaining the math to group members,proving to be ‘‘good teachers.’’




Fall: Observations and Postobservation Interviews

Marcie. In Marcie’s first observation in early November, she made anattempt to move away from her typical whole class instruction. Her goalswere to use questioning strategies while students worked in groups.Teachers had discussed group work several times in the fall, even beforethe November belongingness and KISS meetings, and Marcie was one ofthe first to implement grouping strategies. During the observation, thegroups completed word problems that required the creation of a table.Each group had a different problem; in general, students had to determinehow much money a family would save if they saved a certain amountmore each week than they had the previous week.

During the observation, Marcie did most of the talking and maintainedcontrol over how her students worked. Although she attempted to questionstudents, their responses to her questions were very procedural and sheaccepted them without further comment. She did, at one point, appear totry to elicit more explanation from a student, saying, ‘‘I just want to knowwhy, why you think that.’’ She moved on, however, without waiting fora response from this student. One group was having particular difficulty.She hovered over them, instructing, ‘‘Erase those three, and those four,and those four. OK, so put that number there.’’ At her postobservation inter-view, Marcie noted that she had had trouble with that group because theyhad the hardest problem and ‘‘I tried to go through it so many times,’’ buther students did not resolve their misunderstandings. Without reflection,she told them what to do rather than to help them think through the prob-lem themselves. Marcie was especially frustrated that one student in thegroup resisted her attempted questioning, saying that this student alwayswanted ‘‘answers.’’ During the class, Marcie focused on awarding pointsfor correct work. Her goal was for students to complete the work ratherthan to improve their understanding of the problem. Despite her good inten-tions, Marcie was not yet successful in her goal to improve her questioningstrategies in the fall. She seemed caught between old practices and new, notable to resolve the conflicts.

In Marcie’s observation interview, the first author’s questions revolvedaround implementing the competence principle, but Marcie spent aboutthree fourths of the interview discussing her grouping strategies. This ap-peared to be her new priority. She did briefly link grouping with the com-petence principle, saying that the ‘‘whole reason’’ for putting students ingroups was for ‘‘them to feel a little more competent.’’ She noted both suc-cesses and problems with grouping. On the one hand, Marcie was pleasedthat her students—perhaps because of group work—were now willing totry word problems, which they had been unwilling to do at the beginningof the year. On the other hand, Marcie was not yet satisfied with how stu-dents worked together. She complained that many students still regarded

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group work as an opportunity to copy answers from others rather than tolearn from others. After attending the KISS workshop, she realized thatshe needed to prepare students for group work by focusing on group rolesas well as teaching students how to collaborate successfully.

Marcie’s frustration with her students’ cheating seemed to propel her tobegin to make changes in line with the motivation principles. At theDecember meeting, Marcie reported that she had addressed the issue ofcopying from others by creating ‘‘a rule that everyone had to understandhow we got through the process.’’ This simple rule represented a dramaticchange in Marcie’s apparent mathematics teaching philosophy: from a focuson right answers to giving primary importance to everyone understandingthe problem-solving process. Marcie also noted that, due to frequent use,her groups were functioning more autonomously.

Kate. Kate used two competence strategies during her first observationin November: students led homework review from the overhead projectorand the class had a ‘‘math chat.’’ In general, Kate commented with satisfac-tion about the success of the students leading the class. She noted that stu-dents were using the strategy flexibly, depending on their competenceneeds.

Some students will get up there and want to do it themselves. Otherstudents will go up there and . . . call on people . . . if . . . they realizethat ‘‘oh no, I really don’t know how to do this, well, I’ll just call onsomeone and it’s perfectly acceptable to call on someone.’’

One outcome of the strategy for the students appeared to be less fear ofmaking mistakes in math.

In the first activity, the student methodically led her peers through theproblem 200 3 29, multiplying both zeros in 200 by 29, eventually addingthe three partial products to obtain the product. After this, Kate asked if any-one had done the problem a ‘‘different way,’’ trying to lead students to theshortcut of adding two zeros to the product of 29 times two (hundred). Eventhough she assured students that either approach was correct, she seemed toencourage the short cut as helpful ‘‘for those of us like me that like to dothings the short, easy way.’’

As she viewed the video segment, Kate criticized her emphasis on ‘‘theshort, easy way.’’ She said, ‘‘Kids get frustrated because they can’t get tothings quickly and yet, what am I advocating? Speed. . . . Perhaps I needto consider forgetting about speed and not worrying so much about gettingthings done quickly.’’ She agreed that her emphasis on speed as desirableinadvertently sent the message that working more slowly denoted lessexpertise.

In the second incident from the observation, a student demonstratedhow to evaluate expressions by substituting a value for y in an equation




and then solving it using order of operations [8(y 1 4); y = 2]. The studentwho was teaching said, ‘‘What I did is because you’re supposed to do theparentheses first always in the math problem.’’ Kate asked ‘‘Why?’’ Playingthe devil’s advocate, she declared, ‘‘I think you should be able to do itany way you want.’’ She told students to ‘‘investigate together’’ what wouldhappen if they did not perform the operations inside the parentheses first,but instead multiplied 8 times y, then added 4. Although intended to modelthat one can discover why algorithms work, this challenge confused the stu-dents. The teacher attempted to lead a discussion, asking how they coulddecide which is right, but in the end her attempt fell flat because she couldnot provide a mathematical rationale. Kate’s fledgling attempts showed herwillingness to take risks, despite her weak content knowledge.

As she viewed the video, she expressed displeasure with the way shetaught the lesson and confronted the inadequacy of her content knowledge.

This kind of makes me think a little bit differently . . . in the past I’vealways taught order of operations as ‘‘here are the steps, this is whatyou do.’’. . . I’ve come up with little songs . . . to memorize them. . . .But I think that . . . I’ll need to spend some more time talking aboutthe why—although I have to tell you, I’m not sure if I know theanswer to why you have to do what’s in parentheses [first].

This was another step on Kate’s journey to move from a rule-based to a moreconceptually oriented approach to instruction. It illustrates her developingconviction that students’ competence was related not only to whether stu-dents could do the math but also to how well they understood it, and sherealized that she also needed to discover the ‘‘why’’ herself.

Kate’s original goal of having students enhance competence throughleading homework review, albeit successful, was already evolving at thistime. In the November meeting, she commented that her students’ ‘‘confi-dence ha[d] really soared,’’ perhaps because ‘‘they’re sort of the ones thatare in charge of the way math goes.’’ In addition, she noted changes inthe students’ learning: ‘‘Kids are getting up there . . . and not just explain-ing, ‘this is step one, step two,’ but explaining why you have to do it . . . it’sgood thinking.’’ She added that students thought it was okay to say ‘‘I don’tthink that you did that right’’ and were even ‘‘battl[ing] it out’’ about correctanswers. These student discussions led Kate to offer her students morechallenging opportunities—she taught them to ‘‘justify and defend’’ theirideas.

Kate began to use collaborative groups in her class after the Novembermeeting and KISS workshop. Picking up on the motivational potential ofpeer interactions, she proclaimed, ‘‘Kids are better motivators than weare.’’ Reflecting on the theoretical rationale for group work, she commented,‘‘Any one person doesn’t know everything . . . but if you put . . . peopletogether, you’ll . . . have all those skills.’’ In addition, Kate found that her

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students accepted the responsibility for group work: ‘‘They’re pretty muchfiguring that out on their own.’’

Helen. Several themes were evident in Helen’s fall observation, includ-ing Helen’s tendency to misinterpret the meaning (and thus the enactment)of strategies, her tendency to take over a task rather than support studenteffort and skills, and the use of negative language, both in relation to stu-dents’ abilities and in relation to the value of the mathematics. Helen beganthe observed class with an attempt to use the competence strategy of havingstudents lead homework review. However, she misinterpreted the strategyand assigned review problems as seatwork. When she noticed studentsstruggling, she told them that the problems should be review but that shewould go over the questions if students needed help. She asked studentshow to begin the first problem but then took over, explaining the processherself. She attempted to elicit student participation with, ‘‘Does this makesense?’’ but then confirmed the answers without waiting for student input.Students’ opportunities to develop competence were reduced to answeringsimple computation questions. When students could not answer a questionabout integers, Helen blamed them, saying sarcastically, ‘‘Mr. B. said youalready had [integers], but we’ll go over it again.’’

At the postobservation interview, the first author asked Helen why,after students expressed difficulty, she chose to review all problems onthe board instead of enlisting student participation. Helen answered thatthis was the only way she could think of. When asked how she could sup-port students in doing more of the thinking, Helen reiterated her ‘‘philos-ophy’’ that it was effective to tell students how to do problems. Althoughshe felt that her instruction went in one ear and ‘‘right out the otherside,’’ she expressed the faint hope that at least someone would rememberher instruction.

The second video segment for the observation interview was a lesson onusing manipulatives to demonstrate the concept of an equation (Hands onEquations�; Borenson & Barber, 2008). Helen asked if anyone neededa set of manipulatives ‘‘or wants a set, I should say, because the other classestold me, ‘No thank you, we don’t want them.’’’ Although students had notexpressed negative attitudes in this class, Helen assured them, ‘‘One morelesson and we’re finished with these, in case you’re wondering when you’llever be finished.’’ Contrary to the purpose of the manipulatives, which aredesigned for students to actively construct understanding of equality,Helen executed all the problems in the front of the room with students mim-icking her moves at their seats. Much of what Helen said in the fall observa-tion seemed to communicate that students should be afraid of the math,which she implied they were incapable of doing: ‘‘Those nasty integersare going to come back and bite you every time’’ and ‘‘We’re going to be re-viewing integers for the rest of our lives.’’




At the postobservation interview, when asked why she didn’t have stu-dents solve the problems with manipulatives themselves, she replied thatthey wouldn’t be fast enough, echoing her belief that students would learnbest by watching, not doing. When asked why she had spoken about themanipulatives so negatively, she noted that the students did not like them.When pressed to think of ways to be more positive about them, Helen’sonly solution was to have used the manipulatives less. Helen appeared tohave low efficacy for gaining her students’ cooperation, and it is possiblethat some of her negative comments about the math were meant to showthat she was on ‘‘their side.’’ However, this tactic, and her frequent use ofextrinsic motivators like food, undermined her authority more.

At the November and December meetings, Helen blamed her unsuc-cessful strategy attempts on the students. For example, when discussingher competence strategies, Helen said, ‘‘I’m reviewing what they did in classto see if they even know what’s going on.’’ She said that reviewing old workshould be ‘‘great for competence because it should give [students] that, ‘well,I know this [feeling].’ Well, today nobody knew anything.’’ When otherteachers provided suggestions, Helen’s response was, ‘‘Well, some of thesekids aren’t the smartest . . . they’re just not real [sic] competent.’’ InDecember, she reported that her students ‘‘fought me the whole way’’ aboutgroups. She asked for suggestions to ‘‘force’’ them into groups. All the otherteachers suggested strategies they had used. In addition, they challenged herassertions that her students (whom they had taught in previous years) wereresistant. However, Helen’s repeated response was, ‘‘We tried that.’’ Whenone teacher asked her whether students resisted groups because theywere embarrassed about their low math skills, Helen responded that embar-rassment was not the issue. She said, ‘‘They’re that low; they don’t evenknow what’s going on.’’ However, she also reported that two boys whoworked together were the first students in six classes to understand orderof operations ‘‘because they worked together.’’ Although Helen said thatshe would try the suggestions, she sent mixed messages about her commit-ment to trying strategies.

Autonomy Principle

We introduced the autonomy principle in January, emphasizing thenotion of ‘‘cognitive autonomy’’ support (Stefanou et al., 2004)—treating stu-dents as ‘‘origins’’ of ideas rather than simply recipients of content. Strategiesincluded asking students to justify ideas, to generate original solution paths,and to evaluate their own and others’ solutions. The goal was to change thestudent role from passive listener to active explainer, collaborator, or prob-lem solver and to encourage multiple approaches to problems.

To illustrate how one might facilitate autonomous thinking, we showedthe teachers a video of a seventh grade math class taught by Kathy

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Humphreys (Boaler & Humphreys, 2005). Humphreys turned the teaching ofa routine formula (area of a cylinder) into an opportunity for students todevelop cognitive autonomy. Holding an oatmeal box, Humphreys askedthe class if anything they had learned about rectangular prisms (studied pre-viously) could help them develop ‘‘a theory about how to find the volume ofone of these [cylinders].’’ Students in the video began suggesting ways todevelop the formula while the teacher encouraged them. The teachers inthe collaboration group were stunned that the students in the video couldwork so autonomously. They marveled that students could hypothesizeabout how to measure volume in a cylinder without having first learnedthe formula. The video provided teachers an opportunity to see how teacherand student roles might change. They noticed that students took moreresponsibility for learning, changing the teacher role from sole expert tocoach. They liked the notion that sharing responsibility for learning withthe students lessened the teacher’s ‘‘burden.’’ The meeting ended with teach-ers selecting which strategies they intended to implement in the next month.

Teachers found the autonomy principle somewhat elusive, perhapsbecause the kind of instruction that Humphreys modeled was quite differentfrom their own. Therefore, we devoted the February meeting to a further dis-cussion of autonomy. The first author provided guidelines, including differ-ences among procedural, organizational, and cognitive autonomy support(Stefanou et al., 2004). Additional strategies included generating situationalinterest, offering hints rather than answers, allowing students to work in theirown way, and expressing confidence that students can meet high standards.Teachers were to select an upcoming topic and come to the meetingprepared to ask for and share ideas about how to incorporate autonomy-supportive strategies. Although the agenda was incorporating autonomystrategies, the discussion evolved into how one could make geometry con-cepts meaningful.

Winter Observations and Postobservation Interviews

Marcie. Before her winter observation, Marcie e-mailed her lesson planwith the autonomy strategies she would use: setting a purpose for learning,helping students gain control over their learning, and shifting her role fromteacher to facilitator. Instead of traditional homework review, Marcie invitedstudents to request which problems they would like explained. Marcie notedin the interview, ‘‘I get . . . a lot more [questions] now, than what I did in thebeginning of the year. . . . And now they’re saying ‘well I wanna see this, wellI wanna see this.’’’

In a contrast to the beginning of the year when she emphasized check-ing every homework problem, by winter her goal was to check enough tosee if students ‘‘understand the process of getting it, or where they wentwrong . . . to show that they understand that concept.’’ She asked more




‘‘why?’’ and ‘‘why not?’’ questions (a goal set in October). To emphasize thatstudents could learn from mistakes, she asked for volunteers to explain whatthey had done incorrectly, allowing them to discover their mistakes. Shehypothesized that students were ‘‘more comfortable’’ now because theKISS rules required groups to ‘‘Include everyone,’’ so more students wereparticipating. In the interview, Marcie expressed pleasure at her students’willingness to take control of their learning: ‘‘They were ready to give thatfeedback to me . . . which makes them more of a . . . facilitator.’’ Before,‘‘they just wanted to know if it’s wrong or right and move on.’’

Later in the observation, Marcie gave the student groups differentproblems related to percentage. She had written the problems to be rele-vant (e.g., ‘‘How many discounted video games can you purchase at BestBuy for $100?’’). Marcie explained that she was monitoring the groups ask-ing, ‘‘How did you get this?’’ and ‘‘Why did you get that?’’ This was a con-trast to her fall observation in which she did the work for the groups. In theinterview, she noted that she had ‘‘cut loose a little bit more,’’ reducing hercontrol, but she admitted feeling nervous about this move. For example,she revealed that she had not had time to calculate the answers to the ques-tions and admitted that she ‘‘felt like I was walking in[to] that lesson blind.’’She ended up checking the answers as each group explained, a risky movefor a teacher who did not trust her math knowledge. But she was ‘‘saved’’by her students: ‘‘They had to explain their problem to me. So they had thecontrol of their problem, then they had to teach me their problem.’’ Despiteher discomfort in transferring control, she concluded, ‘‘I like it, I do—I dolike it.’’

This lesson represented a revolution in Marcie’s view about how mathcould be taught; in the fall, talking and fun in math had been foreign con-cepts. Marcie’s use of grouping, along with her competence and autonomystrategies, appeared to have changed the norms of the classroom; studentshad begun to act like more autonomous thinkers and learners, rather thanpassive recipients of Marcie’s instruction. Woven throughout Marcie’s post-observation interview was a focus on the how and why of math learning,ideas that had been absent from her earlier observations and that hadevoked only frustration before. Although she continued to struggle withhow to teach some math concepts conceptually to students (e.g., ‘‘I don’tknow how to [make it relevant] . . . integers are just integers’’), Marcie hadbecome more interested in what understanding meant in math. ByFebruary, Marcie had begun to share control with the implementation ofsmoothly functioning groups and had begun to wrestle with teaching formeaning in mathematics.

Kate. In February, we observed Kate teach a 3-day lesson in which stu-dents were to create line graphs to show numbers of people coming andgoing at a certain location (y-axis) over the space of 12 hours (x-axis). For

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example, if the location were a football game on Saturday afternoon, the linemight not rise until around noon, would peak at game time (2:00), remainsteady until the end of the game, and then start to fall, decreasing to zeroagain by 6:00 p.m. The pattern would differ at a grocery store, which seesmuch more steady traffic over 12 hours. The goal of the lesson was to con-struct understanding of the relation between the x- and y-axes. Although thiswas planned to be a 1-day lesson, students had great difficulty conceptual-izing the change in flow of people over time, and it took 3 days for studentsto construct meaningful graphs. For this lesson, Kate intended to enact com-petence, belongingness, and autonomy strategies, including (a) stating pos-itive reasons for learning, (b) shifting her role to expert problem poser, (c)modeling how to think about problems, and (d) having students explainto others how and why they chose a solution method. She was partlysuccessful.

On the first day, Kate introduced the activity by modeling how onemight graph the comings and goings of people at a bus stop on a weekdaymorning. Because she assumed that only one bus stopped there, her linegraph looked like a series of peaks and valleys, always returning to zeroon the x-axis. Her model did not consider locations where people mightremain for a while or come and go in various numbers rather than everyonerushing in and out at the same time. Students did not understand that theirgraphs might differ from Kate’s, so they mimicked the peaks and valleys ofthe teacher’s graph. At the end of the first day, Kate was dismayed to see thesimilarities among the graphs and asked the students to think about ‘‘what itmeans when a line comes to a point, like this and goes sharply down.’’ Shewas discouraged and ready to abandon the lesson. Instead, with our encour-agement, she continued it the next day.

On the second day, she confessed to students that her model of peopleat the bus stop was ‘‘not real accurate’’ because she forgot to include peoplewaiting. ‘‘Do you see what I did wrong? . . . I need to make a revisionbecause nothing on my graph shows me that the number of people staysthe same.’’ Despite Kate’s efforts, graphs had not changed much by the thirdday. There were still lots of peaks and valleys. The teacher again displayedall the graphs, encouraging students to explain the meanings of the patternsand highlighting the value of effort.

Your graphs are sort of like a rough draft—how many times do youhave to redo a rough draft? . . . So, it doesn’t matter to me if this is thethird time you’ve made a graph. That’s OK. This graph is probablygoing to be better than the last one.

Throughout this discussion she refrained from giving answers and insteadasked questions to help students come up with the answers on their own.However, at the end she was unsure whether or not it was ‘‘3 days wasted.’’




In the postobservation interview several weeks later, Kate reflected onwhat she had learned. Referring to her reluctance to continue the lessonfor 3 days, she said, ‘‘Looking back on my teaching, I am a person who istoo quick to scrap an idea, [if] I think it didn’t work well. . . .’’ She continued,‘‘There is value in taking a long time to do something . . . I didn’t think that atall at first. . . . It is . . . important for me to . . . just stick with it . . . and seewhat comes out of that.’’ In addition, her students had shown remarkablelearning gains. She said these students understood the relation betweenthe x- and y-axes, whereas past students had not.

The issue of Kate’s content knowledge in mathematics emerged again inthe winter observation. During some exercises on multiplying decimals (e.g.,39.56 3 48.1), she noted consistent mistakes in students’ answers. Some stu-dents were placing two decimal points in the products (‘‘bringing down’’both decimals). She commented, ‘‘You can’t just go around adding decimalswherever you want or else people would do that in a bank and everybodywould be millionaires.’’ She asked the class why they were doing this, andthey replied that they were trying to ‘‘line up the decimals.’’ Kate responded,‘‘With multiplication, you just don’t do that. You just leave it alone.’’ She didnot (and could not) provide a mathematical explanation.

During the interview about this incident, Kate bemoaned,

I do that a lot . . . saying, ‘‘you just don’t or you just do’’ . . . and that isnot good enough. . . . I really do need to start spending more time . . .investigating the whys of things myself . . . all I am doing is going bywhat I was taught . . . and it doesn’t fly anymore.

Kate’s challenging experience with the line graph activity had heightenedher awareness of how procedurally focused instruction could backfire, cre-ating misconceptions. In addition, she admitted that she was not prepared toteach so that students would develop conceptual understanding. These inci-dents fuelled her desire to make her mathematics instruction moremeaningful.

Helen. Helen was observed twice in January. She continued to attemptstrategies, occasionally with some success. In contrast to the negative viewsshe expressed about her students in the group meetings, Helen was usuallypolite and solicitous of students during these observations (though with anundertone of sarcasm). However, her behavior seemed less out of respect forthe students than an attempt to plead for cooperation. But the themes notedearlier of low efficacy for teaching and low expectations for students werestill evident.

In the first January observation, as a review of geometry, Helen assignedstudents the task of drawing a boat, ‘‘But since we’re in geometry, we’regoing to use geometric items!’’ The task was much too simple for eighthgraders who already knew the geometric shapes. In addition, Helen had

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brought cereal and licorice sticks to class, presumably to win their coopera-tion. She was apparently well known for bringing food, as several studentsfrom other classes came by during the lesson asking for treats. Helen joked,‘‘I think they can smell food from miles away. They’re like dogs looking fordog food.’’ In addition, Helen participated willingly in students’ (mostly) offtask conversations, sending the message that the task was not very impor-tant. After 90 minutes, students had not completed the project, so Helenextended the due date.

The second observation occurred 2 weeks later because Helen had newstudents in her class. Helen outlined her lesson: (a) ‘‘begin the lesson with anoverhead reviewing congruence and similarity,’’ (b) ‘‘introduce a tangram onthe overhead and create a square (with cheat sheet handy),’’ and (c) havestudents use shapes to construct various figures. She intended to implementtwo autonomy strategies: (a) students (rather than teacher) will help eachother understand the concept and (b) students will self-evaluate. In neithercase did Helen support student autonomy.

In the first activity, Helen grouped the students, offering the incentivethat ‘‘you won’t have to do as much of the work.’’ Asked in the observationinterview why she had given that reason for students to work together, shereplied, ‘‘I don’t even know . . . I was just trying to tell them . . . if there’smore of them they don’t have to do the whole paper.’’ Asked about the im-plications of that statement, Helen replied, ‘‘I don’t know. Nothing positiveI’m sure.’’ Unlike most of the other teachers, Helen had not reflected onher extrinsic theory of motivation and had difficulty doing so even whengiven the opportunity.

On the overhead, Helen had written questions that were intended forindependent review work. Students were to (a) draw two similar figures,(b) draw two congruent figures, and (c) name the corresponding parts ofboth (a) and (b). The last question was, ‘‘Do you think there is more congru-ence or similarity in the world?’’ Even before students had a chance to con-sider the questions and attempt to work together, Helen said, ‘‘If you haveforgotten what congruence is, what is congruence?’’ Although the originalintent was offer autonomy, Helen quickly took the responsibility herself.She justified this tactic by saying that students ‘‘just sit there and stare atyou. And nobody takes their pencil out.’’ The individual review quicklyturned into a whole class lesson. Helen continued, ‘‘And if you forgotwhat Part (c) is, we’ll go over that shortly.’’ Rather than have studentsattempt Part (c), she explained it all. When the first author asked her if therewere other ways she could have handled the situation, Helen replied, ‘‘I fig-ured . . . no one in the room had it, so I didn’t know how I could even havethe kids explain it.’’ With some modeling, Helen was able to propose somequestions she could have asked, but quickly dismissed the alternativesbecause ‘‘the first two classes [had] just sat there and looked at me. . . .That’s why I wrote it on the board [for this class].’’




Helen was more successful scaffolding students’ discussion of whetherthere was more similarity or congruence in the world. When a studentanswered, ‘‘Similarity, because we all want to be like each other,’’ Helenwas able to redirect the discussion by suggesting, ‘‘So you are thinking[about] people, what if we thought [about] things?’’ Several students providedexamples of similarity and congruence, but Helen was not entirely successfulmaking the conceptual point. Although the students were more engaged inthis discussion (‘‘this is fun’’) than they had been before, Helen was not ableto point out the relevance of similarity and congruence to the real world.Rather, she concluded flatly, ‘‘That was just something to think about.’’ Atthe interview, Helen said, ‘‘I didn’t get what I wanted out of it. But someof the kids . . . were thinking more.’’

In the second event discussed during the observation interview, Helendemonstrated how the seven flat tangram shapes could be assembled intoa square. She did not state a reason for the activity. She passed out paperwith problems such as ‘‘seven pieces make a square.’’ The students were toexperiment with their shapes in order to make new shapes, given only theoutline of the shape. Instead of allowing students to attempt the challenge,Helen said, ‘‘Just follow with me and then you can go on your own,’’ discour-aging autonomy and challenge. Helen directed students to copy the one shehad made on the overhead since that was one of their problems. Then shetold the students two other solutions and advised the students to copythem, saying, ‘‘Here’s a two-piece triangle . . . and there is a five-piece triangle. . . that’s three answers right there for you . . . . Just copy what I have there . . .I’m trying to give you some cheats . . . I’m trying to give you the answers upthere, okay?’’ When the first author asked why she gave the students answers,she replied, ‘‘They had to do that whole worksheet where they had to dis-cover all of the things.’’ Asked why she had selected this activity for a ‘‘review’’of geometry since students clearly knew the shapes, she replied, ‘‘I got it rightout of . . . of my curriculum. They claim it is [a review], so I just took their wordfor it.’’ When asked why this was important, she responded, ‘‘I don’t know.Because . . . it was one of the skills I had to teach for math enrichment.’’Among all the teachers, Helen appeared to have the most difficulty reflectingon why she made instructional decisions and how she could improve them.

During the observation interview, the first author asked if Helen hadchanged her mind about using groups, which she had been negative aboutin December. She replied that with ‘‘changing schools I wasn’t sure if it wasgoing to work. . . . I just wasn’t sure how, with the class climate, how thatwas going to work [at the public school].’’ She continued, ‘‘I don’t think mythinking has changed. You know, I try everything you tell us or that anybodyever says.’’ Helen appeared not to know why her strategy attempts wereunsuccessful. She concluded, ‘‘I feel like maybe I’ve reached . . . a handful,but there’s still a whole lot out there that I don’t know if I’ll reach. . . .They don’t care, and I don’t know what we’re going to do to change that.’’

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These examples—tasks that were too simple for students, Helen’s inability toscaffold, and her tendency to tell students how to do problems—seem to dem-onstrate both her low efficacy for instruction and her beliefs that studentscould or would not learn the mathematics. In addition, she seemed to lackthe confidence (or inclination) to make instructional decisions on her own.

Meaningfulness Principle

We introduced meaningfulness at the March meeting. Teachers had begunto raise this issue on their own in January and February. Most were havingmore success fostering student thinking through group work; that successhad led them to critically evaluate the motivational potential of the tasksthey provided. It was becoming clear that students needed something to thinkabout. Teachers had been intrigued by how Humphreys (Boaler &Humphreys, 2005) used the oatmeal box as an opportunity to consider howand why formulas for volume might differ among three dimensionalobjects—she had created a reason to want to know. Furthermore, some ofthe teachers had begun to critique the way their textbooks treated all topics—-major and minor—as equally important. Kate raised the issue of which con-cepts in the fifth grade geometry unit were fundamental for her students.The teachers ultimately concluded that not all topics could be made directlyrelevant to life but that focusing on why formulas and algorithms workedcould be interesting and empowering for students. Other meaningfulnessstrategies included choosing activities that were both complex (e.g., multiday)and interesting as well as showing one’s interest in the topic. This latter strat-egy was alien to most of the teachers; they either were not interested in thetopic or simply regarded the math as something to ‘‘cover’’ rather than value.During the March meeting, teachers discussed how to apply strategies forinterest and meaningfulness to the concept of place value. In addition to dis-cussing meaningfulness, a portion of the meeting was devoted to discussingthe dilemma, raised by Kate, of whether the motivational strategies were anti-thetical (or irrelevant) to the kinds of teaching required to prepare students forstandardized tests. At the end of this meeting, teachers decided that theyneeded focused time to plan a unit that would incorporate all four strategies.Therefore, the April meeting was devoted to individual teacher planning.

At the April planning meeting, teachers worked alone or in groups toplan a lesson incorporating all four strategies. Although Kate focused on cre-ating a fractions unit, other teachers used this time to catch up with theirpeers rather than for the meeting’s stated purpose. We did not videotapethis meeting because teachers were seated far apart.

Spring Observations and Postobservation Interviews

Marcie. Marcie was observed twice toward the end of the year. By thispoint Marcie had begun to successfully address one of her earliest goals,




asking good questions. In addition, she had developed more efficacy forsupporting student autonomy in math. Marcie made explicit her expectationthat students be self-reliant and ask peers for help. Several times, she toldstudents to ‘‘chat with the people’’ in their groups rather than ask her.This emphasis was markedly different from her earlier tendency to do prob-lems for her students. At her interview, Marcie noted that, as a whole, herclass had become much more independent than at the beginning of theyear. ‘‘I think they are doing really well right now in math. Some studentsI call on them and they are more willing to participate and explain. And ex-plaining is coming through in them.’’ She even contrasted the learning be-haviors of students who had been in her class all year to the behaviors ofthose new to her class, who came up to her for help ‘‘all of the time’’ becausethey did not yet understand her expectations.

Before her second spring observation, Marcie noted, in an e-mail to thefirst author, that rather than leading the lesson herself, she planned to havestudents discuss it in groups. She confessed that she was ‘‘a little nervous . . .not really knowing how it’s going to be giving up the control. At the sametime, it could be good to see how they started and how they ended ontape.’’ As an application of the previous day’s review of mean, medianand mode, Marcie told the class, ‘‘The Eagles, a basketball team, in theirlast eight games, they scored 58, 67, 60, 72, 60, 65, 68, and 70. . . . Makea list of everything you know when you see these numbers.’’ She did notspecifically tell them to use the statistics concepts, but students workedtogether and frequently mentioned relationships including mean, median,mode, and range. Groups chose their own roles, and each group’s presenterthen told the class what they had determined about the numbers. Most of thereporting had little explanation, but Marcie did comment on how two groupspresented the concept of range differently (one group gave the range ofnumbers, 58–72, and the other gave the difference, 14). She said, ‘‘Youwere saying the process and they were saying the end result. It’s good tosee it in two different ways.’’ Then Marcie asked the students whether oneneeds to get the mean, median, and mode every time for a group of num-bers. She emphasized, ‘‘I want you to give me your reasons.’’ Although classtime ran out, Marcie emphasized the importance of the question by tellingstudents to write it down, discuss it, and be ready to talk about it the nextday.

Marcie commented that she used all four principles in this lesson. Theopen-ended question was designed to prompt the students to use statisticsin a way that was meaningful and to increase student competence. Shehad used basketball to gain interest and used groups to support belonging-ness. Earlier in the year, Marcie had often struggled to link specific strategiesto principles and did not use our common terminology for the four princi-ples, but in spring she was more confident making these linkages. She

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was specific about her goals and how the strategies would enable her tomeet them. Her theoretical understanding had grown.

At her observation interview, Marcie noted that, in contrast to earlier inthe year, more students were willing and able to discuss their ideas. She said,‘‘It used to be like four people could explain. . . . Now that they do know thatthey’re gonna get called at some point . . . that they need to know how andwhy.’’ She explained the changes in her students’ group work: ‘‘They reallyworked on . . . ‘Well, how did you get that answer?’ Instead of just saying,‘Oh, yes, that’s right, that’s wrong, we’ll move on’ [they say] ‘What did youdo? How did you get that?’ . . . they know that explaining and understandingwhy is more important than actually the answer.’’ Through participation inthe collaboration, Marcie had altered her practice and had changed herview of what students should and could do in mathematics. She was gratifiedwith the changes in her students and felt more successful as a math teacher.

Kate. Kate’s spring observations were set in the context of her anxiety,expressed at the March meeting, about upcoming district-mandated quar-terly assessments.

I feel that all of those [motivation] strategies are extremely valuableand I . . . feel that if I completely taught this way all the time it wouldbe beneficial. But then I just panic . . . and so I end up like, goingright back to a teaching style that I . . . feel like will get me throughthings quickly, even though I know that that’s not meaningfullearning.

By the end of the discussion, Kate concluded that if she could plan a unitfocused on a central concept, she could nest the operations likely to betested within it. The outcome of this discussion was the fraction unit weobserved in the spring.

Observations occurred on 3 separate days during the 3-week unit. Katechose the equivalence of fractions as her central concept. During this time,students worked with partners, in contrast to the fall when students sat inrows of desks and worked individually on computation problems. In addi-tion, Kate reminded students often of their developing expertise (e.g., ‘‘Iwant to use DeLeon’s work for an investigation on Monday’’). A constant fea-ture of her instruction was posing a question, allowing students to discusswith partners, and eliciting their (different) explanations. She used strategiesfrom all of the motivation principles during these lessons.

During the first observation, students used fraction bars to explore part-whole relationships. Kate said, ‘‘Give me four fractions that are the same asone whole and explain it. Show me on your mat.’’ Kate challenged her stu-dents’ answers: ‘‘I guess we could just say, ‘we believe you,’ but we’velearned to be pretty skeptical. So how do you know that it’s equal?’’Showing her fraction bars, a student responded, ‘‘4/8 is this long and 5/10




is the same exact length as it.’’ Rather than providing answers, Kate encour-aged students to confirm their hypotheses themselves: ‘‘You are just as capa-ble as helping [others] solve this problem as I am.’’

During the second observation, Kate led students to translate equiva-lent fractions (shown with fraction bars) into numbers, intended to promptstudents to discover the algorithm. She challenged students to ‘‘see if thereare relationships between the numerators of your fractions and thedenominators of your fractions . . . write down what you and your partnersay.’’ One student explained the relationship between 3/4 and 6/8 this way:‘‘I did 3 times 2 = 6 because the numerator is 6 . . . and I did 4 times 2because the [denominator] is 4 . . . and I got 8.’’ Kate pointed out whatthe students had done and noted appreciatively, ‘‘Wow! . . . They founda relationship with the same number. It’s two. They found that two is theirmagic number.’’ She told students to think about these patterns for theirhomework.

During the third observation, Kate introduced a division pattern thata special education student had discovered (divide the numerator anddenominator in 6/8 by the same number to find 3/4). She used the student’sdiscovery to demonstrate how division could be used to reduce fractions tolowest terms. Although students recalled the terms reducing, simplifying,and putting fractions in lowest terms from fourth grade, they did notunderstand their meaning. Now they had a conceptual basis for these op-erations, based on their experience with fraction bars and patternsbetween numerators and denominators. ‘‘Today, for example,’’ Kate ex-plained in the interview,

we were discussing adding, subtracting, fractions with unlike denom-inators. They just get it more quickly because . . . they have all of thatconcrete foundation . . . so we’ll sort of make up . . . for some losttime. . . . And I’m trying to not think about lost time anymore becauseit’s not lost time.

Kate appeared finally to resolve her dilemma of teaching for understandingversus covering material for the test. She realized that spending (a lot of)time to build understanding at the beginning paid learning dividends inthe end.

As Kate viewed the lessons during the postobservation interview, shecommented with pride on her ‘‘struggling’’ students. She noted changes inmotivation and in achievement: ‘‘I can really tell how they’ve becomemore confident as little mathematicians . . . they’re so much more comfort-able explaining themselves and also when they are wrong, they don’t get asworried or concerned.’’ She noted that students now treated mistakes andpeer feedback as a matter of course. In contrast to the fall when she domi-nated discussion, now, when a student made a mistake, ‘‘five or six peopleraised their hands’’ and explained how a peer’s thinking went wrong.

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Helen. Helen was observed in late April. Before Helen’s observation, shee-mailed plans for her lessons as well as strategies she would use for eachprinciple. For competence, she planned to continue to review known mate-rial and have students keep track of their progress. Belongingness wasenacted through group work with roles, and one autonomy strategy wasto have group members explain their thinking to others. Other autonomystrategies were explaining the purpose of the lesson to students and model-ing her thinking. To enact the meaningfulness principle, Helen planned touse word problems related to students’ lives.

As in the past, Helen did not succeed in using the review to supportcompetence. After telling the class, ‘‘This should be review . . . [converting]decimals to fractions,’’ she immediately explained how to do the problems:

One decimal place, you put one zero in the denominator; if it’stenths, it’s going to be over ten, for hundredths . . . you need toreduce them to lowest terms . . . if they end in zero or five, youcan divide by five . . . there is one there that . . . is already in lowestterms.

This approach seemed to imply to the students that they couldn’t (ordidn’t have to) do the problems on their own (even though it wasa ‘‘review’’).

Helen’s original plan for the day’s lesson, sent in an e-mail, was to pose‘‘three mathematical questions that [could] be used to investigate the con-tents of a box of cereal.’’ This purpose became lost, however, in the elabo-rate process (25 minutes) of distributing the cereal. Helen passed out papertowels to put the cereal on (‘‘These aren’t for blowing your nose, they’re forthe cereal’’), even distributing latex gloves ‘‘to keep your cereal clean if youwant to eat it afterwards.’’ Then Helen distributed Cheetos ‘‘to munch so thatyou don’t eat the cereal.’’ During this extended period, Helen had many off-task conversations (e.g., joking about quality of the gloves from the DollarStore) with the students.

Abandoning her planned question, she introduced the activity this way:‘‘What are some things we could do [with this cereal]?’’ Students responded,‘‘Who knows?’’ ‘‘Open it.’’ ‘‘Count how many reds, yellows. . . .’’ ‘‘See howthey are shaped.’’ ‘‘How many are in there. . . .’’ Helen acknowledged thesuggestions, then said,

Here are some that I suggest: Could you find out how much onepiece weighs? . . . Could you find out if there are more of one colorthan another? Could you find out how many total pieces are in thebox? . . . What I want you to do, within your group, is to divide itup . . . we are going to figure out how many pieces are in thatbox, how many of each color are there, then we will do a percent. . . you are going to have to make a chart with the colors and

then count up the totals.




She encouraged students to devise their own plan but justified her sugges-tion as intended to ‘‘be as easy for you as can be.’’ Despite the lack of a clearpurpose, students offered more suggestions to Helen’s question than usualin this class. However, once they began working, students showed littleinterest or purpose. Because many students were off task, and because therewas so much cereal, students spent the remaining 60 minutes countingcereal. Helen continued to banter with the students, ‘‘That group over there,if they get through half the box without eating it, we’ll be doing good.’’Because all the groups weren’t following directions, Helen changed thegoal for the task to see whether all groups had the same number of cerealpieces. With 5 minutes remaining, Helen began to talk about the results.She stated that the four groups each had a different number of pieces in theirbox. She asked why this could be, given that each box was 15 ounces, butthen answered this question herself. She told them ‘‘good job’’ for theirwork.

Although Helen’s original intent involved asking students to create orig-inal mathematical questions, this goal dissolved quickly both because of thedistraction with the food and also because Helen did not know how to scaf-fold the students in posing problems. She downgraded the task twice, end-ing with the goal of counting the pieces of cereal. It appeared that Helen wasmost concerned with gaining the cooperation of her eighth graders, whichshe did with food, off-task banter, and completing tasks for them. In contrastto Kate and Marcie, who raised standards and challenges by using theinstructional strategies, in Helen’s hands the instructional strategies back-fired. Instead of increasing autonomy, students remained dependent onthe teacher. Helen willingly filled the role of expert, partly because sheseemed to believe that she was helping her students and also because shefeared an uprising if she did not placate them. In the end, Helen had diffi-culty translating the strategies into good mathematics instruction.

For the May postobservation interview, the first author had given Helena copy of the video of the observed class and asked her to write down whatshe noticed. After watching, she commented,

It was just like . . . did I really do that or say that? I don’t know, and I, Itry not to, I mean I never realized how negative I am at times, but youknow when the tenth kid just asked the same thing and it’s just like . . .NOT AGAIN! But like I said, those were just eye openers.

But she did not reflect more on the ‘‘eye openers.’’ When asked how the col-laboration could have been improved for her, she replied, ‘‘I liked the shar-ing, I picked up a lot out of that! Umm, I needed more, ah, ways toimplement the strategies, I think.’’ It appears that she realized that manyof her attempts had not succeeded, but she continued to cite the studentsas the problem:

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If they had a lot of the background for math we could do a lot more,but they don’t. . . . They can’t write sentences, so it is not just math, itis across the board. We need to hit the basics more than we do.

She continued, ‘‘It is hard making things relevant to them. . . . If I were ableto tell them the velocity of a bullet, maybe they’d hang in there, but that’s notappropriate.’’

During the May meeting, the first author asked teachers to reflect onchanges in their students. All the teachers, including Helen, recountedsuccesses.

Discussion

The goal of this study was to understand how and why three teachers’beliefs and practices related to motivation and mathematics developedand changed as a function of participating in a yearlong collaboration.Several factors help explain the patterns we illustrated. First, the interventionoffered the teachers rationales and strategies to foster their students’ motiva-tion in mathematics. The monthly meetings provided opportunities forteachers to problem solve and learn from and help each other. Kate andMarcie took up these opportunities and adapted them for their instruction,whereas Helen reported that she tried but needed more ‘‘ways to implementthe strategies.’’ At the same time, Helen repeatedly blamed students for herdifficulties rather than taking the view espoused in the collaboration that stu-dent motivation was malleable. In addition, the observations and interviewsprovided opportunities for teachers to reflect with a person knowledgeableabout both motivation and instruction. The interviews provided impetus forKate to make significant changes in her instruction and for Marcie to put hergoals into words and to connect theory and practice. Helen did not appear tobenefit from the interviews. She had difficulty reflecting, often answering ‘‘Idon’t know’’ to the first author’s queries. In addition, Helen often misinter-preted the intent of the strategies, making successful enactment difficult.

Second, teacher efficacy—both efficacy for teaching in general and effi-cacy for teaching mathematics in particular—help explain the change pat-terns. Efficacy is central to theories of teacher conceptual change and toteacher motivation. Gregoire’s (2003) cognitive-affective model of concep-tual change (CAMCC) hypothesizes that efficacy helps determine whetherteachers regard opportunities to change instructional practices as a challengeor a threat. The collaboration sought to challenge teachers’ conceptions ofstudent motivation and ability and mathematics instruction.

In reference to the collaboration, the CAMCC proposes that when pre-sented with an opportunity to learn about motivation strategies, teachersmust first implicate themselves as responsible for motivation, even thoughit provokes stress. For example, Kate took responsibility for her students’




low self-competence and avoidance behavior, blaming her instruction (andlater, her content knowledge). Changing instruction was stressful for herbecause of the conflict she saw between teaching for understanding, whichtook time, and teaching to the test, which she thought required efficient cov-erage of the curriculum. Marcie was dissatisfied with her students’ participa-tion and their lack of responsibility for thinking in math; she implicated herquestioning skills and admitted that she needed to enact different strategies.The change was stressful for her because of her admittedly weak math con-tent knowledge and past experiences with math in high school. It was morecomfortable for her to teach math as algorithms, but she knew this approachdid not support student thinking.

The model proposes that teachers will evaluate the stress partly basedon their efficacy for helping students learn as well as whether they havethe time, supportive colleagues, and subject-matter knowledge to approachthe challenge. Efficacy is thought to act as a buffer for stress. Finally, themodel specifies that teachers must process the message systematically,meaning that they must analyze and scrutinize the information as relevantto their instruction. Both Kate and Marcie were efficacious, though less soin math. We believe that their efficacy was further supported by observingthe positive results with their students (e.g., Guskey, 2002). Both Marcieand Kate enacted the strategies and used the teacher group and the firstauthor as a resource; but content knowledge remained an impediment.The teachers responded to the situation as a challenge, though Kate imple-mented change more quickly and with more confidence than did Marcie.Both teachers evaluated the student responses to their instruction andthen changed instruction continuously to achieve their goals.

Helen, by contrast, did not accept responsibility for her students’ moti-vation to learn mathematics, although she was dissatisfied with their partic-ipation. She consistently stated low expectations for her students based onability and attitude. In addition, Helen admitted that she had given up on‘‘changing the world’’ after her first year of teaching, 19 years earlier. Wewere not able to successfully challenge her belief that motivation comesfrom ‘‘within and [from] home.’’ It is not clear whether Helen felt anxiousabout implementing the strategies, though that is possible, given that shewas in a new school, adjusting to the student population, and also wantedto be accepted as part of the teacher group. Her observations appear to indi-cate that she had low efficacy for instruction, despite her relatively strongermath content knowledge. She did not appear to have the interpersonal orthe instructional skills to implement the strategies successfully, nor did sheappear to understand the intent of the strategies. Furthermore, she wasnot able to use the resources offered by her peers or the first author toimprove. Therefore, the strategies may have posed a threat to Helen’s ex-planations of why she was having more difficulty than did other teachers.In this situation, the CAMCC proposes that Helen would avoid the

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motivational strategies, possibly by dismissing their relevance to studentmotivation (‘‘They don’t care’’) or by excusing herself (‘‘I’ve tried’’). Also,Helen had great difficulty reflecting on her instructional decisions and ac-tions, making it unlikely that she would be able to systematically processthe implications of her instruction. Therefore, she may have used a superfi-cial form of processing, in which she appeared to endorse the goals of thecollaboration, but did not enact them. The result was no change in beliefs,and some surface level changes in practices, which were successful witha few students.

Despite the implication drawn from Gregoire’s (2003) model, Marcieand Kate’s content knowledge in math did not appear to prevent theirchanging instructional practices. Perhaps that is because our main focuswas on motivation, not on reform mathematics. We hypothesized that teach-ers might be more interested in changing mathematics practices if they dis-covered that their students could be mathematical thinkers. We wouldexpect teachers to develop differently based both on their motivation tochange and also on their ability to reflect on student learning. Kate madethe most progress in reconceptualizing the mathematics instruction, statingclearly that she needed to know more. Marcie’s change to using open-endedquestions, although not as fundamental, was nonetheless a major transition,given that she took ‘‘a chance’’ of appearing less expert in front of her stu-dents. Helen, by contrast, decreased the challenge and conceptual nature ofher mathematics instruction in her attempts to enact the strategies in herclassroom, despite her better understanding of mathematics. Both Kateand Marcie appeared similar to the pattern noted by Stipek and colleagues(2001) that teachers who believed that ability was malleable and that mathcould be interesting also believed that instruction should foster autonomyand help students understand concepts.

Third, the nature of the school context, with its primarily low-incomeand high African American and Latino population, may have been relatedto teachers’ change patterns. Many of the students had a history of lowachievement and had, in response, developed behaviors associated withlow motivation such as low self-concept of ability (Covington, 1992) andavoidance (Turner et al., 2002). To teachers like Helen, they appeared notto care. Similarly, many came from families undergoing financial and emo-tional challenges, such as frequent evictions. Talbert and McLaughlin(1994) have observed that ‘‘negative judgments within the local communityabout the abilities, character and potential of some of today’s students’’ chal-lenges the ‘‘service ethic in teaching’’ (p. 128). The message of the collabo-ration, that teachers’ instruction can help change these patterns, had littlesupport among the school staff in general, and the principal showed scantinterest in the teachers’ efforts. In addition, the strong drumbeat of raisingstudents’ standardized test scores made instructional change even more dif-ficult for some teachers. Given these common findings, it is all the more




notable that four of the six teachers discerned real changes in their students’motivated participation.

Motivation theory (e.g., Stipek, Salmon, et al., 1998) predicts that in-creases in competence, autonomy, and interest are related to increased stu-dent academic performance. Although we did not collect achievement data,the teachers’ comments implied that they believed that students’ increasedmotivation was related to better learning. Kate stated explicitly that her stu-dents understood the relation of the x- and y-axes (from the graphing les-son) better than her former students and that their conceptual grasp offractions was qualitatively different from other students who learned it ‘‘bythe book’’ (through memorization of rules). Marcie noted that her studentshad begun to explain ‘‘why’’ in problems, reversing the IRE pattern(Nathan et al., 2009) common in traditional math instruction. Helen notedthat it was her low achievers, who were more willing to work with others,who had begun to talk in class and take more responsibility. The researchliterature has found that student talk, especially when teaching others, con-tributes to learning (e.g., Webb et al., 2008; Webb & Palincsar, 1996).

Limitations

This study focused on only three middle grade math teachers, and there-fore the findings are generalizable neither to all teachers nor to different con-tent areas or levels of schooling. Future research, using differentinterventions in different settings might investigate whether there are com-mon concerns related to classroom teaching with specific implications forthe translation of motivation theory. In addition, we made a decision tointroduce the principles in a certain sequence, but other patterns mighthave yielded different findings. This is a question for future research.Finally, there are other factors that likely contributed to teachers’ change pat-terns that we were not able to investigate. They included (among others)personal factors such as teachers’ differential verbal fluency and tendencyto reflect, interpersonal factors such as the influence of teacher (and firstauthor–teacher) interaction during meetings on teachers’ change trajectoriesand the relative status of the teachers, and community factors such as thenorms of the school and the district.

Implications

This study illustrates the value of situating motivation theory in contentarea instruction (Turner & Meyer, 2009). The context of mathematics wasimportant to this study, given the seemingly conflicting efforts to reform math-ematics instruction (and foster motivation) and the pressure for accountabilityin the United States. In addition, mathematics content knowledge is anacknowledged area of weakness for elementary-trained American teachers.Therefore, this study provided one examination of translating motivation

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theory into accessible classroom instructional practices with the goal of under-standing how and why such opportunities might result in teacher change.

The tenets of this intervention, in terms of both motivation theory andinstructional practices, presented substantial challenges to teachers’ beliefsand practices and thus to the goals of the university-school collaboration.Based on our experience in this study and in other work with teachers in dif-ferent districts and on past research (e.g., Richardson, 1990), we suspect thatmost researchers engaged in motivation interventions will need to understandand to address certain widely held views about motivation, learning, andinstruction. They include teacher beliefs that (a) student motivation is relatedmostly to personal characteristics rather than to instructional interactions in theclassroom; (b) students learn by listening and practice, not by talking andworking together; and (c) the risk of changing instructional practices (i.e.,loss of control) may outweigh the (as yet unproven) benefits. These threewere closely related to persuading teachers to try strategies that we assertedwould give teachers and students an opportunity to change. Finally, we dis-covered that linking practice to theory and reflecting on instructional decisionsand the reasons for student outcomes is exceptionally difficult for some teach-ers and thus needs to be a centerpiece of any intervention. Teachers who donot learn to reflect are unlikely to understand the rationale for certain practicesand are therefore less able to select and interpret instructional practices thatsupport motivation and learning. Ultimately, they may not benefit from theintervention. At the same time, researchers must learn how to supportteachers’ efficacy while also respecting teachers’ knowledge, remaining cogni-zant and appreciative of the important work that teachers do. Motivation re-searchers must also be willing to venture into new areas such as learningand instruction, teacher conceptual change, and professional development.We hope that the results of collaborations such as this one can inform futureefforts in translating motivation theory for teachers and their students

Notes

The research reported in the article was made possible (in part) by a grant from theSpencer Foundation. The views expressed are those of the authors and do not necessarilyreflect the views of the Spencer Foundation. Correspondence concerning this articleshould be addressed to Julianne C. Turner, Department of Psychology, University ofNotre Dame, Notre Dame, IN 46556; e-mail: [email protected].

1Teachers agreed to be videotaped, and then teachers and class periods wereselected randomly with no prior notice. Therefore, Stigler, Gallimore, and Hiebert(2000) considered the lessons to be typical of daily instruction.

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Manuscript received March 20, 2009Final revision received August 14, 2010

Accepted August 20, 2010

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