developing a framework of outcomes for mathematics ...4. cobb, wood, and yackel (cobb, wood, &...
TRANSCRIPT
Damon L. Bahr, Eula Ewing Monroe, & Jodi Mantilla
113
Teacher Education Quarterly, Spring 2018
Developing a Framework of Outcomesfor Mathematics Teacher Learning
Three Mathematics Educators Engagein Collaborative Self-Study
Damon L. Bahr, Eula Ewing Monroe, & Jodi Mantilla
AbstractThis article synthesizes the literature on what it means to teach mathematics and science to ELLs and abstract from it a set of knowledge and skills teachers might need to teach ELLs effectively. To this end, the article brings together the socio-cultural and linguistic perspectives identifying three areas of effective teaching practice. One argument is that collaborative learning conditions are beneficial in teaching mathematics and science to ELLs. A second contention is that teachers should be able to engage ELLs in mathematics ‘talk’ and the discourse of scientific concepts by bridging the divide between students’ background experiences and the content of mathematics and science lessons. The third area of effective teaching practice forwards the claim that teachers should engage ELLs in talking and writ-ing the language of mathematics and science. To support this point, the linguistic perspective identifies the shared and distinctive features of the academic languages of mathematics and science. Into this discussion, we integrate the insights of the mathematics and science specialists that participated in our panel.
Damon L. Bahr is an associate professor, Eula Ewing Monroe is a professor, and Jodi Mantilla is a visiting professor, all in the Department of Teacher Education of the David O. McKay School of Education at Brigham Young University, Provo, Utah. Email addresses: [email protected], [email protected], & [email protected]
Developing a Framework of Outcomes
114
Introduction
Numerous reform documents in mathematics education call for significant changes in perspectives about the nature of teaching and learning mathematics. “More than simply minor adjustments in current ways of teaching” (Wood & Turner-Vorbeck, 2001, p. 185) are proposed, and by extension, a rethinking of the outcomes1 for teacher learning (Ball & Cohen, 1999). Such outcomes could create a common conceptual framework of sufficient detail and clarity to benefit mathematics teacher educators not only in their professional practice but also in their study of that practice. As Simon (1995) wrote, “A well-developed conception of mathematics teaching is as vital to mathematics teacher educators as well-developed conceptions of mathematics are to mathematics teachers” (p. 142). Desimone (2009) concurred, suggesting two interrelated benefits: “The use of a common conceptual framework would elevate the quality of professional development studies and subsequently the general understanding of how best to shape and implement teacher learning oppor-tunities for the maximum benefit of both teachers and students” (p. 181). Similarly, Doerr, Lewis, & Goldsmith (2009) wrote, “A shared framework on which to hang existing findings may also help us develop, as a field, a shared theoretical model of teachers’ on-the-job learning so that we can build on one another’s work more productively” (p. 12). (See also Simon, McClain, Van Zoest, & Stockero, 2009.) A major step toward creating a framework of learning outcomes for teachers was taken by the National Council of Teachers of Mathematics (NCTM) with its Professional Standards for Teaching Mathematics (1991). The stated purpose was “to provide guidance to those involved in changing mathematics teaching” (p. 2). Its successor, Mathematics Teaching Today (Martin, 2007), underscored the central message of the first document: “More than curriculum standards documents are needed to improve student learning and achievement. Teaching matters” (p. 3). Nevertheless, as Linda Gojak, recent president of NCTM, expressed, even teaching standards have been insufficient for promoting the teacher learning that results in the “specific actions that teachers . . . need to take to realize [the] goal of ensur-ing mathematics success for all. . . . ” (NCTM, 2014, p. vii). Gojak continued, “We have learned that standards alone . . . will not realize the goal of high levels of mathematical understanding by all students. More is needed than standards” (NCTM, 2014, p. vii). During our work over the years, we have relied on these standards as well as other related documents (e.g., Principles and Standards for School Mathematics, NCTM, 2000; Adding It Up, NRC, 2002) to create several successive renderings of outcomes to guide our work in teacher preparation and professional develop-ment. Although useful, these renderings felt incomplete and incohesive. We needed a comprehensive, structured set of outcomes to frame the teacher learning that would align with these documents and was reflective of current research in our field. Our thinking was corroborated by the authors of The NCTM Research Agenda
Damon L. Bahr, Eula Ewing Monroe, & Jodi Mantilla
115
Conference Report (Arbaugh, Herbel-Eisenmann, Ramirez, Knuth, Kranendonk, & Quander, 2010), a portion of which calls for the identification of “competencies that teachers need to have and prioritizing these competencies as desired outcomes of professional learning opportunities” (p. 18). The purpose of this article is to describe our Framework of Outcomes for Math-ematics Teacher Learning (FOMTL) and the collaborative self-study (LaBoskey, 2007) we used to construct it. The work portrayed in this study reflects our own long-term journey to refine our vision of teacher learning in mathematics teacher education. In creating our framework of teacher learning outcomes, we synthesized the wisdom inherent in the afore-mentioned documents with a combined teaching experience of more than a century. As described below, our work was a collaborative self-study (LaBoskey, 2007) that bridged to our practice and helped us refine our own teaching and learning as mathematics teacher educators. It does not represent a recommendation or set of recommendations but rather a framework we have found useful and therefore desire to share with our field.
Positions and Domains of Teacher Learning
As previously discussed, we engaged in numerous endeavors to define outcomes for our work in mathematics teacher education. Early in our framework development, we had encountered Wood, Nelson, and Warfield’s (2001) work regarding the posi-tions from which teacher learning is studied and developed—Mathematical Content Knowledge, Student Thinking, Child Development, and Social Interaction. Two posi-tions have developed from a psychological perspective; the third from a sociological, or socio-psychological, perspective; and the fourth from a disciplinary one.
1. Ball, McDiarmid, Wilson, and Shulman (Ball, 1988; McDiarmid & Wilson, 1991; Shulman, 1986, 1987), conjectured that teachers change as the nature of their mathematical content knowledge changes.
2. Carpenter and others (Carpenter, Fennema, Peterson, & Carey, 1988; Carpen-ter, Fennema, Peterson, Chiang, & Loef, 1989; Peterson, Carpenter, & Fennema, 1989) suggested that teacher learning is brought about by changes in perspectives concerning student thinking.
3. Schifter, Simon, and Fosnot (Schifter, 1996a, 1996b; Schifter & Fosnot, 1993; Schifter & Simon, 1992) suggested that teachers learn as their perspectives regard-ing the nature of learning and mathematics change, a child development position.
4. Cobb, Wood, and Yackel (Cobb, Wood, & Yackel, 1990; Wood, Cobb, & Yackel, 1991), viewed teacher change as resulting from renegotiation of the social interac-tion that characterizes classroom pedagogy.
We contend that these positions are not necessarily oppositional but, rather, complementary; moreover, we view the research conducted from each of these positions as making valid contributions to the knowledge base on teacher learning.
Developing a Framework of Outcomes
116
Hence we argue that a comprehensive set of outcomes for teacher learning would reflect all four positions. In addition to the four positions, Wood et al. (2001) also affirmed that teacher learning occurs within three domains—beliefs, knowledge, and practice. In our work, we used Rokeach’s (1968) definition of beliefs, cited by Leatham (2006), as “predispositions to action” (p. 92), and contrasted them with knowledge because beliefs possess a relatively stronger, affective component (Abelson, 1979; Speer, 2005). We defined knowledge as truths, facts, principles, or information acquired or constructed via experience, cognition, or association (Merriam-Webster Dictionary, 2012), and practice as the work that characterizes a profession, such as the teach-ing of mathematics. The singular form of the noun practice is used in this paper primarily to refer to a teacher’s customary way of going about teaching. When the intended meaning of practice refers to ways of dealing with specific teaching situ-ations, the form is usually the plural noun, practices. Research findings from all four positions contributed to our understanding of the beliefs, knowledge, and practices teachers need to construct as they learn to teach, thus giving us a more holistic, comprehensive view of teacher learning. To give structure to our understanding, we created a two-dimensional framework of outcomes organized by these four positions and the domains of Beliefs, Knowledge, and Practice. However, despite the high value we placed on the positions and domains as important anchors of our framework, we became less than satisfied with its content and structure. It seemed cumbersome and incohesive, so we felt compelled to start afresh. Almost immediately, we encountered two documents that were pivotal in reshaping our direction. First, a curriculum-oriented collaborative self-study of sci-ence and literacy integration by Hall-Kenyon and Smith (2013) convinced us that engaging in the deep discourse that characterizes self-study research could help us frame and develop a satisfying set of outcomes for our students. Second, NCTM’s Principles to Actions: Ensuring Mathematical Success for All (2014) confirmed and extended our thinking regarding both the necessity for and the content of outcomes for mathematics teacher education. The first document inspired a different approach to developing our outcomes—collaborative self-study, and the second informed our efforts to develop outcomes to guide our own practice. Although our goal was to improve our practice, we hoped to contribute to the national conversation regard-ing prioritizing “ . . . competencies as desired outcomes of professional learning opportunities” (Arbaugh et al., 2010, p. 18) as well.
Creating a Landscape for Teacher Learning
Our self-study is based on a view of learning that is highly influenced by the landscape metaphor. Traditional characterizations of learning, including professional learning, tend to be unidimensional in their orientation. Such characterizations rep-
Damon L. Bahr, Eula Ewing Monroe, & Jodi Mantilla
117
resent learning as moving along a linear path, and by implication, that all learners are assumed to move along the same pathway, the same linear trajectory. Modern characterizations of learning tend to be more multi-dimensional and respectful of individual differences, sometimes employing a landscape metaphor as part of the characterization. “In this . . . metaphor, learning is analogous to learning to live in an environment. . . . Knowing where one is in a landscape requires a network of connections that link one’s present location to the larger space” (Bransford, Brown, & Cocking, 2000, p. 139). Fosnot and Dolk (2001) adopted the landscape metaphor in their descriptions of students’ mathematical learning. According to these authors, a mathematical landscape includes “landmarks” (p. 18) of big ideas, strategies, and representa-tions leading to various points on the mathematical “horizon” (p. 18) for a given topic, e.g., place value or addition and subtraction. Children may follow different trajectories on the learning landscape for a mathematical topic, yet arrive at the same mathematical horizon. We have found this same metaphor useful in describing teacher learning. As Lesh, Doerr, Guadalupe, & Hjalmarson (2003) suggested,
The essence of the development of teachers’ knowledge . . . is in the creation and continued refinement of sophisticated . . .ways of interpreting the situations of teaching, learning and problem solving. We believe that the theoretical constructs that govern the development of useful models by students are the same theoretical constructs that govern the development of useful models by teachers (p. 227).
Thus, we sought to construct a comprehensive, cohesive, clear, and concise framework of teacher outcomes that would serve as landmarks within the teacher learning landscape and allow for individual trajectories within that landscape. We therefore had two specific purposes for engaging in this study: (a) to uncover and analyze our own outcome lists as experienced mathematics educators, and (b) to investigate how those outcomes evolved as we moved toward a shared vision of the outcomes of our work, a vision we eventually named a Framework of Outcomes for Mathematics Teacher Learning (FOMTL). In this paper we present the FOMTL in the context of its creation. Following a discussion of self-study methodology, we describe the framework’s initial conceptualization as a lengthy list of unstructured outcomes, then the series of qualitative analyses resulting in the framework in its current form.
Self-Study Methodology
Self-study, a form of practitioner research that typically employs qualitative methodology, has emerged as a viable means through which teacher educators learn from their own professional practice. Alluding to the potential of self-study as a process for enacting change, Berry and Hamilton (2012) wrote,
In self-study, researchers focus on the nature and development of personal, practi-cal knowledge through examining, in situ, their own learning, beliefs, practices, processes, contexts, and relationships. Outcomes of self-study research focus
Developing a Framework of Outcomes
118
both on the personal, in terms of improved self-understanding and enhanced understanding of teaching and learning processes, and the public, in terms of the production and advancement of formal, collective knowledge about teaching and teacher education practices, programs, and contexts that form an important part of the research literature on teacher education. (Para. 1)
Recognizing the promise that self-study holds in promoting mathematics edu-cation reform (Goodell, 2011), we engaged in collaborative self-study (LaBoskey, 2007) as we sought to reconstruct our earlier framework of outcomes. We examined our own and each other’s perspectives about learning outcomes, and the records of the interactions associated with these examinations then served as data for our study.
Data Sources and Analysis
As a starting point, Eula and Damon each spent time brainstorming lengthy lists of potential teacher learning outcomes— Eula’s list totaled 53 and Damon’s 33. These lists, which served as our initial data sources, were influenced by knowledge gained from our individual and collective research agendas, our own wisdom of practice, and the outcomes constructed during our previous work. We then engaged in a series of dialogues about the lists that served as our first analyses, not unlike constant comparative methodology (Straus & Corbin, 1990). These dialogues, more fully explained below, occurred in the process of six analyses—“passes”—as we coded and recoded our initial outcomes lists using a mixture of a priori and emer-gent coding. Some of these passes were planned ahead of time, and the need for others arose based on what we were learning from the data. This coding enabled us to simplify, refine, and structure our combined lists into a tentative framework as shown in Table 1.The information in the table is more fully described in the Findings section. This tentative framework went through several more revisions as we used it to guide our methods instruction over two semesters. Our primary data sources then consisted of written and video records of numerous additional dialogues about our changing conceptions of the framework, supplemented by our written plans for course sessions, various in-class and homework assignments and assessments, and written post-instruction notes and reflections. These data served as a history of our journey, helping us to make sense of current dialogues by remembering the lessons of the past. The dialogues produced a steady stream of revised outcomes, some of which resulted in modifications to our practice during each semester. Thus we engaged in a series of miniature, design-based inquiries (Design Based Research Collective, 2003) that involved frequent application of our thinking to our classroom practice for testing and refinement (LaBoskey, 2007). Our dialogues, therefore, served as both an ongoing source of data and a major component in the data analysis process. Interspersed among these dialogues were applications of our current thinking to our work with students. Dialogue became a
Damon L. Bahr, Eula Ewing Monroe, & Jodi Mantilla
119
19
Table
1
A F
ram
ew
ork
of
Outc
om
es f
or
Mat
hem
ati
cs T
each
er
Learn
ing
–V
ersi
on 1
1
Teacher
Lea
rnin
g P
osi
tio
ns
Ma
them
ati
cal
Co
nte
nt
Kn
ow
led
ge
Stu
den
t T
hin
kin
gS
ocia
l In
tera
cti
on
Ch
ild
Dev
elo
pm
ent
Teacher Beliefs
As
stu
den
ts c
ycle
th
roug
h T
he
Co
nti
nu
um
of
Ma
them
ati
ca
lU
nd
ers
tand
ing
, th
ey
mak
e
co
nn
ecti
ons
wit
hin
an
d a
cro
ss th
ree d
om
ain
s—C
on
cep
tual,
Pro
cedu
ral,
& R
ep
rese
nta
tio
nal—
and
m
ath
em
ati
cal
un
ders
tan
din
g i
s co
nst
ructe
d.
“C
hil
dre
n c
an
so
lve p
rob
lem
s in
nov
el
way
s b
efo
re b
ein
g
tau
ght
ho
w to
so
lve s
uch
pro
ble
ms.
Th
e w
ay
s ch
ild
ren
th
ink
ab
ou
t m
ath
em
ati
cs
are
gen
era
lly
dif
fere
nt
fro
m t
he w
ay
s ad
ult
s w
ou
ld e
xp
ect
them
to
thin
k a
bou
t m
ath
em
ati
cs.
Th
e
teach
er
sho
uld
all
ow
th
e c
hil
dre
n to
do
as
mu
ch
of
the
thin
kin
g a
s p
oss
ible
” (
Ph
ilip
p e
t al.
, 2
00
7, p
. 47
5),
th
us
hon
ori
ng
th
e w
ay
s st
ud
ents
thin
k.
“…
Th
ink
ing
wit
h o
thers
pro
mo
tes
ch
ild
ren
’ s
co
gn
itiv
e
pro
gre
ss b
ecau
se i
t en
cou
rag
es
. . . in
div
idu
als
to
byp
ass
th
eir
ow
n c
og
nit
ive l
imit
ati
on
s an
d e
nab
les
ch
ild
ren
to
d
ev
elo
p v
irtu
al d
ialo
gu
e, th
e c
ap
acit
y t
o t
hin
k a
lon
e”
(Ro
ch
at,
200
1, p
. 13
9).
Th
is “
. . . d
isco
urs
e e
mb
ed
s fu
nd
am
en
tal
valu
es
abo
ut
kn
ow
led
ge a
nd
au
tho
rity
. . . .”
(N
CT
M, 1
99
1, 2
1, 34
).
Reco
gn
izin
g t
hat
in a
n e
qu
itab
le l
earn
ing
en
vir
on
men
t all
stu
den
ts c
an
learn
, “ . . .
teach
ing
sh
ou
ld b
e g
rou
nd
ed
in
ho
w s
tud
en
ts l
earn
. . .”
(S
ch
ifte
r &
Fo
sno
t, 1
99
3,
p. 1
93
) an
d “
. . . a
ck
no
wle
dg
es
learn
ers
as
acti
ve k
no
wers
” (
No
dd
ing
s, 1
990
, p
. 1
0)
wh
o e
ng
ag
e in
“acti
ve c
onst
ructi
on
, n
ot
mere
ly p
ass
ive a
bso
rpti
on
. . . (
an
d
in)
inv
enti
on
, n
ot
imit
ati
on
” (
Baro
ody
& G
insb
urg
, 1
99
0, p
.52
). T
his
con
stru
cti
on
o
ften
beg
ins
wit
h c
on
cre
te t
hin
kin
g a
nd
mo
ves
tow
ard
th
e a
bst
ract.
TeacherKnowledge
En
gag
ing
in
th
e “
Math
em
ati
cal
Pra
cti
ces”
(N
ati
on
al
Go
vern
ors
Ass
ocia
tio
n C
ente
r fo
r B
est
Pra
cti
ces
&
Co
un
cil
of
Ch
ief
Sta
te S
cho
ol
Off
icers
, 2
01
0)
wh
ile
stu
dy
ing
ele
men
tary
math
em
ati
cs
pro
du
ces
a
“P
rofo
und
Und
ers
tan
din
g o
f F
und
am
en
tal
Math
em
ati
cs”
th
at
is i
ntr
icate
ly i
nte
r-co
nn
ecte
d,
mu
lti-
facete
d, lo
ng
itud
inall
y c
oh
ere
nt,
an
d s
tru
ctu
red
acco
rdin
g to
“b
ig id
eas”
(M
a, 1
99
9, p
. 12
0).
Becau
se c
hil
dre
n’s
in
itia
l m
ath
em
ati
cal
thin
kin
g i
s b
est
su
pp
ort
ed
by
real-
life
co
nte
xts
(P
hil
ipp
et
al.
, 2
00
7),
th
eir
in
itia
l p
rob
lem
so
lvin
g s
trate
gie
s an
d r
ep
rese
nta
tio
ns
mo
del
the a
cti
on
s an
d c
on
tex
ts i
nh
ere
nt
in t
he p
rob
lem
s (C
arp
en
ter
et
al.
, 1
99
9;
Fosn
ot
& D
olk
, 2
00
1).
Teach
ing
in
vo
lves
no
t o
nly
kno
win
g “
the v
ari
ety
of
way
s ch
ild
ren
th
ink
ab
ou
t and
m
ak
e s
en
se o
f m
ath
em
ati
cal
ideas
bu
t als
o h
ow
ch
ild
ren
’s
thin
kin
g a
bou
t th
ese
id
eas
dev
elo
ps
ov
er
tim
e” (
Wo
od
, 20
01
, p
. 9
).
Inte
racti
on
patt
ern
s eli
cit
ing
gre
ate
r in
vo
lvem
en
t fo
ster
mo
re c
om
ple
x l
ev
els
of
thin
kin
g, th
ink
ing
th
at
can
be
hie
rarc
hall
y o
rganiz
ed
and
cate
go
rized
acco
rdin
g to
Th
e
Fra
mew
ork
of
Cog
nit
ive C
om
ple
xit
y (
Bah
r &
Bah
r,
201
7).
Th
ose
cate
go
ries
rela
te t
o t
each
er
pu
rpo
se—
co
mp
reh
en
sio
n, con
necti
on
, and
co
nse
nsu
s—and
co
nn
ect
clo
sely
wit
h t
he L
ea
rnin
g C
ycle
.
Stu
den
t le
arn
ing
pro
gre
sses
thro
ugh
a L
ea
rnin
g C
ycle
co
nsi
stin
g o
f th
ree
ph
ase
s—D
evelo
p, S
oli
dif
y, an
d P
racti
ce—
that
can
be d
isti
ng
uis
hed
on
th
e b
asi
s o
f th
e d
iffe
rin
g p
urp
ose
s, t
ask
s, a
nd
stu
den
t th
ink
ing
th
at
ch
ara
cte
rize t
hem
(H
en
dri
ck
son
, H
ilto
n, &
Bah
r, 2
00
8).
Dom
ains
of
Pra
ctic
e
Less
on
an
d U
nit
Desi
gn
Q
uest
ion
ing
and
Ass
ess
men
t D
iscu
ssio
n O
rch
est
rati
on
T
each
ers
co
nd
uct 3
-sta
ge i
nq
uir
y v
ia t
he T
ea
chin
g C
ycle
(H
en
dri
ck
son
, S
.,
Hil
ton
, S
. C
., &
Bah
r, 2
00
9,
ren
am
ed
Laun
ch
-Ex
plo
re-D
iscu
ss b
y th
ese
au
tho
rs),
wh
ich
en
ab
les
teach
ers
to
th
ink
th
rou
gh
an
inq
uir
y-b
ase
d l
ess
on
, acco
mm
od
ate
s a n
um
ber
of
inst
ructi
on
al
mod
els
, an
d v
ari
es
dep
end
ing
u
po
n w
here
stu
den
ts a
re i
n t
he L
earn
ing
Cy
cle
.
Th
e p
urp
ose
s and
pro
cess
es o
f q
ues
tion
ing
an
d a
ssess
men
t v
ary
acco
rdin
g t
o t
he
stag
es
of
the T
each
ing
Cy
cle
. D
uri
ng
th
e L
aun
ch
, ass
ess
men
ts a
re m
ad
e r
eg
ard
ing
ta
sk c
om
pre
hensi
on
. D
uri
ng
Exp
lore
, o
bse
rved
th
ink
ing
is
co
mp
are
d t
o a
nti
cip
ate
d
thin
kin
g i
n o
rder
to p
rep
are
fo
r th
e D
iscu
ss. D
uri
ng
th
e D
iscuss
, th
ink
ing
is
co
mp
are
d t
o l
ess
on
pu
rpose
in
ord
er
to in
form
th
e n
ex
t te
ach
er
mo
ve.
Teach
ers
co
nd
uct 3
-sta
ge m
ath
em
ati
cal
inq
uir
y, “L
au
nch
, E
xp
lore
, an
d S
um
mari
ze”
(Sch
roy
er
& F
itzg
era
ld, 1
986
) b
y e
stabli
shin
g th
e “
socio
-math
em
ati
cal
no
rms”
(Y
ack
el
&
Co
bb
, 1
99
6, p
. 45
8)
that
mak
e “
. . . p
oss
ible
all
stu
den
ts’
acti
ve p
art
icip
ati
on
in
. . . [
the]
dis
cou
rse a
nd
mak
es
dem
and
s on
stu
den
ts t
o e
ng
ag
e i
n in
cre
asi
ng
ly m
ore
co
mp
lex
th
ink
ing
”
(Wo
od
& T
urn
er-
Vo
rbeck
, 2
001
, p
.19
2).
Practices
Th
e L
earn
ing
Cy
cle
(D
ev
elo
p-S
oli
dif
y-P
racti
ce)
en
ab
les
teach
ers
to
th
ink
th
roug
h a
un
it “
. . . t
o d
ete
rmin
e w
hat
math
em
ati
cs
wil
l b
e t
aug
ht
wh
en b
y
desi
gn
ing
task
s th
at
are
ch
all
en
gin
g y
et
wit
hin
reach
of
stud
en
ts” (
Rey
s &
L
on
g, 19
95
, p
. 2
96
).
Fo
rmati
ve a
nd
su
mm
ati
ve a
ssess
men
ts, u
sin
g b
oth
tra
dit
ion
al
an
d a
ltern
ati
ve
pro
cess
es,
are
im
po
rtan
t v
eh
icle
s fo
r d
ete
rmin
ing
wh
ere
stu
den
ts a
re i
n t
he
lan
dsc
ape, in
th
e L
earn
ing
Cy
cle
, an
d w
here
th
ey n
eed
to
go
, as
well
as
for
est
ab
lish
ing
acco
un
tab
ilit
y f
or
learn
ing
.
Th
e m
ov
es
a t
each
er
mak
es
du
rin
g a
ll t
hre
e s
tag
es o
f th
e T
each
ing
Cy
cle
, b
ut p
art
icu
larl
y
du
ring
th
e d
iscu
ssio
n, d
ete
rmin
e w
heth
er
or
no
t th
e c
og
nit
ive d
em
and
of
the t
ask
is
main
tain
ed
th
rou
gh
ou
t th
e c
ycle
.
“B
ig D
ev
elo
p” c
ycle
s su
rface t
hin
kin
g a
bo
ut
a w
ho
le u
nit
an
d “
Sm
all
D
ev
elo
p” c
ycle
s su
rface t
hin
kin
g a
bou
t sp
ecif
ic o
bje
cti
ves
wit
hin
un
its.
T
hey
are
oft
en
lau
nch
ed w
ith
wo
rd p
rob
lem
s.
Du
rin
g D
ev
elo
pm
en
t, t
he t
each
er
can
exp
ect
stud
en
t th
ink
ing
th
at
is v
ari
ed
, p
oss
ibly
mis
co
nceiv
ed
, fo
cu
sed
on
how
or
wha
t ra
ther
than
wh
y, and
is
co
nte
xt
dep
end
en
t.
Th
e f
un
dam
en
tal
pu
rpose
in
orc
hest
rati
ng
dis
cu
ssio
ns
du
ring
Dev
elo
p c
ycle
s is
to
pro
mo
te
ov
era
ll c
om
pre
hensi
on
.
So
lid
ify
cy
cle
s in
vit
e s
tud
en
ts to
ex
am
ine a
nd
ex
tend
th
ink
ing
su
rfaced
in
D
ev
elo
p c
ycle
s. T
hey
fo
cu
s st
ud
ents
on
loo
kin
g f
or
patt
ern
s; u
sin
g
sym
bo
ls, st
ructu
re, an
d p
recis
e l
ang
uag
e;
co
nst
ructi
ng
arg
um
en
ts;
cri
tiq
uin
g o
thers
’ re
aso
nin
g;
an
d m
ak
ing
con
necti
ons.
Stu
den
t th
ink
ing
du
ring
So
lid
ify
is
less
vari
ed
, w
ell
co
nceiv
ed
, m
ore
fo
cu
sed
on
w
hy
(ju
stif
icati
on
), l
ess
dep
end
en
t o
n c
on
tex
t, m
ore
gen
era
lized
, and
refl
ecti
ve o
f co
nn
ecti
ons.
Dis
cu
ssio
ns
du
rin
g t
he S
oli
dif
y p
hase
fo
cus
on
co
nn
ecti
on
am
on
g s
tud
en
ts’
thin
kin
g a
nd
m
ak
ing
th
ink
ing
ex
pli
cit
Models
Pra
cti
ce c
ycle
s en
co
ura
ge s
tud
en
ts t
o r
efi
ne a
nd
mak
e s
oli
d t
hin
kin
g
flu
en
t, i
.e., a
ccu
rate
, eff
icie
nt,
an
d f
lex
ible
. S
tud
ent
thin
kin
g d
uri
ng
Pra
cti
ce b
eco
mes
refi
ned
an
d f
luen
t, i
.e., a
ccu
rate
, fl
ex
ible
, an
d e
ffic
ien
t. J
ust
ific
ati
on
s m
ay
beco
me g
enera
lized
pro
ofs
. W
hen
mov
ing
to
Pra
cti
ce, th
e f
ocu
s o
f th
e d
iscu
ssio
n i
s u
po
n c
on
sen
sus
an
d g
en
era
lizati
on
. In
Pra
cti
ce, th
e E
xp
lore
an
d D
iscu
ss s
tag
es
oft
en
occu
r si
mu
ltan
eou
sly
. D
ev
elo
p c
ycle
s fo
llo
w a
gu
ided
in
qu
iry
desi
gn
, w
ith
task
s th
at
hav
e a
lo
w
thre
sho
ld b
ut
hig
h c
eil
ing
(easy
en
try
bu
t ch
all
eng
ing
fo
r all
), a
re
co
nte
xtu
ali
zed
, all
ow
mu
ltip
le w
ay
s to
so
lve th
at
are
no
t n
ecess
ari
ly
pre
dic
tab
le, re
qu
ire r
ele
van
t b
ack
gro
und
kno
wle
dg
e, cre
ate
dis
eq
uil
ibri
um
, an
d d
o n
ot
com
mo
nly
pro
du
ce m
ast
ery
.
Du
rin
g D
ev
elo
p, st
ud
ent
ideas
refl
ect
that
they
reco
gn
ize t
here
is
som
eth
ing
to
w
on
der
abo
ut
(im
pli
cit
patt
ern
or
stru
ctu
re),
are
vari
ed
, an
d p
oss
ibly
mis
con
ceiv
ed.
Th
eir
str
ate
gie
s are
pu
rpo
sefu
l, c
on
tex
t-d
epend
en
t, a
nd
th
e s
teps
are
im
pli
cit
(n
ot
ex
pli
cit
). T
heir
rep
rese
nta
tio
ns
are
use
d f
or
ex
plo
rin
g, se
nse
mak
ing
, d
escri
bin
g,
reco
rdin
g, and
refl
ecti
ng
th
e m
ath
em
ati
cs
inh
ere
nt
in th
e p
rob
lem
.
Du
rin
g D
ev
elo
p c
ycle
s, t
he m
ost
co
mm
on
lev
els
of
thin
kin
g w
ith
in t
he D
iscuss
ion
in
clu
de
sho
rt a
nsw
ers
, b
rief
state
men
ts, d
esc
rip
tio
ns,
ela
bo
rati
on
s, c
lari
ficati
on
s, t
ran
slati
on
s, a
nd
re
pre
sen
tati
ons.
So
lid
ify
cy
cle
s m
ain
tain
so
me e
lem
en
ts o
f in
qu
iry
an
d f
oll
ow
th
ese
d
esi
gn
s: C
on
cept
Att
ain
men
t, R
eso
luti
on
of
Co
nfl
ict,
Co
mp
ari
son
and
R
ela
tio
nsh
ips,
Con
cep
t D
ev
elo
pm
en
t, S
ocra
tic M
eth
od
, an
d S
yn
ecti
cs.
Du
rin
g S
oli
dif
y, id
eas
beco
me c
on
cep
ts w
hen
stu
den
ts c
an
art
icu
late
str
uctu
re o
r p
att
ern
an
d i
ts c
om
pon
en
ts a
nd
giv
e n
am
es
to r
ole
s w
ith
in t
ho
se s
tru
ctu
res
or
patt
ern
s. T
heir
str
ate
gie
s b
eco
me a
lgo
rith
ms
wh
en
th
e s
tep
s o
f th
e a
lgo
rith
m a
re
iden
tifi
ab
le a
nd
rep
eata
ble
, an
d n
ot
so c
on
tex
t d
ep
end
ent.
Th
eir
rep
rese
nta
tio
ns
beco
me t
oo
ls w
hen
th
ey a
re u
sed
fo
r re
aso
nin
g a
nd
ex
pla
inin
g.
Dis
cu
ssio
ns
du
rin
g S
oli
dif
y c
ycle
s are
ch
ara
cte
rized
by
co
mp
ari
son
, re
lati
on
, d
iscern
ing
p
att
ern
s and
/or
stru
ctu
re, re
aso
nin
g, and
tra
nsf
er,
th
ink
ing
th
at
sup
po
rts
the e
xam
inin
g a
nd
ex
ten
din
g o
f th
ink
ing
su
rfaced
du
rin
g D
ev
elo
p c
ycle
s.
Strategies
Pra
cti
ce c
ycle
s re
sem
ble
Dir
ect
Inst
ructi
on
at
tim
es
bu
t do
no
t in
trod
uce
an
yth
ing
a s
tud
ent
has
no
t alr
ead
y c
on
stru
cte
d. T
hey
in
clu
de g
am
es,
ro
uti
nes,
pro
gra
mm
ed
rev
iew
s, a
nd
tex
tbo
ok
ex
erc
ises.
Du
rin
g P
racti
ce, co
ncep
ts b
eco
me d
efi
nit
ion
s o
r p
rop
ert
ies—
pre
cis
e, co
ncis
e
state
men
ts t
hat
cap
ture
th
e e
ssen
tial
natu
re o
f th
e s
tru
ctu
re o
r p
att
ern
of
the
defi
nit
ion
or
the c
hara
cte
rist
ic o
f th
e p
rop
ert
y. A
lgo
rith
ms
beco
me p
rocedu
res
wh
en
th
ere
is
a r
ed
ucti
on
in
ste
ps
thro
ugh
co
mb
inin
g o
r eli
min
ati
ng
. T
oo
ls b
eco
me
mo
dels
wh
en
th
ey
are
use
d a
cro
ss m
ult
iple
con
tex
ts a
nd
em
bod
y a
defi
nit
ion
, p
rop
ert
y, o
r p
roced
ure
.
Stu
den
ts a
re p
rep
are
d t
o m
ov
e i
nto
Pra
cti
ce w
hen
, d
uri
ng
th
e d
iscu
ssio
n, th
ey
eng
ag
e in
ch
all
en
gin
g o
r su
pp
ort
ing
, ju
stif
yin
g, re
fin
ing
, g
enera
lizin
g, and
pro
vin
g. D
uri
ng
Pra
cti
ce
their
th
ink
ing
beco
mes
accu
rate
, eff
icie
nt,
fle
xib
le, an
d r
efi
ned
.
Fle
sh o
ut
the m
ath
in
a C
om
mo
n C
ore
-base
d t
ex
tboo
k u
nit
usi
ng
top
ic
lan
dsc
apes;
th
en
cre
ate
or
uti
lize, o
pera
tion
ali
ze, an
d s
equ
ence o
bje
cti
ves.
O
bse
rve s
tud
en
ts a
nd
ask
goo
d, in
itia
l, p
rob
ing
qu
est
ions;
th
en
reall
y l
iste
n t
o
und
ers
tan
d s
tud
ent
thin
kin
g. T
hen
ask
add
itio
nal
qu
esti
on
s th
at
mov
e a
stu
den
t’s
thin
kin
g f
orw
ard
or
that
help
a s
tud
en
t ad
just
a m
iscon
cep
tio
n.a
s th
ey r
ep
rese
nt
their
th
ink
ing
.
Mo
nit
or
thin
kin
g d
uri
ng
th
e E
xp
lore
, lo
ok
ing
fo
r an
ticip
ate
d t
hin
kin
g a
nd
oth
er
wo
rth
wh
ile
ideas.
Use
an
ticip
ate
d th
ink
ing
to
sele
ct
the s
tud
ent
thin
kin
g t
o s
hare
an
d u
se d
ev
elo
pm
en
t se
qu
ences
to o
rder
that
shari
ng
--as
part
of
less
on
pla
nn
ing
.
Fin
d/c
reate
a B
ig D
ev
elo
p t
ask
to
su
rface t
hin
kin
g a
bou
t th
e w
ho
le u
nit
an
d a
nti
cip
ate
th
e t
hin
kin
g t
he t
ask
wil
l eli
cit
base
d o
n r
ese
arc
h o
n
thin
kin
g w
ith
in m
ath
em
ati
cal
top
ics.
Inte
rpre
t th
e u
nd
erl
yin
g m
ath
em
ati
cal
issu
es
wit
h w
hic
h a
stu
den
t is
gra
pp
ling
. D
isti
ngu
ish
betw
een
err
ors
an
d m
iscon
cep
tio
ns
an
d l
iste
n f
or
the s
en
se e
ven
wh
en
m
isco
ncep
tio
ns
app
ear.
Ass
ign
ro
les
to l
iste
nin
g s
tud
en
ts a
nd
ho
ld th
em
acco
un
tab
le f
or
the l
iste
nin
g u
nti
l so
cio
-m
ath
em
ati
cal
no
rms
are
fir
mly
est
ab
lish
ed
.
As
gu
ided
by
th
e L
earn
ing
Cy
cle
and
th
e a
nti
cip
ate
d t
hin
kin
g, d
evelo
p
pu
rpo
se s
tate
men
ts t
hat
ali
gn
wit
h th
e o
bje
cti
ves.
C
om
pare
ob
serv
ed s
tud
en
t th
ink
ing
to
th
e r
ese
arc
h o
n t
hin
kin
g w
ith
in
math
em
ati
cal
top
ics,
th
us
ass
essi
ng
wh
ere
a c
hil
d i
s on
th
e l
and
scap
e a
nd
in
th
e
Learn
ing
Cy
cle
.
At
the “
Mag
ical
Mo
men
ts” w
hen
a s
tud
en
t m
ak
es
a s
eem
ing
ly p
rofo
und
sta
tem
en
t, d
ecid
e
wh
eth
er
pu
rsu
ing
it
wil
l m
ov
e t
he c
lass
to
ward
th
e p
urp
ose
of
the l
ess
on
. If
a s
tud
en
t co
mm
un
icati
on
is
wo
rthy
of
pu
rsu
ing
at
this
tim
e, d
ecid
e o
n w
ho
sh
ou
ld p
urs
ue i
t—th
e
share
r, o
ther
stu
den
ts, o
r th
e t
each
er.
Teacher Practice
Techniques
Fin
d o
r cre
ate
acco
mp
an
yin
g S
mall
Dev
elo
p, S
oli
dif
y a
nd
Pra
cti
ce t
ask
s o
f v
ary
ing
co
gn
itiv
e d
em
an
d t
o p
rom
ote
co
nn
ecti
on
s acro
ss th
e d
om
ain
s an
d
ph
ase
s o
f th
e C
on
tinu
um
of
Ma
them
ati
ca
l U
nd
ers
tan
din
g (
Hen
dri
ckso
n e
t al.
, 2
00
9)
acco
rdin
g t
o u
nit
ob
jecti
ves.
Re-l
au
nch
a t
ask
wh
en
ass
ess
men
ts d
uri
ng
Exp
lore
or
Dis
cu
ss w
arr
an
t a r
e-l
au
nch
. A
dju
st th
e s
eq
uen
ce o
f ta
sks
wit
hin
a l
ess
on
or
un
it a
s in
form
ed
by
ass
ess
ing
st
ud
ent
thin
kin
g.
Dete
rmin
e a
t w
hat
co
gn
itiv
e l
ev
el
to g
uid
e t
he p
urs
uit
acco
rdin
g t
o w
here
stu
den
ts a
re i
n t
he
Learn
ing
Cy
cle
.
1 Th
e Continuum of Mathematical Understanding and the Learning Cycle are two components of the Comprehensive Mathematics Instructional Framework (Hendrickson, Hilton, & Bahr, 2009.) that is designed to provide a “framework for the kinds of
specific, constructive pedagogical moves that teachers might make” (Chazan & Ball, 1999, p. 2). The Framework of Cognitive Complexity provides guidance for teachers as they seek to enhance the engagement of their students in mathematical discussions.
Developing a Framework of Outcomes
120
means for generating new hypotheses about the nature of our framework and then reflecting on the data obtained from testing hypotheses about the framework in multiple cycles throughout the study, enabling us to investigate the credibility of our data while simultaneously furthering our ideas (Pinnegar & Hamilton, 2009). In recurring cycles we investigated our own changing perspectives by engaging in increasingly deeper examinations of the framework, a process that resulted in a string of consensually agreed-upon revisions. During each of the first six passes through the data, we discussed the meanings of the outcomes among ourselves. Dur-ing an additional 12 analyses, as we were using the outcomes to guide our methods coursework and supporting our students in constructing their own meanings for them, what we were learning informed our revisions. These latter cycles consisted of trying to help our students construct their own meanings for the outcomes and develop the learning they defined, then reflecting upon what we learned as we re-examined and revised the framework. As analysis continued, the structure of the framework reflected the naturally occurring themes that typically characterize qualitative research (Guba, 1978).
Findings
In this section we describe the evolution of our framework from two unstructured lists totaling 86 outcomes to its current form. We describe the specific analyses associated with the first six passes through the lists, during which the outcomes themselves served as data. We also describe the revisions resulting from these analyses as the outcomes were transformed into a tentative framework, which is shown in Table 1. We then describe the subsequent 12 analytical passes, which oc-curred while we were implementing the outcomes with our methods students. The revisions resulting from these analyses transformed the tentative framework into the current version of the framework as it appears in Table 2.
First Pass—Domains
Because we sought to create a framework that was both comprehensive and cohesive, we revisited the three domains that commonly comprise studies of teacher learning—Beliefs, Knowledge, and Practice (Wood, Nelson, & Warfield, 2001)—to code our lists. These domains, according to Arbaugh et al. (2010), may well function as “placeholders for an array of . . . outcomes” (p. 18). Thus, beliefs, knowledge, and practice not only served as a priori codes but also became foundational in structuring our framework. We worked together to categorize each of our competency lists, being sure to reach consensus through negotiation rather than capitulation. Both lists contained a reasonable distribution of outcomes across the three domains. Categorizations of ’s list resulted in 17 belief statements, 17 knowledge statements, and 19 statements
Damon L. Bahr, Eula Ewing Monroe, & Jodi Mantilla
121
28
Tab
le 2
A
Fra
mew
ork
of O
utco
mes
for
Mat
hem
atic
s T
each
er L
earn
ing—
Fina
l Ver
sion
2
Tea
cher
Lea
rnin
g P
osit
ions
M
athe
mat
ical
Con
tent
Kno
wle
dge
Stu
dent
Thi
nkin
g C
hild
Dev
elop
men
t S
ocia
l Int
erac
tion
Teacher Beliefs
• M
athe
mat
ical
wor
k is
a s
ense
-mak
ing
ende
avor
bas
ed u
pon
num
eric
al, s
pati
al, a
nd lo
gica
l thi
nkin
g (N
CT
M, 2
000)
. •
Mat
hem
atic
s un
ders
tand
ing
cons
ists
of
know
ing
how
and
w
hy it
wor
ks (
Phi
lipp
et a
l., 2
007)
. •
Mat
hem
atic
al u
nder
stan
ding
is c
onst
ruct
ed in
an
inst
ruct
iona
l env
iron
men
t cha
ract
eriz
ed b
y pr
oble
m s
olvi
ng,
reas
onin
g, g
entl
e ar
gum
enta
tion,
mod
elin
g an
d re
al-l
ife
appl
icat
ion,
str
ateg
ic to
ol u
se, a
tten
tion
to p
reci
sion
, and
lo
okin
g fo
r an
d us
ing
stru
ctur
e (N
atio
nal G
over
nors
A
ssoc
iati
on C
ente
r fo
r B
est P
ract
ices
& C
ounc
il o
f C
hief
S
tate
Sch
ool O
ffic
ers,
201
0).
• S
tude
nts
acqu
ire
info
rmal
mat
hem
atic
al k
now
ledg
e by
in
tera
ctin
g w
ith
thei
r en
viro
nmen
t and
thus
“ca
n so
lve
prob
lem
s in
nov
el w
ays
befo
re b
eing
taug
ht h
ow to
sol
ve
such
pro
blem
s” (
Phi
lipp
et a
l., 2
007,
p. 4
75).
•
“Eac
h ti
me
one
prem
atur
ely
teac
hes
a ch
ild
som
ethi
ng h
e (s
ic)
coul
d ha
ve d
isco
vere
d hi
mse
lf (
sic)
, tha
t chi
ld is
kep
t fr
om in
vent
ing
it a
nd c
onse
quen
tly
from
und
erst
andi
ng it
co
mpl
etel
y” (
Pia
get,
1970
, p. 7
15).
•
Chi
ldre
n’s
thin
king
is “
gene
rall
y di
ffer
ent f
rom
the
way
s ad
ults
wou
ld e
xpec
t the
m to
thin
k ab
out m
athe
mat
ics”
(P
hili
pp e
t al.,
200
7, p
. 475
).
• In
an
equi
tabl
e le
arni
ng e
nvir
onm
ent A
LL
stu
dent
s,
incl
udin
g st
uden
ts o
f va
ried
soc
io-e
cono
mic
sta
tus,
cul
tura
l ba
ckgr
ound
s, e
thni
citi
es, l
angu
ages
, and
wit
h ex
cept
iona
liti
es
can
lear
n (N
CT
M, 2
000)
. All
mea
ns a
ll.
• “T
each
ing
shou
ld b
e gr
ound
ed in
how
stu
dent
s le
arn”
(S
chif
ter
& F
osno
t, 19
93, p
. 193
), a
pro
cess
of
“act
ive
cons
truc
tion
, not
mer
ely
pass
ive
abso
rpti
on .
. . (
and
in)
inve
ntio
n, n
ot im
itat
ion”
(B
aroo
dy &
Gin
sbur
g, 1
990,
p. 5
2).
• M
athe
mat
ical
thin
king
mov
es g
ener
ally
fro
m th
e co
ncre
te
to th
e ab
stra
ct (
Nat
iona
l Gov
., 20
10; P
iage
t & C
ook,
195
2) o
r fr
om le
ss f
orm
al to
mor
e fo
rmal
thin
king
(B
onot
to, 2
005)
.
• In
stru
ctio
nal t
asks
sho
uld
be d
esig
ned
at a
leve
l tha
t all
ch
ildr
en c
an e
nter
wit
h so
me
degr
ee o
f su
cces
s bu
t tha
t w
ill s
tret
ch th
eir
thin
king
via
inte
ract
ion
wit
h th
eir
peer
s
(Vyg
otsk
y, 1
978)
. •
Bec
ause
”th
inki
ng w
ith
othe
rs .
. . e
nabl
es c
hild
ren
to
deve
lop
. . .
the
capa
city
to th
ink
alon
e” (
Roc
hat,
2001
, p.
139)
, tea
chin
g sh
ould
be
base
d up
on s
eeki
ng to
“u
nder
stan
d ra
ther
than
to b
e un
ders
tood
” (C
ovey
, 198
9).
• “M
athe
mat
ical
dis
cour
se e
mbe
ds f
unda
men
tal v
alue
s ab
out k
now
ledg
e an
d au
thor
ity
. . .”
rel
atin
g to
m
athe
mat
ical
cor
rect
ness
(N
CT
M, 1
991,
21)
.
Teacher Knowledge
• A
“P
rofo
und
Und
erst
andi
ng o
f F
unda
men
tal M
athe
mat
ics”
(M
a, 1
999,
p. 1
20)
is c
onst
ruct
ed in
thre
e co
gnit
ive
dom
ains
—C
once
ptua
l, P
roce
dura
l & R
epre
sent
atio
nal
(Fos
not &
Dol
k, 2
001;
Hen
dric
kson
, Hil
ton,
& B
ahr,
200
8)
and
five
mat
hem
atic
al d
omai
ns—
Num
ber
& O
pera
tion
s,
Alg
ebra
ic R
easo
ning
, Geo
met
ry, M
easu
rem
ent,
& D
ata
Ana
lysi
s/P
roba
bili
ty (
NC
TM
, 200
0).
• L
egit
imat
e m
athe
mat
ical
wor
k is
cha
ract
eriz
ed b
y va
ryin
g le
vels
of
thin
king
(e.
g., t
he F
ram
ewor
k of
Cog
niti
ve
Com
plex
ity,
Bah
r &
Bah
r, 2
017)
.
• R
athe
r th
an tr
ansm
itti
ng in
form
atio
n . .
.” (
Ric
hard
son,
20
01, p
. 279
), te
ache
rs u
se th
e L
ES
Inst
ruct
iona
l Mod
el(L
aunc
h-E
xplo
re-S
umm
ariz
e (D
iscu
ss);
Sch
roye
r &
F
itzg
eral
d, 1
986)
to e
ngag
e st
uden
ts in
legi
tim
ate
mat
hem
atic
al in
quir
y.
• T
he d
egre
e of
teac
her
guid
ance
ass
ocia
ted
wit
h th
e us
e of
th
e L
ES
Inst
ruct
iona
l Mod
el is
dep
ende
nt u
pon
the
teac
her’
s as
sess
men
t of
the
loca
tion
of
the
stud
ents
’ th
inki
ng in
the
Lea
rnin
g C
ycle
.
• S
tude
nts’
thin
king
mod
els
the
acti
ons
and
cont
exts
inhe
rent
in
the
prob
lem
s th
ey s
olve
(C
arpe
nter
et a
l., 1
999;
Fos
not &
D
olk,
200
1), a
nd s
erve
s as
land
mar
ks o
n to
pica
l “la
ndsc
apes
” th
at d
efin
e in
divi
dual
lear
ning
traj
ecto
ries
. •
As
stud
ents
mak
e co
nnec
tion
s w
ithi
n an
d ac
ross
“l
andm
arks
,” th
eir
unde
rsta
ndin
g m
oves
thro
ugh
a C
onti
nuum
of M
athe
mat
ical
Und
erst
andi
ng a
ccor
ding
to th
e ph
ases
of
the
Lea
rnin
g C
ycle
(H
endr
icks
on, H
ilto
n, &
Bah
r,
2008
).
• P
arti
cipa
nts
in m
athe
mat
ical
dis
cuss
ions
ass
ume
one
or
mor
e ro
les—
the
stud
ents
who
“sh
are”
thei
r th
inki
ng, t
he
stud
ents
who
list
en a
nd in
tera
ct w
ith
the
thin
king
of
the
shar
ing
stud
ents
, and
the
teac
her,
who
orc
hest
rate
s th
e di
scus
sion
and
als
o pa
rtic
ipat
es a
s a
list
ener
(W
ood
&
Tur
ner-
Vor
beck
, 200
1).
• T
each
ers
esta
blis
h “s
ocio
-mat
hem
atic
al n
orm
s” (
Yac
kel
& C
obb,
199
6, p
. 458
) th
at m
ake
“pos
sibl
e al
l stu
dent
s’
acti
ve p
arti
cipa
tion
in .
. . [
the]
dis
cour
se”
(Woo
d &
T
urne
r-V
orbe
ck, 2
001,
p.1
92).
Cat
egor
ies
of I
nqui
ry-B
ased
Tea
cher
Pra
ctic
es (
refl
ecti
ng c
onne
ctio
ns a
mon
g be
lief
s an
d kn
owle
dge
wit
hin
and
acro
ss T
each
er L
earn
ing
Pos
itio
ns)
Uni
t and
Les
son
Pla
nnin
g B
efor
e In
stru
ctio
n Q
uest
ioni
ng, A
sses
smen
t, an
d D
ecis
ion-
Mak
ing
Dur
ing
Inst
ruct
ion
Usi
ng th
e F
ram
ewor
k of
Cog
niti
ve C
ompl
exit
y, te
ache
rs c
onsi
sten
tly
obse
rve
and
ask
ques
tion
s in
ord
er to
“ad
vanc
e, .
. . a
sses
s” (
NC
TM
, 201
4) a
nd
inte
rpre
t stu
dent
thin
king
, inf
orm
ing
“ . .
. th
e de
cisi
ons
abou
t the
nex
t ste
ps in
inst
ruct
ion”
(W
illi
ams,
201
3, p
. 43)
as
the
less
on p
roce
eds
thro
ugh
the
plan
ned
LE
S In
stru
ctio
nal M
odel
.
Teacher Practices, Nested Strategies, and Techniques
In u
nit d
esig
n, te
ache
rs u
se th
eir
know
ledg
e of
th
e m
athe
mat
ics
and
the
deve
lopm
enta
l la
ndsc
apes
that
are
ass
ocia
ted
wit
h a
part
icul
ar
topi
c, a
long
wit
h an
und
erst
andi
ng o
f th
e L
earn
ing
Cyc
le, t
ode
term
ine
wha
t m
athe
mat
ics
wil
l be
taug
ht w
hen.
• C
reat
e a
list
of
info
rmal
goa
ls b
ased
on
. . .
M
athe
mat
ics
stud
y L
ands
cape
s –
prob
lem
type
s, la
ndm
arks
, nu
mer
ical
or
spat
ial l
evel
s T
extb
ook—
list
s, le
sson
titl
es, l
esso
n ta
sks
and
exer
cise
s, a
nd a
sses
smen
ts
• C
reat
e a
Big
Dev
elop
task
to s
urfa
ce th
inki
ng
abou
t ent
ire
unit
A
ntic
ipat
e th
inki
ng f
rom
the
task
C
ompa
re th
inki
ng to
the
info
rmal
goa
ls
list
•
Des
ign
an e
nd-o
f-un
it p
erfo
rman
ce
asse
ssm
ent
Sel
ect o
r cr
eate
a ta
sk s
imil
ar to
the
Big
D
evel
op
Dec
ide
upon
cri
teri
a an
d cr
eate
a r
ubri
c A
dd p
rom
pts
to e
nsur
e va
lidi
ty
• P
lan
for
spac
ed p
ract
ice
via
rout
ines
and
ga
mes
Tea
cher
s us
e th
e L
ES
Inst
ruct
iona
l Mod
el a
nd th
e L
earn
ing
Cyc
le to
des
ign
less
ons
that
pro
mot
e hi
gh le
vels
of
inte
ract
ive
disc
ours
e an
d re
sult
in
deep
mat
hem
atic
al u
nder
stan
ding
in h
arm
ony
wit
h C
omm
on C
ore
goal
s.
• C
onti
nue
plan
ning
the
Big
Dev
elop
less
on
Lau
nch —
mat
eria
ls, g
roup
ing,
ass
essi
ng ta
sk
com
preh
ensi
on, m
akin
g ta
sk c
ompr
ehen
sibl
e fo
r ea
rly
lang
uage
lear
ners
E
xplo
re—
anti
cipa
ted
thin
king
, ass
essi
ng
task
-rel
ated
thin
king
, lan
guag
e to
ols
for
earl
y la
ngua
ge le
arne
rs
Sum
mar
ize
(Dis
cuss
) —se
lect
ed a
nd
sequ
ence
d sh
arin
g ba
sed
on a
ntic
ipat
ed
thin
king
, lis
teni
ng s
tude
nt r
oles
and
re
spon
ding
pro
cedu
res
usin
g th
e F
ram
ewor
k of
Cog
niti
ve C
ompl
exit
y, s
caff
oldi
ng e
arly
la
ngua
ge le
arne
r pa
rtic
ipat
ion
• D
esig
n su
bseq
uent
less
ons
Fin
d or
cre
ate
Sm
all D
evel
op, S
olid
ify,
End
-of
-sol
idif
y as
sess
men
t, an
d si
ngle
-dom
ain
prac
tice
task
s us
ing
Inst
ruct
iona
l Mod
els
A
ntic
ipat
e th
inki
ng f
rom
task
s w
here
ne
cess
ary
Cre
ate
purp
ose
stat
emen
ts a
nd o
utco
mes
for
ea
ch T
each
ing
Cyc
le w
ithi
n th
e le
sson
P
lan
less
on s
tage
s
Lau
nch
- A
sses
s ge
nera
l cla
ss
com
preh
ensi
on o
f th
e ta
sk g
ivin
g at
tent
ion
to s
tude
nts
wit
h sp
ecia
l nee
ds
• A
sk q
uest
ions
abo
ut ta
sk c
ompo
nent
s •
Inte
rpre
t res
pons
es to
asc
erta
in ta
sk
unde
rsta
ndin
g •
Dec
ide
whe
ther
to p
roce
ed to
E
xplo
re, o
r cl
arif
y th
e ta
sk f
irst
Exp
lore
- A
sses
s an
d ad
vanc
e ta
sk-r
elat
ed
thin
king
whi
le in
tera
ctin
g w
ith
stud
ents
•
Ask
an
open
-end
ed q
uest
ion
• In
terp
ret r
espo
nses
W
hat i
s th
e st
uden
t thi
nkin
g?
Wha
t is
the
mat
h in
here
nt in
the
thin
king
?
Whe
re is
the
thin
king
in th
e L
earn
ing
Cyc
le a
nd o
n th
e la
ndsc
ape?
Wha
t is
the
rela
tion
ship
of
this
thin
king
to
anti
cipa
ted
thin
king
and
pur
pose
? •
Dec
ide
the
next
mov
e w
ith
this
stu
dent
(s)
Wha
t is
the
next
que
stio
n to
be
aske
d?
Con
tinu
e w
ith
this
stu
dent
or
anot
her?
S
houl
d th
e ta
sk b
e re
-lau
nche
d fo
r th
is
stud
ent(
s)?
Sho
uld
this
stu
dent
sha
re in
the
Sum
mar
ize?
•
Dec
ide
the
next
mov
e w
ith
the
clas
s S
houl
d th
e ta
sk b
e re
-lau
nche
d?
Sho
uld
the
clas
s m
ove
to S
umm
ariz
e?S
houl
d S
umm
ariz
e pl
an b
e al
tere
d? H
ow?
Who
wil
l act
uall
y sh
are
and
in w
hat o
rder
?
Sum
mar
ize
(Dis
cuss
) –
Ass
ess
and
adva
nce
the
thin
king
of
shar
ing
and
list
enin
g st
uden
ts
• A
sk f
or c
ontr
ibut
ions
to th
e di
scus
sion
•
Inte
rpre
t con
trib
utio
ns a
s in
Exp
lore
W
hat i
s th
e st
uden
t thi
nkin
g?
Wha
t is
the
mat
h in
here
nt in
the
thin
king
?
Whe
re is
the
thin
king
in th
e L
earn
ing
Cyc
lean
d on
the
land
scap
e?
W
hat i
s th
e re
lati
onsh
ip o
f th
is th
inki
ng to
an
tici
pate
d th
inki
ng a
nd p
urpo
se?
• D
ecid
e w
hen
and
how
to p
ursu
e S
houl
d th
e st
uden
t’s
com
men
t be
purs
ued?
If
so,
who
sho
uld
purs
ue it
? P
ursu
e at
wha
t thi
nkin
g le
vel o
f th
e F
ram
ewor
k?If
not
pur
suin
g st
uden
t com
men
t, w
ho s
houl
d sp
eak
next
? S
houl
d S
umm
ariz
e pl
an b
e al
tere
d, a
nd if
so,
ho
w?
•D
ecid
e w
hich
task
to la
unch
nex
t
2 Th
e Continuum of Mathematical Understanding and the Learning Cycle are two components of the Comprehensive Mathematics Instructional Framework (Hendrickson, Hilton, & Bahr, 2008.) that is designed to provide a “framework for the kinds of specific, constructive
pedagogical moves that teachers might make” (Chazan & Ball, 1999, p. 2). The Framework of Cognitive Complexity provides guidance for teachers as they seek to enhance the engagement of their students in mathematical discussions.
Developing a Framework of Outcomes
122
related to practice, and Eula and Damon’s list consisted of 13 belief statements, 6 knowledge statements, and 14 statements related to practice. This classification process was the first of several analyses that resulted in adding structure to our outcomes lists.
Second Pass—Positions
As introduced in the Positions and Domains of Teacher Learning section, Wood, Nelson, and Warfield (2001) suggested that research investigating the processes by which teacher learning occurs has been conducted from four positions, each with its own theoretical roots—Mathematical Content Knowledge, Student Thinking, Child Development, and Social Interaction. In our second analytical pass, therefore, we recoded the outcomes according to these four positions, using the same process of negotiation and consensus as was used in the first analysis. This classification pro-cess added another structure to our outcomes lists. Combining our first and second classifications produced a two-dimensional structure, a grid with 12 cells—beliefs, knowledge, and practices related to each of the positions: student thinking, child development, social interaction, and mathematical content knowledge.Third Pass—Combining and Eliminating Redundancy
In our third analytical pass, we combined both lists into one by eliminating redundant outcomes. This process required us to be very explicit in our use of vocabulary and in explaining what each of our outcomes meant or did not mean. Seventeen outcomes on Eula’s list did not appear on Damon’s, eight of Damon’s did not appear on Eula’s list, and some of Eula’s were restatements of previous outcomes. Our negotiations yielded a net list of 44 with an average of three or four outcomes per the 12 above-mentioned domain/position classifications. The grid as it relates to beliefs and knowledge outcomes appears as Table 1. Subsequent paragraphs describe a different structure relating to practice outcomes.
Fourth Pass—Beliefs and Knowledge Connections
The nature of the fourth analysis was unanticipated. This analysis arose as a continuation of the previous positions-based categorization. As we examined the outcomes labeled either as beliefs or as knowledge that were categorized within the same position, we noticed that they tended to relate conceptually in a deeper way than by mere position categorization. It became clear that if a teacher pos-sessed a specific belief, that teacher would likely be disposed to act in acquiring related knowledge. Thus the results of this analysis revealed an internal structure that ensured cohesion between two of its three domains, beliefs and knowledge, as shown in Table 1.
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Fifth Pass—Higher Order Position Connections and Resulting Practices
The need for a cross-categories analysis of outcomes in the practice domain arose after we had noticed the connections between beliefs and knowledge during the fourth pass. We observed that each practice outcome seemed to require the application or use of knowledge relating to all four positions, as in the case of the practice of orchestrating discussions. First, a teacher should know the kinds of questions to use to probe student thinking while a discussion is taking place, and second, be able to identify the mathematics inherent within that thinking. Third, the teacher should know where that thinking fits within the developmental landscape of that particular mathematical domain, and fourth, know the kinds of moves to make in order to take full advantage of that thinking when it is shared. Therefore, as Nelson (2001) wrote, making connections among the work represented by the four positions “create[d] a more complete description” (p. 251) of teaching, and a more cohesive one. Because a position level categorization was insufficient for grouping outcomes we labeled as practices, we examined the relevant data for emergent themes we could use, expecting the outcomes to cluster naturally into a small number of groups. This analysis resulted in three categories—curriculum planning; questioning and assessment; and orchestrating discourse during exploration and discussion—that we labeled Domains of Practice as shown in Table 1. Interestingly, these categories are similar to the domains of practice Boerst, Sleep, Ball, & Bass (2011) used to guide their preservice methods course at the University of Michigan, a connection we discovered after creating our own domains.
Sixth Pass—Grain Sizes
Boerst et al. (2011) added their insight into the specification of practice-oriented practices by discussing the decomposition of the work of teaching.
To make the complex work of teaching mathematics more learnable by beginners, we aim to decompose practice into smaller tasks or routines that can be articulated, unpacked, studied, and rehearsed (Grossman et al., 2009; Grossman & Shahan, 2005; Lampert, 2001, 2005). Such decomposition temporarily reduces complexity by holding some aspects of teaching “still” or by routinizing some components of the work so that beginning teachers can attend to and practice particular skills or focus on specific problems. (p. 2849)
They continued by describing varying sizes of decomposition from large do-mains to “intermediate- and technique-level practices (p. 2871).” These hierarchical levels of decomposition indicate successively smaller “grain sizes” (p. 2850), with smaller grain-sized practices nested within larger ones.
The other product of this approach to decomposition is an articulation of connec-tions among nested teaching practices. Moving along a strand from practices of
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larger grain size to those of smaller grain size details with increasing specificity how a practice can be implemented. In turn, starting with a technique and moving up a strand to practices of increasingly larger grain size makes visible the purpose that those techniques are serving. (p. 2854)
We therefore analyzed the practice domain outcomes, looking for an appropri-ate number of grain sizes while categorizing the outcomes according to size. That is, we examined which outcomes conceptually nested within other outcomes to determine the appropriate number of grain sizes. Using our own wisdom of practice, we also examined each category and added outcomes of various grain sizes when a category appeared wanting. This planned analysis revealed four levels of generality/specificity, or grain size, as shown in Table 1, which we labeled as Practices, Models, Strategies, and Techniques, terms commonly used by curriculum and instructional designers (e.g., Gunter, Estes, & Schwab, 1999; Burden & Byrd, 2013). The term practice now refers both to a domain within our framework, along with beliefs and knowledge, and to a grain size of outcomes within the practice domain. Thus, decomposed practices were labeled as nested models, decomposed models were labeled as strategies, and decomposed strategies were labeled as techniques. We also se-quenced the outcomes within and across grains sizes if the outcomes suggested that teacher practices should be performed in some sort of order while designing or implementing instruction.
Four Coordinated Analyses
Because the set of outcomes was now organized by a structure as described above, we referred to it as a framework, albeit far from complete. We continued to revise and improve it while using it to guide our teaching of preservice methods students. In this process, the framework of outcomes no longer served simply as data we were analyzing, but also as a series of results stemming from our analysis of data obtained from other sources as mentioned previously. We performed vari-ous combinations of the following four types of analyses as we passed through the data 12 more times. By this point, Jodi had joined our faculty and had become an integral partner in our study.
1. Distinguishing between outcomes and course content. In some of the passes, our analyses included determining if any of our outcomes were actually descrip-tions of course content.
2. Reviewing the literature. We examined the literature acquired from our early framework development along with additional sources we located.
3. Refining the language of the framework. We continued the refinement process by considering how best to state our outcomes and to structure the language of the outcomes. The former process consisted of ongoing wordsmithing—looking for the right word here, deleting a word there, etc. The latter process considered the
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sentence or phrase structures that would more clearly present outcomes within each teacher learning domain and deliberating about the language needed for structures across domains and grain sizes and within the practice domain.
4. Reorganizing within and across domains and positions. We addressed several issues during this stage of framework development that went well beyond the work of wordsmithing as described in the previous section. We divided some outcomes and combined others. We moved some outcomes from one domain or position to another. Some outcomes were removed from the framework altogether because they were redundant.
Twelve Passes in the Context of Framework Implementation
We used the four types of analyses in varied combinations during 12 analytic passes associated with using the framework to guide our methods instruction. As we discuss the results, we share the data and sources from which they were ob-tained, the processes of analysis we used and how we used them, and the resulting changes in our framework. We do not relate all of the changes but, rather, a few of the most prominent ones, thus providing the reader an overall sense of our analysis and revision efforts. Our early attempts at sharing the framework with our students revealed that they were generally overwhelmed with the sheer amount of information (Session Notes beginning 9/4/14). This experience influenced all of our analyses, but its first effect occurred in the fourth pass as a reduction in the number of grain sizes in the practice domain from 4 to 3. After engaging in a series of activities designed to help our students construct the outcomes for themselves, we shared the framework with them in its then cur-rent form. We referred to specific outcomes when engaging in activities related to those outcomes and even asked students to self-assess their growth using the framework. It became apparent (Session Notes, 9/4/14) that the outcomes in the beliefs and knowledge domains were difficult to interpret and were therefore not as meaningful for student use as we had hoped. We needed to clarify our thinking both for ourselves and for our students. In multiple iterations of wordsmithing and reconceptualizing, we added to, reworded, deleted, split, and combined outcomes. For example, we noticed the need to emphasize the conceptual and procedural aspects of mathematical work (Philipp, et al., 2007; NCTM, 2014) and added this idea as an outcome in the 11th pass. We added outcomes about sense making, the student mathematical practices from the Common Core State Standards—Math-ematics (National Governors Association Center for Best Practices & Council of Chief State School Officers, 2010), and the authority structures associated with discourse patterns. We also changed the positions and/or domains of some outcomes. We moved, in the sixth pass, the outcome about making connections across domains of under-standing to child development, thinking it more precisely defined a belief related to
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how students learn. We also moved the outcome about authority structures to the social interaction position, recognizing that it is more aligned with social interac-tion patterns than with the nature of mathematics content. We consistently asked students for formative feedback at key points throughout the semester. One piece of that feedback (Formative Feedback Summary, 2/12/15) suggested that our students needed more help with unit and lesson design, including how to utilize a traditional textbook in designing inquiry-based lessons. Thinking that improving the way our unit design outcomes in the practice domain were written would help us improve our teaching in this area, we split the curriculum-planning category into two categories, unit planning and lesson planning, in the sixth pass. We added more outcomes of smaller grain size to the unit-planning category in the tenth pass, then discovered we had added too much detail, so we combined, simplified, and reorganized those outcomes. While considering discussion orchestration in the context of the Launch-Explore-Summarize Instructional Model (Schroyer & Fitzgerald, 1986) in class one day, we were greeted by several puzzled facial expressions and one brave student who queried, “What is Launch-Explore-Summarize?” This time we were the ones mystified because we had discussed this model on more than one occasion (Session Notes 10/2/14). To help ourselves correct this problem, we first added a knowledge outcome to the social interaction position in pass seven, and then revised it and moved it to the student thinking position in the 11th pass. We made the Launch-Explore-Summarize model the organizational structure of the lesson planning category in the sixth pass and the questioning and assessment and discus-sion orchestration categories in the ninth pass, all within the practice domain. We also made one more structural change related to the model in conjunction with the next major revision as explained in the following paragraph. In a feedback session following a methods course field practicum (Session Notes 12/2/14), several students expressed interest in learning more about assessment. We had attempted to strengthen the questioning and assessment category in the practice domain in the fourth pass by including outcomes related to fundamental notions of assessment, such as validity, reliability, formative, summative, etc., prior to receiving this feedback. At this time we sought to create outcomes that more clearly reflected the integral role assessment played in the in-the-moment decision-making that occurs while a lesson is implemented, including questioning and interpreting student responses (see Williams, 2013). As we made the 11th pass, we combined two subcategories, questioning and assessment and discussion orchestration, into one category, which we titled questioning, assessment, and decision-making. Then, continuing with the use of the Launch-Explore-Summarize Instructional Model (Schroyer & Fitzgerald, 1986) as an organizing structure, we used the model to structure the entire category, then used the frame ask, interpret, decide to specify the fine grained outcomes related to Launch-Explore-Summarize, all in the 12th pass. During the first semester of implementation and 2 months after our students
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had studied the learning landscapes described in the Progressions for the Common Core State Standards in Mathematics (Core Standards Writing Team, 2013), we were mystified by a student’s remark that he would like to know where he could find research available on how students’ thinking changes over time (Session Notes 12/2/14). We modified our outcomes to show the pivotal role this knowledge should play in the practices of task selection, anticipating student thinking, interpreting student thinking, and in the decisions involved with selecting and sharing student thinking during a discussion. In our earlier versions, the notion of landscapes was part of the unit planning outcomes, but in the 12th pass we made it an integral part of several outcomes within the questioning, assessment, and decision-making category of the practice domain. When watching our students teach each other and teach small groups of chil-dren, we discovered that assigning roles to listening students during the sharing of student thinking in a discussion (see Wood and Turner-Vorbeck, 2001) was often forgotten or not done very well (Session Notes 10/3/14; 2/6/15). After adjusting our instruction as a reflection of changes in the questioning, assessment, and decision-making category of the practice domain, we noticed improvements among our students in this aspect of discussion orchestration. The changes associated with our multiple analyses are a sampling of the many changes that resulted in the framework in its current form (see Table 2). We doubt it will ever be totally complete, but we believe it is now sufficient to provide meaningful guidance to our teaching, and perhaps help other mathematics educators examine their teaching as well.
Content of the Framework
The purpose of this section is to introduce the reader to the specific content of the framework as shown in Table 2. Our purpose is not to give a lengthy treatise on the underlying research relating to each outcome, but rather to provide a narrative that highlights the key notions from which the content of the framework is taken. Because much of the framework is grounded in well-accepted mathematics educa-tion research, we believe our work may be transferable for use by other mathematics educators. We begin by discussing the content of the beliefs and knowledge outcomes that stem from the four positions (Wood et al., 2001) we described previously in the Positions and Domains of Teacher Learning section. We then discuss the practice outcomes, which represent a synthesis of beliefs and knowledge outcomes across multiple positions. Key statements relative to the mathematical content position define mathemati-cal work as a sense-making endeavor based upon numerical, spatial, and logical thinking (NCTM, 2000). These statements define mathematical understanding as learning how and why mathematics works (Philipp et al., 2007) within an instructional environment characterized by problem solving, reasoning, gentle argumentation,
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modeling and real-life application, strategic tool use, attending to precision, and looking for and using structure (National Governors Association Center for Best Practices & Council of Chief State School Officers, 2010). We knew we wanted the teachers we serve to construct a “Profound Understanding of Fundamental Mathematics” (Ma, 1999, p. 120) in three cognitive domains—Conceptual, Proce-dural & Representational (Fosnot & Dolk, 2001; Hendrickson et al., 2008) and five mathematical domains—Number & Operations, Algebraic Reasoning, Geometry, Measurement, & Data Analysis/Probability (NCTM, 2000). We also knew that these teachers’ mathematical work, and the work of their students, should be character-ized by varying levels of thinking (Bahr & Bahr, 2017). Teacher learning in the student thinking position is based on a deep appreciation for the way students think. Because children can construct informal mathemati-cal knowledge by interacting with their environment, they “can solve problems in novel ways before being taught how to solve such problems” (Philipp et al., 2007, p. 475). Indeed, “each time one prematurely teaches a child something he [sic] could have discovered himself [sic], that child is kept from inventing it and consequently from understanding it completely” (Piaget, 1970, p. 715). Their think-ing is “generally different from the ways adults would expect them to think about mathematics” (Philipp et al., 2007, p. 475). Rather than transmitting information . . .” (Richardson, 2001, p. 279), teachers use the LES Instructional Model (Launch-Explore-Summarize [Discuss]); Schroyer & Fitzgerald, 1986) to engage students in legitimate mathematical inquiry. The degree of teacher guidance associated with the use of the LES Instructional Model is dependent upon the teacher’s assessment of the location of the students’ thinking developmentally as described in the Learning Cycle (Hendrickson at al., 2008), a generic set of learning phases during which students surface their thinking in a problematic environment, refine that thinking by connecting it other ideas, strategies, and representations, then work to make it more fluent. A developmental perspective includes the notion that in an equitable learning environment all student can learn and in fact, “teaching should be grounded in how students learn” (Schifter & Fosnot, 1993, p. 193). Learning is a process of “active construction, not merely passive absorption . . . [and in] invention, not imitation” (Baroody & Ginsburg, 1990, p. 52). Mathematical thinking moves generally from the concrete to the abstract (National Gov., 2010; Piaget & Cook, 1952) or from less formal to more formal thinking (Bonotto, 2005). Students’ thinking models the ac-tions and contexts inherent in the problems they solve (Carpenter et al., 1999; Fosnot & Dolk, 2001), and serves as landmarks on topical landscapes that define individual learning trajectories. As students make connections within and across landmarks, their understanding moves through a Continuum of Mathematical Understanding according to the phases of the Learning Cycle (Hendricksonet al., 2008). Honoring the social nature of learning involves allowing children to think ”. . . with others . . . [which] enables children to develop . . . the capacity to think alone”
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(Rochat, 2001, p. 139). Thus teaching should involve designing instructional tasks at a level that all children can enter with some degree of success but that will stretch their thinking via discussions with their peers (Vygotsky, 1978). This interaction “. . . embeds fundamental values about knowledge and authority . . .” relating to mathematical correctness (NCTM, 1991, p. 21). Participants in mathematical dis-cussions assume one or more roles—the students who “share” their thinking, the students who listen and interact with the thinking of the sharing students, and the teacher, who orchestrates the discussion and also participates as a listener (Wood & Turner-Vorbeck, 2001). Teachers establish “socio-mathematical norms” (Yackel & Cobb, 1996, p. 458) that make “possible all students’ active participation in . . . [the] discourse” (Wood & Turner-Vorbeck, 2001, p.192). The practice domain is divided into two parts—outcomes relating to the planning of lessons and units of instruction prior to instruction and outcomes relating to the questioning, assessment, and decision-making that occur during instruction. Prominent in both parts is knowledge related to all four positions. For example, knowledge of student thinking and how that thinking changes over time informs both planning and instruction. The LES Instructional Model (Schroyer & Fitzgerald, 1986) provides a useful framework both for planning lessons and for organizing outcomes relating to instructional implementation. Knowledge of the Learning Cycle (Hendrickson et al., 2008) influences tasks design and sequence in unit planning along with the interpretations and decisions associated with instruc-tional implementation. Mathematical content knowledge informs every aspect of planning and the interpretation of student thinking that is so critical to decision making. Finally, the various roles participants play in a mathematical discussion guides the planning of those discussions and the carrying out of those plans as a lesson unfolds.
Conclusions
Our purpose in conducting this self-study was to create a comprehensive and cohesive set of outcomes to guide the teaching and learning of preservice and in-service mathematics teachers with whom we work and to guide our study of that teaching and learning. It resulted in the Framework of Outcomes for Mathematics Teacher Learning (FOMTL). As three mathematics educators with views that are both remarkably similar and yet appreciably different, we needed to seek to under-stand each other’s perspectives in order to adequately analyze and synthesize our thinking. Thus, we reconsidered our own perspectives in light of our colleagues’ views, a process that resulted in a refinement of our own understandings and in a negotiated shared perspective in the form of our framework. This work has led us to some generalizations that hold great meaning for us. First, we agree more fully than ever with the Linking Research and Practice: The NCTM Research Agenda Conference Report (Arbaugh et al., 2010) regarding the
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need for a specific set of outcomes to guide mathematics teacher education. We sense that adherence to constructivist principles, principles that we believe in, may have led us to struggle with the call for precise specifications of outcomes that define quality mathematics teaching. We recognize the similarity in our struggle with the debate about the role of outcome or objective specification in teaching children and adolescents. Having a clear sense of where we are leading our students does not mean we ignore our constructivist roots. Rather, we seek to help our students construct outcomes for themselves, as well as the specific beliefs, knowledge, and practices that the outcomes define, without having to construct all of its structure. Our work has enabled us to focus our students more clearly and explicitly on a holistic picture that defines exemplary mathematics teaching and to help them schematize the complex work of teaching mathematics. Echoing Simon’s (1995) statement in the beginning of this paper, not having a clear sense of what good mathematics teaching is, as defined by our framework or other documents, is as dangerous to mathematics teacher education as not hav-ing a clear sense of what mathematics is to mathematics teaching. Thus, we honor the NCTM’s Professional Standards for Teaching Mathematics (1991), along with other landmark documents in the mathematics education reform movement, and hope that our framework will enhance the pioneering work of their authors. Second, we now have a more refined view of the interaction among the domains of teacher learning—beliefs, knowledge, and practice—and the four theoretical positions that characterize the study of teacher learning—mathematical content knowledge, student thinking, child development, and social interaction (Wood, Nelson, & Warfield, 2001). We have long believed these domains and positions could guide some sort of framework that defines teacher learning. This work has made it clear to us that there are distinct beliefs and knowledge outcomes associ-ated with each position, that beliefs and knowledge outcomes within a position are conceptually related, and that all practices of any grain size require varied syntheses among positions and domains. Third, we have a sense for the complexities associated with helping novice and inservice teachers conceptualize teaching and learning that aligns with current research, while at the same time using a structure and language that is accessible to them. On the one hand, we wanted to create a framework that was sufficiently comprehensive to guide our students not only in their preservice learning but also in their ongoing professional learning as inservice teachers. On the other hand, we did not want to overwhelm them with excessive amounts of information or with excessively complex information, thus creating negative impressions of the framework and of reform-based teaching in general. Our intent is to help students construct these outcomes as well as the learning that fulfills them rather than our adopting a teaching-as-telling mode. We are also pleased that we have the opportunity to spend considerable time helping inservice teachers pick up where they left off in their preservice learning via our professional development efforts. Indeed, having the same set of outcomes to
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guide our work in both contexts helps us bridge the chasm, at least for the teachers with whom we work, that exists between preservice and inservice teacher education (Zeichner, 2010). For this reason, among others, we particularly enjoy our professional development efforts when we are privileged to teach our former preservice students. Based on our most recent teaching, we are convinced that the structure of our framework facilitates instruction that supports the individual construction of networks of connections in our students’ learning. Indeed, we see it as a landscape that affords any number of learning trajectories as preservice and inservice teach-ers make sense of mathematics teaching in their own ways while helping us have a comprehensive and well-defined focus. We believe our framework, although still a work in process, will aid us and our students as we encourage them “in the creation and continued refinement of sophisticated models or ways of interpreting the situations of teaching, learning and problem solving” (Lesh, Doerr, Guadalupe, & Hjalmarson, 2003, p. 227).
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