Bobis2019_Article_SageOnTheStageOrMeddlerInTheMi

Journal of Mathematics Teacher Education
https://doi.org/10.1007/s10857-019-09444-1

“Sage on the stage” or “meddler in the middle”: shifting
mathematics teachers’ identities to support student
engagement

Janette Bobis1   · Maryam Khosronejad1   · Jennifer Way1   · Judy Anderson1 

© Springer Nature B.V. 2019

Abstract
Mathematics teachers’ identities profoundly influence how they interact with and position
their students to learn mathematics. In this paper, we examine how a year-long teacher
learning intervention supportive of student engagement in mathematics helped shift teach-
ers’ mathematics-related identities. We use an implied identity perspective as a theoretical
lens to explore changes in what teachers perceive as legitimate ways of being as a result of
their participation in the intervention. Data from pre- and post-intervention concept maps
and focus groups with 15 grade 5–7 teachers of mathematics were integrated for this pur-
pose. Teachers reported shifts in their identities, describing themselves as “facilitators,”
“learners” and “co-creators” of knowledge. We argue that such shifts in the mathematics-
related identities of teachers can have practical consequences in terms of improving stu-
dents’ engagement, and in particular, their autonomy for learning mathematics.
Keywords  Student engagement · Implied identity · Mathematics-related teacher identity ·
Professional learning

Introduction

Positive identities are considered critical to mathematics learning because of their “ten-
dency to act as self-fulfilling prophecies” (Sfard and Prusak 2005, p. 19) and their poten-
tial to influence career and higher education aspirations (Black et al. 2010). Research indi-
cates that the mathematics-related identity of teachers can influence how they interact with
their students (Clark et al. 2013; Reay and Wiliam 1999), how they position themselves as
teachers of mathematics and their students as learners of mathematics (Tait-McCutcheon
and Loveridge 2016), thus directly impacting their engagement levels and, over time, shap-
ing their students’ identities with mathematics (Grootenboer and Zevenbergen 2008). For
example, students not only learn about mathematical concepts and skills, but also learn

Janette Bobis and Maryam Khosronejad have contributed equally to the preparation of this paper.
* Janette Bobis
[email protected]
1 The University of Sydney, Sydney, NSW 2006, Australia

1 3Vol.:(0123456789)

J. Bobis et al.

about themselves as capable and autonomous learners when they succeed in solving chal-
lenging problems (Assor 2012). Students’ sense of belonging, autonomy and the mathe-
matical identities they develop arise from direct experiences of mathematics and interac-
tions with their peers and teachers—a process that is facilitated by teachers of mathematics.

The establishment of an engagement-supportive learning environment is very much
shaped by teachers’ mathematics-related identities (Lutovac and Kaasila 2018), their per-
ceptions of how mathematics should be taught and how students become engaged in it
(Bobis et  al. 2016). Therefore, how their identities as teachers of mathematics develop,
shift or might be reshaped to those that are more supportive of student engagement is an
important aspect of teacher development to explore. Such explorations are suggested by
Skott (2019) to provide new understandings that may improve future teacher learning and
development initiatives. The study reported here addresses this intention by investigat-
ing shifts in teachers’ mathematics-related identifies facilitated by a professional learning
program that aimed to establish enhanced engagement-supportive learning environments.
We use Lutovac and Kaasila’s (2019, p. 506) term “mathematics-related teacher iden-
tity” because it refers to the identity of all teachers of mathematics and our study involved
both specialist (i.e., secondary) and non-specialist (i.e., primary) teachers of mathematics.
While recognizing the differences that generally exist between the two groups of teach-
ers in terms of their personal relationships with the subject, our focus was on the shifts in
their professional identities as teachers of mathematics in response to a shared professional
learning experience and therefore consider the term “mathematics-related teacher identity”
to be the most appropriate term for our study.

Identity: a lens to explore teacher professional learning
and development

We see teacher learning as a process of becoming and mathematics-related teacher identity
formation as situated within cultural, social and physical communities of practice (Lave
and Wenger 1991). Sociocultural approaches to identity bring our attention to contextual
factors in the development of teacher identities and help us explore its context-depend-
ent nature through interaction and across various contexts (Lutovac and Kaasila 2018).
Research shows contextual issues hinder or facilitate processes of identity change (Anders-
son 2011) and that teacher educators need to take into account that “it is the teacher who
controls his or her destiny” (Chapman and Heater 2010, p. 457). In this paper, we extend
previous research by exploring teacher responses to the contextual aspects of the learning
environment (a teacher professional learning program) through an implied identity theo-
retical lens, suggesting that programs of professional learning can become resources for
mathematics-related teacher identity formation.
Identity and implied identity
There is little consensus about the construct of identity or how it can be studied (Dar-
ragh 2016), but “with more clarity, the concept of identity can continue to provide helpful
insights into our experiences of learning mathematics” (p. 29). Our use of the term identity
draws on the concept of position from positioning theory (Davies and Harré 1999). We use
position, as opposed to the classical notion of “role” (which is perceived as a fixed set of
responsibilities), to illustrate the dynamicity of identity through which teachers position

13

“Sage on the stage” or “meddler in the middle”: shifting…

themselves as teachers, in relation to other teachers, their students and to the discipline
of mathematics. Therefore, identity practices are characterized as constantly shifting posi-
tions emerging through social interaction. Participation in professional learning programs
involves both practices of self-identification (or self-positioning) and identification by oth-
ers (other-identification or other-positioning) through acts such as “telling stories, joining
groups, and acting in a particular way at a particular time” (Darragh 2016, p. 29). As Dar-
ragh (2016) suggests, perceiving identity as an act that may or may not be recognized as
desired is a useful future direction for identity research. Our investigation is based on the
premise that teachers’ acts of identity in positioning themselves and their students is the
result of what they recognize as legitimate ways of being, in different contexts, including
professional development programs. We use implied identity (Khosronejad et al. 2015) as
a theoretical lens to examine if and how a teacher professional learning program might
facilitate shifts in teachers’ identities as teachers of mathematics to ones that are more sup-
portive of student engagement in mathematics.

Guided by a situative approach, the process of identity formation is the result of engage-
ment in professional learning activities and therefore the interaction between learners and
their learning experience contexts. This interaction is not unidirectional. While teachers’
professional identities predominantly emerge from their experiences, their identities can
also influence the way they perceive these experiences (Noonan 2018). A number of ter-
minologies have emerged to explore the complexity of this interaction. Cobb and Hodge
(2011), for instance, look at the interplay between normative identity (who you should
become to be recognized as a competent professional in a particular context); and core
identity (who you think you are and who you would like to be) in emerging personal iden-
tities. In contrast, Sfard and Prusak (2005) use the terms actual and designated identities
and recognize learning as closing the gap between the two as learners move from who they
think they are, to their designated identity, of who they think they can or should become.

Normative identity and designated identity are expressions of a positioning mechanism
in the sense that they provide teacher–learners with suggestions of legitimate ways of being
a teacher of mathematics. However, in order to understand the process of identity forma-
tion, we need to look into the process of meaning making by teachers. The concept of
implied identity (Khosronejad et al. 2015) refers to what teachers, as learners, perceive as
suggested ways of being in relation to the context of their experience.

Applying the implied identity approach helps explain the complexities involved in the
dialogic interaction between the teacher–learner and their learning environment (Khosrone-
jad et al. 2015). It places emphasis on the interactions between what educational designers
intend for learners and the learner interpretation of it (see Fig. 1). In the current investiga-
tion, the professional learning program was conceptualized as a resource for identity for-
mation to provide teachers with new knowledge, skills and competencies related to student
engagement in mathematics as initially intended by the researchers. It was anticipated that
teachers would perceive implied identities of being a teacher through their participation in
the professional learning activities, including group discussions and their access to new
materials and tasks. Individual or collective responses to different aspects of the learning
program were expected in the form of both practice and reflection, potentially leading to
changes in perceptions of the ideal teacher and perceptions of self. The interaction between
the learner and the learning environment is considered a dialogic process, informing both
the teacher–learners’ future perceptions or practices and the context of experience in shap-
ing normative identities.

The components of teachers’ reflective responses to the learning experiences depicted
in Fig.  1 help unpack the relations between teacher perceptions and their practices. To

13

J. Bobis et al.

Fig. 1  Implied identity framework (Khosronejad et al. 2015 )

elaborate, a teacher’s actions within a particular context of experience is mediated by
implied identities—her perceptions of what it means to be a teacher of mathematics within
that particular context—and her reflection on what she already thinks about the profession
(ideal teacher of mathematics) as well as what she thinks about herself (perception of self).
Furthermore, any changes in the teacher’s perceptions of self and the ideal teacher of math-
ematics are facilitated by her perceived implied identities across different contexts of her
experience over time.

The implied identity approach reconceptualizes our inferences about collected teacher
data—when examining mathematics-related teacher identity formation relating to specific
contexts such as professional learning programs—and directs our attention to implied iden-
tities as the subject of inquiry rather than focusing on identities per se. In this paper, we
address the research question: How did the implied identities of being a teacher of math-
ematics perceived by teachers change during a professional learning program that focused
on supporting student engagement in mathematics?

Methodology

A teacher intervention was conducted as part of a large two-year project that aimed to
enhance the mathematical motivation and engagement of students in grades 5 through 7
(approximately 10–14  years of age). The intervention involved two different cohorts of
teachers and their students drawn from the same school system. The two cohorts experi-
enced the same program of activities, data collection methods and processes. Each cohort
was followed throughout an academic school year from February to December. This paper
reports on findings related only to the teachers who participated in the project. We report
on two data sources, including pre- and post-intervention focus group interviews and
teacher-produced concept maps, to elicit teacher perceptions of legitimate ways of teaching
concerning student motivation and engagement in mathematics. We explore shifts in teach-
ers’ perceptions as indicators of implied identities perceived during their participation in
the program and as facilitators of teachers’ repositioning practices.

13

“Sage on the stage” or “meddler in the middle”: shifting…

Intervention
The intervention was a teacher professional learning program conducted over 11 months
of the school year. The program incorporated seven full-day workshops that were held
at a centrally located school to allow easy access for teachers traveling from neighboring
schools. The workshops focused on challenging teachers’ perceptions of student engage-
ment in mathematics, building their knowledge of motivation and engagement theoretical
frameworks and of evidence-based instructional strategies for promoting student engage-
ment in mathematics. Hence, the intervention aimed to provide teachers with suggestions
(knowledge and resources) for becoming teachers of mathematics who were supportive of
student engagement.

The researchers adopted the role of workshop facilitators. They introduced teachers to
the theoretical frameworks of Fredricks et al. (2004) and Martin (2009) as vehicles to build
their knowledge about the nature of engagement. Neither of these frameworks specifically
relate to mathematical engagement. To address this, the facilitators drew upon mathemati-
cal contexts to help explain and highlight the classroom implications of various aspects
referred to in the frameworks. For instance, self-belief, an aspect in Martin’s Motivation
and Engagement Wheel was explained in the context of students’ beliefs and confidence in
their ability to perform well in mathematics. Mathematical self-belief was discussed and
explored through activities adapted from Leatham and Hill (2010). It was also discussed
in the workshop as to how the activities could be modified for use with their students. In
one such activity, teachers were presented with descriptive words such as interested/bored
and confident/not confident at either end of a series of continua. Teachers were asked to
indicate where they would place themselves according to their perceived relationship with
mathematics. They then discussed the implications of mathematical self-belief for them-
selves and their students.

Teachers completed between-meeting tasks involving the collection of information
about their students’ identities and engagement in mathematics. This information was
used as a stimulus for discussion and formed the basis of progress reports on school-based
action research projects at subsequent workshops.

The role of the researchers was to build teacher knowledge and stimulate reflection
upon their existing perceptions of student engagement and achievement in mathematics
through activities requiring collaboration and discussion. Researchers sought to facilitate
discussion around evidence-based teaching practices linked to the enhancement of student
engagement in mathematics for teachers to trial in their classrooms as part of their school-
based action research projects. For example, the researchers provided teachers with pub-
lished research-based readings (e.g., Bobis et al. 2011; Hattie and Timperley 2007), which
helped stimulate whole-group discussions and challenge their thinking about existing peda-
gogy and how it might change to promote greater student engagement in mathematics.

Participants
Participating teachers were from 4 secondary and 10 primary schools that were part of a
Catholic Education school system located in the suburbs of a capital city on the east coast
of Australia. In this school system, primary school comprises Kindergarten to grade 6 and
secondary school comprises grades 7 to 12. Primary schools were invited to participate
in the project on the basis that they represented the largest “feeder” schools to the invited

13

J. Bobis et al.

secondary schools. It was envisaged that such a “connection” between schools and teach-
ers would assist in developing a shared purpose of strengthening students’ mathematical
engagement. Of the 39 grade 5 to 7 teachers who participated in the study, data from 15
teachers (5 secondary and 10 primary) are reported in this paper. Data from these teachers
were selected because they included a broad representation of teachers who participated in
the project in terms of teaching experience (ranging from 2 years to more than 15 years),
comprised male and female, primary and secondary teachers, comprised all members from
three different focus groups, were present for both pre- and post-intervention data gathering
sessions and gave their consent to participate in the study. The inclusion of data from the
remaining focus groups was considered unnecessary when analysis of transcripts reached
saturation point with no new codes emerging (Saunders et al. 2018).
Data collection methods and procedures
A multistage data collection process involving concept mapping followed by focus groups
occurred pre- and post-intervention. Concept maps are intended to reveal how individu-
als organize and change their perceptions and knowledge (Novak and Caña 2006). They
traditionally consist of hierarchically arranged pieces of information or concepts (nodes)
with the relationship between particular nodes represented by a uni- or bidirectional arrow
called a link or a cross-link if connections are made between different map segments. Prior
to the first concept mapping activity, teachers were provided with an example of a con-
cept map on an unrelated topic. They were then asked to create a concept map with the
prompt: How is student motivation and engagement in mathematics promoted? The con-
cept maps were returned to teachers just prior to the second focus group interview session
at the conclusion of the intervention, where they were asked to study their initial map and
make modifications using a different colored pen. The concept maps provided a graphical
snapshot of each teacher’s unique experiences, understandings and perceptions of student
motivation and engagement in mathematics and how they perceived it was promoted in
their classrooms at two crucial points in time, thus revealing shifts in teachers’ perceptions
over the course of the intervention. The maps also provided an opportunity for teachers to
“collect their thoughts” in preparation for the focus group that immediately followed each
concept mapping session.

Focus groups are a type “of group interview that capitalizes on communication between
the research participants in order to generate data” (Kitzinger 1995, p. 299). They are par-
ticularly suitable for sociocultural research approaches that aim to encourage participants
to explore and clarify their own understandings, perceptions and views of concepts and
issues that are of importance to the whole group (Marshall and Rossman 2006). The pur-
pose of the focus groups in this study was twofold: first, they provided a collective view of
teachers’ pre- and post-intervention knowledge and practices perceived to be supportive of
student engagement in mathematics. Second, they were used to help elaborate on and begin
to extend individual teacher’s initial perceptions of student engagement and how it is pro-
moted in their classrooms represented in their concept maps.

After individually completing the concept map, focus groups consisting of 5 or 6 teach-
ers representing a mix of schools and grades taught, were formed, and ensuring that no
more than two secondary (specialist) mathematics teachers were in each group. The same
group of teachers met again after the post-intervention concept mapping exercise was com-
pleted. Each focus group was facilitated by one of the researchers, who asked a series of
open–ended questions that required teachers to reflect on their own students and teaching

13

“Sage on the stage” or “meddler in the middle”: shifting…

contexts, safeguarding that each member had sufficient opportunity to contribute to the
group discussion. For instance, one question asked teachers if (and how) they could rec-
ognize if a student in their class was engaged in mathematics. Another asked about the
instructional strategies they specifically employed to promote student engagement as per
the prompt used by teachers to construct their concept maps. Focus groups were approxi-
mately 45 min in duration, were audio-recorded and transcribed to assist with analysis.
Analysis
Concept maps were analyzed inductively, using a content analysis approach similar to that
described by Hough et  al. (2007). This process focused on the identification of concepts
(nodes) and the links and crosslinks to reveal relationships between concepts (content) and
teachers’ perceptions. Individual concepts and content linked via arrows drawn by teach-
ers or their written elaborations adjacent major nodes formed the initial themes by which
data were categorized. Two researchers independently identified concepts and the related
themes emerging from time 1 data collection that reappeared, evolved or were absent in
time 2 data. The two sets of themes were then compared until a single set of initial themes
was agreed upon. For instance, the promotion of confidence in mathematics was mentioned
in four teachers’ time 1 and seven teachers’ time 2 concept maps. Confidence was men-
tioned in connection with teachers’ own levels of confidence and students’ feelings of suc-
cess and prior experiences of mathematics. Analysis focused on how the maps revealed
shifts in teachers’ perceptions of what it meant to be a teacher of mathematics who is capa-
ble of promoting student engagement and how they repositioned themselves and their stu-
dents in the mathematics classroom. Three examples of concept maps (constructed by one
teacher from each of the three focus groups) are provided in Figs. 2, 3 and 4. These three
concept maps were selected based on the representativeness of the final set of themes used
to report the data and because of their clarity when electronically reproduced. As is evident
from the three example concept maps, many maps did not conform to a strict traditional
definition of concept map. Nevertheless, they all represented individual teacher-generated
constructions of their experiences, and perceptions of student engagement in mathematics
at two points in time and therefore provided valuable data. In this paper, we report on the
themes that emerged from a cross-case analysis that were reflective of key components
of the theoretical framework guiding the investigation and draw upon aspects of the three
example teachers’ maps to provide evidence of change at the individual teacher level.

Analysis of the focus group transcripts was mostly deductive involving an adaption
of Braun and Clarke’s (2006) process of thematic analysis. We started with the key ideas
(such as “the promotion of student confidence”) identified in the content analysis of the
concept maps and conducted multiple readings across three main phases of analysis. Dur-
ing the first phase, the transcripts were read to gain an overview of the discussion, to
develop familiarity with content and check relevance and consistency of themes between
the concept maps and focus group data. NVivo was then used to assist coding of transcripts
starting with the initial themes emerging from analysis of the concept maps. Finally, an
inductive approach was once again employed to identify significant themes emerging from
the focus groups that were not highlighted from the initial set of themes but were consid-
ered relevant for addressing the research question. The researchers returned to the concept
maps to deductively search for corroborating evidence to support the inclusion of any new
themes. The theme “teacher as a co-creator of knowledge and co-learner” was added using
this iterative approach.

13

J. Bobis et al.

Fig. 2  Example of concept map generated by Abigail, a grade 6 teacher from focus group 1 (time 2 text is
bolded)

Results

Findings from concept maps and focus groups are reported for both time 1 (T1) and
time 2 (T2) data collection, whereby data from one method is used to support and con-
firm those identified in the other. In this way, ideas expressed by individuals via their
concept maps could be directly linked to the same or similar ideas collectively discussed
and elaborated upon during the focus group interviews. While the implied identity
framework draws attention to individual teacher’s perceptions of what it means to be a

13

“Sage on the stage” or “meddler in the middle”: shifting…

Fig. 3  Example of concept map generated by Barb, a grade 5 teacher from focus group 2 (time 2 text is
bolded)

Fig. 4  Example of concept map generated by Colleen, a grade 7 teacher from focus group 3 (time 2 text is
bolded)

teacher capable of supporting student engagement in mathematics, it also emphasizes
the significance of interactions between individual teacher–learners and the broader
community of teachers in the future shaping of normative identities. In the current
study, the voicing of teachers implied identities and the shaping of normative identities

13

J. Bobis et al.

was considered an important outcome by teachers, all of whom were accustomed to col-
laboratively planning their mathematics teaching programs.

In the following section, results are presented according to the five main themes identi-
fied during the analysis process, including: teacher as a co-creator of knowledge and co-
learner; teacher as a facilitator of student interaction; teacher as a promoter of student con-
fidence; teaching mathematics as a process-oriented practice; and teaching as a practice of
linking mathematics to real-life.

Teacher as a co‑creator of knowledge and co‑learner
Time 1 concept maps revealed twelve of the fifteen teachers linking the use of technol-
ogy, physical resources and games to student engagement in mathematics. For example,
the concept maps of Abigail (Fig.  2), Barb (Fig.  3) and Colleen (Fig.  4) at time 1, each
referred to these resources. During focus group interviews, teachers explained their views
of how students learn mathematics while using these resources and described how they
used games and movement to “keep them physically engaged” (FG2, T1) as part of their
teaching practices. They generally described themselves as transferring knowledge to stu-
dents whereby they “give it (knowledge) to them … we spoon-feed them by giving step
by step by step instructions” (FG1, T1). Teachers remarked that it was often difficult to
get students to “realize that they’ve actually got to do something” and that “it’s not just
us (teachers)” doing the thinking (FG1, T1). A grade 6 teacher summarized her teaching
approach as follows:

I explain it and then encourage questions. But if I don’t get questions then I’ll be
moving around the class and helping them and if I find that there are quite a few chil-
dren that I’ve seen haven’t understood a concept then I’ll go back to the board and
maybe go through something again. (FG2, T1)
Teachers in each focus group talked about the importance of building relationships
with their students and “getting to know” their mathematical needs. They described a wide
range of strategies they perceived helped them achieve these things, including “pretesting”
(FG2, T1), monitoring students by “walking around making sure students are on track”
(FG1, T1), “putting kids in alphabetical order so there is no way they can hide at the back”
(FG2, T1), giving “stickers and sweets” to encourage participation (FG3, T1) and randomly
asking questions so that “the kids don’t know who’s going to get asked.”
In the second round of data collection, eight teachers included nodes on their concept
map referring to allowing students to take greater responsibility for their own learning.
For example, Abigail (Fig.  2) linked students “making decisions” to their willingness to
take “risks” in their learning of mathematics while Colleen (Fig.  4) included new nodes
concerning students “aiming for personal bests.” Nodes referring to teacher “questioning”
rather than “telling” them the answer appeared in ten teachers’ time 2 concept maps (e.g.,
see Fig. 3). In focus groups, Barb (grade 5 teacher) elaborated upon how she tried to shift
responsibility for learning mathematics back to the students:
… rather than being the ‘sage on the stage’ - being the teacher that stands up and
saying, ‘yes, that’s it’. More of the ‘meddler in the middle’ and helping them and
questioning them. And saying, “so do you think that really is the right answer? How
do you know it?”… So, in fact they’re still being responsible for providing their own
answer.

13

“Sage on the stage” or “meddler in the middle”: shifting…

A grade 7 teacher from Focus Group 2 reported that he now considered it better if
students “see the connections” for themselves, rather than starting “straight with the
formula.” While acknowledging that it was “more challenging for students to figure out
the connection,” the group agreed that “resilience is important too, … it can’t all be fun
and bells and whistles, sometimes things are hard, and you just have to apply yourself”
(FG2, T2).

Rather than viewing themselves as merely the transmitters of knowledge, teachers
acknowledged themselves as co-learners of mathematics whereby they and their students
“will learn from each other” (FG1, T2). A grade 7 teacher expressed her agreement with
the view that “it’s okay for them (teachers) to make mistakes” too, considering “the whole
process to be a learning experience” (FG3, T2) for them as it was for their students. A
particular shift in practices that teachers from all focus groups emphasized was their reli-
ance on student feedback and how this now informed their teaching. In focus groups, a
grade 5 teacher described how she had “actually been asking the students what motivates
them, how they learn maths or how they perceive maths and why they enjoy it … having
their input on what works for them” (FG2, T2). Additionally, teachers across all three focus
groups acknowledged that the nature of the tasks they now provided their students had
changed quite dramatically to “things that I’d never tried before” (FG1, T2). For instance,
Barb referred to tasks requiring students to reflect on their learning or provide feedback on
the teaching in her time 2 concept map and explained her thinking to her focus group: “I’ve
learnt the value of the kids reflecting at the end of lessons … I like that [exit] ticket idea. I
use that a lot at the end of a lesson, to see what the children have learnt. … I’ve learnt that
they don’t have to be at their desk with pen and paper all the time.”

Teacher as a facilitator of student interaction
Teachers expressed quite diverse views about the merits of student–student interactions
in the mathematics classroom prior to the intervention. Nearly all primary teachers sup-
ported the use of collaborative learning in their mathematics lessons, but two grade 7
teachers from two different focus groups and three grade 6 teachers were initially skepti-
cal of its merits. A grade 7 teacher explained that it “was a tricky area” (FG3, T1) and
“depends on how the group works … if you have it collaboratively where everyone is
involved in it, so each person actually has a task in the group,” it “could be an asset” to
learning mathematics (FG1, T1).

At time 2 data collection, ten out of 15 concept maps included new nodes linked to
the concept of collaborative learning. Colleen, a grade 7 teacher (see Fig. 4) referred to
“working together” and drew links to “sharing problem solving strategies” and “helping
each other.” Similarly, Abigail’s time 2 concept map (see Fig. 2) shows new links among
student engagement, “collaborative” learning and students “taking risks.” Teachers pro-
vided detailed examples of how they incorporated student–student interactions into
their teaching. In addition to the peer design task described earlier, a grade 7 teacher
remarked how she encouraged “peer conversations” and “peer tutoring” so students can
“learn from each other” (FG2, T2). Abigail elaborated upon her concept map explaining
how an emphasis on collaborative learning had had a positive impact on the learning
environment in her classroom; her students were now more willing to “take risks” and
viewed their achievements as a reflection of “a group learning” process that encouraged
greater student “autonomy for their learning.”

13

J. Bobis et al.

Teacher as a promoter of student confidence
Only two teachers at time 1 referred to their own confidence with either mathematics or
the teaching it as having an impact on their students’ mathematical confidence levels in
their concept maps. Barb (Fig.  3) linked her own confidence in mathematics with her
“approach to maths” teaching and a grade 7 teacher in his first year of teaching indi-
cated that greater experience would help build his confidence. In focus groups, teachers
initially considered a lack of student confidence in mathematics as a major reason for
their poor motivation and engagement in the subject area. A grade 5 teacher expressed
the view that “If they don’t feel good about themselves, obviously they’re not going to
be motivated … the biggest problem is that they lack motivation because they feel they
can’t do it” (FG1, T1). Teachers agreed that they could easily identify students who
“believe in themselves” because they are the ones who “will look at the page and sit up
nice and straight” (FG2, T1). Meanwhile, students who lacked confidence in mathemat-
ics were identified by an experienced grade 7 teacher by the “diversionary tactics” they
employed, such as the “need to go to the toilet, get up and look for tissues, can’t find
their books, can’t find their pencils …” (FG3, T1). Teachers also affirmed comments by
a grade 5 teacher that some students with very low confidence levels “might be actually
very anxious and … almost paralysed with fear” (FG2, T1).

Teachers varied in their approaches for addressing poor student confidence, with a
grade 6 teacher believing that confidence would improve if students would “practice it
[mathematics]” because “… the more you practice it, the easier it becomes” (FG1, T1).
A year 7 teacher required her “less confident” students to “sit up the front row” (FG2,
T1), while another recounted how he gave students more “thinking time … It takes that
anxiety and that pressure off” (FG2, T1).

Teachers emphasized the importance of student self-belief and student confidence in
their time 2 concept maps and interview responses with many referring to “noticing how
much of a difference confidence can play with results and how they (students) perform.
So just targeting students’ confidence—and building that up; that’s probably the biggest
thing I have learned” (FG1, T2). Teachers recognized that it was their “responsibility”
to “lead them [students] towards success” (FG3, T2) by building a stronger “belief in
themselves” (FG2, T2). In tandem with this goal, teachers referred to their own feel-
ings of confidence in mathematics; an aspect they felt had been strengthened by “having
more strategies and more understanding” to address student confidence levels in math-
ematics (FG1, T2).

Teaching mathematics as a process‑oriented practice
Prior to the intervention, teachers likened the process of learning mathematics to learning
a language, one word at a time. Barb emphasized students using the “correct mathematical
language” in her time 1 concept map (Fig. 3), and when she raised this point in the focus
group, another group member agreed and then added that in mathematics “you have to
learn the basics … and you build on that … It’s a skill that you need to practice and build
on …” (FG2, T1). However, unlike words that could vary their meaning according to their
context, “the answer in maths is always … precise” (FG1, T1). Mathematics, as depicted in
Abigail’s time 1 concept map (Fig. 2), was more about students mastering “skills” than the
development of understanding.

13

“Sage on the stage” or “meddler in the middle”: shifting…

The increased attention teachers paid to mathematical processes—rather than the “precise
answer” (FG3, T1)—was evident in their responses at the end of the intervention. In her time 2
concept map, Barb (Fig. 3) linked the “idea of process” to her questioning of students to elicit
their reasoning, indicating that the process “can be more important than the answer.” Another
node in her concept map referred to positioning herself as “a meddler in the middle through
questioning” students to stimulate greater reasoning and justification of their answers. During
the follow-up focus group, other teachers affirmed that the process was “just as important, if
not more so important, than the actual answer” (FG2, T2).

Teachers’ choices of assessment tasks also reflected a greater focus on teaching mathemat-
ics as a process. In her time 2 map, Barb linked “assessment” to consideration of “where
can I take that student?” rather than just giving “a mark.” Similar sentiments were expressed
regarding tasks they selected for assessment by focus group members. Focus Group 2 teachers
expressed their agreement of one teacher’s statement that she tries to select assessment tasks
that not only show students can find the correct answer but will reveal the “process that stu-
dents have taken to get to it” (FG2, T2).
Teaching as a practice of linking mathematics to real‑life
The practice of teaching mathematics according to “topics”-dominated conversations in two of
the three focus groups, as teachers referred to “topic tests” (FG3, T1) and the fact that individ-
ual student’s levels of engagement differed because “some topics are drier than others” (FG1,
T1). Despite this, making connections between mathematics and students’ lives or between
one mathematical concept and another was agreed upon as an extremely important aspect of
teaching mathematics by members of all three focus groups. Only four teachers included a
node linking student engagement and making students “own experiences/everyday life” in
their time 1 concept maps (e.g., see Fig. 3). Barb explained her position: “if you’re truly going
to understand something you actually have to be able to … connect it to your understanding
of a whole lot of different things.” However, she considered that the connections she made
during mathematics lessons were mostly “serendipitous to the situation” and not deliberately
planned. A year 6 teacher from the same focus group gave an example of how she deliberately
used “authentic problems … like when we had this building actually built, we’ve got to carpet
this building…, how much carpet are we going to need to get?” to help make connections
between mathematics and students’ lives.

At time 2, 9 teachers included new nodes on their concept maps that referred to “real world
examples” (see Fig. 2), “investigations that stem from students’ interests” or experiences (see
Fig. 4) and “open–ended tasks” that were considered to be more authentic reflecting students’
interests and lives, but that would also maintain their engagement and reveal their understand-
ings. Colleen (Fig. 4) explained how she now used tasks that were open–ended and encour-
aged autonomous learning strategies to help her assess student understanding, such as a peer
design task, whereby “I ask them to design a task … and develop the questions … Then pass it
on [to another student]. That really shows…the sophistication of their understanding through
the questions that they’ve developed” and was perceived to be “more interesting” for students.

13

J. Bobis et al.

Discussion

Understanding identity development is an essential part of studying teacher professional
learning because it is identity that “mediates what makes its way into the classroom”
(Battey and Franke 2008). In the current research, we applied the notion of implied
identity to understand shifts in how teachers positioned themselves and their students.
An analysis of changes in teachers’ self-reported perceptions of improving student
engagement in mathematics was presented as implied identities perceived during teach-
ers’ participation in the program. In this section, we discuss how the main perceived
implied identities are linked to shifts in teachers’ positioning of themselves and their
students. Additionally, we explain what these repositioning practices mean in terms of
students’ learning.

“Sage on the stage” to “meddler in the middle”
The most significant implied identity to emerge from the intervention was that of
teacher as co-creator of knowledge and teacher as facilitator or in other words, teacher
as the “meddler in the middle”—one who actively scaffolds student learning rather than
simply “telling.” Teacher responses prior to the intervention indicate that teachers posi-
tioned themselves as “sages on the stage,” transmitting mathematics content to students
via controlling practices involving “step-by-step” instructions with few opportunities
for students to independently interact with each other or to explore mathematical con-
nections for themselves. Teachers initially positioned their students as mostly unrecep-
tive learners, dependent upon them to deliver content. Their goals of engaging students
were translated into practices whereby teachers felt that they needed to do most of the
higher order thinking. While teachers expressed their views identified with engagement-
supportive teaching practices, such as the importance of making connections between
mathematics concepts and real-life contexts and highlighting the crucial role of student
confidence in developing engaged learners of mathematics, they felt constrained by their
own lack of strategies for enacting these views in the classroom.

As teachers’ identities shifted to being facilitators of student learning, they posi-
tioned their students as autonomous learners who were afforded greater responsibility
for their own learning. The increased significance placed upon autonomous learning
was evidenced by changes in the nature of the tasks teachers provided their students;
there was increased use of open–ended and collaborative tasks, and tasks that required
students to reflect on their own learning. While changes in teachers’ perceptions do not
necessarily lead to changes in practice, a body of research suggests such a relation exists
(Fang 1996; Thompson 1992). For instance, views about mathematics teaching that
position teachers in control of students’ learning were shown by Stipek et al. (2001) to
be linked to classroom contexts that value extrinsic motivational strategies. In contrast,
autonomy-supportive contexts place greater value on intrinsic motivation derived from
task relevance, student confidence and students’ perceived satisfaction of their need for
autonomy (Assor 2012). Perceived autonomy is important for engagement; students
reporting greater autonomy in the classroom were shown by Hafen et al. (2012) to have
increasing levels of engagement in mathematics, whereas students with low autonomy
showed declines in their engagement as the academic year progressed.

13

“Sage on the stage” or “meddler in the middle”: shifting…

“Teacher as knower” to “teacher as learner”
The second main implied identity to emerge as a result of the intervention was “teacher
as learner” which occurred at two levels. First, “teacher as learner” is implied merely by
teachers’ preparedness to participate in the intervention and second, by teachers reposition-
ing themselves and their students in the mathematics classroom. While the intervention
was not purposefully designed to achieve the first outcome, it inherently positioned teach-
ers as learners and co-creators of knowledge through their interaction with the researchers
of the study and with other teacher participants when sharing ideas about, and strategies
to enhance student engagement in mathematics. The “teacher as learner” implied identity,
therefore, acted as a mediator for other repositioning practices such as those associated
with teacher as “co-creator of knowledge.”

In terms of the classroom, teachers initially positioned themselves as the “knowers” of
mathematics, which translated into practices that emphasized mathematics as a solitary
activity where individuals obtained precise answers and students who lacked mathemati-
cal confidence were positioned as procrastinators who employed “diversionary tactics” to
avoid learning it. Later, teachers repositioned themselves as “learners,” as evidenced by
their increased attention to student feedback as a mechanism for improving their teaching.
Much of their learning focused on how to strengthen students’ own mathematical identi-
ties and stemmed from their increased knowledge and awareness of strategies for address-
ing students’ mathematical engagement. Similarly, a study by Bobis et al. (2016) indicates
that professional learning can positively affect teachers’ capacities for improving student
engagement. In the current study, increased knowledge of student engagement gained from
the intervention resulted in teachers paying greater attention to emotional and cognitive
aspects of engagement. Intentions to increase these aspects of engagement were translated
into teaching and assessment practices that placed greater emphasis on student autonomy,
open–ended tasks, student feedback and building student confidence in mathematics.

Limitations and suggestions for future research
It is important to consider the limitations of this study. First, we did not consider teach-
ers’ actual classroom practices or explore students’ responses to the implementation of
such practices in this paper. In other words, little is known for certain about how mani-
festation of teachers’ views in practice will impact on students. Studies of teacher identity
that include longitudinal classroom observations are needed to determine if self-reported
changes in teachers’ views and practices (implied identities) have already been accepted
by teachers as their ideal being and/or perceptions of self. It is also important to explore
the mechanisms that lead to students’ repositioning through classroom interactions and
activities.

Furthermore, teachers’ narratives delivered in a group interview, may have been
influenced by the presence of researchers’ and or the views expressed by other teach-
ers in the focus group. For instance, during the group interviews, teachers regularly
expressed agreement or disagreement with the views voiced by others. On occa-
sions, teachers openly admitted that they had not previously considered the perspec-
tives expressed by others and that such exposure caused them to think differently. As
explained by Kitzinger (1995), a deliberate intention of focus group methodology
is to use group interactions to generate data. In this case, individual teachers were

13

J. Bobis et al.

encouraged to talk, comment on or question the experiences, practices and perspec-
tives of others and exchange narratives—the stories that illustrated their views, prac-
tices and student responses to such practices—so as to clearly explain how and why
they think and act in certain ways. Such level of detail may not have been so easily
obtained if individual interviews were employed. Indeed, it was through these interac-
tions, that teachers began to reflect upon their views and the views of others that gradu-
ally impacted upon their mathematical identities. While future research should explore
changes to individual teachers’ mathematical identities via one-to-one interviews and
individual case studies, investigations that involve groups of teachers interacting with
each other in complex ways will better inform educators about effective strategies for
teacher professional learning.

Conclusion

Teachers’ identities are difficult to investigate, partly due to the complexity of the identity
construct. Nevertheless, such investigations are important—effective teaching is depend-
ent upon teachers possessing well-developed identities (Grootenboer and Zevenbergen
2008). In the current study, we addressed this complexity by drawing on an implied iden-
tity framework to help structure and guide our inferences. Teachers’ experiences of the
intervention, as an identity resource, provided them with suggestions of teacher identity, or
what it means to be a teacher who is capable of supporting student engagement in math-
ematics. Self-reported changes in teachers’ perceptions were conceptualized as perceived
implied identities that mediate teachers’ repositioning of themselves in terms of their math-
ematics-related teacher identity and therefore have potential implications for new teacher
practices. For example, teachers’ adaptations of an autonomy-supportive learning perspec-
tive, led to their repositioning as facilitators of students’ learning and as co-creators of
knowledge. The use of strategies to foster student–student interactions is also indicative of
teachers repositioning their students as responsible agents for their own learning. Further
repositioning practices are evident in other categories of change including the adaptation of
new teaching resources and student assessment approaches, which in turn, positions teach-
ers as learners who are more likely to welcome and adapt new practices in the future.

An increased understanding of the mechanisms that support the development of healthy
identities in teachers is important because of the direct benefits to student learning. The
data showed how teachers’ experiences of the professional learning program, as an identity
resource, provided them with suggestions of what it means to be a teacher who is capa-
ble of supporting student engagement in the mathematics classroom. The findings indicate
that teachers’ repositioning of themselves as “facilitators,” “learners” and “co-creators of
knowledge” can have practical benefits such as improving students’ confidence, engage-
ment and level of autonomy for learning mathematics. The framework was also useful in
deepening our understanding of teachers’ learning—information that can be applied to
support their future development. In particular, the framework directed our attention to the
often-hidden experiences or resources (such as feedback from students and interactions
with other teachers) that enabled teachers to begin reshaping their identities to ones that
could more positively support student engagement in mathematics.

Acknowledgement  This research was funded by the Australian Research Council, in partnership with the
Catholic Schools Office, Broken Bay Diocese, Project ID LP110200596.

13

“Sage on the stage” or “meddler in the middle”: shifting…

References

Andersson, A. (2011). A "curling teacher" in mathematics education: Teacher identities and pedagogy
development. Mathematics Education Research Journal,23(4), 437–454. https​://doi.org/10.1007/s1339​
4-011-0025-0.

Assor, A. (2012). Allowing choice and nurturing an inner compass: Educational practices supporting
students’ need for autonomy. In S. L. Christenson, A. L. Reschly, & C. Wylie (Eds.), Handbook of
research on student engagement (pp. 421–439). New York: Springer.

Battey, D., & Franke, M. L. (2008). Transforming identities: Understanding teachers across professional
development and classroom practice. Teacher Education Quarterly,35(3), 127–149.

Black, L., Williams, J., Hernandez-Martinez, P., Davis, P., Pampaka, M., & Wake, G. (2010). Developing
a ‘leading identity’: The relationship between students’ mathematical identities and their career and
higher education aspirations. Educational Studies in Mathematics,73(1), 55–72.

Bobis J, Anderson J, Martin A, & Way J (2011) A model for mathematics instruction to enhance student
motivation and engagement. In D. Brahier (Ed.), Motivation and disposition: Pathways to learning
mathematics, National Council of Teachers of Mathematics Seventy-third Yearbook (Chap. 2, pp.
31–42). Reston, VA: NCTM.

Bobis, J., Way, J., Anderson, J., & Martin, A. (2016). Challenging teacher beliefs about student engagement
in mathematics. Journal of Mathematics Teacher Education,19, 33–55. https:​ //doi.org/10.1007/s1085​
7-015-9300-4.

Braun, V., & Clarke, V. (2006). Using thematic analysis in psychology. Qualitative Research in Psychol-
ogy,3(2), 77–101.

Chapman, O., & Heater, B. (2010). Understanding change through a high school mathematics teacher’s
journey to inquiry-based teaching. Journal of Mathematics Teacher Education,13(6), 445–458.

Clark, L., Badertscher, E., & Napp, C. (2013). African American mathematics teachers as agents in their
African American students’ mathematics identity formation. Teachers College Record,115(2), 1–36.

Cobb, P., & Hodge, L. (2011). Culture, identity, and equity in the mathematics classroom. In E. Yackel, A.
Sfard, P. Cobb, & K. Graemeijer (Eds.), A journey in mathematics education research (pp. 179–195).
Dordrecht: Springer.

Darragh, L. (2016). Identity research in mathematics education. Educational Studies in Mathematics,93(1),
19–33. https​://doi.org/10.1007/s1064​9-016-9696-5.

Davies, B., & Harré, R. (1999). Positioning and personhood. In R. Harré & L. Langenhove (Eds.), Position-
ing theory (pp. 32–52). Oxford: Blackwell.

Fang, Z. (1996). A review of research on teacher beliefs and practices. Educational Researcher,38(1),
47–65.

Fredricks, J., Blumenfeld, P., & Paris, A. (2004). School engagement: Potential of the concept, state of the
evidence. Review of Educational Research,74(1), 59–109.

Grootenboer, P., & Zevenbergen, R. (2008). Identity as a lens to understand learning mathematics: Develop-
ing a model. In M. Goos, R. Brown, & K. Makar (Eds.), Proceedings of the 31st annual conference of
the mathematics education research group of Australasia (pp. 243–249). Adelaide: MERGA.

Hafen, C., Allen, J., Mikami, A., Gregory, A., Hamre, B., & Pianta, R. (2012). The pivotal role of adoles-
cent autonomy in secondary school classrooms. Journal of Youth Adolescence,41, 245–255.

Hattie, J., & Timperley, H. (2007). The power of feedback. Review of Educational Research,77(1), 81–112.
Hough, S., O’Rode, N., Terman, N., & Weissglass, J. (2007). Using concept maps to assess change in

teachers’ understandings of algebra: A respectful approach. Journal of Mathematics Teacher Educa-
tion,10(1), 23–41. https​://doi.org/10.1007/s10857​ -007-9025-0.
Khosronejad, M., Reinmann, P., & Markauskaite, L. (2015). Implied identity: A conceptual framework for
exploring engineering professional identity in higher education. In IEEE conference frontiers in educa-
tion: Launching a new vision for engineering education, El Paso, TX. October 21–24.
Kitzinger, J. (1995). Qualitative research: Introducing focus groups. British Medical Journal,311(7000),
299–302.
Lave, J., & Wenger, E. (1991). Situated learning: Legitimate peripheral participation. Cambridge: Cam-
bridge University Press.
Leatham, K. R., & Hill, D. S. (2010). Exploring our complex math identities. Mathematics Teaching in the
Middle School,16(4), 224–231.
Lutovac, S., & Kaasila, R. (2018). Future directions in research on mathematics-related teacher identity.
International Journal of Science & Mathematics Education,16, 759–776.
Lutovac, S., & Kaasila, R. (2019). Methodological landscape in research on teacher identity in mathematics
education: A review. ZDM,51, 505–515.
Marshall, C., & Rossman, G. (2006). Designing qualitative research (4th ed.). London: Sage.

13

J. Bobis et al.
Martin, A. J. (2009). Motivation and engagement across the academic lifespan: A developmental construct

validity study of elementary school, high school, and university/college students. Educational and Psy-
chological Measurement,69, 794–824.
Noonan, J. (2018). An affinity for learning: Teacher identity and powerful professional development. Jour-
nal of Teacher Education. https​://doi.org/10.1177/002248​ 7118​788838​ .
Novak, J., & Caña, A. (2006). The origins of the concept mapping tool and the continuing evolution of the
tool. Information Visualisation Journal,5(3), 175–184.
Reay, D., & Wiliam, D. (1999). ‘I’ll be a nothing’: Structure, agency and the construction of identity
through assessment. British Educational Research Journal,25(3), 343–354.
Saunders, B., Sim, J., Kingstone, T., et  al. (2018). Saturation in qualitative research: Exploring its con-
ceptualization and operationalization. Quality and Quantity,52(4), 1893–1907. https​://doi.org/10.1007/
s11135​ -017-0574-8.
Sfard, A., & Prusak, A. (2005). Telling identities: In search of an analytic tool for investigating learning as a
culturally shaped activity. Educational Researcher,4, 14–22.
Skott, J. (2019). Changing experiences of being, becoming, and belonging: teachers’ professional identity
revisited. ZDM,51, 469–480. https​://doi.org/10.1007/s1185​8-018-1008-3.
Stipek, D., Givvin, K., Salmon, J., & MacGyvers, V. (2001). Teachers’ beliefs and practices related to math-
ematics instruction. Teaching and Teacher Education,17(2), 213–226.
Tait-McCutcheon, S., & Loveridge, J. (2016). Examining equity of opportunities for learning mathemat-
ics through positioning theory. Mathematics Education Research Journal,28, 327–348. https:​ //doi.
org/10.1007/s13394​ -016-0169-z.
Thompson, A. G. (1992). Teachers’ beliefs and conceptions: A synthesis of the research. In D. A. Grouws
(Ed.), Handbook of research on mathematics teaching and learning (pp. 127–146). New York:
Macmillan.
Publisher’s Note  Springer Nature remains neutral with regard to jurisdictional claims in published maps and
institutional affiliations.

13