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FROM 'FROWNS AND GROANS' TO 'ASTONISHMENT': INDICATORS OF A (DEVELOPING) MATHEMATICS TEACHERS IDENTITY

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Abstract

This paper reports on a research project based on designing and teaching in-service courses for Non-Specialist Teachers of Mathematics (NSTM). An NSTM is a school teacher who qualified to teach in a subject other than mathematics, yet teaches mathematics in secondary school (11-16 year old students). While the overall aim of our research was to describe what constitutes a trajectory towards a mathematics teacher identity for a NSTM, in this paper we explain how we sought indicators of a mathematics teacher identity. We do so by first describing how we adapted Wenger's notion of identity and advanced our 'Modes of Belonging' Mathematics Teacher Identity framework. After that we exemplify how we used our framework to locate indicators of mathematics teacher identity in the data from a narrative of NSTMs working on a particular piece of mathematics.
FROM ‘FROWNS AND GROANS’ TO ‘ASTONISHMENT’: INDICATORS OF
A (DEVELOPING) MATHEMATICS TEACHERS IDENTITY
Cosette Crisan and Melissa Rodd
UCL Institute of Education , University College London, UK
This paper reports on a research project based on designing and teaching in-service
courses for Non-Specialist Teachers of Mathematics (NSTM). An NSTM is a school
teacher who qualified to teach in a subject other than mathematics, yet teaches
mathematics in secondary school (11-16 year old students). While the overall aim of
our research was to describe what constitutes a trajectory towards a mathematics
teacher identity for a NSTM, in this paper we explain how we sought indicators of a
mathematics teacher identity. We do so by first describing how we adapted Wenger’s
notion of identity and advanced our ‘Modes of Belonging’ Mathematics Teacher
Identity framework. After that we exemplify how we used our framework to locate
indicators of mathematics teacher identity in the data from a narrative of NSTMs
working on a particular piece of mathematics.
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2018. In NNN (Eds.). Proceedings of the 42nd Conference of the International Group
for the Psychology of Mathematics Education, Vol. 1, pp. XX-YY. Umeå, Sweden:
PME.
Crisan, Rodd
INTRODUCTION
In England, the shortage of mathematics teachers is well-recognised with the demand
far outstripping supply. The latest available statistics on teacher supply gathered by
the Department for Education revealed that “79.8 per cent of mathematics lessons
taught to students in year groups 7-13 were taught by teachers with a relevant
qualification; a decrease from 82.7 per cent in 2013” and “75.8 per cent of teachers of
mathematics to year groups 7-13 held a relevant post A level qualification (down
from 77.6 per cent in 2013)” (Ross 2015, p. 13). The crisis in teacher supply means
that subjects like mathematics have to be covered by teachers who are not specialists
in these subjects.
The current on-going need for specialist mathematics teachers is not unique to
England, but well recognised at international level, too. In Eire, for example a
national survey found that 48% of teachers of mathematics at post-primary schools
were not mathematics qualified, while in Germany, research on ‘fachfremd’ (meaning
‘non-specialist’ in German) teachers of mathematics, includes Bosse’s (2014)
findings that these teachers enjoyed teaching mathematics even though they viewed
mathematics as if it was the mathematics of elementary school and they had had little
professional development in mathematics teaching. In the United States, the NSTMs
teachers are referred to as teaching ‘out-of-field’ (e.g., Ingersoll and Curran 2004),
while in Australia, Hobbs (2013) found that teachers who were ‘teaching across
specialisations’ (TAS) experienced discontinuities which can impact negatively on
their confidence and efficiency as a teacher of the new subject.
At the first Teaching Across Specialisation (TAS) Collective convened in August
2014, presentations from countries across the world indicated the wide spread of the
TAS phenomenon and a call for “Research is needed to establish the key features of
effective professional development that leads to such transformation in identity and
practice for out-of-field teachers.” (Hobbs and Törner 2014, p. 46) was launched.
OUR IN-SERVICE COURSE DESIGN
The design principle of our in-service mathematics courses for NSTMs was informed
both by our research that showed that learning to teach a subject without a
background in either content or teaching approaches requires focused re-training
(Crisan and Rodd 2014) and also an appreciation of the fact that it can actually be
quite difficult to teach out-of-field (du Plessis, Carroll and Gillies 2015; Hobbs
2013).
There are two key practices for teachers of mathematics: engaging in (school)
mathematics and being a teacher. In the case of our NSTMs, the latter is an
established practice, while the former is the practice they are developing. On our in-
service mathematics courses, we brokered these two key practices, enabling
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connections between them, through explicit teaching of school mathematics content
within discourses familiar to school teachers.
Our view that effective secondary mathematics teaching is founded on sound subject
knowledge, together with a thorough, interconnected, knowledge of the curriculum
and sympathetic understandings of students’ needs and interests informed the design
of our in-service courses. In these courses, there was emphasis on revisiting and
teaching school mathematics. This served not only to develop the NSTMs’ technical
fluency of some of the more challenging topics taught at different levels of school
education, but also to promote modes of mathematical enquiry such as generalisation,
abstraction, reasoning and proof. We also emphasised precision in mathematical
language, as well as recognition of conceptual structures within mathematics.
Discussions of pedagogical nature, such as common students’ misconceptions,
multiple representations of a concept, or different teaching approaches, were integral
to course delivery.
OUR RESEARCH INTEREST
A prompt for our research came from the NSTMs themselves. One of our NSTMs
(trained to teach humanities), who was applying for a promoted mathematics teacher
post, told us that she cried when she saw simultaneous equations and, when that topic
came up, always asked a colleague to teach it for her. On one hand, this teacher
wanted to be thought of as an expert mathematics teacher, while on the other hand,
she was not able either to fluently solve problems on this standard topic in the
mathematics curriculum within our class or to contemplate teaching the topic to her
students in school.
Such a disjunction confirmed our thinking that issues of identity were relevant to our
work with NSTs. We became particularly interested in how to make sense of our
NSTMs’ mathematics teacher identities formation and development over the duration
of the course. The research presented in this paper is part of a larger research study
rooted in our teaching of four cohorts of NSTMs over the past four years in London,
UK with an overall aim of answering our research question ‘What constitutes a
trajectory towards a mathematics teacher identity for a NSTMs on an in-service
course?’.
However, in order to answer this research questions, we first needed to be able to
recognise indicators of a mathematics teacher identity and in this paper we offer an
insight into how we engaged with theory and our data in seeking such indicators. We
thus proceed to firstly explain how we adapted Wenger’s notion of identity to
mathematics teacher identity, then we put forward a framework accounting for the
three interlinked ‘Modes of Belonging: engagement, imagination and alignment’
(Wenger, 1998, p. 174) in order to make sense of identity formation in the two key
practices of our NSTMs: learning mathematics and being a teacher and lastly, we
illustrate how we explicitly sought indicators of a mathematics teacher identity
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through our engagement with the framework and data from a narrative of NSTMs
working on a particular piece of mathematics.
ENGAGEMENT WITH WENGER’S PERSPECTIVE ON IDENTITY
While a variety of frameworks have been employed by researchers to describe
teachers’ identity development in mathematics teacher in-service courses (e.g.,
Boaler, 2001; Fennema & Nelson, 1997), Graven and Lerman (2003) argued that
Wenger’s (1998) social practice perspective of learning is a suitable framework to use
to analyse the process of becoming a teacher of mathematics.
Hence we engaged with Wenger’s “Social ecology of identity” (Wenger 1998, p. 190)
and adapted it and operationalised it as an analytic tool in the following way: the
general illustrative examples in the table on page 190 (ibid.) were replaced by
mathematics education-specific examples of indicators of aspects of identity, by
drawing on our own teaching experiences at secondary school level and expertise in
research informed teaching of prospective and practicing teachers. In this way,
Wenger’s notion of identity was adapted to mathematics teacher identity by
interpreting the three interlinked “Modes of Belonging: engagement, imagination and
alignment” (Wenger 1998, p. 174) in the two key practices of doing mathematics
(Identification with school mathematics) and being a teacher (Negotiability in
mathematics teaching) as indicated in Table 1 below.
OUR ‘MODES OF BELONGING’ MATHEMATICS TEACHER IDENTITY
FRAMEWORK
In our study, Identification with school mathematics refers to how the NSTMs
constructed identities as learners of mathematics during our in-service course.
Identification through engagement, imagination, and alignment refers to how the
NSTMs invested themselves in learning about and doing school mathematics topics,
how they constructed images about how students learn mathematics and how their
views converged towards an increasing connection with how the mathematics
teaching community views mathematics as a practice.
Negotiability in mathematics teaching through engagement, imagination, and
alignment refers to how the NSTMs negotiated their ways in the mathematics
teaching community, how the NSTMs constructed images of themselves as potential
specialist mathematics teachers and how their views converged towards an increasing
connection with the mathematics teaching community.
MATHEMATICS TEACHER IDENTITY
Identification
with (school) mathematics
Negotiability
in mathematics teaching
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Identities of
participation
Identities of
non-
participation
MOD
E
Identities of
participation
Identities of
non-
participation
e.g. Enjoy
thinking about
the mathematics
to be taught.
e.g. Avoid
mathematical
activity.
Engagement
e.g. Do in-service
courses; facilitate
students’ presenting
partial proofs
which are
discussed
e.g. Rely on text
book or on
downloaded
powerpoint
resources.
e.g. Find new
ideas in standard
topics.
e.g. Act as if there
was only ‘one
correct method’;
avoid thinking
about alternative
approaches.
Imagination
e.g. Share ideas,
applications, etc
about mathematics
with students;
imagine self as a
mathematics
teacher.
e.g. When being
asked by a
student ‘why are
we doing this?’
reply ‘you need
it for exam’.
e.g. Want to
understand why,
expect proof,
work detail.
e.g. Routinely get
answers to
mathematics
problems from
internet/elsewhere
; make errors.
Alignment
e.g. Discuss, with
students what
progression they
have made in
mathematics.
e.g. Only show
methods in
exam mark
scheme; want
certification of
maths
specialism
without
engagement.
Table 1: ‘Modes of Belonging’ Mathematics Teacher Identity framework
SEEKING INDICATORS OF A MATHEMATICS TEACHER IDENTITY
Data
Throughout the delivery of the four year-long in-service courses we collected
biographical data (routes into teaching, subject specialism of their teacher training,
teaching experience: of mathematics, if any, or of their subject specialism,
mathematics-related material (written diagnostic assessment of mathematics subject
knowledge and capacity to diagnose students’ errors/misconceptions; collection of
on-going mathematical work; analysed by us as mathematics education practitioners)
and written reflections (done during and at the end of their course and essay
assignments) from all participating teachers as an integral part of their respective
course. We also conducted interviews and did school observations) specifically for
the research.
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Data analysis
In the following we first explain how we interpreted and hence allocated data from a
narrative related to a particular piece of mathematics as indicators of Identities of
Participation in both Identification with (school) mathematics and Negotiability in
mathematics teaching in the table above.
Identification with (school mathematics): Identities of participation-Engagement: We
designed the course curriculum in order to give the NSTMs opportunities to
investigate number patterns in Pascal’s triangle, at the same time facilitating for
opportunities to identify for themselves patterns with which they were already
familiar. In each cohort there were expressions of astonishment that there was so
much mathematical content represented in ‘Pascal’s triangle’, for instance: “how did
he [‘Pascal’] manage to fit it all in such a simple format?” (Lech, session discussion).
When looking at the mathematics within the Pascal triangle, the teachers were
amazed to discover ‘in the triangle’ many mathematics topics they had previously
studied. “It’s all in there!” exclaimed one participants in disbelief.
The teachers experienced joy and surprise at noticing connections between different
topics, starting to see mathematics in a new light, more than just a set body of
independent knowledge and skills, clearly expressed by one other participant: “I
actually quite like that. I couldn’t grasp it and I can only just touch it but I really
like the fact that it's connected in different ways and we talk about…for example,
Pascal’s triangles here, there and then!”.
Identification with (school mathematics): Identities of participation-Alignment: More
advanced mathematics topics, such as binomial coefficients and combinatorial
identities were also introduced using Pascal’s triangle. However, when the expression
knNkn
k
n
k
i
n
rk
,,
1
1
was projected on the interactive white board, NSTMs’ responses expressed non-
participation–mathematics-engagement through their frowns, groans and comments
of ‘frightening’, ‘scary’, ‘illegible’.
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Looking at the geometrical pattern similar
to a hockey stick, they noticed the
numerical relationship within this hockey
stick, namely 1 + 4 + 10 + 20 + 35 = 70.
Table 2: Pascal’s triangle and the Hockey Stick Theorem
Our role as tutors on the course included helping the participants overcome the
negative affect by identifying particular cases of this identity in visual representations
inside Pascal’s triangle. We thus introduced the NSTMs to the Hockey Stick
Theorem, in its worded, then visual (by colouring in) and numerical representation at
first. The symbolic notation assigned to each number in Pascal’s triangle was then
introduced, and with some help, the blue hockey stick in Table 2 was represented
using the numerical relationship in the shaded hockey stick above could now be
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Crisan, Rodd
expressed as
4
8
4
7
3
6
2
5
1
4
0
3
. After that, using algebra and the helpful notation introduced earlier
(together with the sigma symbol to mean addition), the NSTMs were able to write
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down even the generalisation of this theorem, as
k
kn
i
in
k
i
1
0
. The NSTMs were very
satisfied with being able to ‘see’ the identity in all these formats, and Lech described
the experience as looking at a stereogram, where focussing on a 2D pattern one can
also see a hidden 3D image inside it.
Identification with (school mathematics): Identities of participation –Imagination:
Being able to make sense of an abstract mathematical expression, as above,
contributed to NSTMs’ identity of belonging to the mathematics community through
their participation in doing mathematics and alignment with the mathematics that
specialist teachers know and do. For example, when yet another emergence of
Pascal’s triangle got the whole class excited, we classified this as the NSTMs
participating mathematically by noticing connections between different mathematical
topics, and also as an instance of “joy and satisfaction in undertaking mathematical
practices” (Grootenboer & Zvenberger, 2008, p. 246) that helped to create a positive
group atmosphere, important in building a community of practice within the class of
NSTMs and helping, through development of positive affect, the NSTMs participate
in other communities of mathematics.
DISCUSSION
This was the first stage in our data analysis, namely explicitly seeking indicators of
identity as conceptualised from interpreting Wenger’s structure and adapted to
mathematics teacher identity by us.
Data from the mathematical episode has been classified in terms of indicating
participation in the three different ‘Modes of Belonging’ – engagement, imagination
and alignment’ via the framework tool we put forward in Table 1. Having provided
examples of how data were allocated to some of the cells in the Table 1, in the
conference presentation we engage further with our framework and we will
exemplify how data collected from one NSTM participant was analysed and a
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Crisan, Rodd
narrative produced, once allocated to the table. In our research, a similar exercise
was applied to all the data we collected from our NSTMs, and this enabled us to
produce a narrative and describe our NSTMs positioning on (different) trajectories
towards a mathematics teachers’ identity.
By considering practices central to being a secondary mathematics teacher, namely
Identification with school mathematics and Negotiability in mathematics teaching, we
have offered a way of thinking about mathematics teacher development. In Wenger’s
terms, our NSTMs were newcomers to the mathematics teaching community and as
such they negotiated their trajectories towards becoming a mathematics teachers in
their own ways and their individual ‘Table 1’s looked different from each other’s and
were different at different points in time.
Using this framework at different points in an in-service course provided a way to
evidence mathematics teacher identities emerge and develop. Graven (2005) points to
identity transformation seldom being the focus of in-service courses and the
researcher proposes that identity interacts with teachers learning and thus should be a
focus of the design and provision of any in-service training. As such, our
contribution to knowledge is in drawing attention to how mathematical knowledge is
realised within non-specialist teachers’ mathematics teacher identity and in
developing understandings of non-specialist teacher experience on an in-service
course.
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