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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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Last names of authors in the order as on the paper

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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Crisan, Rodd

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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Crisan, Rodd

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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Last names of authors in the order as on the paper

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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