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Mathematics teacher educators’ critical colleagueshipSuela Kacerja, Rune Herheim

To cite this version:Suela Kacerja, Rune Herheim. Mathematics teacher educators’ critical colleagueship. EleventhCongress of the European Society for Research in Mathematics Education, Utrecht University, Feb2019, Utrecht, Netherlands. �hal-02422559�

Mathematics teacher educators’ critical colleagueship

Suela Kacerja1 and Rune Herheim

1

1Western Norway University of Applied Sciences, Norway; skac@hvl.no; rher@hvl.no

In this paper, we apply the ideas of Lord (1994) about critical colleagueship to understand how

mathematics teacher educators (MTEs) can work together to become more critical in their teaching

practices. There is relatively little research on MTEs’ learning and development from a critical

perspective. Our study examines a group of MTEs working together to develop novel teaching and do

research about initiating critical discussions. During two meetings, the MTEs discussed their

different perspectives after using indices such as the Body Mass Index (BMI) in teaching. Identified

examples of Lord’s elements were a willingness to seek and try out promising ideas, and being open

to share perspectives and ask for arguments. Such collaboration supports reflections for developing

teaching and research.

Keywords: Critical colleagueship, mathematics teacher educators, reflections.

Introduction and previous research

There are several studies concerning mathematics teachers’ knowledge for teaching (e.g. Ball,

Thames, & Phelps, 2008; Rowland, Hucksteps, & Thwait, 2005), as well as mathematics education

courses designed for mathematics teachers’ professional development. Zaslavsky and Leikin (2004)

pointed to the lack of research on becoming a MTE and lack of formal training programs. “Mostly,

teacher-educators are ‘self-made’” (Zaslavsky, 2008, p. 94). Some studies about MTEs focus on the

mathematical knowledge for teaching mathematics teachers (Zopf, 2010), and MTEs’ practices in

providing professional development (Kuzle & Biehler, 2015). In a historical overview, Jaworski

(2008), in line with Zaslavsky & Leikin (2004), found that a very small number of studies reflect on

the MTE’s learning “from engaging in teacher education, through reflecting on their own practice, or

through research into the programs they design and lead” (p. 3). Our study is a contribution to narrow

this gap by focusing on how MTEs can collaborate to become more critical about their teaching

practices.

According to Zaslavsky and Leikin’s (2004) model of MTEs’ professional development, MTEs learn

through learning (facilitated by an experienced MTE) and through teaching (to mathematics

teachers), while collaborating with other colleagues of similar or differing expertise. The need for

MTEs to reflect, individually and collectively, on different aspects of their own practice and

development is pinpointed as important for their learning (e.g. Tzur, 2001; Jaworski, 2008; Zaslavsky

& Leikin, 2004; Garcia, Sanchez, & Escudero, 2007). Individual reflections upon the different stages

of becoming an MTE include reflections on learning mathematics, learning to teach it, learning to

educate mathematics teachers, and learning to mentor educators (Tzur, 2001). In collective

reflections between colleagues when preparing and teaching different courses, practices such as

sharing experiences, reading and conducting research, and continuous efforts to improve courses

(Roth McDuffie, Drake, & Herbel-Eisenmann, 2008), as well as MTEs adopting theoretical

perspectives to examine their teaching practices (Garcia et al., 2007), can influence MTEs’

development.

Zaslavsky (2008) argued that one of the identified practices, from which MTEs can learn, is the

choice and design of tasks and resources by which students can learn specific content or ways of

teaching. In our research group, we collaborate on identifying and implementing new teaching ideas

that can promote critical mathematical discussions (in line with Skovsmose, 1994) amongst

pre-service and in-service teachers. We collectively reflect upon the implementation of these ideas in

our own teaching. One idea we aim to investigate in our project is the use of indices as mathematical

models for in-service teachers to experience initiating and developing critical discussions about the

role of mathematics in society. Indices, such as the BMI, have proved to be fruitful entry points to

such discussions (see Kacerja et al., 2017). Reflecting collectively as MTEs upon our own practice

can help us learn more about being teacher educators (Roth McDuffie et al., 2008; Zaslavsky &

Leikin, 2004).

In the data presented in this paper, we reflect as a group upon our experiences with critical discussions

after two colleagues had tried out a task about the BMI with in-service teachers. We investigate how

we communicate in the group and how we invite and present ideas and perspectives from a critical

colleagueship perspective (Lord, 1994). Critical colleagueship is a particular type of collegiality and

is elaborated on in the next section. The question we pose in this paper is: What aspects of critical

colleagueship can mathematics teacher educators’ collaboration bring about? By focusing on the

critical colleagueship aspects by Lord (1994), we explore how such a collaboration can support our

work as MTEs.

Critical colleagueship among mathematics teacher educators

Lord (1994) emphasised professional development of teachers and discussed critical colleagueship as

one kind of colleagueship to support teachers’ reflections. Colleagues sharing interests and

experiences, being open and respectful, willing to try out new ideas and to be critical, are crucial

conditions for such a colleagueship. The main difference between critical colleagueship and other

kinds of collective reflections is the support of a “critical stance toward teaching” (p. 192). The

critical stance in our study means we do not only share our ideas with colleagues, but we also ask for,

and articulate, the assumptions behind these ideas.

Lord (1994, pp. 192–193) identified six characteristics of critical colleagueship:

1. Creating and sustaining productive disequilibrium through self-reflection, collegial dialogue,

and on-going critique; 2. Embracing fundamental intellectual virtues (e.g. openness to new ideas,

willingness to reject weak practices or flimsy reasoning, accepting responsibility for acquiring and

using relevant information, willingness to seek out the best ideas, greater reliance on organized

and deliberate investigations, assuming collective responsibility for creating a professional record

of teachers’ research and experimentation); 3. Increasing the capacity for empathetic

understanding; 4. Developing and honing the skills and attributes associated with negotiation,

improved communication, and the resolution of competing interests; 5. Increasing teachers’

comfort with high levels of ambiguity and uncertainty; and 6. Achieving collective generativity.

The characteristics for which a colleagueship can be considered critical, according to Lord, have to do

with a sense of responsibility for seeking improvement and accepting that the best solution is not yet

achieved. It requires participants to be comfortable with uncertainty and to make efforts to develop

and accept new ideas by self-reflection and on-going critique. Differences are seen as driving forces

that can facilitate a productive disequilibrium. Our joint interest as MTEs is to learn more about ways

of fostering critical discussions in teacher education about the role of mathematics in society. The

ideas within critical colleagueship support such group reflections.This is the reason why critical

colleagueship is chosen as a framework for analysing the discussions.

While Lord (1994) defined critical colleagueship for groups of teachers reflecting together, Males,

Otten, and Herbel-Eisenmann (2010) used Lord’s framework to study the collegiality of a group of

mathematics teachers and researchers. They identified challenging interactions in which participants

asked questions to push for in-depth reflections, and located elements of Lord’s intellectual virtues in

those interactions. In our paper, we apply critical colleagueship within a group of MTEs. We extend

critical colleagueship as one way of thinking about professional development of MTEs, by reflecting

upon our own discussions. In line with Males et al., we focus on the discussions when different

perspectives about practice come into play. It is in these situations that it becomes more likely for

MTEs to argue for their ideas and invite the colleagues to share their perspectives.

The study, participants, and data analysis

The study focuses on a group of seven teacher educators, including the two authors, collaborating on

developing teaching and research. In this paper, we use data from two meetings in which teaching

about indices and critical discussions was focused upon. All seven of us took part in the first meeting

shortly after TE11 and MTE2 had collaborated on a 3-hour workshop on indices for in-service

teachers. This was part of a course in Numeracy across the curriculum. The in-service teachers

discussed the BMI task for 60 minutes in two groups of six persons. They examined the BMI’s

mathematical components and formula, its use in society, and the appropriateness of using indices in

school teaching. In the next semester, we had a second meeting and continued to discuss ideas for

developing our teaching about indices as part of our project. In addition, we discussed a research

paper we had written about stimulating critical discussions in mathematics where data from in-service

teachers’ discussions were analysed (see Kacerja et al., 2017).

The two meetings were audiotaped and transcribed. In line with Lord (1994), we, the authors,

investigate discussions when different perspectives in teaching and research come into the fore as

driving forces. An example of this is when one MTE argued that we should teach a kind of scheme for

dissecting indices, while one of the others thought it was important for in-service teachers to be free to

investigate. The first author identified such interactions from both meetings, and both authors

examined them from a critical colleagueship perspective. We also looked for the use of words such as

“yes”, “maybe, “but” etc., in order to identify the different characteristics of the critical

colleagueship. The examples presented in the following are representative for the situations where

different perspectives occurred.

1 TE1 is a teacher educator within social science, while the six others are within mathematics education, thus MTE.

Different perspectives

As previously argued, we identified discussions showing different perspectives on two related topics

– the ways to teach critical skills, and the mathematical level in the discussions. We now analyse

aspects of critical colleagueship in the chosen utterances.

Different perspectives on how to teach critical skills

The first meeting begins with TE1 describing the teaching and his experiences from the workshop.

The goal of the lesson was to create an awareness about the importance of being critical to the use and

misuse of numbers in society. While TE1 and MTE2 agree upon the goal, differences came to the fore

about how to achieve that goal. TE1 then said, “It was actually too little time and too much material”

in the teaching session before the group discussions, and continued by saying:

TE1: I wish I had more time, and maybe do something more, dissect an index to really

give them a useful example, a template in principle, a recipe, how one can approach

an index. How one can take the pieces apart and see what those mean, what the

different numbers mean, the different variables in an index.

TE1 explains what he would do differently next time to improve his teaching. He is critical to his

planning of “too much material” and shows by this self-reflection a characteristic of critical

colleagueship. For him, a way to achieve the goal could be to show the teachers an example of how

one could criticise an index, by dissecting it and working with its different components to get a sense

of the numbers. The use of “maybe” indicates an openness in his reflection. This is strengthened by

the wording “I wish”, a choice of words that makes it possible to characterize the whole utterance as a

“what if …?” approach, a focus towards what can be possible to do. A few utterances later, MTE2

presents a different opinion:

MTE2: Yes, and as you said TE1, one can look at this from two sides; how much material

should they be presented with beforehand for a discussion like this, and how much

should they not [be presented with] …

MTE2 starts with a “yes” and acknowledges TE1’s point of view. However, she also wants to bring

into attention another point of view. There is a dilemma about how much guidance the in-service

teachers should get before they start exploring the problem themselves. MTE2 does not comment

upon the goal of the lesson, she is only trying to look at an alternative way for achieving it – she

creates some disequilibrium. From Lord´s (1994) perspective, disequilibrium provides participants

with opportunities to reflect upon other´s ideas and bring their own arguments into the discussion.

MTE2 elaborates afterwards on her argument with a “because” and exemplifies with an episode from

the in-service teachers’ group discussions in which one teacher was fascinated about how much she

had learned. For MTE2, this is an argument that supports the idea of giving teachers the opportunity

to explore the use of mathematics, without necessarily having ready-made schemas, as TE1

suggested. MTE2 argues that the dialogue the in-service teachers had is important for learning to

explore, while a recipe “can limit the dialogue and they [the in-service teachers] can become

preoccupied with doing it the same way [as the MTE]”. The argument concerns potential negative

effects from presenting a recipe; it could hinder the teachers’ explorations and make them adhere too

strictly to the TE’s schema. MTE2 provides arguments to support her idea of giving the teachers some

space to explore the problem themselves, emphasizing dialogue, wondering and exploration. She

presents ideas and counter-arguments, thinks aloud and refers to examples. MTE2 sets some

standards for the level of reasoning required for the group discussions to be fruitful, in line with

Lord’s (1994) emphasis on negotiation and improved communication.

In all of the utterances presented above, as in other cases of disequilibrium in our data, the

participants start their utterance by acknowledging the colleague’s point of view using phrases like

“yes”, and “agree”, and then introduce an alternative view starting by “because” and supported by

examples. Acknowledging colleagues’ ideas and arguments relates to Lord’s focus on “the capacity

for empathetic understanding” (1994, p. 192). By using phrases such as “maybe” and “you can look at

it from two sides”, TE1 and MTE2 apply some fundamental intellectual virtues in their discussions by

opening up for other opinions. They acknowledge the others’ views and seek the best ideas by

looking at the topic from different points of view. TE1 and MTE2 use classroom examples to support

their arguments by using relevant information. This is typical for the participants in both meetings,

and in line with previous research (Males et al., 2010). The MTEs explore together how to initiate

critical discussions in their teaching without having the answers available. They are, as Lord (1994)

put it, coping with uncertainties and ambiguities that TE1 and MTE2 reflect upon.

Different perspectives on the mathematical level in the discussions

Exploring the mathematics of the BMI, and the in-service teachers’ mathematical competence to do

that, also generated different perspectives. In the first meeting, MTE4 stated that the teachers did not

explore in depth the mathematics behind the chosen index. Similarly, TE1 pointed to the lack of

mathematical competence as a barrier that hindered the teachers in doing so. He supported his

argument by referring to what the teachers expressed during the discussions. This fits with his earlier

reflections about how he would organize the teaching differently next time to help teachers overcome

this barrier, showing again signs of self-reflection for improving his teaching.

Another disequilibrium occurs in the second meeting, when discussing an article in which we all

looked at the competence showed by in-service teachers when working with the BMI task. Similarly

to TE1 and MTE4, MTE5 thinks there were “relatively little mathematical discussions”. MTE6 asks

MTE5 what mathematical discussions are in her opinion. MTE5 answers that she is thinking about

discussing mathematical concepts. MTE6 then adds, “Yes … but there is also a broader

understanding of mathematics, of mathematical discussions, to discuss mathematics in use and its

role in society”. As MTE2 also did earlier, MTE6 accepts MTE5’s perspective about what she

regards as mathematical discussions with a “yes”, but he also introduces his perspective by saying

“but there is also”. MTE6 presents his view by arguing, in line with a critical mathematics education

approach (Skovsmose, 1994), that mathematical competence includes being able to use mathematics

and evaluate its use in different contexts. Mathematics goes beyond discussing mathematical

concepts; it also involves a broader perspective of its use in society. This is reflected in the goal of the

lesson formulated by TE1. MTE6 supports his argument by focusing on competences the teachers

showed in their discussions from this extended perspective on mathematics by saying: “They showed

many good reflections connected to challenges about indices, and about indices in a school context.”

It is possible to identify several fundamental intellectual virtues in this discussion. The MTEs are

invited to share ideas and arguments, problems are investigated from different viewpoints, and

viewpoints are elaborated upon with several arguments before deciding the next step. In the two

meetings, the MTEs clarify their expectations about in-service teachers showing mathematical

competence, but also the competence to evaluate and criticize how mathematics is used in society.

They search for better ideas to improve the teaching about critical discussions, as Lord (1994)

emphasized, as they continue to reflect about the tasks. At the end of the second meeting, when MTE4

wonders if she should ask more targeted mathematics questions in the next teaching session in order

to guide the teachers to thoroughly explore mathematics, MTE2 argues:

MTE2: One thing is to go more in depth into the mathematics [of the index], if they are able

to do that. But we would also like them to stay there and understand that there is

something here that could be necessary for them to understand. They see its

meaning …

MTE2 acknowledges again the other colleagues’ idea of more in-depth exploring of the mathematics.

She then continues in line with her earlier ideas about giving the teachers the time and possibility to

discover things by themselves, to get the feeling that they need to learn something. She connects this

to the meaning the teachers themselves would give to an index, and to the mathematics of the index.

So far, the MTEs have shared their ideas, been open-minded for other arguments and ways of doing,

argued for their views, and presented alternative points of view. One could then ask what effect this

exchange of ideas has on the MTEs and their collaboration. By the end of the meeting, MTE2

continues with a proposal for further developing the task about critical mathematics and BMI:

MTE2: As you [MTE4] mentioned with proportionality … Is it possible to design a

teaching session where one first goes through the main mathematical ideas of the

index? But not specify the index. Then talk [teach] about inverse proportionality

without saying that it is the BMI we are talking about …

MTE2 starts with taking into account MTE4’s earlier idea of proportionality. By raising a question,

“is it possible to design …?”, MTE2 tries to negotiate with MTE4 and the others to find a teaching

approach that takes into consideration many of the colleagues’ comments. She proposes one way to

organize the session by starting with some teaching about the mathematical concepts of BMI, such as

inverse proportionality, but without giving any scheme for how to do it. By so doing, the in-service

teachers would be given some mathematical foundation when exploring the mathematics of the

formula, as MTE5, TE1, and MTE4 called for. At the same time, MTE2 takes care of her own idea of

giving the teachers the freedom to explore by adding “then they [the in-service teachers] could get the

BMI formula and see if they connect it [to inverse proportionality]. We haven’t tried that”. MTE5

agrees with MTE2 by saying “I thought the same … teaching about proportionality independently of

the index”. MTE6 also supports MTE2’s idea by saying, “start the teaching with some mathematics”.

Everyone agrees that it is a good idea to try out MTE2’s proposal. The negotiation highlights the

importance of giving the in-service teachers more support to develop and show mathematical

competence, while at the same time giving them freedom to show their competence on reflecting

upon the mathematics’ use in society. The MTEs have together come to an idea while trying to avoid

the pitfalls expressed earlier by the colleagues, a kind of “collective generativity” (Lord, 1994). The

best agreed upon solution in this round of discussions is to teach some mathematics and see if this will

help the teachers in their discussion of the index.

Conclusions

In this paper, we have focused on identifying aspects of critical colleagueship in mathematics teacher

educators’ collaboration on developing teaching about critical discussions. We singled out utterances

when colleagues had different perspectives because it is when we disagree that we ask each other for

more arguments. This is a vital element of the critical colleagueship perspective as it creates

conditions for the participants to dig into the assumptions behind their ideas, and thus adopt a critical

stance in terms of Lord’s framework. We, MTEs, are also in a position to reflect better upon our own

views when challenged to argue and exemplify the teaching, and modify it for better results. In this

aspect, our results fit with findings from Males et al. (2010).

During the collaboration to develop our teaching, there were particularly two aspects that generated

different perspectives. One aspect is about the way of organizing such teaching and the amount of

guidance to give teachers, and the other concerns expectations about the level of mathematics in the

teachers’ discussions. We, MTEs, discuss different perspectives and consider them, showing several

elements of Lord’s framework, especially fundamental intellectual virtues such as openness to new

ideas, respect, and seeking better solutions. We support our points of view with arguments from

classroom examples, as MTE2 and TE1 did, and from theoretical ideas influenced by critical

mathematics education (see Skovsmose, 1994), as MTE2 and MTE6 did. Examining our practice

with theoretical lenses, as with the critical perspective lenses, is a way for us to develop as MTEs

(Garcia et al., 2007).

At the end, the agreed solution covers some of the challenges discussed in the two meetings. The

engagement brings about some collective generativity. The solution indicates that we, MTEs, value

the development of teachers’ mathematical competency, but also the importance of the freedom to

develop their critical competence by not giving them too much guidance. In this way, we move

forward in developing our understanding of what critical mathematical discussions are and how to

support them in our teaching. As Roth McDuffie et al. (2008) pointed out, sharing experiences,

looking for improvement in our practice, and doing research together, facilitate such development. It

can be concluded that we, MTEs, enrich our own views by listening to our colleagues’ arguments and

by trying to make sense of their reasoning, when we collectively reflect upon and analyse our work.

Given the very few possibilities for MTEs to develop their knowledge and skills, and given that

collaboration of MTEs about teaching and research is a common practice, it is important to study

what and how such collaborations can support the MTEs’ work, as we have done in our study. Lord

(1994) discussed critical colleagueship as a way to support teachers’ reflections. In this study, the

concepts in Lord’s elements of critical colleagueship helped us identify and discuss the potential such

collaboration has in supporting MTEs’ reflections.

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