A functional view on language: a methodology for mathematics education to study shifts in prospective teachers’ discursive patterns

This paper set out to contribute to mathematics education research by elaborating on a methodology developed during a study, trying to understand, view and follow shifts in prospective teachers' discursive patterns. The methodology aims to illustrate and describe how prospective teachers adapt to the context of teaching through a flexible process. This flexible process is then described in the result as a narrative. It is argued that the methodology can be used in relation to different theoretical directions, such as research about beliefs, knowledge, or identity. Another contribution is that the methodology presented gives insights into bridging the gap between different analytical levels, micro and macro. With a theoretical foundation in ‘Cultural Worlds’ [Holland, D., Skinner, D., Lachicotte, W., & Cain, C. (1998). Identity and agency in cultural worlds. Harvard University Press.] the Social Semiotic approach of Systemic Functional Linguistics, SFL [Halliday, M., & Hasan, R. (1989). Language, context, and text: Aspects of language in a social-semiotic perspective (2nd ed.). Oxford University Press] is used as a methodological tool. SFL offers a toolkit that allows the analysis of meaning at the clause level to uncover how and why a speaker produces a particular wording rather than any other in a specific social practice. The paper aims to illustrate and describe how to go beyond findings in the micro-analysis and then present the result as a narrative case.


Introduction and background
This paper set out to contribute to mathematics education research on beliefs, knowledge and identity by illustrating a methodology evolving from a PhD study (Ebbelind, 2020) trying to understand, view and follow shifts in prospective teachers' discursive patterns as teachers-to-be. The thesis aimed to contribute insights into how, or even if, experience from teacher education and other relevant past and present social practices and figured worlds matter for prospective generalist teachers' imaginings of themselves as primary mathematics teachers-to-be. These discursive patterns evolve when participating in situations during their teacher education experience. In this paper, discursive relates to how persons transform their experience into the present situation and transform utterances from one situation to another (Fairclough, 2010). I foreground the prospective teacher in research by focusing on and describing the pre-reified processes (Skott, 2015). A reification process, in this case, can be described as how researchers consider or represent something abstract, like identity, knowledge and beliefs, as a concrete object, where researchers give definite content and form to the concept or idea. Like Skott (2015), I am interested in describing moments preceding this reification process by following shifts in discursive patterns in prospective teachers' daily lives.
The first contribution to mathematics education research relates to Hill et al. (2007) selfcritical point that no quantitative instrument responds entirely to practice and Thompson's (1992) objections to the shift in research focus from beliefs as dynamic to more stable. Hill et al. (2007) suggest Discourse Analysis (DA) as an alternative research approach, but the demands of fine-grained discursive frameworks are complicated and time-consuming from their point of view. Frameworks used in such research need to uncover fine-grained discursive patterns. However, even though time-consuming, I argue that fine-grained discursive frameworks, in any form, are important if interested in human lived experience and how that contributes to how we become and evolve as prospective teachers. DA and its methodology complement the existing body of research by closely investigating human lived experience as beliefs, knowledge and identity derive from this source (Skott, 2018).
The second contribution of this paper to the field of mathematics education is that the methodology presented gives insights into how to bridge the gap between different analytical levels, a problem often addressed in discursive and mathematics education research (Jarowskij & Potari, 2009). The bridging problem also relates to how to present research results in various case studies. In this paper, how the methodology leads to a result section presented as a narrative story.

Background for the chosen methodological approach
Through the lens of prospective teachers' discursive patterns, I will try to show that it is possible to consider individual, social, historical, political and/or cultural participation in situations related to prospective teachers experience as teachers-to-be. One can do this by focusing on the micro-interaction related to what is said, the meso-institutional related to how situations shape meaning and the macro-social that relates to the circulation of cultural meaning shaping how we view the world (Anderson & Holloway, 2020). However, as already pointed out, the analytical connection between these levels is a central problem (Jarowskij & Potari, 2009). How in research bridge, theoretically and/or methodologically, the gap between these different levels? Since the focus of this paper is purely methodological, presenting and describing the methodology in Ebbelind (2020), I will address the bridging problem from that perspective and not as a theoretical discussion. However, some theoretical positioning and statements will follow below to understand the research perspective behind the different choices done.
With a theoretical foundation in Holland et al. (1998) 'Cultural Worlds' and Social Semiotics (Halliday & Hasan, 1989), I assume that when researching micro-interaction, one gets access to the meso-institutional and the macro-social. One way to approach and explain this assumption is through Wetherell's (2001) three principles for any discursive perspectives.
By assuming a discursive perspective, one view language as emergent and situated, the first principle (Wetherell, 2001). Any person's experience has evolved over an extended period and is intertextually stratified in a plurality of relations. The notion of intertextuality is commonly used within different fields of DA (Allen, 2000;Gee, 2011), which highlights how language lies on the borderline between oneself and others. Language-inuse is, therefore, the object of study (Wetherell, 2001), here regarded as the text. Text is then the outcome or an instance of language used in a discursive moment (Halliday, 1978). By exploring intertextual relations, a so-called discursive pattern can emerge. Interpret those discursive patterns and discovering their meanings is one of the main aims of DA (Allen, 2000).
While the first principle relates to the text level, the second relates to social practices. Discursive patterns are functional in every situation shaping the social practice one attends. However, the social practices are also dynamically shaping the text. Halliday (1978) describes this as '[t]he context plays a part in determining what we say; and what we say plays a part in determining the context' (s. 3). By analysing the text and assuming a discursive approach, one recognizes that the first principal shapes social practices and vice versa.
However, the co-construction of social practices does not stop there. The text and social practice contribute to and are contributed by the circulation of cultural meaning, the third assumption (Wetherell, 2001). Holland and Eisenhart (1990) highlighted culture as a medium used as material when interpreting the world. They found that it was not the local culture that contributed to life decisions but rather the global culture and circulated meanings of what it meant to be something, in this case, a mathematics teacher, that mattered to the individual. The circulation of cultural meaning affects both the text and social practice and can be identified and interpreted by looking into the text in a fine-grained discursive manner.
The central question related to the two contributions highlighted in this paper is how to bridge the divide between different analytical levels. Micro-analysis as getting access to fine-grained details of the pre-reified process preceding the reification of beliefs, knowledge, or identity in the empirical material and macro-analysis as analysing large-scale patterns, including participation in social practices and cultural meaning.

Aim of the paper
From now on, I will closely describe how I tried to develop meaningful ways of analysing and interpreting the language-in-use of different actors and illustrate and describe how I tried to go beyond findings in the micro-analysis and apply a conceptual framework and then present the result as a narrative story. In doing so, empirical material, selected data and analytical constructs from the tale of Lisa (Ebbelind, 2020) will be used. Lisa was chosen because there is a lack of research related to prospective primary teachers with a particular interest in mathematics, mathematics teaching and mathematics learning and who regard themselves as knowledgeable/proficient in mathematics. She was regarded as a critical case (Flyvbjerg, 2006) in Ebbelind (2020).

Theoretical direction
The theoretical direction is the chosen view used to understand and proceed with the research process, which means that the theoretical perspective has guided the logic within the study as a sort of main criterion for the process of research (Gee, 2011). The theoretical direction includes pre-defined concepts and assumptions that guide ideas and the research design (Eisenhart, 1991). Thus, theory sets standards for the methodology and the result section, as they are intimately connected.

Patterns of participation
With interest in mathematics prospective teacher's professional development, I turned my focus to the conceptual framework of Patterns of Participation (PoP), professional development as an adaptation to the context of teaching through a process of flexibility and autonomy (Hošpesová et al., 2018). PoP 'seeks to understand how a [prospective] teacher's interpretations of and contributions to immediate social interaction relate dynamically to her prior engagement in a range of other social practices' (Skott, 2013, p. 549). When seeking to understand this process, Lerman (2013) emphasizes that the researcher, through the structuring of the empirical material, needs to elicit 'the teacher's interpretation of what she does in classrooms and how that relates to her prior engagement in other social practices (p. 625)'. The above perspective views learning as a form of participation in social practices. When examining professional development, it considers social, cultural and historical systems, past experiences and current participation (Lerman, 2013;Skott, 2018).
Drawing on Symbolic Interactionism and Dialogism, the theoretical direction views humans as actors and reactors in situations and positions meaning as something one engages in when experiencing things in the situation. Humans respond to the situation by interacting with others and the self and by interpreting the role of others (Skott, 2015). Social practices are stratified (Wenger, 1998), and in my interpretation and conceptualization, the PoP framework recognizes that social practices are ordered and stratified across time and space. Discursive engagement is inextricably related to the social practice where it is historically and presently created (Wetherell, 2001).

Illustrating the methodology
In this paper, the methodology can be understood as methods used for creating empirical material and the specific reasons for using such techniques. Methodology concerns aspects of how the created empirical material was transformed into data material and how the selection of data material was used when preparing to narrate the result (Skott, 2018) in a narrative way. The main reason for narratively illustrating the result was to try to understand and present to the reader a perspective of human lived experience (Kaasila, 2007). In Lincoln and Denzin (2000) words, I wanted to provide the reader 'with some powerful prepositional, tacit, intuitive, emotional, historical, poetic, and empathic experience of the Other via the text (p. 1058)' I write.
Lerman (2013) describes PoP as a dynamic research process that elicits a person's interpretation of what they do in social settings and how that relates to prior engagement in other social practices. This inspired me to develop a methodology that enabled me to systematically analyse language-in-use as unmotivated looking (Coles, 2015) to minimize research bias. The process of transforming empirical material into data material was done in steps in Ebbelind (2020). First, the created empirical material was structured and organized, micro-analysis, using a methodological tool, Systemic Functional Linguistics (presented later). Second, the framework, PoP, was used when generating data material for the case and involved selecting relevant information that highlighted discursive patterns in prospective teachers' discursive participation.

Ethnography -the research design
Cultural identity (Holland et al., 1998) directs attention to people's day-to-day lives. To be interested in prospective teachers' day-to-day lives implies methods for gathering information from which it is possible to observe immediate social interaction. One way, among others, is an ethnographic approach (Prus, 1996) that sets out to understand and interpret the meaning people attach to different experiences and how they are related. Holland and Lave (2009) define ethnography as focusing on persons while attending local practices and long-term institutionalized struggle. Although traditional ethnography has been more descriptive, theory plays an essential role in ethnography today. Theoretical insights inform the interpretation of the created empirical material and selected data (Wilson & Anmole, 2009).

Methods for creating empirical material
The different methods for creating empirical material in this study were interviews, field notes from observations, audio recordings from lectures, seminars and working sessions and collected documents concerning the prospective teacher and the teacher education programme. These must together meet the requirements of capturing human meaningmaking. The prospective teachers were interviewed formally on a regular basis during the whole of two years and six months, and conversations were conducted mainly during the internship and the 30 ECTS credits course in mathematics education. All the interviews were transcribed after they were conducted. Österholm (2011) argues that using a methodological tool for structuring the information is essential when using the PoP framework. Halliday's Systemic Functional Linguistics, SFL, was used to structure the information. Through functional analysis, SFL aims to uncover how and why a speaker produces a particular wording rather than any other in a specific social practice. The answer is found in the text itself and through the relationship between it and other texts within the text (Fairclough, 1992). For the study of Ebbelind (2020), SFL offered a toolkit that made possible the analysis of meaning at the clause level to reveal how language choices contribute to a text's overall meaning and purpose. This was to understand how discursive patterns emerged and were reflected in participants' linguistic choices through discursive engagement. According to Halliday and Matthiessen (2014), SFL allows us to explore particular patterns of wordings to the extent that other categorisations generally do not. This is done through three different metafunctions. Every text reflects that it is addressed to someone (interpersonal metafunction), is about something (ideational metafunction) and uses a particular mode -spoken or written language, for example -to express its meanings (textual metafunction) (Halliday & Hasan, 1989). Each of these metafunctions will be further explained, and examples of the analysis will be exemplified in the following section.

The interpersonal metafunction
The interpersonal metafunction concerns the language used to establish relationships, express attitudes and values and negotiate meaning (Halliday & Hasan, 1989). By focusing on interpersonal metafunction, I tried (Ebbelind, 2020) to gain insights into how prospective teachers established relationships to different persons and social practices. In this study, the interpersonal metafunction related to mood, tense, polarity and modality. Mood refers to how the meaning is being exchanged and the speaker's attitude towards the situation. Tense refers to whether the proposition is valid for the past, present, or future. Polarity marks if the proposition has positive or negative validity through, for instance, the use of negations, for example, 'I will' or 'I will not'. And lastly, Modality relates to the degree of certainty in an utterance.
In this part of the analysis, I first focused on the mood of the text by marking pronouns (bold in the citation below), entities and the attitude in the form of a statement, question, offer or command.

Yes, it did (statement). The stencils were very easy (statement -high modality). For very young children, it felt like you did not (polarity) get motivated by stencils (statement -past tense). Stencils and stencils and you have nowhere to have them (statement-). At least everything is in a math book (statement -present tense). You can (modal verb: indicate potential) see that you have done that many pages (statement). You feel more motivated to continue (statement). You only made the stencils because you had to (modal verb: imperative(demand)) (command -past tense).
Then I marked the tense to highlight if the clause was valid for the past, present, or future. For example, the citation above refers to past school-related experiences using did and made even though some of the clauses are in the present tense. After that, I marked the polarity to stress whether the proposition was about positive or negative validity, 'you do not'. Finally, I marked the modality, which reflects the level of certainty a clause has. Specific modal verbs were marked as they control and define common knowledge and refer to the level of certainty associated with particular forms (Herbel-Eisenmann, 2007).

The ideational metafunction
In the next stage of the analytical process, I looked at the ideational metafunction. The ideational metafunction relates to how actions or experiences are articulated in the recorded material by observing the so-called transitivity systems. A transitivity system refers to how a person -in my case, the prospective teacher -relates to activities or objects through language. This can be reflected by observing the primary process verb of the clause and its relation to the participants (Halliday & Hasan, 1989). There are different processes that the transitivity system deals with, for example: Material processes that involve physical actions such as competing and teaching. Mental processes that involve thinking, wanting etcetera. It is the senser and the phenomenon that are in focus. Relational processes emphasize relations between objects. Verbal processes express something that has been said, for example, talk or something being mentioned. Some examples are highlighted in the citation below.
Yes, it did. The stencils were (relational process: the relation between stencils and difficulty) very easy. For very young children, it felt (mental process) like you do not get motivated by stencils. Stencils and stencils and you have nowhere to have them. At least everything is in a math book. You can see that you have done that many pages. You feel (mental process) more motivated (mental process) to continue. You only made (material process) the stencils because you had to.

The textual metafunction
Finally, the specific social practice and language structures used to carry the meanings of the text are components of the textual metafunction. This function concerns the process of structuring the information conveyed and how clauses follow each other in thematic bindings (Halliday & Matthiessen, 2014). I started by marking the conjunctions and subjunctions (examples below in thew next section). Conjunctions are words or phrases -for example and, or and but -that connect parts of a clause and clauses that are the same kind, combining two main clauses or two noun phrases. Subjunctions usually initiate a subordinate clause, for example, because, as and when. A subjunction introduces a dependent clause that often expresses a reason, a condition, a result or purpose, a time, or a contradiction. Finally, the structure of the text was analysed in relation to themes (patterns that are regenerated) and information (for example, new information).

Using patterns of participation
After the microanalysis, a systematic search for patterns occurred. Each pattern observed was made into an illustrative example. Over 100 illustrations, examples used later in this paper, were made related to the case of Lisa. The template for the illustrations was created with inspiration from Skott (2018, p. 613). Figure 1 shows how parts of the structured data from the citation below are used to create the illustrative example. (statement -present tense). You can (modal verb: indicate potential) see that you have done that many pages (statement). You feel (mental process) more motivated (mental process) to continue (statement). You only made the stencils because you had to (modal verb: imperative(demand)) (command -past tense). When you had stencils, they just picked up (material process) a stencil, and (additive) it felt (mental process) endless (statement). Everyone got (relational process: everybody and different stencils) slightly different stencils, and (additive) there was no structure (statement). You followed (material process) a specific pattern when you got a math book (command). If you read (material process) geometry, you did that for a while (statement). The stencils (theme: repetition) were one paper on that and one on something else, so it was very messy (statement -past tense). The different patterns can be contrasted, illustrating shifts in Lisa's participation concerning what she has experienced in different situations. Thus, by using the framework PoP, the tales of the becoming teachers can be disentangled, illustrating elements of teacher education and other relevant practices. However, more importantly, it allows us to see how cultural meanings are related and patterned at a specific place and time. The uniqueness of every pattern can then be used to view shifts in either beliefs, knowledge, or identity during any longitudinal study.

From structured information to generated data material
The PoP framework was used as a focal lens (thinking tool [Gee, 2011]) to generate data material that illustrates shifts in prospective teachers' discursive patterns. The generated data material is the outcome of using the methodological tool. Examples of such outcomes can be a recurrent theme, shifts in validity or shifts in the use of pronouns.
Through the structuring of the information, it was clear that Lisa's contribution to and interpretation of immediate emerging social interaction could be linked in a fine-grained manner to her 'prior engagement in a range of other social practices'. PoP can be used to search for relevant practices and can also be used to explain and problematise the way the re-engagement is conveyed and connected.

Writing the results section systematically
The first step in writing the results was to generate data material by selecting parts of the structured information related to Lisa's development. All these selected parts (parts of the coded transcripts) were entered into a Word document chronologically. The second step concerned the discursive patterns made visible, illustrations, by the methodological tool in the prospective teachers' participation. These illustrations were also incorporated into the Word document to compare and contrast different patterns in their discursive participation. Illustrations related to each other were marked in the document so it would be possible to connect similar patterns over an extended period. Two examples, illustrated below, show how the illustrations are connected and their relation to each other was marked in the initial phase of writing up the results (Figure 2). In the first illustration, Lisa talks about her positive experience in upper primary school and how the mathematics textbook contributed to her interest in mathematics. As a discursive counter-image illustrated in the second figure, Lisa re-engaged in lower primary school and described what she perceived as bad mathematics teaching. Lisa used similar notions and entities to describe these two social practices, but with one significant difference -a negative attitude can be identified when relating her past participation to the lower primary school. In this example (Figure 3), we can interpret how Lisa's role models, from the past and at present, differ or are similar. We can, in some sense, view that teachers from her past and her internship supervisor are used similarly. Her experience is moulded together with her present experience. The Results chapter was then written during an extended period, systematically in each step to produce the tale presented later (Figure 4). This tale was not just a simple summary of information but the result of the analytical process. The intention was to write tales that let us experience how prospective teachers' interpretations of and contributions to immediate social interaction relate dynamically to prior engagements in various practices. This was to contribute with insights about how, or even if, experience from teacher education and other relevant past and present social practices and figured worlds matter for prospective generalist teachers' imaginings of themselves as primary mathematics teachers-to-be.

The results as demonstrating
The following section is a small part of the result (1 page out of 30), the case of Lisa (Ebbelind, 2020, pp. 143-144). Lisa's development is a complex tale, divided into three phases that include both stability and change. This complex tale is characterized by Lisa's search for teaching strategies related to her prior school mathematics experiences. The result section below is from the first phase.

The case of Lisa -an example of the result in Ebbelind (2020)
The first internship was conducted during the second term approximately eight months after entering the teacher education. At the first internship, prospective teachers have some teaching assignments, but they are not expected to teach to a large extent. In this part of the first phase, Lisa experiences teaching that falls in line with her figured world of teaching and learning mathematics. It resembles the teaching she recalls from upper primary school.
Lisa is impressed after her first three weeks attending Higgins' class: During this internship, Lisa is overwhelmed by the teaching she experiences. Her past experience as a prospective who has gladly participated in competitive settings in mathematics is reinforced as valid in her internship school setting. However, she talks about the mathematics textbook slightly differently.
Many students are very competitive, and they want to get as far as possible in the mathematics textbook . . . and they are very eager and do not focus enough on the tasks . . . but I think that the teachers [Higgins and one of his colleagues] are skilled in lowering the working pace and getting students to focus on the tasks.
In Lisa's point of view, it seems that Higgins can master doing everything that is unconventional very well: 'I think he is very spontaneous and has more playful teaching [laughing] and it is accepted'. Higgins does not only pose questions and search for answers but also engages students in some conversation concerning the subject. Lisa wants to become a teacher like Higgins because he has outstanding pedagogical ability and can explain in many different ways so that it suits many different individuals. Everyone learns differently, and all have various ways of understanding. Not just lectures, but he is engaging students and lets them be a part of the situation. They [teachers] should ask questions and control that students keep up with the lecture.
Lisa's participation in Higgins' teaching contributes to this study with more complex discursive patterns to observe in relation to competitive teaching. Competitive teaching remains a significant linguistic choice for Lisa during and after her internship in Higgins's classroom, but a small change can be identified. Lisa starts to align with Higgins' view of competitions in relation to the mathematics textbook and is challenged in her past experience as a student.

Closure
This paper described a methodology developed in a PhD study, the study of Ebbelind (2020), which aimed to explore shifts in the discursive patterns of prospective teachers' language as teachers-to-be. The case of Lisa was used and was, in Ebbelind (2020), presented as a narrative to experience the other via the text (Lincoln & Denzin, 2000). In line with this article's purpose, it has been illustrated and described how to go beyond the findings (unmotivated looking [Coles, 2015]), apply a conceptual framework and present the result as a narrative case. The main reason for narratively illustrating prospective teachers becoming teachers was to try to understand and give the reader a perspective of human lived experience (Kaasila, 2007). Such experience shape beliefs, knowledge and identity (Skott, 2018). I argue that fine-grained discursive frameworks, in any form, are important if interested in human lived experience and how human lived experience contributes to how we become and evolve as prospective teachers.
I have tried to show that combining SFL (Halliday & Hasan, 1989) and the model by Skott (2018, p. 613) might be a productive way to visualize those discursive patterns in the research process. For the study of Ebbelind (2020), SFL offered a toolkit that made possible the analysis of meaning at the clause level to reveal how language choices contribute to a text's overall meaning and purpose. Analysing the text with elements of Halliday's SFL enabled me to highlight, for example, how relationships are established when Lisa expresses attitudes, values and knowledge. SFL offered a way of eliciting the data in a systematic and unmotivated manner, reducing biases and providing a structured method of making intertextual relations visible. I agree with Halliday and Matthiessen (2014) that SFL allows to explore patterns of wording in a fine-grained manner. This paper has hopefully highlighted the methodological challenges in visualizing, identifying and describing discursive patterns to contribute to the field of mathematics education by giving insights into how to bridge the gap between different analytical levels. To present the results narratively, as in Ebbelind (2020), is a method of recapitulating experience by matching sequences of verbal language to the series of events in which they occurred (Gee, 2011). The main reason for narratively illustrating cases was to try to understand and present to the reader a perspective of human lived experience (Kaasila, 2007). Such experience shape beliefs, knowledge and identity.