The Past That Informs the Present
- Top of page
- The Past That Informs the Present
- What Is Really Going on in the Pedagogical Relation?
- Pedagogical Change
Effective mathematics pedagogy, so the saying goes, is a gatekeeper to lifetime opportunities, signifying upward mobility, and meritocracy for the successful individual student. Today, as in earlier times (see Apple, 1995), a significant proportion of the resources of education systems is absorbed in schools to help teachers develop students’ productive relationships with mathematics. Typically, students spend three to five hours with their teacher each week developing such relationships (Clements, Keitel, Bishop, Kilpatrick, & Leung, 2012). Widespread interest in how teachers develop these relationships has focused on their practice “at all levels in (and outside) the educational system in which it is embedded” (Sierpinska & Kilpatrick, 1998, p. 29). More accurately, the interest, whether expressed in research reports, in reform documents, in the media or in everyday conversation, is focused on what constitutes “good” mathematics teaching.
Initial interest in mathematics pedagogy owed a large debt to Piaget's thinking relating to cognitive growth. As a case in point, interest during the 1950s and 1960s centered on cognitive psychology's practice of exploring discrete elements (e.g., Miller, 1956), but with a focus more on understanding the cognitive repertoire of the student rather than the cognitive-oriented practices of the teacher. Indeed, the distinguishing feature of most of the early work was its vision to understand students’ cognitive capacity with mathematics. The vision was understood in terms of quantifiable, observable student behaviors, such as, for example, students’ memory and retention in the process of learning mathematical content (see Miller, 1956). Differences and changes in human learning were explained by Piaget's theory or maturation stages which also took account of the viability and coherence of students’ thinking.
These early studies within the field of cognitive science generated knowledge about the process of learning that would inform the teaching profession and teacher education. Efforts soon became focused on the relationship between teaching practice and student learning (e.g., Dienes, 1960). Teaching was considered a factor to be measured and examined in relation to its impact on student achievement and proficiency. The idea that student behaviors could be observed and shaped through successive reinforcements built on ideas from the behavioral psychologists such as Watson and Skinner who claimed that, within biological limits, development is driven by connections between stimuli, responses and reinforcement. Learning became equated to external observable variables and the interest became centered on behavioral change.
Gagné's (1965) interpretation of learning as behavioral change is exemplary of the classic tendency within mathematics education. For Gagné, learning did not merely imply observable changes in performance; it also involved hierarchical order. Pedagogically, his “building blocks” model of performance placed an emphasis on making skills automatic through practices. As a consequence, drill-and-practice routines were considered crucial for the acquisition of mathematical knowledge and the development of higher-order concepts, and ability groupings, tracking, and learning styles and teaching styles became important terminology to describe classroom life. The model had important consequences for pedagogical practice. Teaching came to mean transmission of instructional strategies which produces new learning. More specifically, the teacher transfers his or her knowledge by direct instruction into the minds of students who, in turn, receive that knowledge in the same way, irrespective of history or culture. Consistent with these understandings, the teacher is actively engaged whilst the student is a passive, yet receptive, learner.
More often than not, the 1960s and 1970s process-product studies or experimental and quasi-experimental studies provided leverage in making causal arguments between mathematics teaching and learning (e.g., Dienes, 1963). Specific teaching practices were named as causes, in much the same way as aspects of learning were named as the effects (see Begel, 1970). The work resulted in the development of an extensive literature on “good” mathematics teaching. Policy makers, educators and researchers then drew on that particular set of knowledge uncritically, to look at how those practices might be applied in the mathematics classroom. In short, research provided prescriptions of practice that served to regulate and govern teachers’ practice.
Building on developments within other discipline areas, from the 1980s constructivism became a key term in the vocabularies of many researchers and a framework for mathematics classroom teaching practice, management, and policy. For example, von Glasersfeld's (1996) radical constructivist position is one in which the behavorists’ proposal of student passive reception of knowledge is no longer credible. Von Glasersfeld advocates an active cognizing subject whose knowledge of the world is in a constant process of individual construction. In his post-Piagetian formulation, learners actively seek out and make mental connections that link aspects of their physical and social environment with certain numerical, spatial and logical concepts. Pedagogically, failure to learn is considered the responsibility of either the individual teacher in not providing the requisite skills, or the individual student's incapacity to grasp conceptual knowledge.
Principally an epistemological position rather than a theory of teaching and learning, constructivism describes how cognitive processes might be acquired and developed, in terms of “cognitive schemes, tacit models, concept images or misconceptions” (Sfard, 2003, p. 355). This is an individualist approach to learning in which the external context is linked yet peripheral to the student. For the constructivist, it is the nature of the student's developing internal representation that remains of primary interest. In short, the individual mind is privileged, while history, circumstances and social conditions are minimized. Pedagogy, in these understandings, is an intervention (e.g., Cobb, 1988). The teacher organizes the conditions for learning by “guiding from the side” rather than acting as a “sage on the stage.” That is to say, the pedagogical relation privileges the learner who is construed as an active constructor of knowledge.
Many soon began to argue that pedagogy needs to be understood in terms of the social and cultural context in which it is embedded. Within the “social turn” in mathematics education (see Lerman, 2000) characteristic of the 1990s until the present day, the identity of the teacher, in addition to its psychological signifiers, was understood as heavily influenced by social dimensions. The factors that initiated the sociocultural interest within mathematics education are multiple. They include influences from wider social and cultural imperatives and developments, including a rapidly changing demographic profile of increasingly diverse groups of learners within mathematics classrooms.
Sociocultural research in mathematics education became the site where the question about the pedagogical relation was most extensively highlighted not as a question of the constructivist teacher and the cognizing learner but as a question of the “inherently social and cultural nature of cognition itself” (Goos, Galbraith, & Renshaw, 2004, p. 91). As a counterpoint to traditional ideas of the pedagogical relation, sociocultural work grounds its understanding of teachers and classroom practices within wider social institutional processes and influences, “promoting less hierarchical, more interactive, [and] more networked forms of communication within the classroom” (Goos, Galbraith, & Renshaw, 2004, p. 91). To that end, researchers have investigated how relationships with mathematics are formed, and the part that the setting, policy, the pedagogic methods, the teacher beliefs, thinking, and practices, as well as the tools and materials play in that formation (see Cobb, 1994). All of these factors are seen as having a direct bearing on the kinds of mathematical identities that students might create and the kinds of proficiencies to which they might aspire.
In recent work within mathematics education, it is Vygotsky-inspired ideas that are invariably invoked to understand and explain teaching practice (see Jablonka, Wagner, & Walshaw, in press). Researchers drawing on these ideas have used the language of, for example, communities of practice, figured worlds and webs of relationships to map out the ways in which instructional practice influences student learning. Drawing on ideas from sociology, anthropology and cultural psychology, they view a mathematical identity as a co-construction that occurs through participation in social practice. Lave's work (e.g., Lave, 1996; Lave & Wenger, 1991) provides clear examples of the role that the social plays in the pedagogical relation. The concept of “apprenticeship” is used to refer to initiation into and increased participation within a learning community, just as “legitimate peripheral participation” is the conceptual tool to bridge and identify, at any one time, both the knowledge of the learner and the community of practice.
In a series of studies, Nunes, Schliemann, and Carraher (1993) proposed that individuals are elements within an encompassing system of social practices even if they act in physical isolation from others. Their studies revealed how students who traditionally fail in school mathematics could be successful undertaking, on the street, money transactions involving mathematics. Mathematical identities, it was shown, develop from knowledge that “is distributed across people and their tools and technologies, dispersed at various sites, and stored in links among people, their minds and bodies, and specific affinity groups” (Gee, 1999, p. 33).
Many (e.g., Lampert & Blunk, 1998; Wood & Yackel, 1990) have argued that classroom teachers help shape these mathematical identities by co-constructing students’ dispositions and competencies through the patterns of interaction and discourse created in the classroom. What these researchers seek to demonstrate is that mathematical knowledge evolves as students take part in the “socially developed and patterned ways” (Scribner & Cole, 1981, p. 236) of the classroom. Teaching is construed as a facilitative act, and it is by scaffolding the development of those patterned ways, that the teacher regulates the mathematical opportunities available in the classroom. As O'Connor and Michaels (1996) argue:
The teacher must give each child an opportunity to work through the problem under discussion while simultaneously encouraging each of them to listen to and attend to the solution paths of others, building on each other's thinking. Yet she must also actively take a role in making certain that the class gets to the necessary goal: perhaps a particular solution or a certain formulation that will lead to the next step. … Finally, she must find a way to tie together the different approaches to a solution, taking everyone with her. At another level—just as important—she must get them to see themselves and each other as legitimate contributors to the problem at hand. (p. 65)
Students’ mathematical identities are co-constructed from their lived experience in the cultural world of and in the social activity within the mathematics classroom created by teachers. Face-to-face teacher–student talk and productive relationships become central elements in mathematical knowledge production. Teachers endeavor to “align … students with each other and with the content of the academic work while simultaneously socializing them into particular ways of speaking and thinking” (O'Connor & Michaels, 1996, p. 65). In this kind of thinking, the individual/social polarity is resolved by the understanding that the personal and the social cannot be distinguished: the personal is always already social.
Sociocultural perspectives, like the others named in this necessarily partial review, provide a view of the nature of pedagogy. Alternative conceptions lead to different views about what teachers ought to do and what the pedagogical relation means. Early cognitivist formulations from the 1950s focused on the central processor model of mind which led to proclamations about the relationship between specific teaching practices and learning. Constructivist ideas emphasized the transmission of information from an active teacher to passive “empty vessel” students. More recent sociocultural formulations pointed to the social basis of the pedagogical relation, providing a complementary account to the (radical) constructivist teacher. In foregrounding the social aspects of the classroom, sociocultural perspectives paved the way for a sharpened focus on how pedagogy develops and a new understanding and explanation of the interactions inherent within the pedagogical relation.
What Is Really Going on in the Pedagogical Relation?
- Top of page
- The Past That Informs the Present
- What Is Really Going on in the Pedagogical Relation?
- Pedagogical Change
Contemporary sociocultural approaches to the issue of pedagogy within mathematics education are wide in scope and vast in number, yet they do not exhaust the range of contemporary approaches to the fundamental question asked by mathematics education researchers: What is going on? A small number of researchers (e.g., Black, Mendick, & Solomon, 2009; Brown, 2008; de Freitas & Nolan, 2008; Walshaw, 2004) express the view that mainstream sociocultural analyses do not sufficiently explain the constantly changing process of teacher construction, and, in particular, the interactivity between agency and subjection exhibited within pedagogical relationships. A shift in emphases is proposed—one that would magnify a view of pedagogy and pedagogical relationships, closely tied to interactions between people—both past and present—and situated in relation to biography, current circumstances, investments and commitments.
These directions were shaped by theoretical developments across the social sciences and humanities that emerged from 20th-century philosophies. Such developments offer a radical departure from the dualistic notions of thinking and being in commonly held theories of human interaction within mathematics education, in which teachers are understood as autonomous individuals with agency to choose what kind of a teacher they might become. In the humanist view, at the heart of an individual teacher is a fixed essence which is unique to her and which defines what she “is.” Given her natural essence, the teacher makes sense of her teaching journey through initial and ongoing teacher education and through firsthand teaching experiences as a progression that proceeds rationally and linearly. Theoretical developments, categorized under the banner of poststructuralism, trouble the subject of humanism with its core essence. In establishing their own terms for the humanist agentic teacher, poststructuralists utilize the notion of discourse in their analyses to collapse the personal and the public in an imaginative way.
Discursive approaches offer tools and an alternative language for looking at, interpreting and explaining pedagogy. Performing the role of conceptual schemes, discourses operate widely through the social milieu. Importantly, for a focus on the pedagogical relation, discourses produce particular kinds of teachers and students as effects of discursive relations. That is to say that pedagogical “truths” emerge through the operation of discursive systems (Brown & Walshaw, 2012). Broadly speaking, then, discourses sketch out, for teachers, ways of being in the classroom. They do that by systematically constituting specific versions of the social and natural worlds for them, all the while obscuring other possibilities from their vision. They are “not about objects; they do not identity objects, they constitute them and in the practice of doing so conceal their own invention” (Foucault, 1981, p. 49).
Employing the notion of discourses allows poststructuralists to shift the focus from examining the nature of identity of the teacher to a focus on how a teaching identity is discursively produced. De Freitas (2010), for example, investigating the discourses operating in classrooms, studied the activity of teacher talk and illustrated the ways in which teachers use speech to arrange social structures, showing how they contribute to the formation of mathematical identities during lesson time. In an examination of narrative sequences presented in a series of lessons of two teachers of senior high school students, she showed how the teachers’ identities were framed and staged, differentially, through the unfolding turns of classroom interaction. Through fine-grained reading of pedagogical interactions in the two classrooms, de Freitas exposed the regulatory power of teacher discourse in providing students with differential access to mathematics, shedding light on those students who are included within and those who are positioned outside of the text. Importantly, de Freitas demonstrated the way in which the discursive practices of the two teachers contribute to the kind of thinking that is possible within the classroom.
Discourses that differentiate between people are a major concern in mathematics education. A small number of researchers (Bibby, 2010; Hardy, 2009; Walshaw, 2007) have drawn attention to the invented character of the concept of “difference” highlighted within many mathematics educational policies and systems. Exploring how structural processes and historical events contribute to the mathematical identities that students develop in the mathematics classroom, Mendick, Moreau, and Epstein (2009) explain: “A position of mathematically able confers an identity as different and special. This has consequences for mathematics and society: it excludes many people from mathematics and disproportionately excludes particular groups” (p. 72). The fabrication of “difference” is also highlighted by Hardy (2009) who, from her study of primary/elementary pre-service teachers’ confidence in mathematics, argued that “it is the very conceptualising of primary teachers’ professional knowledge of maths which, in its articulation, generates and condemns teachers to having faulty knowledge. That is, it is the attempt at better description … that produces the problem” (p. 195).
The notion of discourse invites the notions of power and regulation. According to Foucault (1972), systems of power both produce and sustain the meanings that people make of themselves. Policy directives carry a certain authority that might lead us to believe that the policy text represents the one true statement about teachers and learners. Various researchers have shown (e.g., Hanley, 2010; Walls, 2010) that this is a necessary deceit, designed to prompt us to think, speak and act in the terms of pedagogy offered within official statements. As Morgan (2009) has explained: “The concepts, values, and positions of the official discourse … have particular force because of the roles they play in regulating school practices and, hence, the extent to which they are integrated into the actual experience of teachers and students” (p. 105).
Popkewitz (2004) has been instrumental within the mathematics education community in problematizing received ideas, particularly those that relate to pedagogy within standards-based policy reform statements. His interrogation demonstrates that directives concerning what teachers and students do and say in mathematics classrooms are part of a pedagogical regulatory apparatus that governs teaching and the “moral development and liberation of the individual” (p. 13). In reality, teaching, teachers and students are all caught up in regimes of truth, discourses, meanings and significations, that pre-exist them. As he notes: “Although conceptions of “participatory structures” and a “community of learners” emphasize children's involvement, that involvement directs the children's attention to propositions that have already been confirmed in the a priori world of schooling and mathematics education research” (p. 21).
In the history of mathematics educational research, very few studies have assumed that power operates in anything more than in the conventional “political” sense. Received ideas of pedagogy and pedagogical relations do not typically acknowledge that power relations are as much part of classroom life as they are of civil lives. Prototypically, analyses track power operating in macro structures, conceiving of power as an entity being used for political and personal purposes, such as in curriculum development, or in the construction of mathematics textbooks. With poststructuralism it became possible to construct plausible explanations of how power operates even at the mundane and routine pedagogical levels of everyday classroom life. It allowed researchers, who were interested in fine-grained analyses at the micro-level, to demonstrate how teaching identities are shaped and how pedagogical relations are strategically fashioned in the dynamics of everyday classroom life.
… in the classroom strands of power entangle everyone, governing, regulating, and disciplining teachers as well as students. Power does its work through the classroom's traditions; through its material, discursive, and technological forms; through its mathematical enactments; and through its discourses that relate to categories of class, gender, ethnicity, and other social determinations. (Walshaw, 2010c, p. 4)
It follows that teachers and students are the production of the practices through which they become subjected. At the institutional level, practices exercise control over the meaning of teaching by normalizing and providing surveillance practices to keep such meanings in check. Even in a classroom environment that provides equitable and inclusive pedagogical arrangements, poststructural approaches have shown that power is ever present through the classroom social structure, systematically creating ways of being and thinking in relation to class, gender, and ethnicity and a range of other social categories (see Mendick, 2006; Walshaw, 2001). In addition to operating at the macro-level of the school, power seeps through lower levels of practice such as within teacher–student relations and school–teacher relations. It invades cultures and all social structures and “reaches into the very grain of individuals” (Foucault, 1980, p. 39).
In “Learning to Teach: Powerful Practices at Work During the Practicum” (Walshaw, 2010a) the focus was on the ways in which social and structural processes and material conditions interacted in the shaping of 72 pre-service teachers. A poststructural analysis used the concepts of discourse and power (normalization and surveillance) to investigate pre-service elementary teachers’ reflections of their practicum experiences within a range of New Zealand schools. The teachers’ reflections were used as a means of understanding the local, systemic and flexible conditions of identity construction. In particular, they were used to capture the ways in which an identity as a teacher is produced and reproduced through social interaction, daily negotiations, and within contexts, that are already overlaid with the meanings of others.
The exploration revealed the way in which particular practices are normalized and monitored within the practicum experience, and, in doing this, foregrounded the political and strategic nature of learning to teach. The analysis highlighted the impact of regulatory practices on pre-service teachers’ understanding of themselves as teachers and on their constructions of what it means to teach effectively in the mathematics classroom. Specifically, shifts in pedagogical practice and unknown knowledge were informed by the supervising teacher's constructions of good teaching. At the beginning of the practicum pre-service teachers identified and named the following categories of unknown knowledge: the teacher's mathematical explanations; algebra and problem solving; starting and introducing a unit; how to get it across; how to teach it; how to run the maths lessons; general mathematical knowledge. At the end of the practicum they nominated timing, pacing, resources, activities, workable strategies, management strategies; teaching to groups; teaching different levels as knowledge that they needed and these, in turn, “became the coordinates through which ‘good’ teaching would be mapped” (p. 117).
In the study, the “panoptical gaze” (Foucault, 1977) of the supervising teacher “worked surreptitiously to equalize behavior, actions, and even thinking, in the most seemingly innocuous details of embodied practice” (p. 124). Importantly, it had the effect, for some pre-service teachers, of advancing a love for teaching. For others, it created an unhappy and unproductive relationship between the pre-service teacher and the supervising teacher. From the investigation, the following conclusion was offered:
… learning to teach becomes an issue of micro-political engagement with discursive classroom codes, all of which are set upon providing the pre-service teacher with a sense of identity in the classroom as a teacher. Becoming a teacher is not so much an issue of a personal journey as a barely visible set of highly coercive practices. Teaching ‘know-how’, then, is linked to networks of power, targeting thinking, speech and actions, with a view towards producing particular constructions of identity. It is the result of compliance to a set of practices that have been naturalized for the pre-service teacher in the classroom. (Walshaw, 2010a, p. 126)
In analyses like these, the quest is not to establish the “truth”; nor is it to broaden the scope of inquiry to include more qualitative approaches or more speaking voices. Rather, the aim is to develop a sensitivity to the impact of regulatory practices and discursive technologies on the constructions teachers have of pedagogy and of the pedagogical relation. It is to explore the question: How does a teacher turn herself into a teacher? The poststructural response takes on board the fact that the self is not a center of coherent experience. Teachers (as well as others) are not masters of their own thoughts, speech or actions. Their identities are historically and situationally produced by discourses that are often contradictory. The ways in which they teach in the classroom and the ways in which they give meaning to their interactions with students are influenced by the discourses made available to them and to the political strength and interest, for the teacher, of those discourses.
Exploring contradictory discourses, Nolan (2010) showed how her efforts to implement inquiry teaching approaches as an educator in an undergraduate teacher education program for middle-years teaching, met with resistance, challenge and dissatisfaction from her pre-service students, when her practices ran up against their entrenched understandings of mathematics teaching. Using a Bourdieuian framework, Nolan analyzed the tensions between thought and action as well as between knowledge and experience enacted in the pedagogical encounter. Hers was an honest account of her lived dilemmas in trying to establish meaningful pedagogical relationships and authority in a classroom environment fuelled with contradictory visions and contestation.
In Project 2 discussed within “Poststructuralism and Ethical Practical Action: Issues of Identity and Power” (Walshaw, 2010c), teaching identity formation was explored through the use of the poststructuralist vocabulary. In particular, the Foucauldian concept of “dividing practices” was drawn upon in an example of a pre-service secondary school teacher teaching in three practicum schools. Each school organized physical space and time for teaching and each nurtured particular kinds of pedagogical relations. The specific conditions for practice as well as the forms of supervision at the first practicum school meshed with the pre-service teacher's own understanding of pedagogy and, to that extent, served to enhance her pedagogical practice. Importantly, it also set a benchmark measure for how she would construe pedagogy and pedagogical relations at the other two schools. For example, she named the first school as a “professional place of learning” in which there is “a sense of belonging in the students,” alongside “high levels of performance” and “pride in the school.” She noted that the principal's “business and pedagogical ideals” coincided with her view of a principal's work. Practices that are understood by a pre-service teacher as effective, profoundly influence the sort of teacher she will become.
Those practices identified by the pre-service teacher as effective at the first school were strikingly at odds with the practices at the third school which, in her view, was “under-resourced” where classrooms were “cramped, dark and unwelcoming” and where the students “seem subdued all the time.” She drew attention to “very prescriptive” pedagogical practices, involving “rote learning,” “on-going testing,” and “no scope for inquiry learning.” Such an environment governed the way in which pedagogy was understood and how pedagogical relations could be enacted.
The paper also highlights the ways in which the teacher was caught up in structural and procedural supervision arrangements not of her own making. It draws attention to the highly contradictory realities, from one school to another, that are sometimes experienced by pre-service teachers in their work with their supervisors within mathematics classrooms. For the pre-service teacher reported on in the paper, specific regulatory practices attempted to govern her meaning of pedagogy, and, more fundamentally concerning, profoundly informed the diminishing sense she formed of herself as a teacher of mathematics. In effect the “divisions” operated not only between the schools, but also powerfully within Alicia herself.
Divisions and conflict are often overlooked in mathematics education research, particularly within the body of research that speaks of generalized effective pedagogical practices on behalf of all teachers or all students who have a particular classed, gendered, or ethnic affiliation. Such research fails to take into account the identity of the teacher or the student as heterogeneous, never in one place at one time. “What we are, what we think, and what we do today” (Foucault, 1984, p. 32) as a classed, raced, or gendered (or categorized-otherwise) teacher or student, for example, is a production of the practices and discourses that are part of the regulatory pedagogical apparatus in the schooling context. Discursivity, in short, is not simply a way of organizing what people say and do; it is also a way of organizing actual people and their systems.
Poststructuralist analyses, like those reviewed above, are able to address the question of what is really going on with mathematics teachers and their teaching. Different kinds of language and different tools that rest on different kinds of assumptions about subjectivity and knowledge, provide a means for understanding what structures the conditions of teachers’ existence. To that end, poststructural analyses do not deny teachers’ subjective experience. The ways in which teachers themselves make sense of their classroom lives are crucially important for understanding the forms of governance and the networks of power that shape the very fabric of pedagogy and the “mathematics teacher.” A poststructural analysis that explores the material and the discursive conditions of teachers’ existence, moves us away from blaming when things go wrong. It does this by acknowledging and taking into account competing subjective teaching realities, and by unpacking the social interests on behalf of which those interests work.
- Top of page
- The Past That Informs the Present
- What Is Really Going on in the Pedagogical Relation?
- Pedagogical Change
So far a case has been made for poststructural (and particularly Foucauldian) theoretical tools to provide researchers in mathematics education a way to account for how the teacher is produced and regulated in the pedagogical relation through discourses. In the previous discussion, it was suggested that a poststructural approach allows us to develop a sensitivity regarding the complexity of the relationships that teachers form with mathematics pedagogy. It also provides the means to unearth the sometimes oppressive conditions in which some of those relationships are embedded. However, poststructural concepts, particularly those made available by Foucauldian poststructuralism and those made available by psychoanalysis, offer mathematics education more than a way of describing how the teacher is produced and regulated. They also offer a way of analysing shifts in pedagogical practice.
What is it that makes teachers shift their practice? How does power insinuate itself to make teachers more susceptible to new ideas; to attach themselves to or ignore specific ways of teaching in the classroom? Butler (1997) has turned the question around to ask: “What is the psychic form that power takes?” (p. 2). Through its complex yet well-developed theories of subjectivity, psychoanalysis allows us to address these questions by exploring unconscious modes of operating. It allows us to investigate what it is that shifts the pedagogical experience within the mathematics classroom. Shifts, in both the Foucauldian and psychoanalytic theoretical apparatus, are grounded in an understanding of fragmented selves, on layers of self-understandings, and on multiple positionings within given contexts and time.
The idea that identity is ever changing, contingent on time and place, is particularly compelling for anyone who has ever pondered upon the fact that we act, speak and think differently in the wide range of social communities in which we are members. Why might that teacher act, speak and think in the classroom differently from the ways in which she acts, speaks and thinks in the home, in the staffroom, or in any other social context? For that matter, why might a teacher practice differently in the classroom with a “mathematically able” class compared to a “mathematically challenged” group of students? The notion of a differential practice or “performance” (Butler, 1997) reinforces the understanding that the identity of a teacher as heterogeneous, conflicted in multiple places at any point in time. Teachers, like all others, are constantly making sense of their worlds.
As a way of coming to terms with the notion of the constantly evolving identity of a teacher, most recent analyses within mathematics education draw on the poststructuralist realization that people are processes rather than concrete entities. An important lesson we might take from poststructural theorizing is that pedagogical “truths” are not absolute but are experientially based and embedded in fluid, social interactions. If pedagogical reality can only ever be local, specific, and temporal, how might such reality be investigated? The suggestion here is that it might be explored through a systematic analysis of meaningful social action, where the interest becomes one of figuring out how identities are constructed, negotiated, resisted, and solidified (see Lerman, 2009).
Negotiated and resisted identities were unearthed in the pre-service teacher whose experiences were reported as Project 2 within “Poststructuralism and Ethical Practical Action: Issues of Identity and Power” (Walshaw, 2010c), noted in the previous section. In the study, shifts in the pre-service teacher's sense-of-self as a secondary school teacher were tracked through her interview reflections on three different practicum experiences over the course of her 1-year post-graduate teacher education course. The three schools she had requested were markedly different in terms of student clientele, philosophies of teaching, physical space for student recreation, and geographical location. As the teacher moved from one practicum school to another, she entered different regimes of knowledge concerning effective mathematics pedagogical practice. The first, and, to a lesser extent, the second school's discursive codes were aligned to her own understanding of effective pedagogy as inclusiveness, two-way communication, informed choice of student task and activity and effective pedagogic rituals and routines. Both schools, by and large, produced the parameters of practice by which she assessed the third.
In the first two schools, unlike in the third school, the practices of the supervising teachers “operated through an understanding that she had the capacity to make sensible choices, rather than out of a pathological framework focused on changing her ‘deviant’ practices” (Walshaw, 2010a, p. 14). The poststructural analysis was able to demonstrate that such productive relationships contributed to her own active involvement in self-forming subjectification and, hence, to her enhanced sense-of-self as a teacher. It also showed that the latitude within the set of supportive supervisory practices offered to the pre-service teacher at these two schools was incompatible with the tight set of regulatory practices and articulations at the third school. As a result of the conflicting discourses in operation at the third school, the pre-service teacher's sense-of-self as a teacher decreased rapidly, and led ultimately to her construction of herself as an incompetent mathematics teacher at that school.
Psychoanalytic conceptual tools offer a language that is particularly useful for analyzing difficult and contradictory experiences from the past, the present and even those experiences anticipated in the future. The tools offer a sharpened investigative force and have sometimes been used to explore the ways in which teachers navigate their way through uncertainty within the pedagogical encounter. Appelbaum (2008), for example, use psychoanalytic tools to offer an understanding of pedagogy that builds around tension. His work contrasts with much of the early work within mathematics education that tends to be framed within the notion of teacher-centeredness. It is also at odds with recent work aligned with the notion of student-centeredness that is privileged in recent reform documents. Inquiry-based classrooms promoted in these reform documents, are characterized by student investigation, collaboration, and communication (Leikin & Rota, 2006). They necessitate a shift from traditional teacher-directed approaches to more inclusive student-centered pedagogies.
Appelbaum's (2008) work strives to “shift our focus away from either the position of the teacher or the learner” (p. 52), noting that both these notions that could only emerge within the liberal-humanist discourse. Within the liberal-humanist discourse, the individual's very conception-as-an-individual requires a relationship of difference. Irrespective of whether teacher-centered or student-centered classroom teaching is at issue, difference is explained as a function of knowledge and expertise and is constructed in a hierarchy in which the first term in the teacher–student or the student–teacher relation is dominant. The second term, on the other hand, is conceptualized as Other. In opposition to this understanding, in the psychoanalytic tradition, there can be no concept of “difference” without the concept of “same.” Both student-centred and teacher-centred pedagogies that either confirm or reverse hierarchies are interpreted as not fully capturing the teacher–student relation.
Mathematics education tends not to acknowledge the I-thou inseparability (Appelbaum, 2008) in relation to pedagogy. Although the concept as an integrated unit was expressed in the seminal work of Vygotsky, contemporary sociocultural analyses, more often than not, draw on a limited version of the relation that ensures the “I” is kept well apart from the “other.” As part of the set of modernist dualities that include individual-society, and structure-agency, and I-other binaries presumed in mathematics education are not understood as implicated in and as implicating each other.
Appelbaum's way forward is to propose a move “toward the relationality of the teaching/learning encounter” (p. 52). This new expression of teaching is no longer focused on the teacher or the student but on the relation between both. That is to say, the I-thou duality of modernist discourse dissolves from a focus on itself into a focus on an “it.” In this understanding, pedagogy crystallizes from a relationship that embraces the knowledge, the discourses, and the practices within the space between the teacher and the student. The key purpose of teaching in this formulation is not to diagnose nor remediate but to allow creative energy to emerge that will facilitate both students’ and teachers’ change. The teacher enhances that potential by suspending her own views and assumptions, while endeavoring to accept and, more particularly, to preserve the difference of the student. As Appelbaum explains, “mathematical ideas, objects, strategies, interpretations, and so on, become the objects of the students’ and teacher's intentionality and the basis for their ability to communicate and understand each other, to be together in the world” (p. 57).
The perspective regarding the nature of teaching that Appelbaum puts forward takes into account the different ways of knowing and thinking, as well as the language, emotion, and discursive registers generated within the learning community in which the “it” is located. Shaping students’ mathematical thinking in this way takes the teacher into the unknown. But the unknown is the place where discoveries big and small can be made. For the teacher it means good lesson starting points, more finely tuned classroom listening, questioning, and noticing both the seen and unseen. Psychic and emotional energy are involved in the process as the teacher attempts to keep tensions in place, relinquishing the need for the perfect lesson. Appelbaum reminds us that there are no perfect lessons but there are, of course, ways to create an infrastructure that is conducive to meaning making. Crafting these conditions involves planning “the structures for how work will get done” (p. 33), establishing the “routines, rituals and other features” (p. 51) that will make learning possible. Teaching like this demands an ongoing examination of the discursive methods through which one (both student and teacher) becomes “subjected” in each ongoing learning relation.
Brown and McNamara (2011) also use psychoanalytic tools, but they cast their investigative net over teaching differently. They draw attention to the conditions through which the teacher's “self” emerges within the contemporary environment in which teachers (and their schools) have become objects of scrutiny and critique. In raising the question about the constructed nature of teaching experience and the ways in which teachers see themselves in school mathematics, they challenge the notion that a teaching identity is constituted in exclusively mindful ways. Their argument is that a teaching “identity is created rather than revealed” (p. 92).
For them, a teacher's identity is constituted through an understanding of how others see that teacher. As they maintain, “there are no identities as such. There are just identifications with particular ways of making sense of the world that shape that person's sense of his self and his actions” (Brown & McNamara, 2011, p. 26). By that they mean that teachers’ identifications are not reducible to the identities that teachers construct of themselves. The teacher's self is performed within the ambivalent yet simultaneous operations of subjection and agency on the teacher. In this understanding, there is a level at which the teacher invests, or otherwise, in a discursive position made available. Observations of the way in which teaching identity is constituted are likely to reveal how identities develop through discourses and networks of power that shift continually in a very unstable fashion, changing as alliances are formed and reformed.
That idea that understandings of pedagogy change as alliances are formed and reformed is given a very clear expression in the discussion provided by Brown and McNamara (2011). The official curriculum policy, they showed, in legitimating certain ways of speaking, also provides a language by which the teacher might shift her identification as a teacher. Working with the material of teachers’ narratives, and paying attention to language, Brown and McNamara showed that teachers, often without their full awareness, were identifying with the particular ways of understanding pedagogy and pedagogical relations, as promoted through the language of the official curriculum policy. This point was exemplified by one teacher, in her fourth year of study, describing her work in schools, who used the language of “the mental starter,” “whole class on the carpet,” “with me working between the groups and stopping them now and again,” and the “whole class plenary” to describe her practice. Without her full awareness, the language used in the curriculum statement had structured her understanding of herself and provided the means by which she might identify herself as a teacher.
Similarly, Hanley (2010) has drawn on psychoanalytic theory to reveal the pull of systemic forces on teachers’ lived classroom teaching experiences. In particular, Hanley explored the impact of teachers’ experiences in a research project—the goal of the project being to shift teachers away from the technicist models of teaching promoted in an official curriculum document. Drawing on Žižek's (1998) understanding of Symbolic authority, Hanley revealed that whilst the teachers attempted to put into practice the inquiry approaches they had learned through the research project, for some, teaching practice was merely a performance—always with one eye on what the researcher wants, and the other eye focused squarely on the official national curriculum guidelines. What is learned and practiced in research initiatives competes for attention with understandings that surface from more official teaching models, and are sometimes never fully cashed in as lived experience within the classroom. Lessons learned from this exploration point to the difficulties involved in negotiating pedagogical change when teachers are more heavily invested in other ways of thinking and acting.
A teacher is not the master of her own destiny: identities are formed in a very mobile space. This psychoanalytic idea (see Lacan, 1977; Žižek, 1989) is fundamental to understanding that teachers negotiate their way through layered meanings and contesting perceptions of what “good” teaching looks like. Researchers in mathematics education who draw on psychoanalytic theory (e.g., Bibby, 2010; Brown, 2008; Hanley, 2010) maintain that determinations exist outside of our consciousness and these influence the ways we conceive of ourselves. In the pedagogical relation, they both influence the way teachers understand pedagogy and influence their interactions in the classroom. The identities teachers have of themselves are, in a very real sense, “comprised,” made in and through the activities, desires, interests, and investments of others. They are constructed in spaces and moments and, as Foucault (1994) put it, by “dim mechanisms, faceless determinations, a whole landscape of shadow that has been termed, directly or indirectly, the unconscious” (p. 337).
A number of influential contemporary theorists (e.g., Butler, 1997; Deleuze, 1990; Žižek, 1998) have addressed the “dim mechanisms” and “faceless determinations” that underwrite the investments that a teacher makes in a particular discursive position. “Mathematics Pedagogical Change: Rethinking Identity and Reflective Practice” (Walshaw, 2010b) takes a Lacanian (1977) psychoanalytic position to address the same question, proposing that each discursive position made available invites a different mode of implied obligation. The paper is an attempt to draw attention to the relational aspects of identity construction and the investments that inhere within that relation. More specifically, it is an attempt to provide evidence of an individual teacher constructing his view of “good” pedagogy and the formation of his teaching identity in relation to understandings of and perceived obligations to the researcher. The exploration involved mapping out change processes and analyzing the intersubjective negotiations that took place between the researcher and the teacher as the teacher reflected on a sequence of lessons involving the solution of linear equations. An overriding interest was in demonstrating how those negotiations shaped the teacher's shifts in practice and his changed understanding of effective teaching.
The argument put forward in the paper is that the reflections on the lessons offered by the teacher were made in relation to how that teacher interpreted the way in which the researcher “read” the teacher, at the time when the final interview took place. Importantly, before the final interview, classroom observations and other interviews with the teacher had taken place. In the psychoanalytic analysis provided, the reflections were shaped by the teacher's search for recognition from the researcher and by coming to an understanding of what the researcher wanted or expected to hear. This proposal subtends the idea that “identity claims can never achieve final or full determination” (Lacan, 1977, p. 496). Constructions that teachers, like any others, have of themselves, are “destined to miss the mark, continually subverted within a kind of metaphorical space between people, never fully understood and never fully captured by language” (p. 496).
The psychoanalytic approach to understanding the ways in which a teaching identity is constituted through a teacher's reflections is quite at odds with conventional approaches. Research within mathematics education dedicated to the reflective practitioner (Schön, 1983) has seeded an extensive literature, focused as it is on the important educational goal of teacher development. For example, García, Sánchez, and Escudero (2006) drew on the notion of reflection-on-action to explore the relationship between theory and practice in their work as educators. Reflecting-on-action provided the means to identify and analyse successive steps in the process of looking back on a study they undertook with 130 elementary school student teachers. The teachers’ task was to analyse, in small groups and with guidance, the multiplicative structure within numeracy problems, as depicted within a number of school textbooks. Reflection on the results and the progress of the research made it possible for the researchers to identify a range of factors crucial to their own development as educators and to build a model of the relationship between theory and practice.
Maintaining that reflection-on-action “is an essential component of the learning process that constitutes professional learning” (p. 2), García, Sánchez, and Escudero (2006) noted that the process of reflection allowed them to “distance [them]selves from the context in which [their] study was developed” (p. 15). The researchers and their professionally developing selves are, in this work, understood as stable and given, and not implicated within the context of the research.
Recent theoretical shifts within mathematics education, however, are not founded on guarantees for a stable self or on de-contextualized experiences. In the psychoanalytic tradition, teachers’ reflections of practice and pedagogical relations are enactments of reciprocity and obligation. Such approaches acknowledge the complexity and complicity at work when teachers engage in reflective practice of their work with students. In the psychoanalytic approach, reflections of practice are always constructed from past investments and conflicts, always with a view towards what the Other (e.g., the conventions of mathematics teacher education; regulatory practice; the social context in which the reflection takes place) wants. In that understanding, reflection becomes filtered by inter-subjective negotiations and the psychical dynamics that are part and parcel of those relations. Teachers’ reflections of practice and pedagogical relations, as enactments of reciprocity and obligation, are “instruments of social reproduction” (p. 496). Paradoxically, then, reflective practice can be considered emancipatory only within the framework within which it took place.
The question of investment in a discursive position of pedagogy and pedagogical practice is also explored in “Affective productions of mathematical experience” (Walshaw & Brown, 2012), where affective ties within the teacher's practice come to the fore. In this effort Spinoza's (2000) theoretical tools, as derived from contemporary readings of his work (Badiou, 2009; Deleuze, 1981/1988), provide the grounding for an explanation of the work that teachers do in their classrooms. Arguably, a focus on affect is by no means new within mathematics education. Typically, however, affect is an add-on, and affect and thinking are understood either as in opposition to each other or are drawn together by causal links. Affect, as understood in “Affective productions of mathematical experience,” is not by any measure an attempt to graft affective dimensions onto a model of human cognition, predicated on a unitary and fixed human subject. Rather, it understands the teacher's subjectivity as constantly mobile and, for that matter, never fully rational in the sense that the student is continually “taking up” mathematical objects and images and the like, in relation to how that “take-up” is made possible by the teacher and others.
The affective ties between the teachers and the students played out in two different classroom settings. In both, knowledge of affective states provided a rationale for the teachers to embrace the dilemmas of “unfinished” knowledge production. In the first, the teacher's seemingly slow-paced coverage of the curriculum was deliberate and was shaped by his care of his students. The discourses of student diversity and inclusion that predominated at this school operated “to provide a moral matrix through which and with which he thought about mathematics pedagogy” (p. 196). In the second classroom setting, student learning of how the moon moves relative to the earth was purposely put on hold as students participated in a “joyous experience through time kept going by the very failure of language to sum things up” (Walshaw & Brown, 2012, p. 197).
Affect, in these analyses, is a constitutive rather than derivative quality of classroom life. It is not an interior experience, but rather, a “form of thinking that is often obscure and nonreflective” (Walshaw & Brown, 2012, p. 188). Pedagogy, as enacted by the two teachers, was built on an understanding of learning as neither completely rational nor entirely social. Rather, learning is conceived of as located in a liminal space through an uneven movement, embracing the discourses, knowledge and of both teacher and student. In work, like this, inspired by ideas such as Spinoza's, in the pedagogical relation thinking and affect are at one with the other. Cognition cannot stand alone without affect precisely because affect initiates the purpose of thinking. In mathematics education we are only now beginning to understand the thinking-being relation (where thinking incorporates both conscious and unconscious processes) as a dependency. It is not possible, in the final analysis, to separate one from the other. This inseparability is immensely relevant in any contemporary discussion of pedagogical practice.
The discussion offered of pedagogical change has taken a different line of inquiry than that typically presented within mathematics education. As a counterpoint to habitual thinking of the way we might construe pedagogical change, the primarily psychoanalytically-inspired analyses force us to rethink what lies behind a teacher's receptiveness to change her practice. In particular, they demand attention to processes of obligation and reciprocity, emphasizing the relational and unseen aspects of pedagogical change. Pedagogical change turns out to be a mode of activity circumscribed beyond the rational autonomous teacher. It is mediated by unseen, unspoken, atemporal coordinates, all of which serve to undermine any certain rational basis for change.