Abstract
Keywords
Introduction: Frameworks in Mathematics Education Research
Several theoretical frameworks have been used in mathematics education research (MER) involving teachers to understand the practice of teaching. The expanse of theoretical frameworks can be captured through an understanding of (a) their assumptions about the relation between the individual and the social, and how such assumptions influence the perceived role of teachers in classrooms and teacher development and (b) the relationship between theory and practice, with a particular focus on the impact on instruction and student learning (Skott et al., 2013). In the practice of research, conceptual and theoretical frameworks guide the selection and framing of the problem, methods used to study the problem, the constructs which get foregrounded in the analysis and the contributions that are made to advance the field. In this article, I argue that a deeper analysis of complex problems of practice requires the use of multiple frameworks, even though they might be ontologically incommensurate.
Each framework makes certain assumptions about the constructs to be studied. In the context of MER, for example, the frameworks which propose the classification of mathematics teacher knowledge into different categories assume that teachers’ knowledge can be characterised, classified and measured (Barwell, 2013). Skott et al. (2013) noted that most research in the field of mathematics teacher knowledge and practice follows from the constructivist paradigm. In their words,‘…teaching is understood as an enactment of pre-existing knowledge and beliefs on part of the teacher, and the practices of the mathematics classrooms are indeed ‘the teacher’s practices’ (ibid., p. 502, emphasis in original). The research on mathematics teacher knowledge, emanating from Shulman’s work in 1986, elaborated by frameworks, such as, mathematical knowledge for teaching (Ball et al., 2008), profound understanding of fundamental mathematics (Ma, 2010), and the knowledge quartet (Rowland, 2009) use a psychological or individual-centred perspective of the teacher in a classroom (Rowland, 2009; Skott et al., 2013). Such frameworks have been used to unpack the complex understanding involved in teaching mathematics by using specific constructs as lenses to zoom into specific tasks of teaching, such as connecting teachers’ knowledge of the subject matter with the knowledge of students’ ways of thinking.
The individual-centred frameworks following the cognitivist paradigm were challenged by the social constructivist and sociocultural theorists by proposing the inherently social nature of cognition (Goos et al., 2004). The social learning paradigm emphasised the study of interaction patterns, classroom activity, discourse patterns, participation in the activity and the socially constructed meaning (Lerman, 2013). For example, the sociocultural frameworks have been used to study teachers’ discourse in instruction (Adler & Ronda, 2015), the activity in the classroom (Potari, 2013) and the nature of classroom interactions (Barwell, 2013). In pointing to the major differences between the two predominant frameworks in MER, Skott et al. (2013) observed that,
In these lines of research (constructivist or individual centred paradigm), the teacher is often seen as a major obstacle to change and a major problem of implementation. Consequently, the task for teacher-related research and development work is to solve the problem by changing teachers’ beliefs and providing them with opportunities to develop forms of knowledge that are deemed relevant for the profession…The more social approach seems at odds with the idea of implementation as conceived traditionally, and it places greater emphasis on understanding learning and lives as they unfold at schools and in mathematics classrooms than research on the individual teachers’ knowledge and beliefs. This is not to say that there is generally no interest in reform, but that the immediacy of impact of research on classroom practice is questioned. (Skott et al., 2013, pp. 502–503)
The social paradigm has been critiqued for not addressing the politics of discourse within which mathematics (and mathematics education) is located. However, it paved the way for discussions on creating socially just mathematics classrooms and practices. The socio-political paradigm foregrounds knowledge and power as key constructs in a social discourse (Gutiérrez, 2013) and acknowledges the connection between mathematics and social, economic and political conditions in society (Skovsmose, 2005). For example, Gutstein (2012) suggests that students must learn mathematics to understand the injustice in the world, their own social realities, and find ways of transforming their social situation. Pais and Valero (2012) lament the exclusionary character of the systems, such as schooling, by locating it in the neo-liberal capitalist discourse, which is designed to favour those who are privileged in the system. The socio-political framework is inclined towards problematising the structures that shape teaching and learning, and in some forms calls for action which is transformatory in nature.
All three paradigms (individual, social and socio-political) and their interfaces have contributed to and continue to shape the discourse in MER. Early cognitivist approaches responded to the information processing model of the human mind (de Freitas & Walshaw, 2016) and the constructivist paradigm guided the investigation on active teacher and learner in the teaching-learning settings. Social approaches drew attention to the multitude of processes and interactions that influence the complex work of teaching. The socio-political approach furthered the understanding of processes and people involved in mathematics teaching and learning to ‘illuminate the politics and power dynamics in mathematics education in all its forms’ (de Freitas & Walshaw, 2016, p. 7).
In the practice of MER, the selection of these frameworks attributes a positioning to the research(er), which remains rather fixated. For instance, research on teacher learning often gets associated with the individualist paradigm, collaborative research falls broadly within the sociocultural paradigm, and questions such as mathematics for whom fall within the socio-political paradigm. Such positioning of the research(er) into the pre-existing frameworks influences the selection of the problem, how it is studied, constructs that are foregrounded in analyses, and the presentation of findings. More importantly, the selection of a paradigm and consequently a theoretical framework determines how the participants of the research, particularly teachers and students, are imagined and constructed. The assumptions, decisions and constructs guided by a paradigm often remain strictly framed within the theoretical framework section of research communications. Additionally, researchers remain silent or implicit about the implications of the choice of specific frameworks, the assumptions of the paradigms that guide these frameworks, and how these influence connections between evidence, interpretations and theory.
In the contemporary discourse on MER, there is a resounding use of research to impact educational practice and policy design (Cai et al., 2017). Given this context, the influence of the positioning of the research(ers) on the images of teachers, students and learning, which are guided by the choice and use of theoretical frameworks, needs to be examined carefully. Further, the assumptions of the theoretical perspectives would need to be examined and discussed explicitly so that the research findings can be reported and interpreted within specific frames of inquiry.
In the following section, I will state the questions that guided the reported study, discuss the methodology for data collection, and use the data collected as part of my research on the work of teaching to foreground the tensions that arise from using specific frameworks. I will attempt to demonstrate that the research paradigms, despite their incommensurable ontological assumptions, can be reliably used for analysing classroom practice. We will notice that analysis from each of the standpoints offers useful inroads to making sense of the complexity of practice. In fact, in the struggle of aligning with the frameworks, I realised that analysing complex classroom situations requires the use of constructs from different frameworks, which might be considered radically different otherwise. Rather than filtering episodes of classroom practice through the lens of specific theoretical frameworks, it might be helpful to use a method of confocal analysis for unpacking the multidimensional and multi-layered complexity within the episode. Such an analysis raises some critical questions about the praxis of MER and calls for reflection. I speculate the need for creating dialogical spaces where the use of frameworks is guided by the questions from practice that we intend to make sense of, and not the other way around.
Guiding Questions and Methodology
The objective of the research reported in this article is to understand the affordance of theoretical frameworks in unpacking complex teaching practice. Further, an examination of the role of frameworks will create opportunities for reflexively examining the attributes of practice that remain within and outside the scope of analysis. The data were collected over a period of 2 years (2012–2014), as part of the research on investigating and supporting mathematics teachers in their knowledge of and responsiveness to students’ thinking. The initial research questions of investigating teachers’ cognitive and social sensitivity to students were revised in the process of research to focus on teachers’ knowledge situated in practice and learning from teacher–researcher collaborations. The article uses data from a case study of four teachers’ teaching in eight classrooms from a school which caters to students from mixed socio-economic and linguistic backgrounds, with parents’ occupations ranging from a cleaner to a scientist. The data were collected from the same site but at different time frames during the 2 years of the research study. The episodes discussed in this article are representative of the classroom practices observed in the larger study as well as of the classrooms observed outside the scope of this study. The episodes include (E1) a textbook excerpt from the chapter on multiplication, (E2) a teacher’s teaching of the textbook excerpt in classroom (fairly representative of the same lesson being taught by other teachers), (E3) post-lesson interaction with two teachers and a student on the teaching of the textbook excerpt, and (E4) a teacher’s attempt to support a girl student’s schooling. The first three episodes are connected through the textbook excerpt that is being referred to in the first episode. The fourth episode aligns more closely with the context depicted in the textbook and the lived experience of a girl student from the same classroom. The data are drawn from the transcripts created based on audio-video records and the researcher notes of classroom observations. The transcripts of interactions were prepared using audio records and researcher notes.
Analytical Description of Episodes of Classroom Practice
In this section, the episodes of classroom practice will be described using the analytical constructs from the existing literature. These constructs draw from different theoretical paradigms and hence reveal different aspects of the praxis of mathematics education. The analysis through multiple lenses offers a layered description of the episodes, thus making visible sometimes contradictory or different cultural and social complexities imbued within the process of teaching and learning.
Episode 1: Word Problem on Wages of Farm Workers
One of the multiplication word problems given in the Grade 5 textbook (NCERT, 2007, see Figure 1) is to find the wages of two farm workers for a given number of days using their per-day wage. The problem also gives the information on the minimum wage set by the government for farm workers. As stated in the problem, both workers are being paid less than the government norm, and the female worker (Thulasi) is being paid less than her husband. The task is to find their wages for the given number of days (49 for the female and 42 for the male worker) and then find their total wage. In two speech bubbles just below the tasks on multiplication and addition, the following statements are mentioned: (a) ‘Oh! He does not give them the minimum wage?’ and (b) ‘And why does he pay less to Thulasi and more to her husband? Discuss’. (NCERT, 2007, p. 145, emphasis in original).
Wages of Farm Workers—Multiplication (NCERT, 2007, p. 145).
The word problem involves five tasks: (a) finding Thulasi’s salary for 49 days, (b) finding her husband’s salary for 42 days, (c) finding their total salary, (d) discussing why they are not being paid the minimum wage and (e) discussing why the female worker is being paid less than the male worker. In the presentation, tasks (a)–(c) appear with the main text of the word problem, while tasks (d) and (e) appear in speech bubbles in a different font at the bottom of the page. The work situation depicted in the context is about the differential wages for physical labour.
The inclusion of problems which refer to social issues is not common in Indian mathematics textbooks. The word problem above is a part of the national textbook designed after the reformed National Curriculum Framework (NCF) 2005. NCF 2005 had an explicit goal of using education as a tool for social transformation (NCERT, 2005). The mathematics curriculum, following NCF 2005, emphasised processes such as estimation and approximation, problem solving to make math enjoyable and something to talk about by the students. Also, it mentioned ways in which textbooks could be written for students rather than for adults, for example, through the use of creative visuals, narratives, or through integration of disciplines using activities such as map reading (Rampal & Subramanian, 2012). Although the math curriculum document did not make explicit suggestions on how to use math as a tool for social transformation, the textbook designers, particularly for primary school mathematics, used the opportunity to include social situations as contexts for word problems (an example shown in Figure 1). In the textbooks, themes from work of different kinds were introduced, for example, using Mason’s work for counting, finding patterns and symmetries in bricks, using a woman rag picker’s story to introduce arithmetic operations and so on (Rampal & Subramanian, 2012).
Reflections on Episode 1
The textbook writers are fraught with challenges when using diverse social situations in a standard national textbook. For example, which aspects of diversity get selected and foregrounded, how are the situations represented and for what purposes, and how does the plurality of perspectives get discussed? What finally appears in the textbooks is a negotiation between the dominant practices, where the emphasis was on procedural and rule-based statements, and the ambitious intent of the textbook authors of creating a textbook which foregrounds different kinds of work situations (Takker, 2021). The differential presentation of tasks which are familiar and traditionally qualify as ‘mathematical’ (a)–(c), and those tasks which require a discussion on the social injustice (d) and (e), and their relative weightage in the textbook, represent this negotiated space. With problem-solving being an explicit aim of the mathematics curriculum, what kind of problems are foregrounded and what knowledge is needed to solve these problems remain questions of interest. Also, what kind of negotiations take place in the process of selection of content and activities represented in a national textbook within a multi-layered stratified society needs deliberation. How do textbooks envision the students (in other words, what kind of learners are assumed by the textbooks) and how does it position mathematics to be learnt? (Herbel-Eisenmann & Wagner, 2007).
Another challenge is to understand what students learn by engaging with mathematical problems which represent specific real-life situations of the kind that represent social inequality. Boaler (1993) cautioned that unless students are encouraged to take their own paths in the process of engaging with a real-life problem in mathematics, the possibilities of learning are meek. Consideration of the context of a task, activity, or example seems to show that students do not perceive school mathematics tasks as “real” merely because they have been given real world “veneer” [Maier, 1991], yet their mathematical procedures and performance are largely determined by context. This suggests that students interact with the context of a task in many different and unexpected ways and that this interaction is, by its nature, individual. Students are constructing their own meaning in different situations, and it is wrong to assume their general familiarity with or general understanding of the context. This acknowledgement does not preclude the use of contexts but suggests that a consideration of the individual nature of students’ learning should precede decisions about the nature and variety of contexts to be used. (Boaler, 1993, p. 16)
Following Boaler’s comment, making the selected contexts meaningful for all students depends on opportunities for interaction with the situation being represented. Clearly, the wages context has the potential to invoke discussions about the organisation of work and labour in society, and how the monetary value attributed to labour perpetuates and reproduces social inequalities of gender and class. However, since the integration of issues around injustice and mathematics is uncommon in formal schooling and teacher education, deliberate efforts and prior didactic work are required to utilise these learning opportunities to generate a discourse on inequality through mathematics. Gutstein (2012) reckons that conscious efforts need to be made for a dialectical interweaving of social justice issues and mathematical competencies in mathematics classrooms. He further argues that developing a deeper understanding of the socio-political issues using the tools of mathematics and strengthening the procedural and conceptual proficiencies in mathematics in the process of studying the complexities of issues are crucial aims of mathematics education. However, paying attention to these goals of mathematics simultaneously is both challenging and ambitious. Discussions in mathematics classroom on aspects of real-life that are represented in such contexts and how it links with the larger issues of social inequality in society would be a significant starting point. The next episode is on how the work context is dealt with in a mathematics classroom, which is fairly representative of classroom observations across different teaching and school settings.
Episode 2: Teaching of the ‘Wages’ Problem
In a Grade 5 classroom with students from mixed socio-economic and linguistic backgrounds, an experienced elementary school teacher, Reema (pseudonym), teaches the word problem on wages depicted in Figure 1. In an earlier interview (2 months before the lesson on wages was being observed by the researcher), Reema mentioned that she appreciated the use of real-life contexts in textbooks. In her words, ‘contexts make the math real… meaningful for children… they [students] understand why learn math’. While teaching the wages context as part of the multiplication chapter, Reema asked a girl student to read aloud the problem to the whole class. She then asked the students what the problem was about, to which the students responded ‘salaries’. She then pointed out—‘did you see that, for the same [amount of] work how much is Thulasi getting (pause) and how much her husband is getting?’. Students answer in chorus, ‘yes’. Reema paused and then asked the students to find specific information from the given problem statement, using prompting questions such as, ‘what is Thulasi’s salary for each day?, for how many days do we need to find her salary?, how do we find the total?’ and so on simultaneously recording this information in the format shown in Figure 2. The collected information was then used to solve tasks (a)–(c), using the multiplication algorithm, on the board by the teacher and in discussion with the students.
Board Work on Wages Context.
As revealed from Reema’s comments, she is appreciative of the inclusion of real-life contexts as they make mathematics meaningful for the students. While teaching the wages context, Reema asked the students ‘What is the problem about’, to which the students responded, ‘salaries’. The choice of the word ‘salaries’ by the students instead of the word ‘wages’, which was used in the textbook, indicates a greater familiarity with the former. Reema noticed the gender discrimination mentioned in the wages problem and began talking to the students about it. After a pause, however, she proceeded with a discussion of the mathematically familiar aspects, that is, on finding the information necessary to solve the word problem and using the algorithm to multiply. The less familiar parts of mathematics, that is, injustice due to differential treatment based on social categories, the monetary value assigned to the labour based on the number of days of work and so on, remained undiscussed. Reema’s selection of the mathematically familiar parts of the word problem from the textbook can be justified using her prior experience of schooling and collective experience of teaching over the last 10 years. Identifying with the role of a ‘knower’ in a mathematics classroom, Reema teaches the content that is supported by her training, teaching experience and the wider culture of teaching. Reema supported students in identifying the relevant information from the word problem and using it to solve the multiplication and addition tasks.
We can unpack the second episode (E2) by understanding the role and involvement of teachers in the development of the reformed curriculum and curricular materials. NCF 2005, despite its progressive rhetoric, has been criticised for being silent on how teachers can be supported in engaging with the reform initiatives, and most importantly, on the teachers’ agency in using the content in the classroom (Batra, 2005; Takker, 2021). The reforms suggested in the curriculum and consequently the textbooks were communicated to teachers through academic circulars, directives to the schools, ignoring the long-term support that teachers might need to implement the reformed vision in practice (Subramaniam, 2023). Dewan (2009) argued that beliefs contrastive to NCF 2005 are held not just by teachers but by teacher educators, school leaders and administrators, directors and faculty members of teacher education institutes and so on. Further, the challenges in dealing with demanding mathematical content, including multiple methods and representations, using mathematics to make sense of the injustice in society, and providing students with the mathematical tools to make sense of and eventually change the world around them, remain ambitious but unattempted through the top-down reform agenda. I have tried to articulate the challenges that teachers face in implementing such a reformed curriculum without adequate support from the systemic structures, teacher educators and the larger school community in my previous work (Takker, 2017, 2021).
Reflections on Episode 2
Contextual word problems have been used in mathematics textbooks and classrooms for a very long time. Research on the treatment of word problems distinguishes between dealing with them in purely ‘mathematical’ ways and using them for ‘sense-making’ (Verschaffel et al., 2001). Research following a range of paradigms: cognitive, social and political, concurs that word problems involving real-life contexts are authentic modelling situations (Verschaffel et al., 2020). Chapman (2006) analysed teachers’ narrations on the use of real-life word problems in the classroom and found that while all teachers related to the motivational purpose of connecting mathematics to real-life, they struggled to link the problem-solving aspects to the context of the problem. Chapman distinguished between teachers using a ‘paradigmatic’ and a ‘narrative oriented’ approach to solving word problems. In the former, teachers used the word problem as an object and focused on the structural or mathematical aspects, such as identifying and carrying out the operation, similar to Reema’s approach of dealing with the word problem on wages. In the latter approach, the emphasis is on how contexts are experienced by the students, that is, making sense of the problem and the solution using realistic considerations. Chapman concluded that both these approaches are useful in instruction. We find examples of selecting contextually relevant content in the work of Gutstein (2012) and Skovsmose (2005), where they use mathematics as a window to unpack oppression in different situations of banking and flight booking, respectively.
The role that teachers take in selecting and orchestrating the content that is meaningful and conceptually useful for students is an important consideration. There is a concern about teachers bringing their own biases into the classroom if given a free hand in deciding the content and pedagogy of a classroom. On the other hand, the top-down approach of training teachers to implement the curriculum in their respective classrooms rips teachers of their agency and constrains them, using the knowledge from outside the classroom, which might be useful in supporting students’ learning.
Several questions about teachers’ imagined and ascribed roles can be asked here: why is the teachers’ role envisioned only as implementers of reforms and curriculum?, how can we imagine teachers’ participation in ways so that they become collaborators in thinking about foregrounding issues of social (in)justices in curriculum development?, and what would it mean for a diverse group of teachers, some of whom belong to the marginalised communities themselves, are experiencing marginality through their students, are silently working in their individual capacities to support students, and those who are disengaged with the problems of students for different reasons.
Episode 3: Post-lesson Discussion on the ‘Wages’ Problem
After the lesson observation, the researcher and teacher discussed the teaching of the word problem; a transcript of two such interactions is reproduced in Excerpts 1 and 2. Excerpt 3 depicts the post-lesson discussion with a few female students.
Post-lesson Discussion with Reema.
RS: Researcher, TR: Teacher Reema
In the post-lesson discussion, Reema acknowledged the relevance of the context on wages. She also noticed the differential wages based on gender. Her comment, ‘that is how it is na’ acknowledges the internalisation of such discrimination. It seemed that she justified her struggle on how and whether to deal with it in the classroom, by classifying the gender-based discrimination, manifesting in different wages, as ‘outside’ knowledge. Although she does not consider the discrimination as information for students and acknowledges that it is knowledge, she finds it difficult to handle such situations in the classroom. Her concern is also linked to the age group of students, evident in her questioning the age-appropriateness of this knowledge. Excerpt 2 reflects the response of another teacher from the same school reflecting on (not) dealing with the wage context during teaching.
Post-lesson Discussion with Pallavi.
RS: Researcher, TP: Teacher Pallavi
Pallavi (pseudonym) seems confident in omitting the discussion on the differential wages as part of the lesson. She mentioned ‘avoiding such’ discussions and how the ‘story’ (meaning, context) might take up time that could be spent on practising the multiplication algorithm (see Excerpt 2). She also mentioned the practice of following the textbook while she chooses to do the word problem without dealing with the conflict.
Mathematics teachers have been a part of the culture of teaching and training, where they are made aware of how to use contexts as word problems. The teacher education does not really engage teachers in discussions on the affordance of social contexts or how contexts can serve as opportunities to make sense of larger social realities using mathematics. Also, until the advent of these textbooks (which attempt to make connections between social realities and traditional mathematical content), the teachers were never expected to deal with conflict issues as part of mathematics teaching and learning. By conflict issues, I mean social concerns which refer to intersectional inequalities in society, where opposing perspectives are offered and require discussion. It seems from teachers’ reflections (as also from interactions with other teachers who were struggling in dealing with conflict issues in the mathematics classroom) that they are not prepared to deal with them in the classroom, and their intuition derived from the culture of schooling classifies such knowledge as ‘outside’ school knowledge (Takker, 2017). While Reema acknowledged the inequality, it has been normalised for her, through her own experience of being a female school mathematics teacher and of seeing workers around her being paid unjustly for their work. As a researcher, I intended to understand the teacher’s perspective on dealing with such issues in their classroom and therefore asked questions which helped in unpacking it further. I acknowledge, however, that in that moment, I did not offer any suggestions on how Reema could deal with such issues in the classroom. The third excerpt is an interaction with three girl students from the same school who have just finished doing this word problem in the mathematics class.
The post-lesson interaction with the students (see Excerpt 3) revealed that they attempted to make sense of the context while having further questions about ‘government norm’ and wages for different kinds of ‘jobs’. One of the students (GS3) acknowledged the presence of differential wages, indicating that she is experiencing it. Further interaction with GS3 revealed that her mother was offered the job of cleaning, after the passing away of her father, at a much lower wage. She expressed how her mother had to take the job since the house in which they were living was connected to the job, and the family needed the money for their education and other expenses.
Interaction with Girl Students on the Wage Context.
RS: Researcher, GS: Girl Students (1, 2, 3)
Reflections on Episode 3
We notice the layers of sense-making about differential wages among different social actors engaging with the context. The teachers are confused about how to deal with such concerns in the mathematics classrooms and also point to a major concern of making the algorithmic knowledge accessible to students from the marginalised section. The in-school knowledge of formal procedures, referred to as classical knowledge, helps students, particularly from the marginalised communities, pass the gatekeeping tests and offer opportunities for further education, career and social mobility (Gutstein, 2012). Therefore, dismissing teachers’ conflict would be engaging with a problem without its roots.
It is also important to note that such conflicts are not unfamiliar to the students, particularly students from this school context. Since such realities are a part of teachers’ and students’ lived experiences, their discussion in the classroom has the potential to create a dialogic space within mathematics classrooms to engage with such inequalities and understand them through the lens of mathematics. However, Christiansen (2007) cautions that addressing these conflicts as legitimate knowledge and a part of students and teachers’ everyday lives requires reframing the relation of participants with mathematics. It also requires challenging the dominant forms of knowledge structures that have solely become defining features of mathematics classroom cultures. In their teaching, Wright et al. (2022) argue that one way of dealing with it could be through making progressive pedagogies visible to students while teaching mathematics to discuss and make accessible an equitable mathematics curriculum.
The analysis of teachers’ struggles in handling conflict issues in the classroom raises several questions about the representation of people and their work in a national textbook, integration of the elusively defined mathematical content knowledge and the context in which it is embedded and used, and the inadequacy of the support offered to the teachers in dealing with issues which are challenging while recognising teachers’ demanding work conditions. How can we use the research frameworks to unpack the complex and dynamic situations that teachers encounter while negotiating with the mathematical and social aspects of a word problem in teaching? At this point, it is relevant to mention the remark of a school teacher, who said that ‘for researchers it is a (select) moment, but for me it is one of the thousand things that I need to take care of’.
E4: Teacher Supports a Girl Student to Continue Schooling
In another incident, Teacher Reema invited me to a walk outside the school. She expressed her worry towards a student (GS3 in Excerpt 3), who belonged to a working-class family. Her father was a cleaner at a government facility. Garima’s (pseudonym) education and the house in which the family was living were offered by the government. After her father passed away, Garima and her mother were financially stricken. Reema did not want Garima to drop out of the school and therefore, was brainstorming ways of finding scholarships for her (see Excerpt 4). Despite her efforts, Reema did not succeed in finding Garima a scholarship and therefore decided to pay her fees. She could not have disclosed that she paid the fee, as it may not be acceptable to the school authorities. Apart from offering Garima the needed financial support, she strongly suggested that Garima attend after-school sessions where she (and sometimes I) interacted with Garima.
Teacher’s Concern Towards a Student.
Today Reema invited me to walk with her outside the school. It was surprising that Reema wanted to talk and that too outside the school premises. She began talking about a girl student, Garima, from her class and how she is coping with mathematics. She mentioned that Garima is a single parent child, is hard working and is a student who is improving in her math performance. While Reema has offered her support through the after-school sessions (which Reema took for students who did not perform well in the math exam), Garima could not attend these sessions due to her responsibilities at home. Garima’s father was a worker at a government facility and had passed away due to an exposure to radiation. The father’s death made it difficult for Garima and her mother to survive financially. So, her mother decided to do cleaning chores at neighbouring households, to which Garima would assist after school. Garima was generally a quiet student and on that particular day Garima told Reema about her inability to continue the school as her mother could not afford the school expenses. Reema found it disturbing and was struggling to think of ways of sustaining Garima’s education. Reema expressed her helplessness. We discussed how it would be difficult that the job of the father be passed on to the mother, due to the physical strength and basic qualification needed for the job. Also, even if she did get the same job, she would be paid much lesser. We brainstormed some ways in which Garima could be supported financially and academically. While Reema strived that the school offered Garima some scholarship, it did not materialise. With the increasing pressure on Garima’s mother to leave the house (which was allotted due to her father’s work), Reema decided to pay Garima’s fee and take care of her other school-related expenses. Reema wanted to support Garima without compromising her professional identity, which meant that she could not disclose it officially the financial support she was offering after she was denied such support from the school. (Researcher notes, 20131223)
It is not uncommon for teachers in India to support their students financially in their personal capacities. Education, in India, is perceived as a means for social mobility. Consequently, parents and teachers are known to be harder on students who belong to underprivileged backgrounds, since education is considered to be a way out of their misery. The actual work of teachers is not limited to instruction in the classroom but involves supporting students financially, academically, and emotionally. Much of this emotional labour remains invisible and unnoticed. The prevalence of this unofficial support for educating students from socio-economically weaker backgrounds has taken different forms in different states within the country. Sometimes this effort is exploited by the state and used to shed off the responsibility of enabling students to access their right to education, as teachers are made to internalise this as part of their work.
Reflections on Episode 4
There are several issues for reflection emerging from the fourth episode. The first being the corporeality of this interaction. While Reema had several conversations about the teaching of mathematics with the researcher, it was only outside the physical premises of the school that the concern about supporting Garima was discussed. Further, Reema’s support for Garima’s education is multifaceted. She insists that Garima takes financial and academic support, which goes beyond the expectations of a teacher’s official work. However, what is concerning is the lack of connection between the two situations (Episodes 1 and 4) which are inextricably linked—one depicted in the textbook, the
Revisiting Existing Frameworks to Analyse Practice
In order to make sense of the four episodes, I have borrowed the literature, although not comprehensively, from all three frameworks: psychological, social and socio-political. The frameworks used in the process of analysis focus on teacher knowledge, classroom discourse, teacher-textbook connection, curriculum development and enactment, and the selection and treatment of content in the culture of mathematics teaching and learning. The use of existing frameworks, by breaking boundaries and moving across paradigms, helps in an initial unpacking of a complex classroom situation. I shall now reflect on the affordances of using analytical constructs belonging to the three different frameworks to understand the classroom situations.
The individualistic cognitivist framework would incline us to reflect on the roles of the contexts (and their connection with content), the teacher and the student in distinct ways. The contexts, such as those depicted in Episode 1, serve as tools to create motivation towards mathematics, particularly (and questionably) at the primary school level. As Boaler (1993) suggested, contexts need to be viewed from what kind of engagement they would have with individual learners. Further, the understanding that contexts are merely tools for making mathematics meaningful for students might need re-examination for the hierarchy that might get created between the mathematical content (as the goal) and context (as the means to achieve the goal). The teacher knowledge frameworks, proposed by Shulman (1986), extended and expanded by Ball et al. (2008), Ma (1999) and Rowland (2009) help in identifying demanding teaching situations, which offer an opportunity to understand the complex mathematical knowledge required for teaching. Often, the data on teachers’ responses to teaching situations, in and outside the classroom, is interpreted in terms of the presence or absence of types of knowledge, eventually leading to what Da Ponte and Chapman (2006) classify as deficiencies in teachers’ knowledge. For example, Episode 2 might reveal that the teachers deliberately omitted the inclusion of social justice issues due to fractures in their mathematical knowledge required for teaching. Episodes 3 and 4 would be interpreted using categories of ‘in’ and ‘outside’ school knowledge as attempts to maintain the status quo about mathematics as an abstract discipline untouched by the social dimensions.
The sociocultural framework would raise questions about interactions between teachers, textbooks, content and students. The interactions between the content, teacher and student would unveil patterns that get reified in the classroom space, which constrain knowledge sharing and learning opportunities between the teacher and their diverse students. The lesser opportunities of questioning the authority (content, textbook and teachers) for students further reifies the reproduction of didactical knowledge of mathematics, which remains isolated from the social contexts and lived experiences of students and teachers (as revealed from Episodes 3 and 4). Further, classroom discourse is moderated by artefacts such as textbooks, which (in the reported case) present reform ideas to the teachers. The relevance of the content that is selected and enacted in school mathematical environments for diverse learners requires further probing. The questions about what content is relevant and why (Gutstein, 2012), who does this content serve and what kind of mathematics do learners need to understand and participate critically in the fast-changing society, would need to be posed when designing curriculum.
Using the socio-political perspective, we would need to engage with the realities of the classrooms and social contexts of students, teachers, and the development of content within mathematics education. Going beyond the judgments about teachers’ knowledge, students’ developmental levels, and interactions between teachers and textbook, we would need to ask questions about the nature of complex relationships between the teachers and the mathematics, and with the content selected and represented in the textbook. Instead of using our research to classify or measure teacher knowledge, researchers have used the frameworks on teacher knowledge to unpack the complex intermix of teachers’ knowledge manifested in practice. Further, acknowledging that teacher knowledge (read as
Evidently, the choice of a theoretical framework or a network of frameworks guides an in-depth analysis by foregrounding specific aspects of classroom practice; the images constructed in this manner are limited in understanding the complex and multi-layered realities of classrooms, learning and teaching. A confocal analysis has the potential to offer insights about the complex nature of classroom practice through a flexible yet informed movement across different constructs located within multiple frameworks. Such analysis has the prospect of raising critical questions about the praxis of MER, some of which have been consolidated in the next section.
Concluding Thoughts
The constructs used for analysis of practice emanate from a variety of theoretical frameworks and offer deeper insights into the complex situations arising in teaching, described through the four episodes. An understanding of complexity requires more than reducing slices of reality into manageable, explainable or discernible facts. However, it is also worrisome how each of the narratives (based on the choice of frameworks) would paint a picture of the teachers and students in the discourse on mathematics education, which potentially influences policy and practice at a large scale. Therefore, it is essential to acknowledge that researchers’ theoretical positions make specific descriptions of teaching practice accessible. However, these images of teaching and cultures of teaching are guided by the theoretical lenses selected by the researchers, which are not comprehensive depictions of the complex and multi-layered realities. Here, I also acknowledge the frames that I have selected to make the arguments in this article. Further, it seems that a reflexive analysis of classroom practice can help raise questions about the goals of MER and mathematics teacher education (MTE) research in specific local and global contexts. A deeper engagement with the contexts that researchers are expected to analyse would require examining our own positionalities, paradigms of research praxis that we are a part of, and most importantly, an awareness of the affordances of the frameworks that we select to represent realities of schools and classrooms. The research(ers’) positioning plays an important role in constructing descriptions of practice, which has implications for policy and practice.
The frameworks can also be extended from developing an understanding of teachers’ struggles in dealing with social justice issues, to raising questions about teachers’ involvement in the process of deciding what goes on inside the mathematics classrooms. Such discourses would also need to be made a part of the teacher education. Teachers, having histories of schooling and education, engage with these ideas in a variety of ways, without receiving adequate support and engagement in decisions about content and pedagogy (Batra, 2005). The socio-political lens suggests how teachers are just a
Some further questions which arise from the linking of the four episodes with the literature in MER are: (a) how and why do teachers differentiate between the two situations of differential wages: one that is discussed in the textbook, and the other that is encountered by a student from her classroom?, (b) what does the kind of mathematics that is selected and taught, mean for a student who encounters such conflict situations in real life, for whom the context is not merely a word problem but a part of their lived experience?, (c) what is the role of mathematics education research(er) in engaging with a discourse that challenges the
Footnotes
Acknowledgement
The author thanks Prof Richard Barwell and Prof Srinivasa S. Rao for their suggestions on an earlier draft of this paper.
Declaration of Conflicting Interests
The author declares no potential conflicts of interest regarding the research, authorship and/or publication of this article.
Funding
The author received no financial support for the research, authorship and/or publication of this article.
