STRUCTURE AND CLOSURE OF SCHOOL MATHEMATICAL PRACTICE - THE EXPERIENCES OF KRISTINA
Ann-Sofi Röj-Lindberg
Abo Akademi University
Aroj(at)abo.fi
In the paper I present some preliminary results from a story-telling case study, the main focus of which is to give an account of the school mathematical practice at one particular Finland-Swedish lower secondary school. The school mathematical practice is looked upon through the voice of the student Kristina. Kristina told me about her experiences of participation in school mathematics in several qualitative interviews during the three school years 7, 8 and 9. The analysis of the interviews has so far resulted in the creation of several preliminary themes indicating her experiences. In this paper I limit the presentation to one theme, Structure and Closure. The paper meets the requests for more studies where researches and teachers listen to the voices of students in order to understand and learn from their school mathematical experiences and expectations (see e.g. Burton, 1994; Corbett & Wilson, 1995).
School mathematics as an individually experienced socio-cultural practice
Many researchers within the field of mathematics education nowadays accept a relational view of learning within which the knowing and identity of persons can be seen as social constructions and as results of the person’s experiences in and contributions to social and cultural contexts. The processes of creating, teaching and learning mathematics and the outcomes of the processes can from this viewpoint be considered as social, cultural and interdependent phenomena (see e.g. Atweh, Forgasz, & Nebres, 2001; Lerman, 2000; Nunes, 1995). Hence, school mathematical practice as well as mathematics as a scientific subject are seen as existing within social systems of thought and culture (Ernest, 1998).
In this paper the level of analysis is the school mathematical practice in a particular school where some mathematics teachers intended to change their teaching practices with the support of colleagues. Elsewhere I have pointed at constraints the teachers experienced in trying to free themselves from the safety of habitual social and cognitive patterns (Röj-Lindberg, 2003, 2006). Practice is conceptualized in different ways by different researchers within social constructivist and socio-cultural perspectives. Paul Cobb defines ‘practice’ as an emergent aspect of the communication in the classroom. Within a lesson design that focuses negotiation in communicative interactions Cobb is researching how students’ mathematical conceptions emerge and become established as a taken-as-shared mathematical practice in the classroom. (Cobb, 2000). Jeppe Skott describes classrooms as communities of mathematical practice and discusses practice as an emergent phenomena influenced by the teacher’s school mathematical images, which are unique for each teacher. School mathematical images are in Skott’s study defined as teachers’ personal interpretations of and priorities in relation to mathematics, mathematics as a school subject and the teaching and learning of mathematics in schools. Skott proposes that the school mathematical images contributes to the development of students’ learning opportunities (Skott, 2000). Marilyn Goos and colleagues also discuss the mathematics classroom as a community of practice, but with a focus on interactions of teachers and students to describe the social conduct in a collaborative classroom. Taken together these interactions reveal the emerging classroom culture and can be taken as indications of whether the mathematical practice in the classroom makes sense to students. Goos and her colleagues further argue that both the teacher’s and the students experiences of schooling can act as potential barriers to reform of school mathematical practice (Goos, Galbraith, & Renshaw, 1999). In researching students’ goals Simon Goodchild approaches ‘practice’ as the context of routine tasks that students are engaged in a mathematics classroom (Goodchild, 2001). From a philosophical viewpoint Anna Sfard discusses meta-rules as her lens into ‘mathematical practice’ and uses Wittgenstein to illuminate her points of view. According to Wittgenstein “the person who follows a rule has been trained to react in a given way. Through this training the person learns to respond in conventional ways and thus enters into practice” (Sfard, 2000). Miller and Goodnow propose an epistemological definition of ‘practice’ as actions that are repeated, shared with others in a social group and invested with normative expectations and with meanings or significance that go beyond the immediate goals of the action’ (Miller & Goodnow, 1995, 7). I find their definition of practice as satisfactory if ‘action’ is considered to comprise both cognitive and socio-cultural activities and ‘meaning’ as referring to individually experienced as well as socio-culturally constructed and valued meanings.
The word ‘practice’ is used with different connotation in common language and it is difficult to avoid a certain ambiguity in its use also in this paper. I sometimes refer to practice as the actions of teacher and students in the classroom. Admittedly, most of the individually experienced and socially valued meanings of school mathematics are related to actions in the classroom. However, when I use ‘school mathematical practice’ my intention is to cast the conceptual net more widely and include as well the images and pedagogical intentions of the mathematics teachers as a social group within the collaborative reform work.
My main focus in the paper is on the student Kristina and her experienced meanings of school mathematical practice during lower secondary school. A basic assumption is that the story she tells in interviews is the meaning she imposes on her world. Her story is the truths of her experiences, not an ‘objective’ reflection of an ontologically real reality (von Glasersfeld, 1991). Thus, it is important to note that her experienced meanings might differ from meanings constructed by other students and by the teachers who were participants in the same mathematics classroom. For example, Ben-Chaim, Fresko and Carmeli (1990) write as one conclusion from a survey to teachers and students that the teachers saw the classroom environment as more diverse than the students. Another remark of importance is the use of the concepts ‘knowing’ and ‘knowledge’. ‘Knowing’ refers to personal meanings of socially and culturally constructed ‘knowledge’, like for instance of mathematics in textbooks, of activities in school or of what it is to do mathematics and be a learner in a classroom.
Theoretical considerations
In the theoretical frame I am inspired by socio-cultural perspectives on human learning and development and a cultural-psychological view on the human mind (Bruner, 1990, 1996/2002; Rogoff, 2003).
Within the socio-cultural perspective that I adapt, learning is participation and development is a function of ongoing transformation of roles and understandings in the socio-cultural activities of the communities in which the individual participates. I argue that the students and teachers within the school mathematical practice that is the focus of my study can be interpreted as forming developing communities of learners in this sense. An essential theoretical argument in this perspective is that culture, including mathematics and school mathematical practice, is seen neither as something static that adds on or is gradually internalized by the individual mind nor as a surrounding to the individual which is gradually changed to fit the developing mind. Culture is the common ways that participants in a community share even if they may contest them (Rogoff, 2003). From a cultural-psychological viewpoint Bruner argues that we should think of culture as something that is in the mind of persons (Bruner, 1996/2002, 200).
This perspective on learning and development is radically different from one that describes learning as transmission of knowledge from authorities outside the individual or as acquisition or discovery of knowledge by the individual. Moreover, it challenges the idea of a boundary between internal and external phenomena, as for example between a students knowing of school mathematical practice and the cultural tools used in this practice. Among cultural tools Roger Säljö includes intellectual tools like systems of ideas and discourses and physical tools like textbooks, mathematical symbols and diagrams (Säljö, 2005).
If this perspective is accepted, a researcher may place an individual student’s knowing or the school mathematical practice in the foreground without assuming that they are actually separate elements. It also makes sense to discuss an individual’s school mathematical experiences in terms of situated knowing or situated understanding and to acknowledge the importance of context for the development of these experiences. The research studies by for instance Boaler (1997), Stigler and Hiebert (1999), Boaler and Greeno (2000) and Nardi and Steward (2003) are well grounded examples of the situated nature of school mathematical experiences: it is impossible to separate the knowing of individuals from the social and cultural practices in which the individual participates.
Barbara Rogoff (2003) uses the concept participatory appropriation instead of internalization or appropriation to refer to the change and development in knowing resulting from a person’s guided participation in an activity. As people participate they are making a process their own, they are not taking something from other persons like you take a thing and add it on to something you already possess. But they are creatively taking for their own use and changing how they may treat future situations that they see as related. Processes of guided participation are defined to incorporate guidance in the sense of direction of a shared endeavour. It is participation in meaning that is the key issue, not necessarily in shared actions of the moment. A person who is observing and following without direct contribution to the decisions made by other people is also participating in the activity. Moreover, a person who acts alone is also participating in a shared endeavour as he or she follows and builds on community traditions for the activity. For instance, a student doing homework is participating with guidance provided as well by the teacher and textbook writers, who may have set the tasks and approaches to be used, as by classmates, and family members, who may support some approaches and suppress others. Also, a mathematician like Andrew Wiles, who grappled eight years with proving the Last Theorem of Fermat and without direct involvement from peers, was building on socially validated traditions for mathematical activities.
Within a theory of development as transformation of participation people are seen as contributing to the creation of socio-cultural processes and socio-cultural processes as contributing to the creation of people (Rogoff, 2003, 51). An example of how this kind of mutuality and interdependence might operate within school mathematical practices can for example be seen in what homework means to a student. The students might perhaps imagine it as a questioning activity in the classroom where teacher wants me to show that I know the correct answer. Then by displaying the correct answer, she is participating in the creation of this activity as one where the student is expected to give the one answer that is correct. When the teacher evaluates the answer as a good answer the teacher fulfils her expectations and they both contribute to the emerging implicit agreements about ‘homework actions’, about responsibilities for actions, about when to do what and how to do it in this particular classroom. In this process the students and the teacher together create normative expectations for the kind of mathematical competence valued, what it means to be a good student and an effective teacher in relation to homework actions. Or the student might imagine homework as questioning activity where teacher wants me to argue mathematically for my answers. If this is the case a homework review in the classroom is an activity constituted by a very different participation structure with different expectations and implicit agreements. At the level of formal engagement the student may be doing the same thing in both examples, i.e. answering questions set by a teacher about homework. These examples indicate that the socially valued meaning of homework review might be constructed very differently in different school mathematical practices. They also indicate that a participation perspective on learning and development requires considerations of not only what it is that the individual is participating in but also the experienced meanings of these activities.
Jerome Bruner states that a culturally sensitive psychology should be based not only upon what people say caused them to do what they did. It is also concerned with what people say others did and why. And above all, it is concerned with what people say their worlds are like (Bruner, 1990, 16). Bruner further states that cultural psychology seeks out the rules that human beings bring to bear in creating meanings in cultural contexts of practice (p. 118). Thus, I argue, it becomes important to foreground school mathematical practices as seen through the eyes of the students. Moreover, this research perspective is needed if students are considered as legitimate participants in, not only as beneficiaries of school mathematical practice (Corbett & Wilson, 1995).
Methodological considerations
Methodology is interpreted by Wellington as the activity when the researcher chooses, reflects upon, evaluates and justifies the research methods she uses to answer the research questions (Wellington, 2000, 22). As already indicated the topic of this paper has emerged from a case study which involved a group of mathematics teachers who participated in reform work to develop their mathematics teaching. I took part in the process as research-assistant within a commissioned research work including a complex set of tasks, among them to bring the voices of students into the reform work. I approached the research work from the viewpoint of being a mathematics teacher myself and I used a mixture of research methods to build up a case record. I got to know the teaching traditions and the teachers’ pedagogical intentions from interviews with teachers and students, from surveys and from being a participant as well as a non-participant observer and note-taker at action research meetings and observer of lessons. It is thus obvious that I cannot act as an outsider detached from the story of school mathematical experiences that is the focus of this paper. As the research process started with a very open bottom-up approach it is natural that a retrospective format is used for reporting the research. Bruner (1990, 119) considers retrospective reporting as viable when personal meanings of activities is the focus for research inquiry.
The interest in gaining insights to the views of students was developed into more specific research questions of which I in this paper will discuss the following: