Conflicting Practices In School Mathematics

Author(s):

Despina Potari(submitting)Jeppe Skott(presenting)Chara Papakanderaki

Conference:

ECER 2014

Network:

24. Mathematics Education Research

Format:

Paper

Session Information

24 SES 09 A, Participatory Approaches in Mathematics Education

Paper Session

Time:

2014-09-04

11:00-12:30

Room:

B113 Sala de Aulas

Chair:

Birgit Pepin

Contribution

Classroom research has documented tensions and challenges related to the adoption of reform-based practices in mathematics (Maass, 2011; Skott, 2001; Törner, Rolka, Rösken, & Sriraman, 2010). Often they are viewed as results of discrepancies between reform intentions and teachers’ knowledge and beliefs. This interpretation is based on acquisitionism as a metaphor for teachers’ learning and knowing (cf. Sfard, 2008).

However, research on and with teachers increasingly interprets professional activity in more participatory terms (Potari, 2013; Wagner & Herbel-Eisenmann, 2009; Walshaw, 2010). We also adopt a participatory approach and more specifically we use a Patterns-of-Participation framework (PoP) to understand the situated and dynamic character of classroom interactions and of teachers’ contributions to them (Skott, 2013). Informed by symbolic interactionism (Blumer, 1969; Mead, 1934), PoP views the meaning of the objects towards which a teacher acts as emerging as she takes the attitude of individual and generalized others, e.g. students, colleagues, and the reform discourse. Teachers’ contributions to the interaction are viewed as dynamic relationships between her participation in different social practices and what Holland et al (1998) call figured worlds, i.e. collective as-if worlds that shape and are shaped by the social practices through which they unfold. To be specific, we take it as empirical questions if and how the teacher re-engages in discourses related to mathematics and mathematics education as promoted by her teacher education programme and if and how these discourses are transformed as the teacher engages in classroom practices.

PoP has been used to study teacher identity and the role of teachers in classroom interaction in Sweden and Denmark (Palmér, 2013; Skott, 2013; Skott, Larsen, & Østergaard, 2011). Here, we use the framework to analyse teaching-learning processes in a Greek secondary school. As suggested above, we do so to understand the role of the teacher’s participation in different prior practices and discourses for the ones that emerge in the classroom.

Elena, the teacher in question, has a university degree in mathematics. Also, she has just completed a master’s course in mathematics education and begun her studies for a PhD on mathematics education at university level. Elena works in a public high school that follows the national curriculum and uses a mandatory textbook. High school teaching is generally oriented towards university entrance exams, which are very competitive. In most cases parents pay for private tuition to improve their children’s chance of success in the exams. Elena worked with private tuition in mathematics for eighteen years before moving to secondary school teaching four years ago.

Both Elena and we perceive tensions within the practices and discourses in which she has previously participated and between some of these prior practices and discourses on the one hand and the ones that unfold at the school and in the classroom on the other. The university entrance exams as well as most approaches to private tuition are highly procedural and impose constraints on classroom practices with a different orientation. In contrast the masters’ programme that Elena took is more reform oriented and in conflict with the experiences and expectations of her colleagues, the students, and the parents.

Our first research questions are if and how Elena (re-)engages in the above-mentioned prior practices and discourses during instruction, and if and how her participation in them is transformed in the process. Our next question asks if and how a PoP-interpretation leads to different understandings of the teacher’s contributions to classroom interaction than one that views classroom practice as an enactment of her knowledge and beliefs. This latter question addresses methodological and theoretical issues of general interest in mathematics education.

Method

This is an instrumental case study of Elena (Stake, 1995). She was eager to participate and was selected because of her enthusiasm for mathematics and mathematics teaching as evidenced during her master’s studies (taught in part by the second author) and her commitment to the reform discourse. This makes the case of Elena a critical one when viewed against the backdrop of the traditions the Greek secondary schools, dominated as they are by the entrance exams. The instrumentality of the case consists of the opportunity for theoretical generalizations that it offers. The data include ten audio-taped, transcribed lessons on functions from the spring term in a grade 10 classroom. Elena was interviewed after each lesson and two longer interviews were conducted, one before the data collection began, the other five months after it ended. In the interviews, which were transcribed in full, we discussed general aspects of mathematics education, often beginning with what we as observers considered critical incidents from Elena’s classroom (cf. Skott, 2001). The incidents were selected because they reflected tensions between Elena’s participation in the reform discourse and the students’ engagement with more traditional and procedural approaches to mathematics, because they exemplified Elena’s involvement in a multitude of competing practices and discourses, and because Elena’s instructional decisions significantly influenced the students’ subsequent learning opportunities. The analyses of the observations and interviews were inspired by recent developments of Grounded Theory (Charmaz, 2006). We used methods like constant comparisons and memo-writing without subscribing to the objectivist connotations sometimes associated with grounded theory. From the analysis of the classroom data some conflicts and tensions emerged. In the present paper we focus on one particular conflict, i.e. the one between Elena’s grade 10 students’ algorithmic and procedural approach to functions and her own conceptual emphasis on functions as sets of ordered pairs. In the interviews, Elena was asked to reflect and comment on the emergence, character, and significance of this conflict, and her elaborations were used to develop interpretations of her re-engagement in the different practices and figured worlds mentioned above. Subsequently a meta-level analysis was conducted that compared the PoP interpretations with one that is based on more acquisitionist understandings of the functioning of the teacher.

Expected Outcomes

In the first three lessons on functions, Elena uses contextual and non-contextual examples and non-examples to highlight definitional and other critical aspects of the concept. Students are to provide their own examples, justify their claims, and value other students’ suggestions. Elena claims that her engagement in the mathematical discourse is transformed by her emphasis on conceptual understanding and student communication promoted by the master’s course. This provides a contrast to the procedural focus and modes of communication that dominate the traditions of school mathematics. The students resist these communicative patterns, and are reluctant to accept most of the examples that Elena provides as mathematically relevant. Later the emphasis shifts towards more expository styles of teaching. Also, the examples are replaced by others more in line with curricular traditions and private tuition. In a PoP interpretation, Elena takes the attitude of the students and adjusts her contributions to the interaction. Simultaneously she meets the obligation to “cover the syllabus”, which puts pressure on her as she compares her progress with that of her colleagues. “The tradition” seems to function as a figured world in which reform oriented teaching is not valued and becomes yet another generalized other whose attitude Elena takes. As a consequence, the discourses on conceptual mathematics and student communication are transformed and gradually lose their significance. The developments in Elena’s classroom may be seen as growing incongruities between her beliefs and practice. Elena possesses sets of stable beliefs about mathematics and its teaching and learning that are the default explanation for classroom practice, but the significance of which is overruled in the particular situation by the school culture. According to PoP this interpretation, adopted in mainstream belief research, does not do justice to the interactive character Elena’s meaning-making, while PoP offers a more dynamic view on classroom interaction.

References

Blumer, H. (1969). Symbolic interactionism. Perspective and method. Berkeley: University of Los Angeles Press. Charmaz, K. (2006). Constructing grounded theory. A practical guide through qualitative analysis. London: Sage. Holland, D., Skinner, D., Lachicotte Jr, W., & Cain, C. (1998). Identity and agency in cultural worlds. Cambridge, MA: Harvard University Press. Mead, G. H. (1934). Mind, self, and society from the standpoint of a social behaviorist. Chicago: University of Chicago. Maass, K. (2011). How can teachers' beliefs affect their professional development? ZDM - The International Journal on Mathematics Education, 43(4), 573-586. Palmér, H. (2013). To become- or not to become - a primary school mathemartics teacher. PhD-thesis, Linnaeus University, Växjö, Sweden. Retrieved from http://lnu.diva-portal.org/smash/get/diva2:613396/FULLTEXT02.pdf Potari, D. (2013). The relationship of theory and practice in mathematics teacher professional development: an activity theory perspective. ZDM - The International Journal on Mathematics Education, 45(4), this issue. doi: 10.1007/s11858-013-0498-2 Sfard, A. (2008). Thinking as communicating. Human development, the growth of discourses, and mathematizing. Cambridge, UK: Cambridge University Press. Skott, J. (2001). The emerging practices of a novice teacher: the roles of his school mathematics images. Journal of Mathematics Teacher Education, 4(1), 3–28. Skott, J. (2013). Understanding the role of the teacher in emerging classroom practices: searching for patterns of participation. ZDM - The International Journal on Mathematics Education, 45(4), 547-559. Skott, J., Larsen, D. M., & Østergaard, C. H. (2011). From beliefs to patterns of participation: shifting the research perspective on teachers. Nordic Studies in Mathematics Education, 16(1–2), 29–55. Stake, R. E. (1995). The art of case study research. Thousand Oaks: SAGE. Törner, G., Rolka, K., Rösken, B., & Sriraman, B. (2010). Understanding a teacher'a actions in the classroom by applying Schoenfeld's theory Teaching-in-Context: reflecting on goals and beliefs. In B. Sriraman & L. English (Eds.), Theories of mathematics education (pp. 401-420). Heidelberg: Springer. Wagner, D., & Herbel-Eisenmann, B. (2009). Re-mythologizing mathematics through attention to classroom positioning. Educational Studies in Mathematics, 72(1), 1–15. Walshaw, M. (2010). Mathematics pedagogical change: rethinking identity and reflective practice. Journal of Mathematics Teacher Education, 13(6), 487-497.

Author Information

Despina Potari (submitting)

University of Athens

Mathematics

Athens

Jeppe Skott (presenting)

Linnaeus University, S, & Aarhus University, DK

Department of Education

Copenhagen

Chara Papakanderaki

1st Gymnasium of Vrilission, Athens, Greece

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