Learner identity conditioned/influenced by mathematical tasks in textbooks? Birgit Pepin University of Manchester School of Education Manchester M13 9PL

Background Importance of textbooks and mathematical tasks in textbooks Learner identity Previous projects Learning mathematics with understanding Connectivity

Importance of mathematical tasks in textbooks Textbooks are one of the main sources for mathematical content covered and the pedagogical styles used in classrooms (Valverde et al, 2002) Importance of the nature of the learning task (Hiebert et al 1997) Students need frequent opportunities to engage in dynamic mathematical activity that is grounded in rich, worthwhile mathematical tasks (Henningsen & Stein, 1997) Using the mathematical task as an analytical tool for examining subject matter as a classroom process (rather than simply as a context variable in the study of learning) (Doyle, 1988)

Learner identity Identity construction is situated in specific cultural (and national and local) contexts where interacting factors help or impede the construction of an identity as and for mathematical learning Social identity helps to consider learners as constructed in terms of the groups/classes of which they are members (or not) What does it mean to be a learner of mathematics in England, France and Germany?- interest in the idea that schools and classrooms help to produce social identities which are available to individual students (Osborn et al, 2003)

Previous projects Mathematics teachers pedagogies in England, France and Germany (Pepin, 1997, 1999, 2002) Mathematics textbooks and their use by teachers in England, France and Germany (Pepin & Haggarty, 2001, Haggarty & Pepin, 2002)

Learning mathematics with understanding the tasks in which students engage provide the contexts in which they learn to think about subject matter, and different tasks may place different cognitive demands on students …. Thus, the nature of tasks can potentially influence and structure the way students think and can serve to limit or to broaden their views of their subject matter with which they are engaged. Students develop their sense of what it means to do mathematics from their actual experiences with mathematics, and their primary opportunities to experience mathematics as a discipline are seated in the classroom activities in which they engage … (p.525) (Henningsen & Stein, 1997) Hiebert et al (1997) similarly argue that students also form their perceptions of what a subject is all about from the kinds of tasks they do. … Students perceptions of the subject are built from the kind of work they do, not from the exhortations of the teacher. … The tasks are critical. (p.17/18).

Connections and connectivity …the way information is represented and structured. A mathematical idea or fact is understood if its mental representation is part of a network of representations. The degree of understanding is determined by the number and strength of the connections. (p67) (Hiebert & Carpenter, 1992). Hiebert and Carpenter (ibid) further point out that if mathematical tasks are overly restrictive, students internal representations are severely constrained, and the networks they build are bounded by these constraints. Further, the likelihood of transfer across settings becomes even more problematic (p79).

Connections and connectors Research conducted at Kings College in the United Kingdom (Askew et al, 1997) revealed that highly effective primary teachers of numeracy paid … attention to Connections between different aspects of mathematics: for example, addition and subtraction or fractions, decimals and percentages; Connections between different representations of mathematics: moving between symbols, words, diagrams and objects; Connections with childrens methods: valuing these and being interested in childrens thinking but also sharing their methods. (Askew, 2001, p.114)

Aim In this seminar I would like to investigate different mathematical tasks in terms of connectivity and explore how this might relate to learner identity in the three countries. This might enable us to consider the different identities available, and further to what extent available identities may be different or similar in the three countries.

Questions What are the connections made in mathematical tasks in selected textbooks in England, France and Germany? How is this likely to shape pupil identity as learners of mathematics? What are the differences, in terms of mathematical tasks, and perhaps proposed learner identity, in the three countries textbook tasks that we can identify?

Contextual factors England: whole school ethos; teachers feel they have to attend to the needs of the individual child; setting in mathematics with different mathematics taught to different groups. France: class as a unit; identity of mathematics as a selection (and difficult) subject; mixed ability teaching (egalitarian ideas) and entitlement to the same curriculum. Germany: class as a unit, but within tri-partite system; different streams had different identities reflected in a different curriculum, teachers and school organisation; different approaches to the academic and affective in different streams.

Making connections and seeking links Individual tasks are analysed with respect to context embeddedness, familiar situations- tasks which make connections with what students already know, real life (for example, in introductory tasks/activities); cognitive demand/formal statements/generalisations- tasks which emphasise relational rather than procedural understanding, tasks which make connections with the underlying concepts being learnt; mathematical representations- tasks which make connections within mathematics and across other subjects, tasks which connect different representations.

Learning and learning identity- harmony of forces? Pollard & Filer (1996) argue that effective learners are produced when their strategies and presentation of identities become well-adapted to the social understandings in the learning context. We can describe these as educational cultures, and the conditioning that is happening is likely to be dependent on classroom and mathematics cultures (which in turn is influenced by the tasks pupils are given).

Back to our question What kinds of opportunities are given to students to learn mathematics (and to connect) and how do these relate to their identity construction as a learner of mathematics?

Typology? What are the resources available to students as they seek to identify or develop or construct their identities? Can we identify typologies (of tasks) that are likely to influence students in their construction of identities, such as Instrumentalist Developmentalist Negotiator Conformist Connector …?