Learning Communities in Mathematics

LCM website

Enhancing knowledge of mathematics, and of mathematics learning and teaching through a collaborative inquiry model.

In these paragraphs the mathematics education research group (MERGA-group) of Agder University College presents some background for research we want to set up during the coming years.

Mathematics in Society

Mathematical knowledge, its relevance and importance

In a technological society with complex social needs and structures mathematics plays a crucial role in communication and development. In order to be full members of society all people need to be fluent in mathematics to some degree, and all should have the opportunity to see the power and beauty of mathematics in contributing to richness and fulfilment in their lives (Howson and Kehane, 1990, Black and Atkin, 1996). We focus primarily on mathematics learning and teaching and their development for the wider enhancement of mathematical understanding and its application. We root our study in the social frame of lifelong learning and acknowledge the central role of language: we see mathematics to be firmly related to language as a tool for communication, and also as a language in its own right (Pimm, 1987).

The curriculum in Norway acknowledges the importance of mathematics both as an indispensable tool for daily living and as a source of social emancipation and enrichment (KUF, 1997, 1999) However, the interpretation and implementation of this curriculum is far from straightforward as research has shown: students' mathematical understanding, learning and achievement do not follow simply from curriculum aims and instructions, and approaches to teaching mathematics depend on complex social, philosophical and pragmatic factors (Alseth et al., in press). The success of the curriculum as implemented and attained depends both on a clearer understanding of this complexity and on a carefully associated programme of reform that includes all participants and stakeholders (Hoyles, Morgan and Woodhouse, 1999; Popkewitz, 2002). We see such inclusion to be central to effective development as we explain below.

Learning Mathematics

In order to exploit fully the power of mathematics in their lives students need a relational or principled understanding of mathematical concepts (Skemp, 1976; Edwards and Mercer, 1987). This means that not only should they develop mathematical skills, know number facts, apply arithmetical procedures correctly, recognise and relate shapes and use statistical formulae, they should perceive the meaning and relatedness of concepts and develop connected understandings that they can apply to problems in their everyday world (Askew et al, 2000). They should be able to draw on mathematics knowledgeably in making informed decisions in life and work. Such principled knowledgeability requires understanding of the nature of mathematics itself in generalisation and abstraction (Nardi, 1996).

However, despite innovative and forward looking curricula, research shows that many students find mathematics difficult and boring, beyond their comprehension and irrelevant to their needs (Cockcroft, 1982; Alseth et al., in press). We should, as a matter of urgency, explore further the concepts and attitudes of students both in school and beyond to perceive the roots of such disaffection and address the issues - mathematical, social and didactic - that sustain it. Here, we need to recognise students' thinking, aspirations and participation in society as central to their mathematical learning improvement.

Teaching mathematics

While students are clearly at the centre of the educational issues we have started to sketch above, mathematics teachers are central to the enhancement of the mathematical learning environment (Nolder, 1992). In interpreting the curriculum in their classrooms, teachers take a bewildering complexity of decisions that affect the mathematical education of their students and contribute to students' experiences, feelings and attitudes (Jaworski, 1994). When students are disaffected it is insufficient simply to blame teachers. We need to look carefully, with teachers, at the curricular, societal and pragmatic demands that influence their decisions and fashion their classroom actions. It is the teachers who are in the best position to identify the essential issues, and resources must be channelled to enable teachers to understand the issues and work on the development of decisions and actions (Krainer, in press). Thus teachers develop professionally alongside and throughout their practice of teaching.

Societal issues

Inclusion of both students and teachers in consideration of issues of learning and teaching and their development presupposes levels of communication that enable deep appreciation of human perceptions and educational needs (Gates and Cotton, 1998) Factors that need attention in our current society are (a) language and discourse; (b) exploitation of technological power and (c) wider societal factors.

Language and Discourse. In classrooms and beyond, communication takes place through languages of different kinds, and discourses develop relative to these languages. Language cannot be separated from the social circumstances in which it is used. Students speaking Norwegian and teachers speaking Norwegian are two languages with only some common elements in the words and grammatical structures employed. Student discourse in and out of the classroom, and teacher discourse in and out of the classroom differ and are embedded in the social fabric of the lives of the different groups. It is therefore unsurprising that as teachers and students together develop a classroom mathematical discourse, the perceptions of those concerned vary according to the other discourses in which they participate (Cestari, 1997; Forman and Ansell, 2001). Natural language is clearly central to such concerns, both in terms of the Norwegian language and of others that students use in their out of school lives (Cestari, 1998; Wells, 1999).

Exploitation of technological power. We live in a society in which technological power is taken for granted at a multitude of levels. However, we have been slow in exploiting this power effectively in the classroom, in understanding the contribution that it can make to learning and teaching and making computer software into personal tools for solving problems (Friedlander and Stein, 2001). Research internationally is starting to show modes of hardware and software that can be of value in mathematics learning and teaching but we need a clearer awareness of how students, teachers and computers can interact most effectively, drawing on their particular qualities and potential to interpret curricular aims (Burton and Jaworski, 1995; Fuglestad, 1999; Oldknow and Taylor, 2000). Human interaction with computers in educational settings needs to be understood so that computers can be used 'naturally' by teachers and students wherever they can contribute to or enhance mathematical thinking (Lagrange et al., 2001; Hershkowitz, et al., 2002).

Wider societal factors. Students come from diverse home backgrounds that represent a wide range of cultures and values. Home influences and cultural expectations serve to fashion the systemic and social foundations of education; issues in learning and teaching have to be seen in the context of the various communities as part of which students learn and use mathematics and teachers provide for their learning. Schools have to serve the wider community of which they are a part, and this means understanding social norms and societal pressures. Students and teachers use mathematics in the world around them and such usage needs to be related productively to learning and teaching in schools. (Abreu, 1995; Civil, 1998). The learning of teachers about teaching is also a lifelong process that develops understandings of practices, processes and issues in relating classroom activity to the wider needs of students and society (Fischer, 1999; Kristjansdottir, in press). Teacher education needs to be interpreted broadly to include teachers in the complexities of their own development (Jaworski, 1998).

Historical foundations and indications

As we seek to explore and develop knowledge in these areas we must draw on lessons from history regarding development of mathematical concepts, human interaction with concepts, international comparisons of mathematical understanding and fluency and the nature and development of curricula internationally. Relating scholarship in these areas to current research and its outcomes is designed to provide a historically rooted analysis with firmer foundations to indicate key aspects of current theory and practice and inform future development. (Mosvold, 2001; Siegmund-Schultze, 2001)

Collaborative inquiry as central to effective learning and development

We propose a research project, Learning Communities in Mathematics (LCM), that has mathematics teaching and learning development as its central essence. We work from a basic premise that people learn through responsible participation in a collaborative enterprise whose aims are central to their well-being and aspirations, and in which their aspirations, qualities and potential to contribute are respected and valued. We work from Vygotskian principles of the social rootedness of human learning (e.g., Vygotsky, 1979) and recognise the dialectic of individual in community as a productive force for development (Wertsch, 1991, Cobb, 1994). We draw on Wenger's (1998) notion of identity within community of practice and relate this to metacognitive inquiry as a collaborative discourse (Wells, 1999). We draw on a rich tradition of inquiry in the learning and teaching of mathematics developing from problem solving in mathematics itself, through inquiry approaches to learning mathematics in classrooms to teacher inquiry in which teachers take on the mantle of researcher to explore questions about effective means of stimulating and sustaining their students' mathematical growth (Polya, 1945; Mason, Burton and Stacy, 1982; Schoenfeld, 1992; Lambdin, 1993; Jaworski, 1994, 1998; Mason 2000; Bjuland, 2002).

Research shows that when professionals reflect critically on and inquire into aspects of their professional lives, issues are revealed, questions refined and socially significant action can be taken in clearly directed and knowledgeable ways (Carr and Kemmis, 1986; Schon, 1987; Mathematical Association, 1991, Zack, Mousely and Breen, 1998; Mason, 2002). The LCM project will be a large-scale inquiry involving professionals - teachers, didacticians and researchers - in inquiry at some level. At any level, those involved will engage in inquiry according to their own particular roles, aims and expertise (Jaworski, 2002). Thus, both teachers and didacticians will engage in research relating to their own domains of knowledge. As well as researching aspects of learning and teaching, the project will also study the very nature of inquiry as a means for sustained effective educational development in mathematics.

The LCM Project - an illustration

The programme focuses on human learning involving students, teachers and researchers:

Students' Learning - students engaging in mathematical inquiry related to the mathematics curriculum with teachers and didacticians fostering and researching inquiry approaches;

Teachers' Learning - teachers engaging in inquiry into processes of learning and teaching as they interpret the curriculum in collaboration with their didactician colleagues;

Researchers' Learning - researchers (including teachers and didacticians) developing research knowledge and capacity through clearer understandings of approaches and issues.

A three dimensional dynamic figure can be seen to represent the design of the study drawing on the principles outlined in the introduction above. Imagine a central sphere in which we locate students and teachers engaging in learning and teaching interactions in mathematics classrooms. Here the learning and teaching of mathematics, leading to the generation of mathematical knowledge and understanding, and knowledge about the learning and teaching of mathematics, are the main focuses of research. Imagine, around this central sphere, two ellipsoids rotating orthogonally, one in a horizontal and one in a vertical plane. In the horizontal plane we see a research focus on issues in language and discourse and in technology; in the vertical plane we see a social and historical focus emphasising particularly home-school links and comparative historical analyses. These elements of the study fit within a philosophical and practical perspective of communities inquiry in mathematics learning, teaching and development.

Concluding remarks

An application for a research project on this background is sent to the Research Council of Norway (Norges Forskningsråd) under the KUL program. We will also seek other sources for funding to support the research.

More information and reports will appear on this web page as the work proceeds.

MERGA group

Trygve Breiteig		Raymond Bjuland		Hans Erik Borgersen
Maria Luiza Cestari		Anne Berit Fuglestad		Barbro Grevholm
Barbara Jaworski		Anna Kristjánsdóttir

Other staff and doctoral students will also take part in the project.

References

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