This special issue grew out of a symposium that I organized and chaired at the Tenth Conference of the European Association for Research into Learning and Instruction (EARLI), held in Padova, Italy from the 26–30 August 2003. The special issue was announced and papers called for over the mathematics education networks in the following way:Mathematics education is well established as a field of study in its own right. Researchers in this field draw on literature from psychology, sociology and anthropology to substantiate their theoretical positions but have also contributed to the development of theories and practices within the area of learning and teaching mathematics. Because of the interesting problems encountered in mathematics classrooms, mathematics education has provided a seductive area within which some of those in education, and other cognate disciplines, have sited their research. However, many such are unfamiliar with the substantial literature that now exists within mathematics education itself nor, despite considerable advancement in the field, necessarily cognisant of major disagreements, problems and contradictions that remain unresolved. Papers are invited that air and discuss ‘provocations’, drawing on established research and raising challenges to mathematics education and different approaches to such challenges within the field.I am delighted, therefore, to present this special issue consisting, as it does, of papers from across the spectrum of primary to tertiary education, with author representation from English-speaking countries across the world, and from the non-first language English countries of Italy, The Netherlands and Sweden. Authors have been encouraged to present their provocations in ways that are meaningful to the non-mathematics education community. This is in the hope of building bridges with those outside mathematics education who find the challenges of learning and teaching mathematics engaging but who are unaware of the kinds of debates ongoing within mathematics education and, in part, represented here.The issue consists of six articles together with a commentary paper on these that Phil Hodkinson of the Lifelong Learning Institute at the University of Leeds was kind enough to write for it. By this means, I hope that the issue will not only offer readers of the journal some interesting, indeed provocative, ideas but also that the views on the papers, expressed by Phil Hodkinson, from the position of a generalist, will offer an additional take to engage readers.The first article in the issue is by colleagues Liz Bills and Chris Husbands, at the University of East Anglia in Norwich. The title is ‘Values education in the mathematics classroom: subject values, educational values and one teacher's articulation of her practice’. The authors address the relationship between general social and values education, the development of which they call ‘the most important aim of schooling’, and mathematics education practice, one of the mechanisms through which this development is influenced. For those not engaged in mathematics education, there might be some surprises in the authors' references to the ways in which mathematics bears upon values issues and the ways in which they sensitively discuss one teacher's dilemmas in confronting the tensions between subject-specific, that is mathematical, and general values in her classroom.The second article, ‘Getting political and unraveling layers of gendered mathematical identifications’ is by Margaret Walshaw of Massey University, New Zealand. Margaret uses a post-structural analysis to draw attention to the politicization of knowledge and the ways that focusing upon ‘getting the analysis right’ can possibly lead to a failure to recognize that ‘hidden forms of social relations lie behind what our research participants tell us’. In this paper we meet Rachel, a senior secondary student of mathematics, who is attempting to deal with her mathematical identity and its gendering both in the classroom and in her relations with significant others. Through the format that Margaret has chosen to present her work, a split text strategy using the interview transcript and her analysis of the interview, she provokes the reader to reconsider constructions normally taken for granted in academic writing.Marja van den Heuvel-Panhuizen, who is at the Freudenthal Institute, Utrecht University, The Netherlands but, at the time of writing, was a visiting professor at the University of Dortmund in Germany, poses the question: ‘Can scientific research answer the ‘what’ question of mathematics education?’ Marja begins by clearly articulating a contradiction that emerges from the papers of many who write about mathematics learning and teaching: ‘for outsiders mathematics may be a school subject with an indisputable content, but for those who are involved in mathematics education it is clearly not’. She points out that decisions on content are not even necessarily the same where approaches are shared, never mind where these differ and points to the consequent inadequacies of testing regimes. She asks does research help to find an answer to the question of what should be taught, a question which policy-makers and politicians, as well as the general public, recognize as most important but about which there is apparently little agreement. She develops her argument through published work and concludes that research not only must address this question; she proposes that the European approach to the didactics of mathematics is one way of doing so.A team, consisting of Ferdinando Arzarello, Ornella Robutti and Luciana Bazzini, from the University of Torino in Italy, are the authors of the next paper entitled ‘Acting is learning: focus on the construction of the mathematical concepts’. They move to the primary school where, in a particular classroom with a specific problem, they investigate the interaction between language, gesture and representational artefacts that support pupils' construction and understanding of meaning associated with variable and function. The provocation that they offer to practitioners is to acknowledge and build upon these interactions. To those who assert this important relationship between action, language and cognition, they offer a further provocation to foster ‘its transposition in school practice’. The authors critique the symbolic–reconstructive mode of learning that underlies transmission teaching and support an approach that promotes involvement in activities that help to build an experiential base for learning. They draw attention, in particular, to the role of technological artefacts. However, as they point out, artefacts, alone, are insufficient to promote learning; as we know, learning depends upon the sensitive understanding and use of these artefacts made by the teacher and students in interaction. Herein lies the provocation to practitioners.The fifth paper comes from Ulla Runesson, a member of the phenomenographic school of educational research associated with Gothenburg University, in Sweden. Her paper is called ‘Beyond discourse and interaction. Variation: a critical aspect for teaching and learning mathematics’. She utilizes transcripts from respected published work to demonstrate the different insights that can be obtained by the application of variation theory, a theory of learning that draws attention to the object that is necessary for learning to take place and the need to vary the learner's encounters with this object. She calls for this kind of eclectic approach to broaden perspectives on research data and research publications and points out that ‘by comparing analyses of the same data from different theoretical positions I have addressed the “provocation” created by the relationship between theoretical positions and findings of research in mathematics education. I have been proposing an alternative theoretical position, that of variation theory, for examining classroom learning in mathematics due to its power to reveal what can be learned’.The final paper, ‘Rethinking the tertiary mathematics curriculum’, by Peter Petocz and Anna Reid at Macquarie University in Australia, moves into university studies but focuses on the perspectives of the students, how they understand the nature of mathematics and its role in their future professional lives, and how this impacts upon their learning while at university. The authors' intent is to provoke a debate on the aims and nature of the tertiary mathematics curriculum. After examining a range of data from studies on student perspectives, they conclude by summarizing the impact on curriculum development and review, from the points of view of the institution, the students' professional preparation and learning needs, the coherence of the individual units that comprise a course, and the teaching and learning strategies incorporated in the classroom. They draw the conclusion that ‘students at university level will have some ideas about their future work, but they are not generally aware of the relation between these ideas and their conceptions of their subject, learning in the subject, and their overall approach to their studies. If we discuss these ideas with them, we can make explicit the relationship between narrower and broader views of professional work and limiting and integrated conceptions of learning in a subject’. This is, indeed, a provocation to many of those currently teaching mathematics at tertiary level.In his commentary paper, ‘Learning as cultural and relational: moving past some troubling dualisms’, Phil Hodkinson identifies similarities and differences in the six papers. He finds, not unexpectedly in an issue devoted to provocations, that the papers share an approach to mathematics education that is potentially contentious; it is broad, situated, embraces complexity and the role of values and purposes. Despite being situated within mathematics education, he finds that the papers embrace broad pedagogic perspectives that have the teacher in a central position. But these are not papers within a positivist or neo-realist position. All recognize the multi-variate, relational and complex nature of research in this field and look to interpretive research to help understand the influences at work. Underlining this, Phil Hodkinson reflects upon a number of dualisms that pervade a lot of academic writing, in particular but not only within mathematics education, and ways in which he perceives the papers in this Issue as challenging them. The dualisms he addresses are mind/body, individual/social, process/product and formality/informality of learning. This thought provoking paper concludes with a similar appeal to the one that I made at the outset for us as researchers in education, first, and mathematics education second, to turn our backs on Balkanization and engage with the issues and challenges that we jointly face building upon the contributions that are there to be found in the discipline-specific, as well as the generalist, literatures.While I hoped that Phil Hodkinson would do the kind of job that he has done, I cannot say that I expected the thorough, challenging and extremely interesting paper that he has provided, for which I am extremely grateful. In conclusion, I would also like to thank all the other authors represented here for their willing attention to detail and reactions to critique as well as the colleagues who acted as reviewers of papers for this special issue. Without their work, I would not have been able to produce an issue that, I hope, will be of interest and challenge to readers. In addition to members of the Editorial Board of the journal, thanks to Mike Askew, Mary Barnes, Bill Barton, Alan Bishop, Suzanne Damarin, Ed Dubinsky, Kathleen Lynch, John Mason, Anna Pendry and Hilary Povey and to Paula Peachey for her unstinting administrative support.Notes on contributorLeone Burton, the editor of this special issue, is Professor Emerita in Mathematics Education at the University of Birmingham and Visiting Professor at King's College, London. Her research interests span problem-solving, epistemology, assessment and social justice in mathematics classrooms from pre-school to university. Her most recent book is Mathematicians as enquirers: learning about learning mathematics (2004). [ABSTRACT FROM AUTHOR]