Julie Ryan and Julian Williams have worked together for several years in England exploring children's mathematical errors and misconceptions with a view to informing and developing classroom teaching practice. Both came to research from school teaching careers. Their latest collaborative research draws on children's argument in discussion by provoking reasoned 'changes of mind' in classrooms. Julie Ryan taught mathematics in secondary schools in Melbourne for ten years. Her interest in research grew from children's problems with algebra. Her research led her to believe that the move from arithmetic to algebra involved a new frame of thinking -- creating a challenge for the development of suitable teaching tasks. She was particularly taken by the intelligent 'mistakes' that children made. She has been involved in teacher education in Australia and England and is Senior Lecturer in Mathematics Education at Manchester Metropolitan University. Her research fucuses on pedagogy and teacher education. Julian Williams taught mathematics for eight years before he moved to teacher education and curriculum research and development , more recently focussing on research into learning, assessment and pedagogy. His particular focus on assessment grew from a belief that national testing of all pupils in England and Wales was impoverished -- the most useful formative data are to be found if test items are carefully crafted to uncover common misconceptions well known in the literature. He is Professor of Mathematics Education, and Director of Graduate Research students at the University of Manchester.