The teaching of arithmetic in the primary school has traditionally been dominated by a focus on standard algorithms and this approach is being questioned. Curriculum changes are taking place that promote the development of mental strategies, and provide more opportunities for children to develop their own ways of working. This book shows contrasting influences for change as leading mathematics educators from the USA, the UK and the Netherlands identify the way research is used to develop different classroom practices. In England, changes are taking place through a National Numeracy Strategy which is set to raise standards in every primary classroom while in the Netherlands, this text introduces innovative approaches such as use of an "empty number line" to support the teaching of mental strategies. It explores why we teach in different ways, challenges orthodoxy, and sets the agenda for learning from each other.