Valeria and Emilia are two experienced high-school teachers, who position themselves as enthusiasts of using instructional mathematical videos in their teaching. They both see the videos as adjuvant to students' learning, but their respective lessons turn out to be very different, in particular with respect to students' participation. Teachers'…

We aim to open up discussion about the intertwined roles of teachers and tasks that involve students communicating about mathematics when working in groups. Over many years we have observed, researched and ourselves have taught students working on mathematics in groups and find that it is often easier to pay attention to the forms of communication…

In this article, we question what is an appropriate balance of ordinal and cardinal work in the early learning of number. We see an over-emphasis, in current research and practice, on the cardinal that leads, for example, to only using small numbers. We report on empirical work we have carried out in the UK and Canada that suggests the potential…

Drawing on the work of Gattegno, it is suggested that a powerful way of teaching mathematics is to introduce symbols as relationships between visible or tangible resources. The symbols are abstract (formal) from the beginning and yet there are concrete resources to support their use. Drawing on data from a research project in primary schools in…

It is argued that the discourse currently regulating early years mathematical development in the UK and elsewhere is "producing its own truths" in an unhelpfully narrowing way, restricting the objects, imagery, and symbols involved to those arising within "everyday" or "realistic" situations, and producing an…

The curriculum for eleven-year old students in the United Kingdom, currently adopted by most schools, includes solving linear equations with the unknown on one side only before moving onto those with the unknown on both sides in later years. School textbooks struggle with the balance between developing algebraic understanding and training…

Explains why mathematics lectures have a special role in stimulating the imagination and why attending lectures has a significant role in inducting students into their university's mathematical community. (KHR)