The talk will be in two parts. In the first part, I will discuss and analyse examples of children’s responses (some recorded on video) to mathematical tasks in elementary school. What is surprising about these responses? How do theories in mathematics education account for them? In the second part, I will discuss a few relatively successful approaches to teaching specific concepts that are important in the transition to secondary mathematics. I will also make some remarks about the criteria by which we can understand whether an approach is successful.