This is the second of three articles that draw on findings from Nunes, Watson and Bryant (2009): "Key understandings in school mathematics: a report to the Nuffield Foundation". The report was soundly based on research about how children learn mathematics, much of it done in the UK in the '80s. Most of the findings about algebra are very well-known, but the author uses them to emphasise challenges for teaching. At school level, algebra consists of: (1) manipulation and transformation of symbolic statements about relationships; (2) rules for transforming and solving equations; (3) generalisations of laws about patterns and number; (4) modelling situations; (5) the study of structures and systems abstracted from computations and relations; and (6) learning about variables, functions and expressing change and relationships. Students have to understand the nature of arithmetical operations and inverse operations and understand that algebra represents relations between variables, and structures which combine operations. These understandings are essential in other areas of mathematics. Learning algebra is a mixture of learning to use the notations and learning the underlying meaning. In this article, the author describes what learners already do that can help them learn algebra and what they do naturally that hinders the learning of algebra. She concludes the article with some recommendations for teaching. (Contains 1 footnote.) [For Part 1, see EJ889866.]