These four chapters describe studies of using models for integer addition and subtraction. The models draw principally on the two grounding metaphors of object collection and motion along a path. A strength of all chapters is detailed analysis of how the models are and can be implemented and how they influence student's learning. Together the…

Ginsburg (1977) observed that children typically develop surprisingly powerful informal (everyday) knowledge of mathematics and that mathematical learning difficulties often arise when formal instruction does not build on this existing knowledge. By using meaningful analogies teachers can help connect new formal instruction to students' existing…

The crucial role of teachers in introducing integers to children is highlighted in chapters 8-10, comprising this section. The three chapters discuss (prospective) teachers' conceptions of integer equations, of children's thinking about integer expressions, and of the role of some didactical models used in teaching integer addition and…