Commentary on Chapters 4 to 7: Students’ Learning of Integer Addition and Subtraction Using Models

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 chapters demonstrate that teaching with any model requires care and attention to achieve conceptual clarity. Teachers need to act entirely within in the world of the model, only later adding other models to round out the mathematical picture. The teaching of integers is a key point for maintaining students' confidence that mathematics is a subject that makes sense. These chapters demonstrate how use of models can assist in this goal.