Introduction

All learning environments are designed based on a set of assumptions about what knowledge should be learned. For example, most mathematics classrooms are designed to teach a certain kind of mathematical knowledge that comprises procedures that solve isolated problems quickly, and this implicitly devalues the importance of structural understanding or of developing an appreciation of underlying mathematical models (see Lehrer & Schauble, this volume). This means that all too often, students do not see the need for consistency or rigor, do not notice conflicting strategies or solutions, and therefore cannot learn from them.

Based on our research in a variety of workplace situations, we are convinced that a crucial element of knowledge required by most, if not all, people, is precisely this appreciation of underlying models. A version of mathematics that emphasizes structures also has the potential to help students understand the computational systems that are increasingly critical in today's society, because computer systems are mathematical models – computer software is built out of variables and relationships. As technology becomes more and more advanced, and the underlying models more and more obscure and invisible, it becomes increasingly important that children learn awareness of models; how to build, revise, and evaluate them, and to develop some analytic understanding of how inputs relate to outputs.