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.

FROM IDEA TO TRANSMISSION

Yorkshire Television, one of the larger independent television companies in the U.K., is well known for programmes popularising science. In October 1986, the science department came up with the idea to try to popularise mathematics in a similar manner. As Duncan Dallas, the head of the science department, wrote “As a kamikaze notion it could hardly be better … (Mathematics) is the least popular or accessible of the sciences. It does not sell magazines in the same way as computing, it is not the subject of dinner party conversations as is ecology, nor is it a trendy part of our life style like technology. Indeed, it is universally acceptable to trumpet our ignorance whenever the subject is mentioned. Clearly mathematics is important enough to command our attention but on a list of programme ideas rated by popularity it will probably become bottom” (Dallas 1988).

The crucial question was to find a format, a way into the mathematical perspective that would be entertaining and lively. After considerable discussion, Yorkshire Television came up with the idea to base the programme around puzzles and games. The rationale was to capitalize on people's interest in puzzles, an interest which goes back for many years, and to use these puzzles as a vehicle to think about the embedded mathematical ideas - after all recreational mathematics has been the source of a great deal of mainstream mathematics.