**FEATURES** |

Making Music with Mathematics
*Maria Fernandez* Have you ever wondered how to bring music into your mathematics lessons? Music can be a rich context for facilitating mathematical connections. |

A Star to Guide Us
*Rick Norwood* Perhaps it is time to retire both the times sign and the multiplication dot and to use the star for multiplication from the very beginning. |

The Amazing Octacube
*Michael Naylor* This octacube makes an excellent classroom model that will fascinate and challenge your students to explore further many of the intriguing properties and relationships of the Platonic polyhedra. |

Multiple Representations for Pattern Exploration with the Graphing Calculator and Manipulatives
*Douglas Lapp* This investigation introduces the students to the process of conjecture followed by proof, and the combination of manipulatives and technology allows this process to be approached earlier than traditionally thought appropriate. |

Geometry Problems Promoting Reasoning and Understanding
*Alfred Manaster, Beth Schlesinger* The four related problems presented in this article provide examples of ways to include justifications of interesting mathematics in courses taught before a geometry course. This understanding requires that the students follow chains of reasoning that furnish convincing justifications of the correctness of the general results. The reasoning involves both algebra and geometry, but all the problems can be done before the student takes a formal geometry course. |

Promote Systems of Linear Inequalities with Real-World Problems
*Thomas Edwards, Kenneth Chelst* Because operations researchers solve problems in the real world, operations-research-based problems have rich connections to the world in which students live and work. Drawing on such problem situations is one way in which teachers can let applications of mathematics drive instruction. We believe that doing so will better motivate students to learn the mathematics they encounter in the classroom. |

Area of Spherical Triangles
*Elwyn Davis* Interest in teaching spherical geometry—both as a subject and as a means of increasing interest in geometry—has been growing. In this article, I hope to further this interest by describing a set of classroom activities or steps that can be used to determine the area of a triangle on a sphere. |

Discuss with Your Colleagues: Now Here Is That Authority on Mathematics Reform, Dr. Constructivist!
*Michael Mikusa, Hester Lewellen* Dr. Constructivist is a character who was "born" while we were trying to develop ways of teaching what constructivism is and ways to address the obstacles that teachers might have to overcome in adapting a mathematics-reform and constructivist approach in the classroom. We were both working with in-service and preservice teachers at all levels and found that the ideas of constructivism were still new to them. |

3-D Graphing, Contour Graphs, Topographical Maps, and Matrices Using Spreadsheets
*Louis Feicht* Modern spreadsheets can make powerful mathematical concepts accessible to students at a younger age than ever before. Contours and three-dimensional graphing are topics that were previously reserved until well into the first year of college calculus. Three-dimensional graphing now can be successfully taught to middle school students with the assistance of a computer spreadsheet. This combination of the computer with hands-on activity exposes students to numerical and graphical representations of data on the same spreadsheet "page" and forces them to make connections between the two forms of data. |