**FEATURES** |

Historical Research: How to Fit Minority and Women's Studies into Mathematics Class
*Margaret Sáraco* A lesson for studying minority and women's contributions to the field of mathematics. This article contains the groundwork for teaching to get started on a research project that highlights the accomplishments of famous women and minority mathematicians. An activity sheet and rubric for assessing student work are included |

Fraction Representation: The Not-So-Common Denominator among Textbooks
*Thomas Hodges, JoAnn Cady, R. Collins* Three widely used sixth-grade textbooks were studied to see how fraction concepts were represented. Results of the findings as well as classroom implications are explored with the intention to assist teachers and district curriculum specialists in making educational decisions regarding the use of a chosen textbook. Suggestions for supplementing your current textbook are offered. |

Is 1 a Prime Number? Developing Teacher Knowledge through Concept Study
*Brent Davis* Teachers develop knowledge of content through concept study. This article describes "concept study" as a collaboration of teachers to investigate images, analogies, metaphors, exercises, and other experiences that are used to support students' understandings of concepts. The term concept study borrows from the ideas of "concept analysis" and "lesson study." The discussion is illustrated with the examples of multiplication and prime numbers. For the teachers and students participating in this concept study, productive discussions occurred as all became more aware of the evolutions of mathematical ideas. |

Developing Spatial Sense and Communication Skills Reflect and Discuss
*Kerri Richardson, Catherine Stein* How spatial instruction with preservice teachers can be implemented in a middle-grades mathematics methods class. The preservice teachers noticed that the students started used terms and speaking more "mathematically" because they were making connection with their prior knowledge. A "Reflect and Discuss" section is included for professional development study. |

Beneath the Tip of the Iceberg: Using Representations to Support Student Understanding
*David Webb, Nina Boswinkel, Truus Dekker* The "iceberg model" is introduced, which has been used in the Netherlands to support teacher identification of informal and preformal representations that build students' understanding of formal mathematics. The tip of the iceberg represents the targeted formal procedure or symbolic representation. The bulk of the iceberg that sits under the water is represented by a combination of informal representations. Included are suggestions on how this model can be used to support professional development, collaborative instructional planning, and identification of appropriate interventions. |

Harry Potter and the Coding of Secrets
*Boon Chua* An activity that incorporates an application of functions and their inverses: cryptography. This activity capatilizes on the popularity of the Harry Potter series by J. K. Rowling. The activity begins with messages that need to be sent, but those messages must use a special encryption. To incorporate the concept of composite functions, the encryption process can be revised. Students can enjoy and appreciate the application of mathematics because of its relevance. Includes activity sheets. |

1P plus 4R equals 5D: An Equation for Deepening Mathematical Understanding
*Alan Hackbarth, Margaret Wilsman* Illustrates mathematical discussions that can occur with students in the classroom. Example discussions focus on the relationships of events within a problem, connects the ways chosen to represent those relationships, and shows their equivalence. A deep understanding does not necessarily result from being able to represent a mathematical situation in multiple ways. Even when representations are similar or contain errors, when a group of learners contribute to the pool of possibilities, they enter into discussions that may lead to deeper understanding. |