**FEATURES** |

Optimization of Cubic Polynomial Functions without Calculus
*Ronald Taylor Jr., Ryan Hansen* How to find the relative maximum and minimum values of a cubic polynomial by using algebra or precalculus and calculus. Then, extreme values are computed. |

Are You Connected? Fostering Exploration with Unexpected Graphs
*Michael Edwards, Jeffrey Reinhardt* The importance of unexpected calculator graphs as a vehicle for encouraging critical classroom dialogue. Specific examples are provided along with explanations and fixes for the calculator graphs. Authors give examples of student versus computer versions of graphing, how to change domain or range to make the graph easier to view, the limitations of graphing calculators, and how to eliminate graphing gaps. |

Explorations with 142857: Connecting the Elementary with the Advanced
*Alfinio Flores* Explorations with the cyclic number for students and teachers help connect the mathematics taught in school with algebra and number theory concepts learned in college. Additional ways to manipulate sequential numbers and how they relate create new sequences as well as graphical and visual representations are given. |

Analyzing Online Discourse to Assess Students’ Thinking
*Randall Groth* How the analysis of discussion board conversations can be useful for charting the path instruction should take. This analysis is illustrated within the context of a course for preservice teachers. The use of such analysis as an assessment tool is also considered in relation to mathematics courses for high school students. Online discourse offers an alternative to in-class group work where the teacher cannot monitor or be privy to all discussions and learning. |

Connecting Students’ Informal Language to More Formal Definitions
*Jon Davis* How student-generated terminology for the y-intercept evolved within one Standards-based classroom. It also discusses the teacher’s role in this evolution as well as students’ understanding of this terminology within different function representations, and it presents ways teachers can help students develop mathematically precise definitions. The article is a description of a qualitative research project including video tapes, transcriptions, and student artifacts. |

Poverty: Teaching Mathematics and Social Justice
*Leah McCoy* Three mathematics lessons in a social justice setting of learning about poverty. Student activities include budgeting, graphic data representation, and linear regression, all in the context of connecting, communicating, and reasoning about poverty. The author leads students through defining the poor and poverty, the effects of poverty on education, and what students can do to combat poverty through understanding the mathematical realities. |

Building Intuitive Arguments for the Triangle Congruence Conditions
*Katrina Piatek-Jimenez* A hands-on activity that uses straws and pipe cleaners to explore and justify the triangle congruence conditions. The author uses the activities to explain when and why an idea is a postulate, theorem, accepted without need for proof, which ones need proof. |

Beyond Teachers’ Sight Lines: Using Video Modeling to Examine Peer Discourse
*Donna Kotsopoulos* The author describes her work in mathematics education discourse between student and peer and student and teacher. This article introduces readers to various examples of discourse analysis in mathematics education. Highlighted is interactional sociolinguistics, used in a present study to investigate peer discourse in a middle-school setting. Key findings from this study include the benefits of video modeling as a mechanism for fostering inclusive peer group work and the usefulness of video modeling as a tool for assessing peer communication. Implications for low performing students are discussed. |