**Second Look - Learning from Student Errors** |

Sharing Teaching Ideas: Why Is the square root of 25 Not ±5? The value/s of the square root of a positive integer. How students may incorrectly acquire the belief that the square root of 25 is plus or minus 5 and describes teaching methods to be used to convince students their belief is incorrect. Includes graphs to show how the misconceptions occur. |

A Function or Not a Function? That Is the Question Analyzing
and discussing examples and nonexamples of functions can help address students’
misconceptions about this important concept. |

Connecting Research to Teaching: Quadratic Functons: Students Graphic and Analytic Representations Analyzing one student's work on four translation tasks. Through separate
interviews and attempts to determine a possible reason for the nature of the
student's understanding of the vertex and the coefficients *b*
and *c* of a quadratic function in standard form, the authors
call attention to cognitive obstacles and how to address these challenges. |

Connecting Research to Teaching: Habits in the Classroom Common practices that have the unexpected consequence of narrowing the student's view of the concept of function. |

Illuminations: There Has to Be a System for This Sweet Problem We are confronted with problems on a regular basis. Some of these are easy to solve, while others leave us puzzled. In this lesson, students use problem-solving skills to find the solution to a multi-variable problem that is solved by manipulating linear equations. The problem has one solution, but there are multiple variations in how to reach that solution. |

The Teaching Principle Effective mathematics teaching requires understanding what students know and need to learn and then challenging and supporting them to learn it well. |

The Learning Principle Students must learn mathematics with understanding, actively building new knowledge from experience and prior knowledge. |