A mathematics classroom that reflects the vision of NCTM's "Principles and Standards for School Mathematics" will have the teacher posing problems, asking questions that build on students' thinking, and encouraging students to explore different solutions. In teaching about area, it is not sufficient to give students the…

For many students, making connections between mathematical ideas and the real world is one of the most intriguing and rewarding aspects of the study of mathematics. In the Common Core State Standards for Mathematics (CCSSI 2010), mathematical modeling is highlighted as a mathematical practice standard for all grades. To engage in mathematical…

Today, the Common Core State Standards for Mathematics (CCSSI 2010) expect students in as early as eighth grade to be knowledgeable about irrational numbers. Yet a common tendency in classrooms and on standardized tests is to avoid rational and irrational solutions to problems in favor of integer solutions, which are easier for students to…

The Common Core State Standards for Mathematics (CCSSI 2010) asks students in as early as fourth grade to solve word problems using equations with variables. Equations studied at this level generate a single solution, such as the equation x + 10 = 25. For students in fifth grade, the Common Core standard for algebraic thinking expects them to…

The authors present three strategies for making sense of students' mathematical thinking. These lenses make the abstract idea of "making sense of student thinking" more manageable and concrete. We start by taking an initial look at a student's idea, going deeper, and finally looking across several ideas.

The Common Core Standards for Mathematical Practice (CCSSI 2010) and NCTM's "Focus in High School Mathematics: Reasoning and Sense Making" (2009) present a vision of high school classrooms in which the majority of the activity involves students working on rich mathematical problems and engaging in mathematical discourse. This model…

Most people who are attracted to STEM-related fields are drawn not by a desire to take mathematics tests but to create things. The opportunity to create an algebra drawing gives students a sense of ownership and adventure that taps into the same sort of energy that leads a young person to get lost in reading a good book, building with Legos®,…

After serving as a high school math teacher in Maryland for three years, the author moved to teaching sixth-grade math. His high school background led him to differentiate differently than his colleagues. The article discusses his observations and his conclusions and offers a plan to implement changes in the way mathematics is taught through…

Must two triangles with equal areas and equal perimeters also be congruent? This question was introduced in "Mathematics Teacher" ("MT")by Rosenberg, Spillane, and Wulf in their article "Heron Triangles and Moduli Spaces" (2008), which also described the authors' subsequent investigation of a particular moduli…

Many preservice teachers come into teacher education programs with a rule-based view of mathematics, a view that is to a large extent indicative of their previous school experiences. In teaching preservice elementary school teachers, this author has used his instrumental understanding as the starting point for rich discussions about the…

Learning to play tennis is difficult. It takes practice, but it also helps to have a coach--someone who gives tips and pointers but allows the freedom to play the game on one's own. Learning to act like a mathematician is a similar process. Students report that the process of proving the inscribed angle theorem is challenging and, at times,…

An understanding of proof does not start in a high school geometry course. Rather, attention to logical reasoning throughout a student's school experience can help the development of proof readiness. In the spirit of problem solving, the author has begun to use some Japanese logic puzzles other than sudoku to help students develop additional…

This article provides a historical context for the debate between Georg Cantor and Leopold Kronecker regarding the cardinality of different infinities and incorporates the short story "Welcome to the Hotel Infinity," which uses the analogy of a hotel with an infinite number of rooms to help explain this concept. Wanko makes use of this history and…

This article describes several distinct approaches taken by preservice elementary teachers to solving a classic rate problem. Their approaches incorporate a variety of mathematical concepts, ranging from proportions to infinite series, and illustrate the power of all five NCTM Process Standards. (Contains 8 figures.)

This article describes the A, E, I, O, U technique, designed to help teachers ensure that teaching and learning are not mutually exclusive in the classroom. Most teachers would agree that motivating average teenagers to communicate how they got an answer or justify their problem-solving strategies can be as difficult as teaching a dog to whistle.…