Although high school geometry could be a meaningful course in exploring, reasoning, proving, and communicating, it often lacks authentic proof and has become just another course in algebra. This article examines why geometry is important to learn and provides an outline of what that learning experience should be.

A critical aspect of mathematics study is developing both understanding of quantities and fluency in manipulating them. These abilities are essential for students who intend to enter technical or scientific professions. All students, however, should be exposed to methods of quantitative reasoning in order to function effectively in today's…

Using technology to explore the coefficients of a quadratic equation can lead to an unexpected result. This article describes an investigation that involves sliders and dynamically linked representations. It guides students to notice the effect that the parameter "a" has on the graphical representation of a quadratic function in the form…

Statistical studies are referenced in the news every day, so frequently that people are sometimes skeptical of reported results. Often, no matter how large a sample size researchers use in their studies, people believe that the sample size is too small to make broad generalizations. The tasks presented in this article use simulations of repeated…

In a recent collaboration with an area high school teacher, the authors were asked to develop an introductory sinusoidal curves lesson for a group of second-year algebra students. Because the topic was abstract and unfamiliar to these tenth graders, they looked for hands-on lessons to support their learning. One lesson that they found, which they…

Open-ended questions, as discussed in this article, are questions that can be solved or explained in a variety of ways, that focus on conceptual aspects of mathematics, and that have the potential to expose students' understanding and misconceptions. When working with teachers who are using open-ended questions with their students for the…

Shapedoku is a new type of puzzle that combines logic and spatial reasoning with understanding of basic geometric concepts such as slope, parallelism, perpendicularity, and properties of shapes. Shapedoku can be solved by individuals and, as demonstrated here, can form the basis of a review for geometry students as they create their own. In this…

Concepts in mathematics are often universally applicable to other fields. A critical aspect for success in high school or college is the ability to transfer content knowledge from one discipline to another. This is especially true for material learned in the sciences and mathematics. Several studies have suggested that strong mathematical skills…

Constructing formulas "from scratch" for calculating geometric measurements of shapes--for example, the area of a triangle--involves reasoning deductively and drawing connections between different methods (Usnick, Lamphere, and Bright 1992). Visual and manipulative models also play a role in helping students understand the underlying…

Much of what is taught, especially in college, is designed to support other disciplines. To determine the current mathematical needs of twenty-three partner disciplines, the Mathematical Association of America (MAA) conducted the Curriculum Foundations Project (Ganter and Barker 2004; Ganter and Haver 2011), as discussed in the appendix…

Can one infinity be more than another infinity? Ask students this question, and many will be puzzled; others will insist that "infinity is infinity." The question seems to pique their interest and provides an opportunity to present the beautifully simple but counterintuitive proofs concerning the size of infinity first constructed by…

In line with the Common Core and Standards for Mathematical Practice that portray a classroom where students are engaged in problem-solving experiences, and where various tools and arguments are employed to grow their strategic thinking, this article is the story of such a student-initiated problem. A seemingly simple question was posed by…

Confusion can arise from the subtle difference between proving a general and a particular statement, especially when general statements are presented by textbooks in ways that make them appear particular in nature. The authors discuss the implications for teaching proof in light of the current opportunities in high school geometry textbooks.

Sage is an open-source software package that can be used in many different areas of mathematics, ranging from algebra to calculus and beyond. One of the most exciting pedagogical features of Sage (http://www.sagemath.org) is its ability to create interacts--interactive examples that can be used in a classroom demonstration or by students in a…

When examining transformations of the plane in geometry, teachers typically have students experiment with transformations of polygons. Students are usually quick to notice patterns with ordered pairs. The Common Core State Standard, Geometry, Congruence 2 (G-CO.2), requires students to describe transformations as functions that take points in the…