Since the 1970s, the Mathematical Association of America's (MAA) journals "Mathematics Magazine" and "College Mathematics Journal" have published "Proofs without Words" (PWWs) (Nelsen 1993). "PWWs are pictures or diagrams that help the reader see why a particular mathematical statement may be true and how one…

Constructing viable arguments and reasoning abstractly is an essential part of the Common Core State Standards for Mathematics (CCSSI 2010). This article discusses the scenarios in which a mathematical task is impossible to accomplish, as well as how to approach impossible scenarios in the classroom. The concept of proof is introduced as the…

People are regularly confronted with environmental and science-related issues presented to them in newspapers, on television, or even in their own doctor's office. Often the information they use to inform their decisions on matters of science may be ambiguous and contradictory. This article presents an activity that investigates how students deal…

This article adopts the following classification for a Euclidean planar [triangle]ABC, purely based on angles alone. A Euclidean planar triangle is said to be acute angled if all the three angles of the Euclidean planar [triangle]ABC are acute angles. It is said to be right angled at a specific vertex, say B, if the angle ?ABC is a right angle…

Writing is an important content area that pervades all subject areas and is required for post-school success, yet many students with autism spectrum disorders (ASD) often struggle in written expression. In this article we discuss the characteristics of students with ASD that make writing difficult, and the strengths, such as the use of technology,…

This classroom note is presented as a suggested exercise--not to have the class prove or disprove Goldbach's Conjecture, but to stimulate student discussions in the classroom regarding proof, as well as necessary, sufficient, satisfied, and unsatisfied conditions. Goldbach's Conjecture is one of the oldest unsolved problems in the field of number…

Today's students must be more than good test takers. They must be able to collaborate, innovate, and think critically to solve real-world problems. As content demands increase, how can teachers make time to teach these advanced skills? "Sparking Student Synapses, Grades 9-12" describes how master teacher Nigel Scozzi used Rich Allen's Green Light…

The purpose of this article is to provide examples of "non-traditional" theorems that can be explored in a dynamic geometry environment by university and high school students. These theorems were encountered in the dynamic geometry environment. The author believes that teachers can ask their students to construct proofs for these theorems. The…

While serving in the U.S. Congress, Abraham Lincoln, a self-taught learner, mastered Euclid's Elements (Basler 1953). Most students today do not study mathematics for recreation. Unlike Lincoln, they need a little help in learning how to write a geometry proof. Today's technology--specifically, The Geometer's Sketchpad[R] (GSP)--can help make…

Recognizing that textbooks play a prominent role in fostering students' understanding of reasoning and proof, the authors conducted a study to examine the extent to which textbooks used in U.S. high schools provide opportunities for students to encounter proof-related reasoning and how the nature of proof-related reasoning in textbooks varies by…

The central ideas of postcalculus mathematics courses offered in college are difficult to introduce in middle and secondary schools, especially through the engineering and sciences examples traditionally used in algebra, geometry, and trigonometry textbooks. However, certain concepts in music theory can be used to expose students to interesting…

Many students find proofs frustrating, and teachers struggle with how to help students write proofs. In fact, it is well documented that most students who have studied proofs in high school geometry courses do not master them and do not understand their function. And yet, according to NCTM's "Principles and Standards for School Mathematics"…

The purpose of this article is to present a collection of problems that allow students to investigate magic squares and Latin squares, formulate their own conjectures about these mathematical objects, look for arguments supporting or disproving their conjectures, and finally establish and prove mathematical assertions. Each problem is completed…

In this article, the author presents some interesting geometric fallacies that were used in a problem-solving course for preservice high school mathematics teachers. He states that presenting these fallacies in ordinary math classes can enrich the class by making the students think hard about both the theorems themselves and the steps in the proof…

This guide is intended to be useful to educators in high schools and middle schools, to superintendents and school boards, and to state policymakers in planning and executing dropout prevention strategies. The target audience includes school administrators as well as district-level administrators, seeking to help develop practice and policy…