How students' rigid perceptions of mathematics persist over years has been a conundrum. Students not only think that only one solution exists for a mathematical task (Schoenfeld 1985) but also rigidly fuse mathematical concepts with specific tasks. Students deserve to learn a variety of mathematical strategies to flexibly deploy at will. Teaching…

In first- and second-year algebra classrooms, the all-too-familiar whine of "when are we ever going to use this in real life?" challenges mathematics teachers to find new, engaging ways to present mathematical concepts. The introduction of quadratic equations is typically modeled by describing the motion of a moving object with respect…

It is critical for mathematics tasks to provide students with the opportunity to engage actively in reasoning, sense making, and problem solving so that they develop a deep understanding of mathematics. Learning mathematics while solving a problem can be like entering a dark room with a single small light. The objects are in the shadows, difficult…

How can we make sense of what we learned today?" This is a question the author commonly poses to her algebra students in an effort to have them think about the connections between the new concept they are learning and concepts they have previously learned. For students who have a strong, expansive understanding of previously learned topics,…

"Media Clips" appears in every issue of "Mathematics Teacher," offering readers contemporary, authentic applications of quantitative reasoning based on print or electronic media. Based on "In All the Light We Cannot See" (2014), by Anthony Doerr, this article provides a brief trigonometry problem that was solved by…

Mathematical modeling, a key strand in mathematics, engages students in rich, authentic, exciting, and culturally relevant problems and connects abstract mathematics to the surrounding world. In this, article, the authors describe a modeling activity that can be used when teaching linear equations. Modeling problems, in general, are typically high…

In this article, Jason Taff shares an approach that he presented to advanced seventh-grade prealgebra students. He begins by summarizing some of the shortcomings of equating the order of operations concept with the PEMDAS (often rendered mnemonically as "Please Excuse My Dear Aunt Sally") procedure with the hope of helping teachers at…

Can you solve the following Problem? There are 200 fish in an aquarium, and 99 percent of them are guppies. How many guppies must be removed to reduce the tank's guppy population to 98 percent? The key to this problem is to work backward by using the data given in figure 2 to determine the surface area of the top of the aquarium; then determine…

A Steiner chain is defined as the sequence of n circles that are all tangent to two given non-intersecting circles. A closed chain, in particular, is one in which every circle in the sequence is tangent to the previous and next circles of the chain. In a closed Steiner chain the first and the "n"th circles of the chain are also tangent…

A first-year algebra student's curiosity about factorials of negative numbers became a starting point for an extended discovery lesson into territory not usually explored in secondary school mathematics. In this article, the authors, math teachers in Massachusetts, examine how to solve for factorials of negative numbers and discuss how they taught…

The perpetuation of mathematical rules that expire (Karp, Bush, and Dougherty 2014; 2015), or rules that are taught in previous grades that no longer hold true, suggests that many secondary students harbor misconceptions from their elementary and middle-grades mathematics experiences as they progress to mathematics classes that are more…

The Common Core State Standards for Mathematics (CCSSM) (CCSSI 2010) propose a new vision for the mathematics classroom with updated content standards and Standards for Mathematical Practice (SMP). These practices are founded on NCTM processes (Problem Solving, Reasoning and Proof, Communication, Representation, and Connections) and abilities…

"Mathematical Lens" uses photographs as a springboard for mathematical inquiry and appears in every issue of "Mathematics Teacher." Recently while dismantling an old wooden post-and-rail fence, Roger Turton noticed something very interesting when he piled up the posts and rails together in the shape of a prism. The total number…

The editors of Mathematics Teacher appreciate the interest of readers and value the views of those who write in with comments. The editors ask that name and affiliation including email address be provided at the end of their letters. This September 2016 Reader Reflections, provides reader comments on the following articles: (1) "Innocent…

This analysis of a problem that is frequently posed at professional development workshops, in print, and on the Web--the coffee-milk mixture riddle--illustrates the timeless advice of George Pólya's masterpiece on problem solving in mathematics, "How to Solve It." In his book, Pólya recommends that problems previously solved and put…