The fundamental theorem of calculus (FTC) plays a crucial role in mathematics, showing that the seemingly unconnected topics of differentiation and integration are intimately related. Indeed, it is the fundamental theorem that enables definite integrals to be evaluated exactly in many cases that would otherwise be intractable. Students commonly…

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…

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…

According to the conceptual framework for K-grade 12 statistics education introduced in the 2007 Guidelines for Assessment and Instruction in Statistics Education (GAISE) report, students can be located at one of three developmental levels of statistical literacy: A, B, or C. These levels are independent of age and grade level, so, in theory,…

In this brief article, the author illustrates the flaws of FOIL (multiply the First, Outer, Inner, and Last terms of two binomials) and introduces the box method. Much like FOIL, the box method can become easy to use. Unlike FOIL, however, the box method is a more direct and visible link to using the distributive property to determine area, a…

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…

Both the Common Core State Standards (CCSSI 2010) and the NCTM Process and Content Standards distinguish between Standards for Mathematical Practice (SMP) and standards for mathematical content. We believe this distinction is important and note that students often acquire knowledge of mathematical content without necessarily developing the related…

Mathematics has been taught throughout history without much more than a straightedge, a compass, and an abacus. So there is little question that all the major concepts of mathematics can be delivered without the aid of modern technology. But clearly, just the time-saving aspect of using technology can greatly enhance the art of teaching. In this…

Calculus has frequently been called one the greatest intellectual achievements of humankind. As a key transitional course to college mathematics, it combines such elementary ideas as rate with new abstract ideas--such as infinity, instantaneous change, and limit--to formulate the derivative and the integral. Most calculus texts begin with the…

In this article, Scott Smith presents an innocent problem (Problem 12 of the May 2001 Calendar from "Mathematics Teacher" ("MT" May 2001, vol. 94, no. 5, p. 384) that was transformed by several timely "what if?" questions into a rewarding investigation of some interesting mathematics. These investigations led to two…

Magic captivates humans because of their innate capacity to be intrigued and a desire to resolve their curiosity. In a mathematics classroom, algorithms akin to magic tricks can be an effective tool to engage students in thinking and problem solving. Tricks that rely on the power of mathematics are especially suitable for students to experience an…

One of the foundational topics in first-year algebra concerns the concept of factoring. This article discusses an alternative strategy for factoring quadratics of the form ax[superscript 2] + bx + c, known as "factoring for roots." This strategy enables students to extend the knowledge they used when the leading coefficient was 1 and…

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…

Numerous studies have shown that mathematical knowledge is situational, meaning that students' abilities to transfer and apply skills depends on the conditions in which they were learned (Barab 1999; Boaler 2002, 2016). Given that the real world seldom presents problems that are confined to a single discipline (Barab 1999), it is imperative that…

In this article, the authors describe how a theoretical framework--the modeling cycle of Bliss, Fowler, and Galluzzo (2014)--came to life in their classroom as students struggled with an open-ended modeling task. The authors share their high school students' work--warts and all. They explain how they used their students' ideas and errors to help…