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…

The goal of this article is to raise questions that will promote discussions among secondary mathematics teachers about the concept of inverse functions and how to motivate a more conceptual understanding of them in their classrooms. The authors share data to answer the following questions: (1) What meanings of inverse functions do high school…

This article highlights three of the eight Mathematics Teaching Practices (MTP) published in the National Council of Teachers of Mathematics' (NCTM's) "Principles to Actions: Ensuring Mathematical Success for All" (2014): (1) facilitating meaningful mathematical discourse (MTP 4); (2) posing purposeful questions (MTP 5); and (3)…

Polynomials with rational roots and extrema may be difficult to create. Although techniques for solving cubic polynomials exist, students struggle with solutions that are in a complicated format. Presented in this article is a way instructors may wish to introduce the topics of roots and critical numbers of polynomial functions in calculus. In a…

"Find the extreme values of the function." "At what rate is the distance between A and B increasing after 12 seconds?" Prompts like these can be heard in most first-semester calculus courses. Unfortunately, these cues also tend to prompt students' eyes to glaze over with thoughts of "When will I ever use this?" This…

The correct use of visual perspective is one of the main reasons that virtual reality environments and realistic works of art look lifelike. Geometric construction techniques used by artists to achieve an accurate perspective effect were developed during the Renaissance. With the rise of computer graphics, translating the geometric ideas of 600…

It is no easy feat to engage young people with abstract material as well as push them to greater depths of understanding. Add in the extra pressures of curriculum expectations and standards and the problem is exacerbated. Projects designed around standards and having multiple entry points clearly offer students the best opportunity to engage with…

On a typical day in a university precalculus classroom, as students arrive, they settle into their small groups and begin sharing their ideas and questions from the homework assigned the previous night. It can be quite challenging to create a learning environment in which students engage in mathematical thinking outside class and are prepared to…

Most word processors, including Google Docs™ and Microsoft® Word, include an equation editor. These are great tools for the occasional homework problem or project assignment. Getting the mathematics to display correctly means making decisions about exactly which elements of an expression go where. The feedback is immediate: Students can see…

Guessing, for Pólya, is an important way of getting an initial handle on a mathematical problem. An argument can be made to place guessing in any one of the first three steps of the four-step approach to problem solving as described in "How to Solve It" (Pólya 1945). It could be a part of understanding the problem, devising a plan, or…

One of the primary expectations that the author has for her students is for them to develop greater independence when solving complex and unique mathematical problems. The story of how the author supports her students as they gain confidence and independence with complex and unique problem-solving tasks, while honoring their expectations with…

Calculus, perhaps more than other areas of mathematics, has a reputation for being steeped with procedures. In fact, through the years, it has been noticed of many students getting caught in the trap of trying to memorize algorithms and rules without developing associated concept knowledge. Specifically, students often struggle with the…

Typical treatments of the derivative do not clearly convey the idea that the derivative function represents the original function's rate of change. Revealing the relationship between a function and its rate-of-change function for static values of "x" does not facilitate productive ways of thinking about generating the rate-of-change function or…

Sometimes the best mathematics problems come from the most unexpected situations. Last summer, a Corvette raced down a local quarter-mile drag strip. The driver, a family member, provided the spectators with time and distance-traveled data from his time slip and asked "Can you calculate how many seconds it took me to go from 0 to 60…

In this article, the authors introduce the value of "e" by building on students' prior knowledge of slope and using their abilities to analyze, approximate, and interpret rates of change using graphs, symbols, and numerical data. This approach allows students to construct and interpret the value of "e" while laying the conceptual foundation for…