In a previous article published in the "American Mathematical Monthly," Tucker ("Amer Math Monthly." 1997; 104(3): 231-240) made severe criticism on the Mean Value Theorem and, unfortunately, the majority of calculus textbooks also do not help to improve its reputation. The standard argument for proving it seems to be applying…

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…

To address student misconceptions and promote student learning, use discussion questions as an alternative to reviewing assessments. This article describes how using discussion questions as an alternative to going over the test can address student misconceptions and can promote student learning.

This article describes some misconceptions about random variables and related counter-examples, and makes suggestions about teaching initial topics on random variables in general form instead of doing it separately for discrete and continuous cases. The focus is on post-calculus probability courses. (Contains 2 figures.)

Calculus instructors struggle to teach infinite series, and students have difficulty understanding series and related concepts. Four instructional strategies, prominently used during the calculus reform movement, were implemented during a 3-week unit on infinite series in one class of second-semester calculus students. A description of each…

Instructors are constantly struggling to help students understand mathematical concepts as well as the relevance of mathematics to the real world. In calculus, students possess misconceptions of the limit concept. "Pushing the Limit" refers to a semester-long calculus class project that required students to read about, interview calculus…

The Intermediate Value Theorem (IVT) is not often proved in Calculus I classes because many teachers and students see the theorem as obvious and its proof as impenetrable. This article addresses those two misconceptions, showing "how" the IVT can be proved in Calc I... and "why" it "should" be.

Limits are foundational to the central concepts of calculus. However, the authors' experiences with students and educational research abound with examples of students' misconceptions about limits and infinity. The authors wanted calculus students to understand, appreciate, and enjoy their first introduction to advanced mathematical thought. Thus,…

As a fundamental resource, textbooks shape the way we teach and learn mathematics. Based on examination of secondary school and university textbooks, we describe to what extent, and how, the presentation of mathematics material--in our case study, the concept of the line tangent to the graph of a function--could contribute to creation and…

How do mathematics teachers introduce the concepts of slope, rate of change, and steepness in their classrooms? Do students understand these concepts as interchangeable or regard them as three different ideas? In this article, the authors report the results of a study of high school Advanced Placement (AP) Calculus students who displayed…

I discuss my experience with teaching an advanced undergraduate Real Analysis class using both lecturing and the small-group guided discovery method. The article is structured as follows. The first section is about the organizational and administrative components of the class. In the second section I give examples of successes and difficulties…

Many students, when they take an elementary differential equations course for the first time, bring with them misconceptions from numerical methods that they had learnt in their calculus courses, most notable of which concerns the mesh width in using a numerical method. It is important that we strive to dispel any of these misconceptions as well…

We find infinite series in calculus to be one of the most confusing topics our students encounter. In this note, we look at some issues that our students find difficult or ambiguous involving the Ratio Test, the Root Test, and also the Alternating Series Test. We offer some suggestions and some examples, which could be a supplement to the set of…

Suggests that the cross product of two vectors can be more easily and accurately explained by starting from the perspective of dyadics because then the concept of vector multiplication has a simple geometrical picture that encompasses both the dot and cross products in any number of dimensions in terms of orthogonal unit vector components. (AIM)

This paper describes a sequence of lessons from two Calculus I classes for teaching the epsilon-delta definition of a limit. In these lessons, the author elicited students' misconceptions and perceptions of this definition through a reading/writing lesson and then used these student ideas to design a lesson aimed at addressing these misconceptions…