In this note, some variants of Cauchy's mean value theorem are proved. The main tools to prove these results are some elementary auxiliary functions.

This research deals with a possible use of history of mathematics in mathematics education. In particular, history can be a fundamental element for the introduction of the concept of integral through a problem-centred and intuitive approach. Therefore, what follows is dedicated to the teaching of mathematics in the last years of secondary schools,…

We present results of a research developed with first semester students from a Colombian Public University based on classroom intervention in a precalculus laboratory course mediated by an interactive mathematical software. We characterize and exemplify the cognitive skills of explanation, justification, argumentation and validation, using a…

One desired outcome of introductory physics instruction is that students will develop facility with reasoning quantitatively about physical phenomena. Little research has been done regarding how students develop the algebraic concepts and skills involved in reasoning productively about physics quantities, which is different from either…

In this note, we report on an implementation of discovery-oriented problems in courses on Real Analysis and Differential Equations. We explain a type of task design that gives students the opportunity to conjecture, refute and prove. What is new is that the complexity in our problems is limited and thus the tasks can also be used in homework…

This article reports on the design, development, and validation of a new instrument, the Technology-Enabled Active Learning Inventory (TEAL), to measure students' perceptions of active learning in a technology-enabled learning context. By laying the theoretical foundation, a conceptual framework for technology-enabled active learning was…

This article describes various courses designed to incorporate mathematical proofs into courses for non-math and non-science majors. These courses, nicknamed "math beauty" courses, are designed to discuss one topic in-depth rather than to introduce many topics at a superficial level. A variety of courses, each requiring students to…

The purpose of this study is to analyze the complex argumentative structure in undergraduate mathematics classroom conversations during problem solving by taking into consideration students' and teacher' utterances in the classroom using field-independent Toulmin's theory of argumentation. Analyzing students' and teacher' utterances in the class…

A visual proof that 1 - (1/2) + (1/4) - (1/8) + ... 1/(1+x[superscript 4]) converges to 2/3.

Although dynamic geometry software has been extensively used for teaching calculus concepts, few studies have documented how these dynamic tools may be used for teaching the rigorous foundations of the calculus. In this paper, we describe lesson sequences utilizing dynamic tools for teaching the epsilon-delta definition of the limit and the…

Despite few limitations, GeoGebra as a dynamic geometry software stood as a powerful instrument in helping university math majors understand, explore, and gain experiences in visualizing the limits of functions and the ?-d formalism. During the process of visualizing a theorem, the order mattered in the sequence of constituents. Students made use…

Calculus II students know that many alternating series are convergent by the Alternating Series Test. However, they know few alternating series (except geometric series and some trivial ones) for which they can find the sum. In this article, we present a method that enables the students to find sums for infinitely many alternating series in the…

Proofs that the area of a circle is nr[superscript 2] can be found in mathematical literature dating as far back as the time of the Greeks. The early proofs, e.g. Archimedes, involved dividing the circle into wedges and then fitting the wedges together in a way to approximate a rectangle. Later more sophisticated proofs relied on arguments…

The development, in an introductory differential equations course, of boundary value problems in parallel with initial value problems and the Fredholm Alternative. Examples are provided of pairs of homogeneous and nonhomogeneous boundary value problems for which existence and uniqueness issues are considered jointly. How this heightens students'…

Many mathematics students have difficulty making the transition from procedurally oriented courses such as calculus to the more conceptually oriented courses in which they subsequently enroll. What are some of the key "stumbling blocks" for students as they attempt to make this transition? How do differences in faculty expectations for students…