A derivative is the limit of a quotient. It is an abstraction of division. Since division is harder to understand than multiplication, teachers teach it later, hopefully only after a sound understanding of multiplication has been attained. For the same reason, it may make sense to teach integration first, and move on to differential calculus only…

In the teaching of calculus, the algebraic derivation of the derivative (gradient function) enables the student to obtain an analytic "global" gradient function. However, to the best of this author's knowledge, all current technology-based approaches require the student to obtain the derivative (gradient) at a single point by…

Proceeding like Newton with a discrete time approach of motion and a geometrical representation of velocity and acceleration, we obtain Kepler's laws without solving differential equations. The difficult part of Newton's work, when it calls for non-trivial properties of ellipses, is avoided by the introduction of polar coordinates. Then a simple…

The performance of over 2000 students in introductory calculus-based electromagnetism (E&M) courses at four large research universities was measured using the Brief Electricity and Magnetism Assessment (BEMA). Two different curricula were used at these universities: a traditional E&M curriculum and the Matter & Interactions (M&I)…

Senior secondary students cover numerical integration techniques in their mathematics courses. In particular, students would be familiar with the "midpoint rule," the elementary "trapezoidal rule" and "Simpson's rule." This article derives these techniques by methods which secondary students may not be familiar with…

Historically, calculus has displaced many algebraic methods for solving classical problems. This article illustrates an algebraic method for finding the zeros of polynomial functions that is closely related to Newton's method (devised in 1669, published in 1711), which is encountered in calculus. By exploring this problem, precalculus students…

Problem-Based Learning (PBL) is a problem-centered teaching method with exciting potential in engineering education for motivating and enhancing student learning. Implementation of PBL in engineering education has the potential to bridge the gap between theory and practice. Two common problems are encountered when attempting to integrate PBL into…

This project is one of four that were issued to first semester sophomore undergraduates at the United States Military Academy as part of an integrated learning experience at the end of their Calculus II course work. This project was used during a short, seven lesson block of instruction that was intended to capitalize on their recent academic…

Bootstrap methods and randomization tests are increasingly being used as alternatives to standard statistical procedures in biology. They also serve as an effective introduction to the key ideas of statistical inference in introductory courses for biology students. We discuss the use of such simulation based procedures in an integrated curriculum…

The following article discusses holistic grading as implemented in a first-year undergraduate calculus course. The intent of this manner of grading is to distinguish both students that truly master the material and those that have not even a limited understanding from the majority of the student body. (Contains 6 figures.)

In my Calculus classes I encourage my students to actively reflect on course material, to work collaboratively, and to generate diverse solutions to questions. To facilitate this I use peer instruction (PI), a structured questioning process, and i-clickers, a radio frequency classroom response system enabling students to vote anonymously. This…

This paper proposes modelling based learning as a tool for learning and teaching mathematics. The example of modelling real world problems leading to the exponential function as the solution of differential equations is described, as well as the observations about students' activities during the process. The students were acquainted with the…

In recent years, a considerable number of teachers in Spain have been using DERIVE to teach math subjects in High Schools and Universities. This software has been used by the authors of this work as a support tool in Mathematics courses for Engineering. Since Texas Instruments does not support DERIVE, we were faced with finding an alternative…

In the popularization course of higher education, students entering into the higher colleges are not those excellent students with excellent achievements in the senior school, so the teaching of college mathematics in higher colleges should transform from the learning emphasis to popularization emphasis for traditional excellent higher education.…

To achieve the vision of mathematics set forth in "Crossroads" ("AMATYC," 1995), students must experience mathematics as a sensemaking endeavor that informs their world. Embedding the study of mathematics into the real world is a challenge, particularly because it was not the way that many of us learned mathematics in the first place. This article…