Positive sums count. Alternating sums match. Alternating sums of binomial coefficients, Fibonacci numbers, and other combinatorial quantities are analyzed using sign-reversing involutions. In particular, we describe the quantity being considered, match positive and negative terms through an Involution, and count the Exceptions to the matching rule…

Motivated by centaurs jumping around a circular stadium, we derive Kronecker's Approximation Theorem, which in turn provides elementary solutions to difficult problems in the theory of Diophantine approximations.

Uniform sampling of binary matrices with fixed margins is known as a difficult problem. Two classes of algorithms to sample from a distribution not too different from the uniform are studied in the literature: importance sampling and Markov chain Monte Carlo (MCMC). Existing MCMC algorithms converge slowly, require a long burn-in period and yield…

With a staggering number of students dropping out of school, widespread below-grade reading proficiency, and PISA results showing a demoralizing lag internationally in math, science, and general problem solving, teachers need to exercise all the control available to them to engage students in meaningful learning. Wilhelm suggests six steps…

The standard approach to finding antiderivatives of trigonometric expressions such as sin(ax) cos(bx) is to make use of certain trigonometric identities. The disadvantage of this technique is that it gives no insight into the problem, but relies on students using a memorized formula. This note considers a technique for finding antiderivatives of…

Two original problems in geometry are presented with solutions utilizing to differential calculus: (a) rectangle inscribed in a sector; (b) point on the ray of the angle. The possibility of applying mathematics in general and differential calculus in particular for solution of practical problems is discussed. (Contains 8 figures.)

An acronym is presented that provides students a potentially useful, unifying view of the major topics covered in an elementary calculus sequence. The acronym (CAL) is based on viewing the calculus procedure for solving a calculus problem P* in three steps: (1) recognizing that the problem cannot be solved using simple (non-calculus) techniques;…

We present a derivation for the coordinates of the centre of mass--or centre of gravity--of a homogeneous triangular plate by using scaling and symmetry. We scale the triangular plate by a factor of 2 and divide its area into four plates identical to the original. By symmetry, we assert that the centre of mass of two identical masses lies at the…

This article focuses on the use of students' mathematical representations to build classroom community, promote participation in classrooms, and further equity. An example of a teacher who consciously chose to allow her students to bring their own representations to class is presented. The choice not only allowed the students better access to the…

Unlike traditional lecture-based instruction, where information is passively transferred from instructor to student, problem-based learning (PBL) students are active participants in their own learning, thrust into unknown learning situations where the parameters of the problem may not be well-defined and the task at hand ambiguous--just like in…

Many medical students use exchange programmes to enrich their portfolio. However, such a confrontation with reality can be challenging because custom and practice sometimes differ radically between countries. Exchange students post specific peculiarities, as encountered by them abroad, onto the medical exchange website. They can ask for the…

This research was designed to study the effect of a professional development program in place-based science education on teacher and student learning in rural elementary schools in Thailand's northern Lampang province. Based on the results of a pilot study conducted in 2008 at one school, which was then followed by a needs assessment conducted…

"Latina/os in Rhetoric and Composition: Learning from their Experiences with Language Diversity" explores how Latina/o academics' experiences with language difference contributes to their Latina/o academic identity and success in academe while remaining connected to their heritage language and cultural background. Using qualitative…

Results from the Organisation for Economic Cooperation and Development's (OECD's) Programme for International Student Assessment (PISA) have shown that the degree to which education systems succeed in equipping students with important foundation skills varies significantly. Since the quality of teaching is at the heart of student learning…

The public school superintendent's role has changed substantially over time, and the work of superintendents is more complex, demanding, and challenging than ever before. Recent studies indicate that 70% to 80% of the nation's superintendents will leave their positions over the next five years, suggesting the imminent entry of many new…