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.

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…

The routine jobs of yesterday are being replaced by technology and/or shipped off-shore. In their place, job categories that require knowledge management, abstract reasoning, and personal services seem to be growing. The modern workplace requires workers to have broad cognitive and affective skills. Often referred to as "21st century…

School librarians occupy a space rich in resources and numeracy. They are thus positioned to offer resources and standards that promote inquiry, drawing conclusions, real world connections, and aesthetic enjoyment. Part of their task with teachers is to create those connections with curriculum and students. School librarians' collections are aswim…

The authors developed this experiential exercise to aid students in learning how others perceive them as well as how they perceive others and, ultimately, to begin to help them identify the origins of such perceptions. The exercise's goal is to explore how participants may make inaccurate perceptions of others and how they might pierce through…

From the time they are born, children have a natural curiosity toward the world around them. Even though they can't speak, one can still observe that they are skilled at discovery. Science fair projects can nurture this natural curiosity and give elementary students the opportunity to explore their interests in a rewarding way. Participation in a…

Much has been written about the disconnect between education research produced in graduate schools of education and the practice of school leaders. In this article, the authors share one story of an external partnership that promotes the development of a principal's capacity for complex problem solving and the early research that suggests this…

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…