Integration (here visualized as a pounding process) is mathematically realized by simple transformations, successively smoothing the bounding curve into a straight line and the region-to-be-integrated into an area-equivalent rectangle. The relationship to Riemann sums, and to the trapezoid and midpoint methods of numerical integration, is…

Illustrated is how students can be directed to move from a geometric to an algebraic mode of thought and back again in linear algebra, thereby deepening their understanding. Rotations and line reflections in the plane are used as the example. (MNS)

Presents ideas that are intended to convey new insights on familiar topics and to enhance pedagogy. The mathematical topics addressed are applications of transformations to numerical integration, the golden ratio, Cramer's rule, a counting problem, and the general form of the arithmetic-geometric mean inequality. (PK)