1.1. Introduction

"The problems that exist in the world today cannot be solved by the level of thinking that created them."--Albert Einstein

Introduction

At first glance, coordinate systems may seem a trivial topic, hardly worthy of deeper mathematical inquiry. However, coordinates are indispensible in describing locations, quantifying problems and shapes in geometry, and visualizing functions. In addition, the choice of a coordinate system and/or coordinates can either obfuscate or clarify the solution of a mathematical problem; anyone who has attempted to find the volume of a sphere by integration in cylindrical coordinates will know this intimately. The ability to translate, transform, or convert coordinates thus may streamline problem-solving and visualization. In life, coordinate systems appear in contexts as diverse as navigation and orienteering, algebraic chess notation and computer graphics.

The NCTM Standards suggest that students be able to "use Cartesian coordinates and other coordinate systems, such as navigational, polar, or spherical systems, to analyze geometric situations" (NCTM, 2000, p.308). These systems can also be extended naturally from two to three or more, or even fractal, dimensions. Going in the reverse direction, from higher to lower dimensions, leads into projective geometry, which we see in the tradeoffs in accuracy between different cartographic projections. The existence of distance metrics provides a further window into the geometry of a space: for example, the use of a sphere's great circle as a geodesic appears in the flight paths of airplanes. Furthermore, the existence of alternate coordinate systems for the same underlying space leads to the concept of bases and dimension in vector spaces.

Vectors facilitate modeling the physical world through the representation of magnitudes paired with directions. They are ubiquitous in physics, from problems involving the addition of velocity vectors to Maxwell's equations. They allow us to represent and manipulate quantities as diverse as momentum, velocity, and the wave function in the same notation, which emphasizes certain underlying commonalities of behavior.

According to the NCTM Standards, students should "develop an understanding of properties of, and representations for, the addition and multiplication of vectors and matrices" (NCTM 2000, p.290). Vectors can provide a steppingstone to more general matrix or tensor operations and broaden students' conception of "number" and the importance of mathematical properties such as the existence of inverses.

A concept map (JPEG scan) shows one way of organizing some of the relevant concepts, especially in view of the second essential question, "What is a number?"

Copyright © 2002 Yoon Ha Lee <requiescat@cityofveils.com>
Last updated on 3 June 2002