Some suggestions for helping student understand dimension.
1. Have them read Flatland: A Romance of Many Dimensions by Edwin Abbott Abbott (Princeton University Press). This book is set in a 2-dimensional world where no one (except the hero) believes in 1-dimensional or 3-dimensional worlds. These are descriptions of what life is like in a two dimensional world and what objects from other dimensions world look like. This book also happens to be a social commentary and makes for interesting discussion (women are line segments in Flatland while men are polygons. The more sides each man has, the higher in class he is. Therefore, women are the lowest class. Abbott was commenting on the class distinctions of the late 19th century (among other things).)
2. I have students consider our 3-dimensional world in the following way.
1-dimension: First I take them out to a (straight) sidewalk on our campus running north and south. I clearly mark the four directions north, south, east, and west (our major sidewalks and streets on campus happen to run n, s, e, and w) on the sidewalk with masking tape. The rules are that they can take one step north along the sidewalk (I am giving them one direction vector), they can repeat that step as many times as they like (multiplying the vector by a positive scalar), and they can step south as many times as they like (multiplying the vector by a negative scalar). Then I ask them where they can and can't go using only those rules. I try to pick the longest sidewalk on campus so that we don't get into the walking through building questions. 1-dimension = where you can go using one direction (north) and its opposite(south).
2-dimensions: Now the rules are they can move one or many steps in one of two directions (north and east) and the opposite of those two directions. They may combine any number of steps in any of the directions in any order they want. Where can they go now? 2-dimensions = where you can go using two directions (north and east) and their opposites (south and west).
3-dimensions: I now ask them what they would have to do to get to top of a tall building on campus or into a basement. How many directions (and the opposites of those directions) do they need?
--------------------- Murphy Waggoner Department of Mathematics Simpson College 701 North C Street Indianola, IA 50125 firstname.lastname@example.org ---------------------