Abbott begins his narrative from the perspective of a two-dimensional being (A Square). The dimension in which a character resides determines perspective. For example, the two-dimensional being watching a sphere pass through their plane would perceive it as a line appearing suddenly, increasing in length (quickly and then slowly), and then decreasing in length (slowly and then quickly) before disappearing altogether. Whereas, a three-dimensional being would perceive the same event quite differently.
How does this connect to teaching and learning? How often have you asked a student a question and been completely confused by their response? I have come to understand that students always respond in a way that makes sense - to them. I can't think of it as me being right and them being wrong. I am right from the standpoint of my own perspective. In order to teach them my perspective, I need to be open. I need to understand their perspective first. I need to know how do they see it? What sense are they making of it? What experiences are they drawing from? How do I provide them with opportunities to widen their perspective? I've come to realize that simply asking, What do you see? and then following up with How do you know? is often a great, low floor, high ceiling starting point.
Just for kicks, ask your students (whatever their age) for their perspective on mathematics sometime. What is math all about? I've done this many times with students at various grade levels. Most often they say that math is about answering questions involving numbers and equations. They might mention the importance of efficiency, too. They might mention the importance of "right" and "wrong." They tend to focus in on the quantitative, closed-ended aspects of math rather than seeing creative and exploratory aspects of math. Math is about discovering, describing, generalizing, and explaining patterns in structure, quantity, space, and change. Efficiency is an important perspective, but it's not the only perspective.
There is a myriad of dimensions to mathematics (literally), each offering a unique perspective. Dimensions are (and have always been) elemental to visualizing and conceptualizing math. By exploring dimensionality, we can make concepts and patterns concrete and then extend from what we can experience to the realm of abstract ideas. Dimensions provide us with a set of tools (hand tools, perhaps) that we can use to build understanding and communicate perspective.
So, what could the fourth dimension look like? Is there only one answer? What do you see?
