Is There A Geometric Imperative?

In this article, Dick Tahta discusses how geometry largely “lies in imagery rather than in the words that accompany or describe such imagery.” (p.20)  He does this by breaking down the geometric abilities of people, which are already present in students when they first start school: imagining, construing, and figuring.  It is difficult to fully understand imagination as it presents itself differently amongst every person.  In addition to this, it is near impossible to visualize what someone is imagining, because the moment the image is described in words, the image changes as the words can mean different things for different people.  Construing is defined by Tahta as either “seeing what is drawn” or “saying what is seen” (p.20)  He goes on to discuss how depending on who you are and how much two-dimensional imagery you have seen, a 2D picture may look like many different things, depending on shading, or perspective.  When I read this, I immediately thought of M.C. Escher and his puzzling imagery of never-ending staircases, the hand drawing the hand picture.  These are classic examples of how subtle nuances in imagery can elicit different visualizations.  The last geometric ability, figuring, is defined as “drawing what is seen” (p.20)  Tahta goes on to discuss how many students may be able to visualize numerous shapes, both two dimensional and three dimensional in their head, however if asked to draw the image they will often do so with great difficulty.  For example, a cube before the age of seven or eight is often drawn as the net of a cube (6 squares).  Most often, a student is taught how to draw three dimensional shapes, or two dimensional shapes from a different perspective, by someone else.  Tahta also makes some connections between understanding geometry and the human vestibular system.  Through interactions of our semi-circular canals and the objects in the world that we see and manipulate, a sense of space and perspective is gained.  Multiple perspectives can be achieved through sharing with others. 

In response to this week’s focus question, I must admit that I do not feel that I have enough of a geometry knowledge-base to bring it in whenever it is relevant in mathematical lessons.  This does not mean that I do not bring it in, in other subject areas such as art, or even P.E.  I have gained an appreciation now for students who may have different perspectives in geometry due to their past experiences, and I will attempt to honour those different perspectives whenever possible.