Many people say that they have no spatial vision, that they do not orient themselves well... And they seem resigned to believe that they do not possess these capacities which will limit them, for example, to understand architecture, engineering or art, where we must interpret blueprints, imagine structures or dominate perspectives. However, they are not innate abilities that are acquired spontaneously, but it is possible and necessary to model them as part of the learning of mathematics.
Spatial Sense helps us to understand the world around us: understanding the plane and space, identifying bodies, managing concepts and geometric relationships... In the day to day, it is used when placing a lamp in a room, imagining the house of our dreams on a plane, locating the car in the parking, or looking for a way to fit lumps in the trunk. It also allows you to determine how many squares are in the figure below. In order to distinguish the fourteen hidden squares it is necessary to put into play the perception of the figure and the context. Other skills such as visual discrimination or the perception of position in space allow us to understand that there are four squares formed by two black and white squares, which are all the same if two of them turn 90 degrees.
Although in many cases is forgotten, it is essential to develop this capacity in the classroom of mathematics. This translates into teaching geometry beyond the management of formulas and memorizing terms, and making it present in everyday life, asking why the geometry around us.
For example, to define a circumference, we say that it is the set of points that is the same distance from a central point. This property makes it be drawn with a compass, and used on the sports tracks to have the same distance from the opponents at the time of the sack. The circumferences are also characterized by not having vertices, which may avoid the possibility of cracking the windows of the boats. And they have a constant width, which prevents the lids from falling into the pots and we use them as wheels for bicycles. It also has constant curvature, which makes you build pots in a lathe. And infinite symmetries, which make it easier to distribute the pizza in equal parts. And they maximize the area to equal perimeter, as seen in phenomena of nature, as in the formation of soap bubbles. However, they are not suitable for tiling the plane and depend on Pi to know its area and length, which complicates the exact measurements.
On the other hand, visualization can also be used to understand better mathematical concepts. Although mathematics boasts of abstracts and demonstrations must be independent of images, the power of the use of symbolic language to generalise does not have to be incompatible with a visual support. The images can serve as a basis for completing formal demonstrations, which will then generalize the observed and abstract it from the concrete figures that have been used.
Important scientists have highlighted the role that visualization has occupied in their thinking. Albert Einstein himself, in a letter to the mathematician Jacques Hadamard, commented: "Words or language, written or spoken, I do not believe that they play any role in the mechanism of my thought. Physical entities that seem to serve as elements of thought are certain signs and certain more or less clear images that can be "voluntarily" reproduced and combined. "
Like Einstein, we use geometry to describe the universe, even if it's just the one around us. Spatial sense allows us to convert the images we perceive into mental representations that we can manipulate and from which we can extract properties. And visualization strengthens the connection between the perception of the world around us and the equations that explain it. It's worth exercising.
Rafael Ramírez is a professor in the Department of Didactics of Mathematics at the University of Granada