Mathematical thinking and reasoning begins for students long before it is taught
through any sort of schooling. Beginning as infants, humans are attracted
to patterns, designs and shapes. Parents reinforce this by often
purchasing toys or mobiles with brightly colored shapes, pictures or
designs. Babies are attracted to these items before they are able to
reach, grasp or manipulate them in anyway. Later, toys are manipulated in
such a way as to provide further hands on learning to develop these types of
skills. These shapes and designs are the very foundational level of the mathematical field of geometry.
Geometry is everywhere.
Angles, shapes, lines, line segments, curves, and other aspects of
geometry are every single place you look, even on this page. Letters
themselves are constructed of lines, line segments, and curves! Take a
minute and look around the room you are in, take note of the curves, angles,
lines and other aspects which create your environment. Notice that some
are two-dimensional shapes while others are three-dimensional shapes. These man-made
geometrical aspects please us in an aesthetic way.
An angle
is formed when two rays come together at the same point (end point).
The distance between the two rays is measured in degrees using a tool known as
a protractor. Angles can be found on the human body as well as in the
many structures we have created for living and working. On your body,
each joint as it is moved creates different sized angles based on how far apart
the body parts are located. An example of angles with in a home might
include the brackets holding a shelf to the wall. Angles are created as
shapes come together.
Shapes are unique representations with specific properties to
define them. Shapes can be two- or three-
dimensional. There are numerous defined shapes. Shapes include
things such as polygons, which include squares, circles, rectangles, triangles,
etc.., quadrangles, which include parallelograms, rhombus, trapezoids,
etc…solids, which include cylinders, pyramids, prisms, etc… Each item in
our tangible world is created by combining shapes of some sort together.
Thinking of a soda can as a cylinder or a refrigerator as a combination of
squares, cubes and rectangles provides a deeper understanding of how shapes can
be combined together to create the world around us.
A line is the path, which is
always straight, and extends out infinitely (forever). A line will not necessarily
extend forever, but in order for it to be considered a line, it has the
potential to, if continued on, to never end. Lines are represented by a
straight line with arrows on both ends, indicating that it could extend
forever. Line segments are similar to lines, in that they are
always straight, but they do not extend out forever, instead they end at
specific points, known as endpoints. Line segments are typically
represented by a straight line with two dots at each end, representing the end
points. These end points are generally given a label such as line segment
AB. A curve is similar to a line segment in that it has two
specific end points, however it is never straight. A curve would be
represented in the same manner; however, instead of being straight the portion
between the two end points would be curved.
Nature also has an abundance of
geometry. Patterns can be found on leaves, in flowers, in
seashells and many other places. Even our own bodies consist of patterns,
curves and line segments. It is through the observation of nature that
scientists have begun to explore and explain the more basic principles now
accepted as scientific truths. These observations and realizations have
lead to the progression of new learning in both science and geometry.
This began with the simple repetitive patterns such as the orbiting of the
planets or the back and forth motion of a pendulum. It continues today as
new theorems and natural events are explained and represented through geometric
representations, thinking and principles.
At the most basic level, geometric
principles occur all around us. Mankind craves the geometrical
principles and to explain events occurring within the natural world.
Home builders, interior designers, landscape designers all rely on
geometric principles to attract the eye of prospective customers. In
nature, animals use the patterns and other geometric ideas as part of the
reproduction process, defense mechanisms, and as a method to attract
others. In some cases, the geometry found in nature has provided
inspiration for man-made items. While in other cases, it is the natural
events which have provided the inspiration for further developments and
understanding of geometric principles and ideas.
Homes maximize their geometric
aspects to draw the eye of potential buyers. Curves are added to break up
traditional rectangular patterns. Spiral stairways might be added to
replace the traditional straight staircases. Patterns are
found in every single part of the home including: painting designs,
window placements, carpeting, and numerous other examples. In
addition to the aesthetic principles, geometric thinking is needed to ensure
homes and buildings are structurally sound. Understanding which angles
provide make for stronger and safer buildings has helped to change trends in
construction. Additionally, laws, in this case building codes, have been
changed to include the implementation of the known laws of geometry into
various trades. These changes have significantly increased the
safety of many things in our world.
Therefore, many professions require
at least a foundational understanding of geometry. Sports,
construction, weaving, sewing, decorating, as well as many others require the
use of the concepts learned through the study of geometry. In many of
these professions, the knowledge learned through a complete understanding of
geometric principles has provided not only an increase in safety, but also an
increase in the creation of tools, skill level enhancement, and aesthetically
pleasing arrangements. In sports, an understanding of angles might allow
a baseball player to better catch or hit a ball farther. In weaving and
sewing, aesthetics can of course be increased by using geometric designs, but
other designs can be introduced, perhaps increasing sales for a new clothing
designer.
In fact, almost all professions
require some basic understanding of the more basic principles of
geometry. Whether it includes the idea of shapes or on the more complex
end, the understanding of the process involved in proving a supposition,
geometry is a direct or indirect influence. Part of geometric thinking is
the understanding of thinking in a critical manner and deducing answers given
specific facts. In this way, geometry is a life-long skill. It is
this type of thinking which is used to create new inventions or discover
solutions to a variety of life's problems. In the end,
geometry surrounds all of us, making it a safer and more productive place.
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