Q&A #1560

Teachers' Lounge Discussion: Teaching geometry

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From: Robert Walker

To: Teacher2Teacher Public Discussion
Date: 1999083122:19:37
Subject: Re:Teaching Geometry

Geometry originated when man made the astounding discovery that the
length of a line can be associated with, or represented by, a number.
This was a great discovery at the time, something that we take for
granted now, but really dazzled the people of the past.  It was like
the wheel.  Who would think that a round wheel would be a device that
could actually convert a rotation into a linear distance: when you
think about it the wheel is really a magic device that converts a
rotation into a straight(also called a linear) distance. They say it
took man 1-million years to discover the wheel.  The wheel is a
geometical form that, through it's rotation, and geometric
characteristics, allows a person to sit in a stationary carriage (or
car) while the car is moving along the ground.  When you think
strongly about it, you are amazed.  If you think about it in a light
manner it doesn't mean a thing to you.  But, take the wheel away and
you'll soon appreciate it's value as a geometrical form.
In the latter part of the 19th century the great mathematician Felix
Klein (1849-1925) said "physics is geometry."  By this he meant that
any geometrical form has an equivalent real-life problem that can be
associated with it.  His interpretation of geometry was this: take any
triangle and you can find a problem that can be solved by using that
triangle. In other words, there is an analogous, real-life problem
that can be related to any geometric form.  Any square drawn on a
piece of paper can be so scaled in feet so that it represents the
dimension of one of the square rooms in your house, etc.

Geometry is not just a bunch of lines drawn to aggravate students. 
Every geometric form has real-life problems that can be associated
with it.  That is the meaning of the word "representation".  Every
number between 1 and 10 can be used to represent one of 10 different
cars.  Are the cars numbers? No.  Are the numbers cars?  No.  The
numbers only are used to represent different cars on a car lot.  In
like manner the lines in a geometric form can represent different
physical things.  They can represent distances, heat intensity, force
intensity, light intensity, electrstatic, or magnetic field intensity,
etc.  This is why geometry and trigonometry are studied.  Because the
relationships we learn from the geometry and trigonometry are related
to real life problems and help us solve these.

All students need to know what geometry is, not just the girls.  The
more the students are told about what geometry is and why it is
needed, the more they will have a use for geometry.  Nobody wants to
study a bunch of nonsense syllables.  Students want to learn things
that they can use.  They have a whole world of things to learn when
they study geometry because it relates to everything physical and is
being used in the humanities, also.  So teach the students the
underlying facts behind geometry and they will be invigorated to learn
the details behind the geometry.  

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