A Two Component View of How Geometric Thinking Works
Component #1: Abstract vs. Concrete Thinking. Sometimes students (and their teachers)
have difficult transitions from Algebra 1 to Geometry.
The difficulty begins with how we start problem solving in each type of math.Algebra
usually begins in the imagination with variables and formulas (the abstract); Geometry
begins with what we can see and touch (the concrete).
Now this doesn’t seem like much difference, but it is a big one. Algebra begins with
the abstract, like formulas and variables, where you constantly draw upon your imagination
to remember that letters represent numbers. Geometry deals with the concrete visual
pictures of points, lines, planes, angles, polygons and circles, before any formula
is put on paper. And although geometry formulas are algebraic, thought begins with
a picture of the figure before we begin to reach for any formulas.
Early grade school began with visual pictures of numbers. We learned about the number
3 by seeing pictures of 3 sticks, 3 cats, 3 dogs, etc. We even learned arithmetic
through pictures; we might have learned 3+4=7 by seeing a picture of 3 apples and
another 4 apples grouped to make 7 apples.
Soon, our numbers got bigger; we worked with numbers in the hundreds, thousands and
millions. Because these numbers are too big to draw and count up, pictures get abandoned
for algorithms. Algorithms are those steps we use from memory to: add and multiply
large numbers by carrying digits, or learn subtraction and division through borrowing
Because our late grade school and junior high years are heavily weighted developing
arithmetic, pre-algebra and algebra thinking, we can get into a 6+ year rut of one
way of thinking - that is remember/imagine a formula or algorithm which matches the
problem, and move on to ‘solve’ it.
In geometry thinking, we go BACK to the way we approached math in early grade school
- we go back to looking at pictures first. In geometry as a high school teen, we
are drawing on brain cells and thinking patterns which haven’t been used in years,
unless we are big into art, design, building or tinkering.
So, to be successful in geometry, we have to break an old habit of how we approach
math -that old habit of beginning with imagination and equations, and going straight
to theanswer, rarely drawing a picture.
Our new habitfor successful geometryrequires us to begin with a right understanding
of a what we SEE, BEFORE we ever bring any algebra into it.
Component #2: Logic and Reasoning. In addition to the visual aspects, there’s a
logic component to geometry which is a challenge whether a person is a concrete/visual
learner or abstract learner.
There are 3 undefinded terms, a host of definitions, and a handful of laws which
belong to the world of geometry. How we combine them creates the rules or postulates
allowing us to use geometry; this combination process is the logic and reasoning
component to geometry.
The legal and medical professions constantly draw upon the logic and reasoning processes
of geometry. Research and analysis in science and engineering are also dependent
upon these same processes developed within geometry. And how much improvement could
business, government and personnel management be improved through applying logic?
The flowchart below shows typical ups and downs associated with the logic and reasoning
found within geometry. You can actually see the complexity of thought which resides
in geometry. You don’t just think ‘forwards’ like in algebra; you have to think ‘forwards’
and ‘backwards’ until you get to your solution. This is why just being a visual learner
is not enough to be good at geometry; it requires learning to think clearly and logically
The consequences of not learning geometric processes may lock up a student’s math
effectiveness for a long time. Even if we’re good at math, all of our math knowledge
won’t do us much good if we look at something and can’t relate it to any of the arithmetic,
algebra, geometry, or calculus we may know.
A Flowchart of Geometry Thinking
Logic and Reasoning
x = 13
y = 2x2 - 31
y = ?
“Just when I was getting good at math, you changed it!