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why teach geometry (part 1)
Posted:
Jun 4, 1994 11:41 PM
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To all interested,
Over a month ago I posted a question about why we should and should
not teach geometry in high school. I received many responses and also
messages asking me to post the results. Well here they are. I tried to
put all the reasons why we should teach geometry in high school together
and
at the beginning. These are not all the responses I received (many were
duplicated). I tried to put together the responses that I liked best. I
would like to thank all the people who responded to my question. Their
responses helped out a lot in my discussion.
ENJOY!!
WHY SHOULD WE TEACH GEOMETRY IN HIGH SCHOOL?
WHY SHOULD WE NOT TEACH GEOMETRY IN HIGH SCHOOL?
I am greatly encouraged to see positive and powerful reasons to teach
Geometry at the secondary level in forms and with methods developed over
several thousand years. At times I am inclined to be very humble when
answering such ultimate questions as WHY?. At these moments, at least a
few steps away from my more aggresive personality, I notice the big
picture.
Societies have used the study of Geometry as an interim step toward higher
learning for a long time. (From before Euclid through today...)
It was only after Descartes' efforts helped us join algebra and geometry
that we moved with Newton and Leibnitz through the Calculus into a mental
mode ready to harness something more than human slaves and mighty beasts
and natural forces to perform work.
Without geometric vissualization it is very difficult to perceive the
interactions of complex processes.
That is a little bit of the high end. My view of mathematics is as a part
of our overall thinking akill. Visualization, measurement, rigorous
proof, estimation, intuition and much more is practiced even in the
poorest implementation of a Geometry course. If a student does not
practice these skills in Geometry classes, then we must make sure we can
provide a course better than this one which has (above all others)
withstood the test of time.
What is the best of Geometry teaching? Are any methods correct of all
learners? Is any set of methods sufficent for a classroom full of
learners? These are the questions to which we seek answers. The
rationale which we here provide gives context to our pedagogy.
The question less well answered by time and success is: Do students not
going to college REALLY ned geometry? or What kind of Geometry do non
college bound students need?
My inclination is to point to professions and activities which use
visualization and measurement. The tailor, the carpenter, the navigator,
and so on ... In so far as a learner mightr ever work in such ways, then
it is evident that a study of the underlying "geo"-"metry" can be of
benefit.
<<<<<<<<<<<<<<<<<<<<----------->>>>>>>>>>>>>>>>>
Steven S. Means means@belnet.bellevue.k12.wa.us
Sammamish HS Math Teacher & Tech Coordinator
(206) 455-6162 (wk) (206)TOP DUCK (Lk. Margaret)
Snail mail: 19921 330th NE, Duvall, WA 98019
<<<<<<<<<<<<<<<<<<<<----------->>>>>>>>>>>>>>>>>
Five interesting ideas are given and expanded upon by John Van De Walle on
page 325 of his second edition: Elementary School Mathematics, Longman,
1994. And SOME OF THESE ARE ALSO REASONS WHY NON COLLEGE-INTENDING
HIGH SCHOOL STUDENTS SHOULD STUDY GEOMETRY AS WELL (in my view). Briefly,
he
suggests:
1. Geometry helps people have more complete appreciation of the
world in which they live [he then details instances of both the structure
of the solar system as well as our synthetic universe]
2. Geometric explorations can develop problem solving skills
[spatial reasoning, etc]
3. Geometry plays a major role in the study of other areas of
mathematics [e.g., ratio and proportion with symmetry]
4. Geometry is used by people in their professional lives [e.g.,
artists, land developers]. And also in the home [design a dog house,
decorate a living room, etc.]
5. Geometry is fun and enjoyable [can be used to entice students
into studying more math]
I've ommitted his details due to time, but perhaps the direction
is clear. Among other things, I'd suggest that geometry has been important
since the days of the Babylonians, Egyptians, ancient Chinese & Hindus, not
to mention Thales and all the illustrious Greeks who followed in his path.
It is an integral part of the culture of mathematics. Let's hold onto it,
please.
Ron Ward
Western Washington U.
Bellingham, WA
Just to add my two cents to an interesting discussion...
Geometry does lend itself to the use of manipulative types of excercises
that
are difficult to apply to other branches of mathematics, thus students may
get more chances to "explore" in geometry class than in any other area of
math. The inclusion of Geometry in the curriculum BEFORE algebra, trig,
etc.
may lead to greater interest in mathematics as students go on.
In the devil's advocate category, I must admit that despite the fine
sounding
rhetoric above, it seems that geometry class is instead the place where
students tune out of the math sequence, not become more interested.
Perhaps
the standards have something to say to this (i.e. use manipulatives,
integrate
geometry, algebra, trig, etc. across all grade levels). It seems that most
students find geometry to be their least favorite topic, despite its
potential
for exploration
Finally, to the question of geometry for non-college bound students, I must
ask
the further extension: What good is any secondary coursework to non-college
students? It seems that the professions available to high school diplomas
are
better prepared for by apprentice experiences than by public schooling, but
these experiences are largely unavailable to students. This is, of course,
outside the realm of the NCTM Standards :)
Keep up the good work!
John Pannell
X_PANNELL@VULTUR.EMCMT.EDU
John,
I don't think geometry must be taught in HS. I agree that geometry
is an interesting and important subject but is it suitable for HS students?
When we decide whether a topic is suitable for students, we need to
consider not
just how useful the topic is but we need to consider the necessity and
suitablity.
I think Euclidean geometry is suitable for elementary school and
the
differential geometry is suitable for university level. So, for HS
students
they can learn other interesting and new topics in mathematics such as
factals, choas... instead of OLD Geo. (Euclidean) and HARD Geo. (Others).
David Chung
Hong Kong University of Science and Technology
Davidc@usthk.bitnet
As a student teacher in a city high school in Seattle, WA I have been
struggling with just this question. Here are some thoughts:
- Geometry is important from a historical perspective. I am a strong
proponent of teaching math in a context of history (ancient to present),
and at least among the ancients, geometry dominated that context.
- I would find it hard to argue for or against geometry as a whole. Seems
like the pieces should be taken separately. For instance, in a world of
computer graphics, coordinate geometry is becoming very useful. The
standard constructive geometry on the other hand is downright silly. When
was the last time you were given only a straight-edge and a compass and
asked to construct a perpendicular bisector (for that matter, when was the
last time you were given a compass?)
- Speaking of pieces, the standard way of teaching proofs has to be one of
the largest travesties of modern education. Mathematicians aren't given a
truth and asked to prove it; they use experience to follow their
intuitions toward truths. The way most "proof" geometry is taught should
be called "here's where I ended up; guess how I got there."
Of course, the truth of the matter is that many of us will end up
teaching geometry in high school whether or not we think it worthwhile.
If anyone comes up with ideas for increasing its "worthwhileness," please
let me know. I'm going to need them this spring.
-Dug Steen
dug@seattleu.edu
Dr. Meseke,
I teach math methods for preservice elementary school teachers at
Towson State University (Baltimore, MD). I am often asked this question by
my students. I try to get the students to answer the question on their
own.
The following are a few responses that my students give:
* Since we live in a geometric world, the study of geometry helps
students better understand the world in which they live.
* The study of informal geometry at the elementary school level
helps
students learn geometric concepts at a more formal level (high
school
* Knowledge of geometry can be used to help students learn other
con-
cepts, such as fractions, measurement, functions.
Most of my students don't realize the importance of studying and
understanding
geometry concepts and skills until they have had opportunities to actively
participate in activities that utilize physical materials, cooperative
groups,
and discussion of the concepts inherent in the activities (which is
basically
the way I "teach" my methods course!).
Mike Krach
Associate Professor
Mathematics
TSU
As long as algebra and geometry proceed along separate paths,
their advance was slow and their applications limited. But
when these sciences joined company, they drew from each other
fresh vitality and thenceforce march on at a rapid pace towards
perfection.
--Joseph Louis Lagrange--
Jan. 25, 1736--April 10, 1813
> How will the students apply it in the future?
> What use is it in the future?
> Do students not going to college REALLY need geometry?
> What good is it to the students?
> What is the importance of geometry?
> Are there alternatives to teaching geometry in the High School?
> Overall, why learn geometry?
As another post said, one reason for teaching geometry is that it
takes students into a new area of mathematics, in the sense that it
doesn't have all those nice, neat algebra formulas. It requires
thinking, which scares many students, but it gives those students who
are more visually oriented a chance to shine compared to the more
analytical students.
Another nice thing about geometry is that it lends itself so well
to student investigations. The instructor gives the student a couple
of different triangles and has them construct the medians. What do
the students notice? Hey neat, they all meet at the same point. I
wonder if that always happens. This can also be done in algebra but
it is such a neat thing the way it can be done in geometry.
In summary, geometry makes students think in ways they may never
have thought before. That is a benefit to them for the rest of their
lives even if they never use the actual concepts learned. That is an
answer to all but the "alternatives" question above. I also like
having geometry in high school because it makes my job easier when I
teach Trig in college.
> It would also be great if I got some "devil advocate" responses,
> claiming that geometry is unimportant and so on. So, pros and
> cons on this question would be very beneficial.
>
One argument against geometry in high schools is that it can be
so difficult for students. Many people seem to think that anything
that causes a student to do poorly damages their self esteem and
should therefore be watered down. However, I have personally never
understood how remaining uneducated and ignorant can improve anyone's
self esteem.
Why teach geometry in high school? I won't say the sarcastic "Why not?";
but I do like the fact that geometry tends to be less formula-oriented
that algebra. This is not said to put down algebra, just to describe a
difference between the two subject areas.
Too often, I see mathematics students come to college as
"formulaholics". You know these students. Their first question about a
problem is usually "What formula do I use?" When you tell them that
there isn't a "formula" to solve this problem, they immediately exhibit
withdrawal symptoms and check into the Betty Ford Clinic.
Geometry forces the students out of the "formula-mode" and into the
"thinking-mode". This is one reason to study geometry in high school
that goes beyond the usual nature of the course.
Geometry in HS
Dear John,
I've had the same question on my mind for years now. I'm about to attend
graduate school in education, and have spent some time wondering how I
would
re-design a math curriculum if I could play God.
I'm at work, without the proper references handy, but I'll try to give you
my
thoughts.
First of all, I remember my h.s. geometry teacher telling me it was hard to
find teachers who liked to teach the geometry course. He did, and taught
it
well, but people with a more "symbolic math" orientation tend to shy away
from
the course, or teach it less creatively than they would another math class.
So
I wonder if a lot of teachers at some level are asking the same question.
Another is the definition of the h.s. geometry class. Locally (in a wealty
school district), the Honors Geometry is classical two-column proofs. The
other tracks are more "applied geometry" a la NCTM standards. Interesting
that
they kept the "old form" for the Honors math students, huh?
I don't believe the value of geometry is in proof. You can introduce proof
in
algebra (my algebra instructor did, starting with axioms and having us
prove
with predicate logic that for all x in the real numbers, x*0 = 0. The
uniqueness of the additive and multiplicitive inverses, etc.) Any time we
developed a technique in algebra, we had to first convince ourselves with
proof
that it was sound.
Ian Stewart in "Problems of Mathematics" (I don't have the page number)
documents the fact that the Greeks didn't believe in this "pure geometry
with
only straight-edge and compass" nonsense. They freely integrated analysis
and
geometry all over the place. The existence proof of irrational numbers is
based on a geometric problem, for example. It wasn't until around 500 AD
or so
(???) that some philosopher showed how you *COULD* go back and re-derive
geometry using only straight-edge and compass procedures. That's the form
that's been taught ever since.
It's rather like teaching arithmetic/algebra through the Peano axioms,
though.
Do you really need to derive *everything* before you use it? Although I
learned algebra this way, I'm not sure it needs to be so rigorously
applied.
Where that balance is between hooking a student with new applications, and
needing to backtrack and justify with proof, I don't know. I might mention
that we're doing it backwards in calculus as well, requiring weeks of
delta-epsilon proofs before we taste the meat of derivatives (although
historically, limit proofs didn't come along until a hundred years after
calculus was routinely used.)
Back to your questions...
*Some* form of deductive proof, and the form of axiomatic systems, should
be
taught. It's what separates math from all other subjects. To not include
that
early and often is to offend what Jerome Bruner calls "intellectual
honesty."
Math is not math without this quality, it's just shop-work. However, I'm
not
convinced geometry is necessarily the only vehicle for this.
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