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.
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 firstname.lastname@example.org 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
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 email@example.com
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.