Date: Mar 30, 2002 11:53 PM
Author: Alan
Subject: Follow up: Teacher looking for a good geometry textbook

I have been searching for a Geometry text to use with my students.
There is a current debate over what approach to geometry is best. If I
may request, please read all this post before you respond since I will
try to address many issues at the end of my post.
My criteria for an ideal text are:
1*) Academically rigorous (text targeted towards advanced
placement students)
2*) Must stress formal proof throughout text
3) Exploratory exercises and experiments
4) Has creative thought problems
5*) Text is easy to read
6) Historical and other notes/sidebars
7) Examples of geometry in Art/fractals/etc.
8) Has computer programming examples
The *'s indicate what are crucial criteria for me. The others would be
nice but not necessary since I supplement these areas with my own
materials. I didn't expect to find a text that fit everything :) My
educational goal is to prepare the students for majors in math/hard
sciences/engineering/any mathematically demanding field. I know that
many of my students will not go on in these fields! But what I have
found is that when I set the educational bar at this level my students
rise to the challenge and they have amazing gains in their ability to
use their minds mathematically/creatively. It is truly an awesome
phenomenon to watch 6th and 7th graders debate if one infinity is
larger than another or be able to derive the quadratic equation on
their own (more on this later). And in the end, students who thought
that math was boooring fall in love with the subject.


I finally narrowed down my search to 2 texts
Geometry
Moise, Downs
Hardcover (January 1991)
Addison-Wesley Pub Co; ISBN: 0201253356

Geometry for Enjoyment & Challenge
Rhoad, Milauskas, Whipple
Hardcover (June 1991)
McDougal Littell & Co; ISBN: 0866099654

Geometry by Moise, Downs as far as I could determine is out of print.
Geometry for Enjoyment & Challenge turns out to fit my * criteria very
well. I will go over each one:

1*) Academically rigorous (text targeted towards advanced placement
students)
This text will definitely hone the mathematical mind in a
rigorous way. I am quite pleased.
2*) Must stress formal proof throughout text
Most of the book is devoted to proof
3) Exploratory exercises and experiments
Receives an F since there are none. I will supplement with my own
material and the book:
Discovering Geometry: An Inductive Approach, Serra
4) Has creative thought problems
Some. They have questions that relate to material that hasn't
been covered which I like because it encourages the student to explore
concepts before they encounter them in class. But in general there are
not many creative thought problems. Discovering Geometry has many more
but my students are too advanced for these and would be bored. I have
developed my own set of brain busters that I use. A nice set of easy
thought problems is Visual Brainstorms (1 and 2) By Binary Arts
(www.binaryarts.com)
5*) Text is easy to read
Yes
6) Historical and other notes
Some but I find the notes to be so-so. As far as I can see no
historical notes. This for me is really fluff, but nice fluff to be
sure :)
7) Examples of geometry in art/fractals/etc.
None. There is a wonderful book that lets you create
3-dimensional Escher models from cardboard cutouts that are included:
M. C. Escher: Kaleidocycles, Doris S. Schattschneider,
Wallace Walker
Good books on/with Moiré patterns are:
Visual Illusions in Motion: With Three Different Moire
Screens, Cassin, Craig, ISBN 0486295710
Optical Designs in Motion: With Moire Overlays, Grafton,
Carol Belanger, ISBN 0486232840
A good book with simple "easy to see" stereograms is:
Stereogram, Christopher W. Tyler
Anamorphic Art is a wonderful art that was developed hundreds
of years ago and entails distorting an image so it is recreated in a
silvered cylinder. For examples go to:
www.counton.org/explorer/anamorphic/
A good book on anamorphic art with classical art examples is:
Hidden Images-Games of Perception: Anamorphic Art Illusion,
Fred Leeman
A fun book with anamorphic art set to a Carl Sandburg poem is:
Arithmetic, Sandburg, Carl/ Ted Rand
There are wonderful non-periodic and periodic tiles in the shape
of bugs. Loads of fun. For more information go to
http://www.damert.com/pages/tessellations.html
They are going out of business so order soon. I have found them
at:
http://puzzles.cleversoul.com/mg_brain.html (look at busy
beetles)
Will also use Discovering Geometry to supplement.
8) Has computer programming examples
None. Discovering Geometry has some. For a free logo with ideas
go to http://www.softronix.com/logo.html

So I can say that in all the essential things I am looking for,
Geometry for Enjoyment fits the bill nicely and I am pleased. I will
post when we are completed with the course to tell you how it went!

Now to the wonderful geometry debate that has been raging. One side
says that a book dedicated to proof is dry, turns students off and
isn't the correct approach. For this camp, they prefer an
inductive/exploratory approach. Another side says that since formal
rigorous proof is the foundation of math, it needs to be stressed and
that the inductive approach lacks rigor. But we shouldn't forget that
every teacher is different. An approach that works well for one
teacher may fail utterly for another. The key points are:
Are the students learning the math?
Are the students learning to think logically/mathematically?
Are the students learning to think mathematically out of the box?
Are the students learning to think creatively?
Are the students excited about math?
Do the students feel mastery of the subject?
So from my point of view, if a teacher is doing the above things they
have succeeded. If it turns out that one teacher uses a different
approach from another and achieves basically the same result then why
argue with the approach since it works. I know that in many teachers'
hands, a proof laden book will turn the students off and they will not
learn the math. But for my teaching style this type of book is the
perfect foil for what we do in class. Let me explain. When I teach I
teach Socratically. This means that in class the students derive
EVERYTHING themselves with me as their guide. The class doesn't move
forward unless the students come up with the answers/ideas to my
questions. So class time is 100% creative/exploratory. Then students
go home and do the homework. Just like when you take piano you need to
do scales and practice, I have found that my students need to do
mathematical scales to burn in the math. Repetition of concepts is
essential, without this the math doesn't stick as well. I also have a
ton of creative puzzlers for homework which adds a crucial dimension.
But what I like to say is the proof is in the pudding. I can talk all
day about philosophy of teaching but what are my results? Let me give
you examples from my Algebra class as proof that for me my approach
works. I started with 6th and 7th grade students. The students are
being home-schooled and the parents pay for the class (we are not
affiliated with a school). Most of them had no previous experience
with algebra and most of the class would say they didn't like math
when we started. I choose a textbook that is only used at the college
level (in fact one of the parents had used the book in college). This
book is very difficult and covers all of Algebra I and a fair amount
of Algebra II. This book is totally drill and kill and as dry as a
bone in the summer desert. They were assigned between 150-200 problems
a week for homework. A recipe for disaster? Quite the contrary! I
watched EVERYONE of my students bloom mathematically/intellectually.
How many students at this grade level could solve this problem:
1+1/(1+1/(1+1/(1+...)...))
the answer to this infinite fraction is: (square root of 5 + 1)/2
Or discuss Cantors proof of why the set of fractions is the same order
of infinity as the integers and THEN derive the mathematical formula
that relates the positive integers to the fractions (can you do this?)
When I teach it is not good enough to just do the problem you have to
understand why what you are doing works mathematically. That is, you
have to understand math from a fundamental perspective. For example we
all know that:
2 raised to the 0 power = 1 (I will write this 2^0)
But why is this so? What is the reason for this? Just saying that you
were told this is not enough! In fact, many math teachers don't know
why we choose 2^0 to be defined as 0. My students can give you 2
different logical arguments as to why this should be so. The quadratic
equation is another example. Memorizing a formula won't cut it can you
derive it? My students can and not by memorizing the steps in the
derivation but logically putting the proof together.
Over 60% of my student came in with low math self esteem and everyone
of them now knows that they can do it! When I gave the final exam, the
students could pick a regular 1 hour exam or they could opt to take a
3 hour college level exam. We discussed that there was no advantage in
taking one exam over the other. I explained that I wanted them to have
the opportunity to experience what a college test is like if they so
choosed. They debated which exam they would take. When the day came
everyone of them took the 3 hour exam voluntarily. In fact, when they
finished the section that comprised the 1 hour exam, they exclaimed
that this was too easy! Every student scored 90%+ on the exam. (FYI
the exam was taken from the textbook and was what the author
considered to be a standard exam for the book)
One of the most gratifying things has been to watch the student's
excitement about math grow. In the beginning most of the students
didn't do extra homework or brainteasers. By the end of the class 100%
of them were doing extra work outside of class on a regular basis.
Some students now complain when we have vacation days and miss a
class. They all want to continue during the summer. Students have
announced that they want to become mathematicians. Did I mention that
we meet once a week for a 3 hour math class? That's right! And most of
the time, my students remain sharp for the 3 hours. When I teach
physics at college, my students don't have that kind of attention
span! By word of mouth I have students and parents banging on my doors
to get in.
So after 36 weeks of doing algebra, every student has excelled and is
looking forward to geometry.
As a foot note, at this point the students are 1-2 years ahead of
grade level in math. My plan is to have them finish pre-calculus by
10th grade and then go to the local university (UC Berkeley) to take
calculus and differential equations for the 11th-12th grade. Wish me
luck!

A little about me and the math program. If you talk to many University
professors they will tell you that their foreign students are much
better prepared in math even compared to the ones from private
schools. Routinely, Japanese students take differential equations in
high school, something that is unheard of here. European students in
general have a much higher standard. So I want to close the gap. I am
currently developing a curriculum starting with algebra that will take
average students and accelerate them mathematically to the above
levels. It is important that the math not become a grind and that I
don't overwork the students (in Japan many of the students complain of
overwork and I would agree with them). I believe that what I am
developing could NOT be used effectively in a standard school
environment for these reasons:
1) I graduated summa cum laude from Cornell and have a master in
physics from UC Berkeley. I am constantly using my mathematical
knowledge in class to bring in advanced concepts and analyze student's
responses to find the mathematical gold in a "wrong" response or to
amplify a concept to a higher level. Without this degree of education
my class would lack an essential depth that is a major reason for my
students developing their mental facility. Obviously most math
teachers don't have this level of knowledge.
2) My curriculum takes a tremendous amount of time, energy and
commitment on my part. Teaching math is a calling for me and again you
can't expect the average teacher or any person to have this type of
commitment.
3) I have 100% backing from the parents and they are involved in
cheering the students on. They come to every class and help me when
rough spots occur with their child. Again this kind of support is not
common.
4) By 6th grade, many students are watching loads of TV and have
short attention spans. In general, the students in the home-school
community I deal with have excellent attention spans. This is crucial
to the class being a success.

Does this mean that the curriculum would not be useful to a wider
audience? No, I think that there are many things I am developing that
would be helpful but I would think when applied to a standard
situation you would not get the results I get. What I am hoping to do
is to develop a solid curriculum/game plan for like minded teachers.
Maybe this will start a small revolution!


Alan Sagan