This is an example of the sort of discussion that takes place on the newsgroup geometry.pre-college. The first article, posted by Mark Saul (via his America Online account) asks a question. The subsequent articles are replies to that message, either via mail or news. They all appear in a "thread," which is a string of related articles.
```From: Marksaul@aol.com
Date: Tue, 28 Feb 1995 21:29:01 -0500
To: geometry-pre-college@mathforum.org
Subject: What next for geometry?

I would like some feedback.

I'm teaching an excellent ninth grade honors class (following, more or less,
a tenth year curriculum).  The kids are bright and work hard.  They form
cooperative groups spontaneously and smoothly. When the class is
teacher-centered, they listen and are free to interrupt with questions or
comments.  The whole thing is a joy.

Without going full speed ahead, but just through the kids' own momentum, I
will come to the end of the planned curriculum some time in March.  We've had
axiomatics in the context of simple group and field theorems, and in a more
or less traditional, Euclid-based geometry.  We've looked at Euclid bare, in
the Heath translation, and traced through some of his proofs.  We did an
intensive unit on geometric inequalities (well beyond the usual 10th year
texts).  We did two weeks of informal geometry using Mira.  We did the
standard analytic geometry of the line and parabola.

My question is, what should I do with the time left?  I'd like to do more
geometry, because I know that other teachers in my department won't do as
much.  It's not an emergency: I've got lots of stuff.  I would love to do
more classic geometry (Ceva, Menelaus, Simson, Brahmagupta, Ptolemy,
nine-point circle, and so on).  I've done this often and could do it in my
sleep.  One of my kids is already about to discover the circles of
Appolonius.  Or, I could do the rest of the conics a la Appolonius.  Or, I
could do transformations in space--that'll keep them busy.  Or, I could do
graph theory.  Or combinatorics.  Or begin projective geometry.  Or...

My question is, what's new in geometry?  In algebra, the reforms seem to be
centered around data analysis.  Graphing of lines is treated as related in
some way to lines of regression.  Parabolas and sine curves are also fit to
data.  It's one enormous way in which technology can be applied: getting kids
to think about functions as objects, not as "something that you do with a
number."  It also fits in with the current idea of education as preparing
kids to be productive in the "real world" so we can beat Japan.

Well, now you can guess that I'm not in complete agreement with the above
sort of thing.  I find it as rigid as the old style.  But I'm glad I've seen
it, and will certainly use some of what has been developed through this
movement in my teaching.

But what corresponds to this in geometry?   I'm not talking about just about
technology--I can't get all the kids on the Sketchpad at once (as I'd like
to).  But what would technology, or other re-thinking of curriculum, lead us
to in geometry?   I'd like to hear what ideas others have on this question.

From: bo3b@oui.com (bo3b Overkamp)
Newsgroups: geometry.pre-college
Subject: Re: What next for geometry?
Date: 1 Mar 1995 09:27:43 -0500
Organization: Odyssey Ultraware Inc.

Marksaul@aol.com is trolling for geometry class ideas, asking specifically "
what's new in geometry?". My answers are motivated not by newness but by
recollections of students' excitement in the classroom.

How about a comparison and contrast of Euclidean and nonEuclidean geometries?
and Biperpendicular Quadrilaterals) offer to sort out the differences as well
as the similarities?

In any plane which can measure angles and lengths, the summit angles of a
Saccheri Quadrilateral (SQ) have equal measures, the diagonals of a SQ have
equal lengths, and the line joining the midpoints of the summit & base is
perpendicular to both summit and base. Any side of any triangle in any such
plane has a naturally associated SQ for which the side you chose is the
summit, and the base is collinear with the midpoints of the other two sides
of the triangle; the area of the SQ and the sum of the measures of the summit
angles of the SQ have intimate relationships with the area of the original
triangle, and the angle measure sum of the original triangle. In a Euclidean
plane, there is no difference between a SQ and a rectangle, but in a
Lobachevskian plane or a Riemannian plane, the differences between a
rectangle and a SQ are evident. In order for the differences to be evident to
students, you probably need some models. The 19th century geometers Beltrami,
Klein, and Poincare provided descriptions of models of the nonEuclidean
investigate the geometry of its surface. Construct a SQ there, and observe
that the summit angles are necessarily obtuse. Construct a triangle there,
and consider the SQ associated with one of its sides.

Or what about considering the properties of functions defined by the power
series representations of sin, cos, exp?

Or what about spending some time folding polygons and braiding polyhedra
(_Build Your Own Polyhedra_ by Jean Pederson and Peter Hilton,

What about tessellations of the plane(s both Euclidean and nonEuclidean)?

The fundamental principle of mathematics which is least communicated to
nonmathematicians is "you could have figured this out yourself." The best
chance we have to communicate this to our students is in our geometry
classes.

From: danh@math.ilstu.edu (Dan Hirschhorn)
Newsgroups: geometry.pre-college
Subject: Re: What next for geometry?
Date: 1 Mar 1995 17:41:11 GMT
Organization: Illinois State University

In article <950228212859_35429783@aol.com>, Marksaul@aol.com wrote:
>
> I would like some feedback.
>
>
> My question is, what should I do with the time left?
>
> But what corresponds to this in geometry?   I'm not talking about just about
> technology--I can't get all the kids on the Sketchpad at once (as I'd like
> to).  But what would technology, or other re-thinking of curriculum, lead us
> to in geometry?   I'd like to hear what ideas others have on this question.

As you know Mark, but others may not; finishing Geometry in March is a rare
occurrence and it is clear that you have plenty to do.  I'm speaking and
attending the NCTM Regional in Chicago this weekend & the question you pose
is one I am interested in.  In my talk tomorrow morning "Geometry and
Curricular change"  I mostly focus on transformations as the dynamic
software does them so nicely and the notion that geometry is dynamic leads
to a necessity for transformations.  An example that I do at the end is how
translations and size changes of data relate to the geometryt of
translations and size changes.  My pretty conclusion at the end is to take
some 2-d data and then apply a scale factor of 1.25 to both the x and y
variables.  The resulting figure is a blow-up of the original data & the
regression lines are parallel.  This is visually very nice.  Doing the same
thing with functions nicely connects geometry with future ideas in
mathematics as well as the visual and technological changes that are going
on.  Relating plane transformations with 2 x 2 matrices is another nice
connection.

On a totally different note, I always thought the vector approach to
analytic geometry is very pretty and most college-bound students never get
a chance to see it.

Dan

From: schattdo@catwoman.moravian.edu (Doris Schattschneider)
Newsgroups: geometry.pre-college
Subject: Re: What next for geometry?
Date: 1 Mar 1995 18:38:39 -0500

I don't see any 3-dimensional geometry in your list of topics covered.  Why
not spend some time on geometric relationships in three dimensions?  Study
polyhedra and their nice properties-- Euler's and Descarte's theorems,
special polyhedra (Platonic and Archimedean, deltahedra), symmetry
properties, etc.  You can do it very informally with lots of hands-on
discovery activities; you can also do some real proofs.  Students should
know that geometry is important in the world in which we live-- from the
basics, they can branch out to other topics of interest.  There are
activity books, videos, and manipulatives available from Key Curriculum
Press, Dale Seymour, and other distributors.  Kids love to build models--
their own.

Doris Schattschneider

From: Anthony@ynot1.demon.co.uk (Anthony Hugh Back)
Subject: Re: What next for geometry?
Organization: Myorganisation
Date: Thu, 2 Mar 1995 21:47:56 +0000

In article: <950228212859_35429783@aol.com>  Marksaul@aol.com writes:
>
> My question is, what should I do with the time left?  I'd like to do more
> geometry, because I know that other teachers in my department won't do as
> much.
>
How about transformation geometry or an introduction to topology?
--
Anthony Hugh Back

From: bmarthinsen@keypress.com (Bill Marthinsen)
Newsgroups: geometry.pre-college
Subject: Re: What next for geometry?
Date: Thu, 02 Mar 1995 14:35:14 -0800
Organization: Key Curriculum Press

In article <950228212859_35429783@aol.com>, Marksaul@aol.com wrote:

> My question is, what's new in geometry?

Some possible directions include:

3-D geometry--build and study polyhedra, make them different ways, unit
origami is a nice approach, branch out and look at the history of the
shapes, the etymology of their names, occurrances in nature, applications,
not to mention their properties an related theorems.

Or go the cross-disciplinary route

History--great chance to do some math history, focus on the people who
have done the math. Have your class research and present information about
mathematicians who made major contributions in geometry; then let them
debate who they would judge to be the most important/significant geometer
in history. From the description of your class they might really get into
this.

Literature--search out, have them search out examples of geometry in
literature or geometry-based literature. This is a stretch that might be
intriguing; then again it could go no where.

Debate--Hot topics in geometry, historical hot topics in geometry (you
could ask on the Forum for suggestions for the topics).

More of a straight geometry focus could take you through vectors,
projective geometry, or transformations.

Good luck and enjoy your class.

Bill Marthinsen
Key Curriculum Press

From: appelbaum@zodiac.rutgers.edu (Peter Appelbaum)
Newsgroups: geometry.pre-college
Subject: Re: What next for geometry?
Date: Sat, 04 Mar 95 14:41:10 GMT
Organization: William Paterson College of New Jersey

Here are two open-ended ideas:
1.  Have the students decide, based on what they have done this
past year so far, whatthe students in teh course NEXT YEAR
should learb/do/etc.  They have to talk about what is
IMPORTANT, what made the BEST IMPACT, what areas
of geometry need more or less focus, ...

2. Have them plan THEMSELVES what questions theyhave in
geometry, and make a plan for what they want to learn about.

Peter M. Appelbaum
Curriculum and Instruction
The William Paterson College of New Jersey
(201)595-3123  appelbaum@zodiac.rutgers.edu

From: Marksaul@aol.com
Newsgroups: geometry.pre-college
Subject: Re: What next for geometry?
Date: 5 Mar 1995 22:23:43 -0500

> Have the students decide, based on what they have done this
>past year so far, whatthe students in teh course NEXT YEAR
>should learb/do/etc.

Great idea!  I often give the kids a choice of curriculum, but never in just
these terms.

The problem with a choice of curriculum is that some of the Great Things To
Do are not things that kids would fantasize about, without having seen them
before.

Let's see what happens.
```