Date: Feb 26, 1993 4:19 PM
Author: Michelle Manes
Subject: Connected Geometry


I've been asked for more information about the Connected Geometry
project at EDC by several people, so this is a brief description.

Paul Goldenberg, June Mark, and Al Cuoco are directing a new
geometry curriculum development project at EDC. This is an overview
of what we'll try to do over the next four years. We'd appreciate
your comments and advice.

The project is called ``Connected Geometry,'' and it has three purposes:

1. To develop high school curriculum materials that get at the centrality
of geometry and visualization in almost every field by showing how there
are two-headed arrows between geometry and other parts of mathematics,
science, art ... .

2. To give high school teachers tools and support that will allow them to
assemble the curriculum materials in many different ways so that they can
design a course (or a unit) that is customized for their interests, their
students' interests, and the demands of their schools.

3. To construct the activities in a way that gives students a research
experience in mathematics. This deviates from many traditional geometry
courses in which a student is given a known result and then is
asked to prove it or to apply it. It also differs from ``discovery''
approaches in which a student makes a data-driven conjecture and then
walks away.

To do all this, we'll develop a collection of activities, each one taking
between a day and a month of classtime, and each one getting at some big
ideas in geometry and in other fields (like number theory or mechanics).
Together, these activities will provide 300 days (or so) of things to do,
far more time than is available in a high school course. Teachers will
construct a course or a piece of a course with an electronic tool that we
call the ``Curriculum Map Maker.'' They will be able to access the
activities through some indexes, so that you could ask the map maker for a
list of activities that discuss area, or the pythagorean theorem, or
continuous variation, or mathematical induction, or linkages, or
perspective drawing. Of course, all of the activities will show up on more
than one list (indeed, that's one of the benchmarks we'll use in deciding
which activities to develop), and this fact might be a stimulus for
teachers to rethink their notions about connections among mathematical

Technology will also play an important part in how the students do
mathematics. We plan to use dynamic geometry software (Sketchpad-like
environments), Logo, drawing tools, and more specialized applications that
allow students to work with continuity, iteration, and linear and affine
mappings of R^2. One of the most difficult tasks we face is to find a
match between computational environment and mathematical perspective
that allows and supports both the development of conjectures AND the
ensuing mathematical analysis of the conjectures. For example, when using
a dynamic geometry tool, we'd like students to develop habits of mind that
seek to explain phenomena by reasoning through continuity. When using Logo
to perform an iterated geometric construction, the mathematical
underpinnings should involve inductively defined functions and mathematical
induction. The idea is that we want to make proof and explanation a
central research technique in high school mathematics rather than a
separate and decontextualized chapter in a geometry book.

Some examples of possible activities:

1. Given a polygon (a triangle, say) find a point that minimizes the sum
of its distances from the vertices. Or from the sides. This requires
thinking about continuous (but not necessarily differentiable) functions
from R^2 to R without the need for any algebraic symbolism.

2. Analyze the classic INSPI construction in Logo:

to inspi (s,a,i)
forward (s) rt (a)
inspi (s, a+i, i)

To accurately predict the structure of the resulting path, students need to
develop some number theory to the level of, say, greatest common divisor,
least common multiple, and elementary congruences.

3. Design a complete specification (including costs) for the construction
of a house. This requires three dimensional visualization, perspective
drawing, and a host of calculations with volumes and surface areas.

4. Predict the number of odd integers in the 100th row of Pascal's
triangle. A geometric analysis of this problem leads to the Sierpinski
triangle and cellular automata.

So, what do you think? If you'd like, I can send a couple papers
that describe the project in more detail and that elaborate on these ideas.
And, please circulate this note to anyone you know who might be interested
in our work.


Michelle Manes
Reasearch Assistant, Connected Geometry Project
Education Development Center