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

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Hi,

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

ideas.

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)

end

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

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Michelle Manes

Reasearch Assistant, Connected Geometry Project

Education Development Center