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Connected Geometry
Posted:
Feb 26, 1993 4:19 PM


++++++ 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 twoheaded 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 datadriven 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 (Sketchpadlike 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
  Michelle Manes Reasearch Assistant, Connected Geometry Project Education Development Center



