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Department of Mathematics, LSU Baton Rouge
Math 4005: Geometry

# I.B. Quadrilaterals

### Assignemnt due Monday, February 2, 2004

Spend 1 hour reading this excerpt from a high school textbook. At the end of the hour, write 3 question that you would like to ask to help you understand this stuff, and bring the questions to class on February 2.

### Assignemnt due Monday, February 9, 2004

Write proofs of the equivalence of items i)-iv) I of the Parallelogram Theorem.

### Student Work

Pre-test February 2, 2004. (To appear)

### Readings

Readings on the idea of a definition: In discussing the way that knowledge about quadrilaterals is organized, we had occasion to talk about the nature and role of definitions in mathematics and in teaching. Here are three readings that I think provide useful insights into these issues.

• Poincaré. This reading consists of excerpts form a classic essay on definitions in education that is included in Poincaré's famous book of essays: Science and Method. Poincaré considers the educational problems that arise from the peculiarities of mathematical definitions.
• Murphy. Excerpts form Gregory L. Murphy, The Big Book of Concepts.Cambridge: MIT Press 2002. This is a textbook on cognitive science. This chapter demonstrates that the "natural" way in which concepts work is quite different from the way mathematical concepts function. A mathematical definition for a concept gives necessary and sufficient conditions for a thing to be an instance of that concept. ("Parallelogram: a quadrilateral in which the both pairs of opposite sides are parallel." The conditions are necessary: if something is not a quadrilateral or if it is, but it fails to have opposite sides parallel, then it's not a parallelogram. The conditions are sufficient if something has the properties in the definition, then it's a parallelogram. Nothing more is needed.) But natural concepts (like "dog" or "game") cannot be described by necessary and sufficient conditions.:
• Wu. This is a contemporary essay by a mathematician on the use of definitions in mathematics education.