Return to course home page.

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

**On parallelograms:** Excerpt
from
a high school textbook. | Glencoe
Geometry - summary of contents related to parallelograms.

**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.