Proof writing

Proof writing is often thought of as one of the most difficult aspects of math education to conquer. However, students who spend time writing about increasingly difficult math topics can develop this skill over time.

Getting Started

The fundamental aspects of a good proof are precision, accuracy, and clarity. A single word can change the intended meaning of a proof, so it is best to be as precise as possible.

There are two different types of proofs: informal and formal.

A formal proof is usually in a two-column format. This is favored by many geometry teachers.

An informal proof can be in a wide variety of styles. It is usually not as neat as a two-column proof but is far easier to organize. Except in geometry class, the vast majority of proofs worldwide are informal.

Practice

Art of Problem Solving (AoPS) has many resources to help students begin writing proofs.

• The AoPS forums (which you can get to through the Community tab on the left sidebar) are a great place to practice writing solutions to problems. Do your best to make your explanations both clear and complete. Read solutions by other students to see what you might do better. Listen to the constructive criticisms of others.
• AoPS Blogs (also in the Community area) are a great place to showcase your best solutions.
• The AoPSWiki you are in now is written by members of the AoPS community. Contributing to the AoPSWiki means writing mathematics as clearly as you can.

Proof writing guides

• How to Write a Solution by Richard Rusczyk and Mathew Crawford
• How to Read Mathematics -- Not really proof writing, but a helpful read for those learning to write basic proofs.

See also

• Math books
• Mathematics competitions
• Mathematics competition resources

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