Proof writing is often thought of as one of the most difficult aspects of math education to conquer. Proofs require the ability to think abstractly, that is, universally. They also require a little appreciation for mathematical culture; for instance, when a mathematician uses the word "trivial" in a proof, they intend a different meaning to how the word is understood by the wider population. Students who spend time studying maths can develop proof-writing skills over time.
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.
Formal proof is often introduced using a two-column format, as favored by many geometry teachers. In higher-level mathematics (taken as meaning an advanced undergraduate level of mathematical maturity or above), two methods of formal proof predominate. These are proof by construction (a common example of which is proof by induction), and proof by contradiction (which in its simplest form requires only the demonstration of a counterexample).
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. It is important to note that people trained to a university-level of mathematics do not consider so-called "informal proofs" to be proof of anything at all. Instead, they may be regarded as "heuristics", or teaching tools, at best.
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.
- The Contests button on the top of the page has many Olympiad-level problems from contests such as IMO, USAMO, and many international mathematics competitions.
Proof Writing Guides
- How to Write a Solution by Richard Rusczyk and Mathew Crawford
- How to Read Mathematics -- Not exactly proof writing, but a helpful read for those learning to write basic proofs.
- How To Prove It: A Structured Approach by Daniel J. Velleman -- an excellent primer on methods of proof; train your ability to do proofs by induction, contradiction and more.
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