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[quote=Anonymous][quote=Anonymous][quote=Anonymous]
Another teacher here. Listen to the above teacher, very good advice! A few caveats: While proofs are fantastic for developing strong deductive and logical skills, the artificial two column format proof environment in geometry class can actually hinder their development, and even bore them to death. In the real world, a mathematical proof is written in free form 'essay' natural language format. In geometry class there are certain unnecessary stringent rules forced upon kids. A good teacher knows not to fixate on these and instead focus on the logic and reasoning in a problem. If throughout the year students feel like they do not have any room for creativity in a geometry proof, either the problem is not very interesting, or the 2 column proof format is too rigid.[/quote]

Yet another math teacher here.

Both formal proofs and paragraph proofs have their place.

While I agree that paragraph proofs (i.e., essay proofs) are very useful for free-forming your ideas so you can get to the root of the problem, formal proofs are really valuable, too. They are great for succinctly breaking down a problem into easy steps, and showing those steps - just as math language does. For example, breaking down an English problem - words - into math language - such as algebra symbols - can make it much more easily solved, because of the use of symbolic representation, one of the wonderful values of algebra especially. Two-column, or formal, proofs can be easier to understand for the reader too, than essay proofs, because a formal proof is basically a list, a list of statements and the reasons for each of those statements. Unlike a paragraph, you can understand everything at a glance.

It takes time to learn to use this "tool" - the method of writing a proof - just as it takes time to learn to use any tool. It takes work on my (the teacher's) part to break down the process and make it as interesting and easy to learn as I can, but it will take work on the students' part to learn to use that particular tool of mathematics.

Not everything a student learns should involve "creativity - some things such as learning how to do a formal proof just take work and have to be done, because it will be so helpful in later math classes, and also in life. Learning to make a series of statements that flow logically and sequentially from a hypothesis to a conclusion and being able to back up each step with a "law" or "rule," and being able to write it in a list form to make it easy for your audience to read and understand, applies to other areas besides math - for example, computers and law.

I think that many math teachers don't teach formal proofs because they are both hard to teach and hard for students to "get." But just because somehting is hard doesn't mean it doesn't have great value and isn't greatly useful. [/quote]

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