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When I speak to adults, they tell me that they learned how to think doing two-column proofs. I think what they really mean is that it was in geometry that they learned how to put together an argument to establish something. There are all types of approaches to two-column proofs. Some teachers expect the student to do the proof the same way that the teacher would do the proof. It is rare that two people think exactly alike, especially when there is more than one fact to be established. Some teachers expect the student to do the proof in the most efficient way possible. Students do not always think of the most efficient way the first time they approach a problem. Other teachers believe that a two-column proof is the only way to prove something. Now that I have told you what some teachers will tell you, I will tell you that I think that two-column proofs are a good way for some students to organize their work. However, when I am the teacher I teach students to do proofs in a flow chart manner so that they can see what is a consequence of knowing something. Once they establish two or more facts they can see the consequence of their thinking. In this way they will see that different "flows" can be rearranged but that the final consequence has to come at the end. This methodology establishes good thinking as well. Accompanying a flow chart is a listing of reasons that for a two-column proof would be the righthand column. When doing proofs, work hard on them, do them every day, and collect the work from the students. If their work is not correct, have them do it over again until it is correct. Do not expect that the average student can do many of them on a daily basis. Pick and choose. Once a student shows you that he or she can do a proof, assign fewer but more difficult proofs. However, do not spend more than four weeks doing them. By that time the students who will learn from them will have learned, and the other students who have not learned will probably need more work one-on-one with you, the teacher. What material were you considering covering when you thought of cutting out proofs?
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