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Q&A #769 |
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Hi, Jan, Different text books introduce "doing of proofs" differently. If you do not understand the textbook that the student is using in class try to find another text by going to a local school and asking to see one. In teaching proof I think that the two column proof comes after looking at a flow chart. Most students don't understand that a proof is a logical argument where certain statements flow from what is known to be true. In order for a defense attorney to win a case he has to present the evidence. In a proof the student has to present the evidence. Where do you start? There are only three places to find evidence to start a proof: 1. The statement is given to you as true. 2. The statement is a definition that is known to be always true. 3. The statement is expressing a relationship observable from a diagram that is given with the problem. This could be something such as: B lies on the line segment AC, with the reason being that it given in the diagram. Then as a consequence of the position of B lying on segment AC, AB + BC = AC. W also know that AB < AC and BC < AC. It is not possible to compare AB and BC just from knowing that B lies on AC. However, if it is also known that B is the midpoint of AC then AB = BC by the definition of a midpoint of a segment. Students have to know definitions well. When instructing students in writing a proof, I would ask them to write down one thing that they know to be true. Then I would ask them to write down everything that they know as a result of the one thing being true. When they run out of things to say, ask another thing that they know to be true and everything that they know as a consequence of that new thing being true. With this style you generate more ideas than you need. Having done this, ask the student what does one have to know in order to determine that which they are trying to prove is true. Look back among the various ideas that were generated and see if they have found these necessary statements. Then it is time to construct the order by which the statements will be arranged. You should point out that if two independent ideas are generated, then the order in which these two are generated may be interchanged. In writing the two column proof the statements are put in one column in order by which they are generated. The reasons why the statements are known to be true are placed in the second column. I hope that this has helped you. Look through different text books. The student does have to know the theorems (statements), so that they will know the reasons. Emphasize the "If ..., then ..." nature of the theorems. The "If" part is what is known. The "then" part is the consequence of the "If" part being true. -Marielouise, for the Teacher2Teacher service
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