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Unformatted text preview: Math 100 Counting and Metacognition Martin H. Weissman 1. A Metacognitive Worksheet Writing proofs can be difficult. We have devoted a lot of attention to difficulties in language – how to express a mathematical idea clearly and concisely using academic English as well as symbolic mathematical language. Another difficulty in proof-writing is knowing “what to do next”. Writing proofs is not like solving a linear equation in high-school. Proofs are not mechanical. Computers can solve equations, and do calculus problems quite well. Computers cannot write mathematics papers with proofs (yet..). Writing proofs requires a disciplined thought-process. One must be able to think about one’s own thought- process while writing a proof – this is called metacognition. Metacognitive proof-writing involves the following skills: • You must always be aware of what you know. – You know things that are “given” as hypotheses. – In a case-by-case proof, you know something specific in each case. – You know basic facts about numbers, sets, shapes, etc... For example, you know that squares of numbers are positive, that even numbers cannot have 3 as their last digit, you know that n 2 > n , when n is a nonzero integer. You know many many things, but you might not actively know them. • You must always be aware of your goals. – You must resist the temptation to “solve for x ” (or any other variable), unless that is a goal. – You must be able to set intermediate goals for yourself – “If I only knew that , I could figure out ”. – You must be aware of the final goal (conclusion) of the proof. You must know when you are done! • You must be aware of the large-scale logical structure of your proof. – Are you attempting a long direct proof, which proceeds through many straightforward deduc- tions from hypothesis to conclusion?...
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- Fall '08
- Counting, Natural number, proof-writing