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We want the students to apply logic in the process of reasoning. We know from Almstrum that logic is difficult for students preparing for CS major. It is not logical to assume a general rule is true on the basis of a few examples, yet some students have difficulty with this. There is a more general difficulty with quantifiers. Some students seem to have attempted to memorize when counterexamples suffice, rather than to understand. Sometimes students attempt to substitute solution finding over solution creation. Some students feel that use of logic distinguishes proofs they can attempt procedurally (using steps) from other proofs, which they call "laws of logic proofs". Some students know they should not jump to conclusions, but are not always sure how to avoid doing so. Patterns are recognized in use of rules of inference. Proof by contradiction is one such pattern. Students sometimes begin a pattern, or declare a pattern, but do not follow it to its conclusion. For example, a student might start proof by contradiction but omit to conclude that a variable assumed false must therefore be true. Some students wonder whether the law of the excluded middle is always valid (with boolean logic). Some students say they find it difficult to follow the exposition of a proof. It may be that difficulties with abstraction are connected with difficulties logic, because use of logic, application of rules of inference, makes use of abstraction. The rules are presented at a level of abstraction. Some students do operate at the level of abstraction of rules of inference. They call logic cool and beautiful. Interestingly, some differentiate between proofs and "normal math". |