Topic: Logical Reasoning |
Thinking and Reasoning |
| Standard 1. Understands and applies the basic principles of presenting an argument |
| | Level IV (Grade 9-12) |
| | | Benchmark 2. | Develops logical arguments that are based on quantitative data |
| | | | Knowledge/skill statements |
| | | | | 1. | Develops logical arguments |
| | | | | 2. | Develops arguments based on quantitative data |
| Standard 2. Understands and applies basic principles of logic and reasoning |
| | Level III (Grade 6-8) |
| | | Benchmark 2. | Understands that some aspects of reasoning have very rigid rules but other aspects do not |
| | | | Knowledge/skill statements |
| | | | | 1. | Understands that some aspects of reasoning have very rigid rules |
| | | | | 2. | Understands that some aspects of reasoning do not have rigid rules |
| | | Benchmark 3. | Understands that when people have rules that always hold for a given situation and good information about the situation, then logic can help them figure out what is true about the situation |
| | | | Knowledge/skill statements |
| | | | | 1. | Understands that when people have rules that always hold true for a situation, logic can help them figure out what is true about the situation |
| | | | | 2. | Understands that when people have good information about a situation, logic can help them figure out what is true about the situation |
| | | Benchmark 4. | Understands that reasoning by similarities can suggest ideas but cannot be used to prove things |
| | | | Knowledge/skill statements |
| | | | | 1. | Understands that reasoning by similarities can suggest ideas |
| | | | | 2. | Understands that reasoning by similarities cannot be used to prove things |
| | | Benchmark 5. | Understands that people are using incorrect logic when they make a statement such as "if x is true, then y is true; but x isn't true, therefore y isn't true" |
| | | | Knowledge/skill statements |
| | | | | 1. | Understands incorrect logic |
| | | | | 2. | Knows that the statement "if x is true, then y is true; but x isnâ€™t true, therefore y isn’t true" is an example of incorrect logic |
| | | Benchmark 6. | Understands that a single example can never prove that something is true, but a single example can prove that something is not true |
| | | | Knowledge/skill statements |
| | | | | 1. | Understands that a single example can never prove that something is true |
| | | | | 2. | Understands that a single example can prove that something is not true |
| | Level IV (Grade 9-12) |
| | | Benchmark 1. | Understands the differences between the formal and informal uses (e.g., in everyday situations) of the logical connectors: "if...then," "not," "and," "or" |
| | | | Knowledge/skill statements |
| | | | | 1. | Understands formal uses of logical connectors |
| | | | | 2. | Understands informal uses of logical connectors |
| | | Benchmark 3. | Understands the difference between formal and informal uses (e.g., in everyday situations) of the terms "sufficient" and "necessary" |
| | | | Knowledge/skill statements |
| | | | | 1. | Understands the formal uses of the term "sufficient" |
| | | | | 2. | Understands the informal uses of the term "sufficient" |
| | | | | 3. | Understands the formal uses of the term "necessary" |
| | | | | 4. | Understands the informal uses of the term "necessary" |
| | | Benchmark 4. | Understands the formal meaning of the logical quantifiers: "some," "none," and "all" |
| | | | Knowledge/skill statements |
| | | | | 1. | Understands the formal meaning of the logical quantifier "some" |
| | | | | 2. | Understands the formal meaning of the logical quantifier "none" |
| | | | | 3. | Understands the formal meaning of the logical quantifier "all" |
| | | Benchmark 5. | Understands that formal logic is mostly about connections between statements and that these connections can be considered without attention to whether the statements themselves are true or not |
| | | | Knowledge/skill statements |
| | | | | 1. | Understands that formal logic is mostly about connections between statements |
| | | | | 2. | Understands that in formal logic connections between statements can be considered without attention to whether the statements themselves are true or not |
| | | Benchmark 6. | Understands that people sometimes reach false conclusions either by applying faulty logic to true statements or by applying valid logic to false statements |
| | | | Knowledge/skill statements |
| | | | | 1. | Understands that when using logical reasoning people sometimes reach false conclusions |
| | | | | 2. | Understands that false conclusions may be reached by applying faulty logic to true statements |
| | | | | 3. | Understands that false conclusions may be reached by applying valid logic to false statements |
| | | Benchmark 7. | Understands that a reason may be sufficient to get a result but may not be the only way to get the result (i.e., may not be necessary), or a reason may be necessary to obtain a result but not sufficient (i.e., other things are also required; some reasons may be both necessary and sufficient) |
| | | | Knowledge/skill statements |
| | | | | 1. | Understands that a reason may be sufficient to get a result but may not be the only way to get the result |
| | | | | 2. | Understands that a reason may be necessary to obtain a result but not sufficient |
| | | | | 3. | Understands that some reasons may be both necessary and sufficient to get a result |
| | | Benchmark 8. | Understands that logic can be used to test how well any general rule works |
| | | | Knowledge/skill statements |
| | | | | 1. | Knows that it is important to be able to test how well a general rule works |
| | | Benchmark 9. | Understands that proving a general rule to be false can be done by finding just one exception; this is much easier than proving a general rule to be true for all possible cases |
| | | | Knowledge/skill statements |
| | | | | 1. | Understands that proving a general rule to be false can be done by finding just one exception |
| | | | | 2. | Understands that proving a general rule to be false is easier than proving a general rule to be true |
| | | | | 3. | Understands that proving a general rule to be true must be done for all possible cases |
| | | Benchmark 10. | Understands that logic may be of limited help in finding solutions to problems if the general rules upon which conclusions are based do not always hold true; most often, we have to deal with probabilities rather than certainties |
| | | | Knowledge/skill statements |
| | | | | 1. | Understands that logic may be of limited help in finding solutions to problems if the general rules upon which conclusions are based do not always hold true |
| | | | | 2. | Knows that most often in problem-solving one has to deal with probabilities rather than certainties |
| | | Benchmark 12. | Understands that very complex logical arguments can be formulated from a number of simpler logical arguments |
| | | | Knowledge/skill statements |
| | | | | 1. | Understands complex logical arguments |
| | | | | 2. | Understands simple logical arguments |
| | | Benchmark 13. | Identifies counter examples to conclusions that have been developed |
| | | | Knowledge/skill statements |
| | | | | 1. | Understands the concept of a counter example |