Benchmark 9E: The Mathematical World : Reasoning
(grades 9-12, page 234)
 
  • To be convincing, an argument needs to have both true statements and valid connections among them. Formal logic is mostly about connections among statements, not about whether they are true. People sometimes use poor logic even if they begin with true statements. (1 of 5)
    • Standard 3-4 page 143, Grades 9-12
    Judge the validity of arguments

    Standard 3-5 page 143, Grades 9-12
    Construct simple valid arguments
  • Logic requires a clear distinction among reasons: A reason may be sufficient to get a result, but perhaps is not the only way to get there; or, a reason may be necessary to get the result, but it may not be enough by itself; some reasons may be both sufficient and necessary. (2 of 5)
    • Standard 3-1 page 81, grades 5-8
    Recognize and apply deductive and inductive reasoning

    Standard 3-4 page 143, Grades 9-12
    Judge the validity of arguments

    Standard 3-5 page 143, Grades 9-12
    Construct simple valid arguments
  • Wherever a general rule comes from, logic an be used in testing how well it works. Proving a generalization to be false (just one exception will do) is easier than proving it to be true (for all possible cases). Logic may be of limited help in finding solutions to problems if one isn't sure that general rules always hod or that particular information is correct; most often, one has to deal with probabilities rather than certainties. (3 of 5)
    • Standard 3-2 page 143, Grades 9-12
    Formulate counterexamples
  • Once a person believes a general rule, he or she may be more likely to notice cases that agree with it and to ignore cases that don't. To avoid biased observations, scientific studies sometimes use observers who don't know hat the results are "supposed" to be. (4 of 5)
    • Standard 3-4 page 143, Grades 9-12
    Judge the validity of arguments
  • Very complex logical arguments can be made form a lot of small logical steps. Computers are particularly good at working with complex logic but not all logical problems can be solved by computers. High-speed computers can examine the validity of some logical propositions for a very large number of cases, although that may not be a perfect proof. (5 of 5)
    • Standard 3-3 page 81, Grades 5-8
    Make and evaluate mathematical conjectures and arguments

    Standard 3-3 page 143, Grades 9-12
    Follow logical arguments