Benchmark 9E: The Mathematical World : Reasoning
(grades 6-8, page 233)
 
  • Some aspects of reasoning have fairly rigid rules for what makes sense; other aspects don't. If people have rules that always hold, and good information about a particular situation, then logic can help them to figure out what is true about it. This kind of reasoning requires care in the use of key works such as if, and, not, or, all, and some. Reasoning by similarities can suggest ideas but can't prove them one way or the other. (1 of 6)
    • Standard 3-1, page 81, Grades 5-8
    Recognize and apply deductive and inductive reasoning

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

    Standard 3-4 page 143, Grades 9-12
    Judge the validity of arguments
  • Practical reasoning, such as diagnosing or troubleshooting almost anything, may require many-step, branching logic. Because computers can keep track of complicated logic, as well as a lot of information, they are useful in a lot of problem-solving situations. (2 of 6)
    • Standard 3-5, page 81, Grades 5-8
    Appreciate the pervasive use and power of reasoning as a part of mathematics
  • Sometimes people invent a general rule to explain how something works by summarizing observations. But people tend to over generalize, imagining general rules on the basis of only a few observations. (3 of 6)
    • Standard 7-3 page 41, Grades K-4
    Recognize that a wide variety of problem structures can be represented by a single operation

    Standard 2-3, page 78, Grades 5-8
    Develop common understandings of mathematical ideas, including the role of definitions

    Standard 3-3, page 81, Grades 5-8
    Make and evaluate mathematical conjectures and arguments

    Standard 2-2 page 140, Grades 9-12
    Formulate mathematical definitions and express generalizations discovered through investigations

    Standard 3-1 page 143, Grades 9-12
    Make and test conjectures
  • People are using incorrect logic when they make a statement such as "If A is true, then B is true; but A isn't true, therefore B isn't true either." (4 of 6)
    • Standard 3-3 page 81, Grades 5-8
    Make and evaluate mathematical conjectures and arguments
  • A single example can never prove that something is true, but sometimes a single example can prove that something is not true. (5 of 6)
    • Standard 3-2, page 81, Grades 5-8
    Understand and apply reasoning processes, with special attention to spatial reasoning and reasoning with proportions and graphs

    Standard 3-2 page 143, Grades 9-12
    Formulate counterexamples
  • An analogy has some likenesses to but also some differences from the real thing. (6 of 6)
    • Standard 3-2, page 81, Grades 5-8
    Understand and apply reasoning processes, with special attention to spatial reasoning and reasoning with proportions and graphs