Benchmarks 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.
Benchmarks 9E (The Mathematical World: Reasoning)
Grades 3-5, page 232
One way to make sense of something is to think how it is like something
more familiar.
Benchmarks 9E (The Mathematical World: Reasoning)
Grades 6-8, page 233
An analogy has some likenesses to but also some differences from the
real thing.
Benchmarks 9E (The Mathematical World: Reasoning)
Grades 6-8, page 233
A single example can never prove that something is true, but sometimes
a single example can prove that something is not true.
Benchmarks 9E (The Mathematical World: Reasoning)
Grades 3-5, page 232
Reasoning can be distorted by strong feelings.
Benchmarks 12A
Grades 3-5, page 286
Offer reasons for their findings and consider reasons suggested by
others.
Benchmarks 12E (Habits of Mind: Critical-Response Skills)
Grades 6-8, page 299
Question claims based on vague attributions (such as "Leading doctors
say ...") or on statements made by celebrities or others outside the area
of their particular expertise.
Benchmarks 9E (The Mathematical World: Reasoning)
Grades 6-8, page 233
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.
Benchmarks 2C (The Nature of Mathematics: Mathematical Inquiry)
Grades 6-8, page 37
When mathematicians use logical rules to work with representations
of things, the results may or may not be valid for the things themselves.
Using mathematics to solve a problem requires choosing what mathematics
to use; probably making some simplifying assumptions, estimates, or approximations;
doing computations; and then checking to see whether the answer makes sense.
If an answer does not seem to make enough sense for its intended purpose,
then any of these steps might have been inappropriate.
Benchmarks 2A (The Nature of Mathematics: Patterns and Relationships)
Grades 3-5, page 27
Mathematics is the study of many kinds of patterns, including numbers
and shapes and operations on them. Sometimes patterns are studied because
they help to explain how the world works or how to solve practical problems,
sometimes because they are interesting in themselves.
Benchmarks 9E (The Mathematical World: Reasoning)
Grades 6-8, page 233
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.