- Recognize and apply deductive and inductive reasoning
- Understand and apply reasoning processes, with special attention to spatial reasoning and reasoning with proportions and graphs
- Make and evaluate mathematical conjectures and arguments
- Validate their own thinking
- Appreciate the pervasive use and power of reasoning as a part of mathematics

Grades 3-5, page 299

Recognize when comparisons might not be fair because some conditions are not kept the same.

*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.