- Make and test conjectures

- Formulate counterexamples
- Follow logical arguments
- Judge the validity of arguments
- Construct simple valid arguments

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* 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 9-12, page 234

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.

*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* 12E (Habits of Mind: Critical-Response Skills)

Grades 9-12, page 300

Be aware, when considering claims, that when people try to prove a
point, they may select only the data that support it and ignore any that
would contradict it.

*Benchmarks* 9E (The Mathematical World: Reasoning)

Grades 9-12, page 23

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.

*Benchmarks* 12E (Habits of Mind: Critical-Response Skills)

Grades 9-12, page 300

Notice and criticize arguments based on the faulty, incomplete, or
misleading use of numbers, such as in instances when (1) average results
are reported, but no the amount of variation around the average, (2) a
percentage or fraction is given but not the total sample size (as in "9
out of 10 dentists recommend ..."), (3) absolute and proportional quantities
are mixed (as in "3,400 more robberies in our city last year, whereas other
cities had an increase of less than 1%"), or (4) results are reported with
overstated precision (as in representing 13 out of 19 students as 68.42%).

*Benchmarks* 12E (Habits of Mind: Critical-Response Skills)

Grades 9-12, page 300

Insist that the critical assumptions behind any line of reasoning be
made explicit so that the validity of the position being taken whether
one’s own or that of others can be judged.

*Benchmarks* 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.