### Standard 3: Mathematics as Reasoning

In grades 9-12, the mathematics curriculum should include numerous and varied experiences that reinforce and extend logical reasoning skills so that students can:

• Make and test conjectures
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.

• Formulate counterexamples
• 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.

• Follow logical arguments
• 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.

• Judge the validity of arguments
• 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.

• Construct simple valid arguments
• 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.

and so that, in addition, college-intending students can:
• Construct proofs for mathematical assertions, including indirect proofs and proofs by mathematical induction