Instructional programs from
prekindergarten through grade 12 should enable all students to
 recognize reasoning and proof as fundamental aspects of mathematics;
 make and investigate mathematical conjectures;
 develop and evaluate mathematical arguments and proofs;
 select and use various types of reasoning and methods of
proof.

Systematic reasoning is a defining feature of
mathematics. Exploring, justifying, and using mathematical conjectures are
common to all content areas and, with different levels of rigor, all grade
levels. Through the use of reasoning, students learn that mathematics makes
sense. Reasoning and proof must be a consistent part of student's mathematical
experiences in prekindergarten through grade 12.
Reasoning mathematically is a habit of mind, and
like all habits, it must be developed through consistent use in many contexts
and from the earliest grades. At all levels, students reason inductively from
patterns and specific cases. For example, even a first grader can use an
informal proof by contradiction to argue that the number 0 is even: "If 0 were
odd, then 0 and 1 would be two odd numbers in a row. But even and odd numbers
alternate. So 0 must be even."
Increasingly over the grades, students should
learn to make effective deductive arguments as well, using the mathematical
truths they are establishing in class. By the end of secondary school, students
should be able to understand and produce some mathematical proofslogically
rigorous deductions of conclusions from hypothesesand should appreciate the
value of such arguments.