Mathematical reasoning can serve as an instrument of inquiry for discovering and exploring new ideas, a process that we call the reasoning of inquiry. Mathematical reasoning also functions centrally in justifying or proving mathematical claims, a process that we call the reasoning of justification.

The reasoning of justification, as we see it, rests on two foundations. One foundation is an evolving body of public knowledge — the mathematical ideas, procedures, methods, and terms that have already been defined and established within a given community.

… And the second foundation of mathematical reasoning is mathematical language — symbols, terms, notations, definitions, and representations — and rules of logic and syntax for their meaningful use in formulating claims and the networks of relationships used to justify them.

Three domains of teacher work:

1. Selecting mathematical tasks that need and provide opportunities for mathematical reasoning.
2. Making mathematical knowledge public and scaffolding the use of mathematical language and knowledge. Making mathematical records (through notebooks, public postings) to make the work public and available for public development.
3. Establishing a classroom culture permeated with serious interest in and respect for others' mathematical ideas. Students learn to attend and respond to, as well as use others' solutions or proposals as a means of strengthening their own understanding and the subsequent contributions they can make to the class work.

Ball, 2002