Unit 3: Reasoning and Explaining (MP2 and MP3)

# 3.1.2 Developing a Community of Reasoners

Read the following excerpt (Ball, 2002) for a description of a reasoning community and the three domains of work a teacher must do to develop that community.

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:

- Selecting mathematical tasks that need and provide opportunities for mathematical reasoning.
- 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.
- 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