What This Means for Instruction

Advancing Children's Thinking Framework identifies instructional strategies that promote mathematical reasoning:

1. Elicit Children's Solution Methods.
• Facilitate students' responses by:
1. eliciting multiple solution methods for one problem;
2. listening to students' descriptions of solution methods;
3. encouraging elaboration;
4. conveying acceptance toward students' errors and efforts;
5. promoting collaborative problem solving.
• Orchestrate classroom discussions by:
• using students' explanations for the lesson's content;
• monitoring students' levels of engagement;
• deciding which students need opportunities to report to the class.
2. SupportChildren's Conceptual Understanding.
• Support the describer's thinking by:
1. reminding students of similar problem situations;
2. providing background information;
3. directing groups to help an individual students' thinking;
4. assisting individuals in clarifying solution methods.
• Support listeners' thinking by:
1. providing teacher-led instant replays;
2. demonstrating teacher-selected solution methods.
• Support describers' and listeners' thinking by:
1. recording symbolic representations of solution methods;
2. asking a different student to explain the solution.
• Support individuals in private help sessions.
3. ExtendChildren's Mathematical Thinking.
• Maintain high standards and expectations for all students.
• Encourage mathematical reflection by:
1. encouraging students to draw generalizations;
2. encouraging consideration of conceptual relationships;
3. listing all solution methods on the chalkboard.
• Go beyond initial solution methods by:
1. pushing individuals to try alternative solution methods;
2. promoting use of more efficient solution methods;
3. using student-generated problems.
• Cultivate a love for challenge.

(Fraivillig, Murphy, & Fuson, 1999)