Reasoning & Proving

Reasoning and proving are a mainstay of mathematics and involves students using their understanding of mathematical knowledge, concepts, and skills to justify their thinking. Proportional reasoning, algebraic reasoning, spatial reasoning, statistical reasoning, and probabilistic reasoning are all forms of mathematical reasoning. Students also use their understanding of numbers and operations, geometric properties, and measurement relationships to reason through solutions to problems. Teachers can provide all students with learning opportunities where they must form mathematical conjectures and then test or prove them to see if they hold true. Initially, students may rely on the viewpoints of others to justify a choice or an approach to a solution. As they develop their own reasoning skills, they will begin to justify or prove their solutions by providing evidence.

(Ontario Elementary Math Curriculum, 2020)

Instructional Strategies

• Ask questions that require students to hypothesize and make conjectures.

• Facilitate sharing of hypotheses/conjectures and the reasoning behind them.

• Accept all student suggestions and help them decide what evidence they need to confirm or disprove their hypotheses.

• Model how to adjust a hypothesis that has been disproved by evidence.

• During whole-class discussions, foster behaviours such as active listening to the reasoning of others; legitimizing errors as part of the learning process; and tolerating ambiguity.

• Provide frequent opportunities for students to work in small, mixed-ability groups so that students who are experiencing difficulty can hear and see the reasoning and proofs of their peers.

• Provide frequent opportunities for students to work in homogeneous groups so that differentiated instructional activities target their readiness to reason in different ways.

• Listen to what students say and look at what students write to identify misunderstandings and misconceptions, and then differentiate instruction accordingly.

• Provide opportunities for students to read, hear, question, and discuss explanations of others.

• Ask students to explain the reasoning that accompanies each step of a mathematical argument or proof.

Prompting Questions

• How can you show that this is true for all cases?

• In what case might out conclusion not hold true?

• How can you verify this answer?

• Explain the reasoning behind your prediction.

• Why does this work?

• What do you think might happen if the pattern continues?

• Show how you know this statement is true.

• How could you check your answer?