# Critiquing reasoning for solutions

Students need to be supported to develop their communication of mathematical solutions.

Students will need to communicate a mathematical solution which integrates worded and symbolic reasoning and where the logic is apparent. Having students critique the reasoning of given solutions, whether orally or in writing, helps students to construct their mathematical understanding (Hillman, 2014).

## Understanding this strategy

This strategy, adapted from Swan (n.d), helps students to develop their ability to communicate reasoning for solutions. Teachers can provide opportunities for students to become readers of sample solutions and to critique and improve these solutions.

### Teacher actions

The teacher:

1. provides students with a problem, asking them to solve it individually and record appropriate reasoning
2. asks students to discuss (in pairs) the reasoning for their solutions
3. provides each pair of students with a sample solution, which has some reasoning missing or where the solution is incomplete
4. asks students to critique and improve this sample solution with the goal of producing a polished, correct solution.

### Student actions

Students discuss the reasoning provided, identifying:

• reasoning which is appropriate
• places where reasoning should be added.

To scaffold students, teachers could direct their attention to particular points in the solution where reasoning is provided and ask students to comment on and improve the reasoning.

Emphasise the importance of someone else (i.e. another student or a teacher) being able to read and understand their solution.

## Example of critiquing reasoning in action

The example below demonstrates how this strategy can be used to solve simultaneous equations in a Year 10 class (VCMNA337).

### Scenario

The teacher provides students with the problem below, asking them to solve it individually, recording appropriate reasoning.

### Problem

Solve this pair of simultaneous linear equations:

2x + y = 14

3x + 2y = 23

In pairs, students discuss their reasoning for the solutions they have produced.

### Sample solution and discussion

Teacher provides each pair of students with a sample solution, see example below.

Teacher asks students to critique and improve this sample solution with the goal of producing a polished, correct solution. For this example, teacher questions could include:

• What has been done to the pair of equations to produce this result? How can this step be written?
• How can you tell where this equation came from?
• Which lines of reasoning are clear?
• Which aspects of the reasoning need to be improved?
• What is needed here to help the reader to understand how the problem was solved?
• Does the worded reasoning match the symbolic reasoning?
• Has the problem been fully answered?
• Is the answer clearly communicated?
• Would someone else be able to understand this solution?

### Student critique of provided solution

• Student 1:
It's not clear what they've done to each equation.
• Student 2:
They've multiplied the first one by 3 and the second one by 2
• Student 1:
Yeah, I know that, but they haven't shown what they've done. You need to show your working.
• Student 2:
Ok. And you're also meant to check your answers by substituting back in.

2x + y = 14

3x + 2y = 23

6x + 3y = 42

6x + 4y = 46

y = 4

x = 5