# Explicitly teaching counter-examples

To improve their mathematical reasoning, students need to understand what a counter-example is, and how and when to use them. In secondary school, students will often use counter-examples to disprove given statements (refer to examples below). The use of counter-examples can be either in written or oral format; their purpose is to disprove statements.

## Understanding this strategy

Teachers can explicitly teach counter-examples and provide examples of when and how they can be used to show that the original claim (or conjecture) is false. This supports students' ability to reason and argue. Identifying counter-examples is helpful when responding to multiple choice questions under test conditions.

Teachers can also use counter-examples to surface misconceptions. They can do this by:

• presenting a false conjecture to students (either orally or written)
• asking students to agree or disagree with the conjecture, if students disagree, have them provide a counter-example (see example 1 below).

Alternatively, teachers can encourage students to use counter-examples in their reasoning and justifications. When students are reasoning to one another, a student might use a counter-example to challenge another student's conclusion.

## Examples using counter-examples

### Conjecture: 'All primes are odd'

• It is true that MANY primes are odd (e.g. 3, 5, 7, 11, 13,...)
• 2 is a counter-example as it is prime but not odd
• Hence the original conjecture is false.

### Conjecture: 'All quadrilaterals have four right angles'

• It is true that SOME quadrilaterals have four right angles (e.g. rectangles, which includes squares)
• Some parallelograms have angles which are not 90 degrees, so these are counter-examples
• Hence the original conjecture is false.

Curriculum links for the above examples: VCMNA208, VCMMG262.