# Levels of proof

There are three levels of proof that primary and secondary teachers will see in students' responses.

• The first approach most students use is to test some possibilities. This is called the 'naive empirical' level because no matter how many samples are tried, there is no proof that the proposition is always going to be true. One negative case, however, is enough to prove that a proposition is not always going to be true. Thus this level could not be considered formal proof.
• When students start to use extreme cases (e.g. varied examples such as very large numbers, negative numbers, non-regular shapes), this could be considered a higher level of exploration and reasoning, although it is still not conclusive so also cannot be considered formal proof.
• Formal proof involves conceptual or analytic or deductive reasoning. Students need to learn to make strong, well-reasoned arguments about why a proposition will always be true or an event or consequence will always happen. At this highest level of reasoning, proofs can be quite concise and conclusive.

These levels of proof do not correlate with year levels: students in primary grades are capable of clear deductive reasoning as shown in the video Addition of Odd and Even Numbers (also featured in Proof: The Foundation of Mathematics).

You can download a transcript of the video Addition of Odd and Even Numbers.

Proofs may also be visual.

• Young students unfolding 'snowflakes' in a paper folding activity may either prove or disprove their predictions.
• Older primary students may draw two sequential triangular numbers to make a square number and then come to understand why this is always the case.

While such proofs may not be considered to be formal mathematical proofs, students are developing the type of logic called 'formal proof'.

## Triangular numbers

Year 9 students demonstrate different levels of ability when asked to prove a statement about triangular numbers.