What Counts as Proof in Secondary Mathematics?

Evidence is not automatically proof

Suppose you test the first ten odd numbers and find that each has the form 2k + 1. Those examples illustrate the definition of odd numbers, but checking ten values cannot establish a universal statement about every integer. A proof connects the definitions and valid reasoning so that no permitted case is left outside the argument.

Likewise, a geometry diagram may look as though two lines are parallel. Unless parallelism is given or established, appearance is not proof. A graph drawn by software can strongly support a pattern, but the visible window may omit other behaviour. Mathematical proof controls what the conclusion actually follows from.

A proof begins with definitions and accepted facts

To prove that the sum of two even integers is even, let the integers be 2a and 2b, where a and b are integers. Their sum is 2a + 2b = 2(a + b). Because a + b is an integer, the sum has the form two times an integer. Therefore it is even.

The conclusion is not based on selected examples such as 4 + 8 = 12. It follows from the definition of evenness and applies to arbitrary even integers.

What a school-level proof needs

1. A clear statement of what is given or assumed.
2. Definitions, properties or earlier results that are permitted.
3. Steps that logically follow from the previous information.
4. A conclusion that matches the claim actually being proved.

The exact level of formality depends on the course and question. The goal is not to imitate university notation unnecessarily. It is to make the reason visible and valid.

Example: prove an algebraic identity

Show that (n + 1)² − (n − 1)² = 4n for every real n.

Expand the left side: n² + 2n + 1 − (n² − 2n + 1). Removing the bracket gives n² + 2n + 1 − n² + 2n − 1 = 4n. The left side has been transformed into the right side using algebraic equivalence, so the identity holds for every real n.

Substituting n = 3 and obtaining 12 would be a useful check, but it would prove only that the equality works at that selected value. The algebraic argument establishes the general identity.

Example: geometry needs reasons

Suppose a triangle has two equal sides. To conclude that the opposite angles are equal, cite the relevant isosceles-triangle property rather than writing “they look equal”. If a proof requires congruent triangles, identify the corresponding sides and angles that satisfy the chosen congruence condition.

A labelled diagram is valuable because it organises information. It does not create facts that were never given or derived.

Proof, verification and disproof do different jobs

Verification checks whether a statement works in a particular case or whether a proposed solution satisfies the original conditions. Proof establishes a stated general claim. Disproof shows that a universal claim is false, often through one valid counterexample.

These jobs support each other. Testing examples can reveal a pattern worth turning into a conjecture. A failed example can destroy the conjecture. Surviving many tests can motivate a proof, but it does not replace one.

Three common non-proofs

“I checked several examples.” This may be useful evidence, but a universal claim still needs a general argument.

“It is obvious from the diagram.” A diagram can guide reasoning, but the proof must depend on stated or derived geometric relationships.

“The calculator says both sides are equal.” That verifies the entered case to the calculator’s displayed precision. It does not establish an identity for all permitted inputs.

A proof-reading checklist

• Did I prove the exact statement, or a nearby statement?
• Did I assume what I was supposed to prove?
• Does each step have a reason?
• Did I use an unstated property from the diagram?
• Does the argument cover every case named in the claim?
• Are there restrictions, such as a denominator being nonzero, that I must retain?

Practice

1. Prove that the sum of two odd integers is even. Let them be 2a + 1 and 2b + 1. Their sum is 2(a + b + 1), which is twice an integer.

2. Is checking 1² + 1 = 2, 2² + 2 = 6 and 3² + 3 = 12 enough to prove that n² + n is even for every integer n? No. A proof is n² + n = n(n + 1); consecutive integers include one even factor, so their product is even.

3. A diagram shows two angles looking equal. Is that proof? No. Equality must be given or follow from an applicable theorem or established relationship.

Continue

Next, study How Mathematical Proof Works for the wider proof landscape, then use the Secondary Mathematics proof strand for direct proof, contradiction and disproof. Return to the Secondary Mathematics Master Index.