OT1: Mathematical argument, language and proof: Comprehend and critique mathematical arguments, proofs and justifications of methods and formulae, including those relating to applications of mathematics.

The idea

This objective requires you to read a mathematical argument, proof or justification and judge whether it is valid, complete and correctly reasoned. You must also critique methods and formulae, including in applied contexts. To comprehend, identify the claim, assumptions, definitions and each logical step. To critique, check each step follows from the previous ones, that all cases are covered, and that no hidden assumptions are made. For example, a proof that the sum of two odd numbers is even must define odd numbers, use algebra to represent them, and show the sum is a multiple of two. In applied work, critique whether a model's assumptions are reasonable and whether its formula is used within its valid range. Assessment may ask you to find errors, explain why a step fails, or improve a weak justification.

What earns the marks

• Identify the claim, assumptions and conclusion of an argument.
• Check each logical step for validity and completeness.
• Recognise common errors such as assuming the conclusion or using an invalid converse.
• Evaluate whether a formula or method is appropriate for its context and conditions.
• Suggest improvements or corrections to an argument or justification.

Common mistakes and how to put them right

• Accepting a proof because the conclusion is true, without checking the reasoning.

  Put it right: Evaluate each step independently of the result.

• Confusing a statement with its converse, for example assuming that if a shape is a square then it is a rectangle means all rectangles are squares.

  Put it right: Test converses separately.

• Ignoring domain restrictions when critiquing a formula, such as using a model outside its valid range.

  Put it right: State and check the domain and assumptions.

Exam tips

• Underline each step of a given argument and write a brief reason beside it to spot gaps.
• When asked to critique, name the specific error and explain why it invalidates the argument.
• For applied contexts, comment on assumptions and whether the mathematics matches the situation.