Which of the following statements are appropriate regarding "Argumentation" in…

Concept — argumentation in a mathematics classroom is a connected line of reasoning offered to show or explain why a mathematical claim is true. It does two jobs: justification, which moves a conjecture towards proof, and heuristic exploration, in which learners generate, test and refine conjectures. Mathematics also differs from the empirical disciplines in the warrant it accepts: observation and experiment support an empirical claim, whereas a mathematical claim is settled by deduction from definitions and results already proved.

Application — test each statement against that concept.

• Argumentation as a line of reasoning that shows or explains why a mathematical result is true: this is the standard characterisation of the term. The target of the reasoning is the question why is this true, not the production of the answer. Appropriate.

• Argumentation is always helpful in mathematics in the same manner as in other empirical disciplines: in an empirical discipline, agreement with repeated observation is itself support for a claim; in mathematics, a pattern that holds in a hundred checked cases is still only a conjecture until it is deduced. The role of argument therefore differs in kind, and the absolute claim always and in the same manner fails. Not appropriate.

• Argumentation can also be part of the heuristic approach in mathematics education: heuristic, that is discovery- or inquiry-based, teaching asks learners to conjecture, to justify, and to question one another’s reasoning, and argumentation is the vehicle for exactly that work. Appropriate.

• Argumentation is less useful for elementary grade students because its assessment will be difficult: usefulness to the learner is not fixed by ease of assessment, and reasoning at that stage is assessed through oral explanation and observation of classroom talk. NCF 2005 and NCF 2023 both place reasoning and justification among the goals from the foundational stage onward. Not appropriate.

Cross-check — the difference in warrant is precisely what makes statement B fail.

Result — the appropriate statements are A and C, so the answer is the pairing A and C.