Methodology of Mathematics (Subtest III)
Problem Solving and Modeling
The Methodology domain tests how mathematics is taught, learned, and applied. Polya's four phases-understand the problem, devise a plan, carry out the plan, and look back-remain the standard framework. Modeling follows a cycle: identify a real situation, make assumptions, formulate a mathematical model, solve, interpret, validate, and revise. Good modeling questions ask what assumptions were made and how sensitive the result is to them.
Proof and Reasoning
Know the main proof forms. Direct proof derives the conclusion from definitions and given facts. Proof by contrapositive proves ¬Q → ¬P instead of P → Q. Proof by contradiction assumes the negation and derives a falsehood. Mathematical induction has two parts: a base case and an inductive step showing P(k) → P(k+1); it proves statements about all positive integers but does not explain why they are true. A single counterexample disproves a universal claim, while examples never prove one.
Communication, Representation, and Technology
Effective instruction connects multiple representations-symbolic, graphical, numerical, verbal, and physical-and asks students to explain reasoning, not just report answers. Technology such as graphing calculators and dynamic geometry supports conjecture and exploration, but it does not replace justification. Assessment should align with stated objectives and include both procedural fluency and conceptual understanding; rubrics should reward valid reasoning even when arithmetic slips.
Worked Example
Prove that the sum of the first n odd numbers is n². Base case: n = 1, the sum is 1 = 1². Inductive hypothesis: assume 1 + 3 + … + (2k − 1) = k² for some k ≥ 1. Inductive step: add the next odd number, 2(k+1) − 1 = 2k + 1, to both sides, giving k² + 2k + 1 = (k+1)². This is exactly the statement for n = k + 1, so by induction the formula holds for all positive integers n. The step uses the hypothesis rather than assuming the conclusion.
Traps That Cost Marks
- Circular reasoning in induction: assuming P(k+1) instead of proving it.
- Proving the converse when the original implication was requested.
- Treating several confirming examples as a proof.
- Presenting technology output without interpretation or justification.
- Stating modeling assumptions only implicitly, so conclusions look broader than they are.