Proof and counterexample artifacts
Produce a visible example, counterexample, or short proof artifact that justifies or rejects a mathematical claim.
Variation ladder
1Foundation recognitionNotice the relevant quantity, shape, relation, or constraint before calculating.Attention to the relevant objects, relation, or constraint without solving it.
2Guided strategyUse a visible scaffold, model, or narrowing move to choose a valid next step.One scaffolded move, then completion of the decision.
3Core taskSolve the target skill independently.Progressive hints that preserve independent evidence before reveal.
4Transfer formApply the same idea when numbers, wording, or representation changes.The invariant idea and what changed in the representation.
5Challenge extensionCombine the skill with a second idea, hidden constraint, or explanation requirement.Smaller case, counterexample, or explanation before reveal.
6Advanced enrichmentReason about the general method or a non-routine variant.Method hint, not the answer; no copied external source text.
Common slips
Treats one confirming example as proof of a universal claim.
Gives a counterexample that does not actually satisfy the claim conditions.
Hints and checks
Decision restated without adding a new method.
Common slip nudge: Treats one confirming example as proof of a universal claim.
One useful scaffold after help is requested.
Non-routine reasoning around the same core idea.