Counterexample hunting

Have cases produced that your understanding doesn't cover, and find out that the rule you'd have stated was the wrong shape.

When to use this: When you want to go deeper
Learning impact: Strong change
Learning evidence: Credible with caveats
Where you are with the topic: Applying in new contexts
How often to use it: Worth repeating often

When to use it

You state a rule confidently and want to know where it fails.
A seminar or tutorial is coming and you need edge cases, not slogans.
You want a dress rehearsal for questions that begin with “But what about…?”

What to ask AI

Type something like: “Here's a rule I'd state confidently: [rule]. Give me three cases where it gives the wrong answer.”

Ask it not to explain yet, just give cases, and wait for your attempt to handle them.

Revise the rule yourself before you ask for the canonical fix.

How it helps

A rule that only meets friendly examples is not a rule you understand.

Counterexample hunting produces cases your generalisation does not cover.

The discomfort is the point: you find out the rule was the wrong shape.

Repairing the rule after a failed case builds a more accurate concept than polishing a slogan.

How to run it

State your rule in one sentence, as you would on an exam.
Ask for cases where the rule gives the wrong answer, no spoilers yet.
For each case, try to save the rule or admit it breaks.
Rewrite a narrower or corrected rule in your own words.
Check the rewritten rule against one fresh case before you stop.

Example exchange

You

Here’s a rule I’ve been using: if a function is continuous on an interval, it attains a maximum on that interval.

Try f(x) = x on the open interval (0, 1). What does your rule predict, and what is actually true?

You

The rule says it has a max, but it doesn’t. Values get arbitrarily close to 1 and never reach it. So I need closed and bounded.

Good. State the repaired rule in one sentence.

Copyable prompt

Here's a rule I'd state confidently: [rule]. Give me three cases where it
gives the wrong answer. Don't explain them yet, wait for my attempt to handle
each. After I revise the rule, check whether my revision still fails anywhere.

The Tell

If you can recite three counterexamples and still state the original broken rule on an exam, the hunt did not stick.

If the AI explained the fix before you tried to save the rule, you skipped the learning.

Stop when your rewritten rule survives a new case you have not seen.

Principle evidence

Strength of the underlying learning idea, not a claim about AI products.

The underlying learning idea is rated moderate. Counterexamples are central to conceptual change and to mathematical reasoning pedagogy. Refutation by cases improves the shape of generalisations.

AI delivery evidence

Whether an AI tutor delivers this method well is a separate question.

AI-invented counterexamples can be flawed or off-topic. Verify important ones. Delivery is speculative.