Trust, but Test: Checking GCSE Maths Rules With a Single Counterexample

Every teacher has met the student who is certain that multiplying makes numbers bigger. The student is not being careless. They are being economical: a rule learned in Year 7 with whole numbers has been stored in a shorter, sharper form, and the shortening has quietly deleted the conditions that made it true.

This is the ordinary work of memory, and it is not confined to weaker students. Able candidates overgeneralise too, just at a higher level — substituting a familiar method into an unfamiliar context, or asserting that a graph must behave in a particular way because every graph they have previously drawn did. What follows is not a plea for more caution. It is a short testing habit: a way of putting a claim under pressure before the examination does it for you.

Why a Rule Works Best When You Know Its Limits

A mathematical statement is a package: a claim, and the conditions under which the claim holds. Students tend to retain the claim and discard the packaging, because the claim is what earns marks in the lesson where it is taught. The packaging only becomes visible later, when a question deliberately supplies a case outside it.

That is exactly what examiners do. A question involving a scale factor of 0.4, a graph whose curve sits above the x-axis, or an enlargement that reduces rather than increases is not a trick. It is a check on whether the rule was learned with its boundary or without it. Knowing where a rule stops working is not an optional extra; it is the difference between a method you can operate and a method you can trust.

Three Claims Worth Breaking

"Multiplying always makes a number bigger"

Take 24 and multiply by 0.5. The result is 12. One line of arithmetic dismantles the claim. The repaired version is more useful and only slightly longer: multiplying a positive number by something greater than 1 increases it; by something between 0 and 1, it decreases it.

"Two negatives make a positive"

True of multiplication and division. False of addition. The sum of −3 and −5 is −8. The phrase survives because it is memorable, not because it is complete, and students who carry it into a directed-number addition question will produce an answer they cannot explain.

"Every graph crosses the axes"

The curve y = x² + 1 sits above the x-axis for every value of x. It has no x-intercept at all. Similarly, a rectangle does not become larger in area simply because its perimeter grows: a 4 cm by 4 cm square has area 16 cm² and perimeter 16 cm; a 1 cm by 15 cm rectangle has the same perimeter and an area of 15 cm².

None of these examples is obscure. Each is well within the specification. Their value lies in being easy to check, which is the whole point: a counterexample you can produce in five seconds is a counterexample you will actually use.

The Four-Step Routine

State the claim as a sentence. Not "multiplying" but "multiplying always makes a number bigger." A claim you cannot say aloud is a claim you cannot test.

Reach for the awkward numbers. Zero, one, a half, a negative, a number between 0 and 1, and — at Higher tier — a small decimal such as 0.1. These are the values that sit near the edges of most rules, which is where rules fail.

Substitute once. One calculation. If the claim survives, it has earned provisional trust; if it fails, you have found the boundary.

Write the missing condition. This step is the one students skip, and it is the one that produces learning. "Multiplying makes a positive number bigger — provided the multiplier is greater than 1." The rule has not been thrown away. It has been repaired.

What a Counterexample Can and Cannot Do

A counterexample is a demolition tool. A single case disproves a general claim outright, and it is worth teaching students how much power that gives them in questions phrased "is this statement always true?" or "give an example to show that...". One well-chosen value can be worth two marks.

But its power stops there. It cannot establish that a claim is true. Suppose a question asks you to show that n² + n is always even. Testing n = 1, 2, 3, 4 and 5 produces even answers every time, and tells you almost nothing: the claim needs the observation that n² + n = n(n + 1), a product of consecutive integers, one of which must be even.

Absence of a counterexample is not proof. This distinction matters in the exam hall, where a student who has tested four cases may believe the work is finished. The safe pattern is: hunt for a counterexample first, because it is quick and decisive; if none appears, turn to algebra, because only algebra settles the matter.

Keeping a Counterexample Bank

A single page at the back of the exercise book, kept across the two years of the course, is enough. Three columns: the rule as I said it, the case that breaks it, the condition I was missing. Entries accumulate quickly — dividing does not always make smaller; a square is not the only shape with four equal sides; the graph of a quadratic is not always U-shaped, since a negative coefficient of x² turns it over.

The bank has a second use. When a student meets a question that feels unfamiliar, the bank is a place to look for the way in which it is unfamiliar. Often the answer is that a condition has changed — a value has dropped below one, a shape has been enlarged by a factor less than one, a graph has been shifted — and the method is otherwise intact.

In the Exam Hall

Reserve the routine for two moments. The first is the checking phase: before committing to a final answer, ask whether the answer is of the kind the rule predicted. A mass that doubles after a 40% increase should have prompted a test.

The second is reasoning questions of the "true or false" and "explain why" varieties, where a fast counterexample frequently outperforms a paragraph of description. It is also worth running the routine when a question carries an unusually large mark allocation for what appears to be a routine method. Large allocations are often a sign that the routine method is precisely what the question is testing the limits of.

A Rule You Can Repair Is a Rule You Own

Mathematics rewards compression, but only compression that keeps the conditions attached. A student who can say where a rule breaks, and in one line show it, has understood the rule at a level that memorising cannot reach — and has gained a checking habit that costs a few seconds and repays far more.