Multiplication makes a number bigger
A widespread KS2 and KS3 maths misconception. Pupils believe multiplication always produces a bigger number and division always produces a smaller one. The rule fails for fractions below 1.
- Evidence
- Strong
- Subject
- Maths
- Key stage
- KS2, KS3
- Citations
- 4
“When you multiply two numbers, the answer is always bigger than both. When you divide, the answer is always smaller.”
Multiplying by a number less than 1 makes the result smaller. Dividing by a number less than 1 makes the result bigger. The intuitive rule comes from whole-number examples and does not generalise.
Diagnostic items
Use these to surface the misconception before teaching the corrective sequence. The target distractor is what most pupils with this belief will choose.
- 1
A litre of fizzy drink costs £1.20. How much does 0.8 litres cost? Will the answer be bigger or smaller than £1.20?
- A.Bigger, because we are multiplyingtarget distractor
- B.Smaller, because 0.8 is less than 1
- C.The same, because the drink is the same price
- D.Cannot tell without doing the calculation
Source: Adapted from Bell, Fischbein & Greer, 1984
- 2
12 divided by 0.5 equals ?
- A.6, because dividing always makes the number smallertarget distractor
- B.24, because there are 24 halves in 12
- C.0.5, because the divisor is 0.5
- D.12, because dividing by half leaves it unchanged
If you have ever asked a Year 7 class what 12 divided by 0.5 equals and heard 6 from half the room, you have met this misconception. It is one of the cleanest examples of an over-generalised rule that worked through primary, then breaks loudly the moment decimals below 1 appear.
Why it persists
This is an over-generalisation, not a careless error. Pupils correctly induced the rule from hundreds of whole-number examples and the rule was always right within that world. More drill on the same kind of problem reinforces the rule, not the boundary.
Cognitive scientists call this the boundary-extension problem. The fix is variation: examples where the rule breaks, with the boundary taught explicitly. “This rule works for whole numbers. Here is what happens when the multiplier is less than 1.”
Evidence
Strong evidenceOne of the best-documented misconceptions in mathematics education. Featured in international concept-inventory work and in the EEF guidance reports for KS2 and KS3 mathematics. Persists into adulthood unless directly addressed.
Practice alignment
Research citations
- Bell, Fischbein & Greer(1984)Choice of Operation in Verbal Arithmetic Problems, The Effects of Number Size, Problem Structure and ContextCohortFieldPositivePopulation: KS2 and KS3 pupils
- Greer(1992)Multiplication and Division as Models of Situations (Handbook of Research on Mathematics Teaching and Learning)ReviewMetaPositive
- Fischbein, Deri, Nello & Marino(1985)The Role of Implicit Models in Solving Verbal Problems in Multiplication and DivisionCross-sectionalFieldPositivePopulation: Italian middle-school pupils
- Diagnostic Questions (Eedi)Multiplication and division of decimals item bankReviewField
Caveats
- The misconception is reduced but rarely eliminated by a single corrective lesson.
- Mature mathematicians have been shown to revert to the intuition under time pressure.
Corrective approaches
Pedagogies and tasks with evidence for addressing this misconception.
Concrete-pictorial-abstract sequence
Use bar models or fraction strips to make the result of multiplying by a fraction less than 1 visible. The picture shows the shrinking; the number sentence then matches.
Worked examples with the intuition explicitly named
Present two side-by-side worked examples (whole × whole, then whole × 0.5) and ask pupils what changed. Name the "multiplication always makes bigger" intuition out loud as the thing being broken.
Refutation text
Begin the lesson with "you might think multiplication always makes a number bigger. It doesn't. Here is when it does and when it doesn't." The directness is the mechanism.
Diamond 9 with statements
Use nine multiplication statements (some true, some false) and ask pupils to rank by "most surprising". The discussion surfaces who still holds the misconception.
Try this in Chalk
Related concepts
Questions teachers ask
Why doesn't more practice fix this?
When should I introduce multiplication by fractions less than 1?
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