HOW SINGAPORE EDUCATION WORKS · Examples, Counterexamples and Mathematical Proof
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When a mathematical question asks you to explain why a statement is always true, checking several examples is a useful start. The explanation must then cover every case allowed by the statement. If you think the statement is false, find one permitted case that breaks it and show the calculation clearly.
The first move is to read the claim carefully. “Every integer” is a different promise from “some integers.” “Positive” excludes zero and negative numbers. A valid argument depends on those boundaries.
This guide helps school learners and parents distinguish testing a claim from proving it. The original exercises develop reasoning; they are not a list of topics required in every subject level or examination. Use the actual question and your teacher's instructions for assessed work.
The choices at a glance
| What you notice | Possible issue | Useful next move |
|---|---|---|
| A few examples are called proof | Untested cases remain | Find the shared structure |
| A counterexample uses an excluded input | The domain has changed | Check the starting condition |
| The desired result appears as an assumption | The reasoning is circular | Trace from given facts to the conclusion |
| A reversed implication seems obvious | The direction of the statement changed | Test the new claim separately |
Full chapter index
Read the claim and test permitted examples
Build a general argument and repair its gaps
Practise proof and changed claims
Transfer the reasoning and answer questions
CHAPTER 1 OF 8 · Read the claim and test permitted examples
1. State exactly what the claim promises
Consider the claim: “The product of two consecutive integers is even.” Consecutive means one integer immediately follows the other, such as 4 and 5, or −3 and −2. Even means divisible by 2 with an integer result.
Try 4 × 5 = 20. That example fits. Try −3 × −2 = 6. That fits too. Both help you understand the claim, but neither calculation accounts for every possible pair.
Now consider a different claim: “If the product of two integers is even, the integers are consecutive.” The wording reverses the direction. The numbers 2 and 8 have an even product, 16, but they are not consecutive.
Before calculating, mark the starting condition and the promised result. This prevents a learner from proving a related statement while leaving the actual question unanswered.
A parent can ask, “What are you allowed to choose, and what must happen?” That question is often more revealing than “Which formula do you use?”
CHAPTER 2 OF 8 · Read the claim and test permitted examples
2. Give examples a useful but limited job
Examples help you notice structure, detect errors and decide what to investigate. They can also expose a missing condition. Try zero, negative values or equal numbers when the stated domain allows them.
Suppose someone claims that n² + n + 11 is prime for every non-negative integer n. The first few values look promising: n = 0 gives 11; n = 1 gives 13; n = 2 gives 17. Each is prime.
That pattern is a reason to investigate, not permission to write “therefore always.” At n = 10, the expression equals 100 + 10 + 11 = 121. Since 121 = 11 × 11, it is not prime. The chosen n is permitted, so this is a valid counterexample.
Notice the difference between a failed calculation and a failed claim. Check the arithmetic and the domain before declaring either. If the claim had only concerned n = 0, 1 and 2, the n = 10 example would not address that narrower statement.
The source of confidence should match the scope of the conclusion.
CHAPTER 3 OF 8 · Build a general argument and repair its gaps
3. Build a general argument from the structure
Return to two consecutive integers. Call the first one n, where n is any integer. The next one is n + 1. Their product is n(n + 1).
Exactly one of two consecutive integers is even. If n is even, n supplies a factor of 2. If n is odd, n + 1 is even and supplies that factor. In either case, the product is even.
This argument covers every integer n through the two possible parity cases. It does not rely on the size of n or whether it is positive. That is why it reaches beyond a list of numerical checks.
Write the conclusion in the language of the original claim: the product of any two consecutive integers is even. Do not quietly strengthen it to “a multiple of 4.” For example, 1 × 2 = 2 is even but not divisible by 4.
Cambridge's NRICH discussion of generalising and proof offers a useful international teaching connection: moving from explored examples to a reason that accounts for a whole class of cases.
CHAPTER 4 OF 8 · Build a general argument and repair its gaps
4. Diagnose the gap in an explanation
If a solution says “I tried five examples, so it is true,” ask which untested cases remain. For an unrestricted integer claim, no finite sample alone settles all of them. Help the learner identify a property that all permitted cases share.
If the solution introduces n without defining it, repair that sentence first. “Let n be any integer” supplies a different starting point from “Let n be a positive whole number.” Symbols should preserve the domain, not blur it.
If the learner writes the desired result as an assumption, trace the direction of the reasoning. An argument must start from the given conditions or established facts and reach the conclusion through valid steps. Restating the conclusion in the middle does not establish it.
If a counterexample seems convincing, check that it actually meets the starting condition. Choosing n = 0.5 cannot refute a statement restricted to integers.
If the explanation is verbally sound but difficult to write, ask the learner to say one step at a time. Record the condition, the reason and the conclusion. Formal notation should make the logic easier to inspect, rather than make a simple explanation harder to follow.
CHAPTER 5 OF 8 · Practise proof and changed claims
5. Work through an argument with less prompting
For guided practice, explain why the sum of any two even integers is even. First describe an even integer as twice an integer. Let the two numbers be 2a and 2b, where a and b are integers.
Their sum is 2a + 2b = 2(a + b). Because a + b is an integer, the sum is twice an integer and therefore even. The two numbers need not be equal or positive.
Checking answer: the decisive step is not merely collecting terms. It is explaining why a + b remains an integer, so the expression fits the definition of even.
Now inspect a proposed shortcut: “Any two even numbers add to a multiple of 4.” Choose 2 and 4. Their sum is 6, which is not a multiple of 4. Both inputs are even, so the counterexample meets the condition.
Ask the learner to repair the false claim. “The sum is even” is one correct version. If both numbers are multiples of 4, their sum is a multiple of 4, but that adds a stronger starting condition. Say explicitly which repair you are making.
CHAPTER 6 OF 8 · Practise proof and changed claims
6. Try an independent proof and a changed claim
For independent practice, prove that the product of any two odd integers is odd. Define the two inputs, multiply them, then connect the resulting expression with the definition of odd. Try the work before reading the checking paragraph.
Checking answer: write the inputs as 2a + 1 and 2b + 1 for integers a and b. Their product is 4ab + 2a + 2b + 1 = 2(2ab + a + b) + 1. The expression in parentheses is an integer, so the product is one more than an even integer and is odd.
For a different task, decide whether n(n + 2) is even for every integer n. Do not assume the earlier consecutive-integer proof still applies: these two factors differ by 2.
Checking answer: n = 1 gives 1 × 3 = 3, so the universal claim is false. More fully, even n produces two even factors, whereas odd n produces two odd factors. The changed spacing changes the parity relationship.
At a later study session, return to the original consecutive-integer claim without opening the notes. Ask the learner to explain why the argument covers all integers, then change the promised conclusion to “divisible by 6.” The pair 1 and 2 gives a counterexample because their product is 2.
CHAPTER 7 OF 8 · Transfer the reasoning and answer questions
7. Carry the distinction into schoolwork and global communication
In a geometry task, a measured sketch can suggest a relationship, while the required explanation may depend on stated angle or shape properties. In algebra, a numerical substitution can check a step without proving an identity for every permitted value.
In a finite problem, exhaustive checking can be a proof if you genuinely cover all possible cases and explain why none are missing. This differs from choosing a few examples from an infinite set. The important question is whether the reasoning matches the full scope.
Mathematical notation helps learners communicate across languages, but the conditions still need clear words. In an international study group, define “integer,” “consecutive” and the direction of an implication before comparing solutions. A shared expression can conceal a difference in interpretation.
The guide to mathematical modelling develops another boundary: a conclusion about the real world also depends on the model's assumptions. A correct calculation does not establish that those assumptions describe the situation.
Ask a peer to identify the first step they cannot justify. That is more useful feedback than whether the finished solution looks sophisticated.
CHAPTER 8 OF 8 · Transfer the reasoning and answer questions
8. Common questions about proof and counterexamples
Is one counterexample really enough?
For a claim about every permitted case, yes. One case satisfying the starting conditions but failing the conclusion refutes that universal claim. It does not prove that the conclusion fails for every case.
Does a proof need algebra?
No. A valid explanation can use words, diagrams, cases or other appropriate reasoning. It must account for the claim's scope and justify its steps. The required presentation depends on the task.
Can checking lots of values ever settle a statement?
Yes, when the statement concerns a finite set and every case is checked correctly. Explain why your list is complete. A long but incomplete list is still incomplete.
What should a parent do when the answer is correct but the reason is weak?
Ask what makes the conclusion hold beyond the particular numbers used. If the learner is unsure, work with one familiar example, name its structure, then ask how that structure appears in a general case. Keep the support close to the actual difficulty.
How much explanation is enough in an examination?
Follow the question, current syllabus and teacher's guidance. This guide builds the underlying distinction; it does not predict a marking scheme or replace subject-specific requirements.
Sources and next reading
Source checked on 22 October 2026: Cambridge NRICH, The Challenge of Generalising and Proof. The expressions, worked arguments and practice sequence above are original teaching examples.
Continue with mathematical modelling and real-world problems or return to How Singapore Education Works.
Learning at other stages
For early-years learning, explore fair comparisons and equal measuring units. Comparing Lengths and Measuring in Kindergarten.
For university study, practise retrieving sources and recording search decisions. University Literature Search and Keeping a Search Trail.
For work, volunteering and later-life learning, make a task understandable to another person. Writing Clear Instructions for Adult Learners.