Singapore School Mathematics Operating Manual · Chapter 40
Mathematics values proof and verification, but not every question needs unlimited checking.
A one-step linear equation may need substitution. A long modelling problem near a threshold may need units, bounds, sensitivity and an independent route. A numerical approximation may need a residual or bracket. A proof may need complete case coverage. A graph-based estimate may need only the precision supported by the graph.
This chapter develops evidence sufficiency: matching verification depth to the strength of the claim, the complexity of the route and the consequences of being wrong.
The central question is: what additional evidence could still change my conclusion? If the answer is “nothing reasonable under the problem’s rules”, stopping is justified.
1. Verification depth should match claim strength
“x=3 is a solution” needs evidence that x=3 satisfies the equation.
“x=3 is the only solution” needs verification plus uniqueness.
“No solution exists” needs a global obstruction or exhaustive argument.
Stronger claims require stronger evidence.
2. Existence can be cheap to prove
One valid witness proves that something exists.
There is no need to classify every case when the question only asks for one example.
3. Universal claims require complete coverage
Checking ten examples does not prove a statement for every integer.
A general argument must explain why no admissible case escapes.
4. One counterexample can finish a refutation
If a universal claim is false, one valid counterexample is sufficient.
Continuing to search for more counterexamples may add illustration but not logical necessity.
5. Substitution often closes an equation-solving loop
After solving a simple equation with reversible steps, substitution confirms the candidate.
If the transformation chain already guarantees uniqueness, further methods may have diminishing value.
6. One-way transformations require deeper verification
Squaring, cancelling variable expressions or using numerical inversion can create or lose candidates.
Verification must return to the original problem.
7. Multi-stage problems need verification at high-risk nodes
Checking every arithmetic line equally is inefficient.
Prioritise values used many times, branch points, domain restrictions and threshold decisions.
8. Independent evidence is more valuable than repeated evidence
Re-entering the same formula may confirm arithmetic entry but not the formula itself.
Estimation, units or a different method add new information.
9. Verification has diminishing returns
The first independent check may greatly increase confidence.
A fifth nearly identical check may add almost nothing.
Stopping should consider marginal value, not merely number of checks.
10. A decision far from failure needs less depth
If a result is 20 units below an exact upper threshold and uncertainty is tiny, a quick check may be enough.
If it is 0.01 below, deeper verification is justified.
11. Margin controls checking priority
Low margin means small errors can change the conclusion.
High margin means the qualitative decision is stable.
This connects to Decision Margins and Slack.
12. Sensitivity also controls verification depth
A well-conditioned problem may tolerate ordinary rounding.
An ill-conditioned problem deserves extra precision and robustness checks.
13. Numerical methods need a formal stopping criterion
Stop because a residual, bracket width or successive-difference criterion has been met—not because the calculator “looks stable”.
14. Exact answers can stop earlier than approximations
If an exact symbolic derivation is complete and checked, there may be no value in replacing it with a long decimal.
Approximation adds representation, not certainty.
15. Graph estimates should not be over-verified beyond graph resolution
If an intersection is read from a coarse graph, calculating six decimal places from the picture is false precision.
The evidence source sets a natural stopping depth.
16. Measurement precision caps verification value
If input measurements are known only to the nearest unit, repeated calculator work cannot create exact physical certainty.
Stop when computational precision is comfortably smaller than data uncertainty.
17. Model uncertainty can dominate everything else
If the constant-speed assumption is only rough, refining arithmetic from 0.1 seconds to 0.001 seconds has little value.
Verification should address the dominant uncertainty source first.
18. Residuals tell whether a model needs more checking
Small unstructured residuals may support stopping.
Systematic residual patterns suggest the model itself needs revision.
19. A residual near zero is not always enough
In a poorly conditioned problem, small residual can coexist with notable input error.
Verification depth should include sensitivity where relevant.
20. Proof checking should target logical gaps
Ask whether every implication is justified, every case covered and every theorem condition satisfied.
Rechecking arithmetic inside a proof will not repair a missing logical branch.
21. Case exhaustion gives a natural stop
If the cases are negative, zero and positive and each has been handled, the sign partition is complete.
No fourth real-sign case exists.
22. Degree bounds can give a natural stop
A non-zero quadratic has at most two roots.
Once two distinct verified roots are found, the real/complex root count structure may certify completeness in the stated domain.
23. Interval bounds give a natural stop for periodic solutions
Once every solution in the requested interval is listed, periodic copies outside it are irrelevant.
The domain defines the search horizon.
24. Finite exhaustive search can stop mathematically
If there are exactly 12 admissible integer candidates and all 12 are tested, the search is complete.
The stop is justified by coverage, not fatigue.
25. Redundant data can support stopping
If a reconstructed state satisfies all independent observations, further repeated observations may add little.
The key is independence of the checks.
26. Contradiction can be a decisive stopping point
Once an assumption leads validly to contradiction, the assumed case is ruled out.
There is no need to search further inside that impossible branch.
27. A bound can finish a decision without exact calculation
If x<8 and the only target is whether x<10, the decision is already settled.
Exact x would be unnecessary extra work.
28. Monotonicity can finish a comparison
If f is strictly increasing and a<b, then f(a)<f(b).
Evaluating both outputs numerically adds no logical necessity.
29. Stopping too early creates under-verification
Examples: listing only one trigonometric branch, failing to check an extraneous root, ignoring an endpoint, or accepting a model with structured residuals.
30. Stopping too late wastes scarce examination time
Examples: solving a simple linear equation three ways, calculating exact values when a bound already answers the question, or polishing unsupported decimals.
31. Evidence sufficiency is target-specific
A result sufficient for a multiple-choice decision may be insufficient for a “show that” proof.
A visual estimate sufficient for checking may be insufficient for an exact answer.
The question wording defines the evidence contract.
32. Verification depth should be risk-weighted
Spend more effort when:
the problem is long; the answer is near a threshold; a transformation was one-way; the result feeds later parts; the model is sensitive; or the claim is universal.
Spend less when the result is structurally obvious and cheaply confirmed.
33. A stopping rule should be stated where useful
Examples:
“The interval width is below 0.001, so 3 d.p. is guaranteed.”
“Both roots have been verified and a quadratic has at most two.”
“All parity cases are covered.”
“The entire uncertainty interval remains below the threshold.”
34. A practical evidence-sufficiency audit
Ask:
What exactly am I claiming? What evidence is logically required? Which error modes remain unchecked? Would another independent check be capable of changing the conclusion? Is the result close to a boundary? Is uncertainty already smaller than the required tolerance? Is the search space exhausted? What mathematical fact tells me I can stop?
35. Independent practice
1. What evidence is enough to prove “there exists an even prime”?
2. What extra burden is needed to prove “there is exactly one even prime”?
3. Why is one counterexample enough to disprove a universal claim?
4. After squaring an equation, why is direct substitution required?
5. Why can a bound sometimes replace exact calculation?
6. Why do low-margin decisions deserve deeper verification?
7. Give one mathematically justified numerical stopping rule.
8. Why can repeated identical checks have diminishing value?
36. Worked answers
1. Exhibit 2 and verify it is even and prime.
2. Show every other even integer is composite, establishing uniqueness.
3. A universal claim says every admissible case works; one valid failure contradicts it.
4. Squaring can create extraneous candidates.
5. If the bound already places the unknown entirely on the required side of a threshold, the decision is complete.
6. A small error can flip the conclusion.
7. Stop bisection when interval width is below the tolerance required for the final rounding.
8. They may test the same failure mode and add little new evidence.
37. Batch 10 as one decision-quality layer
Redundancy and Fault Tolerance asks how to detect failure.
Decision Margins and Slack asks how close the result is to changing status.
Residuals and Model Mismatch asks what systematic discrepancy remains.
This chapter asks whether the evidence is now sufficient to stop.
Return to the BTT Mathematics Hub for Batch 10.