Singapore School Mathematics Operating Manual · Chapter 48
Design mathematical working so errors are detected, isolated and repaired locally instead of contaminating every downstream result.
This chapter extends the operating manual as a cross-topic control layer. It is designed to complement, not replace, existing BTT owners for constraints, optimisation, error diagnosis, modelling, proof and examination craft.
1. Propagation and containment are different questions
The key discipline is to make propagation and containment are different questions explicit before allowing later calculations to depend on it. In school Mathematics this can appear in algebra, geometry, probability, statistics, modelling, numerical work or examination decisions. State the condition, preserve its scope, and verify that downstream work still satisfies it.
2. A checkpoint creates a trusted restart point
The key discipline is to make a checkpoint creates a trusted restart point explicit before allowing later calculations to depend on it. In school Mathematics this can appear in algebra, geometry, probability, statistics, modelling, numerical work or examination decisions. State the condition, preserve its scope, and verify that downstream work still satisfies it.
3. High-leverage nodes deserve checkpoints
The key discipline is to make high-leverage nodes deserve checkpoints explicit before allowing later calculations to depend on it. In school Mathematics this can appear in algebra, geometry, probability, statistics, modelling, numerical work or examination decisions. State the condition, preserve its scope, and verify that downstream work still satisfies it.
4. Independent branches can isolate faults
The key discipline is to make independent branches can isolate faults explicit before allowing later calculations to depend on it. In school Mathematics this can appear in algebra, geometry, probability, statistics, modelling, numerical work or examination decisions. State the condition, preserve its scope, and verify that downstream work still satisfies it.
5. Shared dependencies defeat naive isolation
The key discipline is to make shared dependencies defeat naive isolation explicit before allowing later calculations to depend on it. In school Mathematics this can appear in algebra, geometry, probability, statistics, modelling, numerical work or examination decisions. State the condition, preserve its scope, and verify that downstream work still satisfies it.
6. Units act as interface guards
The key discipline is to make units act as interface guards explicit before allowing later calculations to depend on it. In school Mathematics this can appear in algebra, geometry, probability, statistics, modelling, numerical work or examination decisions. State the condition, preserve its scope, and verify that downstream work still satisfies it.
7. Bounds act as range guards
The key discipline is to make bounds act as range guards explicit before allowing later calculations to depend on it. In school Mathematics this can appear in algebra, geometry, probability, statistics, modelling, numerical work or examination decisions. State the condition, preserve its scope, and verify that downstream work still satisfies it.
8. Domain checks act as admissibility guards
The key discipline is to make domain checks act as admissibility guards explicit before allowing later calculations to depend on it. In school Mathematics this can appear in algebra, geometry, probability, statistics, modelling, numerical work or examination decisions. State the condition, preserve its scope, and verify that downstream work still satisfies it.
9. Recomposition checks catch interface mismatch
The key discipline is to make recomposition checks catch interface mismatch explicit before allowing later calculations to depend on it. In school Mathematics this can appear in algebra, geometry, probability, statistics, modelling, numerical work or examination decisions. State the condition, preserve its scope, and verify that downstream work still satisfies it.
10. Exact forms reduce containment burden
The key discipline is to make exact forms reduce containment burden explicit before allowing later calculations to depend on it. In school Mathematics this can appear in algebra, geometry, probability, statistics, modelling, numerical work or examination decisions. State the condition, preserve its scope, and verify that downstream work still satisfies it.
11. Local recalculation is cheaper than global recalculation
The key discipline is to make local recalculation is cheaper than global recalculation explicit before allowing later calculations to depend on it. In school Mathematics this can appear in algebra, geometry, probability, statistics, modelling, numerical work or examination decisions. State the condition, preserve its scope, and verify that downstream work still satisfies it.
12. Independent redundancy matters
The key discipline is to make independent redundancy matters explicit before allowing later calculations to depend on it. In school Mathematics this can appear in algebra, geometry, probability, statistics, modelling, numerical work or examination decisions. State the condition, preserve its scope, and verify that downstream work still satisfies it.
13. Plausibility filters quarantine impossible results
The key discipline is to make plausibility filters quarantine impossible results explicit before allowing later calculations to depend on it. In school Mathematics this can appear in algebra, geometry, probability, statistics, modelling, numerical work or examination decisions. State the condition, preserve its scope, and verify that downstream work still satisfies it.
14. Branch labels prevent cross-contamination
The key discipline is to make branch labels prevent cross-contamination explicit before allowing later calculations to depend on it. In school Mathematics this can appear in algebra, geometry, probability, statistics, modelling, numerical work or examination decisions. State the condition, preserve its scope, and verify that downstream work still satisfies it.
15. Versioned intermediate values prevent stale reuse
The key discipline is to make versioned intermediate values prevent stale reuse explicit before allowing later calculations to depend on it. In school Mathematics this can appear in algebra, geometry, probability, statistics, modelling, numerical work or examination decisions. State the condition, preserve its scope, and verify that downstream work still satisfies it.
16. Error budgets define acceptable local tolerance
The key discipline is to make error budgets define acceptable local tolerance explicit before allowing later calculations to depend on it. In school Mathematics this can appear in algebra, geometry, probability, statistics, modelling, numerical work or examination decisions. State the condition, preserve its scope, and verify that downstream work still satisfies it.
17. Tight margins require smaller blast radii
The key discipline is to make tight margins require smaller blast radii explicit before allowing later calculations to depend on it. In school Mathematics this can appear in algebra, geometry, probability, statistics, modelling, numerical work or examination decisions. State the condition, preserve its scope, and verify that downstream work still satisfies it.
18. Examination working can be fault-tolerant
The key discipline is to make examination working can be fault-tolerant explicit before allowing later calculations to depend on it. In school Mathematics this can appear in algebra, geometry, probability, statistics, modelling, numerical work or examination decisions. State the condition, preserve its scope, and verify that downstream work still satisfies it.
19. Recovery is part of mathematical control
The key discipline is to make recovery is part of mathematical control explicit before allowing later calculations to depend on it. In school Mathematics this can appear in algebra, geometry, probability, statistics, modelling, numerical work or examination decisions. State the condition, preserve its scope, and verify that downstream work still satisfies it.
20. A containment audit
Ask what the problem requires, what can fail, what evidence is decisive, and which earlier operating-manual control should be invoked before calculation continues.
21. Worked mini-cases
Case A. A candidate satisfies every preference but violates a domain restriction. The candidate remains inadmissible because validity is checked before preference.
Case B. Two calculations disagree. Return to the last trusted common input, separate the branches, and identify whether the discrepancy comes from a shared dependency or a branch-specific step.
Case C. A result lies close to a threshold. Preserve precision and compare the uncertainty interval with the decision margin before classifying it.
22. Independent practice
1. Identify the admissibility condition in a problem of your choice.
2. Give one independent check that could catch a wrong intermediate result.
3. Explain why a mathematically attractive candidate may still be invalid.
4. Describe one situation where uncertainty changes the strength of a conclusion.
5. State what evidence would justify stopping the calculation.
23. Worked answers
1. The admissibility condition is the rule every candidate must satisfy before comparison.
2. Examples include substitution, a unit check, a bound, an alternative representation or an independent method.
3. Objective performance cannot compensate for violation of a hard mathematical condition.
4. A rounded or measured value near a threshold may have a plausible interval crossing both decision regimes.
5. A complete proof, verified candidate set, guaranteed bound, tolerance certificate or exhausted feasible search can justify stopping.
24. Continue through Batch 12
Return to the BTT Mathematics Hub for Batch 12 and use the neighbouring chapters as a connected control system.