Singapore School Mathematics Operating Manual · Chapter 49
Trace conclusions through layers of assumptions so learners can distinguish unconditional facts from results that hold only when particular model, domain or data conditions remain true.
This chapter is a cross-topic control layer for Primary, PSLE, SEC G1/G2/G3, E-Math, A-Math and JC Mathematics. It complements existing BTT topic owners rather than replacing them.
1. Conclusions inherit assumptions
The operating principle is to make conclusions inherit assumptions explicit. Use a simple numerical example, state the active domain or scale, preserve units and conditions, and verify that the conclusion still matches the original task before passing it downstream.
2. Definitions sit below derived claims
The operating principle is to make definitions sit below derived claims explicit. Use a simple numerical example, state the active domain or scale, preserve units and conditions, and verify that the conclusion still matches the original task before passing it downstream.
3. Domain conditions form an assumption layer
The operating principle is to make domain conditions form an assumption layer explicit. Use a simple numerical example, state the active domain or scale, preserve units and conditions, and verify that the conclusion still matches the original task before passing it downstream.
4. Model assumptions form another layer
The operating principle is to make model assumptions form another layer explicit. Use a simple numerical example, state the active domain or scale, preserve units and conditions, and verify that the conclusion still matches the original task before passing it downstream.
5. Data assumptions can be independent of model assumptions
The operating principle is to make data assumptions can be independent of model assumptions explicit. Use a simple numerical example, state the active domain or scale, preserve units and conditions, and verify that the conclusion still matches the original task before passing it downstream.
6. Approximation assumptions weaken guarantees
The operating principle is to make approximation assumptions weaken guarantees explicit. Use a simple numerical example, state the active domain or scale, preserve units and conditions, and verify that the conclusion still matches the original task before passing it downstream.
7. Conditional conclusions need conditional language
The operating principle is to make conditional conclusions need conditional language explicit. Use a simple numerical example, state the active domain or scale, preserve units and conditions, and verify that the conclusion still matches the original task before passing it downstream.
8. Stronger assumptions can produce stronger conclusions
The operating principle is to make stronger assumptions can produce stronger conclusions explicit. Use a simple numerical example, state the active domain or scale, preserve units and conditions, and verify that the conclusion still matches the original task before passing it downstream.
9. Weaker assumptions produce more robust conclusions
The operating principle is to make weaker assumptions produce more robust conclusions explicit. Use a simple numerical example, state the active domain or scale, preserve units and conditions, and verify that the conclusion still matches the original task before passing it downstream.
10. Removing one assumption tests necessity
The operating principle is to make removing one assumption tests necessity explicit. Use a simple numerical example, state the active domain or scale, preserve units and conditions, and verify that the conclusion still matches the original task before passing it downstream.
11. Assumptions can be nested
The operating principle is to make assumptions can be nested explicit. Use a simple numerical example, state the active domain or scale, preserve units and conditions, and verify that the conclusion still matches the original task before passing it downstream.
12. Local assumptions should not escape their scope
The operating principle is to make local assumptions should not escape their scope explicit. Use a simple numerical example, state the active domain or scale, preserve units and conditions, and verify that the conclusion still matches the original task before passing it downstream.
13. Contradictory assumptions collapse the branch
The operating principle is to make contradictory assumptions collapse the branch explicit. Use a simple numerical example, state the active domain or scale, preserve units and conditions, and verify that the conclusion still matches the original task before passing it downstream.
14. Hidden assumptions should be surfaced
The operating principle is to make hidden assumptions should be surfaced explicit. Use a simple numerical example, state the active domain or scale, preserve units and conditions, and verify that the conclusion still matches the original task before passing it downstream.
15. Default conventions still need awareness
The operating principle is to make default conventions still need awareness explicit. Use a simple numerical example, state the active domain or scale, preserve units and conditions, and verify that the conclusion still matches the original task before passing it downstream.
16. A theorem carries its hypotheses
The operating principle is to make a theorem carries its hypotheses explicit. Use a simple numerical example, state the active domain or scale, preserve units and conditions, and verify that the conclusion still matches the original task before passing it downstream.
17. A numerical answer can be exact inside an approximate model
The operating principle is to make a numerical answer can be exact inside an approximate model explicit. Use a simple numerical example, state the active domain or scale, preserve units and conditions, and verify that the conclusion still matches the original task before passing it downstream.
18. Sensitivity identifies fragile assumptions
The operating principle is to make sensitivity identifies fragile assumptions explicit. Use a simple numerical example, state the active domain or scale, preserve units and conditions, and verify that the conclusion still matches the original task before passing it downstream.
19. Provenance records the assumption stack
The operating principle is to make provenance records the assumption stack explicit. Use a simple numerical example, state the active domain or scale, preserve units and conditions, and verify that the conclusion still matches the original task before passing it downstream.
20. An assumption-stack audit
Ask what level of claim is required, which conditions support it, how much detail is justified, and whether the conclusion survives the relevant uncertainty or model variation.
21. Worked examples
Example 1. If a conclusion depends on x>0, test what changes at x=0 and x<0 before calling it universal.
Example 2. If a small term is ignored, compare its maximum possible contribution with the decision margin or required tolerance.
Example 3. If data are grouped, do not claim exact within-group values that the representation no longer contains.
Example 4. If a model conclusion survives several plausible parameter ranges and remains on the same side of every relevant threshold, the qualitative conclusion is stronger than one supported by a single point estimate.
22. Independent practice
1. State one assumption behind a familiar school formula.
2. Give one situation where a smaller term cannot safely be ignored.
3. Explain why a grouped-data summary cannot recover exact raw observations.
4. Give one example of a conclusion that remains true under a range of input values.
5. State the check required before an approximation is used for a threshold decision.
23. Worked answers
1. Example: distance=speed×time under a constant or appropriately averaged speed model.
2. Near cancellation or near a decision threshold, a numerically small term can control the sign or classification.
3. Aggregation maps many possible raw data sets to the same grouped representation.
4. If an entire uncertainty interval lies below an upper threshold, the below-threshold conclusion is robust to every value in that interval.
5. Bound the approximation error and confirm the full plausible result remains on the required side of the threshold.
24. Continue through Batch 13
Return to the BTT Mathematics Hub for Batch 13.