Dependency Mapping: Fix the First Weak Link, Not Only the Last Broken Question
The topic where a student loses marks is not always the topic where the weakness began.
Additional Mathematics is highly connected. A differentiation question may fail after the derivative is found because factorisation is weak. A trigonometric identity may fail because algebraic fractions are unstable. A circle equation may fail because completing the square is slow. An integration question may fail because index notation is not secure. Repeating only the visible chapter can therefore waste time.
This guide develops dependency mapping: identify the first capability that the current question requires, locate where the solution route actually breaks, repair that upstream dependency, then return to the original A-Math task and retest transfer.
AI Extraction Box: The Dependency Loop
current question → route decomposition → first point of failure → upstream prerequisite → targeted repair → near transfer → return to original question → delayed mixed retest.
- Current topic: where the error appears.
- Dependency: earlier capability required for that step.
- First failure: earliest invalid or unavailable move.
- Repair scope: smallest upstream skill that restores the route.
- Return test: solve the original question independently after repair.
- Transfer test: changed-surface question without chapter cue.
The A-Math Dependency Spine
arithmetic and signs → algebraic manipulation → equations and factorisation → exact forms and indices → functions and graphs → trigonometry/coordinate geometry → calculus → mixed-topic integration.
This is not a strict teaching order. It is a dependency map. Later topics often call earlier capabilities repeatedly.
Worked Diagnosis 1: Differentiation Error That Is Really Factorisation
A student differentiates f(x)=x³−6x²+9x correctly:
f′(x)=3x²−12x+9.
Then the student cannot solve f′(x)=0. The calculus rule is not the weakness. Factorisation is:
3(x²−4x+3)=3(x−1)(x−3).
Repairing differentiation practice alone would miss the cause. The upstream repair is quadratic factorisation and equation solving.
Worked Diagnosis 2: Trig Identity Error That Is Really Fractions
A learner knows tanθ=sinθ/cosθ but cannot simplify an identity involving secθ and tanθ because algebraic fractions are combined incorrectly.
The trig knowledge is present. The missing dependency is common-denominator and factor-cancellation control.
Visible topic: trigonometry. Root cause: algebraic fractions.
Worked Diagnosis 3: Circle Geometry Error That Is Really Completing the Square
Question:
x²+y²−8x+6y−11=0.
A student knows the centre-radius form but cannot convert the equation. The prerequisite is completing the square:
(x−4)²+(y+3)²=36.
The geometry becomes accessible once the algebraic representation skill is restored.
Worked Diagnosis 4: Integration Error That Is Really Indices
A student struggles with ∫1/√x dx. The integration rule is easier after rewriting:
1/√x=x^{-1/2}.
Then:
∫x^{-1/2}dx=2x^{1/2}+C.
The upstream dependency is fractional and negative index notation.
Error Propagation
An upstream weakness can create downstream errors across many topics:
| Upstream weakness | Possible downstream effects |
|---|---|
| sign control | quadratics, trig identities, quotient rule, integration, kinematics |
| factorisation | roots, inequalities, stationary points, polynomial work |
| fractions | partial fractions, trig identities, rational equations, calculus |
| indices | surds, exponentials, logs, differentiation, integration |
| completing square | quadratic extrema, circle geometry, graph interpretation |
| equation solving | trig equations, calculus stationary points, modelling |
This is why high-spread dependencies deserve early repair.
Find the First Point of Failure
When reviewing a wrong solution, ask:
- Was the mathematical object recognised?
- Was a suitable method selected?
- Was the first transformation valid?
- Where did the first incorrect or unavailable step occur?
- What earlier capability does that step depend on?
The first wrong step gives more diagnostic information than the final wrong answer.
Repair the Smallest Useful Dependency
If a student fails a differentiation question because of one factorisation gap, do not restart the entire Secondary Mathematics curriculum. Repair the specific factorisation structure that blocks the route.
A narrow repair might include:
- one explanation;
- two direct examples;
- two changed examples;
- one return to the original A-Math question.
Repair only what is blocking the current system, then return upstream knowledge to service.
Return to the Original Question
A prerequisite worksheet can create temporary success. The real proof of repair is whether the learner can return to the original A-Math question and complete the full route independently.
This closes the loop:
diagnose → repair → return → retest.
Near Transfer and Far Transfer
After the original question succeeds, use one changed-surface question. If factorisation was repaired inside differentiation, test another calculus problem whose derivative produces a different quadratic. Later, place the same dependency inside an unlabeled mixed set.
This prevents the repair from becoming tied to one remembered question.
Dependency Mapping by Topic
- Quadratics: signs, expansion, factorisation, equations, graph meaning.
- Surds: factors, indices, exact arithmetic.
- Polynomials: algebraic division, factorisation, coefficient control.
- Binomial: indices, combinations, term numbering.
- Logs/exponentials: indices, functions, domains, equation solving.
- Trigonometry: fractions, identities, exact values, equations, intervals.
- Coordinate geometry: linear equations, gradients, completing square.
- Calculus: indices, algebraic structure, functions, equation solving.
Build a Personal Dependency Map
For each repeated error, record:
| Field | Question to ask |
|---|---|
| Visible topic | Where did the error appear? |
| First failure | What was the earliest invalid/unavailable step? |
| Dependency | What earlier capability should have supplied that step? |
| Spread | Which other topics use the same capability? |
| Repair | What smallest exercise set will restore it? |
| Return test | Did the original A-Math route now work? |
Prioritise by Spread, Frequency and Severity
A rare arithmetic slip is lower priority than a repeated algebraic-fraction weakness that damages five topics. Rank repairs by:
- how often the weakness appears;
- how many topics depend on it;
- how many marks/later steps it destroys;
- whether the learner can self-detect it.
High-spread, high-frequency upstream weaknesses deserve the earliest intervention.
When the Problem Is Recognition, Not Knowledge
Sometimes a student can execute a method perfectly when told the topic but cannot recognise when to use it. This is not a prerequisite content gap. It is a routing gap.
Repair with mixed, unlabeled questions and first-line prompts rather than more blocked topical worksheets.
When the Problem Is Representation
A student may know completing the square but not recognise that a circle equation needs it. Or know logarithms but not see that an exponential equation cannot be matched to a common base. This is a representation-selection gap.
Repair by practising transitions between equivalent forms and asking what information each form reveals.
Dependency Repair Decision Tree
- Method unknown? teach topic concept/procedure.
- Method known but not recognised? train mixed-topic routing.
- Method chosen but algebra fails? repair upstream symbolic dependency.
- Representation not seen? practise form-switching.
- Correct work but invalid final answer? repair domain/interpretation discipline.
- Repeated error across topics? prioritise shared dependency.
Common Failure Modes
| Error | Cause | Repair |
|---|---|---|
| repeats whole chapter after one error | repair scope too broad | identify first failure |
| does more calculus when factorisation is weak | visible topic mistaken for root cause | repair upstream algebra |
| prerequisite worksheet improves but original question still fails | return transfer not tested | re-solve original task |
| same error appears across many topics | shared dependency untreated | rank by spread and fix once |
| method works only when chapter is named | recognition gap | interleave unlabeled questions |
A 50-Minute Dependency Repair Session
- 10 minutes: analyse four recent wrong questions and find first failure.
- 8 minutes: name the upstream dependency for each.
- 12 minutes: complete targeted prerequisite repair questions.
- 10 minutes: return to the original A-Math questions.
- 5 minutes: solve one changed-surface transfer question.
- 5 minutes: rank remaining dependencies by spread/frequency/severity.
Useful Repair Routes
When the prerequisite is genuinely a Secondary Mathematics dependency, use the Secondary Mathematics S1–S4 Capability Map. Narrow repair routes include Algebraic Factorisation and Structural Control, Algebraic Fractions and Formula Rearrangement, and Linear Graphs, Coordinates and Relationships.
Use only the repair route that addresses the blocking dependency, then return to the A-Math task.
What Mastery Looks Like
- The learner distinguishes visible topic from root cause.
- The learner finds the first point where a route actually fails.
- The learner identifies upstream prerequisites accurately.
- The learner repairs the smallest useful dependency rather than restarting everything.
- The learner returns to the original question after repair.
- The learner tests transfer on changed surfaces and mixed sets.
- The learner prioritises high-spread weaknesses before low-impact slips.
- The learner enters Secondary 4 with a clearer map of which capabilities support the whole subject.
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