Z0–Z3 Directory (Skills → Failure Modes → Sensors → Repairs) + Pri 1–6 Level Directory (WordPress Paste-Ready)

Definition (Primary Math OS):
Primary Mathematics OS is the capability pipeline that converts a child from counting and procedures into reliable quantitative reasoning under time, with transfer across word problems and novel question skins. Mastery is not “getting the answer.” Mastery is P3 reliability: correct reasoning that survives time pressure, unfamiliar formats, and mixed-topic papers.

0) P3 Mastery Definition (LOCKED)

P3 in Primary Math =
A student can (1) identify the concept, (2) choose the method, (3) execute accurately, and (4) verify—under time—across multiple skins (numbers-only, word problems, bar models, tables/graphs, novel variants), with low variance across days.

1) Primary Math OS Core Output

Primary Math OS must output 4 capabilities:

Meaning primitives (number sense, place value, fractions/ratio meaning)
Routing (choose method before computing)
Transfer (same concept survives new formats/wording)
Timed stability (low easy-error rate under pressure)

2) The 6 Dominant Failure Modes (Top 6, LOCKED)

FM1 — Primitive meaning gap hidden by procedures

Student can follow steps but doesn’t understand:

place value structure
fraction as quantity and operator
ratio as relationship

Symptom: collapses on word problems and novel variations.

FM2 — Word problem parsing failure (language → math translation)

Student misreads:

what is asked
what is given
which relationship matters

Symptom: wrong setup even with correct arithmetic.

FM3 — Misrouting (wrong method chosen)

Student applies the wrong tool:

wrong model type
wrong operation
wrong sequence

Symptom: “I know this topic” but still fails mixed papers.

FM4 — Computational fragility (arithmetic slips)

Even with correct concept:

careless mistakes
weak written working
weak estimation/verification habits

Symptom: loses marks on easy questions; confidence drops.

FM5 — Transfer brittleness (same concept fails in new skin)

Student succeeds in one format (worksheet) but fails in:

bar model
table/graph
reverse/inverse question
multi-step contexts

Symptom: practice volume rises but scores stay unstable.

FM6 — Time pressure variance collapse

Student works slowly or panics:

rushed easy errors
late-paper collapse
inconsistent marks across papers

Symptom: “At home can do; in test cannot.”

3) Z0–Z3 Sensor Pack (Primary Math Instrument Panel)

Z0 Sensors (student micro)

S-MATH-01: easy-error count under time (per 20 marks)
S-MATH-02: recurring error-type rate (same mistake repeats weekly)
S-MATH-03: wrong-method frequency (misrouting count)
S-MATH-04: verification habit rate (does student check?)
S-MATH-05: time-per-mark drift (speed stability)

Z1 Sensors (pattern / topic)

S-MATH-06: word-problem setup accuracy rate
S-MATH-07: fraction/ratio meaning probe score
S-MATH-08: multi-step chain integrity score
S-MATH-09: transfer gap (Format A vs Format B)

Z2 Sensors (paper readiness)

S-MATH-10: mixed-topic vs topical gap
S-MATH-11: mock-to-mock variance spread
S-MATH-12: end-of-paper degradation (last 20% error spike)

Z3 Sensors (systemic)

S-MATH-13: repair loop adherence (logs + retest cycles completed)
S-MATH-14: intervention timing (months before PSLE)
S-MATH-15: stability ladder progression (accuracy→pace→timed→sim)

4) Primary Math Repair Loops (Executable)

RL1 — Error-Type Loop (mandatory)

Log → classify → fix root cause → 48h retest → 7d retest

Classification categories:
(A) misread, (B) misroute, (C) arithmetic slip, (D) concept gap, (E) time panic

Verification: recurrence rate trends down.

RL2 — Primitive Rebuild Loop (meaning first)

Use when FM1 appears.

Concrete meaning → representation (bar/diagrams) → symbols → reverse question

Verification: fraction/ratio probes improve; novel word problems stop collapsing.

RL3 — Parsing-to-Model Loop (word problem translation)

Use when FM2 appears.

Read → underline ask/given → list relationships → choose model → solve → sentence answer

Verification: setup accuracy rises; wrong setup count falls.

RL4 — Routing Drill Loop (decision before execution)

Use when FM3 appears.

10-second concept ID → choose tool → commit method → execute

Tools list:

unitary method
part-whole model
comparison model
fraction operator model
ratio table
speed/time/distance setup
percentage equivalence map

Verification: wrong-method frequency drops in mixed papers.

RL5 — Computation Integrity Loop

Use when FM4 appears.

Neat working → estimate range → compute → check inverse operation

Verification: easy-error count decreases; marks stabilise.

RL6 — Transfer Inoculation Loop (5 skins)

Use when FM5 appears.

For each concept, practice across:

numbers-only
word problems
bar model/diagram
inverse / “what if” variant
novel mixed context

Verification: transfer gap shrinks.

RL7 — Stability Ladder (timed reliability)

Use when FM6 appears.

Accuracy → Pace → Timed micro-sets → Full paper simulation → Post-sim repair

Verification: variance spread narrows across 3–5 mocks.

5) Primary Math Skill Map (Core primitives → tools)

Core primitives (must lock)

place value and estimation
four operations mastery
factors/multiples (basic)
fractions meaning (part-whole, operator)
decimals/percent equivalence
ratio meaning (relationship, scaling)
units and measurement sense

Core toolset (routing library)

bar model (part-whole, comparison, before-after)
unitary method
percentage equivalence + ratio tables
speed-time-distance setup
average concept (where applicable)
interpretation of tables/graphs

6) Pri 1–6 Directory (What to lock at each level)

Pri 1 (P1→P2 primitives)

number bonds, place value basics
addition/subtraction meaning
simple word problems (one-step)
Goal: meaning + clean working habits

Pri 2

regrouping, mental math strategies
simple multiplication/division meaning
two-step word problems begin
Goal: reduce careless errors; build parsing habit

Pri 3

multiplication/division fluency
fractions introduction (meaning)
bar model basics
Goal: bar model becomes default tool

Pri 4

fractions/decimals deeper
factor/multiple basics
multi-step word problems
Goal: transfer across skins starts

Pri 5

ratio basics, percentage, speed intro
heavy multi-step mixed questions
Goal: routing drills + transfer inoculation

Pri 6 (PSLE readiness)

full integration: fractions/ratio/percent/speed
mixed papers + time pressure
stability ladder cycles
Goal: P3 reliability under PSLE conditions

7) Failure Mode Trace (schematic, required)

Procedures rise → meaning gaps persist → word problems misparsed → wrong methods chosen → easy arithmetic slips under time → transfer collapses on novel skins → mock variance widens → PSLE performance becomes unstable.
Repair loops (primitives + routing + transfer + stability ladder) truncate and stitch back to P3.

8) LOCK-WORTHY Primary Math OS statement

Primary Math mastery is not speed or worksheets; it is low-variance reasoning under time across word problems and novel skins. Primary Math OS builds primitives, routing, transfer, and stability—so a child becomes PSLE-reliable.

Primary Mathematics OS (V1.1)