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E-Math OS — Z0–Z3 Directory Table (Skills → Failure Modes → Sensors → Repairs)

Unified Almost-Code Directory Spec (Registry-Clean) v0.2

PAGE-ID: EDK-DIR-EMTH-Z0Z3-v0.2
Role: DIR.TopicMap + SkillGraph + FailureAtlas + SensorPack + RepairLoops
Purpose: a publishable high-traffic reference page that makes Secondary E-Math executable (not topic-chasing).

0) Definition Lock

E-Math OS (Secondary) := the capability system for method selection + representation + mixed-set stability under time.

Z-ladder meaning (LOCK):

  • Z0 = micro-skills (execution atoms)
  • Z1 = method signals + representations (choose + encode)
  • Z2 = transfer (variation + mixed sets)
  • Z3 = timed stability (pace + checking + low drift)

1) Directory Table Schema (copy-paste standard)

Each row is a “node”:

NodeID | Z | Skill / Topic Family | Typical Failures | Primary Sensors | Repair Loops

2) E-Math OS Directory (Z0–Z3)

Z0 — Foundation Atoms (Execution Reliability)

Node: Algebra Manipulation (Core)

  • NodeID: EMTH-Z0-ALG-MANIP
  • Failures: sign slips; expansion errors; collecting like terms wrong (B2/B5)
  • Sensors: Foundation Fluency Sensor; Error Recurrence Sensor
  • Repairs: Fluency Lab (micro-bursts) + Repair & Retest Lab (≤7 days)

Node: Fractions / Ratios / Percentages

  • NodeID: EMTH-Z0-FRP-RATIO
  • Failures: wrong multiplier; ratio misread; percentage change errors (B2/B3)
  • Sensors: Foundation Fluency; Method Selection
  • Repairs: Fluency Lab + Method Signal Lab (signals: “increase/decrease”, “compare”, “rate”)

Node: Arithmetic & Indices Basics

  • NodeID: EMTH-Z0-ARITH-IND
  • Failures: careless arithmetic; wrong index law (B2/B5)
  • Sensors: Foundation Fluency; Pace Budget
  • Repairs: Fluency Lab + Careless Trap mini + timed micro sets

Z1 — Method Signals + Representation (Choose + Encode)

Node: Linear Equations / Simultaneous (Method Choice)

  • NodeID: EMTH-Z1-EQ-LIN-SIM
  • Failures: wrong rearrangement; wrong method selection; messy steps (B3/B2)
  • Sensors: Method Selection; Foundation Fluency
  • Repairs: Method Signal Lab (“equation form → solve tool”) + Chain discipline (one step/line)

Node: Word Problems Encoding (Representation)

  • NodeID: EMTH-Z1-REP-WORD
  • Failures: wrong variables; wrong constraints; misread “at least/at most”; wrong unit (B4/B5)
  • Sensors: Representation Sensor
  • Repairs: Representation/Encoding Lab (no solving first) + re-read “answer asked thing”

Node: Graphs & Relationships (Gradient/Intercept)

  • NodeID: EMTH-Z1-GPH-LIN
  • Failures: wrong gradient; scale reading errors; confusion between point vs intercept (B4/B5)
  • Sensors: Representation Sensor; Error Recurrence
  • Repairs: Graph Encode/Read Lab + “scale audit” checklist

Node: Geometry Reasoning (Angle Facts / Proof Chains)

  • NodeID: EMTH-Z1-GEO-CHAIN
  • Failures: missing theorem; skipping steps; wrong diagram inference (B1/B3)
  • Sensors: Method Selection (theorem signals); Error Recurrence
  • Repairs: “Statement → reason” chain rewrite + theorem flash-signals

Z2 — Transfer & Mixed Sets (Variation Survivability)

Node: Variation Ladder (Topic Family Transfer)

  • NodeID: EMTH-Z2-TRANSFER-VAR
  • Failures: perfect on drills; fails on mixed; freezes when form changes (B3 hidden B1)
  • Sensors: Transfer/Variation Sensor
  • Repairs: Variation Ladder Lab (V0→V4) + mixed weekly retest

Node: Mixed Topic Switching (Paper Reality)

  • NodeID: EMTH-Z2-MIX-SWITCH
  • Failures: method panic; tries random tools; wastes time switching (B3/B6)
  • Sensors: Method Selection; Pace Budget
  • Repairs: Mixed Mini-Paper Lab + “signal→method” one-liner per question

Z3 — Timed Stability (Pace + Checking + Low Drift)

Node: Pace Budget & Completion Stability

  • NodeID: EMTH-Z3-PACE-STABLE
  • Failures: unfinished paper; late collapse; rushing errors (B6→B5)
  • Sensors: Pace Budget Sensor; DriftRisk
  • Repairs: Timed Stability Lab (smaller scope) + skip/return policy + final 10-min audit

Node: Careless Control & Audit Routine

  • NodeID: EMTH-Z3-CARELESS-AUDIT
  • Failures: sign slips; wrong final answer; unit loss; copying wrong numbers (B5)
  • Sensors: Error Recurrence; DriftRisk
  • Repairs: Careless Trap Lab + mandatory check stack:
  • sign scan
  • unit check
  • substitution check (when applicable)
  • “answer asked thing” re-read

Node: O-Level Wrapper Readiness (Exam Load)

  • NodeID: EMTH-Z3-OLEVEL-WRP
  • Failures: collapse only in full paper despite topic ability (hidden drift)
  • Sensors: Math Gauge + Monitoring Math Health (trend)
  • Repairs: O-Level E-Math Wrapper cycle (mixed minis + pace budget + transfer ladder)

3) Minimal Sensor Pack (E-Math page footer)

Every E-Math topic page should link to the 6 sensors and the 6 labs (so it stays operational):

  • Foundation Fluency
  • Method Selection
  • Representation
  • Error Recurrence
  • Pace Budget
  • Transfer/Variation

4) Why this Directory is Important (eduKateSG “completion” logic)

This directory does 3 things your competitors don’t:

  1. Stops topic-chasing by routing failures into buckets/sensors.
  2. Makes transfer measurable (variation ladder + mixed stability).
  3. Makes exam reliability trainable (pace budget + audit routine + drift control).

That’s how Math OS becomes “complete” and replicable across every topic and every year band.

Start here if you want the full sequence:

Vocabulary OS Series Index:
https://edukatesg.com/vocabulary-os-series-index/

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