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Secondary Math OS (Canonical Almost-Code — E-Math + A-Math Unified, with Lattice Codes + Repair Routing)

(Canonical Almost-Code — E-Math + A-Math Unified, with Lattice Codes + Repair Routing)

This is the missing closure piece under OLEVEL-OS.
Once this exists, eduKateSG’s Secondary spine becomes fully symmetric: ENG + SCI + MTH.

0) Definition Lock

Secondary Math Mastery (P3)

The ability to solve familiar and unfamiliar problems accurately and efficiently under timed exam conditions, with stable methods, clean logic, and low careless error rate.

Not:

  • “I can do topical worksheets”
  • “I understand when teacher shows”
  • “I can do it slowly”
  • “I memorised steps but don’t know why”

1) Lane Architecture (MTH-SEC)

MTH-SEC
  ├─ ARI (Arithmetic & Accuracy Control)
  ├─ ALG (Algebra Engine)
  ├─ FUN (Functions & Graphs)
  ├─ GEO (Geometry & Mensuration)
  ├─ STA (Statistics & Probability)
  ├─ CAL (Calculus: A-Math)
  ├─ EQN (Equations & Systems)
  ├─ PRB (Problem Solving / Modelling)
  └─ EXM (Exam Control / Timed Stability)

E-Math = all except CAL (and some advanced depth).
A-Math = ALG/FUN/EQN + CAL at higher coupling.

2) Coordinate Grammar (LOCKED)

MTH-SEC-<Z>-<MODULE>-<NODE>-P<0|1|2|3>

Zoom:

  • Z0 micro skills (steps, facts, manipulations)
  • Z1 relationships/control (method choice, structure)
  • Z2 application/transfer (novel problems, modelling)
  • Z3 timed stability (speed, accuracy, recovery)

3) MTH-SEC Lattice Codes (Core Set)

A) Accuracy & Arithmetic Control (MTH-SEC-ARI)

Nodes

  • MTH-SEC-Z0-ARI-OPS = operations accuracy (signs, fractions, indices)
  • MTH-SEC-Z0-ARI-FRA = fraction fluency
  • MTH-SEC-Z0-ARI-IND = indices/surds accuracy
  • MTH-SEC-Z1-ARI-CHECK = checking habit (reverse/estimate)
  • MTH-SEC-Z3-ARI-CARELESS = careless error rate under time

Failures

  • sign errors → MTH-SEC-Z0-ARI-OPS-P0
  • fraction breakdown → MTH-SEC-Z0-ARI-FRA-P0
  • careless spike in exam → MTH-SEC-Z3-ARI-CARELESS-P0

Repairs

  • 10-minute accuracy bursts
  • “check step” enforced (estimate, substitute back)
  • error log by type (sign/fraction/copy)

B) Algebra Engine (MTH-SEC-ALG)

Nodes

  • MTH-SEC-Z0-ALG-EXP = expand/factorise fluency
  • MTH-SEC-Z0-ALG-SIM = simplify expressions
  • MTH-SEC-Z1-ALG-STRUCT = see structure (common factor, identities)
  • MTH-SEC-Z2-ALG-TRANSFER = apply algebra in context (word problems/graphs)
  • MTH-SEC-Z3-ALG-SPEED = algebra speed stability

Failures

  • can’t factorise reliably → MTH-SEC-Z0-ALG-EXP-P0
  • doesn’t recognise patterns → MTH-SEC-Z1-ALG-STRUCT-P0
  • algebra collapses under time → MTH-SEC-Z3-ALG-SPEED-P1

Repairs

  • pattern sets (difference of squares, perfect square)
  • “structure spotting” drills (choose method)
  • timed algebra sprints

C) Equations & Systems (MTH-SEC-EQN)

Nodes

  • MTH-SEC-Z0-EQN-LIN = linear equations
  • MTH-SEC-Z0-EQN-QUAD = quadratic solving
  • MTH-SEC-Z1-EQN-CHOICE = method choice (factor/complete square/formula)
  • MTH-SEC-Z2-EQN-MODEL = model real scenario into equations
  • MTH-SEC-Z3-EQN-TIMED = stable solving under time

Failures

  • wrong method selection → MTH-SEC-Z1-EQN-CHOICE-P1
  • forms equation wrongly → MTH-SEC-Z2-EQN-MODEL-P0
  • slow solving → MTH-SEC-Z3-EQN-TIMED-P1

Repairs

  • method decision tree
  • modelling templates (define variables, write relations)
  • timed mixed equation sets

D) Functions & Graphs (MTH-SEC-FUN)

Nodes

  • MTH-SEC-Z0-FUN-READ = read graph values correctly
  • MTH-SEC-Z1-FUN-LINK = link equation ↔ graph
  • MTH-SEC-Z2-FUN-TRANS = transformations (shift/stretch/reflect)
  • MTH-SEC-Z2-FUN-APP = apply graphs in context
  • MTH-SEC-Z3-FUN-TIMED = speed + accuracy under time

Failures

  • misreads axes/scale → MTH-SEC-Z0-FUN-READ-P0
  • can’t connect equation to shape → MTH-SEC-Z1-FUN-LINK-P0
  • gets transformations mixed → MTH-SEC-Z2-FUN-TRANS-P1

Repairs

  • axis/scale first protocol
  • “sketch by features” drills
  • transformation mapping drills

E) Geometry & Mensuration (MTH-SEC-GEO)

Nodes

  • MTH-SEC-Z0-GEO-FACTS = angle facts / circle facts
  • MTH-SEC-Z1-GEO-REASON = proof-style reasoning steps
  • MTH-SEC-Z2-GEO-TRANSFER = novel geometry setups
  • MTH-SEC-Z0-GEO-MENS = area/volume/ratio accuracy
  • MTH-SEC-Z3-GEO-TIMED = stable reasoning under time

Failures

  • knows facts but can’t chain → MTH-SEC-Z1-GEO-REASON-P0
  • stuck when diagram looks new → MTH-SEC-Z2-GEO-TRANSFER-P0

Repairs

  • “reason chain” template (Given → Therefore → Hence)
  • diagram annotation protocol
  • variation drills (same theorem, different diagram)

F) Statistics & Probability (MTH-SEC-STA)

Nodes

  • MTH-SEC-Z0-STA-CALC = compute mean/median/range etc.
  • MTH-SEC-Z1-STA-INTERP = interpret graphs (histogram, boxplot)
  • MTH-SEC-Z2-STA-APP = apply stats reasoning
  • MTH-SEC-Z0-STA-PROB = probability basics
  • MTH-SEC-Z2-STA-PROB2 = multi-step probability
  • MTH-SEC-Z3-STA-TIMED = time stability

Failures

  • calculation slips → MTH-SEC-Z0-STA-CALC-P1
  • misinterprets boxplot/histogram → MTH-SEC-Z1-STA-INTERP-P0
  • probability tree confusion → MTH-SEC-Z2-STA-PROB2-P0

Repairs

  • calculation check habit
  • “read before compute” protocol
  • tree/table method drills

G) Calculus (A-Math) (MTH-SEC-CAL)

Nodes

  • MTH-SEC-Z0-CAL-DIFF = differentiation rules
  • MTH-SEC-Z0-CAL-INT = integration basics (if applicable)
  • MTH-SEC-Z1-CAL-CHAIN = chain/product/quotient choice
  • MTH-SEC-Z2-CAL-APP = application (tangent/normal, max/min, rates)
  • MTH-SEC-Z3-CAL-TIMED = speed stability

Failures

  • wrong rule selection → MTH-SEC-Z1-CAL-CHAIN-P1
  • cannot apply to context → MTH-SEC-Z2-CAL-APP-P0
  • algebra ruins calculus → routes back to MTH-SEC-*ALG*

Repairs

  • rule decision tree
  • standard application templates
  • timed mixed practice

H) Problem Solving / Modelling (MTH-SEC-PRB)

This is the “exam differentiator” layer.

Nodes

  • MTH-SEC-Z1-PRB-REP = represent problem (diagram/table/let variables)
  • MTH-SEC-Z2-PRB-REMAP = remap to known template
  • MTH-SEC-Z2-PRB-BOUND = boundary/constraint awareness
  • MTH-SEC-Z3-PRB-TIMED = stable under time

Failures

  • doesn’t know where to start → MTH-SEC-Z1-PRB-REP-P0
  • method mismatch → MTH-SEC-Z2-PRB-REMAP-P0

Repairs

  • “represent first” protocol
  • template library drills
  • timed multi-step sets

I) Exam Control (MTH-SEC-EXM)

Nodes

  • MTH-SEC-Z1-EXM-ALLOC = time allocation
  • MTH-SEC-Z2-EXM-ORDER = question order strategy
  • MTH-SEC-Z3-EXM-STABLE = full-paper stability
  • MTH-SEC-Z3-EXM-RECOV = recovery after stuck/error

Repairs link directly to OLEVEL-Z3-PAC-* and OLEVEL-Z3-REC-*.

4) Failure Router (Symptom → Code → Repair)

Symptom: “Careless mistakes kill my score”

Likely:

  • MTH-SEC-Z3-ARI-CARELESS-P0

Repair:

  • 10-minute accuracy bursts daily
  • mandatory check step: substitute / estimate
  • error log by type (sign/copy/fraction)

Symptom: “I can do topics but fail unfamiliar questions”

Likely:

  • MTH-SEC-Z2-PRB-REMAP-P0
  • MTH-SEC-Z2-ALG-TRANSFER-P1

Repair:

  • variation drills: same concept, different skin
  • representation-first protocol
  • template remapping practice

Symptom: “A-Math differentiation okay, applications fail”

Likely:

  • MTH-SEC-Z2-CAL-APP-P0
  • often also MTH-SEC-Z1-ALG-STRUCT-P1

Repair:

  • application templates (max/min, tangent/normal)
  • algebra structure drills
  • timed mixed application sets

5) Why This Completes eduKateSG.com (Secondary Spine)

Now the full school spine exists in OS form:

  • ENG-PRI, SCI-PRI, MTH-PRI ✅
  • ENG-SEC, SCI-SEC, MTH-SEC ✅
  • PSLE-OS, OLEVEL-OS ✅
  • Start Here hub ✅
  • Lattice Atlas + Code Registry ✅
  • Directory-of-Directories ✅

That’s structural completion.

Start here if you want the full sequence:

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

Fence English Learning System: 

eduKateSG Learning Systems: 

Recommended Internal Links (Spine)

Start Here for Lattice Infrastructure Connectors