6.9 KiB

Raw Blame History

Simulated Student Model

Overview

The SimulatedStudent class models how students learn soroban skills over time. It's used in journey simulation tests to validate that BKT-based adaptive problem generation outperforms classic random generation.

Location: src/test/journey-simulator/SimulatedStudent.ts

Core Model: Hill Function Learning

The model uses the Hill function (from biochemistry/pharmacology) to model learning:

P(correct | skill) = exposure^n / (K^n + exposure^n)

Where:

exposure: Number of times the student has attempted problems using this skill
K (halfMaxExposure): Exposure count where P(correct) = 0.5
n (hillCoefficient): Controls curve shape (n > 1 delays onset, then accelerates)

Why Hill Function?

The Hill function naturally models how real learning works:

Early struggles: Low exposure = low probability (building foundation)
Breakthrough: At some point, understanding "clicks" (steep improvement)
Mastery plateau: High exposure approaches but never reaches 100%

Example Curves

With K=10, n=2:

Exposures	P(correct)	Stage
0	0%	No knowledge
5	20%	Building foundation
10	50%	Half-way (by definition of K)
15	69%	Understanding clicks
20	80%	Confident
30	90%	Near mastery

Skill-Specific Difficulty

Key insight from pedagogy: Not all skills are equally hard. Ten-complements require cross-column operations and are inherently harder than five-complements.

Difficulty Multipliers

Each skill has a difficulty multiplier applied to K:

effectiveK = profile.halfMaxExposure * SKILL_DIFFICULTY_MULTIPLIER[skillId];

Skill Category	Multiplier	Effect
Basic (directAddition, heavenBead)	0.8-0.9x	Easier, fewer exposures needed
Five-complements	1.2-1.3x	Moderate, ~20-30% more exposures
Ten-complements	1.6-2.1x	Hardest, ~60-110% more exposures

Concrete Example

With profile K=10:

Skill	Multiplier	Effective K	Exposures for 50%
basic.directAddition	0.8	8	8
fiveComplements.4=5-1	1.2	12	12
tenComplements.9=10-1	1.6	16	16
tenComplements.1=10-9	2.0	20	20

Rationale for Specific Values

Based on soroban pedagogy:

Basic skills (0.8-0.9): Single-column, direct bead manipulation
Five-complements (1.2-1.3): Requires decomposition thinking (+4 = +5 -1)
Ten-complements (1.6-2.1): Cross-column carrying/borrowing, harder mental model
Harder ten-complements: Larger adjustments (tenComplements.1=10-9 = +1 requires -9+10) are cognitively harder

Conjunctive Model for Multi-Skill Problems

When a problem requires multiple skills (e.g., basic.directAddition + tenComplements.9=10-1):

P(correct) = P(skill_A) × P(skill_B) × P(skill_C) × ...

This models that ALL component skills must be applied correctly. A student strong in basics but weak in ten-complements will fail problems requiring ten-complements.

Student Profiles

Profiles define different learner types:

interface StudentProfile {
  name: string;
  halfMaxExposure: number; // K: lower = faster learner
  hillCoefficient: number; // n: curve shape
  initialExposures: Record<string, number>; // Pre-seeded learning
  helpUsageProbabilities: [number, number, number, number];
  helpBonuses: [number, number, number, number];
  baseResponseTimeMs: number;
  responseTimeVariance: number;
}

Example Profiles

Profile	K	n	Description
Fast Learner	8	1.5	Quick acquisition, smooth curve
Average Learner	12	2.0	Typical learning rate
Slow Learner	15	2.5	Needs more practice, delayed onset

Exposure Accumulation

Critical behavior: Exposure increments on EVERY attempt, not just correct answers.

This models that students learn from engaging with material, regardless of success. The attempt itself is the learning event.

// Learning happens from attempting, not just succeeding
for (const skillId of skillsChallenged) {
  const current = this.skillExposures.get(skillId) ?? 0;
  this.skillExposures.set(skillId, current + 1);
}

Fatigue Tracking

The model tracks cognitive load based on true skill mastery:

True P(correct)	Fatigue Multiplier	Interpretation
≥ 90%	1.0x	Automated, low effort
≥ 70%	1.5x	Nearly automated
≥ 50%	2.0x	Moderate effort
≥ 30%	3.0x	Struggling
< 30%	4.0x	Very weak, high cognitive load

Help System

Students can use help at four levels:

Level 0: No help
Level 1: Hint
Level 2: Decomposition shown
Level 3: Full solution

Help provides an additive bonus to probability (not multiplicative), simulating that help scaffolds understanding but doesn't guarantee correctness.

Validation

The model is validated by:

BKT Correlation: BKT's P(known) should correlate with true P(correct)
Learning Trajectories: Accuracy should improve over sessions
Skill Targeting: Adaptive mode should surface weak skills faster
Difficulty Ordering: Ten-complements should take longer to master than five-complements

Files

src/test/journey-simulator/SimulatedStudent.ts - Main model implementation
src/test/journey-simulator/types.ts - StudentProfile type definition
src/test/journey-simulator/profiles/ - Predefined learner profiles
src/test/journey-simulator/journey-simulator.test.ts - Validation tests

Future Improvements

Based on consultation with Kehkashan Khan (abacus coach):

Forgetting/Decay: Skills may decay without practice (not yet implemented)
Transfer Effects: Learning +4 may help learning +3 (not yet implemented)
Warm-up Effects: First few problems may be shakier (not yet implemented)
Within-session Fatigue: Later problems may be harder (partially implemented via fatigue tracking)

See .claude/KEHKASHAN_CONSULTATION.md for full consultation notes.

6.9 KiB Raw Blame History Unescape Escape