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| 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 | /** * Simulated Student Model - Hill Function Learning * * Simulates a learning student using exposure-based learning with the Hill function. * * Learning Model: * P(correct | skill) = exposure^n / (K^n + exposure^n) * * Where: * - exposure: number of times student has attempted problems using this skill * - K (halfMaxExposure): exposure count where P = 0.5 * - n (hillCoefficient): controls curve shape (n>1 delays onset) * * For multi-skill problems, uses conjunctive model: * P(correct) = P(A) × P(B) × P(C) for skills A, B, C * * Key behaviors: * - Exposure increments on EVERY attempt (not just correct answers) * - Exposure persists across sessions (same student instance for entire journey) * - Help provides additive bonus to probability */ import type { GeneratedProblem } from '@/db/schema/session-plans' import type { SeededRandom } from './SeededRandom' import type { SimulatedAnswer, StudentProfile } from './types' /** * Skill difficulty multipliers for K (halfMaxExposure). * * Higher multiplier = harder skill = needs more exposures to reach 50% mastery. * * Example: If profile.halfMaxExposure = 10: * - basic.directAddition: K = 10 × 0.8 = 8 (easier, 50% at 8 exposures) * - fiveComplements.*: K = 10 × 1.2 = 12 (harder, 50% at 12 exposures) * - tenComplements.*: K = 10 × 1.8 = 18 (hardest, 50% at 18 exposures) */ const SKILL_DIFFICULTY_MULTIPLIER: Record<string, number> = { // Basic skills - easier, foundational 'basic.directAddition': 0.8, 'basic.directSubtraction': 0.8, 'basic.heavenBead': 0.9, 'basic.heavenBeadSubtraction': 0.9, 'basic.simpleCombinations': 1.0, 'basic.simpleCombinationsSub': 1.0, // Five-complements - moderate difficulty (single column, but requires decomposition) 'fiveComplements.4=5-1': 1.2, 'fiveComplements.3=5-2': 1.2, 'fiveComplements.2=5-3': 1.2, 'fiveComplements.1=5-4': 1.2, 'fiveComplementsSub.-4=-5+1': 1.3, 'fiveComplementsSub.-3=-5+2': 1.3, 'fiveComplementsSub.-2=-5+3': 1.3, 'fiveComplementsSub.-1=-5+4': 1.3, // Ten-complements - hardest (cross-column, carrying/borrowing) 'tenComplements.9=10-1': 1.6, 'tenComplements.8=10-2': 1.7, 'tenComplements.7=10-3': 1.7, 'tenComplements.6=10-4': 1.8, 'tenComplements.5=10-5': 1.8, 'tenComplements.4=10-6': 1.8, 'tenComplements.3=10-7': 1.9, 'tenComplements.2=10-8': 1.9, 'tenComplements.1=10-9': 2.0, // Hardest - biggest adjustment 'tenComplementsSub.-9=+1-10': 1.7, 'tenComplementsSub.-8=+2-10': 1.8, 'tenComplementsSub.-7=+3-10': 1.8, 'tenComplementsSub.-6=+4-10': 1.9, 'tenComplementsSub.-5=+5-10': 1.9, 'tenComplementsSub.-4=+6-10': 1.9, 'tenComplementsSub.-3=+7-10': 2.0, 'tenComplementsSub.-2=+8-10': 2.0, 'tenComplementsSub.-1=+9-10': 2.1, // Hardest subtraction } /** * Get the difficulty multiplier for a skill. * Returns 1.0 for unknown skills (baseline difficulty). */ function getSkillDifficultyMultiplier(skillId: string): number { return SKILL_DIFFICULTY_MULTIPLIER[skillId] ?? 1.0 } /** * Convert true probability to a cognitive load multiplier. * * Higher P(correct) → lower multiplier (more automated, less fatiguing) * Lower P(correct) → higher multiplier (less automated, more fatiguing) * * This is the "ground truth" multiplier based on actual skill mastery, * used to measure fatigue independently of what budgeting system was used. */ export function getTrueMultiplier(trueP: number): number { if (trueP >= 0.9) return 1.0 // Automated if (trueP >= 0.7) return 1.5 // Nearly automated if (trueP >= 0.5) return 2.0 // Halfway if (trueP >= 0.3) return 3.0 // Struggling return 4.0 // Very weak } /** * Simulates a learning student using exposure-based Hill function model. */ export class SimulatedStudent { private skillExposures: Map<string, number> private profile: StudentProfile private rng: SeededRandom constructor(profile: StudentProfile, rng: SeededRandom) { this.profile = profile this.rng = rng this.skillExposures = new Map() // Initialize exposures from profile (pre-seeded learning) for (const [skillId, exposure] of Object.entries(profile.initialExposures)) { this.skillExposures.set(skillId, exposure) } } /** * Hill function: calculates P(correct) based on exposure count. * * P = exposure^n / (K^n + exposure^n) * * Curve behavior (example with K=10, n=2): * - 0 exposures → P = 0 * - 5 exposures → P ≈ 0.20 * - 10 exposures → P = 0.50 (by definition of K) * - 20 exposures → P ≈ 0.80 * - 30 exposures → P ≈ 0.90 * * @param exposure - Number of exposures to this skill * @param K - Half-maximum constant (exposure where P = 0.5) * @param n - Hill coefficient (controls curve steepness) * @returns Probability of correct answer [0, 1] */ hillFunction(exposure: number, K: number, n: number): number { if (exposure <= 0) return 0 if (K <= 0) return 1 // Edge case: instant mastery const expN = exposure ** n const kN = K ** n return expN / (kN + expN) } /** * Simulate answering a problem. * * IMPORTANT: Exposure increments BEFORE calculating probability, * simulating that the student is learning from the attempt itself. * This matches real learning where engaging with material teaches you, * regardless of whether you get it right. * * Fatigue is calculated BEFORE exposure increment, representing the * cognitive load of the problem based on the student's state when * they first see it. */ answerProblem(problem: GeneratedProblem): SimulatedAnswer { const skillsChallenged = problem.skillsRequired ?? [] // Calculate fatigue BEFORE incrementing exposure // This represents cognitive load at the moment of problem presentation let fatigue = 0 for (const skillId of skillsChallenged) { const trueP = this.getTrueProbability(skillId) fatigue += getTrueMultiplier(trueP) } // Increment exposure for all skills BEFORE calculating probability // (Learning happens from the attempt, not from success) for (const skillId of skillsChallenged) { const current = this.skillExposures.get(skillId) ?? 0 this.skillExposures.set(skillId, current + 1) } // Determine if student uses help (binary) const hadHelp = this.selectHelpUsage() // Calculate answer probability using Hill function + conjunctive model const answerProbability = this.calculateAnswerProbability(skillsChallenged, hadHelp) const isCorrect = this.rng.chance(answerProbability) // Calculate response time const responseTimeMs = this.calculateResponseTime(skillsChallenged, hadHelp, isCorrect) return { isCorrect, responseTimeMs, hadHelp, skillsChallenged, fatigue, } } /** * Calculate probability of correct answer using conjunctive Hill function model. * * For multi-skill problems, each skill must be applied correctly: * P(correct) = P(skill_A) × P(skill_B) × P(skill_C) × ... * * Help bonus is additive (applied after the product). */ private calculateAnswerProbability(skillIds: string[], hadHelp: boolean): number { if (skillIds.length === 0) { // Basic problems (no special skills) almost always correct return 0.95 } // Conjunctive model: product of individual skill probabilities let probability = 1.0 for (const skillId of skillIds) { const exposure = this.skillExposures.get(skillId) ?? 0 // Apply skill-specific difficulty multiplier to K // Higher multiplier = harder skill = needs more exposures const effectiveK = this.profile.halfMaxExposure * getSkillDifficultyMultiplier(skillId) const skillProb = this.hillFunction(exposure, effectiveK, this.profile.hillCoefficient) probability *= skillProb } // Add help bonus (additive, not multiplicative) // helpBonuses[0] = no help, helpBonuses[1] = with help const helpBonus = this.profile.helpBonuses[hadHelp ? 1 : 0] probability += helpBonus // Clamp to valid probability range // Never 0% (lucky guess) or 100% (always room for error) return Math.max(0.02, Math.min(0.98, probability)) } /** * Select whether student uses help. * Based on profile's helpUsageProbabilities [P(no help), P(help)]. */ private selectHelpUsage(): boolean { const [pNoHelp] = this.profile.helpUsageProbabilities return this.rng.next() >= pNoHelp } /** * Calculate response time based on skill exposure and other factors. */ private calculateResponseTime(skillIds: string[], hadHelp: boolean, isCorrect: boolean): number { const base = this.profile.baseResponseTimeMs const variance = this.profile.responseTimeVariance // Higher exposure = faster response (more automatic) let avgExposure = 0 if (skillIds.length > 0) { avgExposure = skillIds.reduce((sum, id) => sum + (this.skillExposures.get(id) ?? 0), 0) / skillIds.length } // Normalize exposure effect: 0 exposures → 2x time, many exposures → 1x time const exposureFactor = 2.0 - Math.min(1.0, avgExposure / (this.profile.halfMaxExposure * 2)) // Help usage adds time (reading hints, etc.) const helpFactor = hadHelp ? 1.25 : 1.0 // Incorrect answers: sometimes faster (gave up), sometimes slower (struggled) const correctnessFactor = isCorrect ? 1.0 : this.rng.chance(0.5) ? 0.7 : 1.4 // Add randomness const randomFactor = 1.0 + (this.rng.next() - 0.5) * variance return Math.round(base * exposureFactor * helpFactor * correctnessFactor * randomFactor) } /** * Get the current exposure count for a skill. */ getExposure(skillId: string): number { return this.skillExposures.get(skillId) ?? 0 } /** * Get all exposure counts. */ getAllExposures(): Map<string, number> { return new Map(this.skillExposures) } /** * Get the computed P(correct) for a skill based on current exposure. * This is the "ground truth" that BKT is trying to estimate. * * Uses skill-specific difficulty multiplier: * - Ten-complements (multiplier ~1.8) need ~80% more exposures than baseline * - Five-complements (multiplier ~1.2) need ~20% more exposures than baseline * - Basic skills (multiplier ~0.8-0.9) need fewer exposures */ getTrueProbability(skillId: string): number { const exposure = this.skillExposures.get(skillId) ?? 0 // Apply skill-specific difficulty multiplier to K const effectiveK = this.profile.halfMaxExposure * getSkillDifficultyMultiplier(skillId) return this.hillFunction(exposure, effectiveK, this.profile.hillCoefficient) } /** * Get all true probabilities for skills with any exposure. */ getAllTrueProbabilities(): Map<string, number> { const result = new Map<string, number>() for (const [skillId] of this.skillExposures) { result.set(skillId, this.getTrueProbability(skillId)) } return result } /** * Get the student's profile. */ getProfile(): StudentProfile { return this.profile } /** * Ensure a skill is tracked (initialize to 0 exposures if not already tracked). * Useful when journey includes skills not pre-seeded in the profile. */ ensureSkillTracked(skillId: string): void { if (!this.skillExposures.has(skillId)) { this.skillExposures.set(skillId, 0) } } /** * Ensure all skills in a list are tracked. */ ensureSkillsTracked(skillIds: string[]): void { for (const skillId of skillIds) { this.ensureSkillTracked(skillId) } } /** * Assess a single skill WITHOUT learning (no exposure increment). * * This simulates giving the student a test on just this skill. * Used to measure true mastery after practice sessions. * * @param skillId - The skill to assess * @param trials - Number of assessment trials (default 20) * @returns Assessment result with accuracy and probability */ assessSkill(skillId: string, trials: number = 20): SkillAssessment { const trueProbability = this.getTrueProbability(skillId) const exposure = this.getExposure(skillId) // Run trials WITHOUT incrementing exposure let correct = 0 for (let i = 0; i < trials; i++) { if (this.rng.chance(trueProbability)) { correct++ } } return { skillId, exposure, trueProbability, assessedAccuracy: correct / trials, trials, correct, } } /** * Assess multiple skills WITHOUT learning. * * @param skillIds - Skills to assess * @param trialsPerSkill - Number of trials per skill (default 20) * @returns Map of skill ID to assessment result */ assessSkills(skillIds: string[], trialsPerSkill: number = 20): Map<string, SkillAssessment> { const results = new Map<string, SkillAssessment>() for (const skillId of skillIds) { results.set(skillId, this.assessSkill(skillId, trialsPerSkill)) } return results } } /** * Result of assessing a single skill. */ export interface SkillAssessment { skillId: string /** Number of exposures during practice */ exposure: number /** True P(correct) from Hill function */ trueProbability: number /** Measured accuracy from assessment trials */ assessedAccuracy: number /** Number of assessment trials */ trials: number /** Number correct in assessment */ correct: number } |