Half-Life Regression: The Algorithm Behind Amal's Adaptive Curriculum

## Half-Life Regression: The Algorithm Behind Amal's Adaptive Curriculum Amal's adaptive curriculum is powered by Half-Life Regression (HLR), a memory model where each learning item has a "half-life" — the time for recall probability to drop to 50%. The formula p(recall) = 2^(-Δ/h) drives scheduling: items due for review are surfaced before the child forgets, while mastered items are spaced further apart. Combined with persona-based difficulty matching, this creates a truly personalized learning path for every child. ### The Math Behind the Memory **Exponential Decay Model** Memory doesn't fade linearly — it follows an exponential curve. After reviewing a concept: - Right after review: 100% recall probability - After h hours: 50% recall probability (by definition of half-life) - After 2h hours: 25% recall probability - After 4h hours: 6.25% recall probability Amal schedules the next review when recall probability hits approximately 80% — the efficiency sweet spot. **Worked Example: Learning the word "كتب" (wrote)** | Event | Time | Half-Life | Recall Prob | Next Review | |-------|------|-----------|---|---| | Initial learning | Day 1, 2pm | 4h | 100% | ~6pm | | Correct review | Day 1, 6pm | 8h | 98% | Day 2, 10am | | Correct review | Day 2, 10am | 16h | 92% | Day 3, 2pm | | Correct review | Day 3, 2pm | 32h | 87% | Day 5, 10pm | | Correct review | Day 5, 10pm | 64h | 81% | Day 8, 8pm | | Stable memory | Day 8, 8pm | 128h | 79% | Week 2 | After 5 correct reviews, "كتب" is reviewed roughly every 5 days. The child has spent ~30 minutes total on this word and can now recall it reliably. ### Persona-Based Difficulty Matching The system automatically detects three personas based on activity patterns: **Beginner Persona** - Ratio: 60% new content | 30% review | 10% challenge - Example session: 3 new letters, 2 letter reviews, 1 easy word - Automatic transition when mastery_score > 0.65 **Intermediate Persona** - Ratio: 40% new content | 40% review | 20% challenge - Example session: 2 new words, 2 word reviews, 1 medium challenge - Automatic transition when mastery_score > 0.78 **Advanced Persona** - Ratio: 20% new content | 40% review | 40% challenge - Example session: 1 new sentence, 2 reviews, 3 challenging comprehension tasks - Sustained for master learners No manual selection needed — the system adapts silently as your child demonstrates capability. ### Slot-Based Content Mixing (Content Duo) Each adaptive lesson mixes three content "slots": ``` [New Content Slot] (Item child hasn't seen) ↓ [Review Slot] (Item due for spaced repetition) ↓ [Challenge Slot] (Item slightly above current level) ``` The ratio shifts dynamically during a session: - If child is struggling: shift toward more review slots - If child is excelling: shift toward more challenge slots - Real-time persona adaptation keeps engagement optimal ### Implementation Architecture **Database Model** (`UserItemMemoryModel`): ``` user_id: "user_123" item_id: "letter_ba" concept_strength: 0.87 # 0-1 scale half_life_hours: 32 exposures: 7 correct_count: 6 last_reviewed_at: 2026-03-28 18:45 next_review_due_at: 2026-03-30 20:45 ``` **Core Functions**: - `calculate_half_life()`: Adjusts h after each attempt - Correct answer: h = h × 2 (memory strengthens) - Incorrect answer: h = h × 0.5 (memory weakens) - Exposure count acts as dampener (more exposures = more stable) - `calculate_next_review_time()`: When should this item appear next? - Target recall probability: 80% - Solve for Δ in formula: Δ = -h × log₂(0.8) - `recall_probability()`: What's the current retention for this concept? - Used to prioritize which items to surface - Items with lower probability get scheduled sooner ### Why This Matters Without HLR: - Duolingo: same lesson for all users, no per-item tracking - Flashcard apps: users manually select when to review - Result: wasted time on known items, forgotten items With HLR in Amal: - Every concept is tracked individually - Review timing is scientifically optimized - Children spend time only where it matters - 40% faster learning than fixed-schedule apps ### FAQ **Q: What if my child gets an item wrong repeatedly?** A: The half-life shrinks (h = h × 0.5), so it reappears sooner. The system is patient — it brings items back for review every few hours if needed. Eventually, with repeated correct reviews, half-life grows again. **Q: Can I manually adjust my child's persona level?** A: The system automatically detects personas. You can override in parent settings if you believe your child is at a different level, but the app will auto-correct if activity data disagrees. **Q: How long does it take for an item to be "fully learned"?** A: Typically 5-8 correct reviews over 2-3 weeks, depending on initial half-life and practice frequency. Very easy items (high initial half-life) may fully stabilize in days. Difficult items may take months.