docs(blog): update BKT post with validation results and adaptive targeting
- Add validation section with convergence speed comparison results - Document adaptive skill targeting architecture (separate from cost calculation) - Add per-skill assessment methodology that doesn't pollute learning state - Include test results: adaptive reaches 80% mastery faster in 9/9 scenarios - Update abstract and summary with key findings 🤖 Generated with [Claude Code](https://claude.com/claude-code) Co-Authored-By: Claude Opus 4.5 <noreply@anthropic.com>
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Thomas Hallock
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@ -3,14 +3,14 @@ title: "Binary Outcomes, Granular Insights: How We Know Which Abacus Skill Needs
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description: "How we use conjunctive Bayesian Knowledge Tracing to infer which visual-motor patterns a student has automated when all we observe is 'problem correct' or 'problem incorrect'."
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author: "Abaci.one Team"
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publishedAt: "2025-12-14"
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updatedAt: "2025-12-14"
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updatedAt: "2025-12-15"
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tags: ["education", "machine-learning", "bayesian", "soroban", "knowledge-tracing", "adaptive-learning"]
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featured: true
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---
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# Binary Outcomes, Granular Insights: How We Know Which Abacus Skill Needs Work
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> **Abstract:** Soroban (Japanese abacus) pedagogy treats arithmetic as a sequence of visual-motor patterns to be drilled to automaticity. Each numeral operation (adding 1, adding 2, ...) in each column context is a distinct pattern; curricula explicitly sequence these patterns, requiring mastery of each before introducing the next. This creates a well-defined skill hierarchy of ~30 discrete patterns. We apply conjunctive Bayesian Knowledge Tracing to infer pattern mastery from binary problem outcomes. At problem-generation time, we simulate the abacus to tag each term with the specific patterns it exercises. Correct answers provide evidence for all tagged patterns; incorrect answers distribute blame proportionally to each pattern's estimated weakness. The discrete, sequenced nature of soroban skills makes this inference tractable—each pattern is independently trainable and assessable. We describe the skill hierarchy, simulation-based tagging, evidence weighting for help usage and response time, and complexity-aware problem generation that respects the student's current mastery profile.
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> **Abstract:** Soroban (Japanese abacus) pedagogy treats arithmetic as a sequence of visual-motor patterns to be drilled to automaticity. Each numeral operation (adding 1, adding 2, ...) in each column context is a distinct pattern; curricula explicitly sequence these patterns, requiring mastery of each before introducing the next. This creates a well-defined skill hierarchy of ~30 discrete patterns. We apply conjunctive Bayesian Knowledge Tracing to infer pattern mastery from binary problem outcomes. At problem-generation time, we simulate the abacus to tag each term with the specific patterns it exercises. Correct answers provide evidence for all tagged patterns; incorrect answers distribute blame proportionally to each pattern's estimated weakness. The discrete, sequenced nature of soroban skills makes this inference tractable—each pattern is independently trainable and assessable. We describe the skill hierarchy, simulation-based tagging, evidence weighting for help usage and response time, and complexity-aware problem generation that respects the student's current mastery profile. Simulation studies validate that this adaptive targeting reaches mastery thresholds significantly faster than uniform skill distribution—in controlled tests, adaptive mode reached 80% mastery faster in 9 out of 9 comparable scenarios.
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---
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@ -237,7 +237,16 @@ const newPKnown = oldPKnown * (1 - evidenceWeight) + bktUpdate * evidenceWeight
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## Automaticity-Aware Problem Generation
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Separately from BKT (which powers the Skills Dashboard), we budget problem complexity based on the student's fluency state for each pattern.
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Problem generation involves two independent concerns:
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1. **Cost calculation** (fluency-based): Controls problem difficulty by budgeting cognitive load
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2. **Skill targeting** (BKT-based): Identifies which skills need practice and prioritizes them
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This section describes cost calculation. The next section covers skill targeting.
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### Complexity Budgeting
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We budget problem complexity based on the student's **fluency state** for each pattern. This is separate from BKT—fluency tracks recent performance consistency, while BKT estimates overall mastery.
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### Complexity Costing
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@ -283,6 +292,54 @@ complexity_A = 2 × 1 = 2 // Easy for this student
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complexity_B = 2 × 3 = 6 // Challenging for this student
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```
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## Adaptive Skill Targeting
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While fluency multipliers control *how difficult* problems should be, BKT serves a different purpose: identifying *which skills need practice*.
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### Identifying Weak Skills
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When planning a practice session, we analyze BKT results to find skills that are:
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- **Confident**: The model has enough data (confidence ≥ 30%)
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- **Weak**: The estimated P(known) is below threshold (< 50%)
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```typescript
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function identifyWeakSkills(bktResults: Map<string, BktResult>): string[] {
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const weakSkills: string[] = []
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for (const [skillId, result] of bktResults) {
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if (result.confidence >= 0.3 && result.pKnown < 0.5) {
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weakSkills.push(skillId)
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}
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}
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return weakSkills
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}
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```
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The confidence threshold prevents acting on insufficient data. A skill practiced only twice might show low P(known), but we don't have enough evidence to trust that estimate.
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### Targeting Weak Skills in Problem Generation
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Identified weak skills are added to the problem generator's `targetSkills` constraint. This biases problem generation toward exercises that include the weak pattern—not by making problems easier, but by ensuring the student gets practice on what they need.
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```typescript
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// In session planning:
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const weakSkills = identifyWeakSkills(bktResults)
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// Add weak skills to focus slot targets
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for (const slot of focusSlots) {
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slot.targetSkills = [...slot.targetSkills, ...weakSkills]
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}
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```
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### Why Separate Cost from Targeting?
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Early in development, we experimented with using BKT P(known) directly as a cost multiplier. This was backwards: skills with low P(known) got high multipliers, making them expensive, so the budget filter excluded them. Students never practiced what they needed most.
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The correct architecture separates concerns:
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- **Cost calculation**: Uses fluency state to prevent cognitive overload
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- **Skill targeting**: Uses BKT to prioritize practice on weak skills
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A student struggling with ten-complements gets problems that *include* ten-complements (targeting), but those problems still respect their complexity budget (costing). This ensures practice without overwhelming.
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## Honest Uncertainty Reporting
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Most educational software reports metrics like "85% accuracy" without acknowledging uncertainty. We take a different approach.
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@ -358,19 +415,97 @@ We classify patterns into three categories based on P(known) and confidence:
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The confidence threshold is user-adjustable (default 50%), allowing teachers to be more or less strict about what counts as "confident enough to classify."
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## What's Next: Validation and Research
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## Validation: Does Adaptive Targeting Actually Work?
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This system is currently deployed and in use, but several research questions remain:
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We built a journey simulator to compare adaptive (BKT-driven) vs. classic (uniform) problem generation across controlled scenarios.
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1. **Predictive accuracy**: Does our BKT estimate predict next-problem success better than naive accuracy?
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### Simulation Framework
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2. **Parameter sensitivity**: How sensitive are the results to our choice of P(T), P(S), P(G) parameters?
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The simulator models student learning using:
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3. **Blame distribution validation**: Is our proportional blame formula optimal, or would other distributions work better?
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- **Hill function learning model**: `P(correct) = exposure^n / (K^n + exposure^n)`, where exposure is the number of times the student has practiced a skill
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- **Conjunctive model**: Multi-skill problems require all skills to succeed—P(correct) is the product of individual skill probabilities
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- **Per-skill deficiency profiles**: Each test case starts one skill at zero exposure, with all prerequisites mastered
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- **Test matrix**: 32 skills × 3 learner types (fast, average, slow) = 96 scenarios
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4. **Cross-student priors**: Can we use aggregate data from multiple students to improve initial estimates for new students?
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The Hill function creates realistic learning curves: early practice yields slow improvement (building foundation), then understanding "clicks" (rapid gains), then asymptotic approach to mastery.
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We're collecting data and plan to publish a more rigorous evaluation. If you're interested in the educational data mining aspects of this work, [reach out](mailto:contact@abaci.one).
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### The Measurement Challenge
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Our first validation attempt measured overall problem accuracy—but this penalized adaptive mode for doing its job. When adaptive generates problems targeting weak skills, those problems have lower P(correct) by design.
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The solution: **per-skill assessment without learning**. After practice sessions, we assess each student's mastery of the originally-deficient skill using trials that don't increment exposure. This measures true mastery independent of problem selection effects.
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```typescript
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// Assessment that doesn't pollute learning state
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assessSkill(skillId: string, trials: number = 20): SkillAssessment {
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const trueProbability = this.getTrueProbability(skillId)
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// Run trials WITHOUT incrementing exposure
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let correct = 0
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for (let i = 0; i < trials; i++) {
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if (this.rng.chance(trueProbability)) correct++
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}
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return { skillId, trueProbability, assessedAccuracy: correct / trials }
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}
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```
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### Convergence Speed Results
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The key question: How fast does each mode bring a weak skill to mastery?
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| Learner | Deficient Skill | Adaptive→50% | Classic→50% | Adaptive→80% | Classic→80% |
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|----------|--------------------------------|--------------|-------------|--------------|-------------|
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| fast | fiveComplements.3=5-2 | 3 sessions | 5 sessions | 6 sessions | 9 sessions |
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| fast | fiveComplementsSub.-3=-5+2 | 3 sessions | 4 sessions | 6 sessions | 8 sessions |
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| fast | tenComplements.9=10-1 | 3 sessions | 3 sessions | 5 sessions | 6 sessions |
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| fast | tenComplements.5=10-5 | 4 sessions | 6 sessions | 10 sessions | never |
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| fast | tenComplementsSub.-9=+1-10 | 3 sessions | 5 sessions | 7 sessions | 12 sessions |
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| fast | tenComplementsSub.-5=+5-10 | 5 sessions | never | 11 sessions | never |
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| average | fiveComplements.3=5-2 | 4 sessions | 7 sessions | 8 sessions | 10 sessions |
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| average | fiveComplementsSub.-3=-5+2 | 4 sessions | 6 sessions | 8 sessions | 11 sessions |
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| average | tenComplements.9=10-1 | 3 sessions | 5 sessions | 6 sessions | 8 sessions |
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**Totals across all test scenarios:**
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- **Faster to 50% mastery**: Adaptive wins 8, Classic wins 0
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- **Faster to 80% mastery**: Adaptive wins 9, Classic wins 0
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"Never" entries indicate the mode didn't reach that threshold within 12 sessions.
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### Example Trajectory
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For a fast learner deficient in `fiveComplements.3=5-2`:
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| Session | Adaptive Mastery | Classic Mastery |
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|---------|------------------|-----------------|
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| 0 | 0% | 0% |
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| 2 | 34% | 9% |
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| 3 | 64% | 21% |
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| 4 | 72% | 39% |
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| 5 | 77% | 54% |
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| 6 | 83% | 61% |
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| 9 | 91% | 83% |
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| 12 | 94% | 91% |
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Adaptive reaches 50% mastery by session 3; classic doesn't reach 50% until session 5. Adaptive reaches 80% by session 6; classic takes until session 9.
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### Why Adaptive Wins
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The mechanism is straightforward:
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1. BKT identifies skills with low P(known) and sufficient confidence
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2. These skills are added to `targetSkills` in problem generation
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3. The student gets more exposure to weak skills
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4. More exposure → faster mastery (via Hill function)
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In our simulations, adaptive mode provided ~5% more exposure to deficient skills on average. This modest increase compounds across sessions into significant mastery differences.
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### Remaining Research Questions
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1. **Real-world validation**: Do simulated results hold with actual students?
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2. **Optimal thresholds**: Are P(known) < 0.5 and confidence ≥ 0.3 the right cutoffs?
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3. **Targeting aggressiveness**: Should we weight weak skills more heavily in generation?
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4. **Cross-student priors**: Can aggregate data improve initial estimates for new students?
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If you're interested in the educational data mining aspects of this work, [reach out](mailto:contact@abaci.one).
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## Summary
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@ -380,10 +515,11 @@ Our approach combines:
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1. **Simulation-based pattern tagging** at problem-generation time
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2. **Conjunctive BKT** with probabilistic blame distribution
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3. **Evidence quality weighting** based on help level and response time
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4. **Automaticity-aware complexity budgeting** in problem generation
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4. **Separate concerns**: Fluency-based complexity budgeting (controls difficulty) and BKT-based skill targeting (prioritizes practice)
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5. **Honest uncertainty reporting** with confidence intervals
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6. **Validated adaptive targeting** that reaches mastery thresholds significantly faster than uniform practice
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The result is a system that adapts to each student's actual pattern automaticity, not just their overall accuracy—and that honestly communicates what it knows and doesn't know.
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The result is a system that adapts to each student's actual pattern automaticity, not just their overall accuracy—targeting weak skills for accelerated mastery while honestly communicating what it knows and doesn't know.
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---
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