Selector in action

Algorithm at a glance

flowchart TD
  S["StudentState · P(L) vector"] --> Loop["Each task in pool"]
  Loop --> P["Joint P(solve) geometric mean over skills"]
  P --> C["Closeness — Gaussian kernel around 0.7"]
  P --> R["Rarity share of skills with P(L) below 0.4"]
  C --> Score["score = closeness + 0.15 * rarity"]
  R --> Score
  Score --> Sort[Sort by descending score]
  Sort --> Recent[Exclude tasks from last 5 attempts]
  Recent --> TopN[Top-N recommendations]

That’s the moving parts. No neural nets, no gradients, no training loop — just weighted ranking.

Step by step

1. Student state

state = {
  student_id: "S-12",
  mastery: {
    "linear_eq.expand_brackets":     0.166,  // dropped after errors
    "arith.signs":                   0.831,  // strong
    "arith.distributive_law":        0.45,
    "linear_eq.move_to_one_side":    0.62,
    "linear_eq.divide_by_coefficient": 0.78,
    // ...
  },
  history: [/* recent answers */]
}

2. Score each task’s fit

Task T-147 with skills [expand_brackets, signs]:

• $P(\text{solve}_1) = 0.166 \cdot 0.9 + 0.834 \cdot 0.2 = 0.317$
• $P(\text{solve}_2) = 0.831 \cdot 0.9 + 0.169 \cdot 0.2 = 0.782$
• $P_{\text{joint}} = \sqrt{0.317 \cdot 0.782} \approx 0.498$

Closeness: $\text{closeness} = \exp\left(-\frac{(0.498 - 0.7)^2}{0.03}\right) \approx 0.260$

Rarity (1 of 2 skills below 0.4): $\text{rarity} = 0.5$

Score: $\text{score} = 0.260 + 0.15 \cdot 0.5 = 0.335$

3. Top-N after sorting

The selector scans the pool, scores each task, sorts descending, drops tasks seen in the last five attempts (avoid repeats), returns top-5.

Real code

export function recommend(
  state: StudentState,
  pool: Task[],
  topN = 5,
  opts?: SelectorOptions,
  params?: BktParams
): ScoredTask[] {
  const recentIds = new Set(
    state.history.slice(-5).map((h) => h.task_id)
  );
  const scored = pool
    .filter((t) => !recentIds.has(t.id))
    .map((t) => scoreTaskForStudent(state, t, opts, params));
  scored.sort((a, b) => b.score - a.score);
  return scored.slice(0, topN);
}

When every task is a bad fit

Suppose Ivan’s $P(L)$ for parentheses is 0.166 and the pool lacks “parentheses + gentle arithmetic” items — everything bundles harder skills. Then what?

The selector still returns the least bad option — but top pick shows low closeness. That’s a signal:

• to the teacher: “item bank doesn’t cover real class gaps”;
• to Andri (content lead): “need easier drills on this skill.”

Teacher UI can surface “weak recommendation” when closeness < 0.3 — part of explainability.

Tie-breaks and rare-skill bonus

When closeness ties, rareSkillBonus favors tasks training weak skills. Why it matters:

• without bonus the selector may stall in comfort zone;
• with bonus it explores weaknesses for remediation.

rareSkillBonus = 0.15 balances:

• too small (0.05): never pushes into weak skills — stagnation;
• too large (0.5): always brutal difficulty — frustration.

0.15 roughly says: “when closeness ties, prefer tasks mixing one weak skill among two.”

When the selector looks wrong

When smoke-testing weird tops, reasons usually boil down to:

1. Mastery vector uninitialized — missing skill defaults to params.pInit = 0.2. Fine for cold start.
2. Stale history — two-week gap; mastery should decay. Out of MVP scope.
3. Taxonomy imbalance — Andri tagged skill A on 80% of items, skill B on 5%; selector skews to A. Fix via tagging policy.

See explainability — surfacing reasons simplifies debugging and earns teacher trust.