P(solve) and the zone of proximal development

Given $P(L)$ per micro-skill we can predict the probability a student solves a specific problem right now — and pick one in the sweet spot.

The $P(\text{solve})$ formula

For a task requiring one skill:

$$P(\text{solve}) = P(L) \cdot (1 - P(S)) + (1 - P(L)) \cdot P(G)$$

“Knows and doesn’t slip” plus “doesn’t know but guesses.”

Numerically:

$P(L)$	$P(\text{solve})$
0.0	0.20
0.2	0.34
0.5	0.55
0.7	0.69
0.9	0.83
1.0	0.90

So P(solve) always sits between $P(G) = 0.20$ and $1 - P(S) = 0.90$ — never 0 (guess chance) and never 1 (slip chance).

What is ZPD

Lev Vygotsky, Soviet psychologist of the 1930s, crystallized the guiding idea:

Students grow fastest on tasks just beyond current ability — not what they already ace (no growth), not what’s far away (frustration, dropout).

That’s the Zone of Proximal Development (ZPD).

Quantitatively: “just beyond” in studies often maps to $P(\text{solve}) \approx 0.7$ — tough enough to stretch, not hopeless.

Applying it in the selector

Task selection algorithm:

1. For each candidate problem compute the student’s $P(\text{solve})$.
2. Pick the task minimizing $|P(\text{solve}) - 0.7|$.
3. Recommend it.

In code we use a Gaussian “distance to target”:

$$\text{closeness}(p) = \exp\left(-\frac{(p - 0.7)^2}{0.03}\right)$$

  // Closeness to target — Gaussian-ish, peaks at target=0.7
  const closeness = Math.exp(-Math.pow(pSolveJoint - target, 2) / 0.03);

Closer $P(\text{solve})$ is to 0.7, higher closeness; peak exactly at 0.7.

show |p − 0.7|

Gaussian: smooth, symmetric, penalises extremes fast. Absolute difference is sharp and lumps «almost 0.7» with «way off».

Gaussian vs absolute difference

Gaussian:

• symmetric around 0.7;
• punishes far-off tasks quickly;
• smooth — no kinks, stable selector behavior.

Denominator 0.03 controls width:

• 0.03 ⇒ tasks roughly $[0.55, 0.85]$ still get closeness > 0.5;
• 0.01 — too tight (“exactly 0.7 ± 0.05”);
• 0.10 — too loose (“anything between 0.4 and 1.0 feels fine”).

Tune if needed; 0.03 works well in practice.

What if ZPD pushes toward a strong skill only

Teacher scenario: Ivan’s arithmetic sits at 0.85, parentheses at 0.40. Which task?

Naïvely chasing $P(\text{solve}) \approx 0.7$ might favor arithmetic-heavy items (combined with other micro-skills) and never train parentheses.

So the selector adds a rare-skill bonus:

  // Rarity bonus: how many of this task's skills are below 0.4?
  const undertrained = task.microskills.filter(
    (s) => (state.mastery[s] ?? params.pInit) < 0.4
  ).length;
  const rarity = undertrained / task.microskills.length;

  const score = closeness + rareBonus * rarity;

Translation: if many required skills have $P(L) < 0.4$, bump the task slightly in ranking — exploration so we don’t stall in comfort zone.

Try it: ZPD curve

Drag target and σ². Watch how the ZPD “window” narrows — tighter σ² ⇒ pickier selector around target.

target0.70σ² (width)0.030

closeness = exp(−(p−target)²/σ²). Highest at the peak, drops fast to the sides. The smaller σ², the narrower the «ZPD window».

What the selector ends up choosing

Back to Ivan after six tasks (previous chapter):

• $P(L) = 0.166$ for parentheses.
• Single-skill $P(\text{solve}) = 0.166 \cdot 0.9 + 0.834 \cdot 0.2 = 0.317$.

Too hard — closeness ≈ 0; selector won’t lead with that.

Better:

• mix two skills (parentheses + familiar arithmetic);
• joint $P(\text{solve})$ lands ~0.55–0.65 — nearer ZPD;
• trains the weak spot without total frustration.

See multi-skill tasks.