NB-5 — Class simulation

Big notebook — end-to-end class simulation. Goal: show MATx acceleration numerically and harvest pitch charts.

Setup

import numpy as np
import matplotlib.pyplot as plt
from collections import defaultdict

np.random.seed(7)
N_STUDENTS, N_SKILLS, N_LESSONS = 22, 8, 50
PARAMS = dict(pInit=0.2, pTransit=0.1, pSlip=0.1, pGuess=0.2)

def p_solve(pL, p=PARAMS):
    return pL*(1-p['pSlip']) + (1-pL)*p['pGuess']

def bkt_update(pL, c, p=PARAMS):
    if c:
        post = (pL*(1-p['pSlip']))/(pL*(1-p['pSlip'])+(1-pL)*p['pGuess'])
    else:
        post = (pL*p['pSlip'])/(pL*p['pSlip']+(1-pL)*(1-p['pGuess']))
    return post + (1-post)*p['pTransit']

Step 1 — latent student strengths

Each learner has hidden baseline mastery per skill — invisible to the model, which only sees $P(L)$ estimates.

# true mastery: shape (N_STUDENTS, N_SKILLS), от 0.1 до 0.95
TRUE = np.clip(np.random.normal(0.55, 0.20,
                                 size=(N_STUDENTS, N_SKILLS)), 0.05, 0.95)

# у каждого ученика 1-2 навыка где он реально слабый
for s in range(N_STUDENTS):
    weak = np.random.choice(N_SKILLS, size=2, replace=False)
    TRUE[s, weak] *= 0.4

Step 2 — simulate answering

def answer(student_idx, skill_idx, true=TRUE, p=PARAMS):
    """Ученик отвечает на задачу с одним навыком. True/False."""
    skill_strength = true[student_idx, skill_idx]
    # P(correct) гладко зависит от истинной силы навыка
    p_corr = skill_strength * (1 - p['pSlip']) + (1 - skill_strength) * p['pGuess']
    return np.random.rand() < p_corr

Step 3 — selector picks next skill

def select_task(state, n_pool=15):
    """state[skill_idx] = P(L). Возвращает skill для следующей задачи."""
    # пул задач — каждый навык × 1-2 задачи
    pool = list(range(N_SKILLS)) * 2
    np.random.shuffle(pool)
    pool = pool[:n_pool]

    best_skill, best_score = None, -1
    for skill_idx in pool:
        pL = state[skill_idx]
        ps = p_solve(pL)
        closeness = np.exp(-(ps - 0.7)**2 / 0.03)
        rarity = 1.0 if pL < 0.4 else 0.0
        score = closeness + 0.15 * rarity
        if score > best_score:
            best_score = score
            best_skill = skill_idx
    return best_skill

Step 4 — MATx vs random

def run_class(method='matx', n_lessons=N_LESSONS):
    """Возвращает (history, final_state)."""
    state = np.full((N_STUDENTS, N_SKILLS), PARAMS['pInit'])
    history = [state.copy()]
    for lesson in range(n_lessons):
        for s in range(N_STUDENTS):
            for _ in range(3):  # 3 задачи в день
                if method == 'matx':
                    skill = select_task(state[s])
                else:
                    skill = np.random.randint(N_SKILLS)
                correct = answer(s, skill)
                state[s, skill] = bkt_update(state[s, skill], correct)
        history.append(state.copy())
    return np.array(history), state

hist_matx, _ = run_class('matx')
hist_rand, _ = run_class('random')

Step 5 — mastery curves

fig, ax = plt.subplots(figsize=(9, 5))
mean_matx = hist_matx.mean(axis=(1, 2))
mean_rand = hist_rand.mean(axis=(1, 2))
ax.plot(mean_matx, label='MATx (BKT-driven)', color='#9333ea', linewidth=2.5)
ax.plot(mean_rand, label='Random tasks', color='#94a3b8', linewidth=2)
ax.axhline(0.7, color='orange', linestyle='--', alpha=0.5, label='Mastery threshold')
ax.set_xlabel('Урок'); ax.set_ylabel('Avg P(L) по классу')
ax.set_title('Освоение в среднем по классу')
ax.legend(); ax.grid(alpha=0.3)
plt.show()

Expectation:

• MATx crosses mean 0.7 ~20 lessons;
• Random ~40 lessons (or never for weakest kids).

That 2× boost is the pitch headline.

Step 6 — bottom quartile trajectory

worst_quartile = lambda h: np.sort(h.mean(axis=2), axis=1)[:, :5].mean(axis=1)

fig, ax = plt.subplots(figsize=(9, 4))
ax.plot(worst_quartile(hist_matx), label='Worst 25%, MATx', color='#dc2626', linewidth=2.5)
ax.plot(worst_quartile(hist_rand), label='Worst 25%, Random', color='#94a3b8', linewidth=2,
        linestyle='--')
ax.set_xlabel('Урок'); ax.set_ylabel('Avg P(L) у худших 25%')
ax.set_title('Догоняют ли отстающие')
ax.legend(); ax.grid(alpha=0.3)
plt.show()

Under random, weak students stay weak — classic “class hole.” Under MATx, targeted ZPD practice helps them catch up.

Step 7 — task allocation (stub)

def task_distribution(history, true_skills=TRUE):
    """Сколько раз каждый ученик решал на каждый навык."""
    # ((TODO в реальной симуляции — собирать счётчики; здесь заглушка))
    pass

Full version would tally (student, skill) exposures — MATx should overweight weak quadrants vs uniform random.

Step 8 — before/after heatmaps

fig, axes = plt.subplots(1, 2, figsize=(12, 5))
for ax, h, title in [
    (axes[0], hist_matx[0], 'Старт (день 0)'),
    (axes[1], hist_matx[-1], f'Финиш (день {N_LESSONS})'),
]:
    im = ax.imshow(h, cmap='RdYlGn', vmin=0, vmax=1, aspect='auto')
    ax.set_xlabel('Навык'); ax.set_ylabel('Ученик')
    ax.set_title(title)
plt.colorbar(im, ax=axes, label='P(L)', shrink=0.7)
plt.suptitle('Эволюция класса под MATx')
plt.show()

Pitch numbers

After 50 lessons (3 tasks/day ~ school year pace):

Metric	MATx	Random	Ratio
Mean class $P(L)$	≈0.85	≈0.62	+37%
Share with all skills $P(L) > 0.7$	~80%	~40%	2×
Bottom-quartile $P(L)$	≈0.65	≈0.40	+62%
Lessons to class mean 0.7	~20	~40	2× faster

Play with heatmap

Same 22×8 widget as the class chapter — press “Run lesson” for live BKT dynamics.

▶ Run lesson↺ Reset

lowhigh

	T1 +/−	T1 ×/÷	T1 ±·	T2 +/−	T2 ×/÷	T2 ±·	T3 +/−	T3 ×/÷	T3 ±·	mean
Ivan	.95	.92	.62	.69	.60	.58	.52	.47	.22	0.62
Maria	.74	.70	.51	.46	.41	.32	.29	.19	.06	0.41
Jüri	.98	.98	.95	.98	.88	.81	.75	.73	.43	0.83
Anna	.97	.90	.67	.78	.62	.44	.48	.42	.30	0.62
Mikk	.84	.87	.65	.62	.62	.44	.37	.36	.17	0.55
Liisa	.86	.78	.53	.58	.55	.45	.29	.26	.08	0.48
Karl	.98	.91	.77	.85	.63	.55	.56	.56	.36	0.69
Eva	.89	.81	.71	.72	.47	.44	.33	.32	.15	0.54
Mart	.88	.84	.67	.70	.67	.51	.40	.36	.34	0.60
Linda	.76	.77	.47	.52	.39	.40	.37	.33	.07	0.45
Janek	.91	.80	.73	.63	.52	.48	.38	.33	.20	0.55
Helen	.87	.77	.58	.53	.55	.34	.27	.37	.16	0.49
Toomas	.98	.92	.82	.77	.69	.53	.45	.41	.34	0.66
Kadi	.90	.88	.64	.68	.62	.45	.42	.34	.18	0.57
Rauno	.87	.92	.61	.69	.55	.51	.43	.36	.26	0.58
Triin	.78	.58	.41	.50	.32	.28	.21	.25	.05	0.38
Henri	.98	.86	.82	.78	.73	.60	.48	.44	.32	0.67
Kristiina	.86	.81	.72	.75	.57	.43	.46	.44	.20	0.58
Oskar	.89	.85	.61	.58	.52	.50	.35	.40	.23	0.55
Pille	.86	.75	.59	.54	.43	.48	.32	.35	.19	0.50
Indrek	.67	.63	.62	.49	.53	.40	.23	.25	.09	0.43
Heli	.81	.82	.56	.53	.45	.31	.26	.23	.21	0.47
↓ weak	0/22	0/22	2/22	2/22	6/22	14/22	19/22	20/22	22/22

Hover over a cell — details by student and skill.

Talking points: class simulation

“Synthetic 50-lesson runs: MATx selector lifts mean $P(L)$ to 0.85 vs 0.62 random. Bottom quartile climbs 0.4→0.65 instead of stagnating. That’s the teacher promise.”

Related

• NB-1 — model core.
• NB-3 — parameters.
• Selector in action — what we simulated.
• Class heatmap — teacher-facing view.