🧠 Adaptive Learning Engine — Knowledge Tracing Live

BKT assumes a student can learn from the act of attempting a question regardless of whether they got it right — from feedback, from working through the problem, or simply from repeated exposure. After computing the Bayesian posterior from the observed correctness, the model applies one more step: P(L_next) = P(L_post) + (1−P(L_post))·T, where T is the learning-rate (transition) probability that an unmastered student crosses into mastery on this attempt. Critically, BKT assumes learning is one-directional — mastery is never assumed to be forgotten — so T only ever pushes the estimate upward. This is exactly what the learning-rate slider in this simulation controls, and you can watch higher settings pull mastery estimates toward 100% much faster.

The "Weakest skill first" policy simply asks a question from whichever skill currently has the lowest P(L) estimate, concentrating practice where the model believes the student needs it most — this is the core idea behind mastery-based adaptive systems like Khan Academy's practice engine and ALEKS. "Weighted by uncertainty" does the same thing probabilistically, sampling skills with a chance proportional to (1 − P(L)), so weak skills are favoured but not deterministically forced. "Round robin" cycles through skills in a fixed order regardless of mastery, and "Random" picks uniformly at random as a baseline. Switching between these policies while watching the mastery bar chart makes the efficiency gain from adaptivity directly visible: weakest-first reaches full mastery across all five skills in noticeably fewer questions than the random baseline.

This is exactly what the guess and slip parameters model, and it is why BKT tracks a probability rather than a single observed fact. A student with true mastery state "not mastered" can still answer correctly with probability g (a lucky guess, or partial knowledge that happens to work this time) — the simulation actually flips this coin every time an unmastered simulated student is asked a question. Symmetrically, a truly mastered student answers incorrectly with probability s (a momentary slip, misreading the question, or a typo). Because both of these are simulated as genuine random draws — not scripted outcomes — you will see occasional lucky correct answers on weak skills and occasional unlucky misses on skills the model already estimates as highly mastered, exactly as would happen with a real student.

BKT's assumptions are strong simplifications: it treats each skill's mastery as fully binary and independent of every other skill, ignores partial credit and response time, and assumes a fixed guess/slip rate rather than one that varies by question difficulty or student. Modern research systems such as Deep Knowledge Tracing (DKT, using recurrent neural networks) and Bayesian extensions with per-item difficulty parameters (closer to Item Response Theory) relax these assumptions and often fit real student data more accurately, at the cost of being far less interpretable — you cannot read off a single transparent probability that the student has mastered fractions from a neural network's hidden state the way you can from BKT. That interpretability is precisely why BKT, despite being three decades old, remains widely deployed in real production tutoring systems today.