What BKT does in Interlaza

In short

A child answers correctly five times running, choosing between two options each time. Five independent, equally likely correct guesses landing in a row happen about once in every thirty-two such five-answer blocks — that measures how surprising the streak is, not the odds that this particular child was guessing, which BKT works out from the whole history instead.
The app keeps one number per target that answers that question. It updates on an unaided, independent answer — never on a no-feedback check, a correction trial, guided-practice mode, or a maintenance review, which measure something else.
Four things shape the calculation: what the child already knew, what they learn between one opportunity and the next, the chance of being right by luck, and the chance of slipping when they knew it.
Prompted trials stay in the record, but they do not move that number: being right with help does not say the same thing as being right alone.
The threshold for calling something mastered is set by the program. Crossing it means a criterion was reached under those conditions, not that every form of learning is proven — and the Instructor can inspect it and override it.

Two learners can both finish today at 80% accuracy. One arrived after several strong sessions and made two late slips. The other is on a first exposure, where some of those correct answers may simply have been lucky. A session percentage treats them alike; their histories do not.

Bayesian Knowledge Tracing (BKT) gives Interlaza a compact way to carry that history forward.

The four parts of the estimate

For each target, the model starts with a prior: an initial probability before the next eligible response. It then updates that probability using:

learn: the modeled chance that knowledge changes between opportunities;
guess: the chance of a correct response without the modeled knowledge;
slip: the chance of an incorrect response despite it.

Interlaza varies defaults by task type and adjusts guessing to the number of comparisons. A two-choice identity task and an equivalence probe — a few trials with no help and no feedback, on relations that were never trained — should not begin from identical assumptions. The service does carry a default for that probe type, with its learn rate set to zero since a no-feedback check cannot teach — but in practice the engine never sends a probe trial to BKT at all, so that default sits unused rather than steering an estimate.

The worked example below is illustrative, not a claim about any real child: it shows how one independent correct or incorrect answer moves the estimate under one task’s own parameters.

Figure — One concept, nine trials, one estimate

independent correct error correct with a prompt

Vertical scale, read outside the chart: 0 to 1 (P(mastery)). This example starts at 0.10 and its criterion line sits at 0.95.

The three moments that explain the curve

The error: 0.62 → 0.20. In this example one error subtracts more than one hit adds, because the model assumes slips are rare (5%) for this task, so a miss is informative — that ratio is a property of these parameters, not a general rule about errors and hits. It does not collapse to zero, and two independent hits later the estimate is already back above where the run began.
The prompted trial: the line holds still. A correct answer with a prompt is evidence that the prompt works, not that the child knows the concept — so the engine records it and scores nothing. The estimate only moves on what the child does alone.
The threshold: crossing 0.95. This worked example uses 0.95 as its configured criterion. Crossing it supports the program decision shown here; it does not prove generalization, transfer or maintenance.

Computed with the engine's own update rule and its published identity-matching parameters (prior 0.10, learn 0.10, guess 0.33 with three choices on screen, slip 0.05). Not a sketch of the idea — the same arithmetic the app runs.

Bayesian Knowledge Tracing across one worked example: each eligible independent answer moves the estimate, an error moves it down without erasing the history, and a prompted trial moves it nowhere. The 0.95 line is the criterion configured for this example, not a universal definition of learning.

Which responses move the model

The trial record can contain independent, prompted, incorrect and no-response outcomes. Teaching information should remain visible, but the estimate only moves on an independent response — a prompted correct one is kept as teaching evidence, never scored as if the child produced it alone.

That separation lets an Instructor see both the success of the teaching procedure and the evidence relevant to the configured criterion.

Four kinds of trial never reach the model at all: probes, correction trials, anything run in guided-practice mode, and a maintenance review re-presenting material the child already mastered. A probe is meant to test an untrained or unsupported relation, and it would defeat the point if the model updated from one just because it produced a response.

How the estimate changes practice

Interlaza combines the estimate with recent streaks and error patterns. When evidence is strong and stable, the engine can increase the number of comparisons. When it is weak or errors repeat, it can reduce the array or preserve more support.

That is pacing, not diagnosis. The same estimate can be affected by poor stimulus design, motor access, motivation or a prompt rule that does not fit the learner. The Instructor can inspect those conditions and override the automated choice.

Why the threshold is not “learned”

A program can define the probability threshold at which its criterion is considered reached. The label should be read literally: this model, for this target, under these evidence rules, crossed the configured threshold.

Generalization to new examples, transfer to another performance and maintenance after time require their own observations. BKT helps decide when to ask those questions; it does not answer them in advance.

What makes it useful

BKT removes two common sources of waste: overreacting to one mistake and overcelebrating one short streak. It gives the next session a memory.

What this buys you is not a magical “learning score.” It is a pacing system whose inputs and limits an Instructor can examine.