Teaching Machine: Branching Programmed Instruction

Learner

Aptitude 50

Avg. frame difficulty 50

Auto-run speed 1.0×

Current Frame

Frame 1 — main line

P(correct) = 50%P(wrong) = 50%

Readouts

Frames completed

0 / 12

Total attempts

Remedial detours

Accuracy

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How it works

This models Norman Crowder's "intrinsic" (branching) programmed instruction from the 1950s–60s teaching-machine era mentioned in the article: instead of one fixed sequence of frames (Skinner's linear model), each frame's multiple-choice response routes the learner to a different next frame.

P(correct) = 1 / (1 + e^(-k·(aptitude - difficulty + practice)))
practice(frame) = γ · ln(1 + visits_to_frame)

Aptitude vs. difficulty — the logistic response model: a learner is more likely to answer correctly the further their aptitude sits above a frame's difficulty, exactly the intuition behind item-response models used in real adaptive courseware.
Practice term — every time a frame (main or remedial) is revisited, a diminishing-returns bonus is added — a discrete power-law learning curve, the same shape found in real skill-acquisition data.
Branching ON — a wrong answer detours the token down to that frame's remedial branch (a simpler restatement) before looping back to retry the main frame, as Crowder's machines did. Branching OFF reproduces Skinner's linear machine: a wrong answer just repeats the same frame in place.
Auto-run — the machine answers itself by sampling from P(correct) each step, so you can watch many learners' worth of trajectories accumulate at different aptitude/difficulty settings.

Real-world relevance: this branch-on-error idea is the direct ancestor of every modern adaptive tutoring and mastery-based e-learning system referenced later in the article's "Digital Integration" section.

Main-line frame Remedial frame Learner token

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