Learning Curves: How Ebbinghaus and Spaced Repetition Beat Forgetting

Ebbinghaus and the shape of forgetting

In the 1880s Hermann Ebbinghaus ran one of the first quantitative psychology experiments on himself: he memorised lists of meaningless three-letter syllables (chosen specifically to avoid any existing association that could help recall) and measured how much longer it took to relearn a list after various delays. The result, published in 1885, was the forgetting curve — retention drops steeply within the first hour, then the rate of loss slows and the curve flattens, an approximately exponential decay rather than a straight-line fade.

R(t) = e^(−t / S)

R(t) = fraction retained after time t
S    = "stability" of the memory — larger S, slower decay

each successful review, spaced near the point of forgetting,
increases S — the same interval of time produces less forgetting
the next time around, which is the entire justification for
spacing reviews out rather than repeating them close together

The spacing effect and desirable difficulty

The spacing effect — that distributed practice produces stronger, more durable memories than the same number of repetitions massed together — is one of the most replicated findings in memory research, going back to Ebbinghaus himself and confirmed repeatedly since, including large meta-analyses by Cepeda and colleagues in the 2000s. The mechanism most often cited is desirable difficulty, a term from Robert Bjork's research: retrieving a memory that has partly faded requires more effortful reconstruction than retrieving one still fresh, and that extra effort appears to strengthen the trace more than an easy retrieval does. Cramming produces mostly easy, low-effort retrievals in a short window; spacing deliberately waits until recall is a little harder, which is uncomfortable in the moment but produces a flatter forgetting curve afterward.

SM-2: turning the curve into a schedule

SM-2, published by Piotr Wozniak in 1987 as the algorithm behind SuperMemo, is the ancestor of the scheduling logic in nearly every modern flashcard app, including Anki. After each review the learner grades their recall (commonly 0–5, with 3+ counting as a pass), and the algorithm updates two per-card numbers: the next interval, and an ease factor that controls how quickly future intervals grow:

on a passing grade q (3, 4 or 5):
  interval[1] = 1 day
  interval[2] = 6 days
  interval[n] = interval[n-1] × EF          (n > 2)

  EF = EF + (0.1 − (5−q) × (0.08 + (5−q) × 0.02))
  EF is clamped to a minimum of 1.3

on a failing grade (0, 1 or 2):
  reset the repetition count and interval to the start,
  EF is reduced but the accumulated card history is kept

A card that is consistently easy accumulates a high EF and its intervals stretch rapidly — days, then weeks, then months — because the model has evidence that decay is slow for that card. A card that keeps getting missed has its EF pushed down toward the 1.3 floor, keeps a short interval, and reappears often until it, too, earns a longer gap. In effect, SM-2 is fitting a rough, per-card estimate of the decay parameter S from the forgetting-curve equation using nothing but pass/fail history, and scheduling the next review to land close to the point where retention would otherwise cross below a target threshold.

What came after SM-2

SM-2 treats every card's forgetting curve as a single scalar (the ease factor) updated by a fixed formula, which is crude but cheap and works well enough that it became a de facto standard for decades. Later schedulers — SuperMemo's own SM-15/SM-17, and the FSRS (Free Spaced Repetition Scheduler) adopted by newer Anki versions — instead fit an explicit two-parameter memory model (separate estimates of "stability" and "difficulty" per card) to the learner's entire review history using statistical optimisation, which more accurately captures that some material decays faster than others independent of how many times it has been reviewed. The underlying idea, testing near the edge of forgetting and lengthening the gap after every success, has not changed since Ebbinghaus; only the precision of estimating where that edge actually is has improved.

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