Spaced Repetition & Memory Decay

Forgetting is a feature, not a bug

Your brain deletes information that is not accessed repeatedly, and it does so for a good reason: keeping every fact you ever encountered permanently at full strength would flood working memory with noise. In 1885 Hermann Ebbinghaus set out to measure this deletion process quantitatively. Using himself as the only subject, he memorised lists of meaningless syllables and timed how quickly he lost them, producing the first data-driven description of human forgetting — the forgetting curve.

The modern formulation treats retention as an exponential decay:

R(t) = e^(-t / S)

R(t)  probability of successful recall at time t after last review
S     memory stability — strength of the memory trace

A fresh item might have S ~ 1 day  → after 1 day, R ~ e^-1 ~ 37%
After 5 correct reviews, S ~ 30 days → after 1 month, R is still ~37%
                                        (the interval got 30x longer)

The key observation is that stability is not fixed — each successful reactivation raises it, and it rises more when the recall was difficult (low R at the moment of review) than when it was effortless. That asymmetry, the spacing effect, is why reviewing a card the day before you would have forgotten it produces a much bigger stability gain than reviewing it while it is still fresh in mind.

SM-2: the algorithm behind the flashcard revolution

Knowing the shape of the forgetting curve is only useful if you can turn it into a review schedule, and that's exactly what Piotr Woźniak's SM-2 algorithm does. Released in 1987 as part of SuperMemo 2, it still powers Anki — the most widely used spaced repetition app — and dozens of derivatives. Every card carries an ease factor (EF), starting at 2.5, and after each review the learner rates their recall quality from 0 to 5, from complete failure to instant, effortless recall.

EF' = EF + (0.1 - (5 - q) * (0.08 + (5 - q) * 0.02))   // EF floor: 1.3

I(1) = 1 day        // first review
I(2) = 6 days       // second review
I(n) = I(n-1) * EF  // n >= 3

If q < 3: reset — interval returns to day 1, sequence restarts.

The compounding is what makes SM-2 powerful. A card reviewed correctly six times at a stable EF of 2.5 reaches an interval of roughly 6 × 2.5⁴ ≈ 234 days — nearly eight months between reviews. The same card scored at the bare minimum passing grade every time drags EF down toward 1.3 and only reaches 6 × 1.3⁴ ≈ 26 days at the same point in its history — roughly nine times shorter. The algorithm is, in effect, an automatic difficulty tracker: easy material gets reviewed rarely, hard material gets reviewed often, and the schedule adapts card by card without anyone having to plan it by hand.

Active recall versus passive rereading

Spaced repetition only works as well as the retrieval it embeds. Highlighting a textbook or rereading your notes feels productive because the material starts to look familiar — but familiarity is an illusion of fluency, not the ability to produce an answer cold. Active recall, generating the answer before checking it, engages a fundamentally different mechanism: the retrieval attempt itself, whether it succeeds or fails, strengthens the underlying memory trace through a process called reconsolidation. A widely cited 2011 study by Karpicke and Blunt found that students using retrieval practice outscored students who simply restudied by roughly 50% on a later test, and beat students who built elaborate concept maps by a similar margin.

A closely related result is the testing effect: Roediger and Karpicke's 2006 study split students into a group that restudied a passage repeatedly and a group that took recall tests on it instead. A week later, the tested group remembered 61% of the material against 40% for the restudied group — despite spending less total time studying. The mechanism appears to be that successful retrieval activates a wider network of cortical regions than simple restudy, effectively reconstructing and re-storing the memory in a stronger form each time it is pulled up.

Sleep finishes the job

A spaced repetition session only pays off fully if sleep follows it. During slow-wave sleep the hippocampus replays the day's newly formed memory patterns, gradually transferring them into distributed cortical storage — a process called systems consolidation. REM sleep does something complementary, strengthening associative and procedural memories and helping new material connect to what you already know. Studies comparing same-day recall with and without an intervening sleep period have found roughly a 20-30% performance advantage for the group that slept, which is the practical reason a review session the evening before bed tends to outperform the identical session done first thing in the morning with no sleep in between.

Where spaced repetition stops helping

The technique is built for declarative memory — facts, vocabulary, dates, formulas — where the goal is exact retrieval of a fixed answer. It does not substitute for conceptual understanding, which needs elaborative questioning and problem-solving rather than repetition, and it does not substitute for procedural skills like an instrument or a sport, which require deliberate practice with feedback rather than recall. Used within that scope — maintaining a large base of accurate, instantly retrievable facts — it remains one of the most efficient learning interventions cognitive science has validated, and the simulator on this page lets you watch the stability curve and the SM-2 interval grow side by side as you vary recall quality.

Frequently asked questions

Try it live

Everything above runs in your browser — open Spaced Repetition & Memory Decay and adjust recall quality to see how the ease factor and review interval respond in real time.