Spacing algorithms explained: Leitner, SM-2, FSRS and a fixed ladder

A spacing algorithm decides when a thing you have learnt comes back for review. The classic ones are Leitner's boxes, SuperMemo's SM-2, Duolingo's half-life regression and the open-source FSRS, and they differ in how much they fit to the learner. The studies say the big wins are spacing at all and delaying the first retrieval; the exact shape of the gaps matters less. Peabody (peabody.kids), a free AI-driven adaptive learning platform for kids, uses two fixed ladders you can read off a table: items come back after 1, 3, 7 days and then a 21-day gap, and story words need three cold reads on three days, then keeper checks at 7, 30 and 90 days. AI writes the stories; code decides the rest. Peabody is free through the end of 2026.

This post is the mechanics. For the habit side, read spaced practice with AI.

The schedulers, with their actual mechanics

Leitner (1972)

The simplest. Cards live in numbered boxes. Answer one right and it moves up a box; miss it and it goes back to box 1. Box 1 is reviewed often, box 2 less, box 5 rarely. The parameters are the number of boxes and each box's gap. Every learner and every card gets the same ladder, and that is both its weakness and why it is the baseline every later system is measured against.

SM-2 (Woźniak 1987)

The algorithm behind most flashcard apps for three decades (documented at https://www-v1.supermemo.com/archives1990-2015/english/ol/sm2). Each card has an easiness factor EF, starting at 2.5. The first gap is 1 day, the second 6 days, and each later gap is the previous gap times EF. After each review the learner rates the recall quality q from 0 to 5, and EF changes by 0.1 − (5 − q)(0.08 + (5 − q) × 0.02).

EF never falls below 1.3. SM-2 depends on the learner grading themselves, and it never predicted a probability of recall; it only produced the next date.

Half-life regression (Settles and Meeder 2016, Duolingo)

Published at ACL (https://aclanthology.org/P16-1174) with code. Recall probability after Δ days is p = 2^(−Δ/h), where h is the half-life of the memory. The half-life is learned: h = 2^(Θ · x), where x is a feature vector (how many times right, how many wrong, total exposures, which word) and Θ the fitted weights. Fitted on 13 million Duolingo traces, it reported 45% or more reduction in error at predicting recall against Leitner and Pimsleur baselines. The caveat is in the authorship: company data, company authors, and a prediction result rather than a learning result.

FSRS (open source, an Anki option)

FSRS keeps three state variables per card (https://github.com/open-spaced-repetition/fsrs4anki/wiki/The-Algorithm): Difficulty D from 1 to 10, Stability S, the number of days until recall drops to 90%, and Retrievability R, the current chance of recall. Recall decays as R(t, S) = (1 + FACTOR × t/S)^DECAY, with DECAY = −0.5 and FACTOR = 19/81 in versions 4.5 and 5; version 6 learns the decay. Initial difficulty comes from the first grade; difficulty drifts towards a mean after each review. The next gap is S/FACTOR × (R_desired^(1/DECAY) − 1), with R_desired usually 0.9, so you ask for a 90% chance of recall and the formula hands you the date. It has 17, 19 or 21 learned weights depending on version (the research report flags the counts as needing a check in the primary source).

The reported gain, from the authors' own benchmark (https://github.com/ankitects/fsrs-benchmark, August 2025 snapshot):

FSRS-6 beats SM-2 on 99.5 to 99.6% of collections. Four caveats, all from the report: the benchmark was built by FSRS's own authors; SM-2 never predicted probabilities, so extra formulas were bolted on to score it; the metric is prediction quality, not learning; and the "20 to 30% fewer reviews" claim comes from simulation, not a trial.

What the gap studies actually found

Cepeda 2008: the gap as a share of the horizon

Cepeda, Vul, Rohrer, Wixted and Pashler (2008, Psychological Science, https://pubmed.ncbi.nlm.nih.gov/19076480/) ran 1,354 people through gaps up to 3.5 months and tests up to a year later. The best gap grows with how long you want to remember, but shrinks as a share of it.

Two things follow. There is no single best gap, only a best gap for a horizon. And too-short gaps hurt more than too-long ones, so if you must err, err long. The "10 to 20%" rule of thumb often quoted comes from Rohrer and Pashler's 2007 summary, not from this paper.

Karpicke and Roediger 2007: it is the first delay that matters

Karpicke and Roediger (2007, Journal of Experimental Psychology: Learning, Memory and Cognition, https://www.gwern.net/doc/psychology/spaced-repetition/2007-karpicke.pdf) compared expanding schedules (short gap, then longer, then longer) with uniform ones (the same gap each time). Expanding won on a test ten minutes later; uniform won two days later. What drove the result was not the shape but how long the learner waited before the first retrieval: a later first retrieval was harder and stuck better.

Kang, Lindsey, Mozer and Pashler (2014, Psychonomic Bulletin and Review) trained vocabulary for four weeks and found expanding and uniform tied on the eight-week final test, though expanding kept recall much higher during training, which matters for a child's mood. Küpper-Tetzel, Kapler and Wiseheart (2014) found the best shape depends on the horizon: contracting gaps for a test within a week, equal or expanding for one at 35 days.

The practical read, straight from the report: a fixed expanding schedule is fine; a fitted model personalises the gap and is better at prediction; both beat cramming by a wide margin.

What is known for children aged 5 to 12

Less than you would hope, and the studies are small.

Fazio and Marsh (2019, Current Directions) sum up the field: retrieval practice helps from infancy on, works best when it is effortful but successful, and the developmental trajectory is poorly mapped. For sight words specifically, Reitsma (1983) found four to six exposures to a new string before sight identification, and Ehri and Saltmarsh (1995) 4.4 exposures for strong readers against 9.3 for weak ones (secondary citations in the report). Nation, Angell and Castles (2007) found learning rising with one, two and four exposures, with some one-trial learning.

Peabody's two ladders

Peabody uses fixed schedules. Not one scheduler for everything but two, because facts and story words are different things.

The item ladder. Every fact, phonics word, sight word and first sound a child practises in a game sits at a level 0 to 5. Once it reaches level 3 (right first try) it is known and gets a due date. The gap grows with each separate day the child answers it quickly and right: 1 day, then 3, then 7, then a 21-day gap. A miss sends it to level 1 and the ladder starts over. "Quick" is a fixed threshold per strand, from 2.5 seconds for a times fact to 6 seconds for a number-sense question. A question answered right first try is never asked again at all, so the ladder only runs on questions still live.

The story-word ladder. The practice words carried by a story (five vocabulary words in a paragraph story, each used at least three times) follow a stricter Leitner-style rule. Before each story there is a cold warm-up: the child sees the word with no hint and says whether they can read it. A word counts as known only after three right cold reads on three different days. Then come keeper checks at 7, 30 and 90 days. A miss puts the word back to practising and missed words come back first. The details are in how Peabody tracks words.

Set against the studies:

What a per-child model would and wouldn't change

An FSRS-style or half-life model, fitted per child, would do one thing better: it would set the gap for each item from that child's own history instead of from a table. The benchmark numbers say that is a real gain at predicting recall, and the research report's own recommendations list it, starting from FSRS default parameters (which already reach 0.366 log loss) and fitting per child after about 100 reviews.

It would not change the things the studies say matter most. Spacing at all, delaying the first retrieval, retrieval instead of re-showing, and most answers being wins are all properties of the fixed ladders already. It would not change the never-twice rule, the taught-then-rehearsed rhythm, or the frustration guard. And it would cost the one property Peabody values: a parent, a teacher or a test can read the schedule off a table, and the learning invariants test checks that the gaps strictly increase.

The honest position is that nobody has shown in a trial that a fitted scheduler makes a child learn more than a fixed expanding one. If Peabody ever adopts one, it will be measured against the ladders first, on the probe cards, over a held-out month.

What this doesn't show

• Every scheduler comparison above is on prediction, not learning. FSRS predicts recall better than SM-2 on its authors' benchmark; whether a learner remembers more under it is untested.
• The child studies are small (39, 88 children) and the maths evidence is weakest for procedures, where the testing effect is not robust (Murray 2025).
• Peabody has no published efficacy study. The ladders are design facts from its code, not outcome claims.
• Peabody cannot hear a child read, so the story-word ladder measures taps on a cold read, not fluency. It does not diagnose, and it doesn't measure Lexile.

What a parent can do

Spacing at home needs no algorithm. Ask tonight's new word again at breakfast, then at the weekend, then in a fortnight. Make the ask a retrieval (the book shut) rather than a re-read. Err long when in doubt. And treat a stumble on the third day as the point of the exercise, not a failure of the first two. Peabody runs the same ladder for you across games and devices; you can start free on any browser with no account.

Learn more elsewhere

• Nicky Case: how to remember anything forever-ish: an interactive comic about the forgetting curve and the Leitner box, good to share with an older child.
• ABC of FSRS: a plain explainer of the open scheduling algorithm Anki now uses.
• Duolingo: how we learn how you learn: how one big app predicts when each word is about to be forgotten. A company blog, but it reports real tests.

For about seventy more, checked and sorted by who they're for, see where to learn about AI.

Questions parents ask

Why fixed gaps instead of an algorithm that learns my child?

Because the studies say the big wins are spacing at all and delaying the first retrieval, which a fixed expanding ladder does, and because a fixed ladder can be read, tested and changed in one line. A per-child model predicts recall better on its authors' benchmarks; whether it teaches more is untested.

What happens when my child misses a review?

In the item ladder the item drops to level 1 and the gaps start again at 1 day. In the story-word ladder the word goes back to practising and comes up first next time. Nothing earned elsewhere is lost.

Why are story words stricter than facts?

A word has to be read cold, with no story around it, on three separate days before it counts, because a word recognised once inside a story it arrived in could be context. Facts in games have speed as a second signal, so two quick rights on two days do the same job.

Is a 21-day gap too long for a six-year-old?

Cepeda 2008 says that for a horizon of months the best gaps are weeks, and that too-short gaps hurt more than too-long. The item only reaches that gap after quick right answers on four separate days, and a miss brings it straight back.