How a Spaced Repetition Algorithm Decides When You See a Word Again

You spelled "rhythm" correctly this morning, and the app told you it would ask again in eleven days. Last week the same word came back after two. Keep getting it right and the next gap will be well over a month, and the one after that closer to half a year. Miss it once and it is back within the week, somewhat humbled.

From the receiving end, these dates can look arbitrary, like a slot machine that pays out in calendar days. They are not. Each one is the answer to a precise question the software asks about every word you know: on what day will your chance of remembering this fall to an acceptable level? People have been asking that question since the 1880s. What has changed is how good the answer has become.

This is an account of how the answer gets worked out, from a shoebox of index cards to the open-source model that now sits inside several popular study apps. By the end you should be able to look at a review date on any screen and know what it means, why it falls where it does, and when you can safely ignore it.

The problem every scheduler is trying to solve

Memory is not a light switch. A word you have just learned is not simply in or out; it has a probability of being recalled, and that probability drifts downward with time. Hermann Ebbinghaus was the first to measure the drift. Working alone, he memorized lists of nonsense syllables, relearned them after delays of minutes, hours and days, and recorded how much effort the relearning saved. The result, published in 1885, was the first forgetting curve: steep at the start, then flatter, as though memory sheds its loosest material quickly and holds on harder to what remains. Our piece on the forgetting curve tells that half of the story in full.

He noticed a second thing, easy to miss beside the first. Lists he had relearned several times were forgotten more slowly than lists he had learned once. Each review did not just top the memory back up. It changed the rate of the leak.

Put those two facts together and you have the whole design problem. Review too early and you spend time on a word you would have remembered anyway. Review too late and you are relearning rather than reviewing, which costs more and gains less. Somewhere between the two is a moment that gives the most memory for the least effort.

Every scheduler in this story is an attempt to find that moment, word by word.

A box of cards on the kitchen table

The first practical answer came from a German science journalist, Sebastian Leitner, in a 1972 book called So lernt man lernen, "how to learn to learn". It needed no computer, only a box divided into compartments.

Every new card starts in the first compartment, which you go through daily. Answer a card correctly and it moves one compartment along, into a section you visit less often: every few days, then weekly, then monthly. Miss it, and it goes back to the start, however far it had traveled.

The beauty of the Leitner box is that it spends your time where your memory is weakest. Easy cards drift to the back and stop bothering you. Hard cards circle near the front until they settle, and anyone who has used one knows the small satisfaction of watching a stubborn card finally escape compartment one.

Its weakness is that the compartments are fixed. Every card in the third section waits the same length of time, whether it says "cat" or "conscientious", and a single slip on a card you have known for a year throws it back among the beginners. The box knows where a card is. It has no idea how well you know it.

SM-2: the gaps begin to multiply

In the late 1980s Piotr Wozniak, a Polish student trying to hold on to English vocabulary and biology at the same time, began running careful experiments on himself with a computer. The program he built became SuperMemo, and one early version of its rules, called SM-2, went on to become the ancestor of most flashcard software, including the original Anki.

SM-2 replaced compartments with arithmetic. After you learn a card it comes back in one day, then in six. From there, each new gap is the previous gap multiplied by an ease factor, a number that starts at 2.5. A card you keep getting right therefore drifts outward roughly like this: one day, six, fifteen, 38, 95. After each answer you rate how hard it felt, and the ease factor shifts, up a little for easy answers and down for hard ones, never below 1.3.

For the first time each card had its own pace, and the scheme worked well enough to carry spaced repetition for three decades. But its rules were chosen by hand. The 2.5 is a good guess, not a measurement, and nothing in SM-2 actually estimates the probability that you will remember a word on a given day. It assumes that multiplying the last gap by the right number will land somewhere sensible.

Usually it does. Sometimes, for a genuinely hard word, it sends you into a two-month gap you will not survive.

FSRS: a model of the memory itself

The newest step replaces the rule of thumb with a model fitted to data. The Free Spaced Repetition Scheduler, FSRS for short, has been developed since 2022 by Jarrett Ye and a community of volunteer contributors, growing out of research Ye and his colleagues published that year on optimizing review schedules. It is open source, it became a built-in option in Anki in late 2023, and it is what a growing number of apps now use to decide when you next see a word. Our comparison of spaced-repetition apps for spelling shows which of them offer it, and which make you type the word rather than just look at it.

FSRS tracks three quantities for every word you study.

Stability is the durability of the memory, measured in days. Precisely, it is the number of days it takes for your chance of recalling the word to fall from near certainty to 90 percent. A word you met this morning might have a stability of two days. A word you have spelled correctly for a year might have a stability of several hundred.

Difficulty is how hard this particular word is for you, on a scale from one to ten. It rises when you struggle and falls, slowly, when you do not. Its main job is to limit how fast stability can grow, so that hard words gain durability in smaller steps.

Retrievability is the probability that you could spell the word correctly if someone asked you right now. It sits close to 100 percent straight after a review and then declines along a forgetting curve whose steepness is set by the stability.

With those three numbers in hand, the scheduler's task is easy to state. You choose a target, the desired retention, usually 90 percent, and when a word's retrievability is predicted to reach that target, the word is due. This is what "the right time to review" means in practice. Not the instant before the memory vanishes, but the day on which your odds of success have slipped to nine in ten.

The curve FSRS uses is also more honest than the one in most textbooks. Real forgetting slows down more than a neat exponential would suggest. Under the current default model, a word with a stability of ten days, left alone for a whole month, still has roughly an 80 percent chance of being spelled correctly. Memories fade. Once they have a little age on them, they fade slowly.

Why the gaps grow so fast

What surprises people most is how quickly the intervals run away. Using the open-source FSRS code with its default settings and a 90 percent target, a new word that you answer correctly, at an ordinary pace, every time it comes back is scheduled roughly like this.

Your own numbers will differ, because the word's difficulty, the speed of your answers and a little deliberate randomness all nudge them. The shape is the point. Five correct recalls, spread over a little under two years, buy you a word you will probably keep for much longer than that.

The reason is the second thing Ebbinghaus saw, and the thing the psychologist Robert Bjork later named a desirable difficulty: recall that takes some effort does more for memory than recall that is effortless. A review timed for the moment your chance has slipped to 90 percent is, by design, slightly effortful. It resets retrievability to near certainty, and it multiplies stability. Multiply a few times and the gaps compound, the way interest does.

None of this is an artifact of software. In a 2008 study that tested people at delays of up to a year, Nicholas Cepeda and his colleagues found that the best gap between study sessions grew with the delay. For a test a week away, a day between sessions worked well; for a test a year away, a gap of about three weeks worked better. Harry Bahrick's long study of his own family learning foreign vocabulary found the same thing on a scale of years. Sessions spaced eight weeks apart produced memories that outlasted those from sessions two weeks apart.

What happens when you get a word wrong

A miss tells the scheduler two things at once: the memory was weaker than predicted, and the word may be harder for you than it thought. FSRS responds to both.

Difficulty goes up, so future stability will grow more slowly. Stability falls sharply, but not to zero. And the word comes back soon. In the default model, a word that had reached a five-month interval and then slipped is rescheduled a few days out rather than returned to day one. If you get it right from there, the gaps rebuild in steps of roughly twelve days, then four weeks, then two months. That is slower than the first climb, because of the higher difficulty, and far faster than starting from nothing.

This is both kinder and more accurate than the Leitner box. A word you knew for months and briefly lost is not the same as a word you have never seen, and relearning it is quicker, a fact Ebbinghaus measured with his own "savings" scores. The model gives credit for what remains.

It helps to be clear about what a miss is not. It is not a malfunction and it is not a penalty. At a 90 percent target, roughly one due review in ten is supposed to fail; that is the price of not over-reviewing the other nine. And a wrong attempt followed promptly by the right answer turns out to be a strong way to learn, as the research on why you remember a word better after getting it wrong explains.

Why a fixed schedule is worse

Plenty of study guides recommend a tidy fixed plan: review on day one, day three, day seven, day fourteen, day thirty. It is far better than cramming. It is also wasteful in two directions at once.

For easy words it is too frequent. If you have spelled "because" correctly for twenty years, five reviews in a month are five wasted minutes, and across a list of several hundred words those minutes become hours.

For hard words it is too slow. If "accommodate" is going to slip on day five, a review on day seven is not a review at all. It is relearning, and relearning costs more than a well-timed reminder would have.

An adaptive scheduler moves time from the first group to the second. There is nothing mysterious about it; it simply refuses to treat "cat" and "conscientious" as the same problem, which is exactly what a fixed calendar has to do. The practical result is that the same ten minutes a day covers more words and keeps them longer, which was always the argument for spaced repetition in the first place.

No model sees inside your head, and FSRS is no exception. It infers your memory from your answers, and its estimates improve as the answers pile up. Many implementations also add a day or two of random "fuzz" to each gap, so that fifty words learned on the same Monday do not all fall due on the same Thursday.

Reading the numbers on your screen

Most apps expose some of this machinery. A few notes on how to read it.

A next review date is a prediction, not a deadline. It marks the day your chance of recall is expected to reach the target. Come back a day late and nothing breaks. Come back three weeks late and you will miss more words than usual; the scheduler will register that, and the gaps will adjust.

A due count is simply the number of words that have reached their review point. After a week away it can look alarming. Do an ordinary session each day and let the pile shrink over several days, rather than clearing it in one grim sitting.

A retention setting, if your app shows one, is the target itself, and it is a genuine trade-off. Run the same default model at different targets and the difference is stark. At 85 percent, a word you keep getting right comes back after about four days, then a month, then six months. At 95 percent, it comes back after one day, then three, eight, nineteen and 43. The higher target buys a few more correct answers at the cost of several times as many reviews, which is why 90 percent is the usual middle and the default Spelling.School uses.

The one thing to do next

Show up on the days the schedule asks you to, and keep your sessions short enough that you always can. The mathematics is only as good as the timing. Ten minutes a day lets the scheduler catch each word near its moment; an hour on Sundays does not.

Keep the flow of new words modest, too. Every new word sets off a trail of future reviews, and taking on thirty a day feels heroic for a week and miserable for a month. The honest arithmetic is in our piece on how many new words a day you should learn. If you want to see a scheduler at work on your own spelling, five words on the home page will show you the first gap with no sign-up. A shoebox with five compartments will teach you the same principle more slowly, and it has never once crashed.

Questions people ask

What is a spaced repetition algorithm?

A spaced repetition algorithm is a set of rules, or a statistical model, that decides when you should next review each item you are learning. Its aim is to bring every item back at the point where you are likely to still remember it, but only just. Items you know well come back after long gaps and hard items come back sooner. Well-known examples include the Leitner box, SuperMemo's SM-2 and the newer open-source FSRS.

How does a spaced repetition app know when I will forget a word?

It does not know for certain; it estimates. Modern schedulers such as FSRS keep a running estimate of how durable your memory of each word is, based on your past answers, and use a forgetting curve to predict your chance of recall on any given day. The word is scheduled for the day that chance is expected to reach a target, usually 90 percent. The estimates get better as you answer more.

Why do spaced repetition intervals get so long so quickly?

Each successful recall at the right moment makes the memory more durable, often by a factor of several, so the next gap can be several times longer than the last. Under the default FSRS settings, a word answered correctly every time goes from a gap of about two days to about eleven, then about 46, then about five months. The growth compounds, which is why a word can be kept for years with only a handful of reviews.

What happens when I get a word wrong in spaced repetition?

The word comes back soon, and the scheduler lowers its estimate of how durable the memory is and raises its estimate of how hard the word is for you. In FSRS, a miss does not wipe out all progress, because some memory usually remains, so the gaps rebuild faster than they did the first time. Occasional misses are expected; at a 90 percent target, about one due review in ten should fail.

Is spaced repetition better than a fixed review schedule?

An adaptive schedule is usually better than a fixed one such as day one, three, seven, fourteen and thirty. A fixed plan reviews easy words too often and hard words too late, so it wastes time in both directions. An adaptive scheduler moves effort from what you already know to what you are about to lose. Both approaches are far better than cramming everything in one sitting.

What does desired retention mean in FSRS?

Desired retention is the chance of recall at which a word becomes due for review, and so roughly the share of due reviews you should get right. At 90 percent, each word comes back when your predicted chance of spelling it correctly has fallen to about nine in ten. Raising the target to 95 percent means far more reviews for a small gain in accuracy, and lowering it to 85 percent means fewer reviews and more misses.

What is the difference between SM-2 and FSRS?

SM-2, from SuperMemo in the late 1980s, multiplies each gap by a hand-chosen ease factor that starts at 2.5 and is nudged by how hard each answer felt. FSRS, developed from 2022, instead models each memory's stability, difficulty and probability of recall, with parameters fitted to large amounts of real review data. The practical difference is that FSRS schedules each word for a predicted level of recall, rather than for an interval chosen by a fixed rule.

Fix the words you actually get wrong

A few minutes a day, spaced repetition, built only for spelling. Try five words without signing up.