The Scheduler: SM-2, Plainly
SM-2 turns each self-graded recall into a next review date: intervals start at 1 day, then 6, then multiply by an ease factor near 2.5 that drifts down on hard recalls and resets after a lapse — placing every review just before its forgetting curve would fail. · 12 min
The spacing effect leaves one job undone. It tells you that gaps help and roughly how wide to make them, but you carry a hundred different facts, each fading at its own rate, each needing its own next review. No person can track that by hand. An algorithm can. In the early 1980s a student named Piotr Wozniak, frustrated with his own forgetting, worked out a set of rules for when to review each item; the 1987 version, SM-2, is simple enough to run with a pencil and good enough that this Archive still runs it on your memory today. This folio takes SM-2 apart and traces it, one step at a time, on a single card.
Guess before you learn
A review scheduler shows you a card, you recall it correctly, and it sets the next review. You keep getting it right. Roughly how does the wait before each next review change over the first several correct recalls?
It multiplies. SM-2 sets the first interval to 1 day and the second to 6 days, then multiplies each following interval by an ease factor near 2.5. After a handful of successful recalls a card will not return for months. Most people picture reviews marching at a steady pace, which is why an efficient schedule feels almost lazy — you see each card far less often than intuition expects, precisely because each success is evidence you need it less.
9–12
3–5
A good practice schedule does not show you every card the same amount. When you get a card right, it waits longer before asking again — first a day, then several days, then a week or more.
When you get a card wrong, it comes back quickly so you can mend it. Easy cards drift far apart; hard cards stay close. That way your time goes to the cards that actually need it, not the ones you already know.
6–8
SM-2 keeps three numbers for every card: how many times you have recalled it, the gap until its next review, and an ease factor that starts at 2.5. Get the card right and the gaps grow — 1 day, then 6 days, then the previous gap multiplied by the ease factor. Get it wrong and the card resets to a 1-day gap.
After each recall you grade yourself from 0 to 5. A perfect recall nudges the ease factor up; a slow, difficult one nudges it down — so hard cards return sooner and easy cards drift further apart. That honest self-grade is the only thing you feed the algorithm.
9–12
The intervals are chosen to fit your forgetting curve (folio 3): each review is timed to land just before recall would fail, and each success pushes that moment further out. The first two intervals are fixed at 1 and 6 days. From the third on, the next interval is the current one times the ease factor — a number near 2.5 that rises a little after an easy recall and falls after a hard one, but never below 1.3.
A lapse — any grade below 3 — sends the card back to a 1-day interval, because a forgotten card has proven its gaps grew too fast. The ease factor carries the card's whole history: a card you keep finding hard slowly earns shorter gaps, without your having to decide anything beyond an honest grade each time.
K–2
You have a card with a hard word on it. You get it right, so you wait one sleep before trying again. You get it right again, so now you wait almost a whole week.
Every time you get the card right, you wait longer before the next try. If you get it wrong, the card comes back very soon so you can fix it.
Undergrad
SM-2 is a heuristic controller, not a model of memory. Its state is two numbers per item — interval and ease factor — updated by one self-reported grade q in 0 to 5. The ease-factor update, EF' = EF + (0.1 − (5 − q)(0.08 + 0.02(5 − q))), leaves EF unchanged at q = 4, raises it at q = 5, and lowers it for q at 3 or below, with a floor of 1.3 to stop runaway shrinkage.
The design encodes the Bjorks' logic: a longer interval forces a harder, more strengthening retrieval, so intervals expand as storage strength accrues. Its weaknesses are equally plain — a single grade conflates difficulty with confidence, and every lapse is treated alike. Successors such as SM-17 and FSRS replace the fixed rules with fitted forgetting curves; SM-2 endures because it is transparent and runs on almost nothing.
Postgrad
Formally, SM-2 approximates optimal review scheduling under an exponential-forgetting assumption: hold recall probability near a target by expanding the interval geometrically, with the ease factor serving as a per-item estimate of decay rate. It is greedy and myopic — each update depends only on the current grade and state, with no explicit probability model and no lookahead — which is why it is cheap, and why it is beatable.
Modern schedulers make the latent variables explicit. FSRS fits a difficulty-stability-retrievability model to review logs and picks intervals for a chosen retention target; SuperMemo's later algorithms fit two-component curves per item. Each buys accuracy at the cost of interpretability. This Archive runs SM-2 in the Fading Ink — review what's fading — precisely because a reader can trace every interval it chooses by hand, which is the point of this folio.
ease factor
The multiplier SM-2 keeps for each card — starting at 2.5 — that sets how fast its review gaps grow. It rises after easy recalls and falls after hard ones, and never drops below 1.3.
Why is this true?
Why does SM-2 reset a card to a one-day interval the moment you forget it?
Because a lapse is evidence that the gaps grew faster than that card's memory could hold. Sending it back to one day rebuilds the interval from a length you have just proven you can manage, and then lets it grow again from there.
That explosion is the whole point of the schedule. A card you know well should cost you almost nothing, so its gaps stretch until it barely appears. A card you keep missing should stay close, so its interval keeps resetting to a single day. The ease factor is the dial between those two fates, and you turn it without thinking: every time you grade a recall honestly, you tell the algorithm which kind of card it is holding — one that has earned a longer rest, or one that still needs watching.
Trace a card through four correct recalls (ease factor 2.5) — the steps fade as you master them
interval = 1 day
interval = 6 days
6 × 2.5 = 15 days
15 × 2.5 ≈ 38 days
Notice what SM-2 does not need: it never measures your brain, never times you to the millisecond, never knows the material. It asks one question after each recall — how did that go, on a scale of 0 to 5 — and turns your answer into a date. That is also its honest limit. A single number cannot tell difficulty from low confidence, and every lapse looks the same to it. Newer schedulers such as FSRS fit a forgetting curve to your actual review history and do better. SM-2 survives anyway, because you can trace every date it picks with nothing but arithmetic — which is exactly what you just did.
You now hold the arithmetic behind every review this Archive schedules for you. From here the unit turns from timing to protocol. Folio 9 shuffles problem types within a session; folio 10 combines retrieval, spacing, and a success criterion into the single most powerful study routine on record. SM-2 is the engine; the next two folios are what you drive with it.
Note
Every folio you finish returns in the Fading Ink — review what's fading — on exactly the intervals you just traced. Watching SM-2 run on your own memory is half of what this course is for.
Practice — new ink and old, interleaved
1.From folio 3: SM-2 aims each review to land just before recall would fail. What does a review do to the forgetting curve?
2.Which grade leaves a card's ease factor unchanged?
3.From memory: what three things does SM-2 store for each card, and which one does your self-grade change?
It stores the recall count, the current interval, and the ease factor (near 2.5). Your grade adjusts the ease factor — up for easy recalls, down for hard ones — and a lapse resets the interval to one day.
How close were you? Grade yourself honestly — it sets your review date.
4.From folio 5: each SM-2 review asks you to recall the card before checking. Why does that make the review itself worth more than simply rereading the card?
5.Which single event sends a card's interval all the way back to 1 day?
6.Without looking back: what do judgments of learning mostly ride on, and why does rereading inflate them?
They ride on how fluently the material processes right now; rereading raises fluency without adding much to memory, so the judgment climbs while recall stays flat.
How close were you? Grade yourself honestly — it sets your review date.
7.From folio 7: by the 10-to-20-percent rule, for a test 30 days away, a gap near the middle of the range is about how many days?
8.Order the life of a reviewed memory, first to last.
- Learn the list to full strength
- The curve falls steeply through the first day
- A review restores full strength
- The new curve falls more slowly than the first
9.Judgments of learning are built mainly on:
10.From folio 4: SM-2 depends on your honest 0-to-5 grade. Why is over-grading a card you recalled only shakily a mistake?
11.A meta-analysis reports retrieval practice at g ≈ 0.61. What does that number mean?
12.A card passes at an interval of 15 days with an ease factor of 2.5. What is its next interval, in days?
13.You just read that the mitochondrion produces the cell's usable energy. Write a card question whose answer is 'usable energy'.
14.Match each self-grade to what SM-2 does with it.
15.Your notes on one topic feel completely familiar. In one sentence: what should you do before trusting that feeling?