Spaced retrieval: why the flashcards come back later¶

Level: 101 · for anyone who has used flashcards, or crammed the night before and lost it all a week later

One line: Memory fades fast and then slowly, and pulling a fact back out of memory, just as it starts to fade, strengthens it far more than reading it again; a schedule whose gaps grow by a fixed factor makes this cheap, because the number of reviews grows only with the logarithm of how long you want to remember.

The idea¶

Two findings from memory research are behind every flashcard program, and behind the Anki decks on the pages of this library.

Memory fades, fast at first. Hermann Ebbinghaus measured it on himself in 1885, learning lists of nonsense syllables and testing how much he had kept: most of the loss comes in the first day, and what survives fades more and more slowly. A common model is R = e^(−t/S): the chance of recall R after t days, with S the memory's stability, the time it takes to fall to about 37%.

Retrieval strengthens, and effortful retrieval strengthens most. Answering a question from memory changes the memory: it lasts longer afterwards than it would have if you had read the answer again. This is the testing effect: in Roediger and Karpicke's 2006 experiment, students who tested themselves remembered more after a week than students who reread, while the rereaders felt surer (see metacognition). And a review does most when the fact has partly faded, so that getting it back takes effort. This is the spacing effect: the same number of reviews, spread out, beats the same number crammed together.

Cramming feels productive for the very reason it works badly. Every rereading an hour after the last succeeds, so it feels like progress, but the memory was never in danger and has little reason to get stronger.

The arithmetic of spacing¶

If every review strengthens memory, you might expect to need a review every week forever. You do not, because each successful review makes the next gap longer. SuperMemo's SM-2 algorithm, published by Piotr Wozniak in 1987 and the starting point of Anki's scheduler, uses gaps of 1 day, 6 days, and then each gap 2.5 times the one before, rounded up, for as long as you keep answering right.

The review days then form a geometric series, so after k reviews the card has been kept for roughly 2.5^k days. Turning that round: keeping a card for T days takes about log(T) / log(2.5) reviews. Keeping it 2.5 times as long costs one more review. A fixed weekly review would cost T / 7. This is the same geometric growth as compound interest ↗, working for you.

What the program prints¶

Sections 1 and 2 are a toy model: the forgetting curve above, plus a rule that a review multiplies stability by 1 + 6 × (share forgotten). The constants are invented to give the right shape, and the numbers are not measurements. What the model does show honestly is the mechanism: under any rule where a review helps more the more has faded, four crammed reviews lose to four spaced ones. Sections 3 and 4 are exact arithmetic on the SM-2 schedule.

Verified output of spaced_retrieval.py — regenerated by tools/run_examples.py, never hand-typed.

1. THE FORGETTING CURVE: R = e^(-t/S), WITH S = 1 DAY
   after  0 days  recall 100.0%  ########################################
   after  1 days  recall  36.8%  ###############
   after  2 days  recall  13.5%  #####
   after  3 days  recall   5.0%  ##
   after  7 days  recall   0.1%  
   after 30 days  recall   0.0%  
   Half is gone in less than a day. Without a review, a fact met once is
   as good as lost within a week.

2. FOUR REVIEWS EACH, THE SAME EFFORT, DIFFERENT TIMING (toy model)
   plan                                           day  1 day  7 day 30 day 90   stability
   crammed: four times on day 0, an hour apart     56.2%   1.8%   0.0%   0.0%      1.7 days
   spaced: days 0, 1, 3 and 7                      97.2%  82.1%  43.1%   8.0%     35.6 days
   spaced wider: days 0, 2, 7 and 16               98.6%  90.8%  66.2%  29.0%     72.6 days
   (Each test day is counted from the last review, so no plan gets credit for
   having reviewed later.) Cramming feels productive because every re-read
   succeeds, and that is exactly why it teaches the memory little: it was never
   in danger. Spacing lets a fact fade a little before each review.

3. HOW A FLASHCARD PROGRAM SPACES REVIEWS: SM-2 (Wozniak, 1987)
   First gap 1 day, second gap 6 days, then each gap is the last one x 2.5,
   rounded up, for a card you keep answering right.
   review  1   gap     1 days   on day     1  (year  0.0)
   review  2   gap     6 days   on day     7  (year  0.0)
   review  3   gap    15 days   on day    22  (year  0.1)
   review  4   gap    38 days   on day    60  (year  0.2)
   review  5   gap    95 days   on day   155  (year  0.4)
   review  6   gap   238 days   on day   393  (year  1.1)
   review  7   gap   595 days   on day   988  (year  2.7)
   review  8   gap  1488 days   on day  2476  (year  6.8)
   review  9   gap  3720 days   on day  6196  (year 17.0)

4. THE COST OF REMEMBERING GROWS LIKE A LOGARITHM
   After the first two reviews the gaps grow by a factor 2.5, so the days
   covered form a geometric series: 7 + 15 + 37.5 + ... ~ 7 + 15(2.5^k - 1)/1.5.
   Keeping a card for T days therefore takes about log(T) / log(2.5) reviews:
   keep it for | SM-2 reviews | one review a week
        1 year |            6 |                53
       2 years |            7 |               105
       5 years |            8 |               261
      10 years |            9 |               522
   Keeping a fact 2.5 times as long costs one more review, not 2.5 times as many.
   For a deck of 1000 cards kept a year: 16 reviews a day on SM-2,
   145 a day on a weekly round. That is the arithmetic behind the Anki
   decks in this library.

The study cycle: spacing built into a week¶

Saundra McGuire's Teach Yourself How to Learn (Figure 4.6, from Louisiana State University's Center for Academic Success, based on F. L. Christ's 1997 learning cycle) turns these findings into a routine of five steps:

Preview before class: skim the chapter, note headings and bold words, and come up with questions you would like the lecture to answer.
Attend class: go, answer and ask questions, take meaningful notes.
Review as soon as possible after class: read the notes, fill gaps, note questions. Explaining the class to yourself is a first retrieval, made a few hours after the class and not the night before the exam.
Study in intense study sessions: set specific goals (1–2 minutes), do active learning tasks with focus (30–50 minutes), take a break or have a reward (10–15 minutes), then review what you just studied (5 minutes). Sessions can be as short as 15–20 minutes or as long as 75–90, 50–60 being typical. Even a 10-minute one works: a minute of goals, eight of work, a minute of review, and some students do 10 to 20 of them a day. One student who turned his grades round credited "that Power Hour thing". Add a weekend review to connect the week's material, and questions like "why?", "how?" and "what if?".
Assess: ask whether your methods work and whether you understand the material well enough to teach it to others. If flashcards did not do it, try something else, such as concept maps.

Seen through this lesson, the cycle meets each idea several times at growing intervals: preview, class, review the same day, a study session within days, the weekend review, the exam. Each meeting is a retrieval rather than a rereading, if you make it one, and step 5 is the metacognition check.

Questions¶

1. In the model R = e^(−t/S) with S = 1 day, how much is left after one day? After three?

Answer

e^(−1) ≈ 36.8% after one day, e^(−3) ≈ 5.0% after three.

2. Why does a review an hour after the last one do so little?

Answer

The fact has hardly faded, so retrieving it takes no effort and the memory is barely strengthened. The spacing effect: reviews help most when some forgetting has happened.

3. What is the testing effect?

Answer

Retrieving something from memory makes it last longer than reading it again for the same time.

4. SM-2 gaps: 1, 6, then ×2.5 rounded up. List the first five gaps.

Answer

1, 6, 15, 38, 95 days.

5. Why does keeping a card for ten years take only a few more reviews than keeping it for one?

Answer

The gaps grow geometrically, so the time covered grows like 2.5^k and the number of reviews like log T. Ten years is about 2.5^2.5 times one year, so it costs two or three more reviews (the program counts 6 for one year and 9 for ten).

6. Name the five steps of McGuire's study cycle and the four parts of an intense study session.

Answer

Preview, attend, review, study, assess. Set a goal, study with focus, take a break or reward, review.

7. The toy model's numbers are invented. What in its output can you still trust?

Answer

The direction, not the sizes: if a review helps more when more has been forgotten, cramming loses to spacing for the same number of reviews. The size of the effect has to come from experiments, not from this program.

How to practise¶

Answer before you look, every time: a flashcard read front-then-back without an attempt is rereading.
Review the same day, then let it fade a little. After class, then in a few days, then at the weekend.
Use the decks. Every lesson here with a Flashcards section has an Anki deck; Anki does the spacing.
Do not cram what you want to keep. Cramming can get you through tomorrow's test; the program's day-30 column is what it costs.

Flashcards¶

The page as a deck of Anki cards: spaced_retrieval.txt. Import with File → Import. Tags: memory, schedule, study-cycle, trap.

Po polsku, w skrócie¶

Pamięć zanika szybko na początku, a potem coraz wolniej (krzywa zapominania Ebbinghausa). Dwa zjawiska pomagają ją utrwalić. Efekt testowania: przypomnienie sobie czegoś z pamięci wzmacnia ją bardziej niż ponowne przeczytanie. Efekt rozłożenia w czasie: powtórka działa najlepiej, gdy materiał już trochę wyblakł i przypomnienie wymaga wysiłku. Dlatego cztery powtórki w jeden wieczór dają mniej niż cztery powtórki rozłożone na tydzień.

Programy do fiszek, takie jak Anki, wydłużają odstępy między powtórkami: 1 dzień, 6 dni, a potem za każdym razem 2,5 raza dłużej. Odstępy tworzą ciąg geometryczny, więc liczba powtórek potrzebnych, żeby pamiętać coś przez T dni, rośnie jak logarytm z T. Rok to sześć powtórek, dziesięć lat to dziewięć. Przy powtórce co tydzień byłoby ich ponad pięćset.

Saundra McGuire proponuje „cykl nauki": przejrzyj materiał przed zajęciami, bądź na zajęciach, powtórz jeszcze tego samego dnia, ucz się w krótkich, intensywnych sesjach (cel, 30–50 minut skupienia, przerwa, 5 minut powtórki) i sprawdzaj, czy twoje metody działają. To jest właśnie rozłożone w czasie przypominanie, wbudowane w tydzień.

Spaced retrieval: why the flashcards come back later¶

The idea¶

The arithmetic of spacing¶

What the program prints¶

The study cycle: spacing built into a week¶

Questions¶

How to practise¶

Flashcards¶

Po polsku, w skrócie¶

See also¶