Theoretical aspects of spaced repetition in learning

Dr Piotr Wozniak, 1990-2000

This text was derived from P.A.Wozniak, Optimization of learning : Simulation of the learning process conducted along the SuperMemo schedule (1990) and has been updated with revised figures (original text included additional figures related to the forgetting rate which has been significantly overestimated due to an error in the implementation of the simulation model)

This article should help you plan your learning and better understand your lifetime capacity for learning new things. Most of the figures and formulas have been theoretically derived. However, over the last ten years, these theoretical constructs have been confirmed many times by exact measurements taken during an actual learning process

A simple simulation model makes it possible to predict the outcome of a long-term learning process based on spaced repetition. Probability of forgetting at each repetition is determined by the forgetting index. By using a Spaced Repetition Algorithm and a real distribution of element difficulty (A-Factor Distribution), it is possible to predict the course of learning over many years by means of computer simulation (note that you can run a similar simulation of your own learning process based on your own real learning data in SuperMemo 98 and later with Tools : Statistics : Simulation)

The simulation model takes the following assumptions:

  1. Learning proceeds along a standard repetition spacing algorithm (e.g. Algorithm SM-11)
  2. A bell-shaped distribution of A-factors is taken from a generic knowledge system created with SuperMemo
  3. The matrix of optimal factors is taken from a generic knowledge system and does not change in the course of the learning process
  4. At repetitions, a specified portion of items, determined by the forgetting index, is taken as forgotten and reenters the process without a change to their A-factors

The above assumptions eliminate the following problems that might otherwise be encountered while trying to estimate the results of a long-term learning process:

  1. The variability of individual mnemonic skills can be entirely encompassed by the distribution of A-factors (Point 2). After all, the same knowledge system used by a skilled student will show a greater proportion of higher A-factors
  2. The variability of the difficulty of the studied material, which again, can entirely be reflected by the distribution of A-factors (Point 2)
  3. The variability of the mnemonic capability of the brain as a result of training, which is discounted by using a constant distribution of A-factors (Point 2)
  4. The variability of the mnemonic capability of the brain with aging, which can be discounted by using a constant value of the matrix of optimal factors (Point 3). A significant loss of memory with aging can be observed only as a result of a pathological process or because of lack of training (Restak 1984). Otherwise, the mnemonic capability of the brain is likely to increase with age as a result of training!

For simplicity of the description, in the following paragraphs I will use the term generic material, meaning a learning material with a typical distribution of A-factors. It is important to notice that the term reflects also the mnemonic capability of the student. This comes from the fact that good students tend to exhibit a greater proportion of high A-factors in their collections.

Here is the short summary of conclusions that could be drawn from simulation experiments based on the discussed model:

Figure 1 Learning curve for a generic material, forgetting index equal to 10%, and daily working time of 1 minute

NewItems=aar*(3*e- 0.3*year+1)


NewItems - items memorized in consecutive years when working one minute per day,

year - ordinal number of the year,

aar - asymptotic acquisition rate, i.e. the minimum learning rate reached after many years of repetitions (usually about 200 items/year/min)

time = 1/500 * year-1.5 + 1/30000


time - average daily time spent for repetitions per item in a given year (in minutes),

year - year of the process.

Figure 2 Workload, in minutes per day, in a generic 3000-item learning material, for the forgetting index equal to 10%

Retention = -FI/ln(1-FI)


Retention - overall knowledge retention expressed as a fraction (0..1),

FI - forgetting index expressed as a fraction (forgetting index equals 1 minus knowledge retention at repetitions).

The above formula can be derived from the formula for the exponential decay of memory traces (R=e-d*t where R - retention, d - decay constant, t - time)

Figure 3 Dependence of the knowledge acquisition rate on the forgetting index

Figure 4 Trade-off between the knowledge retention (forgetting index) and the workload (number of repetitions of an average item in 10,000 days)

Length of interval Percent of elements Percent of workload
1-60 days 5% 63%
61-300 days 13% 23%
301-1000 days 19% 7%
over 1000 days 63% 7%
Number of lapses Percent of elements Percent of workload
0 62% 42%


2 9% 15%
3 5% 9%
4 3% 6%
5 and more 5% 12%