Forgetting curve
The previous text offered the hypothesis that we forget in a geometrical sequence. Expressed in the formula with the newly implemented terms, we can write:
- m – for example the number of days from the last repetition.
- RET – Retrievability, the ability to retrieve a fact, or to think of it.
The formula shows that in the day, when we review the fact, RET=100%. In case of simple facts, such as words or historical dates, this can perhaps be accepted. However, if we want to extend our model to more complicated knowledge, such as driving a car or comprehending the Theory of Relativity, it would have to be much more complicated and, among others, it would have to consider some level of comprehension of the curriculum. We can avoid this in our considerations.
The speed at which we forget facts varies. In the suggested model, it depends on the Fact Stability (STAB). So far, we did not say anything about specific STAB values, therefore it is quite irrelevant whether we involve it in the formula.
Maybe, for example this:
RET = 1/2^{m/STAB}
Incorporating the Stability in the formula is not as coincidental as we pretend. We are aiming so that the Stability can be interpreted as a number of days for which we remember the fact well.
The figure shows the forgetting curves for various STAB values.
For what do we need the forgetting curve?
Implementing the terms Retrievability (RET), stability (STAB) and Optimal Forgetting Index (OFI) leads to the basic rule “review in 2n days” being replaced by: “review when the ability to recollect a Fact drops under the recommended level.”.
RET < 1 - OFI
Example: Lets determine OFI=30%. (Therefore we admit the RET decrease to 70%). Then the figure shows that the Fact with STAB=50 must be repeated after approx. 16 days.
How much will our model improve by implementing the forgetting curve? If we only work with the rule “repeat in 2n days” and the user does not take our advice and reviews later, we can not deduct anything from monitoring the results.
If the student in the example from the figure repeats the Fact after thirty days instead of the recommended 16, we can, according to the new method, still compare the expected Retrievability (deducted from the chart) and the Retrievability (deducted from the points the student gets). From here, we can see how much our estimate of the curve shape was correct, and eventually adjust the next repetition according to this.
Stability is the time during which I remember a Fact (Stability Normative)
The example in the previous paragraph implicated, the “the Fact with STAB=50 must be repeated in approximately 16 days”.
This does not satisfy us from a purely esthetic angle. We would like to get the Stability directly as the number of days for which we will remember the Fact. That would make this abstract number a bit more human and understandable.
We will achieve this by adding a suitable constant to the formula.
We will call it the Stability Normative (SN).
RET = 1/2^{m*SN/STAB}
Maybe it is not that important how to calculate it exactly. The reader who is only interested in understanding the principle may skip over the entire rest of this paragraph.
For the others we state:
We wish that at the time of the recommended repetition applies:
STAB = m
The recommended moment for repetition is when the Retrievability drops by the Forgetting Index:
RET = 1 - OFI
By entering these two premises in the formula for calculation of the Retrievability, we get:
1 - OFI = 1/2^{SN}
And if we remember our high school material on logarithms, after several lines of boring adjustments, we will get:
But even now, we are still not completely happy. The SN constant depends on the Optimal Forgetting Index value. The opinions on when it is best to review can vary. It is even possible that every student has this value somewhere else. It is therefore possible that we will want to let the student to choose his/her OFI value, or we will even want to calculate this value from his/her long-term history of learning. The existing implementation of the RE-WISE method does not allow this; but what if we would like to improve the algorithm sometimes in the future? The change of OFI will influence, via the SN constant, the calculation of the Stability. When changing the OFI parameter, we would have to recalculate the Stability of all the Facts. The same Stability in two students could mean various levels of learning. We do not like this very much. How do we get out of this? We will calculate the SN as a generally valid constant from some “natural” OFI value. We have chosen 30%. Whoever chooses a different value of OFI, will loose the neat interpretation of the Stability as the number of days before the recommended repetition.
Stability and Stability Normative
If we summarize the previous thoughts, we get the following results:
Retrievability is calculated pursuant this formula:
RET = 1/2^{m*SN/STAB}
- m is the number of days from the last repetition.
- SN (Stability Normatively) is an esthetic constant which ensures that we can neatly interpret the Stability as a number of days.
It can be calculated like this:
SN = log_{2}(1/(1-0.3))
STAB (Stability) then means:
The number of days during which the Retrievability drops by 30%.
A 30% drop in Retrievability is the recommended moment for repetition and the usual value of the Optimal Forgetting Index parameter.
We composed a formula which models the forgetting process. But so far, we did not say anything about learning. And that is actually what we are interested in. Let’s proceed to the paragraph Growth of Stability during repetition.