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During the repetition, it can be estimated to what extent the student really remembered the Fact. The software can evaluate the correctness of his/her answer automatically, or it can ask the student for the evaluation. This will provide a feedback which evaluates the correctness of the suggested process of tutoring. How will we use this feedback for refining the method?
This can be achieved by declaring some of the numbers in the formula as parameters, whose exact value we don’t know, but that we are searching for. The previous text suggests that the Initial Stability and Base Factor are the candidates for the adapted parameters.
The Initial stability in the formula indicates how well we have known the Fact even before we started using the RE-WISE method. At the beginning, we choose the iniStab=1. This is interpreted as we are seeing the Fact for the first time and we will only remember it for one day.
The Base Factor states the difficulty of a Fact for the student. If the Fact is difficult, or the student has a weak memory, the Base Factor is low. With repetition, the Stability improves less than in the case of an easy Fact, and therefore it is necessary to repeat more often.
We will remember the evaluation the student gets during the individual repetitions. We will try to set the Initial Stability and the Base factor so that the student’s evaluation corresponds as much as possible to the forgetting curve.
We have two ideas how to do it.
- A simple Deductive method which deduces the Stability value from only one last repetition.
- The more complex Prognostic method, which predicates the course of learning according to the entire history of repetitions.
We will describe both ideas in more detail and describe their advantages and insufficiencies. Then, we will suggest their compromising combination, which is implemented in the RE-WISE method.
Let’s assume for the beginning, that we will learn the accurate evaluation from the user, so we will be able to determine the Retrievability value in percentage, thus rather accurately. Only later we will consider that the student is giving us this information on a very rough scale.
Deductive method
Let’s presume that the student will tell us the Retrievability at the moment of the repetition. In other words, that he/she will say to what extent he/she remembers the fact. If we remember the interval from the last repetition, we can deduct from this data according to the proposed formula of forgetting the following:
Stability after the past repetition:
RET = 1/2m*SN/STAB => STAB = m*SN/log2(1/RET)
Detail deduction for fans of logarithms:
RET = 1/2m*SN/STAB
We logarithm:
log2(RET) = log2(1/2)*m*SN/STAB
We multiply STAB and divide log 2(RET):
STAB = log2(1/2)*m*SN/log2(RET)
From high school, we remember that:
log2(1/2) = -1
-log(x) = log(1/x)
Therefore we get the advised:
STAB = m*SN/log2(1/RET)
Based on the Retrievability in this repetition and the stability achieved in previous repetition, we are able to forecast the new Stability value:
STABi+1 = STABi * CHNG(RET)
The deductive method only needs the data from the last repetition. It does not consider the entire previous course.
It sensitively reacts to any immediate changes. We consider it a desirable flexibility. But it can also be an insufficiency, because the study plan will sharply deviate, even if the worsened memory is caused by some momentary outage, for example tiredness or the slight consumption of alcohol.
Another weakness is in the fact that it requires accurate information on the Retrievability value. However, one can realistically only expect from a student an evaluation from 0-4 points, or even only the information “I remember/I forgot”.
Besides, the Prognostic method does not tell us anything about whether we have correctly chosen the Base Factor value. And, it only considers the Initial Stability during the first repetition.
Prognostic method
The prognostic method compares the Retrievability value as forecasted by the formula with the evaluation achieved by the Student.
The method tries to choose the formula parameters for calculation of Retrievability – Base Factor and Initial Stability – so the deviation is as small as possible.
The figure lists the course of the Retrievability at the chosen Initial Stability and base Factor (black line).
A student is evaluated on a scale of 0-4 points. Therefore, we will only learn at a rather rough interval what the real Retrievability is. This interval is indicated by the thick green line.
The difference between the prognosis and evaluation is indicated by the thick red lines. They show, to what extent the values of the iniStab and Base Factors conflict with the reality. We can calculate a penalty for a certain combination of the parameters. We will use, for example, the mean square deviation, thus we sum the squares of the lengths of the red lines. Then, we are searching for such a combination of the parameters, which gets the smallest penalty.
The chart of the penalty depending on two variables is a two dimensional surface, as a carpet floating in space. We are only interested in the lowest points the carpet is in. (Let’s hope any mathematicians reading will forgive us.) In our example on the picture, a part of our carpet is even lying on the floor – the penalty value equals zero.
From these points, we will conservatively choose the one which is closest to the default parameter values, as we determined them in the last repetition. This point is indicated by a red square on the figure. Because in this case, the carpet lies in this point the closest to, or even directly on, the floor, we arrived via this complicated procedure to the decision, that this time we will not change anything.
The implementation of the RE-WISE method examines only the values from some surrounding of the existing iniStab and Base Factor. That’s how the algorithm is protected against dramatic changes.
The advantage of the prognostic method is that it works with the entire existing history of repetition. We also learn the iniStab and Base Factor values from it, which provide us interesting information for the evaluation of Fact difficulty and the student’s memory.
On the other hand, the weakness is that this method is not very sensitive to a memory failure in a later repetition. If I remember a Fact four times, but forget it in the fifth attempt, the method will still offer quite a long period before the next repetition.
Compromise determination of Retrievability
In the previous explanation, we have repeatedly seen the trouble that we would need to know a quite accurate real value of the Retrievability at the moment of repetition.
The student will only give us rough information. The RE-WISE method uses evaluation on a scale of 0-4 points. The table shows the relationship between the point and the Retrievability:
| Points |
Retrievability |
|
4
|
80 - 100%
|
|
3
|
60 - 80%
|
|
2
|
40 - 60%
|
|
1
|
20-40%
|
|
0
|
0-20%
|
We will use the forecasted value when determining the precise value of Retrievability. It will be provided by the forgetting curve model.
We will consider the predicted Retrievability correct, provided it does not disagree with the achieved evaluation. If we predicted RET=82% and the student gave him/herself 4 points, it is OK and we believe the RET value=82%. The deviation from the expectation is zero.
In case our forecast varies from the awarded grade, let’s determine a compromise Retrievability so it suits the grade, but is as close as possible to the expected value. If we forecasted RET=82% and the student got evaluation 1 point, we will work with the compromise RET=40%. The deviation of the expectation is 42%.
Crossed deductively-prognostic method
The RE-WISE combines the Deductive and Prognostic methods so we can use the advantages of both approaches and minimize their insufficiencies as much as possible. First, we will use the Prognostic method. We will adapt the values of the Initial stability and Base Factor so that the forecasted Retrievability value in all repetitions corresponds as much as possible to the real evaluation the student achieved.
This calculation also gives the forecast of the Retrievability for the existing repetition. We will compare it to the real evaluation and determine a compromise Retrievability.
Here, we will use as the input parameter in the formulas of the Deductive method:
STAB = m*SN/log2(1/RET)
STABi+1 = STABi * CHNG(RET)
If the predicted value of the Retrievability corresponds to the evaluation, the Deductive method will calculate the same stability as we calculated in the Prognostic method.
If the Prognostic method did not estimate the last evaluation correctly, the Deductive method will temporarily deviate the Stability up or down.
The combination of both methods flexibly responds to changes, but if the memory deviations stabilize, it returns back to normal.
Default Values iniStab, Base Factor
We said that unless we know something more accurate, it is reasonable to use the following initial values:
IniStab = 1
Base Factor = 2
If we were able to estimate which Fact is more difficult and which is less difficult, we could adjust the initial values and the student would get faster to the optimum values of these parameters. We could, for example, monitor the learning results of the same set of Facts in various students. The RE-WISE method does not use this idea so far, but the data structures are ready for it.
Similarly, we could monitor whether the student has an iniStab and Base Factor that is average or higher than the initial values. If, for example, Base Factor was 3 on average, we would assume that the pupil is 1.5 x smarter than the average. In the case of new facts just implemented into the learning process, immediately increase the value of the Base factor by 1.5 x. The RE-WISE method uses this improvement.
This is basically the end of our thoughts which deal with how many days to recommend the next repetition of the fact. One should also mention what happens within one day when studying using the RE-WISE method. This is covered by the paragraph Daily Plan.