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## Abstract

There is a dispute between scholars what determines success. For instance, some psychology books for psychology 101 writes, “aptitude is more important than attitude on reaching a success.” Another hand, Zeek Zeekler says,” Attitude^{[1]} not aptitude determines altitude.” In general, we hear often people credit successful individuals by using words gifted. My point here is to show that the effort is the main factor on reaching a significant goal. In order to make my point clear, I will go over some elements of the work that has been done by Herman Ebbinghaus at least before a century ago. Retention Curve of Ebbinghaus has a meaningful content because it does not rely just on intuition, but it details with mathematical language. Ebbinghaus sought a relationship between philosophy, psychology, and mathematics. The speculation about forgetting respectively memory retention has gotten in the spot inspected with mathematical strategies. Nevertheless, there are many factors, which cannot be adjusted in to formulas. So, Ebbinghaus knew his work couldn’t be precise. Therefore, he relied in probability by using Law of Errors. This technique enabled Ebbinghaus aiming accuracy on forgetting curve (retention curve).

## Preliminaries

**Retention Curve**

Abbildung in dieser Leseprobe nicht enthalten

**Figure 1. The ^{[2]} green curve presents the function as a relationship of percentage retention and time (hours respectively days) that occurs when we learn new material for the first time.**

**Retention: Learning for First Time:**

**A – 0 minutes: 100%**

**B – 20 minutes: 58%**

**C – 1 hour: 44%**

**D – 9 hours: 36%**

**E – 1day: 34%**

**F – 2 days: 28%**

**G – 6 days: 25%**

**H – 31 days: 21%**

## Stability of Memory Might NOT be under control

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**Figure 2 Strength of memory or stability of memory. Strength of memory is a factor in learning that we have a small or none control over it.**

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**Figure 3a. Repeating three times a lesson (Internet Work).**

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**Figure 3b Comparing the exponential functions with above Figure 3a [2]. Repeating a lesson changes the forgetting curve.**

**Repeating the lesson is under our control.**

## Repetition makes huge difference in Forgetting Curve

There are some expressions that have a close relationship with reviewing learning:

*Practice makes perfect – American Expression.*

*Repetition rings a bell –Ernest A. Yebaght*

*Experience has NOT substitution-Old Expression*

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**Figure 4. Repetition influences on the forgetting curve to change it from a function of exponential graph in to a function of constant graph. In other words, the exponential function tends towards the positive constant function that crosses y-axis at the point 100%.**

## Main Idea

**Several Facts on Retention**

**(Based on Herman Ebbinghaus Views)**

*It is not important how much we know, but how do we know - Aristotle*

**Herman Ebbinghaus furthered Aristotelian ideas in to the empirical experiment. He submitted his study on retention of syllables as function in 1885.**

**Fact # 1 - Ebbinghaus connected Philosophy, Phycology, and mathematics together by producing relevant results on shedding lights on retaining curve.**

**Fact # 2 – Retention is a decay function with respect to time that is stated in percentage units.**

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**Fact # 3 – Stability of memory: Retention curve may increase if stability of memory is greater.**

**Fact # 4 – Repetition: The curve of retention will increase the great percentage by reviewing learning.**

**Fact # 5 –[1] Spaced time: According to Ebbinghaus’ experiment, the retention of 38 repetitions of syllables distributed for a given time by one day has the same effect as 68 repetitions by the same given time distributed for three days. Spaced practice is a saving time and active increment in qualitative learning. **

**Fact # 6 – [1] Learning time: Mental vigor and receptivity are less active during the late of the day. The morning hours are more productive for study.**

**Fact # 7 - The lesson that is important to learner has a greater retention than the lesson that is not important to learner. **

## Fact # 1 Relationship among Subjects

*Learn how to see. Realize that everything connects to everything else. –Leonardo Da Vinci*

The closer relationship between different subjects should be viewed in the context of history. If we look at a short period of time on the past, we already might see the development of technology, social, economic, and natural sciences were less advanced. However, if we go back in a far distance of the time, there were just few subjects that contributed in the human knowledge. However, according to Aristotle [5] there are a number of different sciences, but they are not independent one of another.

In the ancient Greek philosophy included many subjects in its scope. The main source of those academic subjects was the necessity. People needed to build buildings, watering system for their plants and farms, engineering fortifications against enemy, counting their goods and property, etc. As a result of their needs established geometry. Whole substance of mathematics was solely condensed in Euclidean Geometry. If we go further in the time at Before Christ, whole knowledge was displayed in one subject. The bottom line is that all academic subjects came from one tiny subject; therefore, they are all connected with each other because academic subjects have the same root.

Herman Ebbinghaus investigated Aristotelian less concerned question such as how much we know, but he was not able to shade light on the other part of Aristotelian question, “how do we know?” Even though second question is very hard to answer, Neurology has done some progress [6] by explaining learning with processes of neurons, pathways of neurons, and synapses. There Learning is based on neural budding and new connections stabilized by repeated use.

Ebbinghaus created various short words in form of syllables that some of them did make sense and some of them did not. The only subject for the study he was himself. He took data about his learning respectively retention of the words after different point of times. According to his study he has gotten some of the results that are representing in the page 1 under the Figure 1. However, a disadvantage of a single-subject design is that it remains unclear what the shape of forgetting would be with other subjects [3].

The advantage of Ebbinghaus’ work is that he took data carefully and diligently created graphs related to the data. All dots of the data connected with the straight lines. The shape of the graph resembles decay exponential function. The whole experimental work communicates through the bridge of different subjects with intention to find out the truth about learning curve.

## Fact # 2 Exponential Decay

According to Ebbinghaus Learning curve, after thirty-one day the retention memory is 21%. The question rises, what happened after one year or two? For instance, [1] a poem is learned by heart and then not again repeated. Let suppose after half-year it has been forgotten: no effort of recollection is able to call it back again in to consciousness. At best only isolated fragments return. Even though, we forget learning after a long period of time, still some percentage of memory retention is present with the value grater than zero.

The best way to describe the memory retention is by using exponential decay equation. Exponential decay is described by the first-order ordinary differential equation

illustration not visible in this excerpt^{4}

## Fact # 3 Stability of Memory

In the figure 2 are presented five sequences of the stability of memory. Intuitively, we can see the spaces between curves of each function with higher strength of memory are declining. Let’s try to write a sequence that follows these function and input numbers in each sequence to verify the intuitively observation is conform with the analytical evaluation.

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In expression (1) we switch negative exponents in to positive exponents.

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Consider the time is constant, and for easy calculation let x=1. Then the expression (2) yields,

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Figure 5. The differences between two sequences have different values.

As we see in the Figure 5 the differences between any two sequences has a unique value. In other words, their differences are not equal, and they become smaller with tendency to zero. However, the sequence of the expression (3) is going to converge.

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Even if we take x = n or time t = n, the sequence [illustration not visible in this excerpt] is going to converge to less than 1 respectively less than a half.

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Regardless how high is stability of memory; the factor ** time is ** going to diminish retention to a certain degree.

The same curves we may elaborate from the perspective of areas by using integrals.

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Comparing or contrasting differences between every two integrals we will see the differences of integrals will corresponds with the differences of sequences.

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As a conclusion the difference between different strength of memory weakens as the strength of memory goes to a higher rating. Retention still increases with lower units.

## Fact # 4 Repetition of a Lesson

*Repetition is the mother of the study – Latin Expression*

After any lesson regardless of the academic subject, teachers give homework. Some teachers grade homework, another hand, other teachers do not. Nonetheless, homework counts as an accelerator of repeating a lesson in different ways, furthermore, helps students with expending their understanding about a specific topic. The more homework a student does, the greater is his grasp about the lesson.

Most of the students do not have a complete picture concerning a lesson when they encounter for the first time as Ebbinghaus underlined in his work. However, by repeating the lesson, they will get an enhanced image of the lesson. Modules differ one from another. Some of them are harder, and some are not. A complex unit requires more work. Anyway, most of the students spend about a week to prepare for an exam. If a student repeats his study guide material for six days, and he or she is able to pass successfully the exam, mathematically we can express with the following formula in the Equation 7.

Abbildung in dieser Leseprobe nicht enthalten

**Figure 6. Learning curve that occurs in six days study before an exam.**

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We can argue if the formula is right or not. There are many elements that are not adjusted [4] in the Equation 7; we will communicate with mathematical formula that will describe in the best possible way the learning curve.

Let analyze the graph of the learning curve. According to the Equation^{[5]} 7, 100% of vertical line number # 6 corresponds to the number value 4.24; the vertical line 0 corresponds to 0%. Rendering the other vertical lines of repetitions is going to emerge as follows.

**Vertical lines of repetitions with respect to numerical values:**

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**Vertical lines of repetitions with respect to percentage values:**

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The green lines are negative slopes from very steep in the beginning tending to the horizontal slope by reaching maximum retention curve.

## Fact#5 Spaced Time Learning

Spaced time learning respectively repeating a lesson it influences on extending retention memory about the lesson, and extends time duration of the lesson in memory. According to the study published in 2008 of the Author, Piotr Vozniak, optimal spaced time learning should be establish as follows:

- First repetition within the day 1

- Second repetition within the day 7

- Third repetition within the day 16

- Fourth repetition within the day 35.

Observing the space-time between repetitions we see an approximated sequence that signifies weeks.

illustration not visible in this excerpt

Estimating time between first and second repetition is one week, between second and third is two weeks, between third and fourth is three weeks. This method works well to memorize formulas for chemistry or other subjects, but the test must be after ten weeks. Usually, tests are given randomly and do not give a chance students to apply this model in their learning.

Whenever we learn something for the first time, afterward we think about what we have learnt. If the passage was harder, we think about the harder parts of the passages. Before the second repetition we think and prepare ourselves for questions. Spaced interval repetition gives a chance to the learner on thinking about the given material. Brain processes that information and cracks complications part of the lesson or it goes over the ways that do not posses the right answer.

Academic, Minir Dushi^{[6]}, used to say “ you are in the study mode if you think most of your spare time about your study.” When I was studying Physics as an undergraduate student, I was struggling to find out the proof of formula for double time when an object goes up and drops dawn. I did not have a chance to see how did the formula [illustration not visible in this excerpt] makes sense.

An early morning, I started to walk to my school that is thirty minutes far from my home. While I was walking, I thought about the formula that was bugging me for a period of time. I was walking closely for ten minutes, and the bell rang in my head. I went back home, and I solved the riddle. During this frame of time I was not at an active study form, but thinking about what I study before made a huge difference. In my opinion, brain works all the time throughout its life span. However, we should feed the brain with information that is essential and valuable to us; moreover, the brain will process them continually.

## Fact # 6 Biological Learning Time

**Think in the morning. Act in the noon. Eat in the Evening. Sleep in the night. William Blake**

Peak of learning time depends on individuals since there are some morning people and some are evening people. Usually morning people are old; another hand, evening people are young. The fact # 6 (Ebbinghaus fact) is not consistent even though [7] morning is more effectual time to study. Nonetheless, there are people who do not prefer morning as a study time.

It is obvious in the morning people are energized after the good sleep, and they are restful to tackle any task. There is inertia law (law of Physics) that contributes to the substances to keep their physical state (spot). During the sleep human activities are in the minimum; in order to increase the activity with tendency to maximum, it takes time and energy.

Researchers, Burrus, Robert T., and Edward Graham from North Caroline University did a significant research on the morning study time and its benefits agree the best time for study is morning. Nevertheless, they agree with the fact that morning study time is not convenient to everyone.

Personally, I study usually during the night. As a young student, I studied as a mining engineer in Kosovo. Most of the classes were in the morning, then after noon we studied the material that we explored in the classroom. I made as a routine to study from nine pm to two am. I woke up at nine am in the morning, and I started the day with my daily schedule. When I asked my classmates what was their favorite study time, most of them preferred the night study time.

Humans can adjust on their life circumstances. Life style orders many of our behaviors. Students who come from families with low income must work a part time job to support family and their college expenses; thus, their only spare time for study is night study time. Once students make a habit to study at night, furthermore, they will fill more comfortable to study at night than in the morning.

We are dependent on our habits. When we make a habit then we usually tend to follow it. Let say we have in the disposition a group of students and test their learning. Then, the same group of night study time learners changes their habit. Assume, they agree and they learn during the morning. After, a period of time, we give them a similar test with the previous test that measures their learning. Afterward, we compare results of their tests. This would give an accurate answer when is the peak learning time. Until, we come up with a proficient device to measure learning study time we `will stick with an inconclusive solution about the study time.

## Fact # 7 Learning that makes Sense

**My style is unique and random. But I think it’s important that it makes sense**

**Jess Glynne**

Whatever subject or lesson is important to us, we pay attention and time to it. There must be the reason why a lesson is important to us. For instance, a topic in teaching and learning that was important to me was creativity. During this lesson in the class, I paid whole attention. Then, I prepared a presentation about creativity in this class. In the process of my presentation, I collected lots of material and I synthetized. While I was preparing my presentation was everything clear. When I addressed the presentation in the class, it did not make sense to some of my classmates. Creativity was not important to those students; therefore, it did not make sense to them.

Let take a simple example with adding positive integers,

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Equation (a) is a simple example and it males sense because it was important to me adding integers when I was in elementary school and high school. Let try another equation,

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Equation (b) is the sum of arithmetic series that makes sense to lots of high school students and college students. Let try another equation,

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Does equation (c) makes sense. All the integers are positive integers [illustration not visible in this excerpt]. How it is possible to get a negative sign. Also, all the integers are whole numbers. How it makes sense the result is a rational number that is smaller than the first integer.

By inspecting the equation (c) intuitively, it does not make sense. However, there are mathematician who agree with this result is correct. Also, there exists the proof of this equation that requires mathematical skills to prove and understand. Mathematicians and scientists to whom is important quantum mechanics or Physics the equation (c) makes sense. Things that are important to us make sense, hence, we think often about them and we wont forget them fast.

## References

[1] Memory a Contribution to experimental Psychology by Herman Ebbinghaus Private docent in Philosophy at University of Berlin 1885

Translated by Henry A Ruger, PhD and Clara Bussenius

Publisher: Teachers College, Columbia University NYC, 1913

[2] Series Influential Educators, Herman Ebbinghaus & Forgetting curve, McGraw Hill Education, www.mheducation.ca

[3] Replication and Analysis of Ebbinghaus’ Forgetting Curve, Jaap M. Murre; Joeri Dros, July 6, 2015, www.Plos.org

[4] Real Numbers don’t Cut in Real World, This Physicist (Nicolas Gisin) Argues, written by Emily Conover, www.sciencenews.org

[5] Aristotle on Education: Extracts from Ethics and Politics, Aristotle Translated and edited by John Burnet Professor of Greek, Cambridge at the university press 1913

[6] Thinking About Teaching and Learning, Robert Leamnson, Stylus Publishing Inc. Sterling, Virginia 1999

[7] “Early Morning Classes and Finance Student Performance.” Burrus, Robert T., and Edward Graham. (n.d.): n. pag. Print. University of North Caroline

^{[1]} Prodigies in a field show enormous success for a short period of time. Obviously, they are inclined (attitude) in a specific field.

^{[2]} Above green curve signifies forgetting curve, while under curve is retention curve. The green curve is learning curve.

^{[3]} In the long run, crystal intelligence surpasses fluid intelligence.

^{[4]} Exponential functions have horizontal asymptote; therefore, it describes the best why learning will not disappear completely.

^{[5]} The Equation 7 may be enhanced if we include more parameters or use a more sophisticated equation. The learning curve depends on complexity of lesson and quantity of repetition. In other words equation of learning curve fluctuates.

^{[6]} Minir Dushi PhD is a professor of mining engineering in Kosovo.

- Quote paper
- Duli Pllana (Author), 2018, Repetition Diminishes the Curve of Forgetting, Munich, GRIN Verlag, https://www.grin.com/document/427645

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