 Open Access
 Total Downloads : 443
 Authors : Gideon. K. Gogovi, Francis. T. Oduro, Gabriel O. Fosu
 Paper ID : IJERTV3IS10500
 Volume & Issue : Volume 03, Issue 01 (January 2014)
 Published (First Online): 20012014
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Performance Analysis of Memorization Rate Models
Gideon. K. Gogovi1, Francis. T. Oduro2 and Gabriel O. Fosu3
Department of Mathematics, Kwame Nkrumah University of Science and Technology, Kumasi, Ghana1,2,3
Abstract
This paper is to corroborate the validity of the rate of memorization model without forgetfulness factor and with forgetfulness factor. These models wereanalyzed and solved and were practicallyexperimented on students.We showed the possibility that short term and long term memory is different but essential for the establishment of long term memory.
Keywords Forgetfulness, Memorization model, memory storage

Introduction
Although humanmemory is usually robust and accurate, different disease processes can disrupt memory and cause either distortions or outright failure [4]. Forgetfulness can be attributed to both physical andpsychological causes. Some causes are reversible while others can be managed with medication [5]. We seek to perform anexperimental analysis on the rate of memorization model with and without forgetfulness.

The Models
A linear differential equation model of memorization without forgetfulness is given as[6]:
dL k (1 L)
Due to the complexity of the human mind and its ability to store memory, the information received from shortterm studies is limited. While memory is
crucial for all of us, there is no time during which
where
dt 1
memory demands are greater than the school years. The
k1isa parameter that measures the retention after
school environment, however, is not often a memory friendly one. Children are presented with new information all throughout the school day and given little opportunity to consolidate new information before more new information is presented to them [1]. Contrary to popular belief, being smart is not synonymous to having a good memory or good retention but lies in the lifestyle of a person, attitude, diet and habits [2]. Things learnt can also be forgotten just as for memorization, if they are not constantly revised or practiced. According to Cowan [3], new information must make contact with the longterm knowledge stored in order for it to be categorically coded. The person who actually learns, rather than merely memorizing is not only able to relate existing
memorization, L(t) Fraction of list learned at time t , L(t) 0 Knowing none of the list, L(t) 1 Knowing the entire list, (1 L) says that the entire list
learned is subtracted by a fraction of the list learned.
The model is based on the assumption that the rate of learning is proportional to the amount left to be learned.
dL k (1 L) dt 1
1 dL k dt
1 L 1
knowledge and apply it to new situations but more
importantly, he can critically judge ideas of learned people.
k1
ln 1 L
(1)
t
Equation (1) is used to measure the rate of retention. The solution to the differential equation gives a continuous function of the amount of the list learned with respect to time.
The fraction of items learnt at any time t is then calculated by plugging the value of k1 into the equation and solving as in;
So that
And
2 1 = 0
1 + 2
2 = 1
eln(1L) e k1t () =
1((1 +2 )) 1
1 + 2
L(t) 1 e k1t (2)
The limit as time approaches infinity is given as:
The parameter that measure the retention after memorization k1is different for each individual.
lim
1((1 +2 )) 1
1 + 2 =
1
1 + 2
When forgetfulness is taken into account, the rate of memorization of a subject is given by[6]:
= 1
A 0 k1 k2 1is the rate of absorption and
C k1 is the constant of retention of the subject.
k1 k2
1 2
This implies A k1 which means that, the retention
Where k1, k2 0, and again, () is the fraction of material memorized in time , and 1 is the fraction remaining to be memorized.
Assuming that 0 = 0, solving for () and finding the limiting value of as , we have:
1 2
= 1
C
constant is inversely proportional to the rate of absorption. One can increase the rate of retention only (without increasing the absorption rate in the short term) by increasing k2 . This will result in the fact that the person is absorbing extra amount of information that will eventually not be retained [6].

Experimentation and Discussion of Results
=
(
+ )
The real data was compiled by testing two
1 1 2
students with list1 (list of integrals) and list 2 (three
digit numbers). The list was studied at oneminute
1 (1 + 2)
=
intervals and the students were then required to
reconstruct the list from memory. The first part of the experiment did not take into account, the fact that a
ln( 11 (1 + 2)) = +
(1 + 2)
ln( 1 (1 + 2)) = 1 + 2 + 1
() = 2 ( 1+ 2) 1(3)
1 +2
Now, it is assumed that 0 = 0, thus
student can forget a number after memorizing. The process was repeated ten times or until the list learned in its entirety for each of the students. After compiling data, the experiment was repeated with the rate of forgetfulness factor taken into consideration. The results are shown in the figures below.
Figure 1: Rate of memorization for student A on list 1 and 2 without forgetfulness factor.
From Figure 1, it is observed thatstudent A has low retention rate compared to studentB in Figure 2. The curves show a good approximation of the rate for this persons memory. It will take student A and B 20 minutes and 11.25 minutes to study a list of 50 three digits numbers, and 40 minutes and 22.5 minutes to study a list of 100 three digits numbers respectively. We now assume forgetfulness in the model.
Figure 2: Rate of memorization of student B on list 1 and 2 without forgetfulness factor.
Figure 3: Rate of memorization of student A on list 1 and 2 with forgetfulness factor.
Figure 4: Rate of memorization of student B on list 1 and 2 with forgetfulness factor.
From figure 3 and 4 it can be seen clearly that, the model predictions with forgetfulness did not start from zero (0). The rate of memorization of student A on the list with forgetfulness factor taken into consideration started from a fraction of about 0.6 while that of student B on list 1 and 2 with forgetfulness factor started from a faction of about 0.2. This means that
taking into consideration forgetfulness; it has reduced the proportion of the total amount L to be memorized. The explanation to this is that with time, the student can memorize a fraction of the list of L information available due to the fact that he will forget some of the information learned.

Conclusion
Two differential equations are practically analysed in this study, the rate of memorization model with and without forgetfulness. These models were solved and were experiment on two students. The model works better than one might reckon, notwithstanding the complexity of the human mind.These differential equations are good model for describing the rate of remembering without taking into accounts forgetfulness on a short term basis and taking into account forgetfulness on a long term basis.
References

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W. R. Klemm, What good is learning if you don't remember it? The Journal of Effective Teaching, 64, 2007.

I. K. Dontwi et al.,Modeling Memorization and Forgetfulness Using Differential Equations. Progress in Applied Mathematics, 6 (1), 2013. 111