=Paper=
{{Paper
|id=Vol-2387/20190286
|storemode=property
|title=Educational Analogy Dedicated for Didactical Process Simulation
|pdfUrl=https://ceur-ws.org/Vol-2387/20190286.pdf
|volume=Vol-2387
|authors=Paweł Plaskura
|dblpUrl=https://dblp.org/rec/conf/icteri/Plaskura19
}}
==Educational Analogy Dedicated for Didactical Process Simulation==
https://ceur-ws.org/Vol-2387/20190286.pdf
Educational Analogy Dedicated
for Didactical Process Simulation
Dr Eng. Paweł Plaskura[0000−0002−8257−781X]
Faculty of Social Sciences, Jan Kochanowski University,
Słowackiego 114/118, 97-300 Piotrków Trybunalski, Poland
p.plaskura@unipt.pl
Abstract. The article presents a new method of modelling the didactical pro-
cess using a developed educational network and the microsystems simulator. The
didactical process can be represented in the intuitive form of a network of con-
nected elements in a similar way to the electrical circuits. The network represents
the differential equations describing a dynamic system which models the infor-
mation flows as well as learning and forgetting phenomena. The solutions of the
equations are more adequate than the direct formulas used in modelling ie. the
learning and forgetting curves known from the literature. The network variables
and their meaning are relative to generalized variables defined in the generalized
environment. This enables using any of the microsystems simulators and gives
access to many advanced simulation algorithms. The paper can be interesting for
those who deal with modelling of the systems which incorporates the learning and
forgetting process, in particular, in production processes or learning platforms.
Keywords: Didactical process simulation · Didactical process modelling · Edu-
cational analogy · Learning and forgetting curves · Educational environment
1 Introduction
The paper presents a new approach to modelling of the didactical process using the ed-
ucational network defined by the author. The main reason for dealing with the subject is
the apparent lack of use of modern methods of description and simulation in the didac-
tics [37]. However, the mathematical models using direct mathematical formulas were
previously created [28].
The initiation of the work was a series of works from various fields of science
[12,28,25,19] and knowledge in the field of numerical methods and microsystems sim-
ulation methods [31,29]. There are many scientific works in each area. The most impor-
tant of them, from the point of view of the article, are cited. Detailed issues raised in the
article can also be found in the articles [36,32,34,33].
The analysis of the learning and forgetting process is based on forgetting curves
[12,46,19] represented by the direct formulas. It was studied in the nineteenth century
by Hermann Ebbinghaus [12] who first proposed forgetting curve (FC). The forgetting
curve is very fast and after about 5 days about 25% of knowledge remains, then the fall
is slower. After 30 days about 20% of the knowledge remains. This curve is still the
subject of research. FCs are the solution to the respective differential equations. Their
form and coefficients allow matching values to measured data obtained during the exper-
iments. FC describe a dynamic model of brain activity in the sphere of learning. Different
functions are used to describe the forgetting curves such as power or exponential [19].
Superpositions of functions (e.g. superposition of exponential functions [49,48]) and
the more complicated models such as Memory Chain Model [25]) are also used. The
learning curves are implemented in repetitive algorithms in many programs i.e. Super-
Memo [4], Anki [2]. The learning platforms also support the didactical process (Smart
Learning Platforms) [14]. The main problem is collecting the user activity data [45,6].
Currently, the analysis of the didactical process is more often based on the large amounts
of data (BigData) [13,10]. Although, the modelling of forgetting is very important not
only in the didactical process. The models are used to describe the efficiency of repetitive
operations on production lines [21,7,20]: hyperbolic and exponential models [23,44,5],
multiparameter and multidimensional models [8,24,47]. The mentioned above models
are mainly based on the analytical equations. Their values of parameters are very sensi-
tive and not intuitive. Even small changes in parameters values strongly affect the result.
The better approach to the problem is to find a model based on a differential equation.
However, differential equations are difficult to arrange and solve. One should look for
methods of more intuitive representation of equations and methods of their effective so-
lution. Example of the model of the brain activity at the level of neurons described as
the electrical circuit can be found in [16].
The universal educational environment presented in this work allows the modelling
of information flows and their collection in the learning and forgetting processes. It en-
ables the representation of network equations in the form of a schematic (diagram) of
connected elements at different levels of abstraction. The network is created by trans-
forming the equations of the generalized network into the educational environment by
using analogy [40].
The aim of the article is to present the developed educational environment and the
educational network and its applications to modelling and simulating (monitoring) of
the didactical process.
The educational environment allows a relatively simple description of various com-
plex phenomena by using elements described by their mathematical models. The net-
work can be represented in the form of a block diagram or connection diagram of net-
work elements (schematic). The network is an intuitive representation of the set of dif-
ferential equations describing the didactical process, here in a similar way to the elec-
trical schematics. The solutions of the network equations are, in particular, equations
describing the forgetting curves known from the literature [12,46,19]. The network can
be generated manually (simple didactical process) or automatically (complex didactical
process) and can be easily analysed and optimized by using microsystems simulator. It
enables the analysis of very complicated didactical processes.
The network equations are formulated automatically thanks to the use of templates
[18,27,30] discussed below. It is possible to select the values of the elements parameters
and/or change the structure of the network in terms of design constraints. Behavioural
modelling enables the use of direct formulas in the element models. In the paper, devel-
oped by the author Model Definition Language (MDL) implemented in the Dero simu-
lator was used [30].
The educational environment and models described below were implemented on
the Quela [42] platform. The platform was design based on the DIKW [9,38,22] model
(Data, Information, Knowledge, Wisdom). The presented approach models the didacti-
cal process at the first, second and third level of the DIKW model. The 4th level can be
also modelled by user profiling what is not discussed here.
An example of network simulation will be shown later in this work. The didactical
process mathematically modelled by using direct mathematical formulas [28] can also
be implemented using described below network as well.
2 The Theoretical Backgrounds
The circuit simulators are specialized programs that solve differential equations. The
equations are described in various forms, in particular, as mentioned above the network
of connected elements. The simulator in the field of electronics implements the nodal
approach to the network analysis [18]. Development of the hardware description lan-
guages [1] lead to the microsystems simulators. The microsystems simulators are able
to analyse the systems which belong to different environments. Each environment has its
own restrictions due to the simulation process i.e. the variables should be analysed with
individual accuracy, the model inputs and outputs values differ significantly. The edu-
cational environment defined below has also its specification, i.e. long time simulation
has to be taken into account. The way to overcome difficulties is to define a general-
ized environment that defines i.e. nodal (effort) and branch (flow) variables. In each new
environment, you can define nodal and branch variables and their accuracy of analy-
sis [41,11]. Equations that occur in different environments are often the same. Thanks
to environments, it is possible to use analogies and simulations of systems from dif-
ferent environments using any microsystem simulator. This enables the analysis of e.g.
electrical-mechanical systems. This approach was used to create an educational environ-
ment. The most important issue is to define network and branch variables and give them
their meanings (Table 1). The educational environment variables correspond to the gen-
Table 1. Generalized variables for the electrical and educational environment.
Generalized variables Electrical env. Educational env.
e effort v voltage k knowledge
f flow i current i information flow
p state q charge q information
W energy E energy E workload
W work W work W work
eralized variables and electrical variables as well. Three basic variables are information,
information flow and knowledge. The variables can be shown as the vector (1).
x = [k, i, q]T (1)
where: k - variables related to knowledge, i - information flows, q - variables describing
unit information. According to the Modified Nodal Equations [18], let us define the basic
branches of the network and its equations:
1. branch describing information flow i = fi (x, ẋ, t),
2. branch describing the level of knowledge with the flow of information as unknown
k = fk (x, ẋ, t),
3. branch describing the level of knowledge with the flow of information as unknown
q = fq (x, ẋ, t),
where ẋ is a time derivative of variable x. Let us use the electrical schematics to repre-
sent the elements equations. Other graphical representation is also possible but is not so
intuitive and well known. Formulation of network equations is accomplished by apply-
ing so-called templates [18,27,30] described below.
ma mb
ia
kb
ma mb
Raa nb
ia kb
ka Kx (ic , kb )
nb
ka Raa Iaa Ix (kb , ic )
Kaa
ic ic
na na
(a) branch of information (b) branch of knowledge
Fig. 1. The network branches.
The branch of information The branch of information is described by the (2) and its
template by the (3).
ia = k a /Raa + Ixb k b + Ixc ic + Iaa (2)
kma
kna
+1/Raa −1/Raa +Ixb −Ixb +Ixc kmb = −Iaa (3)
−1/Raa +1/Raa −Ixb +Ixb −Ixc
knb +Iaa
ic
The equation can be represented in the form of the schematic (Fig. 1a). The meaning of
the elements results from their equations. The Raa element models losses in the trans-
mission of information between nodes ma and na . The Iaa is the source of information.
The Ix∗ is the controlled source of information which is used in the modelling of the
didactical process.
The branch of knowledge The branch of knowledge is described by the (4) and its tem-
plate by the (5).
k a = Raa ia + Kxb k b + Kxc ic + Kaa (4)
kma
kna
+1
kmb
−1
knb =
(5)
−1 +1 +Kxb −Kxb Kxc Raa −Kaa
ic
ia
The equation (4) can be represented in the form of the schematic (Fig. 1b). The mean-
ing of the elements results from their equations. The Raa element models losses in the
transmission of information as described above. The Kaa is the source of knowledge.
The Kx∗ is the controlled source of knowledge which is used in the modelling of the
didactical process.
2.1 The Basic Network Elements and Their Meaning
The use of predefined element models allows to easily create a network description us-
ing graphical symbols. Network equations can be formulated automatically by using the
Modified Nodal Equations [18] as shown above. Let us use again the electrical schemat-
ics to represent the elements equations. The basic elements of the network and its equa-
tions are shown in Fig. 2. The elements equations can be represented according to the
ma ma ma ma
ia
ka Ra ka Ca ka Ka ka Ia
ia ia ia
na na na na
ka
ia = R a
i = Ca · dka
dt
Ka = f (t) Ia = f (t)
(a) R (b) C (c) K (d) I
Fig. 2. Basic elements of the network.
branch equations described above. The template for R element using the information
branch is represented by (6).
+1 kma 0
−1 kna = 0 (6)
−1 +1 Ra ia 0
Template for element C (Fig. 2b) depends on the analysis. In the transient time anal-
dt is calculated using differential formula (7) [27,31,30] -
ysis, the derivative k˙a = dk a
trapezoidal method [26,27] or better Gear formulae [15,27,31].
ẋ = γ x + dx (7)
where: dx is a vector of the past of x. Substituting to the equation in Fig. 2b and per-
forming the transformations we get (8).
γCa ka = −Ca dka (8)
where dka = dkma − dkna is a vector of the past of ka . By saving the matrix equation,
we get a template of the form (9).
+γCa −γCa kma −Ca dkma
= (9)
−γCa +γCa kna +Ca dkna
2.2 Solving Network Equations - Classical Time Analysis
The network equations presented above are algebraic differential equations (10).
f (x, ẋ, t) = 0 (10)
where x is an unknown, ẋ = dx dt is the time derivative of x. The classical time analysis
makes it possible to determine the time response of the network. The solving process
begins with calculating the derivative ẋ (discretization) using a differential scheme (11).
ẋn = γ xn + dxn (11)
where: n is the identifier of the variable, dx is a vector of the past, γ a coefficient depend-
ing on the type of differential scheme. In this way, we will obtain a system of non-linear
(possibly linear) algebraic equations (12).
f (xn , γ xn + dxn , tn ) = 0 (12)
It can be solved, for example, by the Newton-Raphson method [18,27,31] by linearizing
equations, i.e. expanding into the Taylor [43] series to the first order.
" #
δf (xp−1
n , γ xn
p−1
− dxn , tn ) δf (xtp−1 , γ xnp−1 − dxn , tn )
+γ n
(xpn − xnp−1 ) =
δxtn δ ẋtn
| {z }
δf p−1
δxn
= − f (xnp−1 , γ xnp−1 − dxn , tn )(13)
| {z }
f p−1
where p is the number of Newton’s iteration. After grouping, it takes the form of equa-
tions solved iteratively (14).
δf p−1 δf p−1 p−1
xn p = −f p−1 + xn (14)
δx δx
| {zn } x {z n
|{z}
| }
Y B
The system of linear equations Y x = B can be solved by using i.e. the LU method [27]:
L(U x) = B.
Further information on the methods and algorithms of network analysis can be found
in the literature [18,27,31,30]. The process of solving network equations can be found
in [27]. Implementation in the Dero simulator is described in details in [30,31].
3 Didactical Process Modelling
As mentioned above, the educational environment enables describing the didactical pro-
cess in the form of the network of connected elements. Each element is described by its
model (equation or set of equations). The didactical process needs several models, in
particular, model of the didactical unit, learning and forgetting, exam, and evaluation.
The models were developed due to the DIKW [9,22] describing relationships between
parts of the educational process.
Didactic process in the context of learning objectives The information can be classified
in terms of the didactical objectives. According to Bloom’s Taxonomy [3], the main
categories of learning objectives (o) can be distinguished: knowledge, comprehension,
application, analysis, synthesis, evaluation (Table 2). A simplified model can use for
Table 2. Categories of learning objectives.
K knowledge
C comprehension
P application
A analysis
S synthesis
E evaluation
example concepts, procedures, achievements. Thus the learning and forgetting can be
simulated in relation to the categories of learning objectives (LO). The LO can be treated
as a vector of coefficients (15) used to profile student knowledge.
o = [oK , oC , oP , oA , oS , oE ]T (15)
The values of network variables for each category of learning objectives can be used
to monitor every part of the didactical process. The LO coefficients can also be used in
the evaluation or optimization process. Let’s rewrite the vector (1) including objectives
(16).
xo = [ko , io , qo ]T (16)
where: ko , io , qo are the corresponding vectors of variables respect to the objectives cat-
egories (Table 2). This changes elements models and makes models more complicated.
Every learning objective has to be modelled separately.
Model for knowledge management and data value extraction The DIKW [9,22] model
shows the relationship between parts of the educational process. It connects data with
information, knowledge, and wisdom. The basis of the model is data that come, for
example, through research or discovery. Data is converted into information by presenting
it. Information is converted into knowledge. Knowledge changes dynamically and allows
to look at a given issue from different perspectives. Knowledge is difficult to transfer to
another person. Wisdom is the ultimate level of understanding where main rule plays:
analysis, synthesis, and evaluation (in terms of Bloom’s taxonomy [3]). Experiences can
be shared with other persons. Conversion between individual levels of the DIKW [9,22]
model occurs in the system described here. Data, in general, are represented by didactical
materials included in the didactical units. The amount of data and form determines the
ease of data acquisition at the stage of conversion to information. Information presents
data. The parameter here is the amount of information being transmitted per unit of time.
Knowledge is the amount of processed information stored in the learning process. It can
be described by appropriate numerical measures. Wisdom is experience in the use and
processing of knowledge in practice. The numerical values of network variables related
to the relevant didactical objectives are the measure of knowledge. The interrelationships
between values for the category of the learning objectives form the student’s profile
(describing wisdom).
Didactical units and course The learning process creates engrams [49] or sets of en-
grams. However, the process of engrams creation still requires research. Taking into
account the DIKW model, it was assumed that each engram is created by the set of
information connected to the set of data, which may not be the case in general. The
learning process is modelled as a collection of independent (learning) paths creating the
sets of engrams [49]. Taking this into account, it was assumed that the didactical unit
(DU) can be decomposed into the collection of pieces of information (parts) connected
with the data (didactical material DM) and forming the appropriate set of engrams. The
didactical unit usually uses many different didactical materials. The course is the col-
lection of the didactical units. The knowledge level of the DU is the superposition of
the knowledge level of its parts (DM). The presented modelling method corresponds to
the method evaluation of knowledge for individual categories of learning objectives. All
topics must be represented in the exam test to correct evaluation of the whole process.
The whole didactical process The result of the whole didactical process is the superpo-
sition of the knowledge levels for each course (DC) as shown in Fig. 3. In this way, the
total level of knowledge over time is determined. The total level of knowledge changes
over time due to material repetitions.
DC1
k1
(t )
DC2 k2 (
t)
P k(t)
DC3
k3 (t) DCgain
..
.
(t )
kN
DCM
Fig. 3. Total level of knowledge of the didactical process.
Material retention Every repetition of the single DM is represented separately as shown
in Fig. 4. The total knowledge level of the DU is the superposition of the knowledge level
Didactical Unit
Model of
learning &
forgetting
DM1 (t1,n+1 ) DM1 (t1,n+1 )
1
0.95
0.9
0.85
kD
0.8
0.75
0.7
0.65
0.6
0.55
0.5
0.45
M
,1
0.4
(t )
0.35
0.3
0.25
0.2
0.15
0.1
5 · 10−2
1
0
time 0.95
0.9
1
0.85
0.8
0.75
0.7
0.65
k(t)
0.6
P
0.55
0.5
0.45
0.4
0.35
0.3
0.25
0.2
DUgain
0.15
0.1
5 · 10−2
0
time
(t)
1
Model of , 2 kDU (t)
M
learning & kD
.
forgetting
DM2 (t2,n+1 ) DM2 (t2,n+1 ) .
1
0.95
0.9
0.85
0.8
0.75
.
0.7
0.65
0.6
0.55
0.5
0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
5 · 10−2
0
time
1
Fig. 4. Retention of the materials in the single didactical unit.
of its parts (DM). The total knowledge level of the didactical course is the superposition
of the knowledge level of every DU.
3.1 Didactical Process As a Network
The information (2nd) level of the DIKW [9,38,22] model is responsible for the presen-
tation of data and is related to the flow of information. The information flow is modelled
by the elements of the previously discussed educational environment. The parts of the
DU studied in the paper are modelled as the network of information sources. The in-
formation can be collected by the student who transforms them into knowledge. The
model is composed of an information source and an information resistance correspond-
ing to difficulties in providing information (Fig. 5). The source of information is related
to data. The efficiency of the information source is related to the speed of information
transfer per unit of time. Information sources may be related to different categories of
i(t)
ma RK...E
i PW
I2
R
TR TF
i(t) k
TD iK...E (t) kK...E (t)
I1
PERIOD
I0
na t
(a) one objective category (b) A . . . K objective categories
Fig. 5. Model of didactical material.
learning objectives (Table 2). Models for one category of didactical objectives is shown
in Fig. 5a. More complicated model for A . . . K categories (Table 2) is shown in Fig. 5b
(bolded). It means that the system has many variables describing information flow and
levels of knowledge for particular categories of learning objectives. The didactical unit
described above can be modelled using MDL language of the Dero [30] simulator.
Model of learning and forgetting As mentioned above, different functions are used to
describe the forgetting curve [19], in particular power, exponential, and superposition
of functions. As shown in the [49] the superposition of the exponential function proba-
bly the better describe the forgetting curve than the power function. As shown in [35],
none of such relatively simple functions can fit the forgetting curve in all its points. In
the work, the piecewise linear model (Fig. 6) was used. The model is relatively simple
RI (mode)
ik (t)
i(t)
kin R 1 · kin CL kout
Fig. 6. Piecewise linear model of learning and forgetting.
and in this form do not fit all points on the forgetting curve. The more complex model
is required. The model takes into account every category of learning objectives. Pa-
rameters of elements can be dependent, for example, on time and/or mode. The model
enables flexible modelling of the learning and forgetting by changing the time constant
τ = RI CL . It is realized by changing the RI parameter. The mode denotes learning or
forgetting. The repetition of the material is also taken into. The initial values of τ are
as follow [34]: about 20..31minutes for learning, about 9..12hours for short time for-
getting, about 31..43.4days for a long time forgetting. The values changes in time due
to i.e. the material repetition. The model describes the learning process for the isolated
part of the material which creates single engram or set of engrams. The model is similar
to the Memory Chain Model [17,39].
Evaluation The evaluation of the courses in term of knowledge level for each objective
category can be modelled as a comparator. The results of tests or examinations for in-
dividual learning objectives and results of simulations should be provided at the input
of the model. The appropriate differences will be obtained at the output of the model.
Depending on the results, there may be three cases:
– results of the simulation overlap with test results,
– the results of the simulation are worse,
– the results of the simulation are better than the results obtained.
Evaluation of the didactical process can also be carried out by determining the appropri-
ate user profile for each category of learning objectives. This is possible by setting the
appropriate parameters. When comparing exam results and/or values obtained as a result
of the simulation, it is possible to determine whether a given user profile is preserved or
not.
Network representation of material repetitions In the example, there are 3 repetitions of
the same didactical material as shown in Fig. 7. The network describes the flow of infor-
Model of
forgetting
1
0.95
0.9
kDM,1 (t)
DM1 (t1 ) DM1 (t2 ) DM1 (t3 )
0.85
0.8
0.75
0.7
0.65
0.6
0.55
0.5
0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
5 · 10−2
0
time
1
DM (t3 ) DM (t2 ) DM (t1 )
RI (mode)
ik (t) k(t)
iin (t)
Ra Ra Ra
R k(t) CL km,out
ia (t3 ) ia (t2 ) ia (t1 )
Fig. 7. Network representation of the retention of the didactical material.
mation at the specific time points represented by the information sources. The learning
and forgetting are modelled using the model described earlier (Fig. 6). The input file for
Dero simulator is presented on the Listing 1.1.
Listing 1.1. Example of the input file describing material retention.
1 .TASK ”EXAMPLE OF THE DIDACTICAL PROCESS SIMULATION”
2 .LIB ” t y p e s . m d l ” # Data t y p e s
3 .LIB ” u n i t s . m d l ” # Units ( minuts , hours , e t c )
4 .LIB ” math / o n e . m d l ” # Math f u n c t i o n one
5 .LIB ” math / p u l s e . m d l ” # Math f u n c t i o n p u l s e
6 .LIB ” edu / v a r s . m d l ” # Types o f n e t v a r i a b l e s
7 .LIB ” edu / b u s . m d l ” # Bus v a r i a b l e s d e f i n i t i o n s
8 .LIB ” edu / i n p u t s . m d l ” # I n p u t s d e f i n i t i o n s
9 .LIB ” edu / o u t p u t s . m d l ” # O u t p u t s d e f i n i t i o n s
10 .LIB ” edu / exam.mdl ” # Exam model
11 .LIB ” edu / d i d a c t i c a l u n i t . m d l ” # D i d a c t i c a l u n i t model
12 .LIB ” edu / e x e r c i s e . m d l ”# e x e r c i s e model
13 .LIB ” edu / s t u d e n t . m d l ” # l e a r n i n g and f o r g e t t i n g model
14 .MODEL ”DM” DIDACTICALUNIT ;
15 .MODEL ” s t u d e n t ” ”STUDENT ” ;
16 # d i d a c t i c a l m a t e r i a l r e p e t i t i o n s
17 ”DM” . ”DM1” ” OBJID. ”K 0 ” OBJID.” I 0 =0 . 0 I 1 =0 . 0 I 2 =1 . 0 W=1 . 0 TD=2017−01−01T14 : 0 0 : 0 0 TR=10 TF=10 PW=45 min
,→ PERIOD=1000 d a y s ;
18 ”DM” . ”DM2” ” OBJID. ”K 0 ” OBJID.” I 0 =0 . 0 I 1 =0 . 0 I 2 =1 . 0 W=1 . 0 TD=2017−02−01T14 : 0 0 : 0 0 TR=10 TF=10 PW=45 min
,→ PERIOD=1000 d a y s ;
19 ”DM” . ”DM3” ” OBJID. ”K 0 ” OBJID.” I 0 =0 . 0 I 1 =0 . 0 I 2 =1 . 0 W=1 . 0 TD=2017−05−01T14 : 0 0 : 0 0 TR=10 TF=10 PW=45 min
,→ PERIOD=1000 d a y s ;
20 # l e a r n i n g and f o r g e t t i n g model
21 ” s t u d e n t ” . ”ST” ” OBJID.” 0 ” OBJID.” RIN=1 OKN10=0 OCO20=0 OAP30=0 OAN40=0 OSY50=0 OEV60=0 CO=100 RL
,→ =27 RLDIV=0 . 5 RF=4320 RF5=3116 RF30 = 9 6 7 4 4 ;
22 .CMD
23 .PROBE ADD TR( ” ∗ ” ) ;
24 .PROBE ADD TR( ” ∗ ” ) ;
25 .OPTI ALLP=1 , HMIN= 1 ;
26 .OP ; # i n i t i a l v a l u e s
27 .TR STEP = 2 4 : 0 0 : 0 0 T0=2017−01−01T00 : 0 1 : 0 0 TMIN=2017−01−01T00 : 0 1 : 0 0 TMAX=2017−05−15T00 : 0 1 : 0 0 TFMT=REAL TITLE =”
,→ DIDACTICAL PROCESS SIMULATION ” ;
28 .END
The file can be automatically generated based on student activity. In line 1, the network
simulation task starts and the title is set. Model libraries are loaded in lines 2..13. Model
lines storing common parameters were placed on lines 14 and 15. Three didactical ma-
terials were included in the network description (DM1, DM2 and DM3 - lines 17 . . . 19).
The learning and forgetting process is described by the ST element (line 21). The com-
mand block starts on line 22. The commands for deriving the results of the analysis
are on lines 23 and 24. Initial values for differential equations are calculated in line 26.
The time analysis command is on line 27. The system description ends with the .END
directive (line 28).
4 Results and Discussion
The system described above was used in practice. Several assumptions have been made.
The output gain is a superposition of the gains for every part of the course. The courses
were simulated individually for each student based on their activities. The results of
the simulation are compared with the real results obtained during the exams. Questions
cover all topics what means the uncertainty of the assessment will not occur. The min-
imal score of the test was set to 0.51 (51%). Final grades are within limits (0.51..0.60,
0.61..0.70, 0.71..0.80, 0.81..0.90, 0.91..1.00). Example of the student activities during
the didactical process for Information Security (IS) course is shown in Fig. 8. The
BI − Knowledge represents simulated results of the designed course. The simulated
level of student knowledge represents the variable of Knowledge. The real level of stu-
dent knowledge obtained during the exam represents Evaluation. The initial level of
knowledge was set to 0. The default parameters of the learning and forgetting process
were set. The simulation does not include activities that took place outside the regis-
tered didactical process. Because of this, there are differences between the simulated
and real process. The simulation results are all the more accurate the more activity data
1
BI - Knowledge
Knowledge (simulation)
Evaluation (exam)
0.8
0.6
0.4
0.2
0
0
0
00
0
0
0
:0
:0
:0
:0
:0
:
00
00
00
00
00
00
1
0
30
20
09
9
.2
.1
.2
4.
5.
6.
03
04
06
.0
.0
.0
.
.
4.
14
14
14
14
14
1
20
20
20
20
20
20
Fig. 8. Didactical process simulation for IS course based on the student’s activity.
is available. The expected level of knowledge differs from the real level at around 10%.
As can be seen, there are no visible activities before the exam. It means that the student
did not use the system - worked off-line on his own materials. Further information and
simulation results, including long-term simulations after completing the course, can be
found in literature [32,34,35].
5 Conclusions and Prospects for Further Research
As shown in the article the didactical process can be described by the differential equa-
tions represented in the intuitive form of schematics (here electrical like). Other nota-
tions can also be used or developed (further work). The developed educational analogy
enables defining basic types of equations as element models. Behavioural modelling al-
lows creation models based on mathematical functions, including nonlinear ones. The
model’s parameters can be easily adjusted to the measurement data in the optimization
process. The network describing the complicated didactical process should be gener-
ated automatically taking into account students activities. The learning and forgetting
can be modified using both behavioural modelling or the circuit models (not discussed
in the article). The elements models are very sensitive to the input parameters. The most
important are τ constants in the model of learning and forgetting, which values have a
real-life interpretation (not discussed here).
The presented approach allows the use of microsystems simulator and gives access
to many advanced simulation and optimization methods and algorithms implemented
in the simulators. It enables designing more ergonomic didactical processes. The tool
gives the opportunity to reduce expenditure on the teaching and learning process through
more effective management of the process structure, time spent on the learning and the
number of material repetitions. It also allows for detecting critical parts of the process.
The approach allows reducing the time and costs of designing the didactical process.
The practical implementation of the system on the Quela platform is still used in the
research. The issues described above can be used in many areas, e.g.: on the learning
e-learning platforms, in medical research, in psychology.
Further work will focus on modelling of the simulation process, using faster simula-
tion techniques, the use of advanced techniques (i.e. ODOS [26]) to analyze the process.
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