# Multicriteria Model of Students’ Knowledge Diagnostics Based on the Doubt Measuring Level Method for E&M Learning

Authors: V. V. Serbin; K. S. Duisebekova; A. M. Smaiyl; L. M. Alimzhanova; G. D. Zakirova; A. N. Moldagulova; R. J. Satybaldiyeva; Y. S. Maulenov
Journal: Mobile Information Systems
 (2019)

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## Abstract

This research is devoted to the development of a multicriteria model, valid only for testing, which takes into account the number of responses, the level of question complexity depending on the time spent on their response, time of the whole testing, doubts, number of passes, and the use of additional programs. Also the developed method of measuring the level of the user doubts is described, which gives a clearer and “transparent situation picture” for a more objective decision-making. The method makes it possible to reduce the probability of guessing the correct answer, which increases the objectivity of assessing the knowledge level in the diagnostic systems for E&M learning.


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## Body

## 1. Introduction

Distance education and e-learning have laid the foundation for a new global phenomenon, Smart Education, which is not so much about technology as it is about the new philosophy of education.Developers of distance learning systems cannot always fully take into account all the laws of the learning process and therefore use not quite correct models. And even the most successful solutions in the field of online training rarely get widespread.Undoubtedly, distance learning is by no means a complete substitute for traditional training, because it is difficult to replace live communication with a teacher or the educational atmosphere that develops in the classroom between subjects of the learning process. At the same time, modern ICT (information and communication technologies) can minimize the “narrow” places of distance learning. Modern pedagogy is also changing, and nowadays transfer of part of the training load to the online mode is quite an acknowledged educational scenario. This can be done when developing educational content on a modern IT basis, for example, using MOOCs (a mass open educational course).Today, we can confidently state the existence of a new digital Z-generation of people for whom a mobile phone, a computer, and the Internet are natural elements of their living space. In modern conditions, effective education is education without reference to time and place. This education teaches through everyday life. The transition to this mobile learning technology involves the application of new methods, approaches, and principles of the learning process organization. When developing content for mobile devices, it is necessary to take into account that it is intended for young people of the Z-generation, as the method of preparing training content for them differs from the traditional learning content.So, smart technologies in education, such as mobile applications, are of great importance as they allow to optimize the costs of the university logistics and also to raise the quality of educational services and products to a new level. It is smart technology that allows to develop revolutionary teaching materials, as well as to form individual trajectories of training.In 2015, UNESCO published Recommendations on Mobile Learning Policies [1], which fully justifies the need for the introduction of mobile technologies in the educational process. According to these recommendations, “In a world in which dependence on means of communication and access to information is growing, mobile devices will not be a transient phenomenon. As the capacity and capabilities of mobile devices are constantly growing, they can be used more widely as educational tools and take a central place, both in formal and informal education.”History shows that the competitiveness of the national economy as a whole depends on the development of information technologies. According to the World Economic Forum, the competitiveness index of the economies of states has a high level of correlation with the countries’ information and communication technologies. According to the 2012 rating of the competitiveness of 142 countries of the world by the World Economic Forum, countries that are actively developing information technologies are ahead of Kazakhstan which ranks 51st in terms of creating demand for information technology (the USA-13, Germany-19, India-63, and Egypt-96) and occupies the 104th place in the information technology business conditions place (USA-21, Germany-38, and India-72).Scientific novelty consists in developing a method of objective measurement of students’ knowledge on the basis of a multicriteria decision-making model.


## 2. Development of a Multicriteria Model for Diagnosing Knowledge

To solve the problem, an intellectual multicriteria model for assessing students’ knowledge has been developed, which makes it possible to take into account the following parameters: (1) time spent on the answer, (2) the number of correct answers, (3) time of all testing, (4) the doubts of the test person, (5) omission of questions, (6) use of additional programs, and (7) psychological characteristics of the tested person.Given in the following sections are the coefficients of the model and the methods for measuring the coefficients.


### 2.1. Coefficient of Accounting for Time Spent on the Answer

Tests contain questions of different levels of complexity (50% of them are easy (levela), 30% are average (level b), and 20% are difficult (level c)). Therefore, the score for each correct answer should depend on the complexity of the question, as well as on the time spent on it.The maximum scoreGmax (subject to a quick response in less than 15 s) can be calculated from the formula(1)Gmax=A⋅amax+B⋅bmax+C⋅cmax,where A,B,C is the number of questions on levels of complexity а,b,c, amax is the maximum score for answering easy questions (level a), bmax is the maximum score for answering average questions (level b), and cmax is the maximum score for answering difficult questions (level c).(2)A+B+C=V,where V is the number of questions in the test.The maximum scoreGmax can be typed with a different number of questions in the test.Points scored for the elapsed timeGcost can be calculated by the formula(3)Gcost=∑i=1Aai+∑i=1Bbi+∑i=1Cci,where ai refers to the points for the correct i-question level а, bi refers to the points for the correct i-question level b, and ci refers to the points for the correct i-question level c.The coefficientK1 characterizing the share of knowledge at score “5” under the condition of a quick response (for a time not exceeding 15 seconds) is determined by the formula(4)K1=GcostGmax,where Gcost is the number of points scored for the elapsed time, subject to the choice of the correct answer.Provided that all the answers are correct and the time for making a decision of each answer will not exceed 15 seconds, the knowledge coefficient of the tested “excellent” is 1.


### 2.2. Coefficient of the Number of Correct Answers

(5)
K
2
=
N
answers
V
,where K2 is the coefficient that characterizes knowledge of all issues regardless of time, Nanswers is the number of correct answers, and V is the total number of questions.


### 2.3. Coefficient for Time of All Testing

To take account of the time spent on the test (regardless of anything), we need to use the following formula:(6)K3=1−TcostTmax,where K3 is the coefficient characterizing the passage of the test, Tcost is the time spent on testing, and Tmax is the maximum allowable time that can be spent on testing.


### 2.4. Coefficient of Doubt

The mathematical description of doubt as a significant component of the rational choice model, based on the axioms of independence, transitivity, convexity, and monotony of individual preferences, is represented in some way in the study, as discussed by Vinogradov and Kuznetsov [2]. A person builds a sequence of conclusions about the adequacy of his ideas in accordance with his subjective level of conviction. He is convinced of the adequacy of his views in the situation of choosing the type of Ω with respect to goal G, if he believes that the choice of the mode of action C on their basis will allow it to be achieved. At the same time, he perceives some of the characteristics of the X situation Ω; with respect to the other part, he makes assumptions and shows an intention to prove (to verify their plausibility).The assumption is the default value of the observed characteristic or a description of the cause-effect relationship between the observed characteristics.Representations of a person are characterized by a level of conviction. The level of his belief in his ideas about the situation of choosing type Ω with respect to goal G is determined by the frequency of its achievement when choosing the mode of action C based on them. The assessment of the level of conviction changes from zero to one. If the number of unsuccessful attempts to reach goal G when choosing the mode of action C based on human representations increases, then the level of conviction of a person decreases (and vice versa), which becomes an incentive for him to apply efforts for their modification or complete reconstruction due to the growing doubt in the plausibility of the assumptions made. The desire to verify the correctness of assumptions is a measure of a person’s doubts.Thus, the efforts that a person expends to prove (refute) assumptions characterize the degree of his doubts about the ideas about the situation of choosing type Ω as he strives for goal G. According to the provisions of the theory of behavioral psychology, if the level of conviction that depends on the number of confirmations of the correctness of the choice based on representations increases, then the person’s desire for verification falls, since he does not see the point in this. The increase in the degree of doubt is an incentive for finding additional arguments (counterarguments).A parameter that takes into account these two characteristics is the degree of conviction considered as(7)SU=Uuα⋅1−SSβ∈0,1,α+β=1,where SU is the degree of conviction; Uu is the level of conviction (past experience); SS is the degree of a person’s doubt about the correctness of his ideas about the situation of choice; α and β are the coefficients of significance that a person gives to his experience and the need to find evidence. Thus, from the preceding expression, one can derive a formula expressing the degree of doubt.The degree of doubt in modern tests, in our opinion, is a latent parameter and can be measured only indirectly. If the degree of doubtSS and the degree of confidence U is estimated on a 100-point scale and expressed in %, then the formula for the degree of doubt SS will acquire the following form:(8)SS=100%−U.Quantitative characteristics are needed to measure latent parameters. The quantitative parameters for measuring the degree of doubt of the user are the following variables: the number of missed operations, the amount of unconfirmed information, the amount of heterogeneous information (ambiguous from the first time), and the state of the logical chain (the sequence of actions, levels of complexity, etc.).To determine the level of doubt, we used a superposition ofn models, each of which determines the latent parameter of doubt on the quantitative parameters.Thus, we can postulate the following:(1)
At the initial level of complexity, the measured level should be the lowest level of doubt.(2)
On the average level of complexity, the level of doubt should be greater for the advanced level, but less for the initial level.The quantitative parameters for measuring the level of doubt of the user are the following:(i)
The number of missed operations(ii)
The amount of unconfirmed information(iii)
The amount of heterogeneous information (ambiguous from the first time)(iv)
The state of the logical chain (the sequence of actions, levels of complexity, etc.)To take into account the user’s level of doubt, the following method is suggested:(1)
A test is organized, containingi-questions, under the condition i > 0.(2)
Eachi-question contains j variants of answers under condition 2 < j ≤ 5.(3)
The testing questions are divided into training elements. Each training element contains questions ofS different levels of complexity. The question contains only one correct variant Gi for the i-question, Gi > 0, Gi ≥ nij. Each j variant can be preselected and subsequently confirmed with the only accepted variant of the answer kij under the condition j = 1. Each variant of the question has the ability to switch m times. Each question contains a pih ability to skip an i-question only once, that is, P=i/pih. Then, the amount of heterogeneous information (ambiguous from the first time) can be determined by the number of switching options m≤jj−1.Consequently,(9)Di=ϕ1Srw+ϕ2Swr+ϕ3Swwm,underφ1>φ2>φ3,where Di is the coefficient per doubt of i-question, m is the number of variants of answers in the question, φ1 is the coefficient that characterizes the weight of switching from the correct to the wrong answer, φ2 is the coefficient that characterizes the weight of switching from wrong to correct answer option, φ3 is the coefficient that characterizes the weight of switching from wrong to wrong answer option, Srw is the number of switches from right to wrong answer, Swr is the number of switching from wrong to the correct answer, and Sww is the number of switching from wrong to wrong answer option.Let us find the average coefficients of doubtSaavg,Sbavg,…,Ssavg to questions of level a,b,…,s, respectively,(10)Saavg=∑i=1naDina,Sbavg=∑i=1nbDinb,…,Ssavg=∑i=1nsDins,where s is the number of difficulty levels, na is the number of level a questions, nb is the number of level b questions, and ns is the number of level s questions.The final coefficient, which characterizes the average level of doubt of the learning element, is calculated by the formula(11)Сn=Saavg+Sbavg+⋯+Ssavgs,where s is the number of difficulty levels.When S = 3 (3 levels: initial, intermediate, and advanced) of each educational element and states 2 (1, answered; 0, did not answer), there are 23 = 8 various states of the logical chain, each of which has its own level of doubts in the matrix states of the level of complexity logical chain. The level of doubt in the matrix depends on the subject area, compiled by the expert.The weight of the doubt level ats = 3 for this model corresponds to A-50%, B-30%, and C-20%. The user is given a portion of information of A, B, and C levels of this training element. After that at Table 1, the truth table is analyzed, and the decision is made.Table 1
Matrix of states of the logical chain of the complexity level.
State
Doubt level of logical chainL (%)
A
B
C
0
0
0
0
0
0
1
80
0
1
0
20
0
1
1
50
1
0
0
20
1
0
1
30
1
1
0
20
1
1
1
0Amount of unconfirmed information isF. If a negative answer was received to the question, then this question is randomly formed even m times during the check-up phase. If the answer is correct the second time, then the chain is simulated a second time, taking into account the level of complexity.This model defines a latent parameter that is a response to a decision taking into account the level of complexity. The essence of the model is to confirm the doubt of the decision taken by comparing it with a decision of this type.The latent parameter of this model determines the following quantitative characteristics:(i)
Subjects(ii)
Complexity(iii)
DecisionWe propose the following method. Testing ofx questions is organized. All x questions are divided into t-groups on a certain topic. Each question has one correct answer.In testing, allx questions are divided into n complexity levels. Each question has its own level of complexity. The proportion of questions on levels of complexity must satisfy the following condition: for n=3 easy-50%, medium-30%, and complex-20%. Each level of complexity corresponds to the weighting factor F (Fmax-the most complex, Fmax−1-less complex, etc.).Doubt questions are exposed to which the correct answer was given at all levels of complexity, except the first. Doubts when answering a question of levelF are calculated from the answers to questions of the same subject (i.e., the same group) whose complexity is lower than the level of complexity of the current question. Figure 1 shows graph of the dependence of the coefficient Ki of the weight of doubts on the current level of complexity of the question (logarithmic).(12)SF=∑i=1mkiWi,where m is the number of difficulty levels below that for which the calculation is carried out:(13)m=F−1,where F is the weighting factor of the current difficulty level; ki is the weight of doubts per question below the current level:(14)ki=−i−m−1∑j=1mj.Figure 1
Graph of the dependence of the coefficientKi of the weight of doubts on the current level of complexity of the question (logarithmic).W
i is the total weight of doubt level i-th question:(15)Wi=1g∑l=1gVi,where g is the number of questions of the i-th level of complexity of the same subject as the question for which the doubt is calculated; Vi is the decision taken (the answer to the question).(16)Vi=0,if correct answer given,1,if incorrect answer given..Proceeding from the fact that there can be several questions of the same level of complexity in the test, there is a need to find the mean-square value of the doubt at each level:(17)Ss=∑t=1qSFt2q.The final coefficient characterizing the degree of doubt is calculated by the formula:(18)S=∑F=2FmaxfF−1⋅Ss,where Ss is a doubt in the answers to the question of complexity F and fF is the doubt weight factor in the answers to the question of the complexity of F:(19)fF=F∑j=1mj,where Fmax is the maximum weight of the highest level. Figure 2 shows graph of dependence of the coefficient fF of total doubts on the current level of complexity of the question (logarithmic).Figure 2
Graph of dependence of the coefficientfF of total doubts on the current level of complexity of the question (logarithmic).Despite the fact that this method uses a small number of characteristics (subject, complexity, and decision) for the calculation, it gives an estimate of the doubts of correctly accepted decisions, which is unquestionably significant for the interpretation of the final test result.For understanding the developed method of calculating the level of doubt, it is suggested to consider the following example with specific input data.Let the number of questions = 10, the number of levelsF = 3, the coefficients of the complexity levels of the question be A (easy) = 1, B (medium) = 2, and C (complex) = 3. Correct answers of the user in difficulty levels: NA = 4, NB = 2, and NC = 1. It is required to determine the coefficient of doubt on the average and complex levels of complexity of the questions on which the correct answers were given.Solution:
(1)
The number of questions from 3 levels of complexity (according to the method) must satisfy the condition:
Questions of the level of complexityA = 50%, level B = 30%, and level C = 20%
Then,
Number of questions of levelA (simple) = 5
Number of levelB questions (averages) = 3
Number of questions of levelC (complex) = 2
(2)
Calculation of the weight coefficients of doubts of the levelC in accordance with formula (13):
M = F − 1 = 3 − 1 = 2
The weight of doubt for levelA in accordance with formula (14):
(20)
k
1
А
=
−
i
−
m
−
1
∑
j
=
1
m
j
=
−
1
−
2
−
1
1
+
2
=
2
3
.
The weight of doubt for levelB in accordance with formula (14):
(21)
k
2
В
=
−
i
−
m
−
1
∑
j
=
1
m
j
=
−
2
−
2
−
1
1
+
2
=
1
3
.
(3)
Calculation of the total weights of doubt for levelC:
The user answeredNA = 4 out of 5 questions, NB = 2 of 3, NC = 1 of 2 questions, means number of questions on which the wrong answer was given:
g
A = 5 − 4 = 1, gB = 3 − 2 = 1, gC = 2 − 1 = 1
Let us find the total weight of doubts of levelA (i = 1) in accordance with formula (15):
(22)
W
i
=
1
g
∑
l
=
1
g
V
i
=
1
5
.
Let us find the total weight of doubts of levelB (i = 2) in accordance with formula (15):
(23)
W
i
=
1
g
∑
l
=
1
g
V
i
=
1
3
.
(4)
Calculation of doubt on the levelC questions in accordance with formula (17):
(24)
S
3
с
=
∑
i
=
1
m
k
i
W
i
=
2
3
⋅
1
5
+
1
3
⋅
1
3
=
0.24.
Percent of doubts on the issue of theC level = 24%
(5)
Calculation of the weight coefficients of doubts of levelB in accordance with formula (13):
M = F − 1 = 2 − 1 = 1
The weight of doubt for levelA in accordance with formula (14):
(25)
k
1
А
=
−
i
−
m
−
1
∑
j
=
1
m
j
=
−
1
−
1
−
1
1
=
1.
(6)
Calculation of the total weights of doubt for levelB:
Let us find the total weight of doubts of levelA (i = 1) in accordance with formula (15):
(26)
W
i
=
1
g
∑
l
=
1
g
V
i
=
1
5
.
(7)
Calculation of doubts on the levelB questions in accordance with formula (17):
(27)
S
3
с
=
∑
i
=
1
m
k
i
W
i
=
1
⋅
1
5
=
0.2.
Percentage of doubts about the level ofB = 20%
(8)
The meaning of doubt in accordance with formula (19) = 20%Answer: Doubts arose in 3 correct answers of the student, because questions are raised only on the average (B) and complex (C) levels of complexity.(1)
In 2 responses of levelB, where the correct answers were given, the level of doubt is 20%.(2)
In the question of levelC, where the correct answer was given = 24%.


### 2.5. Coefficient of Missing Questions (Savings/Waste of Time)

If the test program is able to skip the question, then it is logical to keep a record of the gaps that affect the probability of nontripping:(28)K5=1−12PaPamax+13PbPbmax+15PcPcmax,where K5 is a coefficient characterizing nontransmissions and Pa, Pb, and Pc are the number of passes in the levels a, b, and c.In order to keep the coefficient correctly recorded, it is necessary to enter restrictions, the maximum permissible number of passes in the levelsa, b, and c.P
a
max, Pbmax, and Pcmax are the maximum number of omissions in questions on levels a, b, and c. Provided that Pamax≤15, Pbmax≤9, Pcmax≤6.


### 2.6. Coefficient of Using Additional Programs

If, for example, there are problems in testing that require mathematical calculations, then it is necessary to include the “Calculator” in the testing interface. Thus, keep a record of the amount of use of the calculator:(29)K6=minLused,LmaxLused,L,where K6 is the utilization rate of additional programs, L is the number of tasks requiring calculation, and Lused is the amount of use of the calculator.In this example, the calculator is a useful element of encouragement. There are undesirable but accessible elements of use-hints.The final formula for calculating the total coefficient characterizing knowledge on “5” is as follows:(30)K=a1K1+a2K2+a3K3+a4K4+a5K5+a6K6,where a1K1 is the coefficient of knowledge which is excellent under the condition of rapid response, a2K2 is the coefficient of knowledge of all questions regardless of time, a3K3 is the test passing ratio, a4K4 is the coefficient of confidence, a5K5 is the nonrolling ratio, and a6K6 is the coefficient of using additional programs.It was experimentally deduced thata1 = 0.23, a2 = 0.5, a3 = 0.07, a4 = 0.1, a5 = 0.07, a6 = 0.03, and (a1 + a2 + a3 + a4 + a5 + a6) = 1.By calculating the coefficientK, the coefficient of knowledge for “excellent,” we can compile Table 2 and determine the assessment.Table 2
Evaluation depending onK.
Condition
Evaluation
Rating
If 0 ≤K > 0.6
2 (unsatisfactory)
K
∗
100
If 0.6 ≤K > 0.8
3 (satisfactorily)
If 0.8 ≤K > 0.9
4 (good)
If 0.9 ≤K ≥ 1
5 (excellent)The final formula forK:(31)K=a1∑i=1Aai+∑i=1Bbi+∑i=1CciA⋅amax+B⋅bmax+C⋅cmax+a2NanswersA+B+C+a31−TcostTmax+a41−12⋅SaSamax+13⋅SbSbmax+15⋅ScScmax+a51−12⋅PaPamax+13⋅PbPbmax+15⋅PcPcmax+a6minLused,LmaxLused,L.The multicriteria model is valid only for testing, which takes into account the number of answers, the level of complexity of questions depending on the time spent on their response, the time of all testing, doubts, the number of omissions, and the use of additional programs.


## 2.1. Coefficient of Accounting for Time Spent on the Answer

Tests contain questions of different levels of complexity (50% of them are easy (levela), 30% are average (level b), and 20% are difficult (level c)). Therefore, the score for each correct answer should depend on the complexity of the question, as well as on the time spent on it.The maximum scoreGmax (subject to a quick response in less than 15 s) can be calculated from the formula(1)Gmax=A⋅amax+B⋅bmax+C⋅cmax,where A,B,C is the number of questions on levels of complexity а,b,c, amax is the maximum score for answering easy questions (level a), bmax is the maximum score for answering average questions (level b), and cmax is the maximum score for answering difficult questions (level c).(2)A+B+C=V,where V is the number of questions in the test.The maximum scoreGmax can be typed with a different number of questions in the test.Points scored for the elapsed timeGcost can be calculated by the formula(3)Gcost=∑i=1Aai+∑i=1Bbi+∑i=1Cci,where ai refers to the points for the correct i-question level а, bi refers to the points for the correct i-question level b, and ci refers to the points for the correct i-question level c.The coefficientK1 characterizing the share of knowledge at score “5” under the condition of a quick response (for a time not exceeding 15 seconds) is determined by the formula(4)K1=GcostGmax,where Gcost is the number of points scored for the elapsed time, subject to the choice of the correct answer.Provided that all the answers are correct and the time for making a decision of each answer will not exceed 15 seconds, the knowledge coefficient of the tested “excellent” is 1.


## 2.2. Coefficient of the Number of Correct Answers

(5)
K
2
=
N
answers
V
,where K2 is the coefficient that characterizes knowledge of all issues regardless of time, Nanswers is the number of correct answers, and V is the total number of questions.


## 2.3. Coefficient for Time of All Testing

To take account of the time spent on the test (regardless of anything), we need to use the following formula:(6)K3=1−TcostTmax,where K3 is the coefficient characterizing the passage of the test, Tcost is the time spent on testing, and Tmax is the maximum allowable time that can be spent on testing.


## 2.4. Coefficient of Doubt

The mathematical description of doubt as a significant component of the rational choice model, based on the axioms of independence, transitivity, convexity, and monotony of individual preferences, is represented in some way in the study, as discussed by Vinogradov and Kuznetsov [2]. A person builds a sequence of conclusions about the adequacy of his ideas in accordance with his subjective level of conviction. He is convinced of the adequacy of his views in the situation of choosing the type of Ω with respect to goal G, if he believes that the choice of the mode of action C on their basis will allow it to be achieved. At the same time, he perceives some of the characteristics of the X situation Ω; with respect to the other part, he makes assumptions and shows an intention to prove (to verify their plausibility).The assumption is the default value of the observed characteristic or a description of the cause-effect relationship between the observed characteristics.Representations of a person are characterized by a level of conviction. The level of his belief in his ideas about the situation of choosing type Ω with respect to goal G is determined by the frequency of its achievement when choosing the mode of action C based on them. The assessment of the level of conviction changes from zero to one. If the number of unsuccessful attempts to reach goal G when choosing the mode of action C based on human representations increases, then the level of conviction of a person decreases (and vice versa), which becomes an incentive for him to apply efforts for their modification or complete reconstruction due to the growing doubt in the plausibility of the assumptions made. The desire to verify the correctness of assumptions is a measure of a person’s doubts.Thus, the efforts that a person expends to prove (refute) assumptions characterize the degree of his doubts about the ideas about the situation of choosing type Ω as he strives for goal G. According to the provisions of the theory of behavioral psychology, if the level of conviction that depends on the number of confirmations of the correctness of the choice based on representations increases, then the person’s desire for verification falls, since he does not see the point in this. The increase in the degree of doubt is an incentive for finding additional arguments (counterarguments).A parameter that takes into account these two characteristics is the degree of conviction considered as(7)SU=Uuα⋅1−SSβ∈0,1,α+β=1,where SU is the degree of conviction; Uu is the level of conviction (past experience); SS is the degree of a person’s doubt about the correctness of his ideas about the situation of choice; α and β are the coefficients of significance that a person gives to his experience and the need to find evidence. Thus, from the preceding expression, one can derive a formula expressing the degree of doubt.The degree of doubt in modern tests, in our opinion, is a latent parameter and can be measured only indirectly. If the degree of doubtSS and the degree of confidence U is estimated on a 100-point scale and expressed in %, then the formula for the degree of doubt SS will acquire the following form:(8)SS=100%−U.Quantitative characteristics are needed to measure latent parameters. The quantitative parameters for measuring the degree of doubt of the user are the following variables: the number of missed operations, the amount of unconfirmed information, the amount of heterogeneous information (ambiguous from the first time), and the state of the logical chain (the sequence of actions, levels of complexity, etc.).To determine the level of doubt, we used a superposition ofn models, each of which determines the latent parameter of doubt on the quantitative parameters.Thus, we can postulate the following:(1)
At the initial level of complexity, the measured level should be the lowest level of doubt.(2)
On the average level of complexity, the level of doubt should be greater for the advanced level, but less for the initial level.The quantitative parameters for measuring the level of doubt of the user are the following:(i)
The number of missed operations(ii)
The amount of unconfirmed information(iii)
The amount of heterogeneous information (ambiguous from the first time)(iv)
The state of the logical chain (the sequence of actions, levels of complexity, etc.)To take into account the user’s level of doubt, the following method is suggested:(1)
A test is organized, containingi-questions, under the condition i > 0.(2)
Eachi-question contains j variants of answers under condition 2 < j ≤ 5.(3)
The testing questions are divided into training elements. Each training element contains questions ofS different levels of complexity. The question contains only one correct variant Gi for the i-question, Gi > 0, Gi ≥ nij. Each j variant can be preselected and subsequently confirmed with the only accepted variant of the answer kij under the condition j = 1. Each variant of the question has the ability to switch m times. Each question contains a pih ability to skip an i-question only once, that is, P=i/pih. Then, the amount of heterogeneous information (ambiguous from the first time) can be determined by the number of switching options m≤jj−1.Consequently,(9)Di=ϕ1Srw+ϕ2Swr+ϕ3Swwm,underφ1>φ2>φ3,where Di is the coefficient per doubt of i-question, m is the number of variants of answers in the question, φ1 is the coefficient that characterizes the weight of switching from the correct to the wrong answer, φ2 is the coefficient that characterizes the weight of switching from wrong to correct answer option, φ3 is the coefficient that characterizes the weight of switching from wrong to wrong answer option, Srw is the number of switches from right to wrong answer, Swr is the number of switching from wrong to the correct answer, and Sww is the number of switching from wrong to wrong answer option.Let us find the average coefficients of doubtSaavg,Sbavg,…,Ssavg to questions of level a,b,…,s, respectively,(10)Saavg=∑i=1naDina,Sbavg=∑i=1nbDinb,…,Ssavg=∑i=1nsDins,where s is the number of difficulty levels, na is the number of level a questions, nb is the number of level b questions, and ns is the number of level s questions.The final coefficient, which characterizes the average level of doubt of the learning element, is calculated by the formula(11)Сn=Saavg+Sbavg+⋯+Ssavgs,where s is the number of difficulty levels.When S = 3 (3 levels: initial, intermediate, and advanced) of each educational element and states 2 (1, answered; 0, did not answer), there are 23 = 8 various states of the logical chain, each of which has its own level of doubts in the matrix states of the level of complexity logical chain. The level of doubt in the matrix depends on the subject area, compiled by the expert.The weight of the doubt level ats = 3 for this model corresponds to A-50%, B-30%, and C-20%. The user is given a portion of information of A, B, and C levels of this training element. After that at Table 1, the truth table is analyzed, and the decision is made.Table 1
Matrix of states of the logical chain of the complexity level.
State
Doubt level of logical chainL (%)
A
B
C
0
0
0
0
0
0
1
80
0
1
0
20
0
1
1
50
1
0
0
20
1
0
1
30
1
1
0
20
1
1
1
0Amount of unconfirmed information isF. If a negative answer was received to the question, then this question is randomly formed even m times during the check-up phase. If the answer is correct the second time, then the chain is simulated a second time, taking into account the level of complexity.This model defines a latent parameter that is a response to a decision taking into account the level of complexity. The essence of the model is to confirm the doubt of the decision taken by comparing it with a decision of this type.The latent parameter of this model determines the following quantitative characteristics:(i)
Subjects(ii)
Complexity(iii)
DecisionWe propose the following method. Testing ofx questions is organized. All x questions are divided into t-groups on a certain topic. Each question has one correct answer.In testing, allx questions are divided into n complexity levels. Each question has its own level of complexity. The proportion of questions on levels of complexity must satisfy the following condition: for n=3 easy-50%, medium-30%, and complex-20%. Each level of complexity corresponds to the weighting factor F (Fmax-the most complex, Fmax−1-less complex, etc.).Doubt questions are exposed to which the correct answer was given at all levels of complexity, except the first. Doubts when answering a question of levelF are calculated from the answers to questions of the same subject (i.e., the same group) whose complexity is lower than the level of complexity of the current question. Figure 1 shows graph of the dependence of the coefficient Ki of the weight of doubts on the current level of complexity of the question (logarithmic).(12)SF=∑i=1mkiWi,where m is the number of difficulty levels below that for which the calculation is carried out:(13)m=F−1,where F is the weighting factor of the current difficulty level; ki is the weight of doubts per question below the current level:(14)ki=−i−m−1∑j=1mj.Figure 1
Graph of the dependence of the coefficientKi of the weight of doubts on the current level of complexity of the question (logarithmic).W
i is the total weight of doubt level i-th question:(15)Wi=1g∑l=1gVi,where g is the number of questions of the i-th level of complexity of the same subject as the question for which the doubt is calculated; Vi is the decision taken (the answer to the question).(16)Vi=0,if correct answer given,1,if incorrect answer given..Proceeding from the fact that there can be several questions of the same level of complexity in the test, there is a need to find the mean-square value of the doubt at each level:(17)Ss=∑t=1qSFt2q.The final coefficient characterizing the degree of doubt is calculated by the formula:(18)S=∑F=2FmaxfF−1⋅Ss,where Ss is a doubt in the answers to the question of complexity F and fF is the doubt weight factor in the answers to the question of the complexity of F:(19)fF=F∑j=1mj,where Fmax is the maximum weight of the highest level. Figure 2 shows graph of dependence of the coefficient fF of total doubts on the current level of complexity of the question (logarithmic).Figure 2
Graph of dependence of the coefficientfF of total doubts on the current level of complexity of the question (logarithmic).Despite the fact that this method uses a small number of characteristics (subject, complexity, and decision) for the calculation, it gives an estimate of the doubts of correctly accepted decisions, which is unquestionably significant for the interpretation of the final test result.For understanding the developed method of calculating the level of doubt, it is suggested to consider the following example with specific input data.Let the number of questions = 10, the number of levelsF = 3, the coefficients of the complexity levels of the question be A (easy) = 1, B (medium) = 2, and C (complex) = 3. Correct answers of the user in difficulty levels: NA = 4, NB = 2, and NC = 1. It is required to determine the coefficient of doubt on the average and complex levels of complexity of the questions on which the correct answers were given.Solution:
(1)
The number of questions from 3 levels of complexity (according to the method) must satisfy the condition:
Questions of the level of complexityA = 50%, level B = 30%, and level C = 20%
Then,
Number of questions of levelA (simple) = 5
Number of levelB questions (averages) = 3
Number of questions of levelC (complex) = 2
(2)
Calculation of the weight coefficients of doubts of the levelC in accordance with formula (13):
M = F − 1 = 3 − 1 = 2
The weight of doubt for levelA in accordance with formula (14):
(20)
k
1
А
=
−
i
−
m
−
1
∑
j
=
1
m
j
=
−
1
−
2
−
1
1
+
2
=
2
3
.
The weight of doubt for levelB in accordance with formula (14):
(21)
k
2
В
=
−
i
−
m
−
1
∑
j
=
1
m
j
=
−
2
−
2
−
1
1
+
2
=
1
3
.
(3)
Calculation of the total weights of doubt for levelC:
The user answeredNA = 4 out of 5 questions, NB = 2 of 3, NC = 1 of 2 questions, means number of questions on which the wrong answer was given:
g
A = 5 − 4 = 1, gB = 3 − 2 = 1, gC = 2 − 1 = 1
Let us find the total weight of doubts of levelA (i = 1) in accordance with formula (15):
(22)
W
i
=
1
g
∑
l
=
1
g
V
i
=
1
5
.
Let us find the total weight of doubts of levelB (i = 2) in accordance with formula (15):
(23)
W
i
=
1
g
∑
l
=
1
g
V
i
=
1
3
.
(4)
Calculation of doubt on the levelC questions in accordance with formula (17):
(24)
S
3
с
=
∑
i
=
1
m
k
i
W
i
=
2
3
⋅
1
5
+
1
3
⋅
1
3
=
0.24.
Percent of doubts on the issue of theC level = 24%
(5)
Calculation of the weight coefficients of doubts of levelB in accordance with formula (13):
M = F − 1 = 2 − 1 = 1
The weight of doubt for levelA in accordance with formula (14):
(25)
k
1
А
=
−
i
−
m
−
1
∑
j
=
1
m
j
=
−
1
−
1
−
1
1
=
1.
(6)
Calculation of the total weights of doubt for levelB:
Let us find the total weight of doubts of levelA (i = 1) in accordance with formula (15):
(26)
W
i
=
1
g
∑
l
=
1
g
V
i
=
1
5
.
(7)
Calculation of doubts on the levelB questions in accordance with formula (17):
(27)
S
3
с
=
∑
i
=
1
m
k
i
W
i
=
1
⋅
1
5
=
0.2.
Percentage of doubts about the level ofB = 20%
(8)
The meaning of doubt in accordance with formula (19) = 20%Answer: Doubts arose in 3 correct answers of the student, because questions are raised only on the average (B) and complex (C) levels of complexity.(1)
In 2 responses of levelB, where the correct answers were given, the level of doubt is 20%.(2)
In the question of levelC, where the correct answer was given = 24%.


## 2.5. Coefficient of Missing Questions (Savings/Waste of Time)

If the test program is able to skip the question, then it is logical to keep a record of the gaps that affect the probability of nontripping:(28)K5=1−12PaPamax+13PbPbmax+15PcPcmax,where K5 is a coefficient characterizing nontransmissions and Pa, Pb, and Pc are the number of passes in the levels a, b, and c.In order to keep the coefficient correctly recorded, it is necessary to enter restrictions, the maximum permissible number of passes in the levelsa, b, and c.P
a
max, Pbmax, and Pcmax are the maximum number of omissions in questions on levels a, b, and c. Provided that Pamax≤15, Pbmax≤9, Pcmax≤6.


## 2.6. Coefficient of Using Additional Programs

If, for example, there are problems in testing that require mathematical calculations, then it is necessary to include the “Calculator” in the testing interface. Thus, keep a record of the amount of use of the calculator:(29)K6=minLused,LmaxLused,L,where K6 is the utilization rate of additional programs, L is the number of tasks requiring calculation, and Lused is the amount of use of the calculator.In this example, the calculator is a useful element of encouragement. There are undesirable but accessible elements of use-hints.The final formula for calculating the total coefficient characterizing knowledge on “5” is as follows:(30)K=a1K1+a2K2+a3K3+a4K4+a5K5+a6K6,where a1K1 is the coefficient of knowledge which is excellent under the condition of rapid response, a2K2 is the coefficient of knowledge of all questions regardless of time, a3K3 is the test passing ratio, a4K4 is the coefficient of confidence, a5K5 is the nonrolling ratio, and a6K6 is the coefficient of using additional programs.It was experimentally deduced thata1 = 0.23, a2 = 0.5, a3 = 0.07, a4 = 0.1, a5 = 0.07, a6 = 0.03, and (a1 + a2 + a3 + a4 + a5 + a6) = 1.By calculating the coefficientK, the coefficient of knowledge for “excellent,” we can compile Table 2 and determine the assessment.Table 2
Evaluation depending onK.
Condition
Evaluation
Rating
If 0 ≤K > 0.6
2 (unsatisfactory)
K
∗
100
If 0.6 ≤K > 0.8
3 (satisfactorily)
If 0.8 ≤K > 0.9
4 (good)
If 0.9 ≤K ≥ 1
5 (excellent)The final formula forK:(31)K=a1∑i=1Aai+∑i=1Bbi+∑i=1CciA⋅amax+B⋅bmax+C⋅cmax+a2NanswersA+B+C+a31−TcostTmax+a41−12⋅SaSamax+13⋅SbSbmax+15⋅ScScmax+a51−12⋅PaPamax+13⋅PbPbmax+15⋅PcPcmax+a6minLused,LmaxLused,L.The multicriteria model is valid only for testing, which takes into account the number of answers, the level of complexity of questions depending on the time spent on their response, the time of all testing, doubts, the number of omissions, and the use of additional programs.


## 3. Experimental Part

### 3.1. Setting the Experiment

To test the developed model, a large-scale experiment was conducted during two academic years. The experiment involved about 1000 students of 29 courses. All students were divided into 2 groups.All students took midterm control tests (Att), which the teacher administered and evaluated during the semester.The first experimental group of students (500 people) passed the final exam (Exm) at the end of the course in the format of computer testing (5 answers and 1 correct one). For the first group, the final grade (F) for the course is calculated using(32)F=0.6∗Att+0.4∗Exm.The second experimental group of students (500 people) at the end of the same courses passed the exam in the format of computer testing on the basis of the multicriteria model (M), proposed in this study. The results of their average estimates (M) are given in Table 3.Table 3
Experimental data.
No course
Average estimation of intermediate control, Att
Exam estimation, Exm
Final estimation,F
Model final estimation,M
1
68.20
84.30
74.64
71.50
2
66.90
67.80
67.26
69.80
3
51.80
58.40
54.44
55.80
4
53.80
66.30
58.80
59.50
5
78.90
93.00
84.54
80.20
6
81.00
91.00
85.00
75.00
7
55.20
78.40
64.48
65.00
8
75.90
85.80
79.86
80.10
9
87.90
88.10
87.98
89.10
10
50.20
69.10
57.76
58.30
11
65.00
85.90
73.36
75.20
12
67.90
88.40
76.10
71.30
13
85.40
96.00
89.64
90.20
14
69.00
69.50
69.20
71.00
15
74.70
81.20
77.30
77.30
16
91.60
85.10
89.00
88.40
17
74.30
87.30
79.50
80.20
18
69.60
85.80
76.08
75.00
19
59.80
75.10
65.92
66.80
20
61.30
81.60
69.42
68.60
21
73.00
78.60
75.24
75.60
22
68.20
88.90
76.48
78.60
23
56.10
74.60
63.50
64.00
24
72.20
88.50
78.72
75.60
25
82.20
74.70
79.20
75.00
26
86.10
77.30
82.58
83.40
27
58.80
72.20
64.16
62.50
28
56.30
58.20
57.06
64.20
29
73.50
85.70
78.38
76.40
Avg.
69.48
79.89
73.64
73.23As a result of the experiment, it is required to determine the quality of the developed multicriteria model and its suitability for an objective assessment of the level of knowledge.The hypothesis of the experiment is that the average values of the final assessment (F) based on the boundary control according to Formula (32) correlate with the average values of the estimates obtained on the basis of the model under study (M). In the experiment, the hypothesis will also be tested that there is a correlation between the mean values of the boundary control (Att) and the model estimates (M).The results of the experiment will make it possible to determine the quality of the proposed multicriteria model, which is not worse than the totality of teacher evaluations (60%) and computer testing (40%).However, the proposed multicriteria model for computer testing, which takes into account the number of answers, the level of questions complexity depending on the time spent for answer, the time of the entire test, doubts, the number of omissions, and the use of additional programs, allows us to bring the % of successful random answers to the minimum, thereby increasing the objectivity of the level of knowledge assessment, having made them close to an assessment of the teacher, but not of the computer program.During the experiment, the following data were obtained (Table3).


### 3.2. Experiment Processing

The null hypothesis H0 is a statistical hypothesis to be verified by statistical data, the results of observations included in the sample. Of the possible statistical hypotheses, choose the one that assumes equity, and it is the most important for further conclusions.An alternative hypothesis is a statistical hypothesis that is considered valid if the null hypothesis is incorrect.Statistical criterion is a rule according to which on the bases of observation results, a decision is made whether to accept or reject the null hypothesis. In this article, the experimental test of the model adequacy is chosen as the criterion or the rule.The article puts forward the null hypothesis that the coefficient of doubt, introduced into the model as one of the factors, affects the objective assessment of the knowledge of the person being tested.In this regard, for a more objective assessment of test results, a multicriteria knowledge assessment model is proposed. This model was compiled and tested in previous works of the authors.This article experimentally proves the effect of the additionally introduced factor, which indicates the level of doubt of the person being tested on an effective assessment of his knowledge. This shows the adequacy of the proposed model. In the article, this type of system evaluates according to the following criteria:(1)
Time spent to respond(2)
Number of correct answers(3)
Time of the entire test(4)
Level of doubt(5)
Skipping questions(6)
Using the additional functionsOne of the most difficult tasks is to take into account consideration of doubts depending on the complexity of the issue. In order to keep records of doubts depending on the level of question complexity, it is necessary that the program be with a preliminary choice of the answer, and subsequently with its confirmation.Consequently, it all comes down to the number of preliminary answers, i.e., to the number of doubts. In this regard, the authors came to the conclusion that the coefficient of doubt level included in the model depends on the following variables:(i)
Number of missed operations(ii)
Amount of unconfirmed information(iii)
Reaction to decision-making(iv)
Level of difficulty(v)
Number of logical chain interrupts(vi)
Doubtful user actionsThe multicriterial model is valid only for adaptive testing, which takes into account the number of answers, the level of complexity of questions depending on the time spent on their answer, the time of the entire test, doubts, the number of omissions, and the use of additional functions.According to the experimental data, the multicriteria model of knowledge testing described in the article objectively assesses the level of knowledge of the person being tested and sets the appropriate assessment.As a result of processing the obtained data, the Pearson correlation coefficients were calculated:(33)Kxy=∑xi−x¯⋅yi−y¯∑xi−x¯2⋅yi−y¯2,where xi refers to values of the variable X, yi refers to values of the variable Y, x¯ is the average value for variable X, and y¯ is the average value for variable Y.Figure3 shows the relationship between the values of the intermediate control (Att) and the examination evaluation of computer testing (Exm).Figure 3
The relationship between the values of the boundary control (Att) and the examination evaluation of computer testing (Exm).The correlation coefficient between Att and Exm = 0.65 (K1).The figure shows that the results of the assessments delivered by the teacher (Att) are weakly correlated with the assessments of the computer testing exam.Figure4 shows the relationship between the values of boundary control (Att) and computer testing based on the multicriteria model (M).Figure 4
The relationship between the values of the boundary control (Att) and computer testing based on the multicriteria model (M).The correlation coefficient between Att andM = 0.94 (K2).Figure5 shows the relationship between the final grade for the course (F) and computer testing based on the multicriteria model (M).Figure 5
The relationship between the final grade for the course (F) and computer testing on the basis of the multicriteria model (M).The correlation coefficient betweenF and M = 0.95 (K3).


## 3.1. Setting the Experiment

To test the developed model, a large-scale experiment was conducted during two academic years. The experiment involved about 1000 students of 29 courses. All students were divided into 2 groups.All students took midterm control tests (Att), which the teacher administered and evaluated during the semester.The first experimental group of students (500 people) passed the final exam (Exm) at the end of the course in the format of computer testing (5 answers and 1 correct one). For the first group, the final grade (F) for the course is calculated using(32)F=0.6∗Att+0.4∗Exm.The second experimental group of students (500 people) at the end of the same courses passed the exam in the format of computer testing on the basis of the multicriteria model (M), proposed in this study. The results of their average estimates (M) are given in Table 3.Table 3
Experimental data.
No course
Average estimation of intermediate control, Att
Exam estimation, Exm
Final estimation,F
Model final estimation,M
1
68.20
84.30
74.64
71.50
2
66.90
67.80
67.26
69.80
3
51.80
58.40
54.44
55.80
4
53.80
66.30
58.80
59.50
5
78.90
93.00
84.54
80.20
6
81.00
91.00
85.00
75.00
7
55.20
78.40
64.48
65.00
8
75.90
85.80
79.86
80.10
9
87.90
88.10
87.98
89.10
10
50.20
69.10
57.76
58.30
11
65.00
85.90
73.36
75.20
12
67.90
88.40
76.10
71.30
13
85.40
96.00
89.64
90.20
14
69.00
69.50
69.20
71.00
15
74.70
81.20
77.30
77.30
16
91.60
85.10
89.00
88.40
17
74.30
87.30
79.50
80.20
18
69.60
85.80
76.08
75.00
19
59.80
75.10
65.92
66.80
20
61.30
81.60
69.42
68.60
21
73.00
78.60
75.24
75.60
22
68.20
88.90
76.48
78.60
23
56.10
74.60
63.50
64.00
24
72.20
88.50
78.72
75.60
25
82.20
74.70
79.20
75.00
26
86.10
77.30
82.58
83.40
27
58.80
72.20
64.16
62.50
28
56.30
58.20
57.06
64.20
29
73.50
85.70
78.38
76.40
Avg.
69.48
79.89
73.64
73.23As a result of the experiment, it is required to determine the quality of the developed multicriteria model and its suitability for an objective assessment of the level of knowledge.The hypothesis of the experiment is that the average values of the final assessment (F) based on the boundary control according to Formula (32) correlate with the average values of the estimates obtained on the basis of the model under study (M). In the experiment, the hypothesis will also be tested that there is a correlation between the mean values of the boundary control (Att) and the model estimates (M).The results of the experiment will make it possible to determine the quality of the proposed multicriteria model, which is not worse than the totality of teacher evaluations (60%) and computer testing (40%).However, the proposed multicriteria model for computer testing, which takes into account the number of answers, the level of questions complexity depending on the time spent for answer, the time of the entire test, doubts, the number of omissions, and the use of additional programs, allows us to bring the % of successful random answers to the minimum, thereby increasing the objectivity of the level of knowledge assessment, having made them close to an assessment of the teacher, but not of the computer program.During the experiment, the following data were obtained (Table3).


## 3.2. Experiment Processing

The null hypothesis H0 is a statistical hypothesis to be verified by statistical data, the results of observations included in the sample. Of the possible statistical hypotheses, choose the one that assumes equity, and it is the most important for further conclusions.An alternative hypothesis is a statistical hypothesis that is considered valid if the null hypothesis is incorrect.Statistical criterion is a rule according to which on the bases of observation results, a decision is made whether to accept or reject the null hypothesis. In this article, the experimental test of the model adequacy is chosen as the criterion or the rule.The article puts forward the null hypothesis that the coefficient of doubt, introduced into the model as one of the factors, affects the objective assessment of the knowledge of the person being tested.In this regard, for a more objective assessment of test results, a multicriteria knowledge assessment model is proposed. This model was compiled and tested in previous works of the authors.This article experimentally proves the effect of the additionally introduced factor, which indicates the level of doubt of the person being tested on an effective assessment of his knowledge. This shows the adequacy of the proposed model. In the article, this type of system evaluates according to the following criteria:(1)
Time spent to respond(2)
Number of correct answers(3)
Time of the entire test(4)
Level of doubt(5)
Skipping questions(6)
Using the additional functionsOne of the most difficult tasks is to take into account consideration of doubts depending on the complexity of the issue. In order to keep records of doubts depending on the level of question complexity, it is necessary that the program be with a preliminary choice of the answer, and subsequently with its confirmation.Consequently, it all comes down to the number of preliminary answers, i.e., to the number of doubts. In this regard, the authors came to the conclusion that the coefficient of doubt level included in the model depends on the following variables:(i)
Number of missed operations(ii)
Amount of unconfirmed information(iii)
Reaction to decision-making(iv)
Level of difficulty(v)
Number of logical chain interrupts(vi)
Doubtful user actionsThe multicriterial model is valid only for adaptive testing, which takes into account the number of answers, the level of complexity of questions depending on the time spent on their answer, the time of the entire test, doubts, the number of omissions, and the use of additional functions.According to the experimental data, the multicriteria model of knowledge testing described in the article objectively assesses the level of knowledge of the person being tested and sets the appropriate assessment.As a result of processing the obtained data, the Pearson correlation coefficients were calculated:(33)Kxy=∑xi−x¯⋅yi−y¯∑xi−x¯2⋅yi−y¯2,where xi refers to values of the variable X, yi refers to values of the variable Y, x¯ is the average value for variable X, and y¯ is the average value for variable Y.Figure3 shows the relationship between the values of the intermediate control (Att) and the examination evaluation of computer testing (Exm).Figure 3
The relationship between the values of the boundary control (Att) and the examination evaluation of computer testing (Exm).The correlation coefficient between Att and Exm = 0.65 (K1).The figure shows that the results of the assessments delivered by the teacher (Att) are weakly correlated with the assessments of the computer testing exam.Figure4 shows the relationship between the values of boundary control (Att) and computer testing based on the multicriteria model (M).Figure 4
The relationship between the values of the boundary control (Att) and computer testing based on the multicriteria model (M).The correlation coefficient between Att andM = 0.94 (K2).Figure5 shows the relationship between the final grade for the course (F) and computer testing based on the multicriteria model (M).Figure 5
The relationship between the final grade for the course (F) and computer testing on the basis of the multicriteria model (M).The correlation coefficient betweenF and M = 0.95 (K3).


## 4. Discussion

The hypotheses that have been advanced are confirmed by the results of processing the experimental data, which show that the model evaluation (based on the multicriteria approach) and the final assessment, as well as the model and assessment of the boundary control obtained on the basis of standard approaches, have a high level of coincidence, which is confirmed by a high degree of pair correlation ofK2, K3, and visual confirmation (Figures 4 and 5).Thus, it can be argued that there is a sufficiently high functional connection (close to linear) between the final assessment for the course and computer testing on the basis of the multicriteria model, which makes it possible to exclude the boundary control conducted directly by the instructor, without violating the high degree of objectivity in assessing student knowledge. This result is of great importance for the organization of distance learning with the automated assessment process.


## 5. Related Work

Tojo et al. [3] aim to implement the prelearning and group discussion in a consistent e-learning system, which develops the human-robot interaction for the purpose of creating a social, autonomous robot capable of conversing using various human-like methods such as body language, hand gestures, facial expressions, gaze, and touching.Joksimovic et al. [4] put forward a model, which intended to guide the future work studying the association between contextual factors, student engagement, and learning outcomes.According to Firat et al. [5], motivation that initiates and sustains behavior is one of the most significant components of learning in any environment.Farhan et al. concluded that students’ interaction and collaboration using the Internet of Things (IoT) based on interoperable infrastructure is an effective learning practice [6]. Measuring student attention is an essential part of educational assessment.Eom and Ashill argue that a significant reduction in dependent and independent variables and their measures are necessary for building an e-learning success model, and such a model should incorporate the interdependent (not independent) nature of e-learning success [7].Bhattacharya et al. present an intelligent recognizer of the cognitive state of an e-learner as an integral part of the confidence-based e-learning (CBeL) system [8]. It addresses the problem of providing technology-driven pedagogical support to an e-learner to achieve the desired cognitive state of mastery, which is endowed by high levels of both knowledge and confidence.Marcelino et al. offered Learning Computational Thinking concepts which have gained an importance in the last years [9].Debiec mentioned that, in this paper, a case study is presented based on one offering of an introductory digital system course taught with a combination of learner-centered strategies selected to overcome these barriers and improve student’s performance [10].Gul et al. identified various issues and challenges faced by massive open online courses (MOOCs) while offering open online courses to vast number of learners [11].Farid et al. propose a sustainable quality assessment approach (model) for the e-learning systems keeping software perspective under consideration [12].Bradac and Walek present the proposal, design, and implementation of a new approach to adaptive e-learning systems [13]. Adaptivity is considered as an ability of the system to adapt to student’s knowledge and characteristics.Ren et al. present an online course applicability assessment (OCAA) to assist learners in course selection [14]. Three main characteristics are taken into consideration, including “learning style,” “learning behavioral type,” and “prior knowledge” which considerably affect the e-learning effectiveness.Cohen and Anat focus on the fact that identifying the changes in student activity during the course period could help in detecting at-risk learners in real time, before they actually drop out from the course [15].In recent years, designing adaptive e-learning systems has become one of the striking topics of discussion in the literature. Hamada and Hassan describe a case study on the integration of the learning style index into an adaptive and intelligent e-learning system [16].Levina et al. studied the continuous information development of all spheres of education: integration of new knowledge, accessibility of information technologies and computer facility aids, professionalization, and computerization of educational activities [17].Sweta and Lal proposed learner model mines and learner’s navigational access data and found the learner’s behavioral patterns which individualize each learner and provide personalization in the learning process according to their learning styles [18].Bralic and Divjak blended a learning model where a MOOC has been integrated in a traditional classroom [19]. The learning outcomes-based approach was implemented that supported a balanced student workload.Gregori et al. aim to design and analyze the implementation of a number of guidelines that allow us to effectively unify a high-quality teaching methodology and the use of new technologies in distance learning [20].Quintana and Perez have analyzed massive, open, and online model course, describing their advantages and disadvantages, besides the formative and motivating factors that guide the students, not only to enroll in particular courses but also to request for their own space to be able to create a MOOC [21].For the first time, a multicriteria method for measuring the level of knowledge of students during computer testing based on the level of doubt for E&M learning has been described by the authors in a number of works, as discussed by Serbin et al., [22–24] and applied for distance training.


## 6. Conclusion

The multicriteria model allows to objectively assess the student’s knowledge level and, accordingly, to estimate the evaluation during computer testing. The model includes an original method of measuring the level of doubt, as a function of the time, and complexity of issues differentiated by thematic blocks. In addition, the assessment model itself being multicriteria, besides the function of doubt, includes the number of correct answers, the testing time, the use of additional programs, spent time on each response, the number of periodic passes in the testing process, and more. In diagnosing the level of knowledge based on computer testing, use of this method will reduce the probability of the influence of an accidental correct answer (“guessing”) on the final result and to obtain the most objective assessment.The proposed model includes the following factors:(i)
Number of answers(ii)
Level of complexity of the questions depending on the time(iii)
Spent on their response(iv)
Time of all testing(v)
Doubt(vi)
Number of passes(vii)
Use of additional programsThe practical significance of this model lies in the objectivity of assessing the level of knowledge, where computer testing is applied, especially relevant for the organization of distance learning using E&M learning. Additional effects of the application of this model may be the reduction of the time for organization, intermediate control procedures, labor costs, operating costs, processing and analysis costs, and others, research planned in subsequent publications.In conclusion, we can make a statement that the null hypothesis put forward by the authors is proven experimentally and the study demonstrates the original methodology for measuring the level of doubt of the tested students in the information and training system. The use of this methodology allows making the most objective decision on the assessment of students’ knowledge.In diagnosing the level of knowledge based on computer testing, the use of this methodology makes it possible to reduce the probability of the random factor influence, such as “guessing” on the final result and to obtain the most objective assessment. Moreover, to control the learning process, measuring the level of doubt allows taking into account the psychological characteristics of human behavior during training.



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