possibilities of fuzzy logic application in creating ... · the application of the fuzzy logic to...

ISSN (p) 2303-4890, ISSN (o) 1986–518X ISTRAŽIVANJE MATEMATIČKOG OBRAZOVANJA

DOI: 10.7251/IMO1901001B http://www.imvibl.org/dmbl/meso/imo/imo2.htm

Vol. XI (2019), Broj 20, 1–10

POSSIBILITIES OF FUZZY LOGIC APPLICATION IN CREATING

ONLINE MATH-TESTING MODELS FOR STUDENTS

Naida Bikić

PhD., Assistant Professor

University of Zenica

[email protected]

Nevzudin Buzadjija

PhD., Associate Professor

University of Zenica

[email protected]

Summary: The work proves the assumption that students who are not trained for online testing will not make a

statistical distinction compared to the testing in a classroom. The students attend the Department of Primary

Education Teaching and they only had a single semester of computer science. The testing was conducted from

three areas of mathematics. Online testing was conducted in a controlled environment without the possibility of

students to help each other. After having tested a specific number of examinees, the statistical processing was

carried out and the Mamdani fuzzy logic was applied. Primarily, the goal is to create a model that could predict

the desired results with the creation of an adequate environment and assumptions for appropriate inputs. In this

way, we would have the capability to control the factors that are important for students to achieve adequate

results in particular areas of mathematics.

Keywords: online testing, fuzzy logic, model, areas of mathematics

1. Introduction

The strategy of building excellence in students is an attempt to find the best solution that would

motivate students to master topics in mathematics. It is very difficult to achieve it nowadays due to a

huge amount of unverified information available online those students are exposed to on a daily basis.

Students lose a lot of time browsing the web content because they are often distracted by irrelevant

and useless information at social networking sites. This is especially noticeable in students who are not

motivated enough to explore problems in mathematics on their own. This is why it is necessary to

focus their attention on online testing in order to achieve adequate learning outcomes, which would

motivate the same students to achieve better results and a passing grade.

However, in order to avoid turning this into mechanical learning, this research aims to find a model

that would enable the best balance between classical and online testing, which primarily depends on

the area of mathematics that is taught and tested. In fact, the goal is to assess whether online testing is

a solution to achieve excellence in students and in our opinion, it is not the case, regardless it is a word

of the Internet generation of students.

Although many educational institutions are introducing e-learning and have certain plans concerning

the way it should be introduced and implemented, on the other hand, it is estimated that less than 20%

of all of these institutions have strategic plans and IT strategies for e-learning. This is particularly

pronounced in designing creative online tests, without it being a mechanical mastering in order to

fulfill the conditions set out.

2. Up-to-date research

Online testing is increasingly being introduced at higher education institutions, but have not achieved

statistically substantial results yet, and it may become a case that they will find their best use as an

‘innovation platform’, as a place for institutions to learn and experiment with pedagogy in an

environment very different from their traditional institutional educational provision [3] (Brown et al.,

mailto:[email protected]

IMO, XI (2019), Broj 20 N. Bikić and N. Buzadjija

2

2015). Therefore, teachers who hold traditional pedagogical beliefs may be compelled to transform

their practices through teaching.

As the focus in the future revolves from informal and free to formal, accredited, and paid education –

there will certainly be more focus on the quality of the courses themselves. There have been empirical

researches into the quality of online testing overall [7](Lowenthal & Hodges, 2015), including the

various assessment methodologies within individual subjects [9](Sandeen, 2013), [2](Balfour, 2013),

[5](Conole, 2016). Even today, we have numerous attempts to create models to create automated

systems that would profile students in order to determine the best model for online testing.

Individual grading by teachers or teaching personnel becomes unfeasible, and so other approaches are

applied, such as automated grading and peer-assessment, greatly relying on students’ active

participation [1](Admiraal et al., 2014), [2](Balfour, 2013), [8](Meek et al., 2017). However, there is a

significant gap in the fact that the previous research did not examine the applicability options of

obtained results in real practice and use, which is the key component of achieving online testing

effectiveness and reducing the cost of investment in human resources.

The application of the fuzzy logic to determine human knowledge in a certain domain of knowledge

using a ‘grade’ (a number of correct answers to questions) as possible measuring data is increasingly

being used. The outcome of the paper would be an intelligent examination system that would be used

to assess students’ knowledge of the subjects taught [6](Fahad S., Shah A., 2007).

3. Model

While conducting the research, students were tested according to the model portrayed in Figure 1.

Specifically, in the first part of the semester, the students were solely subjected to classical teaching in

mathematics and tested after seven weeks using the traditional method. In the second part of the

semester, the students were classically taught and they were using tutorials via Google classroom,

applying the model in the Figure, and they were tested online in the 15th week of the same semester.

Figure 1 – Student testing model in the research

In order to better understand the issues that are present in this research and how students were

subjected to teaching in the second part of the semester using the Google Classroom LMS platform to

master the materials that are listened to in the second part of the semester, it is best seen from the

Activitiesstudents

Checkedstudents

Successfulstudents

Semester

Selectionthe following

lessons

Lessonin the

classroom

Lessonin digital

form

Other sources(books...)

Internetsources

Classiccheck

knowledge

15. week

It's notsatisfied

satisfied

after the 15th week

onlinetesting

8.

week

9.-15. week


3

Figure 1. Specifically, after the teaching unit taught in a classroom, students are referred to additional

research set up on the LMS platform with the ability to undergo independent testing without a

professor monitoring the independent testing activities and results of students who voluntarily do so.

4. Sample

The research was conducted among the students of the Primary Education Teaching Department who

are listening to the 7th semester within the subject of ‘Teaching methods of mathematics’. These are

students who will work with six to ten-year-old children. It is very important for students to be able to

solve mathematical problems in logic, algebra, and geometry so that they could successfully convey

mathematical problems to their pupils.

The respondents tested online and in-classroom belongs to the same group. It should be noted that the

group was not prepared or trained for online testing. Thus, students were not sufficiently familiar with

the online testing environment and they do not belong to a population sufficiently computer-educated.

Namely, it is a word of four years of studying at the Primary Education Teaching Department, and

they only had the subject of computer science in the first year in one semester. The number of students

included in this research is 22. This is a small sample because there is that number of students

studying at the 4th year of the Primary Education Teaching Department.

5. Hypothesis

The focus of the research has been on creating a model for determining the proper ratio between

traditional and online testing that students should undergo depending on the area taught in

mathematics. Mamdani Fuzzy logic will be used to create this model. Using this method, it will be

possible to simulate the representation of classical and online testing and find the best ratio between

classical and online testing, with the aim of achieving the best results by students and therefore,

learning outcomes.

There are two hypotheses set up in the research:

- H1: Students do not achieve better results via online testing compared to classical testing if

they are not sufficiently treated with an online platform.

- H2: Test results are not affected by the knowledge assessment form, depending on the area

taught in mathematics.

6. Research stream

It is necessary to create an online course in Google Classroom that will be used for learning materials,

and will also serve for testing according to the principle ‘in application – testing in realistic conditions

as if they were using an application on a desktop computer’ in the areas of mathematics: arithmetic,

logic and geometry. This additional system will enable students to test themselves online

independently after each classical lecture and to independently research materials provided by a

professor. A professor is in no way involved in overseeing and monitoring what students are doing

within the online platform and what results they are accomplishing in self-examination. This will be

available to students only after the first classical test has been held, i.e. after 8 weeks of teaching.

The respondents will be subjected to the testing without prior preparation, and the testing consists of

two parts:

1. After 7 weeks of lectures in a classroom, students will be tested classically in a classroom on a

paper, and after the remaining 7 weeks, they will be tested online in the 15th week of the

semester. Both tests will be scored with a total of 40 points.

2. Based on the number of points in different areas of mathematics, the model will be created

using the Mamdani fuzzy logic, which would determine which testing system, classical or

online, would be used to test students in specific areas of mathematics.


4

The obtained results will be processed by certain statistical methods, i.e. using the T-test, and an effect

of the results in particular areas of mathematics will be appointed to aggregate results by regression

analysis. In addition to proving the hypothesis by statistical methods, a model will be created to help

us choose the appropriate type of testing depending on what students prefer while applying the method

of Mamdani fuzzy logic.

7. Research Results

The testing of the students was conducted in three areas of mathematics: arithmetic, logic, and

geometry, and the testing within these areas was conducted online. Testing on a paper was done in a

classroom in the same areas from the standpoint of methodical principles of problem-solving.

Different types of tasks with equal difficulty level were included in these two tests. Namely, the aim of

the subject Teaching methods of mathematics is to help students master the methodical principles of

solving mathematical problems. Students listen to different areas of mathematics within this subject by

learning one methodical principle of solving problems in the above-mentioned areas in the first 7

weeks, and then after the first partial exam, they have lectures on other methodical ways of solving

problems in the same areas as in the first 7 weeks. The goal is to teach students to solve problems in

these three areas in different ways.

The T-test will be used for nonparametric variables, which is good as it allows us to determine

whether the data is below the normal curve, which corresponds to the parametric estimate, or not,

which is the nonparametric statistic.

Table 1 Checks the normality of distribution curvature

Kolmogorov-Smirnova Shapiro-Wilk

Statistic Df Sig. Statistic Df Sig.

Logic .160 22 .147 .945 22 .255

Arithmetic .244 22 .001 .900 22 .030

Geometry .231 22 .003 .841 22 .002

In Table 1, we can read that the Kolmogorov-Smirnov test shows that our distribution for the Logic

results is at the limit of significance (Sig. = 0,147), but that the second test (Shapiro-Wilk = 0,255)

shows that this distribution cannot satisfy more rigorous criteria of normality; whereas for the results

in geometry and arithmetic, it is subject to the normal distribution as the significance limit is Sig. <

0.05.

Table 2 – Correlation between the results in specific areas and overall results

N Correlation Sig.

Logical & aggregate 22 .944 .000

Arithmetic & aggregate 22 .494 .020

Geometry & aggregate 22 .369 .091

Table 2 shows the impact of the results achieved by students in the particular areas of mathematics. It

can be seen that the results of the students tested online within the logic category have the greatest

influence on the overall results and that the least impact of the results is from geometry. The

correlation degree is highest in the mathematical logic and it had the greatest overall score influence

on the online test.

Table 3 – Multiple correlations (Model Summary)

Model R R Square

Adjusted R

Square

Std. The error of

the Estimate

1 .944a .892 .886 1.36348


5

In Table 3, we can notice that the contribution of the predictor variable (logical) to our dependent

variable (aggregate) is R = 0,944 and that it explains about 94% of this contribution variance. It can be

seen that the students’ achieved results in logical tasks have the greatest influence on the aggregate

results in these three areas and it can be explained by 94%.

Table 4 – Standardized beta coefficients

Model

Unstandardized Coefficients

Standardized

Coefficients

B Std. Error Beta

1 (Constant) .000 .000

Logical 1.000 .000 .811

Arithmetic 1.000 .000 .314

Geometry 1.000 .000 .215

Table 4 shows the relative strength of the predictor influence on the criterion or dependent variable

(aggregate). We see that the Logical variable has the most pronounced influence (0.811), followed by

Arithmetic (0.314), and finally Geometry (0.215). The results achieved by students in online logic

testing have the greatest impact on the aggregate results achieved by the students at the end of the

semester. The weakest impact on the aggregate results is noticeable in the results achieved by students

in geometry. This can be explained by the fact that students did the geometry tasks using the ‘in

application’ principle using the GeoGebra applets. This can help us in training students and choosing

how to test students.

Table 4 – Comparison of in-class and online Pearson Chi-Square test results

Value df

Asymp. Sig.

(2-sided)

Pearson Chi-Square 130.350a 135 .597

Likelihood Ratio 75.183 135 1.000

Linear-by-Linear

Association 6.186 1 .013

N of Valid Cases 22

In Table 5, Pearson Chi-Square is hi

2 = 130,350 and is not as significant as the asymptotic (sig. =

0,597) is much larger than the boundary one (0,05). Therefore, in units, there is no statistically

significant difference between the aggregate (the results obtained by online testing) and the

methodology (the results obtained by conducting testing on a paper). The results in the Table show

that there is no statistically significant difference between the results obtained by online testing

(aggregate) and the results obtained in the classical way (methodology). This confirms the first

hypothesis H1: Students do not achieve better results via online testing compared to classical testing if

they are not sufficiently treated with an online platform.

7.1. Mamdani Fuzzy Inference System

Mamdani method represents an approach to the problems of managing nonlinear systems. Fuzzy

systems are most commonly used for modelling by using the deduction method in the presence of

uncertainty. When taking into account the characteristics of the fuzzy variables and the overall fuzzy

system, as well as the results obtained by conducting the testing on real numerical data, the approach

chosen has proved to be adequate [10](D. Tiro et al., 2015). Fuzzy logic does not precisely define the

affiliation of a single element to a given set, it is rather that the affiliation is mostly measured in

percentages. Fuzzy management provides a formal methodology for representation, manipulation, and

implementation of human heuristic foreknowledge of how to control a particular system.


6

Figure 1 – The Fuzzy controller structure ([4] (G. Chen & T. T. Pham, 2001)).

The fuzzy controller components are visible from the Figure. The rule base contains the knowledge of

how to control the system in the best way and in the form of a set of logical (if – then) rules. The

fuzzification modifies the input signals so that they can be compared to the rules in the fuzzy rule

base. The deduction mechanism is a mechanism for evaluating which control rules are relevant to the

current state of the system and decides by the logic frame what the control signal will be, that is, the

input to the process. Defuzzification transforms the conclusion of the deduction mechanism into a

signal form so that this may be a signal that represents the input to the process [10](D. Tiro et al.,

2015).

MATLAB R2009b was used to conduct the research. The first step in designing is to select the inputs

and outputs of the controller. The variables that carry information about the behaviour of the system

should be the controller inputs. The fuzzy controller is Mamdani controller, the accumulation of

triggered conclusions uses the maximum operator. Language variables that are defined: Results,

OnLine testing, and InClass testing, all three have discrete values from the range [0, 40]. The list of

variables used in the fuzzy system is given in Table 2.

Table 6 – List of variables used in the fuzzy system

Linguistic variables Variable

type

Linguistic

conditions

RL – possible number of test

scores conducted in-class and

online

Output RInClass

ROnLine

OnLine – points scored in

arithmetic (A), logic (L) and

geometry (G)

Input

OnLineA

OnLineL

OnLineG

InClass – points scored in

arithmetic (A), logic (L) and

geometry (G)

Input

InClassA

InClassL

InClassG

The next step is to select a control variable, i.e. the input to the process. In order for the controller to

make a decision about the value of the control variable, it must receive enough information through

the input signal. Also, the controller must have an output that will control the system so as to bring it

to the required state with the desired performance. All this is defined in the FIS editor in which the

basic input and output signals are entered, their properties and values that can be received by

individual variables.


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Figure 2 – FIS Editor

Figure 3 – The affiliation function editor for the

InClass variable


OnLine variable


RL variable

Since fuzzy logic is constructed from structures that rely precisely on qualitative descriptions used on

a daily basis, in natural language, the ease of use of fuzzy logic imposes itself. Linguistic variables

should also have linguistic values. What the affiliation function will be depends on the conditions and

behaviour of the system. In this case, the Gaussian distribution curve was used for the InClass variable

and all related variables, and for the OnLine variable, different changes were applied with respect to

the difficulty level of the material (OnLineL - the Gaussian distribution curve was applied, while for

OnLineA and OnLineG - the Gaussian curve Gaussmf was applied). Practice shows that student

grades are subject to a normal distribution for areas of mathematics where students are more motivated

and where they can use logic to solve problems. While the practice has shown that students' grades for

more difficult material are subject to the Gaussian Gaussmf curve, that is, we have a higher number of

students with poor results than those who have achieved outstanding test results in these areas. This

process is also called encoding. It is often used in the process of determining a fuzzy set of fuzzy

controller input variables. The inputs then accept a finite number of discrete and specified values.

The goal of the fuzzy controller is to plot the input mapping to the controller outputs by using the

fuzzy logic. The primary mechanism for this is a list of if-then statements, called rules. All rules are

executed in parallel and their order does not matter. This kind of rule list is called the rule base. The

rules apply to linguistic variables and their properties. If all the terms and all the properties that define

those terms, i.e. variables, are defined previously, one can approach projecting a system that interprets

the rules. The rules are defined in the Rule Editor in Figure 7.


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Figure 7 – Rule editor

Figure 8 – Rules viewer for students who have

undergone online and classical testing in three areas

of mathematics

The rules defined in the Rule Editor represent the rule base on which the controller makes certain

decisions, i.e. conclusions based on which certain actions are recommended towards achieving a

certain value of the desired output signal.

The rules are as follows:

1. If (InClass is InClassA) and (OnLine is OnLineG) then (RL is ROnLine) (1)

2. If (InClass is InClassL) and (OnLine is OnLineA) then (RL is RInClass) (1)

3. If (InClass is InClassG) and (OnLine is OnLineL) then (RL is RInClass) (1)

For example, the third rule is: If the InClass result in geometry is excellent, then the output based on

the results from logic is not excellent, which is why it will be excellent on the online test. It can be

seen from the rule base that the method of choice was used when defining the rules. Defuzzification is

essentially a process opposite to the process of fuzzification and is also called decoding. This is, in

fact, a process that needs to convert the result of aggregation, which is basically a cross-section of a

surface, into a signal that is understandable to the process.

The setting of the initial phase of the system can be described by the following example. Figure 8

gives the rules for respondents who are equally inclined towards online and classic testing. These

students earn points in the range of 20 - 26. The results obtained with the modified fuzzy system, with

the centre of gravity as a method of defuzzification, showed that these students achieved better results

by classical testing. This category of respondents achieves significantly better results from paper

testing because the result was obtained by the Mamdani fuzzy system 23.5, which is largely

attributable to the fuzzy set of "medium" efficiency.

Figure 9 shows that students who have a tendency to solve arithmetic problems using the traditional

method prefer the traditional methods of knowledge assessment and that, with the same, the ratio of

classical tests should be more represented than online tests. This category of students in arithmetic and

logic achieves better test results using the classical method than using OnLine testing, as the result of

29.9 points was obtained by the Mamdani fuzzy system, which largely belongs to the fuzzy set of

"high" efficiency, and overall score is 24.0 (number of points scored on online tests).


9

Figure 9 – Rules viewer for respondents with a

better preference for mathematical areas: arithmetic

and geometry versus logic tasks

Figure 10 – Rules viewer for students who have a

better preference for geometry and online testing

Figure 10 shows the rules for the respondents who have a tendency towards online testing and who

show a tendency towards geometry, that is, have a solid visual perception, the score of 26 points. The

results obtained with the modified fuzzy system, with the centre of gravity as a method of

defuzzification, showed that these students achieved better results based on the results obtained from

geometry. A score of 26 points was obtained by the Mamdani fuzzy system, which belongs largely to

the fuzzy set of “high” efficiency, with an overall average of 25.3 points. For these students, online

tests should be more prevalent than classic tests.

7.2. Discussion

Based on the results obtained and on the basis of the fuzzy logic model, it can be concluded that the

choice of testing methods for students who are not sufficiently computer literate should be balanced

between classical and online testing. Students who prefer the field of arithmetic in mathematics should

be more exposed to classical testing, and students who show an affinity for the field of geometry in

mathematics should be more exposed to online testing. This research shows that students' preferences

for online and classical testing can be predicted based on initial tests conducted at the beginning of the

semester.

How can it be operationalized? - Please specify the specifics.

It is possible to predict students' preferences based on initial testing and using fuzzy logic to determine

the type of testing (classical or online), depending on which area of mathematics is being tested. The

model developed demonstrates the ability to predict student outcomes depending on the area being

tested and that this model can be tailored to the individual student approach so that the model shown in

Figure 1 can be adapted to achieve the best student outcomes. Based on the above, we can reject the

second hypothesis – H2: Test results are not affected by the knowledge assessment form, depending

on the area taught in mathematics.

Future research should show all the entropies of this model in the specific cases to which it may be

exposed. This would improve the model by introducing new variables to complement the model.

8. Conclusion

At the very beginning of the paper, the hypothesis was given that says: "Students do not achieve better

results via online testing compared to classical testing if they are not sufficiently treated with an online

platform." In this paper, based on the results of the Hi square test, it was found that the difference was


10

not statistically significant. That is, the students who were tested did not show better results in online

testing because the results primarily depend on the students' affinity for classical and online testing.

First of all, we need to take into account what students individually prefer and what areas of

mathematics they prefer.

When referring to testing conducted in a controlled environment and considering the reason that they

are not sufficiently familiar with the online platform, how and on what basis does this help you form

this conclusion? Students, on the other hand, achieve different results in both classical and online

testing depending on the area being tested.

To confirm the conclusion reached on the basis of the statistical method, hypothesis testing was

performed by the Mamdani fuzzy method. It was concluded that students in different areas of

mathematics achieve different results in both classical and online testing. In some areas such as logic,

respondents achieve better results using the classical method, while in arithmetic and geometry,

respondents achieve better results on online testing. Therefore, the tested model was developed with

fuzzy logic which was intended to identify areas that would be tested online so that students could get

the best results. Based on the above, the second hypothesis H2 was not confirmed: Test results are not

affected by the knowledge assessment form, depending on the area taught in mathematics.

Based on the research results, it is necessary to apply the developed model to large groups of students

in order to test it. In this way, the model could be improved by introducing all the variables that are

essential to achieving excellence in students. Fuzzy logic can integrate all the parameters that have

been researched in the world, which have proven to be essential for achieving the best possible

motivation and therefore results for students. The results of the research show that there is a difference

in results between the areas of mathematics and that the proposed model can be used to create a

balance between classical and online testing.

References

[1] ] Admiraal, W., Huisman, B., & Van De Ven, M. Self-and peer assessment in massive open online courses.

International Journal of Higher Education, 3(3) (2014): 119–128.

[2] Balfour, S. P. Assessing writing in MOOCs: Automated essay scoring and calibrated peer review ™.

Research & Practice in Assessment, 8(2013): 40–48.

[3] Brown, M., Costello, E., Donlon, E., & Giolla-Mhichil, M. N. A strategic response to MOOCs: How one

European university is approaching the challenge. The International Review of Research in Open and

Distributed Learning, 16 (6)(2015): 98 – 115.

[4] G. Chen & T.T. Pham. Introduction to Fuzzy Sets, Fuzzy Logic and Fuzzy Control Systems. CRC Press,

Boca Raton, FL, 2001.

[5] Conole, G. MOOCs as disruptive technologies: Strategies for enhancing the learner experience and quality

of MOOCs. RED: Revista de Educacion a Distancia, 50(2016), Article 2.

[6] Fahad S. & Shah A. Intelligent Testing using Fuzzy Logic. In: Iskander M. (ed.) Innovations in E-learning,

Instruction Technology, Assessment, and Engineering Education (pp. 95-98). Springer, Dordrecht, 2017.

[7] Lowenthal, P., & Hodges, C. (2015). In search of quality: Using quality matters to analyze the quality of

massive, open, online courses (MOOCs). The International Review of Research in Open and Distributed

Learning, 16 (5)(2015): 83 – 101.

[8] Meek, S. E. M., Blakemore, L., & Marks, L. Is peer review an appropriate form of assessment in a

MOOC? Student participation and performance in formative peer review. Assessment & Evaluation in

Higher Education, 42(6)(2017): 1000–1013.

[9] Sandeen, C. Assessment's place in the new MOOC world. Research & practice in assessment, 8(2013): 5–

12.

[10] D. Tiro, A. Poturiović & N. Buzadjija. The possibility of the hologram pyramid applying in the rapid

prototyping. In 2015 4th

Mediterranean Conference on Embedded Computing (MECO) (pp. 25-30).

Budva, Monte Negro, 2015.

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