Analysis the Barrier of E-Learning in Mathematics Using Type-2 Fuzzy Data

E-Learning is mean to learn all round educational subjects taking assist of modern technology of the online. E-learning is also a big platform to learn mathematics. But in the current public health crisis, we are all working quickly to move our classes out of the classroom. Fortunately, even if online teaching and learning are new to all of us, some uncertainties are there exists. According to modern view uncertainty is considered essential to science and technology, it is not only the unavoidable plague but also it has impact a great utility. Generally, fuzzy sets are used to analyse fuzzy system reliability. To analyse the fuzzy system reliability, the reliability of each component of the system is considered as a Triangular intuitionistic Type 2 fuzzy number (TIT2FN). Keyword: Triangular intuitionistic Type 2 fuzzy number (TIT2FN), System reliability, Parallel system, Series system, e-learning, e-learning mathematics INTRODUCTION Necessity of E-Learning: In present busy world many a student cannot attend the classroom but in the online system they need not to do so. They can acquire their subjects matter through technology completely online. There are lot of advantages. A student can complete his course, take his degree in different subjects from online or internet where lot of trained and experienced teachers, professors deliver lecture as per the demand of learners. Here is no need of DVD, CD or a T.V. but although students have chance to talk to your teachers, professors and students. Sometimes the lectures are recorded to hear in future. The teacher can easily justify students ability, brightness, lack of knowledge etc and their grade will be given accordingly to their ability. It is proved that online training is a well method for modern time to get a successful life. Review on Reliability: It is known that conventional reliability analyses using probabilities have been found be inadequate in handling uncertainty of failure data and modelling. Onisawa and Kacprzyk (1995) used the concept of fuzzy approach f evaluation of the reliability of a system to overcome the problem. The discipline of the reliability engineering encompasses a number of different activities is pointed out by Kaufmann and Gupta (1988). Cai et al. (1991) represented the following two fundamental assumptions in the conventional reliability theory like Binary state assumptions: Probability assumptions. The system failure engineering and its use of the fuzzy logic was described by Cai et al. (1993). Cheng and Mon (1993) presented a method for analysing fuzzy system reliability using fuzzy number arithmetic operations. Using the concept of probist reliability as a triangular fuzzy Rahul Kar, et al (Analysis The Barrier of E Learning) Journal of Education and Learning Mathematics Research | Volume 1, Number 2, 2020 59 number in dynamic reliability evaluation of deteriorating systems presented by Verma et al. (2004). Review on Type-2 Fuzzy Number: The theory and design of interval type-2 fuzzy logic systems (FLSs) had been introduced by Liang and Mendel in the year of 2000. The concept of type-2 intuitionistic fuzzy sets under type-2 fuzzy sets and intuitionistic fuzzy sets was introduced by Tao and Jian [2012]. Hu et. al [2013] had proposed a new approach to solve multi-criteria decision making (MCDM) problems based on the form of interval type-2 fuzzy number. Mazandarani and Najariyan[2014] had defined a differentiability of the type-2 fuzzy number-valued functions. A new approach had been applied by Wang et. al [2015] for solving multi-criteria group decision-making (MCGDM) problems, which is described by trapezoidal interval type-2 fuzzy numbers (IT2FNs). An easy to approach to the problems of transportation had described by Chhibber et. al[2019] where incentre of centroids has been employed to convert trapezoidal fuzzy transportation problem of type 1 and type-2 both into crisp one. Senthil Kumar [2020] had designed a transportation problem where he used intuitionistic fuzzy number for supplies, demands. Motivation: Many research papers are available today in this world but discussion on barrier of e-learning especially on mathematics is not available using fuzzy data. Today e-learning is well developed discipline and has branched out into specialised areas such as mathematics, science, social-science, language group etc. In this paper we mainly utilise the uncertainty properties of fuzzy data to represent the failure of e-learning in mathematics by reliability system. Novelties: Some new interest and new work have done by our self which is mentioned below: i. Represent all most of the major barrier in e-learning of mathematics by reliability system using fuzzy data. ii. Try to utilise the properties of Triangular intuitionistic Type 2 fuzzy number to solve the above mentioned reliability problem. iii. Described imprecise reliability both of series and parallel systems using Triangular intuitionistic Type 2 fuzzy number. Frame a problem of reliability of major barrier in e-learning of mathematics with imprecise reliability components by Triangular intuitionistic Type 2 fuzzy number. Preliminaries E-Learning Learning includes the physical, social and pedagogical context in which learning will be taken place. A well environment increases the interest of our learners. It offers all students, teachers the strength of their minds with connectedness. It also offers to talk directly or indirectly to all members of these fields to practise more and study more. For Students: Every student can contact one by one with their teachers. They can get assist of reference books, school basis teaching and advance education in mathematics. For Parents: their sons and daughters can get a best theory for learning with minimum expenditure without sending them out of doors. Their H.W and C.W can be done and in the nick of time examinations are taken steps by steps. For Teachers: A teacher loves invaluable collections of papers in mathematics and he gives a net picture to his students who desire to know more and more about mathematics and take a challenge in math-world. A teacher is well trained about e-learning system. Rahul Kar, et al (Analysis The Barrier of E Learning) Journal of Education and Learning Mathematics Research | Volume 1, Number 2, 2020 60 E-learning in Mathematics In mathematics e-learning education provide interactive videos, can make mathematics easier and more enjoyable for solving any complex mathematics. Problems are displayed through visual and audio formats on a stimulating web-based interface. All these are done by our qualified and experienced teachers. Learning mathematics online has a range of advantages. Students can spend more time on specific subjects than they could in the traditional classroom, allowing them to learn mathematics at the perfect pace for information retention. Diagram 1 Learning Resources Challenges in E-Learning: In the present condition of the word this system is very needful all over the earth and this one of ways to learn perfectly. But some bars are also present now days. Challenges must be taken by the teachers, students and everyone who are linked with on-line education, extraordinarily who wants to get a perfect knowledge in math. Many points are applicable but most of them can be rejected by taking the challenge both teachers and students. We have already showed the issues and can be divided four types and they are followed: Students: The objective is to have online learning as one of the important ways for students to become successful in learning how to do mathematics as well as understand, appreciate and apply mathematics. This creates many challenges. Here the issues are as follows: Table 1 Student Issues 1 Serial no. Issues 1. It is dependent on student’s skill, understanding and thoroughly knowledge of online technology. Arithmetic, algebra, graphs and spreadsheets software is used extensively in this system according to age groups. Teaching is given in a group but thoroughly knowledge of medium is needed. 2. Students display different learning styles when they learn mathematics. Understanding his merit real judgment is required for a teacher, how to guide him a difficult sums. 3. Students want different versions of on-line system. They may require working on their math course. 4. They need to ready, equal access to the internet network in this system. 5. There are bars in this environment but they must contact with their teachers and other same class students. Some e-learning resources for teaching learning mathematics education Future school Online Mathematics Purple maths E-Book and ETelevision: E-Sound Book: S.O.S. Mathemati scs Rahul Kar, et al (Analysis The Barrier of E Learning) Journal of Education and Learning Mathematics Research | Volume 1, Number 2, 2020 61 6. All students desire a healthy environment to take their courses. Teacher: The teacher’s role changes to more of a facilitator and the degree of change varies depending on the type of online learning. Teachers need to maintain their professionalism of quality fundamental pedagogical practices in teaching mathematics online. Here the issues are as follows: Table 2 Issues 2 Serial no. Issues 1. It is paramount that an online mathematics course be thoroughly organized and developed ahead of time. 2. Assessment strategies are altered to fit online learning in mathematics. 3. It is essential to have research into matters that arise for discussion such as the ideal class size, the teacher’s role and the use of textbooks. 4. : Instructional design in a technological environment takes on a new role in remote teaching. 5. A sharing of information with other teachers becomes even more important in this new environment. 6. There could be an issue with society’s perception that the role of students and their teachers has completely changed. Learning Environment: Partnership of experts in discipline, thoroughly skill etc is essential in this system. There is a great need from student side to support the authority for their own requirement. Mathematics students are also need to know various ways how to solve different questions in an ideal environment with joy and happiness. Here the issues are as follows: Table 3 Issues 3 Serial no. Issues 1. There is a monopoly of certain online technological tools that may harm the system. 2. Decisions are made as to the degree on which a course provides. 3. Given the time for the ideal environment for teachers as well as for students. 4. In the nick of the time needful materials are needed for the demand of time. 5. Everyone always be helpful to each others. Mathematics: The educational system needs to stay true to the subject matter of mathematics. The role of pedagogy must be in control not technology or the pursuit of efficiency. The subject matter of mathematics should continue to stay all-important and not be diminished by the necessity of technology in on-line learning. Here the issues are as follows: Rahul Kar, et al (Analysis The Barrier of E Learning) Journal of Education and Learning Mathematics Research | Volume 1, Number 2, 2020 62 Table 4 Issues 4 Serial no. Issues 1. Rich learning tasks are important in developing mathematical concepts. 2. Students should be able to easily communicate mathematics electronically. 3. Best practices in a classroom may not adapt easily to online learning of mathematics. 4. Appropriate software is needed to translate captured graphics. 5. Inadequate different designing mathematics e-learning courses for different Triangular Intuitionistic Type 2 Fuzzy Number Definition: A TIT2FN i IFN A  is an IFN in R with the following membership function AIFN i x          and non membership function AIFN i x         


Necessity of E-Learning:
In present busy world many a student cannot attend the classroom but in the online system they need not to do so. They can acquire their subjects matter through technology completely online. There are lot of advantages. A student can complete his course, take his degree in different subjects from online or internet where lot of trained and experienced teachers, professors deliver lecture as per the demand of learners. Here is no need of DVD, CD or a T.V. but although students have chance to talk to your teachers, professors and students. Sometimes the lectures are recorded to hear in future. The teacher can easily justify students ability, brightness, lack of knowledge etc and their grade will be given accordingly to their ability. It is proved that online training is a well method for modern time to get a successful life.

Review on Reliability:
It is known that conventional reliability analyses using probabilities have been found be inadequate in handling uncertainty of failure data and modelling. Onisawa and Kacprzyk (1995) used the concept of fuzzy approach f evaluation of the reliability of a system to overcome the problem. The discipline of the reliability engineering encompasses a number of different activities is pointed out by Kaufmann and Gupta (1988). Cai et al. (1991) represented the following two fundamental assumptions in the conventional reliability theory like Binary state assumptions: Probability assumptions. The system failure engineering and its use of the fuzzy logic was described by Cai et al. (1993). Cheng and Mon (1993) presented a method for analysing fuzzy system reliability using fuzzy number arithmetic operations. Using the concept of probist reliability as a triangular fuzzy

Motivation:
Many research papers are available today in this world but discussion on barrier of e-learning especially on mathematics is not available using fuzzy data. Today e-learning is well developed discipline and has branched out into specialised areas such as mathematics, science, social-science, language group etc. In this paper we mainly utilise the uncertainty properties of fuzzy data to represent the failure of e-learning in mathematics by reliability system.

Novelties:
Some new interest and new work have done by our self which is mentioned below: i. Represent all most of the major barrier in e-learning of mathematics by reliability system using fuzzy data. ii. Try to utilise the properties of Triangular intuitionistic Type 2 fuzzy number to solve the above mentioned reliability problem. iii. Described imprecise reliability both of series and parallel systems using Triangular intuitionistic Type 2 fuzzy number. Frame a problem of reliability of major barrier in e-learning of mathematics with imprecise reliability components by Triangular intuitionistic Type 2 fuzzy number.

Preliminaries E-Learning
Learning includes the physical, social and pedagogical context in which learning will be taken place. A well environment increases the interest of our learners. It offers all students, teachers the strength of their minds with connectedness. It also offers to talk directly or indirectly to all members of these fields to practise more and study more. For Students: Every student can contact one by one with their teachers. They can get assist of reference books, school basis teaching and advance education in mathematics. For Parents: their sons and daughters can get a best theory for learning with minimum expenditure without sending them out of doors. Their H.W and C.W can be done and in the nick of time examinations are taken steps by steps. For Teachers: A teacher loves invaluable collections of papers in mathematics and he gives a net picture to his students who desire to know more and more about mathematics and take a challenge in math-world. A teacher is well trained about e-learning system. Kar, et al (Analysis The Barrier of E Learning) Journal of Education and Learning Mathematics Research | Volume 1, Number 2, 2020 60

E-learning in Mathematics
In mathematics e-learning education provide interactive videos, can make mathematics easier and more enjoyable for solving any complex mathematics. Problems are displayed through visual and audio formats on a stimulating web-based interface. All these are done by our qualified and experienced teachers. Learning mathematics online has a range of advantages. Students can spend more time on specific subjects than they could in the traditional classroom, allowing them to learn mathematics at the perfect pace for information retention.

Challenges in E-Learning:
In the present condition of the word this system is very needful all over the earth and this one of ways to learn perfectly. But some bars are also present now days. Challenges must be taken by the teachers, students and everyone who are linked with on-line education, extraordinarily who wants to get a perfect knowledge in math. Many points are applicable but most of them can be rejected by taking the challenge both teachers and students. We have already showed the issues and can be divided four types and they are followed: Students: The objective is to have online learning as one of the important ways for students to become successful in learning how to do mathematics as well as understand, appreciate and apply mathematics. This creates many challenges. Here the issues are as follows: It is dependent on student's skill, understanding and thoroughly knowledge of online technology. Arithmetic, algebra, graphs and spreadsheets software is used extensively in this system according to age groups. Teaching is given in a group but thoroughly knowledge of medium is needed.

2.
Students display different learning styles when they learn mathematics. Understanding his merit real judgment is required for a teacher, how to guide him a difficult sums.

3.
Students want different versions of on-line system. They may require working on their math course. 4.
They need to ready, equal access to the internet network in this system.

5.
There are bars in this environment but they must contact with their teachers and other same class students.

6.
All students desire a healthy environment to take their courses.
Teacher: The teacher's role changes to more of a facilitator and the degree of change varies depending on the type of online learning. Teachers need to maintain their professionalism of quality fundamental pedagogical practices in teaching mathematics online. Here the issues are as follows: Table 2 Issues 2 Serial no.

1.
It is paramount that an online mathematics course be thoroughly organized and developed ahead of time.

2.
Assessment strategies are altered to fit online learning in mathematics.

3.
It is essential to have research into matters that arise for discussion such as the ideal class size, the teacher's role and the use of textbooks.

4.
: Instructional design in a technological environment takes on a new role in remote teaching.

5.
A sharing of information with other teachers becomes even more important in this new environment.

6.
There could be an issue with society's perception that the role of students and their teachers has completely changed.
Learning Environment: Partnership of experts in discipline, thoroughly skill etc is essential in this system. There is a great need from student side to support the authority for their own requirement. Mathematics students are also need to know various ways how to solve different questions in an ideal environment with joy and happiness. Here the issues are as follows: Table 3 Issues 3 Serial no.

1.
There is a monopoly of certain online technological tools that may harm the system.

2.
Decisions are made as to the degree on which a course provides.

3.
Given the time for the ideal environment for teachers as well as for students.

4.
In the nick of the time needful materials are needed for the demand of time.

5.
Everyone always be helpful to each others.

Mathematics:
The educational system needs to stay true to the subject matter of mathematics. The role of pedagogy must be in control not technology or the pursuit of efficiency. The subject matter of mathematics should continue to stay all-important and not be diminished by the necessity of technology in on-line learning. Here the issues are as follows:

Some arithmetic operations of Type-2 Intuitionistic Fuzzy Number based on cuts method:
Properties

Imprecise reliability of series and parallel systems using arithmetic operations or Triangular Intuitionistic Type2 Fuzzy Numbers:
The imprecise reliability of a series and a parallel system present here. Triangular Intuitionistic Type2 Fuzzy numbers are used to represent the reliability of each component of the systems.

Series System
Let us consider Application with Discussion:

Calculation of system barrier in e-learning of Mathematics using Triangular Intuitionistic Type2 Fuzzy Numbers:
Barrier in e-learning of Mathematics depends on different facts. The facts are described below. There are five sub factors of each of facts. The fault-tree of barrier in e-learning of Mathematics is shown in the figure 1.  The intuitionstic type-2 fuzzy barrier in e-learning of Mathematics can be calculated when the barrier of the occurrence of basic fault events are known. Barrier in e-learning of Mathematics of a truck can be evaluated by using the following steps. Step1:

Result of Barrier in E-Learning of Mathematics Using TIT2FN
Numerical of barrier in e-learning of Mathematics using fault tree analysis with intuitionistic type-2 fuzzy failure rate. The components failure rates as TIT2FN are given by