Establishing Effective Remedial Instruction Grouping Using the Rough Set Theory and Grey Structural Modeling
Abstract
:1. Introduction
- (1)
- What tasks attribute influences students most in their overall English performance?
- (2)
- How can students be clustered into same level groups and given effective remedial instruction?
2. Methodology
2.1. Cluster Analysis Methods
2.2. Rough Set Theory
2.3. Object of Study by Rough Sets
- (1)
- Information system (IS):
- (2)
- Information function:
- (3)
- Discretization:
- (a)
- The spatial dimension following discretization should be reduced as much as possible (i.e., each attribute after discretization should be reduced to the minimum).
- (b)
- The amount of information lost after the discretization of attributes must be minimized.
- (4)
- Upper and lower approximation sets
- (a)
- The lower approximation set, which represents the set of elements that are “fully (definitively)” classified as equal under all y decisions.
- (b)
- The upper approximation set, which represents the set of elements that are “possibly (any one of them)” classified as equal under all y decisions.
- (5)
- (a)
- The positive domain of : the positive domain or the lower approximation set of , that is, , is the set of elements in that can be fully classified into set according to .
- (b)
- The negative domain of : the negative domain is the set of elements in that, according to , cannot be fully identified as belonging to the set , and it is the complement of .
- (c)
- Boundary: boundary is the set of elements that, according to knowledge and , cannot be fully identified as belonging to set nor to set , expressed in the following equation.
- (6)
- Attribute dependency
- (7)
- Significance of attributes
2.4. The Grey Relational Grade and Grey Structural Modeling
- (1)
- When represents a hierarchical structure, this hierarchy is composed of a group of structural elements, and the formula is:
- (2)
- Elements in set have mutual homogenous relationships. The principles of formation are described as follows:
- (a)
- For each random , the least number of elements is selected to form the set , , where .
- (b)
- For all , , and .
3. Analysis of a Practical Example
3.1. Calculation of the Attributes Using the Rough Set Theory
- (1)
- Online discussion board (): students had to finish the assigned writing homework online weekly.
- (2)
- Grammar exercises (): students did the grammar exercises during class time to check their understanding.
- (3)
- Listening exercises (): students did the listening exercises in the textbook during class time.
- (4)
- Vocabulary tests (): students took the NGSL vocabulary quiz at the beginning of the class weekly.
- (5)
- Oral interview (): a structured oral interview was given to each student after the midterm exam.
- (1)
- Discrete the data into three grades, and they are shown in Table 2
- (2)
- Then, reach condition attributes and decision attributes , the positive posc(D) can be found, and substitute into Equation (11) to obtain the dependent = 0.6207.
- (3)
- Omit the attributes of a1, and by using Equation (12), the significance of σ(a1) = 0.1111.
- (4)
- Then, omit , respectively, the significant of others attribute are:σ(a2) = 0.3333, σ(a3) = 0.1667, σ(a4) = 0.2222 and σ(a5) = 0.0556, respectively.
3.2. GSM Clustering Generation
- (1)
- LGRG calculation
- (a)
- Establish the standard sequence.
- (b)
- According to Table 1, each sequence is as shown below:
- (c)
- Calculate the difference sequence.
- (d)
- Substitute the data mentioned above into Equation (14), then the LGRG value can be obtained.[0.7171, 0.6897, 0.7779, 0.6945, 0.6780, 0.7819, 0.7369, 0.8154, 0.9059, 0.9142, 0.9126, 0.7408, 0.6217, 0.8720, 0.7176, 0.8707, 0.7879, 0.6820, 0.8851, 0.8004, 0.7515, 0.7853, 0.6747, 0.6573, 0.7972, 0.7995, 0.8003, 0.8414, 0.6896]
- (2)
- GGRG calculation
- (a)
- According to Table 1, each sequence is as shown below:
- (b)
- Calculate the difference sequences to obtain Δmax. and Δmin.
- (c)
- Calculate the grey relational matrix.The maximum lambda is 16.9732, and corresponding eigenvector are GGRG values: [0.1683, 0.1755, 0.1978, 0.1721, 0.1646, 0.1968, 0.1864, 0.2061, 0.1696, 0.1878, 0.1898, 0.1768, 0.1102, 0.1980, 0.1753, 0.1994, 0.2034, 0.1665, 0.1873, 0.2132, 0.2048, 0.1991, 0.1491, 0.1503, 0.1923, 0.2122, 0.2073, 0.2120, 0.1733]
4. Results and Discussion
5. Conclusions and Implications
- (1)
- Using rough set theory and GSM, a teacher can calculate the weighting of attributes and identify students’ learning problems. This insight allows him or her to adjust the curriculum design and teaching strategies to fit students’ needs. Before final grades are sealed at the end of the semester, students can follow their teacher’s remedial instruction to strengthen their weak points and achieve higher scores at the end of semester.
- (2)
- In previous studies, determining who needs remedial instruction has basically been based on whether a student passes or fails the subject. Students who had scored below 60 would be asked to take the remedial instruction class. Usually, the remedial class would be offered after school, and the teacher would ask the students to do many exercises. The problem with that approach is that the students would have to do exercises in every section instead of just doing the section that they really did not understand. Basically, it is a waste of time. However, through GSM, a teacher can easily cluster students who are all struggling with the same point at the same level. Then the teacher can provide customized remedial instruction to each group in class. Due to them being low-level students, it is very important for the teacher to guide them in “how to learn.” Besides, letting them learn with their peers is also crucial to their learning experience.
- (3)
- In this study, GSM objectively presents the distribution of students’ actual English performance. With LGRG showing the teacher’s expectation score on the vertical axis and GGRG showing the students’ actual performance on the horizontal axis. The results can conveniently be presented visually.
- (4)
- Together, the rough set and GSM can analyze small samples objectively. With only 29 students in the example analyzed in this study, it would have been difficult to find objective research tools to cluster such small samples. However, due to the characteristics of grey theory, small samples can be analyzed easily.
Funding
Institutional Review Board Statement
Informed Consent Statement
Conflicts of Interest
References
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Students | Scores | |||||
---|---|---|---|---|---|---|
s1 | 74 | 85 | 76 | 76 | 82 | 32 |
s2 | 78 | 64 | 74 | 39 | 62 | 64 |
s3 | 92 | 88 | 85 | 66 | 53 | 89 |
s4 | 84 | 73 | 85 | 71 | 44 | 42 |
s5 | 60 | 64 | 79 | 48 | 45 | 72 |
s27 | 72 | 82 | 76 | 88 | 78 | 72 |
s28 | 84 | 82 | 86 | 80 | 76 | 87 |
s29 | 76 | 82 | 60 | 50 | 44 | 72 |
Students | Scores | |||||
---|---|---|---|---|---|---|
1 | 2 | 2 | 2 | 3 | 3 | 1 |
2 | 2 | 1 | 2 | 1 | 2 | 2 |
3 | 3 | 3 | 3 | 2 | 2 | 3 |
4 | 3 | 1 | 3 | 2 | 1 | 1 |
5 | 1 | 1 | 2 | 2 | 1 | 2 |
27 | 2 | 2 | 2 | 3 | 3 | 2 |
28 | 3 | 2 | 3 | 3 | 3 | 3 |
29 | 2 | 2 | 1 | 2 | 1 | 2 |
Attributes | Online Discussion Board | Grammar Exercises | Listening Exercises | Vocabulary Tests | Oral Interview |
---|---|---|---|---|---|
Weighting |
LGRG | Group | Student Number | English Level |
---|---|---|---|
0.75–1 | 1 | 9, 10, 11, 19, 14, 16, 28 | CEFR A2 |
0.50–0.75 | 2 | 8, 20, 25, 27, 26, 22, 17, 6, 3, 21, 12 | CEFR A2 |
0.25–0.50 | 3 | 7, 1, 15, 4, 2, 29, 18, 5 | CEFR A2 |
0–0.25 | 4 | 23, 24, 13 | CEFR A2 |
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Wang, B.-T. Establishing Effective Remedial Instruction Grouping Using the Rough Set Theory and Grey Structural Modeling. Axioms 2021, 10, 299. https://doi.org/10.3390/axioms10040299
Wang B-T. Establishing Effective Remedial Instruction Grouping Using the Rough Set Theory and Grey Structural Modeling. Axioms. 2021; 10(4):299. https://doi.org/10.3390/axioms10040299
Chicago/Turabian StyleWang, Bor-Tyng. 2021. "Establishing Effective Remedial Instruction Grouping Using the Rough Set Theory and Grey Structural Modeling" Axioms 10, no. 4: 299. https://doi.org/10.3390/axioms10040299