The first module has four procedures. Sequentially, the Test Item Analysis is the first procedure, which calculates the difficulty and the discrimination of each item from students’
testing result. The Learning Response Index (LRI) is generated in the second procedure. The LRI of each item indicates the students’ learning response base upon Item Response Theory.
The third procedure handles the concept decomposition and aggregation, while Item–Concept Relationship Table (ICRT) is applied in concept decomposition of each item with the weight of response and the Sugeno Fuzzy Measure Function is applied in concept aggregation with Weight Learning Response Index (WLRI) that is dissociated by ICRT. Several WLRI of the same concept are aggregated as the value of Aggregated Concept Learning Response (ACLR),
which indicates the concept learning status of the student. The final procedure is to transform the ACLR from numeric into symbolic H/L by the Fuzzy membership function.
1) Test Item Analyzer
The Test Item Analyzer is the one who calculates the difficulty and the discrimination of each item from the result of students’ testing. First of all, we build up the Testing Result Table (TRT) according to the students’ answer sheet. Let Am n× be the matrix of TRT, the element
a is the answered results of the test items ji T , j=1,2,…,m, from students j S , i=1,2,…,n. i
The elements a =1 and ji a =0 denote the ith student having right or wrong to the jth test ji item, respectively. Table 2 shows the example of TRT with six students tested by seven items.
Table 2Testing Result Table (TRT)
Test item Student ID
T1 T2 T3 T4 T5 T6
S1 1 0 0 1 1 0 S2 0 1 1 0 1 1
S 3 0 1 1 0 1 1
S4 1 0 1 1 0 1
S 5 1 0 1 1 0 1
S 6 1 1 0 1 1 1
Definition 2 Student’s Learning Ability
We standardize the score of student S as the student’s learning ability. i
i
Example 2: Student’s Learning Ability
If the student has the score of 92, with the class average of 62 and standard deviation of 15, then the standardized score of the student would be 2. We use the standardized score value 2 as the learning ability of the student.
Definition 3 Difficulty and Discrimination of Test Item
Based upon the theory of instruction, let B and P be the set of high achievement students (the best 27%) and low achievement students (the last 27%), respectively.
y Difficulty of Test Item T : j
2
B P
j j
j
R R
P +
= , j=1,2,…,m,
y Discrimination of Test Item T : j Dj =RBj -RPj , j=1,2,…,m,
where RBj : the ratio of answering right of test item T in set B. j
P
Rj : the ratio of answering right of test item T in set P. j
The value of difficulty is between 0 and 1, which 0 means hard and 1 means easy of the test item. Also, the value of discrimination is between 0 and 1, which 0 means the low discrimination and 1 means the high discrimination of the test item.
2) LRI Generator
The difficulty and discrimination of each test item are computed after the Test Item Analyzer. Let Tm n× be the matrix of Learning Response Mapping Table, where the student
S , i=1,2,…,n is column variable and the test item i T , i=1,2,…,m, is the row variable. The j entries t of the matrix ji Tm n× are defined as the Learning Response Index of the test item T . j
Definition 4 The Learning Response Index of the Test Item
Let Tm n× be the matrix of the Learning Response Mapping Table, and t be the ji entries Tm n× , which indicates the student’s learning status obtain by the test item. We define the Learning Response Index of the Test Item as
( , )( ) denotes the student S ’s learning status response through test item i T . j
Example 3: Student’s Learning Response Index
The difficulty and discrimination of the test item are 0.813 and 0.375 as given, respectively. If the student’s ability is 1.8 and has right answer of the test item, then the student’s LRI through the test item is
( , ) (0.813,0.375) 1.7 0.375(1.8-0.813)
( )= (1.8)= 1 =0.65
The value 0.65 indicates the learning status of the student while having the right answer of the test item.
3) Concept Decomposition and Aggregation
Intuitively, a test item may include several concepts, so we have to decompose the concepts included in a test item with the attribute called WLRI. Later, we aggregate the WLRI of the same concept involved in several test items. The aggregated value of attribute is treated as Aggregation Concept Learning Response index (ACLR).
i. Concept Decomposition :Item Concept Relationship Table
First of all, we decompose the concepts performed in the test item by the test Item - Concept Relationship Table (ICRT). If the test sheet has m test items T , j=1,2... m, with p j concepts C tested, namely k=1,2... ,p. Let k Bm p× be the matrix of ICRT, and the element
b is the weight between 0~1, which indicates the relativity of concept jk C involved in test k
item T . j
Example 4: Test Item - Concept Relativity Table (ICRT)
Table 3 shows the ICRT of seven test item include five concepts, where the concept is the row variable and the test item is the column variable. By referring to the ICRT, we can see that the Test item T includes Concepts 3 C 、3 C4and C with different 5 relativity weight 1, 0.3, 0.4, respectively. Simultaneously, test items T2and T have 6 included the same concepts C2and C , but with different weight of relativity, say 0.8, 1 3
T with the weight of relativity, 0.75, 1, 1, 0.8, respectively. 6
Table 3 Test Item-Concept Relation Table (ICRT)
Concept
Test item C1 C2 C3 C4 C5
T1 0.9 0 0.75 0 0
T2 0 0.8 1 0 0
T3 0 0 1 0.3 0.4
T4 1 0 0 0 0
T5 0 1 0 0 0
T6 0 1 0.8 0 0
T7 1 1 0 0 1
ii. Concept Aggregation
The concepts learning response of a student through each test item are given according to the LRMT with the weight in ICRT.
Definition 5 The Weight Concept Learning Response Index
Since b and jk t are the entries of matrixes ji Bm p× and Tm n× , respectively. Recall that Bm p× and Tm n× is the matrixes of ICRT and Learning Response Mapping Table, respectively. Let Wn m p× × be the matrix of Weight Concept Learning Response mapping Table (WCLRT), and the values of entries w is between 0~1. We obtain ijk w by the ijk following.
ijk jk ji
w =b × , i=1,2,…,n, j=1,2,…,m, k=1,2,…,p, t
where w indicates the concept ijk C ’s learning status of student k S , which the i concept C is involved in test item k T . j
Example 5: Weight Concept Learning Response
Suppose the student S has the LRI=0.6 obtain from the test item i T which involves j
concepts C1 and C2 with the weight of relativity 0.75 and 1, respectively. Then the learning status indexes of the student concerning concept C1 and C2 are
1 1
ij j ji
w =b × = 0.75×0.6=0.45 and t wij2 =bj2× =0.75×1=0.75, respectively. tji
Aggregation The aggregation WCLR of the set concerning concept C is defined as the Aggregation j
Concept Learning Response index (ACLR), which indicates the status of the student in learning concept C . j
Fig. 7 ACLR index is aggregated from item’s WLRI
So far, it is important to figure out the function to achieve the aggregation mention above.
In this thesis, we apply the Sugeno fuzzy measure function as our aggregation function. In order to set the value between 0~1, the aggregation function must satisfy the boundary condition with (1) gλ( )φ = , (2) ( ) 10 gλ X = , and the properties, (3) if A B⊆ , then
( ) ( )
g A ⊆g B , where A and B are subsets of X.
Definition 6 The Function of Aggregation Concept Learning Response index (ACLR)
The Function aggregates the element of the set Xkof WCLR concerning concept Ck, and define the ACRL of concept Ck as
where wijk indicates the WCLR defined in Definition 5.
Example 6: Aggregation Concept Learning Response index (ACLR)
According to the ICRT, the test items T1, T2, T3 are decomposed into {C1,C2,C3}, {C1,C3}, {C3}, respectively. The learning response index of concepts in each test item is obtained by the product of the test items’ LRI and the weight in ICRT. Each concept
Cjhas a set WLRI from several test items Tk concerning concept Cj.
Table 4 The WLRI and the ACLR of concepts
Item C1 C2 C3
T1 0.64 0.80 0.35 T2 0.69 0 0.87 T3 0 0 0.69 ACLR 0.902 0.800 0.996
Suppose student Si has such a table shown in Table 4 after testing. T1 includes the
concepts of C1, C2 and C3 with the WLRI of 0.64, 0.80 and 0.35; T2 include the concepts of C1 and C3 with the WLRI of 0.69 and 0.87, and T3 include the concepts of C3 with the WLRI of 0.69. By looking through the columns, concept C1, C2 and C3 will have the sets of WLRI {0.64,0.69,}, {0.80,} and {0.35,0.87,0.69}through T1, T2 and T3, respectively. Then the ACLR of student Si concerning concepts C1, C2 and C3 are calculated as below.
i
(1- 0.97 0.64)(1- 0.97 0.69) -1
({0.64,0.69}) 0.902
-0.97
g × ×
= =
(1- 0.97 0.80) -1
({0.80}) 0.800
-0.97
gi ×
= =
(1- 0.97 0.35)(1- 0.97 0.87)(1- 0.97 0.69) -1
({0.35,0.87,0.69}) 0.996
-0.97
gi × × ×
= =
4) Fuzzy ACLR Generator
In order to mine association rule further, it is necessary to transform the numeric data into symbolic data. We achieve the transformation by applying fuzzy Theory. Here we have two membership functions shown in Fig. 8 to transform the students’ numeric ACLR into symbolic notation. The symbolic notations obtain by the fuzzification is “L” and “H”, which denote “Poor Learning” and “Well Learning”, respectively. Ci.L and Ci.H denote the value obtain from the LOW Fuzzy Function and the HIGH Fuzzy Function of the concept Ci.’s ACLR, respectively. The value of each concept’s ACLR will transform into the notation “L”
and “H” if Ci.L>Ci.H and Ci.L ≤ Ci.H, respectively. For example, if ACLR=0.65, by the given membership functions, we have Ci.L=0.2 and Ci.H=0.66. Since Ci.L ≤ Ci.H the Fuzzy ACLR is denoted as “H”. Completely, fuzzification of ACLR is described in Example 4.6.
ACLR
Fig. 8 The given membership functions of students’ numeric ACLR.
Example 7: Fuzzy ACLR Generator
Suppose six students are tested with five concepts. Students’ ACLR of each concept is shown in Table 5.
Table 5 The Students’ ACLR of each concept
C1 C2 C3 C4 C5
By the given membership functions, ACLR of each student has the degree value of Ci.L and Ci.H. Table 6 shows the fuzzy degree values of ACLR obtain by the LOW/HIGH fuzzy membership functions.
Table 6 The degree value of ACLR translated by the LOW /HIGH fuzzy membership functions
C1 C2 C3 C4 C5
We then transform the numeric ACLR into the symbolic notation of “H” if Ci.L ≤ Ci.H and “L” if Ci.L>Ci.H. The notation “H” and “L” represent the meaning of “well learning”
and “poor learning”, respectively. Table 7 shows the fuzzification result of ACLR.
Table 7 The Students’ fuzzy SCCI of each concept.
C1 C2 C3 C4 C5
S1 H L H H L
S2 H L L H H
S3 H L L L H
S4 H L H H H
S5 H L L L L
S6 H L L H H
Concept FACLR
Student