We define the Fuzzy Learning Response Membership Function with two parameters, the difficulty and discrimination of the item, and the variable of the function is the ability of the student. The Fuzzy Learning Response Membership Function is denoted as
( , ) 1.7 ( - )
The graph of the Fuzzy Learning Response Membership Function is shown in Fig. 3.
Fig. 3 The curve of the Fuzzy Learning Response Membership Function
The difficulty effect in the Fuzzy Learning Response Membership Function is shown In Fig. 4. With the same learning ability and discrimination given, the difficulty of the test item decreases cause the function’s curve shift to the right, thus the student’s LRI decreases as the difficulty decreases. (P > ⇒P LRI >LRI ).
1
Learning Response Index x
Learning ability
3 -3
Fig. 4 The difficulty effect in the Fuzzy Learning Response Membership Function
Fig. 5 shows the discrimination effect in the Fuzzy Learning Response Membership Function. The same difficulty of the test item and the difference of students’ ability were given; the curvature of the function increases while the discrimination of the test item increases. Thus the difference of the LRI increases (D2 >D1⇒ ∆LRI2 > ∆LRI1).
Fig. 5 The discrimination effect in the Fuzzy Learning Response Membership Function
The Fuzzy Learning Response Membership Function includes the parameter of difficult and discrimination. It is useful in indicating the actual degree of wellness in concepts learning responded from the test item, and the bias caused by the difficulty and discrimination can be
Example 1: Measures Function Of LRI
If a student has correctly answered a test item, based on the result of the testing, we say that the student learns well but no further information about how well it is. With the same situation, suppose the difficulty and the discrimination of the item is 0.813 and 0.375, respectively. If the student with learning ability of 1.8, we would have the following LRI to indicate the learning performance of the student.
1.7 0.375(1.8-0.813)
(1.8) 1 =0.65
LRI 1
e− ×
= +
3.2 System Architecture
The Concept Effect Relation Map of a course is quite useful as mentioned above.
However, mining with naïve data preprocessing may not reflect the physical effect relations of concepts. Therefore, in this thesis, we propose an IRT-Based Data preprocessing approach to construct the Concept Effect Relation Map, which is a map of directional graph with influence weights among cognition learning concepts of a course. Fig.7 shows the IRT-Based Data Preprocessing Concept Effect Relation Map Construction System with two modules: Data Preprocessing Module and Data Mining Module.
Fig. 6 IRT-Based Data Preprocessing Concept Effect Relation Map Construction System
In the module of Data Preprocessing, four procedures are held: Test Item Analysis, LRI Generation, Concept Decomposition/Aggregation and ACLR Fuzzification. In the Test Item Analysis procedure, Instruction Theory is applied to generate the difficulty and discrimination of test item and define the learning ability of the student. In the second procedure, Item Response Theory is applied to LRI in order to indicate the students’ learning status responded
Test Item
Concept Effect Relation Map
Associatio
from the test items. Concept Decomposition/Aggregation is the third procedure, which the Item-Concept Relationship Table is applied in concepts decomposition from items with Weight Learning Response Index (WLRI) and each concept has the attribute value called Aggregation Concept Learning Response index (ACLR) after the aggregation of the same concept separate in different items. The final procedure is the fuzzification of ACLR, where Fuzzy Theory is applied in transforming the numeric ACLR into symbolic “H” and ”L” to indicate well learning and poor learning, respectively.
The second module, Data Mining Module, has two procedures. In the former, applying Apriori Algorithm of data mining discovers four association rule types, L-L, L-H, H-H and H-L. In the latter, CERM is constructed based upon the scenario explanation of the mining association rule we proposed.
Base upon the historical testing records of students, we are able to preprocess the testing records with IRT-Based. Later, the embedded association rules are discovered by Data mining process. Finally, the procedure of Concept Effect Relation Map Construction Generates the Concept Effect Relation Map by scenario explanation of association rules. The procedures of the construction module are described as follows.
1) Data Preprocessing Module
y Test Item Analyzer:
Difficulty and discrimination of the test item are analyzed, and the students’ score are normalized by the normal reference as the relative learning ability of the students’.
y LRI Generator:
We adopt the two-parameter Logistic model function of Item Response Theory as our Learning Response Index of item (LRI), where the difficulty and discrimination of the test item is the parameters of our Learning Response measure Function, and the relative learning ability of the student is the variable of the function. Each test item answer by a student will have a value of LRI with valve between 0 and 1. The LRI of the test item individually responses the student’s learning status of the involved concepts.
y Concept Decomposition/Aggregation:
Usually, a test item may include several concepts; we separate the involved concepts of the test item by the test Item Concept Relationship Table (ICRT). Also, Concept may be involved in several test items. Concepts included in each test item can be separated by weight according to the entries of ICRT. The attributed value of decomposition concept is called Weight Concept Learning Response (WCLR), which are the multiple of the
decomposition weight of concept and the LRI of item. Same concepts’ WCLR will be aggregated by applying Sugeno Fuzzy Measure Function and are defined as the Aggregation Concept Learning Response index (ACLR).
y Fuzzy ACLR Generator:
In order to mine further association rules, we translate the students’ ACLR into the notation of “H”(Well Learning) and “L”(Poor Learning) by the Fuzzy membership function.
2). Data Mining Module y Association Rule Mining
The association rules are mined from the Fuzzy ACLR by using Apriori Algorithm. Four types of association rules L-L, L-H, H-L and H-H are used as the model to discover the assimilation and mis-concept effect relations among concepts.
y Concept Map Constructor
We define the direction and the weight of edge by the effect relationship and the value of support and confidence, respectively. Concept Effect Relation Map of the students’ is constructed based on the scenario explanation of association rule.
4. IRT-Based Data Preprocessing CERM Construction System
IRT-Based Data Preprocessing CERM Construction System includes two modules, the Data Preprocessing Module and the Data Mining Module. There are four procedures included in the Data Preprocessing Module: Test Item Analyzer, Learning Response Index (LRI) Generator, concept decomposition/aggregation and Fuzzy ACLR Generator. The second Module includes two procedures: Association Rule Mining and Concept Map Constructor.
4.1 Data Preprocessing Module
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