IRT-Based Concept Learning Response Function

The quality of CERM construction deeply depends on the method of data preprocessing before data mining. For example, ratio of incorrect/correct answers is a naïve way of data preprocessing to represent the learning status of concept. However, the ratio value mentioned above may be affected by the difficulty and discrimination of the test item. In order to eliminate the bias affect of concept learning response cause by the difficulty and discrimination, we propose an IRT-Based Concept Learning Response (CLR) Function.

Easy Difficult Easy Difficult

1

CT C_T2 C Score T3 CT1 C_T2 C ScoreT3

S1

2 3 3

S2

3 3 2

IRT-Based CLR Function

S2 0.24 0.53 0 0.77

Fig. 1 The difficulty effect of IRT-Based CLR Function

In Fig. 1, Two students S1 and S2 are tested by three items C_T1, C_T2andC _T3 concerning the same concepts. As we can see, S2 has answered the C_T1 correctly but S1 doesn’t. Contrarily, S1 has answered the C correctly but S2 doesn’t. Identically, both S1 _T3 and S2 have answered C_T2 correctly. On the left side of Fig. 1, we may say the “weight” of learning response is the same in C_T1, C_T2 and C . In other words, since S1 and S2 have _T3 the same score = 2, without the consideration of difficulty, it is hard to distinguish the learning status of the two students in learning concept C. But the situation changes while we apply the IRT-Based CLR Function. That is, S1 would have higher CLR than S2 has. Since

the difficulty ofC_T3 is higher than C_T1. Totally, S1’s CLR=0.97 is higher than S2’s CLR=0.77, and we are able to say that S1 has learned the concept better than S2.

Low High Low High

1

CT C_T2 C Score T3 CT1 C_T2 C ScoreT3

S1

2 3 3

S2

3 3 2

IRT-Based CLR Function

S2 0.17 0.53 0 0.70

Fig. 2 The discrimination effect of IRT-Based CLR Function

In Fig. 2, Two students S1 and S2 are tested by three items C_T1, C_T2andC _T3 concerning the same concepts. As we can see, S2 has answered the C_T1 correctly but S1 doesn’t. Contrarily, S1 has answered the C correctly but S2 doesn’t. Identically, both S1 _T3 and S2 have answered C_T2 correctly. On the left side of Fig. 2, we may say the “difference”

of the learning response is the same in C_T1, C_T2 and C . In other words, since S1 and S2 _T3 have the same score = 2, without the consideration of discrimination, it is hard to distinguish the learning status of the two students in learning concept C. Again the situation changes while we apply the IRT-Based CLR Function. That is, S1 would have higher CLR than S2 has.

Since the “difference” of CLR in concept C is larger than the “difference” of CLR in _T3 concept C_T1. Finally, S1’s CLR=1.20 is higher than S2’s CLR=0.70, Thus we are able to say that S1 has learned the concept better tha S2.

As mentioned above, it is necessary to have a fuzzy membership function of learning status which is obtained from the testing item with the value between 0~1. To build up such a

Fuzzy Learning Response Membership Function, difficulty and discrimination of test item should be considered. Meanwhile, the Fuzzy Learning Response Membership Function must have the characteristics below:

1. Positive relative to the difficulty of the test item.

2. Positive relative to the discrimination of the test item.

With the goals and characteristics mentioned above, we consider the Two-parameter Logistic Model function

1.7 ( )

( ) 1

1 ^{D x P}

P x = e^{− −}

+ ^

 , e =2.719

of the Item Response Theory (IRT). Originally, IRT is used to estimate the aptness of the test item. The ability x of the student is the variable of the Logistic function, which includes two parameters, difficulty P and discrimination D. The aptness of the test item is indicated by distribution of the answering probabilities of different abilities of students.

We adopted the Two-parameter Logistic model as our Fuzzy Membership Function of learning response to indicate the student’s learning status responded from the testing items with the consideration of difficulty and discrimination. The definition of the Fuzzy Learning Response Membership Function is described as follows.

( )_i

Definition 1 Fuzzy Membership Function of Learning Response

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^{− ×}

= +