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participat<strong>in</strong>g students have taken the test, their test results for each test item are given, as shown <strong>in</strong> Table 10.<br />

Table 7. Illustrative example of degree of difficulty each test item<br />

Degree of Difficulty of Test Item<br />

I1 I 2<br />

Test Item<br />

I 3 I 4 I 5<br />

Difficulty Degree 0.2 0.4 0.8 0.2 0.4<br />

Test Item<br />

Table 8. Illustrative example of the relationships among test items and concepts<br />

Concept<br />

C1 C2 C3 C4 C5<br />

I1 1 0 0 0 0<br />

I2 1 0 1 0 1<br />

I3 0 1 0 0 0<br />

I4 0 1 0 0 1<br />

I5 0 0 0 1 0<br />

Concept<br />

Table 9. Illustrative example of relationship among concepts<br />

Concept<br />

C1 C2 C3 C4 C5<br />

C1 1.0 0.0 0.4 0.0 0.0<br />

C2 0.0 1.0 0.6 0.0 0.2<br />

C3 0.4 0.6 1.0 0.2 0.6<br />

C4 0.0 0.0 0.2 1.0 0.4<br />

C5 0.0 0.2 0.6 0.4 1.0<br />

Table 10. Illustrative example of the relationship among students’ answers and test items<br />

Test Item<br />

S1 S 2<br />

Student<br />

S3 S4 S5<br />

I1 1 1 0 1 1<br />

I2 1 0 1 1 1<br />

I3 0 0 0 0 1<br />

I4 0 1 0 1 1<br />

I5 0 0 1 0 0<br />

Based on the association among Tables 8, 9, and 11, the highest difficulty level of the five concepts answered by the<br />

five students can be measured with Equation (5). In order to illustrate this clearly, Table 11 shows the association<br />

among the five test items and five concepts related to the fourth student (S4).<br />

Table 11. Illustrative example of relationship of test items, concepts, and the fourth student’s answers<br />

Concept<br />

Test Item<br />

C1 C2 C3 C4 C5<br />

I1 1 0 0 0 0<br />

I2 1 0 1 0 1<br />

I3 0 1 0 0 0<br />

I4 0 1 0 0 1<br />

I5 0 0 0 1 0<br />

The gray rows means that the fourth student answered the first, second, and fourth test items (I1, I2, and I4) correctly,<br />

as seen <strong>in</strong> Table 10. Based on Table 11, therefore, the highest difficulty level of the five concepts answered by the<br />

five students can thus be measured. For <strong>in</strong>stance, the highest difficulty level of second concept answered by the<br />

fourth student is:<br />

HDL( S4, C 2)<br />

= 0.2<br />

Furthermore, based on Tables 9,10, and11 the relevant level of five concepts answered by the five students correctly<br />

can be measured us<strong>in</strong>g Equation (2). Similarly, as shown <strong>in</strong> Table 12, the relevant level of second concept answered<br />

133

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