Chapter 8 Knowledge Miner (KMin)
8.2 Two-Phase Concept Map Construction (TP-CMC)
In the last five years, many adaptive learning and testing systems have been proposed to offer learners customized courses in accordance with their aptitudes and learning results [5] [21] [22] [38] [41] [49] [51] [54] [122] [126]. For achieving the adaptive learning, a predefined concept map of a course, which provides teachers for further analyzing and refining the teaching strategies, is often used to generate adaptive learning guidance. However, it is difficult and time consuming to create the concept map of a course. Thus, how to automatically create a correct concept map of a course becomes an interesting issue.
Therefore, in this dissertation, we propose a Two-Phase Concept Map Construction (TP-CMC) algorithm to automatically construct a concept map of a course by historical testing records. In TP-CMC, the Test item-Concept Mapping Table records the related learning concepts of each test item. As shown in Table 8.7, five quizzes contain these related learning concepts A, B, C, D and E, where “1” indicates the quiz contains this concept, and “0” indicates not. Moreover, a concept set of quiz i is denoted as CSQi, e.g., CSQ5={B, D, E}. The main idea of our approach is to extract the prerequisite relationships among concepts of test items and construct the concept map.
Based upon assumptions, for each record of learners, each test item has a grade.
Table 8.7: Test Item–Concept Mapping Table
A B C D E
Q1 0 0 0 1 0
Q2 1 0 1 0 0
Q3 1 0 0 0 0
Q4 0 1 1 0 0
Q5 0 1 0 1 1
As shown in Figure 8.8, our Concept Map Construction includes two phases:
Grade Fuzzy Association Rule Mining Process Phase and Concept Map Constructing Process Phase. The first phase applies fuzzy theory, education theory, and data mining approach to find four fuzzy grade association rule types, L-L, L-H, H-H, H-L, among test items. The second phase further analyzes the mined rules based upon our observation in real learning situation. Even based upon our assumptions, constructing a correct concept map is still a hard issue. Accordingly, we propose a heuristic algorithm which can help construct the concept map.
Figure 8.8: The Flowchart of Two-Phase Concept Map Construction (TP-CMC)
8.2.1 Grade fuzzy association rule mining process
In [126], the Look Ahead Fuzzy Association Rule Miming Algorithm (LFMAlg) has been used to find the associated relationship information embedded in the testing records of learners. In this phase, we propose an anomaly diagnosis process to improve LFMAlg and reduce the input data before the mining process.
1. Grade Fuzzification:
Firstly, because the numeric testing data are hard to analyze by association rule mining approach, we apply Fuzzy Set Theory to transform these into symbolic. Thus, after the fuzzification, the grade on each test item will be labeled as high (H), middle (M), and low (L) degree, which can be used as an objective judgment of learner’s performance.
2. Anomaly diagnosis:
Based upon Item Analysis for Norm-Referencing of Educational Theory [89], the discrimination of item can tell us how good a test item is, i.e., item with high degree of discrimination denotes that the item is well designed. If the discrimination of the test item is too low (most students get high score or low score), this item as redundant data will have no contribution to construct the concept map. For decreasing the redundancy of test data, we propose a fuzzy item analysis, called Anomaly Diagnosis, to refine the test data.
3. Fuzzy Data Mining:
Then, we can apply LFMAlg [126] to find the grade fuzzy association rules of test items from the historical testing data. In this dissertation, we analyze the prerequisite relationships among learning concepts of quizzes according to 4 association rule types, L-L, L-H, H-L, H-H, generated from Large 2 Itemset. Qi.L notation denotes that the ith question (Q) was tagged with low (L) degree, e.g., Q2.L→Q3.L means that learners get low grade on Q2 implies that they may also get low grade on Q3.
8.2.2 Concept map constructing process 1. Concept map constructor:
Firstly, the result of analyzing four association rule types, L–L, L–H, H–H, and H–L, are used to construct the prerequisite relationships between concept sets, which are used to define the edge between nodes of concept set and provide teachers with information for further refining the test sheet, of learning concepts of test items. Then, based on the prerequisite relationships of concept set and the Test item-Concept Mapping Table, we propose a Concept Map Constructing (CMC) Algorithm to find the corresponding learning concepts of concept set to construct the concept map according to the join principles of concept-pair.
8.2.3 Grade fuzzy association rule mining process 1. Grade fuzzification:
As described in Section 8.2.1, we apply fuzzy concept to transform numeric grade data into symbolic, called Grade Fuzzification. Three membership functions of each quiz’s grade are shown in Figure 8.9. In the fuzzification result, “Low”, ”Mid”, and
“High” denote “Low Grade”, “Middle Grade”, and “High Grade” respectively. Qi.L, Qi.M, and Qi.H denote the value of LOW fuzzy function, MIDDLE fuzzy function, and HIGH fuzzy function for the quiz i, respectively. By given membership functions, the fuzzification of testing records is described in Example 8.1.
Example 8.1:
In Figure 8.10, assume there are 10 testing records with 5 quizzes of learners and the highest grade on each quiz is 20.
Figure 8.9: The given membership functions of each quiz’s grade.
Figure 8.10: The Fuzzification of Learners’ Testing Records
2. Anomaly diagnosis:
For refining the input testing data, we propose the anomaly diagnosis, called Fuzzy Item Analysis for Norm-Referencing (FIA-NR) by applying Item Analysis for Norm-Referencing of Educational Theory, shown in Figure 8.11. A test item will be deleted if it has low discrimination.
Figure 8.11: Fuzzy Item Analysis for Norm-Referencing (FIA-NR)
Example 8.2:
Table 8.8 shows the fuzzified testing grades of learners on Q4 sorted in the descending order of each learner’s total score in the test sheet. For example, in Figure 8.10, because the result of fuzzification of learner ID 4 is (0.3, 0.5, 0.0), her/his Grade Level can be tagged with M by the Max(L, M, H) function.
Table 8.8: Sorted Fuzzified Testing Grade on Q4
Group High Middle Low
Learner ID 1 2 3 4 6 5 7 8 9 10
Total (100) 77 54 53 48 44 36 35 28 26 21 Grade Level
=Max(L,M,H) H L L M L L L L L L
Then, by applying FIA-NR algorithm, we can get the Difficulty and Discrimination of every quiz. For example, the P4H and P4L of Q4 are
3
P L , respectively. Therefore, its Difficulty P4 and Discrimination D4 are 1 4 + 4 =1−1/3+0=5=0.83
−
= P P
P4 H2 L 2 6
and 0.33 respectively.
Thus, learners’ grade on Q4 will be deleted because its Discrimination is too low to use during the mining process and the construction of the concept map. Accordingly, the test sheet can be redesigned. All evaluated results are shown in Table 8.9.
Table 8.9: Difficulty and Discrimination Degree of Each Quiz Q1 Q2 Q3 Q4 Q5
Difficulty (0 to 1) 0.25 0.42 0.42 0.83 0.75 Discrimination (-1 to 1 ) 0.5 0.83 0.83 0.33 0.5
3. Fuzzy Data Mining:
After filtering out these useless quizzes, we can apply Look Ahead Fuzzy Association Rule Mining Algorithm [126] as shown in Figure 8.12 to find the fuzzy association rules of test items. In LFMAlg Algorithm, the support value of every itemset x in candidate can be evaluated by the support(x) function, where x={A, B} , A∩B=
Cl ⊆Cl−1
φ . Then, the support(x) = support(A∪B) = Min(A, B), where n is the number of learners. For example, in Figure 8.10, support(Q1.L, Q3.H)=Min(1.0, 0.7)+Min(1.0, 0.7) =1.4.
1n
∑
Figure 8.12: Look ahead Fuzzy Association Rule Mining Algorithm (LFMAlg)
Example 8.3:
For the data shown in Examples 8.1 and 8.2, Figure 8.13 shows the process of finding the association rules with large 2 itemset by LFMAlg algorithm.
Figure 8.13: The Mining Process of Large 2 Itemset
Thus, Table 8.10 shows the grade fuzzy association rules with minimum confidence 0.8 generated from large 2 itemset into L-L, L-H, H-H, and H-L types. The Confi
(Confidence) is used to indicate the important degree of ith mined association rule. For
example, the Confidence (Conf1) of rule Q2.L→Q3.L can be obtained as follows.
Table 8.10: The Mining Results (Confi > 0.8) Rule Types Mined Rules Confi
Q2.LÆQ3.L 0.95
8.2.4 Concept Map Constructing Process 1. Concept Map Constructor:
Before constructing the concept map, we can get the prerequisite relationship among concepts of quiz from analyzing four association rule types, L-L, L-H, H-L, and H-H, based upon our observation obtained by interviewing the educational experts, in real learning situation. Therefore, we can conclude the Heuristic 1: given two quizzes Q1 and Q2, if concepts of Q1 are the prerequisite of concepts of Q2, Learner gets low grade on Q1 implies that s/he may also get low grade on Q2 or Learner gets high grade on Q2 implies that her/his grade on Q1 is high. As shown in Table 8.11, for each rule type, we use Heuristic 1 to get its prerequisite relationships among concept sets of quizzes with parameterized possibility weight, which are used to construct the concept map. The definition of the symbols used in Table 8.11 is described as follows.
Symbol Definition:
CSQi : indicate concept set of quiz i
Wi : indicate the possibility of the possible scenario of the rule
Table 8.11: Prerequisite Relationship of Association Rule
Rule Wi Prerequisite
Relationship Qi.LÆQj.L 1.0 CSQi⎯⎯→pre. CSQj
Qi.LÆQj.H 0.8 CSQj⎯pre⎯→⎯. CSQi Qi.HÆQj.H 1.0 CSQi⎯⎯→pre. CSQj Qi.HÆQj.L 0.8 CSQi⎯⎯→pre. CSQj
In this dissertation, association rules generated from Large 2 Itemset are firstly used to analyze the prerequisite relationships between learning concepts of quizzes.
Therefore, by looking up Table 8.11, we can obtain the prerequisite relationships of concept set of quizzes with the possibility weight (Wi) for each mined rule in Table 8.10.
The possibility Wi is a heuristic parameter of CMC algorithm because it can be modified according to different domains and learners’ background. Moreover, the related explanations of the analysis in Table 8.11 are shown in Table 8.12. Table 8.13 shows the result of transforming association rules in Table 8.10 by analyzing the prerequisite relationships in Table 8.11.
Table 8.12: The Explanations of Rule Types
Rule Description of Learning Scenario
LÆL
If the association rule Qi.LÆQj.L is mined, it means that the CSQi is the prerequisite of CSQj, represented as . That is why getting low grade on Qi might imply getting low grade on Qj. prerequisite of CSQi because CSQi may be not learned well resulting from CSQj.
HÆL If the association rule Qi.HÆQj.L is mined, it means that the CSQi is the prerequisite of CSQj.
Table 8.13: Result by Analyzing the Prerequisite Relationships in Table 8.11 Rule Type Association rules of
quiz
For example, in Figure 8.14, the mined rules, Q1.LÆQ2.H and Q1.HÆQ2.L, can be transformed into corresponding prerequisite relationship of concept set, resulting in a confused relation as a cycle between concept sets, called circularity. That is to say, concepts of Q1 and concepts of Q2 are prerequisite of each other, which is a conflict in our analysis. Therefore, during creating the concept map, we have to detect whether a cycle exists or not, e.g., CSQ1ÆCSQ2ÆCSQ1.
Figure 8.14: The Transforming of Association Rules.
Because each concept set may contain one or more learning concepts, we further define a principle of joining two concept sets and then generate corresponding concept-pair, (Ci, Cj), that is, if CSQ1= and CSQ2= , the set of concept-pair is CSQ1 JOIN CSQ2 = , where
}
{U1nai {U1mbj}
)}
, (
{U1k ai bj ai ≠ and k≦n×m. For example, if bj
CSQ1={a1,a2} and CSQ2={b1,b2}, CSQ1 JOIN CSQ2 ={(a1,b1),(a1,b2),(a2,b1)}, where a2=b2
is deleted. The related definition used in creating the concept map is given as follows:
Concept Map CM = (V, E), where
z V = {Ci | the node is unique for each i}
z E = {CiCj | i ≠ j }
The node, Ci, denotes the learning concept and the edge, CiCj , which connects Ci
and Cj, denotes that Ci is the prerequisite of Cj. The CiCj has an Influence Weight, IWk, denotes the degree of relationship between learning concepts. The formulation of IWk is
, 1≦k≦n, where n is the amount of k
/ ) Conf W
IW ) 1 k
(( − × k-1+ k× k CiCj.
The proposed Concept Map Constructing (CMC) algorithm is shown in Figure 8.15.
Figure 8.15: Concept Map Constructing (CMC) Algorithm
For the CMC algorithm shown in Figure 8.15, the main purpose of Cycle Detection Process is to detect the unreasonable prerequisite relationship as a cycle among concept sets. It should be noted that the prerequisite relationship in the concept set map also
fulfills the indicator f12 > f12 in Table 8.14, which is an extension of [5] after cycle detection. The indicator denotes that if concepts of Q1 are prerequisite of concepts of Q2, it is reasonable that f12 > f12 , where f12 =Count(Q1.H∩Q2.L) and
f = ∩ . In addition, the Influence Weight, IWk, denotes the degree how the learning status of concept Ci influences Cj. Therefore, the number of CiCj will enhance the value of Influence Weight. In the formulation of influence weight, the Wi denotes the possibility of the learning scenario of the association rule in our analysis.
Thus, the educational experts can assign different value of Wi to the algorithm according to different domains and learner’s backgrounds.
Table 8.14: Relative Quizzes Frequency (Q1) Higher (Q1) Lower (Q2) Higher f 12 f 21 (Q2) Lower f 21 f 21
For the association rules given in Table 8.13, the process of CMC algorithm is shown in Figure 8.16. In Figure 8.16b, the edges drawn as dash line have the lowest confidences in cycles will be deleted in Cycle Detection Process. Moreover, Table 8.15 shows the example of computing the Influence Weight of Concept-Pair (B, E) in Figure 8.16f. Because the Concept-Pair (B, E) has two edges between CSQ5 and CSQ1, we have to compute the Influence Weight twice.
Table 8.15: The Result of Computing the Influence Weight of Concept-Pair (B, D) in Figure 8.16.f
Rule Prerequisite Relationship Confi Wi IWi
Q1.LÆQ5.H CSQ5ÆCSQ1 0.90 0.8 W1× Conf1=0.9*0.80≅0.72
Figure 8.16: The Process of Concept Map Constructing Algorithm
8.2.5 Evaluating the redundancy and circularity of concept map
In this dissertation, creating a concept map without Redundancy and Circularity is our concern. As shown in Figure 8.17, we create three concept maps by using different approaches and evaluate their difference in terms of Redundancy and Circularity. Thus, we use three processing steps including anomaly diagnosis, the prerequisite relationship based upon analyzing L-L or L-L, L-H, H-L, H-H rule types, and cycle detection to create different concept maps. As shown in Figure 8.17, the prerequisite relationship between concept sets in Figure 8.17a is created based upon analyzing L-L rule type only, and Figure 8.17c is created based upon analyzing L-L rule type and anomaly diagnosis
we proposed. Then, the concept maps as Figure 8.17b and d are transformed according to the Test Item-Concept Mapping Table. Figure 8.17e and f are created by our proposed approach.
Figure 8.17: The (a) and (b) created based up analyzing L-L rule type only. The (c) and (d) are created based upon Anomaly Diagnosis and analyzing L-L rule type only.
The (e) and (f) created by our approach.
Based upon these results of different approaches, the characteristics of approach are concluded as follows.
z Non-redundancy: the anomaly diagnosis can filter many useless test items with low discrimination for refining the input data. For example, in Figure 8.17a, the Q4 with low discrimination results in generating many co-prerequisite links as a
cycle in Figure 8.17b.
z Non-circularity: the cycle detection process can delete these cycles, e.g., the cycle between A and C in Figure 8.17d, to make the concept map un-ambiguous.
Moreover, analyzing association rule with L-L, L-H, H-L, and H-H types can refine the concept map, e.g., the edges ED and BD connect the node D only in Figure 8.17f.