Chapter 2. Related Work
2.1 Concept Map
Concept map, developed by Novak [16] in 1984, is a technique for organizing or representing knowledge as networks. Networks consist of nodes (points/vertices) and links (arcs/edges). Nodes represent concepts and links represent the relations among concepts. Links can be non-, uni- or bi-directional. According to Jonassen et al. [7], concept maps are “representations of concepts and their interrelationships that are intended to represent the knowledge structures that humans store in their minds”.
Concept map has been widely applied in the evaluation of students’ learning in the school system, policy studies, and the philosophy of science to provide a visual representation of knowledge structures. In many disciplines various forms of concept map are already used as formal knowledge representation systems, for example: semantic networks in artificial intelligence, bond graphs in mechanical and electrical engineering, CPM and PERT charts in operations research, Petri nets in communications, and category graphs in mathematics.
An example of a concept map is shown in Figure 1. In this example, each node is a learning concept, and each uni-directional link denotes relationship “prerequisite of “.
Figure 1. An example of concept map
Concept maps are very useful in that they can be considered from different points of views and a wide variety of different forms of concept map have been defined and applied in various domains and can be done for several purposes [4][10][20]. The functions of concept map are shown as follows.
(1) Knowledge representation (to design a complex structure, etc.)
(2) Knowledge acquisition/communication (to communicate complex ideas, etc.) (3) Meaningful learning for education (to help brain storming in mapping process, etc.)
(4) Navigation tools in adaptive learning system (to find learning paths by assessment and diagnosis for understanding of learning concept)
2.2 Uses of Concept Maps as Navigational Tools in Education
As described in Section 2.1, the concept map has many potential roles in education on hypermedia learning. They can be used to help designers in designing hypermedia or, as navigational tools, for helping learners to find an appropriate learning path. After the test results of students were analyzed based on concept maps, the students were given guidance on concepts needing improvement to enhance their learning performance. To model the learning effect relationships among concepts, a conceptual map-based notation, Concept Effect Relationships (CER), is proposed by Hsu [14]. Consider two concepts, Ci and Cj, if Ci is prerequisite to the more complex and higher level concept Cj, then a concept effect relationship Ci Æ Cj exists. A single concept may have multiple prerequisite concepts and can also be a prerequisite concept of multiple concepts. In the computer-assisted instructional environment [3][4][6][9][10][14][23], for achieving the adaptability of learning, the predefined CER-like concept map of the course is often used to demonstrate how the learning status of a concept can possibly be influenced by learning status of other concepts and give learners adaptive learning guidance to improve their learning performances.
2.3 Construction of Concept Map
Since the use of concept map as education tools in the hypermedia is widespread, the eliciting of concept maps from domain expert or experienced teachers becomes very important. However, the job for
constructing the concept maps is still hard and very time consuming.
Therefore, due to the usefulness of the concept map, many approaches are proposed to construct the concept map. The construction of concept maps can be generally classified into manual [2], semi-automatic [21] and nearly automatic [13][17] three categories. In semi-automatic or nearly automatic construction of concept maps, the technique of extracting predicates from a text file or dialog using syntactic and discourse knowledge is often used. Moreover, the browsing behavior and testing records of learners can even be analyzed to construct the concept map.
The following approaches proposed to construct prerequisite relationships among learning concepts of the concept maps are built by analyzing the testing records of students.
Appleby, et al. [3] proposed an approach to create the potential links among skills in math domain.
The direction of a link is determined by a combination of educational judgment, the relative difficulty of
skills, and the relative values of cross-frequencies. Moreover, a harder skill should not be linked forwards to an easier skill. As shown in Table 1, f
B
A represents the amount of learners with wrong answers of skill A and right answers of skill B. If
B A B
A f
f >> , a skill A could be linked to a harder skill B, but backward link is not permitted
Table 1. Relative Skills Frequency A is right A is wrong B is right fAB f AB B is wrong fAB f AB
Later, based upon statistical prediction and approach of Hsu, et al. [14], a CER Builder was proposed by Hwang [12]. Firstly, CER Builder finds the test item that most students failed to answer correctly and then collects the other test items which were failed to answer by the same students. Thus, CER Builder can use the information to determine the relationships among the test items. Though the CER Builder is easy to understand, only using single rule type is not enough to analyze the prerequisite relationships among concepts of test items, which may decrease the quality of concept map.
Tsai, et al. [23] proposed a Two-Phase Fuzzy Mining and Learning Algorithm. In the first phase,
Look Ahead Fuzzy Mining Association Rule Algorithm (LFMAlg) was proposed to find the embedded association rules from the historical learning records of students. In the second phase, the AQR algorithm is applied to find the misconcept map indicating the missing concepts during students learning.The obtained misconcept map as recommendation can be fed back to teachers for remedy learning of students.
However, because the creating misconcept map, which is not a complete concept map of a course, only represents the missing learning concepts, its usefulness and flexibility are decreased. In addition, their approaches generate many noisy rules and only use single rule type to analyze the prerequisite relationship among learning concepts.
Thus, in this thesis, we propose a Two Phase Concept Map Construction (TP-CMC) to construct the complete concept map with influence weights among learning concepts of a course. For improving [23], we apply anomaly diagnosis process to reduce the noise rules and then we take multiple rule types into account to further analyze the mined rules for refining the quality of concept map. Therefore, according to the analysis results, we propose an algorithm to automatically construct the concept map of a course.
Chapter 3. Two Phase Concept Map Construction (TP-CMC)
As mentioned above, the concept map of a course is quite useful. However, the construction is time consuming. Therefore, in this thesis, we propose an approach to automatically construct the concept map as a directional graph with influence weights among learning concepts of a course.
In TP-CMC, the Test item-Concept Mapping Table records the related learning concepts of each test item. As shown in Table 2, 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 2. 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 2, 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.
Historical Testing
Grade Association Rule Mining Process
Mined Association Rules
Figure 2. The Flowchart of Two Phase Concept Map Construction (TP-CMC)
3.1 Grade Fuzzy Association Rule Mining Process
In [23], the Look Ahead Fuzzy Association Rule Miming Algorithm (LFMAlg) can be used to find the associated relationship information embedded in the testing records of learners. In the first phase, we propose an anomaly diagnosis process, a preprocessing, to improve LFMAlg and reduce the input data before the mining process.
In the following, three steps of Phase 1 shown in Figure 2 will be briefly described.
(1) Grade Fuzzification
Firstly, we apply Fuzzy Set Theory to transform these numeric testing data into symbolic form.
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. Then, the association mining approach can be used to find the association rule among these testing items.
(2) Anomaly Diagnosis
Based upon Item Analysis for Norm-Referencing of Educational Theory [1][17], 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 using difficulty and discrimination of test item, called Anomaly Diagnosis, to refine the test data.
(3) Fuzzy Data Mining
Then, we apply LFMAlg [23] to find the grade fuzzy association rules of quizzes from the historical testing data. In this thesis, we analyze the prerequisite relationships among learning concepts of
quizzes according to 4 association rule types, L-L, L-H, H-L, H-H. We use Qi.L notation to denote 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.
3.2 Concept Map Constructing Process
In the second phase, based upon the heuristic of our observation in real learning situation, we further analyze the mined rules. According to the analysis result, we propose a heuristic algorithm to construct the concept map of a course.
In the following, the process of Phase 2 shown in Figure 2 will be briefly described.
(1) Association Rule Analyzer
Firstly, we analyze the four association rule types, L-L, L-H, H-H, and H-L, to generate related prerequisite relationships among concept sets of test item based on our observation in real learning situation. The result of analysis is used to define the edge between nodes of concept set and provide teachers with the possible learning scenario of students for further refining the test sheet.
(2) Concept Map Constructor
Then, based on the prerequisite relationships of concept sets described above 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 mapping.
Chapter 4. Grade Fuzzy Association Rule Mining Process
4.1 Grade Fuzzification
As described in Section 3.1, 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 3. In the fuzzification result, “Low”, “Mid” and “High” denotes “Low Grade”, “Middle Grade”
and “High Grade” respectively. Qi.L denotes the value of LOW fuzzy function, Qi.M denotes the value of MIDDLE fuzzy function i, and Qi.H denotes the value of HIGH fuzzy function for the quiz i. By given membership functions, the fuzzification of testing records is described in Example 1.
10% 20% 40% 60% 70%
Figure 3. The Given Membership Functions of Each Quiz’s Grade Example 1:
In Figure 4, assume there are 10 testing records with 5 quizzes of learners and the highest grade on each quiz is 20.
Fuzzification
Figure 4. The Fuzzification of Learners’ Testing Records
4.2 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 5. By the anomaly diagnosis process, a test item will be deleted if it has low discrimination.
Algorithm: Fuzzy Item Analysis for Norm-Referencing (FIA-NR)
Symbol Definition:
RiH /RiL: The sum of the fuzzy grades ( H=1, M = 0.5, L=0 ) on test item i of each student in the high(H)/Low(L) group.
NiH /NiL:The number of learners in high/low group.
PiH & PiL:The ratios of RiH to NiH and of RiL to NiL , respectively.
Input : Fuzzified testing records of learners
Output : The Difficulty index (Pi ) and the Discrimination index (Di) of each test item Step1: Sort Scores in descending order and divide it into High, Middle, and Low groups,
each has 1/3 learners.
Step2: Let
Step 4: Delete the test items with low Discrimination (<0.5).
Figure 5. Fuzzy Item Analysis for Norm-Referencing (FIA-NR) Example 2:
Table 3 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 4, 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 3. 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.
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 4.
Table 4. 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
4.3 Fuzzy Data Mining
After filtering out these useless quizzes, we can apply Look Ahead Fuzzy Association Rule Mining Algorithm [23] as shown in Figure 6 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=
C
−1
C φ . Then, the support(x) = support(A∪B) = Min(A, B), where n is the number of learners. For example, in Figure 4, support(Q
1n
∑
1.L, Q3.H) = Min(1.0, 0.7) + Min(1.0, 0.7) = 1.4.
Algorithm: LFMAlg Algorithm
Symbol Definition:
α : The minimum support threshold in the -large itemset.
C : The -Candidate itemset.
L : The -large itemset
λ: The minimum confidence threshold.
Input: The test records of learners after Fuzzification and Anomaly Diagnosis.
The minimum support threshold α1 and λ.
Output : The fuzzy association rules of test records of learners.
Step1: Repeatedly execute this step until C = NULL.
1.1: Generate and insert the -itemset into C
1.2: α =max(α2−1 , α −1−( α−1−)1×c), where >1 and c is constant.
1.3: L =
{
x|support(x)≥α , for x∈C}
1.4: = +1
Step2: Generate the association rules according to the given λ in L .
Figure 6. Look ahead Fuzzy Association Rule Mining Algorithm (LFMAlg)) Example 3:
For the data shown in Examples 1 and 2, Figure 7 shows the process of finding the association rules by LFMAlg algorithm.
Figure 7. Mining Process of LFMAlg algorithm
Thus, Table 5 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.
95
Table 5. The Mining Results (Confi > 0.8) Large 2 Itemset
Rule Types Mined Rules Confi
Q2.LÆQ3.L 0.95
Chapter 5. Concept Map Constructing Process
In Phase 2, the rules mined in Phase 1 will be analyzed based upon the heuristics of our observation in real learning situation. Accordingly, we propose a heuristic algorithm to automatically construct the concept map of a course. The Concept Map Constructing Process shown in Figure 2 is described as follows.
5.1 Association Rules Analyzer
(1) Analysis of association rules generated from Large 2 Itemset
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 as follows.
Heuristic 1 :
Given two quizzes Q1 and Q2, if concepts of Q1 are prerequisite of concepts of Q2, we summarize the possible learning scenarios of students as follows.
z Illustrations of rule Q1.LÆQ2.L
Scenario 1) Learners get low grade on Q1 implies that they must get low grade on Q2.
Scenario 2) Learners get low grade on Q2 implies that their grade on Q1 might be bad.
z Illustrations of rule Q1.HÆQ2.H
Scenario 3) Learners get high grade on Q1 implies that they may also get high grade on Q2 Scenario 4) Learners get high grade on Q2 implies that they must get high grade on Q1.
z llustrations of rule ( Q1.HÆQ2.L or Q2.LÆQ1.H)
Scenario 5) Learners get higher grade on Q1 (an easier quiz) but get lower grade on Q2 (a harder quiz).
As shown in Table 6, for convenience to explain the following process in this thesis, we adopt Scenario 1, 4, and 5 of Heuristic 1 to get prerequisite relationships among concept sets of quizzes with parameterized possibility weight for each rule type, which are used to construct the concept map. The definition of the symbols used in Table 6 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 6. 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 CSQj⎯⎯→pre⎯. CSQi Qi.HÆQj.L 0.8 CSQi⎯⎯→pre. CSQj
In this thesis, 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 6, we can obtain the prerequisite relationships of concept set of quizzes with the possibility weight (Wi) for each mined rule in Table 5. 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 6 are shown in Table 7. Table 8 shows the result of transforming association rules in Table 5 by analyzing the prerequisite relationships in Table 6.
Table 7. The Explanations of Rule Type Rule Description of Learning Scenario
Qi.LÆQj.L CSQi is the prerequisite of CSQj. That is, learners get low grade on Qi
implies that they must get low grade on Qj.
Qi.HÆQj.H CSQj is the prerequisite of CSQi. That is, learners get high grade on Qj
implies that they must get high grade on Qi
Qi.LÆQj.H CSQj is the prerequisite of CSQi That means learners get higher grade on Qj (an easier or simpler quiz) but get lower grade on Qi (a harder or more complex quiz).
Qi.HÆQj.L
CSQi is the prerequisite of CSQj That means learners get higher grade on Qi (an easier or simpler quiz) but get lower grade on Qj (a harder or more complex quiz).
Table 8 shows the result of transforming association rules in Table 5 by analyzing the prerequisite relationships in Table 6.
Table 8. Result by Analyzing the Prerequisite Relationships in Table 6
Rule Type Association rules of quiz Prerequisite relationship of Concept Set Conf i Wi
Q2.LÆQ3.L CSQ2⎯pre.⎯→CSQ3 0.95 1.0 Q3.LÆQ2.L CSQ3⎯pre.⎯→CSQ2 1.00 1.0 Q2.LÆQ5.L CSQ2⎯pre.⎯→CSQ5 0.86 1.0 L-L
Q3.LÆQ5.L CSQ3⎯pre.⎯→CSQ5 0.90 1.0 Q1.LÆQ5.H CSQ5⎯⎯→pre. CSQ1 0.90 0.8
L-H Q5.LÆQ1.H CSQ1⎯pre.⎯→CSQ5 0.82 0.8
H-H Q2.HÆQ3.H CSQ2⎯pre.⎯→CSQ3 0.91 1.0
H-L Q5.HÆQ1.L CSQ5⎯⎯→pre. CSQ1 1.00 0.8
(2) Analysis of association rules generated from Large n (>= 3) Itemset
In addition to Large 2 itemset, Large 3 itemset may also help refining learning strategies. The possible scenario is as follows.
z We may find and generated from large 2 itemset, but
we are not sure if concepts of Q1 and Q2 must be learned together to ensure concepts of Q3 well learned. However, we can clarify the uncertainty if we find generated from large 3 itemset.
CS
The scenarios of the larger n ( >3 ) itemset are the same as that described above. Therefore, in this section, we extend Heuristic 1 and only adopt the L-L, H-H rule types to help analyzing the prerequisite relationships between learning concept sets for further analyzing and refining the teaching strategies. For
not losing focus on the analysis of Large n itemsets, now we only adopt rules with prerequisite
relationship of N:1 (CS CS CS CS
Qh Qk
Qj
Qi∩ ∩...∩ ⎯⎯→pre. ) format after applying Heuristic 1. As shown in Table 9, Scenario 1 and Scenario 4 of Heuristic 1 are adopted here as an example.
Qi∩ ∩...∩ ⎯⎯→pre. ) format after applying Heuristic 1. As shown in Table 9, Scenario 1 and Scenario 4 of Heuristic 1 are adopted here as an example.