Development of an adaptive learning case recommendation approach for problem-based e-learning on mathematics teaching for students with mild disabilities

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Development of an adaptive learning case recommendation approach

for problem-based e-learning on mathematics teaching for students

with mild disabilities

Hui-Chuan Chu

a

, Tsung-Yi Chen

b

, Chia-Jou Lin

c

, Min-Ju Liao

d

, Yuh-Min Chen

c,* a_{Department of Special Education, National University of Tainan, Tainan, Taiwan, ROC}

b_{Department of Electronic Commerce Management, Nan-Hua University, Chia-Yi, Taiwan, ROC} c

Institute of Manufacturing Information and Systems, National Cheng Kung University, Taiwan, ROC d

College of Humanity and Social Science, National Chiao-Tung University, Taiwan, ROC

a r t i c l e

i n f o

Keywords:

Problem-based e-learning Case-based reasoning Clustering analysis

Probability latent semantic analysis

a b s t r a c t

Most e-learning platforms offer theoretical knowledge content but not practical knowledge required for problem solving. This study proposed a problem-based e-learning (PBeL) model which incorporates the problem-based learning (PBL) theory, social constructivism, and situated learning theories to assist reg-ular and special education teachers in effectively developing knowledge for mathematics teaching for students with mild disabilities. To support adaptive case-based learning in the proposed PBeL and to ade-quately address the real complexity and diversity of the learning problems of students’ with mild disabil-ities, this research also developed an adaptive case recommendation approach which identiﬁes the most suitable authentic learning cases based on the characteristics of learners (teachers), the strengths, weak-nesses, and types of disabilities of their students, the teaching problems of various mathematical topics, and the teaching context in order to facilitate adaptive case-based learning in the context of problem-based e-learning for regular and special education teachers’ knowledge development. Clustering and information retrieval techniques were used to construct the context and content maps for case-based reasoning with the capability of semantics identiﬁcation. The adaptive recommendation approach not only enables the realization of adaptive PBeL, but also enhances teachers’ practical knowledge and assists them to solve students’ learning problems.

1. Introduction

e-Learning provides learners with another learning channel that enables the learner to break free from constraints on time and space, and to engage in distance-based, non-synchronized learning activities. However, most e-learning platforms emphasize the con-venience offered by digital knowledge content, without integrating suitable learning theory into the learning. Therefore, these e-learning platforms degenerate into knowledge dissemination tools, and neglect learning theory and practice. Additionally, as far as the development of knowledge and competence of pre-service and in-service teachers of special education are concerned, current e-learning platforms provide only knowledge and teaching materials related to formal teaching knowledge, but not sufﬁcient practical knowledge required for solving students’ learning problems.

Problem-solving is knowledge intensive. It involves acquiring relevant knowledge to identify the core causes of a problem,

devel-oping solutions, and taking appropriate actions to solve the

prob-lem (Liu & Ke, 2007). Although e-learning easily provides

learning resources, without taking into account the characteristics of problems being encountered, the large amount of learning resources will result in cognitive overload or disorientation.

Case-based reasoning (CBR) method has been widely used to provide knowledge for problem solving by adapting solutions from

that of historical problems (Carrascosa, Bajo, Julian, Corchado, &

Botti, 2008; Liu & Ke, 2007; Yang, Han, & Kim, 2004). However, conventional case-based reasoning approaches have their limita-tions in handling semantics of knowledge, thus decreasing possi-bilities for knowledge dissemination. Moreover, as oppose to conventional case-based reasoning that merely focuses on the characteristics of the problems being encountered, in the areas of teaching or students’ learning problem solving, characteristics of both students and learners (i.e. teachers) are being considered in the learning case recommendation.

In the trend of integrating students with mild disabilities into regular education classes and non-categorical resource programs, advancing mathematics achievement of the students with mild

*Corresponding author.

E-mail address:ymchen@mail.ncku.edu.tw(Y.-M. Chen).

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disabilities, including the students with learning disabilities, high-functioning autism, Asperger syndrome, mild mental retardation, emotional and behavior disorders, or Attention Deﬁcit Hyperactiv-ity Disorder, becomes even more challenging for regular education and special education teachers. Research suggest that the mathe-matics underachievement of students with mild disabilities is the result of a complex interplay of cognitive, emotional/behavioral, physical, sensory, communication, social factors and mathematics (Montague & Applegate, 2000).

To adequately address the real complexity and diversity of learning problems of students with mild disabilities, this study (1) designed an e-learning model that featured the situated learn-ing as a theoretical basis to integrate learnlearn-ing theories of social constructivism and case-based learning along a problem-based learning approach, and (2) developed an adaptive case recommen-dation approach which identiﬁes the most suitable authentic learning cases based on the characteristics of learners (teachers), the strengths, weaknesses, and types of disabilities of their stu-dents, the teaching problems of various mathematical topics, and the teaching context. Clustering and information retrieval tech-niques were employed for construction of context and content maps for case-based reasoning with the capability of semantics identiﬁcation.

In particular, this study focused on reasoning of learning cases on the mathematics teaching for students with mild disabilities, and considered the relevant needs for knowledge development and for improving problem-solving capabilities of teachers in real-istic teaching situations. The results of this research not only strengthened teacher’s knowledge for practical teaching, but also assisted teachers to solve students’ learning problems, which would in turn to improve teaching quality.

2. Problem-based e-learning

This section presents a proposed problem-based e-learning model with the capability for adaptive learning case recommenda-tion. This model incorporates problem-based learning, contextual learning and social constructivism as the underlying theoretical bases.

2.1. Theoretical framework

This study referred to the ‘‘course of cognitive skill acquisition” (Renkl & Atkinson, 2003; Patel, Kinshuk, & Russell, 2000; VanLehn,

1996) as the basis of the proposed e-learning process which

in-cludes stages of concept knowledge development, problem-solving knowledge development, and overall professional judgment knowledge development.

Problem-based learning (PBL) is to solve real-world problems by assembling and focusing problems, and guide the learner in uti-lizing his or her understanding, problem solving skills, and

judg-ments (Barrows, 1996; Levin, 2001). It provides the learner with

pragmatical experiences, allowing the user to achieve both

‘‘know-ing” and ‘‘knowing how” (Delisle, 1997) through the learning

con-text that much resembles an authentic learning concon-text for the

learner, as emphasized by the situated learning theory (Souders

& Prescott, 1999). It is believed that problem-based learning may augment the structure of the learner’s understanding, as well as his or her ability to integrate new knowledge (Robertson et al. 2000).

Nevertheless, Fenwick and Parsons (1998) mentioned that

while PBL is able to improve the learner’s motivation, solely utiliz-ing such pedagogy may sometimes cause knowledge gaps. Advo-cates of situated learning also stressed the need for professional cognitive apprenticeship, and contended that the learner must

en-gage in an active, participatory learning process within a scaffold provided by a teacher, an expert or a more experienced learner, who plays the role of a coach or facilitator to the learner.

Pedagogies based on social constructivism perceives learning as a collective thinking process that involves teacher–student or stu-dent–student interactions to solve problems, learn new knowledge and concepts, and make appropriate decisions. Students have to discuss with each other, in order to negotiate meanings or forge

a consensus (Rogoff, 1990). Learning through discussions is highly

valued and supported by social constructivists. When the learner interacts with other learners or the teacher, concepts are formed in a natural way because through talking to each other, they have collectively created a world that can be described and discussed as well as a common framework under which communication takes

place (Solomon, 1987).

The case-based learning approach used in the ﬁeld of teacher professional development involves narration of teaching practices based on a real classroom case, and helps the learner to link

theo-ries with practice (Chin & Lin, 2000; Merseth, 1996; Richardson,

1993) to stimulate introspections (Richert, 1991) and effective

con-structions of teaching knowledge.

According to the above discussions, we believe that providing a learner with a learning context similar to the teaching problems encountered by the learner may better enhance teachers’ profes-sional development. Therefore, this study has chosen the situated learning as a theoretical basis to integrate learning theories of so-cial constructivism and case-based reasoning along a problem-based learning approach. The theoretical framework of this study

is illustrated asFig. 1.

2.2. Problem-based e-learning model

In accordance with the previous theoretical framework, this study designed an e-learning model with problem-based learning as its core and social constructivism and situated learning as its auxiliary theories. In the spirit of PBL, this model includes the stages of analysis, design, development, and practice (refer to Fig. 2).

The analysis stage involves assessing a learner’s (i.e. teacher’s) knowledge of the students with mild disabilities, pedagogical con-tent knowledge of mathematics, knowledge of modiﬁcations of curriculum, teaching methods, materials, techniques, and learning environments for teaching students with mild disabilities, mathe-matical content knowledge, and then diagnosing the students’ learning problems. The learning goal for the learner is then trans-lated into ‘‘solving students’ learning problems”. The design stage identiﬁes the learner’s background information and teaching objectives in order to outline a personalized learning plan. The

Cognitive Skill Acquisition

Problem Solving Knowledge Development Overall Professional Judgment Knowledge Development Case-based Learning Social Constructivism Problem-related Concept Knowledge Development Problem-based Learning Situated Learning

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development stage develops contents, such as concepts and cases, for the personalized learning plan. Finally, the practice stage guides the learner to initiate learning activities, such as concept learning, case studies, practical teaching, feedback on teaching experience and knowledge sharing. After the learner has completed the con-cept learning and case studies, he/she is required to begin realistic teaching, by applying learned knowledge to realistic teaching con-text. Lastly, the system knowledge content can continue to expand and update as the learners would share their knowledge and thoughts.

A learner undertaking case studies may select either ‘‘individual learning” or ‘‘group learning”. The ‘‘group learning” takes the lear-ner to a learning mode based on social constructivism, where the learner may initiate a group discussion and direct questions to ex-perts or learners with related experience in any phase of the case study. During Q&A sessions or online discussions in this forum, an experienced teacher or expert plays the role of an e-consultant to guide the learners to complete their learning processes. 2.3. Learning case structure

Development of an e-learning platform requires not only the design of functional learning activities, but also the provision of suitable learning materials. Moreover, a successful design of learn-ing objects necessitates incorporation of instructional design and

learning theories (Roderick & Baden, 2005).

Learning cases which were developed by expert teachers according to their own teaching narrations and were analyzed through actual teaching, observation, discussion, and assessed by experts were used to store practical teaching knowledge. Each case contained the parts of ‘‘teaching context” and ‘‘teaching narration”. The teaching context contained the learner’s demographic data, the

student’s personal data, and the statement of student’s learning problems. The teaching narration included three sections of

teach-ing objectives, teachteach-ing units and learnteach-ing assessments. Fig. 3

shows the learning case model deﬁned in terms of UML (Booch,

Rumbaugh, & Jacobson, 1999) notations, where a box represents a class of learning objects and a diamond indicates a composite class, which is composed of its component classes.

To store, organize, manage and use the case contents effectively, this study deﬁned the instances of each class in the case model as learning objects. The bold rounded squares stand for the retriev-able learning objects, while plain squares denote non-retrievretriev-able learning objects.

3. Adaptive case recommendation

This section describes the framework of proposed adaptive case recommendation approach for supporting case-based learning in problem-based e-learning.

3.1. Requirements

The requirements for the proposed adaptive case recommenda-tion approach are identiﬁed as follows to ensure that the recom-mended learning cases conform to the learner’s learning requirements.

(1) Adaptability: Research on adaptive learning indicated that learner’s interests, ability, and cognitive characteristics

greatly inﬂuence learning effectiveness (Kalyuga, 2007).

Fur-thermore, the nature and scope of the problem, the type of disability, and the characteristics of students should be taken into consideration for developing solutions to

Case Study Development Design Analysis Identification(teacher&student)

Theoretical Knowledge Content Development Problem Identification

Problem-based e-learning Situated Learning

Case and Practical Knowledge Cntent Development Social Constructivism

Counsultant

Learning Map Planning e-Counsultnat & e-Tutoring Assignment

Practice

Concept Knowledge Learning

Teaching Practicing

Tutoring Q&A

Online Discussion

Learning Effect Assessment Case Study

Knowledge Sharing Learning Objective Definition

Individual Learning Group Learning

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students’ learning problems (Steele, 2006). Therefore, identi-ﬁcations of the characteristics of both the learner (i.e. teacher) and the student are required for case recommenda-tion to facilitate learning adaptarecommenda-tion and thus maximize the beneﬁt to the learner and the student.

(2) Natural language processing: The accuracy of case retrieval is highly dependent on the keys for searching. Convention-ally, keywords are commonly used in information as well

as case retrieval (Hsu & Wang, 2004). However, since

stu-dents’ learning problems can not be fully described in terms of keywords, a better way for students’ learning problem deﬁnition that can be in turn used for case retrieval is required. This research allowed learners to describe stu-dents’ learning problems using natural language so as to improve the accuracy of case retrieval. Therefore, the adap-tive case recommendation approach needs to analyze the content of learning problems described in natural language and develop the keys for case retrieval.

(3) Semantic searching: Since both teaching problems and learning cases were stated in natural language, synonyms and homographs appeared frequently to cause semantic variations in learning case searching. Hence a method to handle semantic variations was required to promote the search accuracy.

3.2. The framework of adaptive case recommendation

The adaptive recommendation framework included layers of

functional tasks, knowledge map and database as shown inFig. 4.

By referencing the cycle of case-based reasoning (Agnar & Enric,

1994), tasks in the functional task layers included characteristic

identification, student’s learning problem analysis, learning case retrieval and learning case recommendation. Characteristic identi-fication included analysis and identiidenti-fication of characteristics of teachers and students. The learning problem analyses, including

functions for teaching context identiﬁcation and learning case con-tent matching, were responsible for identifying the learner’s requirements. The layer of learning case retrieval retrieved rele-vant learning cases according to the results of characteristic

iden-tiﬁcation and learning problem analysis. Learning case

recommendation layer selected the most suitable learning case from the retrieved learning cases.

The knowledge map, which was a structured knowledge repre-sentation, depicted the knowledge structure of learning cases and was used as a knowledge search guide to facilitate learning cases retrieval.

The database layer contained a user model base and a learning case repository. According to the learning case structure, the learn-ing case repository was divided into areas for teachlearn-ing contexts, teaching plans, and teaching cases. The user model was designed based on the personal characteristics, abilities, and preferences to deﬁne the learner’s demographic data and background informa-tion. The user models can be further categorized into learner

mod-els and student models. The former contained learner’s

demographic data, characteristics, and learning proﬁle, while the latter contained student’s demographic data and characteristics information.

4. Deﬁnition and development of knowledge map

This section deﬁnes and develops the knowledge map of learn-ing case repository. Accordlearn-ing to the content of learnlearn-ing cases, the knowledge map is split into a context map and a content map as discussed below.

4.1. Deﬁnition and development of context map 4.1.1. Deﬁnition of context map

The context map model contains clusters and occurrences as

shown inFig. 5. A cluster aggregated learning cases with the

sim-XXX Retrievalable Learning Object Legend: Non-retrievalable Learning Object Mathematics Abilities Intelligence Cognitive Abilities Strength& Weakness Disabilities Preferences XXX Learning Characteristic Error Pattern Learning Style 01030202 01030203 01030204 01030205 01030206 01030207 01030208 01030209 Teaching Aids Teaching Objectives 0201 General Assessment Unit Assessment Learning Attitude 01030210 Teaching Procedures 0102 Problem Statement 0101 Teacher Profile 0103 Student Profile 010101 Personal Data 010102 Characteristics Characteristics

Competency Teaching _Style Personal Data 010301 010302 01010201 01010202 01030201 0202 0203 02020104 02020105 Teaching Objectives 02020101 Teaching Strategies 02020103 Student Error Pattern 02020102 00 Case 01 Teaching Context 02 Teaching Plan 020201 Teaching Case

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ilar teaching contexts. The occurrence was the index of a corre-sponding learning case.

The schema of context maps was designed based on the learn-ing case structure and the context map model, as presented in Fig. 6. The cluster ID distinguished each cluster, while the occur-rences recorded the IDs of learning cases in this cluster. In other words, the occurrence linked a cluster with its related learning cases. The centroid recorded the characteristics of clusters and con-tains the characteristics of teachers and students. The characteris-tics of a teacher consist of his/her competence and teaching style. The characteristics of a student included age, grade, mathematic abilities, types of disabilities, strengths, weaknesses, learning style, and learning preferences.

4.1.2. Development of context map

The learning cases were clustered according to the context fea-ture tuples, which contained the attributes of the teacher and

stu-dent proﬁles in leaning cases.Table 1presented the context feature

tuple of the learning cases. The context feature tuple included nu-meric and categorical attributes.

Huang (1998)proposed a k-prototypes algorithm, which inte-grates the k-means and k-modes processes to cluster data with mixed numeric and categorical values. However, the attributes can only be a single value. In this study, certain context featured with categorical values was multi-values. For example, the values of disability were dyslexia, reading disability and mathematic dis-ability. The context featured with categorical values in this study

Context Map

_{Content Map}

Student Characteristic Identification

Learner Characteristic Identification

Student Learning Problem Analysis

Teaching Context Identification

Learning Case Content Matching

Learning Case Recommendation Learner

Characteristic Identification

Learning Case Retrieval

Case Learning Student Model Teacher Model Teaching Case Problem

Statement Learning Plan

User Model Learning Case _Repository

Functional Layer Knowledge Map

Layer

Database Layer

Fig. 4. The adaptive case recommendation framework.

Legend Cluster Occurrence

Context Map

Learning Case _Learning Case Learning Case Learning Case Learning Case Learning Case Learning Case

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can be classiﬁed into single-value and multi-value categorical attributes. Therefore, this study proposed an enhanced algorithm that used the k-prototypes paradigm to cluster data with numeri-cal, categorical single-values and categorical multi-values.

Let X = {x1, x2, . . . , xn} denote a set of n learning cases, and let k be

a positive integer. The partition of X into k clusters to minimize the within-groups sum of dissimilarity can be discovered by the fol-lowing optimization model:

Minimize EðW; MÞ ¼X k j¼1 Xn i¼1 wijDðXi;MjÞ Subject to X k j¼1 wij¼ 1; 1 6 i 6 n wij2 f0; 1g; 1 6 i 6 n; 1 6 j 6 k

where W is an n k partition matrix; M is the centroid of Cluster,

and M = {M1, M2, . . . , Mk}, D(Xi, Mj) is the dissimilarity between two

objects.

To solve the above model, the enhanced k-prototypes algorithm (EKP) applies the new dissimilarity measure approach, which not only clusters multi-type data, but also gives equal importance to each context feature. This is because the dissimilarity value with every context feature is between 0 and 1. Furthermore, the new dissimilarity measure approach also considers the relative fre-quencies of categorical attributes in each cluster in order to

en-hance the accuracy of the clustering results (He, Xu, Deng, &

Dong, 2004). The dissimilarity measure approach of IKP is de-scribed below:

Let X = {x1, x2, . . . , xp, . . . , xm} and Y = {y1, y2, . . . , yp, . . . , ym} be two

mixed-type objects described by m attributes, where the ﬁrst p

Cluster ID Cluster Centroid Learning Case Occurrence Teacher Teaching Style Teacher Competence Teacher Characteristics Student Characteristics Age Grade Mathematics Abilities Disbilities Strengths Weaknesses Learning Style Preferences Attribute Have Link Legend

Fig. 6. The teaching context map schema.

Table 1

The context features

User Attribute name Value Type

Teacher characteristics Teaching style (T1) Symbolic language teaching and individual CS

Learning style V CS

Teaching seniority 6 N

Teaching course Special education CM

Student characteristics Grade 3 N

Age 9 N

Math abilities – number 35.5 N

Math abilities – fractions 39.9 N

Math abilities – ﬁgures and space 39.6 N

Math abilities – addition and subtraction 37.3 N

Math abilities – multiplication and division 37.7 N

Math abilities – the four fundamental operations of arithmetic 0 N

Math abilities – measure 36 N

Math abilities – time and Calculation 34.5 N

Math abilities – statistics and statistical chart 37.7 N

Math abilities – a calculating machine and strategies on math word problems 31.8 N

Disabilities (D4) Dyslexia CM

(D7) Reading disability (D11) Math disability

Learning style (L2) Diverger CS

Preferences (P4) Computer game, CM

(P5) Liking work of images and drafting, (P6) Singing

Strengths (A1) Superior observer CM

(A14) Good ability to express in speech

Weaknesses (w8) Limited vocabulary CM

(w14) Difﬁculty in grasping important information *

N – Numerical data type.

CS – categorical single-value data type. CM – categorical multi-value data type.

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attributes are numeric values, and the rest are categorical values.

Suppose Y is the centroid of the cluster Cq. The dissimilarity

mea-sure between X and Y can be described as

DðX; YÞ ¼ Dn þ Dc

The ﬁrst term, namely the summation of the numeric attributes, can be calculated as Dn ¼X p i¼1 xi yi imax imin 2

The second term, which is the summation of the categorical attri-butes, is computed as

Dc ¼ X

m

i¼pþ1

dðxi;yiÞ

If the ith attribute is single-value category, then it can be derived as

dðxi;yiÞ ¼

1 f ðji¼ xijCqÞ ðxi¼ yiÞ

1 ðxi–yiÞ

where f(ji= xijCq) represents the frequency of the data objects, of

which the ith attribute is equal to xiin Cq. This function is calculated

as

f ði ¼ AkjCqÞ ¼

nAk;i Nq

where Nqis the number of objects in clusterCq, and nAk;i is the

num-ber of objects which the category value of the ith attribute is Ak.

If the ith attribute is a multi-value category, xi= {xi1, xi2, . . . , xin}

and yi= {yi1, yi2, . . . , yim}, then it is obtained as dðxi;yiÞ ¼ 1

P_{f ðy}

im¼ xinjCqÞ

jxi[ yij

where f(yim= xinjCq) represents the frequency of the element of

ob-jects, of which the mth element in the ith attribute equals xinin Cq,

and is obtained as

f ði ¼ AmjCqÞ ¼

nAm;i Nq

where nAm;iis the number of objects in which the category value of

the ith attribute contains Am.

Additionally, the IKP algorithm also deﬁnes an approach to ﬁnd

centroid M from a given set. Let X = {x1, x2, . . . , xn} be a set of

mixed-type objects containing attributes A1, A2, . . . , Ap, . . . , Am, where the

ﬁrst p attributes are numeric, and the rest are categorical, and X

be-longs to cluster Cj. The values of the ﬁrst p attributes in centroid Mj

are calculated by the mean approach, and can be determined by

mz¼

Pn i¼1xzi

Nj 1 6 z 6 p

where Njis the total number of objects in cluster Cj, and xziis the zth

attribute in the ith object.

The categories single-value of attributes in centroid Mjare

cal-culated by the mode approach. Objects with the same multi-value category attributes do not necessarily have the same number of attribute values. Therefore, the mean is ﬁrst calculated to deter-mine the number of attribute values, and the mode is then com-puted to select the elements of attribute values. For instance,

x1i= {a, b, c}, x2i= {a, c, d, f}, x3i= {a, f} are the ith attribute in object

x1, x2and x3, respectively of cluster Cj. The ith attribute in Mjhas

three elements according to the mean approach, 3 + 4 + 2/3 = 3. Then the mode approach is applied to calculate the frequency of each element, f(a) = 1, f(b) = 1/3, f(c) = 2/3, f(d) = 1/3, f(f) = 2/3.

Hence, the ith attribute in Mjis {a, c, f}.

The IKP algorithm was utilized to construct the context map. Let

X = {x1, x2, . . . , xn} denote a set of n learning cases, and xi= [xi1,

xi2, . . . , xip, . . . , xim]T be a learning case represented by mvalues of

attributes, where the ﬁrst p elements are numeric values, and the rest are categorical values. To conﬁrm the optimal clusters number, the experience rule was used to determine the range of numbers of

cluster as kmax6

ffiffiffi n p

(Ramze, Lelieveldt, & Reiber, 1998), where n is the amount of learning cases. Finally, the number of clusters k which minimizes the within-groups sum of dissimilarity is the

optimal result of the cluster.Table 2 describes the context map

construction algorithm.

4.2. Deﬁnition and development of content map 4.2.1. Deﬁnition of content map

The content map presents the knowledge structure of the prob-lem statements, the teaching plans and the teaching cases in learn-ing cases. It facilitates semantic searchlearn-ing of the adaptive recommendation approach. Components of the content map model

include concepts and the occurrences, as shown inFig. 7. The

con-cepts are the knowledge concon-cepts appearing in the learning prob-lem statements, the teaching plans and the teaching cases in learning cases. The occurrences link concepts and learning cases, meaning that an occurrence indexes a concept appearing in a cer-tain learning case. Since the importance of a concept varies with learning cases, learning problems, and teaching plans, each

occur-rence has three weights, WPi, WNiand WCi. WPiis the importance of

theith concept in the learning problem, WNis the importance of the

ith concept in a teaching plan, and WCis the importance of the ith

concept in a teaching case.

Fig. 8 illustrates the content map schema, which is designed based on the learning case structure and the proposed content map model. The Concept name indicates the name of a knowledge concept in the learning case. The occurrence links a concept with a learning case. The occurrence contains three weight attributes, the weight in problem statement, the weight in teaching plan and the weight in teaching case.

Table 2

Context map construction algorithm

Context map construction algorithm Input: a set of n learning cases

Output: leaning cases were assigned into their corresponding clusters begin

let case features Xi= {x1, . . ., ,xp, . . . , xh}, i = 1 n, the ﬁrst p elements are numeric values and the rest are categorical values;

let k be a positive integer; for 15k5pffiffiffindo begin

select k as the number of initial cluster centroids randomly, each centroids corresponds to one and only one cluster;

while new cluster centroids are not equal to existing cluster centroids do begin for 15i5n do begin DijðXi;MjÞ ¼Ppt¼1 xitmjt tmaxtmin 2 þPht¼pþ1dðxit;mitÞ for 15j5k do select min Dij(Xi,Mj);

let case i belong to cluster j; end;

calculate new cluster centroids for each cluster; end;

EkðW; MÞ ¼ Pk

j¼1 Pn

i¼1DijðXi;MjÞ for each Xi 2 Cj end;

take minEkas the optimal clustering result; end;

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4.2.2. Development of content map

The ﬁrst step in developing the content map is pre-process, which includes tasks of sentence breaking, word breaking, word tagging and concept parsing.

Sentence breaking: As learning case is described in natural lan-guage and the sentence is the basic unit of natural lanlan-guage, the sentence breaking techniques is ﬁrst performed to break the con-tent of learning cases into sentences. This study deﬁned sentence breaking as periods, commas and semicolons.

Word breaking and tagging: The purpose of this step was to re-trieve the core concepts in the learner’s teaching problem for adap-tive learning case recommendation. This study used the AutoTag techniques developed by CKIP projects in Academia Sinica to seg-mentalize phrases, to transform each sentence into a series of words, and to tag each word to determine its grammatical attribute.

Concept parsing: Concept parsing was performed primarily to retrieve the core concepts from the tagged word in each sentence. Since the meaning of a sentence was usually determined by nouns, concept parsing selected nouns in the problem statement, the teaching plan and the teaching cases of a learning case as core knowledge concepts to represent the learning case.

Unstructured learning cases were transformed into structured data after pre-process. The collection of learning cases were

capsu-lated in three m n co-occurrence matrices Mp, Mnand Mc, where

Mp(c, lc) represents the number of occurrences of a concept c in the

problem statement of learning case lc; Mn(c, lc) represents the

num-ber of occurrences of a concept c in the teaching g plan of learning

case lc, and Mc(c, lc) represents the number of occurrences of a

con-cept c in the teaching case of learning case lc.

To deal with polysemous words, and to distinguish explicitly between different meanings and word usages, the Probability

La-tent Semantic Analysis (PLSA) was applied to model the relation-ships between concepts and learning cases.

PLSA is a statistical latent class model or aspect model

(Hofmann, 1999b; Hofmann, 1999a). The model was ﬁtted to a

training corpus by the Expectation Maximization (EM) algorithm (Dempster, Laird, & Rubin, 1977). PLSA obtained the joint probabil-ity of a document d and a word w based on a latent class variable z: The model assumes that word w and document d are indepen-dent if the latent class z is given, i.e., P(wjz, d) = P(wjz).

Table 3depicts the content map construction algorithm with PLSA. This study initially set z = 100 latent classes. The content map can gauge very accurately which learning cases are relevant to student’s learning problem features, even for features do not ap-pear in a learning case.

5. Adaptive learning case recommendation approach

This section describes the process of adaptive learning case rec-ommendation as the basis for technological development. The pro-cess retrieves adaptive learning cases for the learner from the learning case repository based on student’s learning problem of

the learner. As shown inFig. 9, the adaptive recommendation

pro-cedure includes three phases, namely (i) analysis and identification of teaching context, (ii) definition and establishment of teaching problem features for the student’s learning problem, and (iii) searching, matching and ranking of the adaptive learning cases. 5.1. Analysis and identification of teaching context

This phase includes steps of characteristic assessment and adaptive teaching context identiﬁcation as discussed below. 5.1.1. The assessments

The assessments which include teacher assessment and student assessment aimed to identify the characteristics of both teacher and his/her student for adaptive teaching context identiﬁcation. The results of the assessments are presented in a context feature tuple stored in user model.

5.1.1.1. The teacher assessment. The teacher assessment conducts identiﬁcation of learner’s teaching style, leaning style and

compe-tence. This study adopted the VARK Questionnaire(VARK)to assess

the learner’s learning style, and the CORD Questionnaire (CORD,

2005) to assess the learner’s teaching style. Scores on the

pedagog-ical content knowledge of mathematics, education of students with mild disabilities, and mathematics knowledge tests were adopted as criteria to assess the teacher’s competence.

Fig. 7. The content map model.

Concept Name Learning Case Occurance the weight in problem statement the weight in teaching plan the weight in teaching case Attribute Have Link Legend

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5.1.1.2. The student assessment. The student assessment identiﬁes a student’s disabilities, learning style, strengths and weaknesses, and mathematics abilities. These characteristics information can be provided by the learner, or determined though referral forms of student with special needs and the templates of Universal Design

for Learning (UDL) (Rose & Meyer, 2002), respectively. A learning

style measurement test (Kolb, 1999) and academic ability

assess-ment were then performed to identify the student’s learning style and to determine the current math ability level.

5.1.2. Teaching context identiﬁcation

Based on the assessment results which are presented as a con-text feature tuple, a learner’s suitable teaching concon-text can be iden-tiﬁed through the context map. In other words, the new context

feature tuple is classiﬁed into the most similar cluster by compar-ing each cluster’s centroid with the dissimilarity measure approach of EKP. Smaller degree dissimilarity between two objects indicates greater similarity. Therefore, the cluster with the smallest degree of dissimilarity is identiﬁed as the most suitable teaching context for the learner.

5.2. Deﬁnition and establishment of teaching problem features The second phase transformed an unstructured description of a learner’s teaching problem into structured problem features for learning case searching and matching. The problem analysis in-cluded ﬁve steps, i.e. sentence breaking, word breaking, word tag-ging, concept parsing and problem features weight calculation. The

ﬁrst four steps were as in Section4.2.2. The weight of problem

fea-tures was calculated as the probability of appearing in the teaching problem as

wpfi¼ PðpfiÞ ¼ npfi Npf

where P(pfi) is the frequency of teaching problem feature i in the

learner’s teaching problem; npfiis the number of teaching problem

feature i in the learner’s teaching problem and Npfis the number

of teaching problem features in the learner’s teaching problem. 5.3. The searching, matching and recommendation of the adaptive learning case

The third phase retrieved the adaptive learning case according to the results of the previous two phases. This phase included steps of searching learning case candidates, matching learning content, ranking the learning cases, recommending and re-recommending. 5.3.1. Searching learning case candidates

This step searched relevant learning cases as learning cases can-didates through the context map based on the teaching context identiﬁed by the ﬁrst phase, and then proceeded to content matching.

5.3.2. Matching learning content

The matching model provided global matching and local match-ing accordmatch-ing to the learner’s preference. The global matchmatch-ing computed content similarity on all learning cases, while local matching computed content similarity on problem statement, teaching plan and teaching cases, respectively.

The matching types determined by setting control factors

a

p,

a

n

and

a

cwere set for the student’s learning problem statement, the

teaching plan and the teaching case respectively. A control factor value was set to 1 if the learner selected local matching, and 0 otherwise. If global matching was chosen, all control factor values were set to 1.

The similarity measurement was then performed to calculate the degree of similarity between a learning case and the learner’s

teaching problem using the cosine-measure (Gerard, 1989). The

learning case similarity, denoted as Sim(j), is deﬁned by

SimðjÞ ¼

a

P CosSimðLPj;PÞ þ

a

N CosSimðLNj;PÞ þ

a

C CosSimðLCj;PÞ

CosSimðLPj;PÞ ¼

Pt

i¼1ðwPji wpfiÞ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pt i¼1w2Pji Pt i¼1w2pfi q ¼ Pt i¼1ðPðcPjijlpjÞPðpfiÞÞ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pt i¼1ðPðcPjijlpjÞÞ 2Pt i¼1ðPðpfiÞÞ 2 q Table 3

Content map construction algorithm

Input: concepts in the problem statement of learning case matrix TP(c,lc)c = 1:T, lc = 1:N

concepts in the teaching plan of learning case matrix TN(c, lc) c = 1:T, lc = 1:N concepts in the teaching case of learning case matrix TC(c, lc) c = 1:T, lc = 1:N The number K of latent class z

Output: WPithe weight of theith concept in the problem statement of learning case WNithe weight of theith concept in the teaching plan of learning case WCithe weight of theith concept in the teaching case of learning case begin set M1=matrix TP set M2=matrix TN Set M3=matrix TC for m=1 to 3 do begin set M(c,lc) = Mm

initialize P(cijzk) randomly with numbers between [0 and 1] where columns should be normalized to 1; initialize P(zk) = 1/k initialize P(lcijzk) = 1/k for i: = 1 to N begin for j: = 1 to T begin Pðlci;cjÞ ¼PKk¼1PðzkÞPðcjjzkÞPðlcijzkÞ end; end;

while a convergence condition is not met do begin for i: = 1 to N begin for j: = 1 to T begin for k: = 1 to K begin E-Step Pðzkjlci;cjÞ ¼ PðzkÞPðcjjzkÞPðlcijzkÞ PK k¼1PðzkÞPðcjjzkÞPðlcijzkÞ M-Step PðcjjzkÞ ¼ PN i¼1nðlci;cjÞPðzkjlci;cjÞ PT j0 ¼1 PN i¼1nðlci;cjÞPðzkjlci;cjÞ PðlcijzkÞ ¼ PT j¼1nðlci;cjÞPðzkjlci;cjÞ PN i¼1 PT j¼1nðlci;cjÞPðzkjlci;cjÞ PðzkÞ ¼ PN i¼1 PT j¼1nðlci;cj0ÞPðzkjlci;cj0Þ PN i¼1 PT j¼1nðlci;cjÞ end; end; end; end; for i: = 1 to N begin for j: = 1 to T begin wji¼ Pðlci;cjÞ ¼PKk¼1PðzkÞPðcjjzkÞPðlcijzkÞ end; end; end; end;

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CosSimðLNj;PÞ ¼

Pt

i¼1ðwNji wpfiÞ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pt i¼1w2Nji Pt i¼1w2pfi q ¼ Pt i¼1ðPðcNjijlpjÞPðpfiÞÞ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pt

i¼1ðPðcNjijlpjÞÞ2Pti¼1ðPðpfiÞÞ2

q

CosSimðLCj;PÞ ¼

Pt

i¼1ðwCji wpfiÞ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pt i¼1w2Cji Pt i¼1w2pfi q ¼ Pt i¼1ðPðcCjijlpjÞPðpfiÞÞ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pt

i¼1ðPðcCjijlpjÞÞ2Pt_i¼1ðPðpfiÞÞ2

q

In the above equations, wPji is the weight of concept i

in the problem statement of learning case j; wCji is the

weight of concept i in the teaching case of learning

case j; wNji is the weight of concept i in the teaching

plan of learning case j, and wpﬁ is the weight of

problem feature i in the learner’s teaching problem.

5.3.3. Re-recommendation the adaptive learning case

If the recommended learning case did not ﬁt the learner’s requirements, an adjustment was conducted and which is describe below.

The learning case with the most similarity to the learner’s teaching context was measured by the dissimilarity approach of IKP from the specific cluster which was identified in the first phase. The learning case with the most similarity with the lear-ner’s teaching problem was measured by the equation of learn-ing case similarity through the content map. These two learning cases were re-recommended to assist learning for the learner.

6. Implementation and demonstration

An adaptive learning case recommendation mechanism for problem-based e-learning, based on the proposed adaptive learn-ing case recommendation approach, was implemented at the Enterprise Engineering and Integration Laboratory (EEIL) of National Cheng Kung University, Taiwan, ROC.

6.1. Case representation

To store, organize, manage, and use the case contents effec-tively, the instances of each class in the case model were deﬁned as learning objects by Extensible Markup Language (XML). XML is a simple, cross-platform, extensible and ﬂexible text-based

stan-dard for representing data (Sun Microsystems, 2002, Zhang, Sheng,

Li, & Yao, 2002). Practical teaching knowledge can be represented and stored in XML documents by deﬁning tags and the structural relationships among them. The PBeL platform can display knowl-edge content adaptively by enabling the same data to be published in different media.

In this study, a learning case was composed of three XML doc-ument classes, i.e. the teaching context, teaching plan, and teach-ing case. Part of the XML schema is shown as follows:

<Teaching Procedure> 1–1 able to create numbers up to 2000 by adding 1, 10, 100 and 1000, and arrange them in a cor-rect sequential order. 1–2 able to read corcor-rectly numbers up to 2000. 1–3 able to write correctly numbers up to 2000. 1–4 able to comprehend the correspondence between digits and place

Learning Style Assessment Teaching Style Assessment Competence Assessment Learning Style

Assessment Disability Assessment Strength & Weakness

Assessment Mathematics Ability Assessment Teaching Context Identification Sentence Breaking Word Breaking Word Tagging Concept Parsing Concept Weight Calculation Learning Case Candidates Search Matching Type Selection

Learning Case Similarity Calculation

Learning Case Ranking

Learning Case

Recommendation Teaching Context Re-Matching Teaching Content Re-Matching Content Map Context Map Teacher Identification Student Identification Adjustment Learner Teaching Problem Learning Case Teacher User Model Student User Model

Learning Case Content Matching

First Phase

Second Phase

Third Phase

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values of numbers up to 2000. 1–5 able to operate additions or subtractions for numbers up to 2000. 1–6 able to compare two numbers up to 2000 and to express such a relationship with <, >, or =. 1–7 able to operate additions and subtractions with coins for the sum of monetary values up to 2000. 1–8 able to make correct payment according to prices of the objects concerned.

</Teaching Procedure>

<Error Pattern EPID=”1”>(1) Having difficulty counting 10 continuous numbers in the correct sequential order; unable to identify the sequence for 1080-D1090-D( ) due to problem with numbers from 1090 to 1100. (2) Having incorrect comprehen-sion of place values: missing ‘‘thousand, hundred, or ‘-thy”’ when reading numbers, e.g. 1685 was spoken as ‘‘one thousand six hundred eight five” and 1035 as ‘‘one zero thirty five,” while one thousand and five was written as 105. </Error Pattern>

<Error Pattern EPID=”2”> able to compare numbers but cannot distinguish sign > from sign <. </Error Pattern>

<Teaching Approach and Strategy TASID=”1”>(1) Hands-on operatiHands-ons with visual aids – using decimal digit table, coins, and place value board for hands-on operations. When student places one more 10 on the TEN digit while counting from 1080 to 1090, teacher must emphasize the digit being added has a ‘‘TEN” place value to strengthen student’s understanding of place values. Have student add one more ‘‘10” to 1090, mak-ing ‘‘ten 10s,” which can be replaced by ‘‘one ‘100”’. Through such hands-on operations with the aid of decimal digit table, coins, place value board, student can see clearly the changes in decimal digit values, thus facilitating the students’ under-standing of place values as an abstract concept through physi-cal, visual stimulation. (2) Verbal hints: e.g. teacher says ‘‘one thousand ‘and’ seventy” for 1070, ‘‘one thousand ‘and’ eighty” for 1080, ‘‘one thousand ‘and’ ninety” for 1090, and ‘‘one thou-sand ‘and’ one hundred” for 1100. The emphasized ‘‘and” helps students avoid possible confusions about concepts related to a decimal digit system. </Teaching Approach and Strategy >

<Teaching Approach and Strategy TASID=”2”> Work anal-ysis (1) h > : Have students produce visual cards p>yand p<y,

with the hint that Ap-xB means A is ‘‘ bigger” than B. likewise, in h>, h is a bigger and is smaller; then, have students to ﬁll in two numbers of their choosing, one before and the other after the sign, to personally experience the use and concept of p>y. (2) Have students choose either > or < for a pair of arbitrary numbers, e.g.p8 h 5y. Teacher provides visual cards of both > and< for students to choose from. Based on students’ choice, teacher will be able to see if student understands > and <. (3) Teacher gives direct explanations for > and <, and ends with telling students directly that 8 > 5 means ‘‘8 is bigger than 5,” and p>yis read as ‘‘is bigger,” whereas 5 < 8 means ‘‘5 is smaller than 8,” and p<yis read as ‘‘is smaller”. </Teaching Approach and Strategy >

<Teaching Aids TAID=”3”> decimal digit board</Teaching Tool >

<Objective>Hands-on operations of decimal digit board enhances students’ comprehension of decimal values as a con-cept, and helps train students for the conversion between num-bers and Chinese numerals as well as for the reading and speaking of numbers. </Objective >

<Instructional Practice>decimal digit board.jpg</Instruc-tional Practice>

<Explanation> Chinese words for decimal digits are made into visual cards. Hands-on operations and arranging these word cards in the correct order improve students’ conceptual understanding of decimal values and strengthen students’ interchanging capability in between reading and writing. </ Explanation>

</Teaching Case>

6.2. Example

Learners were required to enter their ID and password via the interface to access the PBeL platform. If a learner logged into the system at the ﬁrst time, a user model would be built to contain the learner’s demographic data, background information, and the teacher assessment results.

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The platform then guided the learner to assess his or her stu-dent’s academic abilities, and analyzes the types of disabilities, learning style, and strengths and weaknesses of the students. A student model was then built based on the student assessment results.

The learner was then required to describe his/her student’s learning problem in Chinese natural language, and chose the

matching type as shown Fig. 10. The adaptive recommendation

mechanism then recommended the most similar learning case according to the characteristics of the learner and the student, and the student’s learning problem. The content of the

recom-mended learning case displays as shown inFig. 11.

7. Conclusion and future work

This study designed an e-learning model, with problem-based learning as its core and social constructivism and situated learning as its auxiliary theories, to assist teachers to effectively develop knowledge of teaching mathematics for students with mild disabilities.

Additionally, an adaptive recommendation approach was de-signed for the problem-based e-learning model to achieve the goal of adaptive learning. This approach recommended adaptive learn-ing cases based on the characteristics of both the learner and the student, and has semantic searching capability to avoid informa-tion mistake and loss caused by semantic variainforma-tions. To fulﬁll the requirements of clustering data with numerical, categorical sin-gle-values, and categorical multi-values, an enhanced k-prototypes algorithm was proposed.

Although IKP used in the recommendation approach can cluster data with different data types, the clustering results allow only one learning case in a cluster. Future work will combine fuzzy theory into the clustering method. Since fuzzy clustering applies member-ship degrees between 0 and 1, instead of crisp assignments, to rep-resent the degree of membership of data to each cluster, it can identify more suitable teaching contexts for the learner.

Acknowledgement

This research is ﬁnancially supported by National Science Coun-cil of Republic of China under Contract Nos: NSC94-2524-S-024-002, NSC94-2524-S-006-005, NSC94-2524-S-006-006, 95-2524-S-024-001, 95-2524-S-006-002, 95-2524-S-006-95-2524-S-024-001, 96-2524-S-006-95-2524-S-024-001, 96-2524-S-006-002 and 96-2524-S-024-001.

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