Learning Spaces (eBook)

Jean-Paul Doignon is a professor at the mathematics department of the Université Libre de Bruxelles, Belgium. His research covers various aspects of discrete mathematics (graphs, ordered sets, convex polytopes, etc.) and applications to behavioral sciences (preference modeling, choice representation, knowledge assessment, etc.). Jean-Claude Falmagne is emeritus professor of cognitive sciences at the University of California, Irvine. His research interests span various areas, focusing on the application of mathematics to educational technology, psychophysics, choice theory, and the philosophy of science, in particular measurement theory.

Preface 4
Contents 10
1 Overview and Basic Mathematical Concepts 15
1.1 Main Constructs 16
1.2 Possible Limitations 23
1.3 A Practical Application: The ALEKS System 24
1.4 Potential Applications to Other Fields 25
1.5 On the Content and Organization of this Book 26
1.6 Basic Mathematical Concepts and Notation 27
1.7 Original Sources and Main References 31
2 Knowledge Structures and Learning Spaces 36
2.1 Fundamental Concepts 36
2.2 Axioms for Learning Spaces 39
2.3 The nondiscriminative case* 43
2.4 Projections 44
2.5 Original Sources and Related Works 51
3 Knowledge Spaces 55
3.1 Outline 55
3.2 Generating Knowledge Spaces by Querying Experts 56
3.3 Closure Spaces 57
3.4 Bases and Atoms 59
3.5 An Algorithm for Constructing the Base 61
3.6 Bases and Atoms: The In nite Case* 64
3.7 The Surmise Relation 66
3.8 Quasi Ordinal Spaces 68
3.9 Original Sources and Related Works 70
4 Well-Graded Knowledge Structures 73
4.1 Learning Paths, Gradations, and Fringes 73
4.2 A Well-Graded Family of Relations: the Biorders? 78
4.3 Infinite Wellgradedness? 81
4.4 Finite Learnability 84
4.5 Verifying Wellgradedness for a U-Closed Family 85
4.6 Original Sources and Related Works 89
5 Surmise Systems 92
5.1 Basic Concepts 92
5.2 Knowledge Spaces and Surmise Systems 96
5.3 AND/OR Graphs 98
5.4 Surmise Functions and Wellgradedness 101
5.5 Hasse Systems 103
5.6 Resolubility and Acyclicity 107
5.7 Original Sources and Related Works 110
6 Skill Maps, Labels and Filters 113
6.1 Skills 113
6.2 Skill Maps: The Disjunctive Model 116
6.3 Minimal Skill Maps 117
6.4 Skill Maps: The Conjunctive Model 120
6.5 Skill Multimaps: The Competency Model 122
6.6 Labels and Filters 123
6.7 Original Sources and Related Works 126
7 Entailments and the Maximal Mesh 128
7.1 Entailments 129
7.2 Entail Relations 133
7.3 Meshability of Knowledge Structures 134
7.4 The Maximal Mesh 136
7.5 Original Sources and Related Works 139
8 Galois Connections* 141
8.1 Three Exemplary Correspondences 141
8.2 Closure Operators and Galois Connections 142
8.3 Lattices and Galois Connections 146
8.4 Knowledge Structures and Binary Relations 149
8.5 Granular Knowledge Structures and GranularAttributions 152
8.6 Knowledge Structures and Associations 155
8.7 Original Sources and Related Works 157
9 Descriptive and Assessment Languages* 159
9.1 Languages and Decision Trees 159
9.2 Terminology 163
9.3 Recovering Ordinal Knowledge Structures 165
9.4 Recovering Knowledge Structures 168
9.5 Original Sources and Related Works 169
10 Learning Spaces and Media 171
10.1 Main Concepts of Media Theory 172
10.2 Some Basic Lemmas 176
10.3 The Content of a State 177
10.4 Oriented Media 182
10.5 Learning Spaces and Closed, Rooted Media 187
10.6 Original Sources and Related Works 191
11 Probabilistic Knowledge Structures 
194 
11.1 Basic Concepts and Examples 194
11.2 An Empirical Application 198
11.3 The Likelihood Ratio Procedure 202
11.4 Learning Models 205
11.5 A Combinatorial Result 207
11.6 Markov Chain Models 210
11.7 Probabilistic Projections 213
11.8 Nomenclatures and Classi cations 216
11.9 Independent Projections 216
11.10 Original Sources and Related Works 220
12 Stochastic Learning Paths* 222
12.1 A Knowledge Structure in Euclidean Geometry 222
12.2 Basic Concepts 223
12.3 General Results 228
12.4 Assumptions on Distributions 231
12.5 The Learning Latencies 232
12.6 Empirical Predictions 235
12.8 Simplifying Assumptions 240
12.9 Remarks on Application and Use of the Theory 242
12.10 An Application of the Theory to the Case n = 2 243
12.11 Original Sources and Related Works 246
13 Uncovering the Latent State: A Continuous Markov Procedure 248
13.1 A Deterministic Algorithm 248
13.2 Outline of a Markovian Stochastic Process 250
13.3 Basic Concepts 253
13.4 Special Cases 256
13.5 General Results 260
13.6 Uncovering the Latent State 262
13.7 A Two-Step Assessment Algorithm 266
13.8 Refining the Assessment 272
13.9 Proofs* 274
13.10 Original Sources and Related Works 278
14 A Markov Chain Procedure 280
14.1 Outline 280
14.2 The Stochastic Assessment Process 284
14.3 Combinatorial Assumptions on the Structure 286
14.4 Markov Chains Terminology 290
14.5 Results for the Fair Case 291
14.6 Uncovering a Stochastic State: Examples 294
14.7 Intractable Cases 299
14.8 Original Sources and Related Works 302
15 Building a Knowledge Space 304
15.1 Background to the QUERY routine 305
15.2 Koppen's Algorithm 309
15.3 Kambouri's Experiment 317
15.4 Results 322
15.5 Cosyn and Thi ery's Work 331
15.6 Refining a Knowledge Structure 335
15.7 Simulations of Various Refi 
337 
15.8 Original Sources and Related Works 339
16 Building a Learning space 341
16.1 Preparatory Concepts and an Example 342
16.2 Managing the Surmise Function 351
16.3 Engineering a Learning Space 361
16.4 Original Sources and Related Works 362
17 Analyzing the Validity of an Assessment 364
17.1 The Concept of Validity for an Assessment 364
17.2 The ALEKS Assessment Algorithm 366
17.3 The Methods 367
17.4 Data Analysis 372
17.5 Summary 378
18 Open Problems 380
18.1 Knowledge Spaces and U-Closed Families 380
18.2 Wellgradedness and the Fringes 381
18.3 About Granularity 382
18.4 Miscellaneous 382
Glossary 383
Bibliography 401
Index 413