Falmagne / Doignon Learning Spaces

Interdisciplinary Applied Mathematics
1. Auflage 2010
ISBN: 978-3-642-01039-2
Verlag: Springer
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 417 Seiten

Learning spaces offer a rigorous mathematical foundation for practical systems of educational technology. Learning spaces generalize partially ordered sets and are special cases of knowledge spaces. The various structures are investigated from the standpoints of combinatorial properties and stochastic processes. Leaning spaces have become the essential structures to be used in assessing students' competence of various topics. A practical example is offered by ALEKS, a Web-based, artificially intelligent assessment and learning system in mathematics and other scholarly fields. At the heart of ALEKS is an artificial intelligence engine that assesses each student individually and continously. The book is of interest to mathematically oriented readers in education, computer science, engineering, and combinatorics at research and graduate levels. Numerous examples and exercises are included, together with an extensive bibliography.

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.

Autoren/Hrsg.

Falmagne, Jean-Claude

Doignon, Jean-Paul

Inhaltsverzeichnis

1;Preface;4
2;Contents;10
3;1 Overview and Basic Mathematical Concepts;15
3.1;1.1 Main Constructs;16
3.2;1.2 Possible Limitations;23
3.3;1.3 A Practical Application: The ALEKS System;24
3.4;1.4 Potential Applications to Other Fields;25
3.5;1.5 On the Content and Organization of this Book;26
3.6;1.6 Basic Mathematical Concepts and Notation;27
3.7;1.7 Original Sources and Main References;31
4;2 Knowledge Structures and Learning Spaces;36
4.1;2.1 Fundamental Concepts;36
4.2;2.2 Axioms for Learning Spaces;39
4.3;2.3 The nondiscriminative case*;43
4.4;2.4 Projections;44
4.5;2.5 Original Sources and Related Works;51
5;3 Knowledge Spaces;55
5.1;3.1 Outline;55
5.2;3.2 Generating Knowledge Spaces by Querying Experts;56
5.3;3.3 Closure Spaces;57
5.4;3.4 Bases and Atoms;59
5.5;3.5 An Algorithm for Constructing the Base;61
5.6;3.6 Bases and Atoms: The In nite Case*;64
5.7;3.7 The Surmise Relation;66
5.8;3.8 Quasi Ordinal Spaces;68
5.9;3.9 Original Sources and Related Works;70
6;4 Well-Graded Knowledge Structures;73
6.1;4.1 Learning Paths, Gradations, and Fringes;73
6.2;4.2 A Well-Graded Family of Relations: the Biorders?;78
6.3;4.3 Infinite Wellgradedness?;81
6.4;4.4 Finite Learnability;84
6.5;4.5 Verifying Wellgradedness for a U-Closed Family;85
6.6;4.6 Original Sources and Related Works;89
7;5 Surmise Systems;92
7.1;5.1 Basic Concepts;92
7.2;5.2 Knowledge Spaces and Surmise Systems;96
7.3;5.3 AND/OR Graphs;98
7.4;5.4 Surmise Functions and Wellgradedness;101
7.5;5.5 Hasse Systems;103
7.6;5.6 Resolubility and Acyclicity;107
7.7;5.7 Original Sources and Related Works;110
8;6 Skill Maps, Labels and Filters;113
8.1;6.1 Skills;113
8.2;6.2 Skill Maps: The Disjunctive Model;116
8.3;6.3 Minimal Skill Maps;117
8.4;6.4 Skill Maps: The Conjunctive Model;120
8.5;6.5 Skill Multimaps: The Competency Model;122
8.6;6.6 Labels and Filters;123
8.7;6.7 Original Sources and Related Works;126
9;7 Entailments and the Maximal Mesh;128
9.1;7.1 Entailments;129
9.2;7.2 Entail Relations;133
9.3;7.3 Meshability of Knowledge Structures;134
9.4;7.4 The Maximal Mesh;136
9.5;7.5 Original Sources and Related Works;139
10;8 Galois Connections*;141
10.1;8.1 Three Exemplary Correspondences;141
10.2;8.2 Closure Operators and Galois Connections;142
10.3;8.3 Lattices and Galois Connections;146
10.4;8.4 Knowledge Structures and Binary Relations;149
10.5;8.5 Granular Knowledge Structures and GranularAttributions;152
10.6;8.6 Knowledge Structures and Associations;155
10.7;8.7 Original Sources and Related Works;157
11;9 Descriptive and Assessment Languages*;159
11.1;9.1 Languages and Decision Trees;159
11.2;9.2 Terminology;163
11.3;9.3 Recovering Ordinal Knowledge Structures;165
11.4;9.4 Recovering Knowledge Structures;168
11.5;9.5 Original Sources and Related Works;169
12;10 Learning Spaces and Media;171
12.1;10.1 Main Concepts of Media Theory;172
12.2;10.2 Some Basic Lemmas;176
12.3;10.3 The Content of a State;177
12.4;10.4 Oriented Media;182
12.5;10.5 Learning Spaces and Closed, Rooted Media;187
12.6;10.6 Original Sources and Related Works;191
13;11 Probabilistic Knowledge Structures
;194
13.1;11.1 Basic Concepts and Examples;194
13.2;11.2 An Empirical Application;198
13.3;11.3 The Likelihood Ratio Procedure;202
13.4;11.4 Learning Models;205
13.5;11.5 A Combinatorial Result;207
13.6;11.6 Markov Chain Models;210
13.7;11.7 Probabilistic Projections;213
13.8;11.8 Nomenclatures and Classi cations;216
13.9;11.9 Independent Projections;216
13.10;11.10 Original Sources and Related Works;220
14;12 Stochastic Learning Paths*;222
14.1;12.1 A Knowledge Structure in Euclidean Geometry;222
14.2;12.2 Basic Concepts;223
14.3;12.3 General Results;228
14.4;12.4 Assumptions on Distributions;231
14.5;12.5 The Learning Latencies;232
14.6;12.6 Empirical Predictions;235
14.7;12.8 Simplifying Assumptions;240
14.8;12.9 Remarks on Application and Use of the Theory;242
14.9;12.10 An Application of the Theory to the Case n = 2;243
14.10;12.11 Original Sources and Related Works;246
15;13 Uncovering the Latent State: A Continuous Markov Procedure;248
15.1;13.1 A Deterministic Algorithm;248
15.2;13.2 Outline of a Markovian Stochastic Process;250
15.3;13.3 Basic Concepts;253
15.4;13.4 Special Cases;256
15.5;13.5 General Results;260
15.6;13.6 Uncovering the Latent State;262
15.7;13.7 A Two-Step Assessment Algorithm;266
15.8;13.8 Refining the Assessment;272
15.9;13.9 Proofs*;274
15.10;13.10 Original Sources and Related Works;278
16;14 A Markov Chain Procedure;280
16.1;14.1 Outline;280
16.2;14.2 The Stochastic Assessment Process;284
16.3;14.3 Combinatorial Assumptions on the Structure;286
16.4;14.4 Markov Chains Terminology;290
16.5;14.5 Results for the Fair Case;291
16.6;14.6 Uncovering a Stochastic State: Examples;294
16.7;14.7 Intractable Cases;299
16.8;14.8 Original Sources and Related Works;302
17;15 Building a Knowledge Space;304
17.1;15.1 Background to the QUERY routine;305
17.2;15.2 Koppen's Algorithm;309
17.3;15.3 Kambouri's Experiment;317
17.4;15.4 Results;322
17.5;15.5 Cosyn and Thi ery's Work;331
17.6;15.6 Refining a Knowledge Structure;335
17.7;15.7 Simulations of Various Refi
nements;337
17.8;15.8 Original Sources and Related Works;339
18;16 Building a Learning space;341
18.1;16.1 Preparatory Concepts and an Example;342
18.2;16.2 Managing the Surmise Function;351
18.3;16.3 Engineering a Learning Space;361
18.4;16.4 Original Sources and Related Works;362
19;17 Analyzing the Validity of an Assessment;364
19.1;17.1 The Concept of Validity for an Assessment;364
19.2;17.2 The ALEKS Assessment Algorithm;366
19.3;17.3 The Methods;367
19.4;17.4 Data Analysis;372
19.5;17.5 Summary;378
20;18 Open Problems;380
20.1;18.1 Knowledge Spaces and U-Closed Families;380
20.2;18.2 Wellgradedness and the Fringes;381
20.3;18.3 About Granularity;382
20.4;18.4 Miscellaneous;382
21;Glossary;383
22;Bibliography;401
23;Index;413