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Learning Spaces Interdisciplinary Applied Mathematics

:	Jean-Claude Falmagne, Jean-Paul Doignon
:	Learning Spaces Interdisciplinary Applied Mathematics
:	Springer-Verlag
:	9783642010392
:	1
:	CHF 154.50
:

:	Sonstiges
:	English

:	417
:	Wasserzeichen
:	PC/MAC/eReader/Tablet
:	PDF

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.

	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	191
	194	191
	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<