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Reverted edits by 103.255.170.8 (talk) to last version by Salix alba

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← Older revision		Revision as of 06:18, 17 March 2013
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	~~Wilber Berryhill~~ is ~~what his spouse enjoys~~ to ~~call him~~ and ~~he totally loves this~~ title. ~~Doing ballet~~ is ~~something she would never give up~~. ~~My wife~~ and ~~I reside~~ in ~~Mississippi~~ and I ~~adore every day living right here~~. ~~Distributing manufacturing has been his occupation~~ for some time.<br><br>~~Feel free~~ to ~~visit my weblog; tarot card readings~~ - [http://1.~~234~~.~~36.240~~/~~fxac~~/~~m001_2~~/~~7330 1~~.~~234~~.36.~~240~~] -		{{about\|knowledge spaces in mathematical psychology\|the concept studied by philosopher Pierre Lévy\|Knowledge space (philosophy)}}

			In [[mathematical psychology]], a '''knowledge space''' is a [[antimatroid\|combinatorial structure]] describing the possible states of knowledge of a human learner.<ref>{{citation\|title=Knowledge Spaces\|last1=Doignon\|first1=J.-P.\|last2=Falmagne\|first2=J.-Cl.\|publisher=Springer-Verlag\|year=1999\|isbn = 3-540-64501-2}}.</ref>
			To form a knowledge space, one models a domain of knowledge as a [[set (mathematics)\|set]] of concepts, and a feasible state of knowledge as a [[subset]] of that set containing the concepts known or knowable by some individual. Typically, not all subsets are feasible, due to prerequisite relations among the concepts. The knowledge space is the family of all the feasible subsets. Knowledge spaces were introduced in 1985 by [[Jean-Paul Doignon]] and [[Jean-Claude Falmagne]]<ref>{{citation\|last1=Doignon\|first1=J.-P.\|last2=Falmagne\|first2=J.-Cl.\|year=1985\|title=Spaces for the assessment of knowledge\|journal=International Journal of Man-Machine Studies\|volume=23\|issue=2\|pages=175–196\|doi=10.1016/S0020-7373(85)80031-6}}.</ref> and have since been studied by many other researchers.<ref>A [http://wundt.uni-graz.at/kst.php bibliography on knowledge spaces - doesn't work anymore; pls correct] maintained by Cord Hockemeyer contains over 300 publications on the subject.</ref> They also form the basis for two computerized tutoring systems, [http://wundt.kfunigraz.ac.at/rath/ RATH] and [[ALEKS]].<ref>[http://wundt.uni-graz.at/MathPsych/cda/overview_sokrates.htm Introduction to Knowledge Spaces: Theory and Applications], Christof Körner, Gudrun Wesiak, and Cord Hockemeyer, 1999 and 2001.</ref>

			It is possible to interpret a knowledge space as a special form of a restricted latent class model.<ref>{{citation\|title=About the connection between knowledge structures and latent class models \|last1=Schrepp \|first1=M. \|journal=Methodology\|volume=1\|issue=3\|pages=92–102\|year=2005\|doi=10.1027/1614-2241.1.3.92}}.</ref>

			==Definitions==
			Some basic definitions used in the knowledge space approach -
			*A tuple <math>(Q, K)</math> consisting of a non-empty set <math>Q</math> and a set <math>K</math> of subsets from <math>Q</math> is called a ''knowledge structure'' if <math>K</math> contains the empty set and <math>Q</math>.
			*A knowledge structure is called a ''knowledge space'' if it is closed under union, i.e. if <math>S, T \in Q</math> implies <math>S\cup T \in Q</math>.
			*A knowledge space is called a ''quasi-ordinal knowledge space'' if it is in addition closed under intersection, i.e. if <math>S, T \in Q</math> implies <math>S\cap T \in Q</math>. Closure under both unions and intersections gives (''Q'',∪,∩) the structure of a [[distributive lattice]]; [[Birkhoff's representation theorem]] for distributive lattices shows that there is a one-to-one correspondence between the set of all [[preorder\|quasiorders]] on Q and the set of all quasi-ordinal knowledge spaces on Q. I.e., each quasi-ordinal knowledge space can be represented by a quasi-order and vice versa.

			An important subclass of knowledge spaces, the ''well-graded knowledge spaces'' or ''learning spaces'', can be defined as satisfying two additional mathematical axioms:
			# If <math>S</math> and <math>T</math> are both feasible subsets of concepts, then <math>S\cup T</math> is also feasible. In educational terms: if it is possible for someone to know all the concepts in ''S'', and someone else to know all the concepts in ''T'', then we can posit the potential existence of a third person who combines the knowledge of both people.
			# If <math>S</math> is a nonempty feasible subset of concepts, then there is some concept ''x'' in ''S'' such that <math>S\setminus\{x\}</math> is also feasible. In educational terms: any feasible state of knowledge can be reached by learning one concept at a time.
			A set family satisfying these two axioms forms a [[mathematical structure]] known as an [[antimatroid]].

			==Construction of knowledge spaces==
			In practice there exist several methods to construct knowledge spaces. The most frequently used method is querying experts. There exist several querying algorithms that allow one or several experts to construct a knowledge space by answering a sequence of simple questions.<ref>{{citation\|title=Extracting human expertise for constructing knowledge spaces: An algorithm \|last1=Koppen \|first1=M. \|journal=Journal of Mathematical Psychology\|volume=37\|pages=1–20 \|year=1993 \|doi=10.1006/jmps.1993.1001}}.</ref><ref>{{citation\|title=How to build a knowledge space by querying an expert \|last1=Koppen \|first1=M. \|last2=Doignon \|first2=J.-P. \|journal=Journal of Mathematical Psychology\|volume=34\|issue=3\|pages=311–331 \|year=1990 \|doi=10.1016/0022-2496(90)90035-8}}.</ref><ref>{{citation\|title=A simulation study concerning the effect of errors on the establishment of knowledge spaces by querying experts \|last1=Schrepp \|first1=M. \|last2=Held \|first2=T. \|journal=Journal of Mathematical Psychology\|volume=39\|issue=4\|pages=376–382 \|year=1995\|doi=10.1006/jmps.1995.1035}}</ref>

			Another method is to construct the knowledge space by explorative data analysis (for example by [[Item tree analysis]]) from data.<ref>{{citation\|title=Extracting knowledge structures from observed data \|last1=Schrepp \|first1=M. \|journal=[[British journal of mathematical and statistical psychology]]\|volume= 52\|issue=2 \|pages=213–224 \|year=1999\|doi=10.1348/000711099159071}}</ref><ref>{{citation\|title=A method for the analysis of hierarchical dependencies between items of a questionnaire \|last1=Schrepp \|first1=M. \|journal= Methods of Psychological Research Online\|volume=19\|pages=43–79 \|year=2003\|url=http://www.dgps.de/fachgruppen/methoden/mpr-online/issue19/art3/mpr106_04.pdf }}</ref>
			A third method is to derive the knowledge space from an analysis of the problem solving processes in the corresponding domain.<ref>{{citation\|title=Knowledge Spaces: Theories, Empirical Research, Applications\|last1=Albert\|first1=D.\|last2=Lukas\|first2=J.\|publisher=Lawrence Erlbaum Associates, Mahwah, NJ\|year=1999}}</ref>

			==References==

			{{reflist}}

			[[Category:Cognition]]
			[[Category:Knowledge representation]]

149.169.239.125 at 17:07, 15 March 2012

2012-03-15T17:07:52Z

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Wilber Berryhill is what his spouse enjoys to call him and he totally loves this title. Doing ballet is something she would never give up. My wife and I reside in Mississippi and I adore every day living right here. Distributing manufacturing has been his occupation for some time.<br><br>Feel free to visit my weblog; tarot card readings - [http://1.234.36.240/fxac/m001_2/7330 1.234.36.240] -

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149.169.239.125 at 17:07, 15 March 2012