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Knowledge space
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<h2 id="motivation">Motivation</h2> <p>Knowledge Space Theory attempts to address shortcomings of <a href="/facts/Standardized_test/PkunTjfB">standardized testing</a> when used in educational <a href="/facts/Psychometrics/56X26MDI">psychometry</a>. Common tests, such as the <a href="/facts/SAT/59352V9D">SAT</a> and <a href="/facts/ACT_(test)/SWeufeHH">ACT</a>, compress a student's knowledge into a very small range of <a href="/facts/Ordinal_data/kgLhAaCn">ordinal</a> <a href="/facts/Ranking/9unRtwbc">ranks</a>, in the process effacing the conceptual dependencies between questions. Consequently, the tests cannot distinguish between true understanding and <a href="/facts/Guessing/9Kq3CIym">guesses</a>, nor can they identify a student's particular weaknesses, only the general proportion of skills mastered. The goal of knowledge space theory is to provide a language by which <a href="/facts/Exam/CjK7g99B">exams</a> can communicate<a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a> </p> <ul><li><i>What the student can do</i> and</li> <li><i>What the student is ready to learn</i>.</li></ul> <h2 id="model-structure">Model structure</h2> <p>Knowledge Space Theory-based models presume that an <a href="/facts/Subject_(documents)/CtYMRwTQ">educational subject</a> S can be modeled as a <a href="/facts/Finite_set/za1newlP">finite set</a> Q of <a href="/facts/Concept/VWB6yhQp">concepts</a>, skills, or topics. Each <i>feasible state of knowledge</i> about S is then a <a href="/facts/Subset/SJGERmbA">subset</a> of Q; the set of all such feasible states is K. The precise term for the information (<i>Q</i>, <i>K</i>) depends on the extent to which K satisfies certain <a href="/facts/Axiom/mLjP14Mc">axioms</a>: </p> <ul><li>A knowledge structure assumes that K contains the <a href="/facts/Empty_set/ffEk3eA6">empty set</a> (a student may know nothing about S) and Q itself (a student may have fully mastered S).</li> <li>A knowledge space is a knowledge structure that is closed under <a href="/facts/Union_(set_theory)/KVVXKfWl">set union</a>: if, for each topic, there is an expert in a <a href="/facts/Form_(education)/L0BEkwMo">class</a> on that topic, then it is possible, with enough time and effort, for each student in the class to become an expert on all those topics simultaneously.</li> <li>A quasi-ordinal knowledge space is a knowledge space that is also closed under <a href="/facts/Intersection_(set_theory)/RPfUNJh9">set intersection</a>: if student a knows topics A and B; and student c knows topics B and C; then it is possible for another student b to know only topic B.</li> <li>A well-graded knowledge space or learning space is a knowledge space satisfying the following axiom: <blockquote><p>If <i>S</i>∈<i>K</i>, then there exists <i>x</i>∈<i>S</i> such that <i>S</i>\{<i>x</i>}∈<i>K</i></p></blockquote> In educational terms, any feasible body of knowledge can be learned one concept at a time.</li></ul> <h3>Prerequisite partial order</h3> <p>The more contentful axioms associated with quasi-ordinal and well-graded knowledge spaces each imply that the knowledge space forms a well-understood (and heavily studied) mathematical structure: </p> <ul><li>A quasi-ordinal knowledge space can be associated with a <a href="/facts/Distributive_lattice/97DFz1cu">distributive lattice</a> under set union and set intersection. The name "quasi-ordinal" arises from <a href="/facts/Birkhoff%2527s_representation_theorem/Mg51p9wz">Birkhoff's representation theorem</a>, which explains that distributive lattices <a href="/facts/Bijection/j4gTuTmW">uniquely correspond</a> to <a href="/facts/Partial_order/qb4a3FAX">partial orders</a>.</li> <li>A well-graded knowledge space is an <a href="/facts/Antimatroid/hYBJb89z">antimatroid</a>, a type of mathematical structure that describes certain problems solvable with a <a href="/facts/Greedy_algorithm/alfsekco">greedy algorithm</a>.</li></ul> <p>In either case, the mathematical structure implies that <a href="/facts/Set_inclusion/SJGERmbA">set inclusion</a> defines <a href="/facts/Partial_Order/qb4a3FAX">partial order</a> on K, interpretable as an <a href="/facts/Curriculum/FEAlV1PH">educational prerequirement</a>: if <i>a</i>(⪯)<i>b</i> in this partial order, then a must be learned before b. </p> <h3>Inner and outer fringe</h3> <p>The prerequisite partial order does not uniquely identify a <a href="/facts/Curriculum/FEAlV1PH">curriculum</a>; some concepts may lead to a variety of other possible topics. But the <a href="/facts/Covering_relation/VW5JsGnh">covering relation</a> associated with the prerequisite partial does control curricular structure: if students know a before a lesson and b immediately after, then b must cover a in the partial order. In such a circumstance, the new topics covered between a and b constitute the outer fringe of a ("what the student was ready to learn") and the inner fringe of b ("what the student just learned"). </p> <h2 id="construction-of-knowledge-spaces">Construction of knowledge spaces</h2> <p>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.<a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a><a class="footnote-ref" id="fnref:10" href="#fn:10"><sup>10</sup></a><a class="footnote-ref" id="fnref:11" href="#fn:11"><sup>11</sup></a> </p><p>Another method is to construct the knowledge space by explorative data analysis (for example by <a href="/facts/Item_tree_analysis/4vvbVSBN">item tree analysis</a>) from data.<a class="footnote-ref" id="fnref:12" href="#fn:12"><sup>12</sup></a><a class="footnote-ref" id="fnref:13" href="#fn:13"><sup>13</sup></a> A third method is to derive the knowledge space from an analysis of the problem solving processes in the corresponding domain.<a class="footnote-ref" id="fnref:14" href="#fn:14"><sup>14</sup></a> </p> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Doignon, J.-P.; Falmagne, J.-Cl. (1999), Knowledge Spaces, Springer-Verlag, ISBN 978-3-540-64501-6. <a href="978-3-540-64501-6" target="_blank">978-3-540-64501-6</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Doignon, J.-P.; Falmagne, J.-Cl. (1985), "Spaces for the assessment of knowledge", International Journal of Man-Machine Studies, 23 (2): 175–196, doi:10.1016/S0020-7373(85)80031-6. <a href="/wiki/Jean-Claude_Falmagne" target="_blank">/wiki/Jean-Claude_Falmagne</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Falmagne, J.-Cl.; Albert, D.; Doble, C.; Eppstein, D.; Hu, X. (2013), Knowledge Spaces. Applications in Education, Springer. <a href="/wiki/Jean-Claude_Falmagne" target="_blank">/wiki/Jean-Claude_Falmagne</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>A bibliography on knowledge spaces maintained by Cord Hockemeyer contains over 400 publications on the subject. <a href="http://kst.hockemeyer.at/kst-bib.html" target="_blank">http://kst.hockemeyer.at/kst-bib.html</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Introduction to Knowledge Spaces: Theory and Applications, Christof Körner, Gudrun Wesiak, and Cord Hockemeyer, 1999 and 2001. <a href="http://wundt.uni-graz.at/MathPsych/cda/overview_sokrates.htm" target="_blank">http://wundt.uni-graz.at/MathPsych/cda/overview_sokrates.htm</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>"Homepage of RATH". Archived from the original on 2007-06-30. <a href="https://web.archive.org/web/20070630200637/http://wundt.kfunigraz.ac.at/rath/" target="_blank">https://web.archive.org/web/20070630200637/http://wundt.kfunigraz.ac.at/rath/</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>Schrepp, M. (2005), "About the connection between knowledge structures and latent class models", Methodology, 1 (3): 93–103, doi:10.1027/1614-2241.1.3.93. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> <li id="fn:8"><p>Jean-Paul Doignon, Jean-Claude Falmagne (2015). "Knowledge Spaces and Learning Spaces". arXiv:1511.06757 [math.CO]. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li> <li id="fn:9"><p>Koppen, M. (1993), "Extracting human expertise for constructing knowledge spaces: An algorithm", Journal of Mathematical Psychology, 37: 1–20, doi:10.1006/jmps.1993.1001. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li> <li id="fn:10"><p>Koppen, M.; Doignon, J.-P. (1990), "How to build a knowledge space by querying an expert", Journal of Mathematical Psychology, 34 (3): 311–331, doi:10.1016/0022-2496(90)90035-8. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></p></li> <li id="fn:11"><p>Schrepp, M.; Held, T. (1995), "A simulation study concerning the effect of errors on the establishment of knowledge spaces by querying experts", Journal of Mathematical Psychology, 39 (4): 376–382, doi:10.1006/jmps.1995.1035 <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></p></li> <li id="fn:12"><p>Schrepp, M. (1999), "Extracting knowledge structures from observed data", British Journal of Mathematical and Statistical Psychology, 52 (2): 213–224, doi:10.1348/000711099159071 <a href="/wiki/British_Journal_of_Mathematical_and_Statistical_Psychology" target="_blank">/wiki/British_Journal_of_Mathematical_and_Statistical_Psychology</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></p></li> <li id="fn:13"><p>Schrepp, M. (2003), "A method for the analysis of hierarchical dependencies between items of a questionnaire" (PDF), Methods of Psychological Research Online, 19: 43–79 <a href="http://www.dgps.de/fachgruppen/methoden/mpr-online/issue19/art3/mpr106_04.pdf" target="_blank">http://www.dgps.de/fachgruppen/methoden/mpr-online/issue19/art3/mpr106_04.pdf</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></p></li> <li id="fn:14"><p>Albert, D.; Lukas, J. (1999), Knowledge Spaces: Theories, Empirical Research, Applications, Lawrence Erlbaum Associates, Mahwah, NJ <a href="#fnref:14" class="footnote-back-ref">↩</a></p></li> </ol>