<?xml version="1.0" encoding="ISO-8859-1"?><article xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance">
<front>
<journal-meta>
<journal-id>2007-0705</journal-id>
<journal-title><![CDATA[Nova scientia]]></journal-title>
<abbrev-journal-title><![CDATA[Nova scientia]]></abbrev-journal-title>
<issn>2007-0705</issn>
<publisher>
<publisher-name><![CDATA[Universidad de La Salle Bajío A. C., Coordinación de Investigación]]></publisher-name>
</publisher>
</journal-meta>
<article-meta>
<article-id>S2007-07052022000100108</article-id>
<article-id pub-id-type="doi">10.21640/ns.v14i28.2947</article-id>
<title-group>
<article-title xml:lang="en"><![CDATA[Mathematical modelling of student&#8217;s cumulative learning]]></article-title>
<article-title xml:lang="es"><![CDATA[Modelo matemático del aprendizaje acumulativo de un estudiante]]></article-title>
</title-group>
<contrib-group>
<contrib contrib-type="author">
<name>
<surname><![CDATA[Sagaceta Mejía]]></surname>
<given-names><![CDATA[Alma Rocío]]></given-names>
</name>
<xref ref-type="aff" rid="Aff"/>
</contrib>
<contrib contrib-type="author">
<name>
<surname><![CDATA[Fresán Figueroa]]></surname>
<given-names><![CDATA[Julián Alberto]]></given-names>
</name>
<xref ref-type="aff" rid="Aff"/>
</contrib>
<contrib contrib-type="author">
<name>
<surname><![CDATA[Martín González]]></surname>
<given-names><![CDATA[Ehyter Matías]]></given-names>
</name>
<xref ref-type="aff" rid="Aff"/>
</contrib>
</contrib-group>
<aff id="Af1">
<institution><![CDATA[,Universidad Iberoamericana Departamento de Física y Matemáticas ]]></institution>
<addr-line><![CDATA[Ciudad de México ]]></addr-line>
<country>Mexico</country>
</aff>
<aff id="Af2">
<institution><![CDATA[,Universidad Autónoma Metropolitana Departamento de Matemáticas Aplicadas y Sistemas ]]></institution>
<addr-line><![CDATA[Unidad Cuajimalpa ]]></addr-line>
<country>Mexico</country>
</aff>
<aff id="Af3">
<institution><![CDATA[,Universidad de Guanajuato Departamento de Matemáticas ]]></institution>
<addr-line><![CDATA[ ]]></addr-line>
<country>Mexico</country>
</aff>
<pub-date pub-type="pub">
<day>00</day>
<month>00</month>
<year>2022</year>
</pub-date>
<pub-date pub-type="epub">
<day>00</day>
<month>00</month>
<year>2022</year>
</pub-date>
<volume>14</volume>
<numero>28</numero>
<copyright-statement/>
<copyright-year/>
<self-uri xlink:href="http://www.scielo.org.mx/scielo.php?script=sci_arttext&amp;pid=S2007-07052022000100108&amp;lng=en&amp;nrm=iso"></self-uri><self-uri xlink:href="http://www.scielo.org.mx/scielo.php?script=sci_abstract&amp;pid=S2007-07052022000100108&amp;lng=en&amp;nrm=iso"></self-uri><self-uri xlink:href="http://www.scielo.org.mx/scielo.php?script=sci_pdf&amp;pid=S2007-07052022000100108&amp;lng=en&amp;nrm=iso"></self-uri><abstract abstract-type="short" xml:lang="en"><p><![CDATA[Abstract In this paper we propose a model to study the learning process of one student during a course. We formulate a stochastic model based on the quality of the teacher&#8217;s class and the affinity of the student to understand the sessions, under the assumption that previous sessions have some influence in the understanding of the next sessions. The afore mentioned assumption implies that the process is not a Markov process. We derive some recursive expressions for the distribution of the number of sessions that the student comprehends. Furthermore, we study the convergence of this distribution and illustrate its speed of convergence through some numerical examples. Finally, we apply these results to propose a methodology to estimate the quality of this kind of courses.]]></p></abstract>
<abstract abstract-type="short" xml:lang="es"><p><![CDATA[Resumen En este artículo se propone un modelo para estudiar el proceso de aprendizaje de un alumno durante un curso. Se estableció un modelo estocástico basado en la calidad de la clase del profesor y la afinidad del alumno para comprenderla, bajo el supuesto de que las clases que éste tuvo previamente influyen en su comprensión de las posteriores. La suposición antes mencionada implica que el proceso no es un proceso de Markov. Se obtuvieron algunas expresiones recursivas para la distribución del número de clases que el alumno comprende. Además, se estudió la convergencia de la distribución obtenida y se ilustró la velocidad de convergencia a través de algunos ejemplos numéricos. Finalmente, se aplicaron estos resultados para dar una estimación de la calidad de este tipo de cursos.]]></p></abstract>
<kwd-group>
<kwd lng="en"><![CDATA[stochastic model]]></kwd>
<kwd lng="en"><![CDATA[learning]]></kwd>
<kwd lng="en"><![CDATA[non-markovian process]]></kwd>
<kwd lng="en"><![CDATA[quality of a course]]></kwd>
<kwd lng="en"><![CDATA[mathematical models]]></kwd>
<kwd lng="en"><![CDATA[learning processes]]></kwd>
<kwd lng="en"><![CDATA[courses]]></kwd>
<kwd lng="en"><![CDATA[educational quality]]></kwd>
<kwd lng="en"><![CDATA[sessions]]></kwd>
<kwd lng="en"><![CDATA[distribution]]></kwd>
<kwd lng="en"><![CDATA[convergence]]></kwd>
<kwd lng="en"><![CDATA[education]]></kwd>
<kwd lng="en"><![CDATA[formation]]></kwd>
<kwd lng="es"><![CDATA[modelo estocástico]]></kwd>
<kwd lng="es"><![CDATA[aprendizaje]]></kwd>
<kwd lng="es"><![CDATA[proceso no-markoviano]]></kwd>
<kwd lng="es"><![CDATA[calidad de un curso]]></kwd>
<kwd lng="es"><![CDATA[modelos matemáticos]]></kwd>
<kwd lng="es"><![CDATA[procesos de aprendizaje]]></kwd>
<kwd lng="es"><![CDATA[cursos]]></kwd>
<kwd lng="es"><![CDATA[calidad educativa]]></kwd>
<kwd lng="es"><![CDATA[sesiones]]></kwd>
<kwd lng="es"><![CDATA[distribución]]></kwd>
<kwd lng="es"><![CDATA[convergencia]]></kwd>
<kwd lng="es"><![CDATA[educación]]></kwd>
<kwd lng="es"><![CDATA[formación]]></kwd>
</kwd-group>
</article-meta>
</front><back>
<ref-list>
<ref id="B1">
<label>[1]</label><nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Bower]]></surname>
<given-names><![CDATA[GH]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[Application of a model to paired-associate learning]]></article-title>
<source><![CDATA[Psychometrika]]></source>
<year>1961</year>
<volume>26</volume>
<numero>3</numero>
<issue>3</issue>
<page-range>255-80</page-range></nlm-citation>
</ref>
<ref id="B2">
<label>[2]</label><nlm-citation citation-type="book">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Busemeyer]]></surname>
<given-names><![CDATA[JR]]></given-names>
</name>
<name>
<surname><![CDATA[Diederich]]></surname>
<given-names><![CDATA[A]]></given-names>
</name>
</person-group>
<source><![CDATA[Cognitive modeling]]></source>
<year>2010</year>
<publisher-name><![CDATA[Sage]]></publisher-name>
</nlm-citation>
</ref>
<ref id="B3">
<label>[3]</label><nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Cajueiro]]></surname>
<given-names><![CDATA[DO]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[Enforcing social behavior in an Ising model with complex neighborhoods]]></article-title>
<source><![CDATA[Physica A: Statistical Mechanics and its Applications]]></source>
<year>2011</year>
<volume>390</volume>
<numero>9</numero>
<issue>9</issue>
<page-range>1695-703</page-range></nlm-citation>
</ref>
<ref id="B4">
<label>[4]</label><nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Chechile]]></surname>
<given-names><![CDATA[RA]]></given-names>
</name>
<name>
<surname><![CDATA[Sloboda]]></surname>
<given-names><![CDATA[LN]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[Reformulating Markovian processes for learning and memory from a hazard function framework]]></article-title>
<source><![CDATA[Journal of Mathematical Psychology]]></source>
<year>2014</year>
<volume>59</volume>
<page-range>65-81</page-range></nlm-citation>
</ref>
<ref id="B5">
<label>[5]</label><nlm-citation citation-type="">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Coombs]]></surname>
<given-names><![CDATA[CH]]></given-names>
</name>
<name>
<surname><![CDATA[Dawes]]></surname>
<given-names><![CDATA[RM]]></given-names>
</name>
<name>
<surname><![CDATA[Tversky]]></surname>
<given-names><![CDATA[A]]></given-names>
</name>
</person-group>
<source><![CDATA[Mathematical psychology: An elementary introduction]]></source>
<year>1970</year>
</nlm-citation>
</ref>
<ref id="B6">
<label>[6]</label><nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Dufresne]]></surname>
<given-names><![CDATA[RJ]]></given-names>
</name>
<name>
<surname><![CDATA[Gerace]]></surname>
<given-names><![CDATA[WJ]]></given-names>
</name>
<name>
<surname><![CDATA[Leonard]]></surname>
<given-names><![CDATA[WJ]]></given-names>
</name>
<name>
<surname><![CDATA[Mestre]]></surname>
<given-names><![CDATA[JP]]></given-names>
</name>
<name>
<surname><![CDATA[Wenk]]></surname>
<given-names><![CDATA[L]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[Classtalk: A classroom communication system for active learning]]></article-title>
<source><![CDATA[Journal of Computing in Higher Education]]></source>
<year>1996</year>
<volume>7</volume>
<numero>2</numero>
<issue>2</issue>
<page-range>3-47</page-range></nlm-citation>
</ref>
<ref id="B7">
<label>[7]</label><nlm-citation citation-type="">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Estes]]></surname>
<given-names><![CDATA[WK]]></given-names>
</name>
<name>
<surname><![CDATA[Suppes]]></surname>
<given-names><![CDATA[P]]></given-names>
</name>
</person-group>
<source><![CDATA[Foundations of stimulus sampling theory]]></source>
<year>1974</year>
</nlm-citation>
</ref>
<ref id="B8">
<label>[8]</label><nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Huang]]></surname>
<given-names><![CDATA[YM]]></given-names>
</name>
<name>
<surname><![CDATA[Huang]]></surname>
<given-names><![CDATA[TC]]></given-names>
</name>
<name>
<surname><![CDATA[Wang]]></surname>
<given-names><![CDATA[KT]]></given-names>
</name>
<name>
<surname><![CDATA[Hwang]]></surname>
<given-names><![CDATA[WY]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[A Markov-based Recommendation Model for Exploring the Transfer of Learning on the Web]]></article-title>
<source><![CDATA[Journal of Educational Technology and Society]]></source>
<year>2009</year>
<volume>12</volume>
<numero>2</numero>
<issue>2</issue>
<page-range>144-62</page-range></nlm-citation>
</ref>
<ref id="B9">
<label>[9]</label><nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Klemm]]></surname>
<given-names><![CDATA[K]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[Nonequilibrium transitions in complex networks: A model of social interaction]]></article-title>
<source><![CDATA[Physical Review E]]></source>
<year>2003</year>
<volume>67</volume>
<numero>2</numero>
<issue>2</issue>
</nlm-citation>
</ref>
<ref id="B10">
<label>[10]</label><nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Laming]]></surname>
<given-names><![CDATA[F]]></given-names>
</name>
<name>
<surname><![CDATA[Straile]]></surname>
<given-names><![CDATA[R]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[Photoresist cleaning]]></article-title>
<source><![CDATA[Journal of The Electrochemical Society]]></source>
<year>1973</year>
<volume>120</volume>
<numero>2</numero>
<issue>2</issue>
<page-range>292</page-range></nlm-citation>
</ref>
<ref id="B11">
<label>[11]</label><nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Lee]]></surname>
<given-names><![CDATA[YA]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[Third turn position in teacher talk: Contingency and the work of teaching]]></article-title>
<source><![CDATA[Journal of Pragmatics]]></source>
<year>2007</year>
<volume>39</volume>
<numero>6</numero>
<issue>6</issue>
<page-range>1204-30</page-range></nlm-citation>
</ref>
<ref id="B12">
<label>[12]</label><nlm-citation citation-type="">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Levine]]></surname>
<given-names><![CDATA[G]]></given-names>
</name>
<name>
<surname><![CDATA[Burke]]></surname>
<given-names><![CDATA[CJ]]></given-names>
</name>
</person-group>
<source><![CDATA[Mathematical model techniques for learning theories]]></source>
<year>1972</year>
</nlm-citation>
</ref>
<ref id="B13">
<label>[13]</label><nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Li]]></surname>
<given-names><![CDATA[C]]></given-names>
</name>
<name>
<surname><![CDATA[Liu]]></surname>
<given-names><![CDATA[F]]></given-names>
</name>
<name>
<surname><![CDATA[Li]]></surname>
<given-names><![CDATA[P]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[Ising model of user behavior decision in network rumor propagation]]></article-title>
<source><![CDATA[Discrete Dynamics in Nature and Society]]></source>
<year>2018</year>
<page-range>1-10</page-range></nlm-citation>
</ref>
<ref id="B14">
<label>[14]</label><nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[McCroskey]]></surname>
<given-names><![CDATA[JC]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[Classroom consequences of communication apprehension]]></article-title>
<source><![CDATA[Communication Education]]></source>
<year>1977</year>
<volume>26</volume>
<numero>1</numero>
<issue>1</issue>
<page-range>27-33</page-range></nlm-citation>
</ref>
<ref id="B15">
<label>[15]</label><nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Nakayama]]></surname>
<given-names><![CDATA[S]]></given-names>
</name>
<name>
<surname><![CDATA[Nakamura]]></surname>
<given-names><![CDATA[Y]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[A fashion model with social interaction]]></article-title>
<source><![CDATA[Physica A: Statistical Mechanics and its Applications]]></source>
<year>2004</year>
<volume>337</volume>
<numero>3</numero>
<issue>3</issue>
<page-range>625-34</page-range></nlm-citation>
</ref>
<ref id="B16">
<label>[16]</label><nlm-citation citation-type="book">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Ostilli]]></surname>
<given-names><![CDATA[M]]></given-names>
</name>
<name>
<surname><![CDATA[Yoneki]]></surname>
<given-names><![CDATA[E]]></given-names>
</name>
<name>
<surname><![CDATA[Leung]]></surname>
<given-names><![CDATA[IXY]]></given-names>
</name>
<name>
<surname><![CDATA[Mendes]]></surname>
<given-names><![CDATA[JFF]]></given-names>
</name>
<name>
<surname><![CDATA[Lió]]></surname>
<given-names><![CDATA[P]]></given-names>
</name>
<name>
<surname><![CDATA[Crowcroft]]></surname>
<given-names><![CDATA[J]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[Ising model of rumour spreading in interacting communities]]></article-title>
<source><![CDATA[]]></source>
<year>2010</year>
<publisher-name><![CDATA[University of Cambridge, Computer Laboratory]]></publisher-name>
</nlm-citation>
</ref>
<ref id="B17">
<label>[17]</label><nlm-citation citation-type="">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Restle]]></surname>
<given-names><![CDATA[F]]></given-names>
</name>
<name>
<surname><![CDATA[Greeno]]></surname>
<given-names><![CDATA[JG]]></given-names>
</name>
</person-group>
<source><![CDATA[Introduction to mathematical psychology]]></source>
<year>1970</year>
</nlm-citation>
</ref>
<ref id="B18">
<label>[18]</label><nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Richmond]]></surname>
<given-names><![CDATA[VP]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[Communication in the classroom: Power and motivation]]></article-title>
<source><![CDATA[Communication Education]]></source>
<year>1990</year>
<volume>39</volume>
<numero>3</numero>
<issue>3</issue>
<page-range>181-95</page-range></nlm-citation>
</ref>
<ref id="B19">
<label>[19]</label><nlm-citation citation-type="">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Roberts]]></surname>
<given-names><![CDATA[FS]]></given-names>
</name>
</person-group>
<source><![CDATA[Discrete mathematical models, with applications to social, biological, and environmental problems]]></source>
<year>1976</year>
</nlm-citation>
</ref>
<ref id="B20">
<label>[20]</label><nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Stauffer]]></surname>
<given-names><![CDATA[D]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[Social applications of two-dimensional ising models]]></article-title>
<source><![CDATA[American Journal of Physics]]></source>
<year>2008</year>
<volume>76</volume>
<numero>4</numero>
<issue>4</issue>
<page-range>470-3</page-range></nlm-citation>
</ref>
<ref id="B21">
<label>[21]</label><nlm-citation citation-type="book">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Wickens]]></surname>
<given-names><![CDATA[TD]]></given-names>
</name>
</person-group>
<source><![CDATA[Models for behavior: Stochastic processes in psychology]]></source>
<year>1982</year>
<publisher-name><![CDATA[WH Freeman &amp; Co Ltd]]></publisher-name>
</nlm-citation>
</ref>
</ref-list>
</back>
</article>
