<?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>1665-2436</journal-id>
<journal-title><![CDATA[Revista latinoamericana de investigación en matemática educativa]]></journal-title>
<abbrev-journal-title><![CDATA[Relime]]></abbrev-journal-title>
<issn>1665-2436</issn>
<publisher>
<publisher-name><![CDATA[Comité Latinoamericano de Matemática Educativa A.C.]]></publisher-name>
</publisher>
</journal-meta>
<article-meta>
<article-id>S1665-24362019000300333</article-id>
<article-id pub-id-type="doi">10.12802/relime.19.2234</article-id>
<title-group>
<article-title xml:lang="en"><![CDATA[Multiplication and division of fractions: numerical cognition development and assessment procedures]]></article-title>
</title-group>
<contrib-group>
<contrib contrib-type="author">
<name>
<surname><![CDATA[Díaz-Cardenas]]></surname>
<given-names><![CDATA[Alfonso F.]]></given-names>
</name>
<xref ref-type="aff" rid="Aff"/>
</contrib>
<contrib contrib-type="author">
<name>
<surname><![CDATA[Díaz-Furlong]]></surname>
<given-names><![CDATA[Alfonso]]></given-names>
</name>
<xref ref-type="aff" rid="Aff"/>
</contrib>
<contrib contrib-type="author">
<name>
<surname><![CDATA[Díaz-Furlong]]></surname>
<given-names><![CDATA[Hector Adrian]]></given-names>
</name>
<xref ref-type="aff" rid="Aff"/>
</contrib>
<contrib contrib-type="author">
<name>
<surname><![CDATA[Sankey-García]]></surname>
<given-names><![CDATA[M. Rayo]]></given-names>
</name>
<xref ref-type="aff" rid="Aff"/>
</contrib>
<contrib contrib-type="author">
<name>
<surname><![CDATA[Zago-Portillo]]></surname>
<given-names><![CDATA[Gemma]]></given-names>
</name>
<xref ref-type="aff" rid="Aff"/>
</contrib>
</contrib-group>
<aff id="Af1">
<institution><![CDATA[,Universidad Autónoma de Puebla  ]]></institution>
<addr-line><![CDATA[ ]]></addr-line>
<country>Mexico</country>
</aff>
<aff id="Af3">
<institution><![CDATA[,Universidad Autónoma de Puebla  ]]></institution>
<addr-line><![CDATA[ ]]></addr-line>
<country>Mexico</country>
</aff>
<aff id="Af2">
<institution><![CDATA[,Universidad Autónoma de Puebla  ]]></institution>
<addr-line><![CDATA[ ]]></addr-line>
<country>Mexico</country>
</aff>
<aff id="Af4">
<institution><![CDATA[,Secretaría de Educación Pública  ]]></institution>
<addr-line><![CDATA[ Puebla]]></addr-line>
<country>México</country>
</aff>
<aff id="Af5">
<institution><![CDATA[,Secretaría de Educación Pública  ]]></institution>
<addr-line><![CDATA[ ]]></addr-line>
<country>México</country>
</aff>
<pub-date pub-type="pub">
<day>00</day>
<month>00</month>
<year>2019</year>
</pub-date>
<pub-date pub-type="epub">
<day>00</day>
<month>00</month>
<year>2019</year>
</pub-date>
<volume>22</volume>
<numero>3</numero>
<fpage>333</fpage>
<lpage>362</lpage>
<copyright-statement/>
<copyright-year/>
<self-uri xlink:href="http://www.scielo.org.mx/scielo.php?script=sci_arttext&amp;pid=S1665-24362019000300333&amp;lng=en&amp;nrm=iso"></self-uri><self-uri xlink:href="http://www.scielo.org.mx/scielo.php?script=sci_abstract&amp;pid=S1665-24362019000300333&amp;lng=en&amp;nrm=iso"></self-uri><self-uri xlink:href="http://www.scielo.org.mx/scielo.php?script=sci_pdf&amp;pid=S1665-24362019000300333&amp;lng=en&amp;nrm=iso"></self-uri><abstract abstract-type="short" xml:lang="en"><p><![CDATA[Abstract The number and its basic operations can be conceptualised within a general system of relations. Children need to construct a system of numbers within which they can add, subtract, multiply and divide any rational number. Products and quotients can be defined in terms of general relational schemes. In this study, we examine whether elementary school children can construct a system of numbers such that fraction multiplication and division are based on the construction of general relational schemes. Groups of students are not homogeneous and children progress at different rates. For reliable assessment teachers need methods to examine developmental and individual differences in cognitive representations of mathematical concepts and operations. A logistic regression curve offers a visualisation of the learning process as a function of average marks. The analysis of fraction multiplication and division items shows an improvement on correct response probability, especially for students with a higher average mark.]]></p></abstract>
<abstract abstract-type="short" xml:lang="es"><p><![CDATA[Resumen El número y sus operaciones básicas se pueden conceptualizar dentro de un sistema general de relaciones. Los niños necesitan construir un sistema de números dentro del cual puedan sumar, restar, multiplicar y dividir cualquier número racional. Los productos y los cocientes se pueden definir en términos de esquemas relacionales generales. En este estudio, examinamos si los niños de escuela primaria pueden construir un sistema de números tal que la multiplicación y división de fracciones se basan en la construcción de esquemas relacionales generales. Los grupos de estudiantes no son homogéneos y los niños progresan a diferentes ritmos. Para una evaluación confiable, los maestros necesitan métodos para examinar las diferencias individuales y de desarrollo en las representaciones cognitivas de los conceptos y operaciones matemáticos. Una curva de regresión logística ofrece una visualización del proceso de aprendizaje como una función de las notas promedio. El análisis de elementos de multiplicación y división de fracciones muestra una mejora en la probabilidad de respuesta correcta, especialmente para estudiantes con una calificación promedio más alta.]]></p></abstract>
<abstract abstract-type="short" xml:lang="pt"><p><![CDATA[Resumo O número e suas operações básicas podem ser conceituados dentro de um sistema geral de relações. As crianças precisam construir um sistema de números dentro do qual possam somar, subtrair, multiplicar e dividir qualquer número racional. Produtos e quocientes podem ser definidos em termos de esquemas relacionais gerais. Neste estudo, examinamos se as crianças do ensino fundamental podem construir um sistema de números de tal forma que a multiplicação e divisão de frações são baseadas na construção de esquemas relacionais gerais. Grupos de estudantes não são homogêneos e as crianças progridem em taxas diferentes. Para uma avaliação confiável, os professores precisam de métodos para examinar diferenças individuais e de desenvolvimento nas representações cognitivas de conceitos e operações matemáticas. Uma curva de regressão logística oferece uma visualização do processo de aprendizagem como uma função das notas médias. A análise dos itens de multiplicação e divisão de frações mostra uma melhoria na probabilidade de resposta correta, especialmente para os alunos com uma nota média mais alta.]]></p></abstract>
<abstract abstract-type="short" xml:lang="fr"><p><![CDATA[Résumé Le nombre et ses opérations de base peuvent être conceptualisés dans un système général de relations. Les enfants ont besoin de construire un système de nombres au sein duquel ils peuvent additionner, soustraire, multiplier et diviser n&#8217;importe quel nombre rationnel. Les produits et les quotients peuvent être définis en termes de schémas relationnels généraux. Dans cette étude, nous examinons si les enfants des écoles élémentaires peuvent construire un système de nombres tel que la multiplication et la division des fractions sont basées sur la construction de schèmes relationnels généraux. Les groupes d&#8217;étudiants ne sont pas homogènes et les enfants progressent à des rythmes différents. Pour une évaluation fiable, les enseignants ont besoin de méthodes pour examiner les différences de développement et individuelles dans les représentations cognitives des concepts et des opérations mathématiques. Une courbe de régression logistique offre une visualisation du processus d&#8217;apprentissage en fonction des moyennes. L&#8217;analyse des éléments de multiplication et de division des fractions montre une amélioration de la probabilité de réponse correcte, en particulier pour les étudiants ayant une note moyenne plus élevée.]]></p></abstract>
<kwd-group>
<kwd lng="es"><![CDATA[Esquemas relacionales]]></kwd>
<kwd lng="es"><![CDATA[Enseñanza de multiplicación de fracciones]]></kwd>
<kwd lng="es"><![CDATA[Curva de regresión logística]]></kwd>
<kwd lng="es"><![CDATA[Evaluación del desarrollo educativo]]></kwd>
<kwd lng="en"><![CDATA[Relational schemes]]></kwd>
<kwd lng="en"><![CDATA[Fraction multiplication teaching]]></kwd>
<kwd lng="en"><![CDATA[Logistic regression curve]]></kwd>
<kwd lng="en"><![CDATA[Educational development assessment]]></kwd>
<kwd lng="pt"><![CDATA[Esquemas relacionais]]></kwd>
<kwd lng="pt"><![CDATA[Ensino de multiplicação de frações]]></kwd>
<kwd lng="pt"><![CDATA[Curva de regressão logística]]></kwd>
<kwd lng="pt"><![CDATA[Avaliação do desenvolvimento educacional]]></kwd>
<kwd lng="fr"><![CDATA[Schémas relationnels]]></kwd>
<kwd lng="fr"><![CDATA[Enseignement de la multiplication de fractions]]></kwd>
<kwd lng="fr"><![CDATA[Courbe de régression logistique]]></kwd>
<kwd lng="fr"><![CDATA[Évaluation du développement de l&#8217;éducation]]></kwd>
</kwd-group>
</article-meta>
</front><back>
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