<?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>2238-121X</journal-id>
<journal-title><![CDATA[Comunicações]]></journal-title>
<abbrev-journal-title><![CDATA[Rev. Comunic.]]></abbrev-journal-title>
<issn>2238-121X</issn>
<publisher>
<publisher-name><![CDATA[Universidade Metodista de Piracicaba]]></publisher-name>
</publisher>
</journal-meta>
<article-meta>
<article-id>S2238-121X2017000100055</article-id>
<article-id pub-id-type="doi">10.15600/2238-121X/comunicacoes.v24n1p35-68</article-id>
<title-group>
<article-title xml:lang="pt"><![CDATA[TRANSFORMAÇÕES DE REPRESENTAÇÕES VISUAIS DE MÚLTIPLOS E DIVISORES DE UM NÚMERO]]></article-title>
<article-title xml:lang="en"><![CDATA[TRANSFORMATIONS OF VISUAL REPRESENTATION OF MULTIPLES AND DIVISORS OF A NUMBER]]></article-title>
<article-title xml:lang="es"><![CDATA[CAMBIO DE REPRESENTACIONES VISUALES DE LOS MÚLTIPLOS Y DIVISORES DE UN NÚMERO]]></article-title>
</title-group>
<contrib-group>
<contrib contrib-type="author">
<name>
<surname><![CDATA[MONTENEGRO]]></surname>
<given-names><![CDATA[PAULA]]></given-names>
</name>
<xref ref-type="aff" rid="Aff"/>
</contrib>
<contrib contrib-type="author">
<name>
<surname><![CDATA[COSTA]]></surname>
<given-names><![CDATA[CECÍLIA]]></given-names>
</name>
<xref ref-type="aff" rid="Aff"/>
</contrib>
<contrib contrib-type="author">
<name>
<surname><![CDATA[LOPES]]></surname>
<given-names><![CDATA[BERNARDINO]]></given-names>
</name>
<xref ref-type="aff" rid="Aff"/>
</contrib>
</contrib-group>
<aff id="A01">
<institution><![CDATA[,Universidade de Trás-os-Montes e Alto Douro  ]]></institution>
<addr-line><![CDATA[Vila Real ]]></addr-line>
<country>Portugal</country>
</aff>
<aff id="A02">
<institution><![CDATA[,Universidade de Trás-os-Montes e Alto Douro  ]]></institution>
<addr-line><![CDATA[Vila Real ]]></addr-line>
<country>Portugal</country>
</aff>
<aff id="A03">
<institution><![CDATA[,Universidade de Trás-os-Montes e Alto Douro  ]]></institution>
<addr-line><![CDATA[Vila Real ]]></addr-line>
<country>Portugal</country>
</aff>
<pub-date pub-type="pub">
<day>00</day>
<month>04</month>
<year>2017</year>
</pub-date>
<pub-date pub-type="epub">
<day>00</day>
<month>04</month>
<year>2017</year>
</pub-date>
<volume>24</volume>
<numero>1</numero>
<fpage>55</fpage>
<lpage>68</lpage>
<copyright-statement/>
<copyright-year/>
<self-uri xlink:href="http://educa.fcc.org.br/scielo.php?script=sci_arttext&amp;pid=S2238-121X2017000100055&amp;lng=en&amp;nrm=iso"></self-uri><self-uri xlink:href="http://educa.fcc.org.br/scielo.php?script=sci_abstract&amp;pid=S2238-121X2017000100055&amp;lng=en&amp;nrm=iso"></self-uri><self-uri xlink:href="http://educa.fcc.org.br/scielo.php?script=sci_pdf&amp;pid=S2238-121X2017000100055&amp;lng=en&amp;nrm=iso"></self-uri><abstract abstract-type="short" xml:lang="pt"><p><![CDATA[RESUMO A resolu&#231;&#227;o de problemas &#233; uma das capacidades matem&#225;ticas fundamentais e uma base para a aprendizagem de conceitos, procedimentos matem&#225;ticos e representa&#231;&#245;es. N&#227;o obstante, os alunos evidenciam dificuldades na resolu&#231;&#227;o de problemas e na multiplicidade de representa&#231;&#245;es para um dado conceito matem&#225;tico. Este estudo procurou sistematizar a relev&#226;ncia e a efic&#225;cia da introdu&#231;&#227;o frequente de representa&#231;&#245;es visuais na leciona&#231;&#227;o dos conte&#250;dos relativos &#224;s no&#231;&#245;es de m&#250;ltiplo e divisor de um n&#250;mero natural, e do seu impacto na resolu&#231;&#227;o de problemas com essas no&#231;&#245;es. Segue uma metodologia qualitativa analisando uma interven&#231;&#227;o de ensino num grupo de 17 alunos (10-11 anos de idade). Os resultados mostraram que (i) as representa&#231;&#245;es visuais permitiram transforma&#231;&#245;es de representa&#231;&#245;es (tratamentos e convers&#245;es), como as representa&#231;&#245;es simb&#243;licas e verbais, (ii) a introdu&#231;&#227;o de representa&#231;&#245;es visuais proporcionou um ambiente de aprendizagem com representa&#231;&#245;es m&#250;ltiplas, com uma maior variedade de representa&#231;&#245;es na resolu&#231;&#227;o de problemas, e (iii) esse efeito prolongou-se no intervalo de tempo considerado. Conclu&#237;mos que a utiliza&#231;&#227;o frequente de representa&#231;&#245;es visuais potenciou as transforma&#231;&#245;es de representa&#231;&#245;es tendo como consequ&#234;ncia uma melhoria no desempenho geral dos alunos.]]></p></abstract>
<abstract abstract-type="short" xml:lang="en"><p><![CDATA[ABSTRACT Problem solving is one of the most important mathematical capacities and bases for learning of mathematical concepts, procedures and representations. However, the students show difficulties in problem solving and in a multiplicity of representations of a given mathematical concept/object. The aim of this study is to determine the relevance and the impact of using visual representations and their transformations in the teaching process of divisors and multiple of a number, and their impact in problem solving with these notions. It follows a qualitative methodology and in this paper we analyze an educational intervention in a group of 17 students (10-11 years old). The results show that (i) visual representations allowed transformations of representations (treatments and conversions) with the same characteristics of the symbolic and verbal representations (ii) these visual representations provided a learning environment with multiple representations that diversified the transformation of representations (treatments and conversions), and (iii) this effect was extended in the considered interval of time (three months). We conclude that the use of visual representations enhances the transformations of representations which contributed to the improvement of the students overall performance.]]></p></abstract>
<abstract abstract-type="short" xml:lang="es"><p><![CDATA[RESUMEN La resoluci&#243;n de problemas &#233; una das capacidades matem&#225;ticas fundamentales y una base para el aprendizaje de conceptos matem&#225;ticos, procedimientos y representaciones. No obstante, los estudiantes muestran dificultades en la resoluci&#243;n de problemas y en la multiplicidad de representaciones para un dado concepto matem&#225;tico. Este trabajo trat&#243; de sistematizar la relevancia y la importancia de la introducci&#243;n frecuente de representaciones visuales en la ense&#241;anza de los contenidos relativos a nociones de m&#250;ltiple y divisor de un n&#250;mero natural, y del su impacte en la resoluci&#243;n de problemas. En este estudio se sigue la metodolog&#237;a cualitativa y se analiza una intervenci&#243;n educativa en un grupo de 17 estudiantes (10-11 a&#241;os). Los resultados muestran que (i) las representaciones visuales permite transformaciones (tratamientos y conversiones), iguales a transformaciones de las representaciones simb&#243;licas y verbales, (ii) esta intervenci&#243;n proporcion&#243; un ambiente de aprendizaje con m&#250;ltiples representaciones que conllev&#243; a una diversificaci&#243;n de la transformaci&#243;n de representaciones en la resoluci&#243;n de problemas y (iii) este efecto se prolong&#243; en el intervalo de tiempo considerado. Se concluy&#243; que el uso de representaciones visuales potencia las transformaciones y contribuy&#243; para mejorar el rendimiento general de los estudiantes.]]></p></abstract>
<kwd-group>
<kwd lng="pt"><![CDATA[DIVISORES]]></kwd>
<kwd lng="pt"><![CDATA[MÚLTIPLOS]]></kwd>
<kwd lng="pt"><![CDATA[REPRESENTAÇÕES VISUAIS]]></kwd>
<kwd lng="pt"><![CDATA[RESOLUÇÃO DE PROBLEMAS]]></kwd>
<kwd lng="en"><![CDATA[DIVISORS]]></kwd>
<kwd lng="en"><![CDATA[MULTIPLES]]></kwd>
<kwd lng="en"><![CDATA[PROBLEM SOLVING]]></kwd>
<kwd lng="en"><![CDATA[VISUAL REPRESENTATIONS]]></kwd>
<kwd lng="es"><![CDATA[DIVISORES]]></kwd>
<kwd lng="es"><![CDATA[MÚLTIPLES]]></kwd>
<kwd lng="es"><![CDATA[REPRESENTACIONES VISUALES]]></kwd>
<kwd lng="es"><![CDATA[RESOLUCIÓN DE PROBLEMAS]]></kwd>
</kwd-group>
</article-meta>
</front><back>
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