<?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>1517-9702</journal-id>
<journal-title><![CDATA[Educação e Pesquisa]]></journal-title>
<abbrev-journal-title><![CDATA[Educ. Pesqui.]]></abbrev-journal-title>
<issn>1517-9702</issn>
<publisher>
<publisher-name><![CDATA[Faculdade de Educação da Universidade de São Paulo]]></publisher-name>
</publisher>
</journal-meta>
<article-meta>
<article-id>S1517-97022022000100703</article-id>
<article-id pub-id-type="doi">10.1590/s1678-4634202248235587</article-id>
<title-group>
<article-title xml:lang="pt"><![CDATA[Erros em esquemas de demonstração com números complexos]]></article-title>
<article-title xml:lang="en"><![CDATA[Errors in proof schemes with complex numbers]]></article-title>
</title-group>
<contrib-group>
<contrib contrib-type="author">
<name>
<surname><![CDATA[Beites]]></surname>
<given-names><![CDATA[Patrícia Damas]]></given-names>
</name>
<xref ref-type="aff" rid="Aff"/>
</contrib>
<contrib contrib-type="author">
<name>
<surname><![CDATA[Branco]]></surname>
<given-names><![CDATA[Maria Luísa]]></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-group>
<aff id="Af1">
<institution><![CDATA[,Universidade da Beira Interior  ]]></institution>
<addr-line><![CDATA[Covilhã Castelo Branco]]></addr-line>
<country>Portugal</country>
</aff>
<aff id="Af2">
<institution><![CDATA[,Universidade de Trás-os-Montes e Alto Douro  ]]></institution>
<addr-line><![CDATA[Vila Real Vila Real]]></addr-line>
<country>Portugal</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>48</volume>
<copyright-statement/>
<copyright-year/>
<self-uri xlink:href="http://educa.fcc.org.br/scielo.php?script=sci_arttext&amp;pid=S1517-97022022000100703&amp;lng=en&amp;nrm=iso"></self-uri><self-uri xlink:href="http://educa.fcc.org.br/scielo.php?script=sci_abstract&amp;pid=S1517-97022022000100703&amp;lng=en&amp;nrm=iso"></self-uri><self-uri xlink:href="http://educa.fcc.org.br/scielo.php?script=sci_pdf&amp;pid=S1517-97022022000100703&amp;lng=en&amp;nrm=iso"></self-uri><abstract abstract-type="short" xml:lang="pt"><p><![CDATA[Resumo As produções escritas de estudantes podem ser objeto de análise com vista à investigação e ao ensino. Neste trabalho, um dos poucos sobre erros e números complexos, analisam-se as produções, elaboradas por estudantes do ensino superior, para uma tarefa de demonstração de um caso particular da identidade do paralelogramo. Num trabalho que pode dar pistas para a prática letiva de professores do ensino secundário e do ensino superior, procura-se classificar os esquemas de demonstração nessas produções, identificar os erros neles cometidos e relacionar estas duas vertentes de análise. A natureza da investigação é qualitativa, recorrendo à análise de conteúdo empregando, por um lado, critérios prévios de categorizações de esquemas de demonstração e, por outro, critérios novos de categorizações de erros. Constatou-se que os estudantes utilizaram, principalmente, esquemas de demonstração de convicção externa, incluídos numa subcategoria nova designada por não válidos no universo, e esquemas de demonstração dedutivos, incluídos num nível novo que se caracteriza por discernimento incipiente da compreensão da tarefa, do contexto matemático, de hipóteses e tese, e dos conhecimentos prévios a mobilizar. Os erros dominantes nas produções estão relacionados à compreensão dos conceitos de linearidade de uma função e de módulo de um número complexo, apresentando-se possíveis explicações para os mesmos. Apesar de não se identificar uma relação propriamente significativa entre os esquemas de demonstração produzidos pelos estudantes e os erros cometidos, obteve-se material para construção e discussão de questões conceituais no âmbito do tópico números complexos.]]></p></abstract>
<abstract abstract-type="short" xml:lang="en"><p><![CDATA[Abstract The written productions of students can be subjected to analysis for research and teaching purposes. In the present work, one of the few on errors and complex numbers, the productions presented by Higher Education students for a proof task of a particular case of the parallelogram identity are analysed. In a work that may give clues for the teaching practice in High School and Higher Education, we attempt to classify the proof schemes of those productions, to identify the errors and to connect these two strands of analysis. The nature of the research is qualitative, making use of content analysis with, on the one hand, known criteria of categorization of proof schemes and, on the other hand, new criteria of categorization of errors. The students produced, mainly, external conviction proof schemes, included in a new subcategory called not valid in the universe, and deductive proof schemes, included in a new level that is characterized by the inchoate discernment of the comprehension of the task, of the mathematical context, of the hypotheses and thesis, and the previous knowledge that was needed. The dominant errors in the productions, for which possible explanations are presented, are related to the understanding of the concepts of linearity of a function and of modulus of a complex number. Although a significant relation between the proof schemes produced by the students and the errors is not identified, material for the construction and the discussion of concept questions in the context of complex numbers is obtained.]]></p></abstract>
<kwd-group>
<kwd lng="pt"><![CDATA[Educação matemática]]></kwd>
<kwd lng="pt"><![CDATA[Esquema de demonstração]]></kwd>
<kwd lng="pt"><![CDATA[Erro]]></kwd>
<kwd lng="pt"><![CDATA[Identidade do paralelogramo]]></kwd>
<kwd lng="pt"><![CDATA[Número complexo]]></kwd>
<kwd lng="en"><![CDATA[Mathematics education]]></kwd>
<kwd lng="en"><![CDATA[Proof scheme]]></kwd>
<kwd lng="en"><![CDATA[Error]]></kwd>
<kwd lng="en"><![CDATA[Parallelogram identity]]></kwd>
<kwd lng="en"><![CDATA[Complex number]]></kwd>
</kwd-group>
</article-meta>
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